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+{"text":"\\section{Introduction}\nThe relationship between phase transitions away from thermal equilibrium and open systems in quantum optics was first addressed in early laser days \\cite{degiorgio&scully_1970,graham&haken_1970,grossmann&richter_1971}. The theme was then carried forward by work on optical bistability \\cite{bonifacio&lugiato_1976,bonifacio_etal_1978,drummond&walls_1980,rosenberger_etal_1991}, the degenerate parametric oscillator \\cite{carmichael_2008}---with loss added to the quantum theory of parametric amplification \\cite{mollow&glauber_1967}---and collective radiative phenomena like cooperative fluorescence \\cite{drummond&carmichael_1978,carmichael_1980}, to name just a few of the examples. As counterpoint to these phase transitions of light away from equilibrium, and contemporary with the early laser work, Hepp and Lieb  introduced the celebrated Dicke-model phase transition \\cite{hepp&lieb_1973a,dicke_1954}---a phase transition for photons in thermal equilibrium.\n\nWhile the dissipative platforms provided by the laser, optical bistability, and parametric oscillator encouraged wide experimental activity, Hepp and Lieb's proposal lay\ndormant on the experimental front. Its call for a dipole coupling strength between light and matter in excess of atomic transition frequencies posed an extreme technical challenge, and also undermined approximations adopted in the Dicke model \\cite{rzazewski_etal_1975,bialynicki-birula&rzazewski_1979,gawedzki&rzazewski_1981,rzazewski&wodkiewicz_1991}.  The long wait ended in 2010, however, with the experimental work of Baumann \\emph{et al.} \\cite{baumann_etal_2010,baumann_etal_2011}, who realized the $T=0$ phase transition of Hepp and Lieb with a superfluid gas in an optical cavity.  The key to success was their engineering of the Dicke-model Hamiltonian as an effective Hamiltonian, by employing an external Raman drive to realize the phase transition in a dissipative setting \\cite{dimer_etal_2007,baden_etal_2014}.\n\nIn a separate development rooted in research on open systems in quantum optics, cavity and circuit QED have shown that where many material particles and photons might traditionally be required for the strong  interaction of matter and light, it is now possible to achieve strong interactions, sufficient to access nonlinearities, with one particle (e.g.\\ two-state system) and photon numbers that range from just one to the relatively low tens, hundreds, or thousands.  Thus, with regard to the laser\nand optical bistability, there are cavity QED versions of both \\cite{rice&carmichael_1994,savage&carmichael_1988}, and even realizations of the parametric oscillator where single photons are enough to access the nonlinearity \\cite{leghtas_etal_2015}. Although the thermodynamic limit of equilibrium phase transitions does not apply under these conditions, it still remains that a mean-field treatment and phase transition perspective can guide much of the phenomenology; albeit with the caveat that fluctuations might add more than just minor corrections.\n\nIn this paper we unify the dissipative extension of the Dicke-model quantum phase transition \\cite{baumann_etal_2010,baumann_etal_2011,dimer_etal_2007,baden_etal_2014}\nwith the recently reported  breakdown of photon blockade \\cite{carmichael_2015,fink_etal_2017}. The former is addressed by Hepp and Lieb through the thermodynamic limit ($N\\to\\infty$ two-state systems), while the latter has been approached in a cavity or circuit QED setting \\cite{carmichael_2015,fink_etal_2017,alsing&carmichael_1991,kilin&krinitskaya_1991,armen_etal_2009} (one two-state system). Both phenomena may be engineered, however, in either of the two ways, and we therefore first consider mean-field results for both (Sec.~\\ref{sec:mean-field}) before turning to results specific to one two-state system (Sec.~\\ref{sec:quantum_fluctuations}).\n\nWe achieve the proposed unification within the framework of a generalized Dicke-model Hamiltonian, where two extensions of the analysis in Ref.~\\cite{dimer_etal_2007} are made: first, we allow for rotating and counter-rotating interactions of independently adjustable coupling strength (see Ref.~\\cite{dimer_etal_2007}, Eq.~(12)); and, second, we add external coherent driving of the field mode. The first extension was made by Hepp and Lieb \\cite{hepp&lieb_1973b}, in a quick followup to their original paper; the generalized interaction Hamiltonian is also featured in a number of recent publications \\cite{stepanov_etal_2008,schiro_etal_2012,tomka_etal_2014,xie_etal_2014,tomka_etal_2015,wang_etal_2016,moroz_2016,kirton_etal_2018}. A key link in our unification is a phase that went unreported by Hepp and Lieb. Beyond this, though, the added coherent drive is also key, since the breakdown of photon blockade is organized around a critical drive strength, identified, to date, in the driven Jaynes-Cummings model (no counter-rotating interaction) alone \\cite{alsing&carmichael_1991,alsing_etal_1992,carmichael_2015}. We show that the critical drive is a feature of the generalized Hamiltonian, rotating and counter-rotating interactions included, and thus links the Dicke model quantum phase transition to the breakdown of photon blockade.\n\nWe begin in Sec.~\\ref{sec:background} with a detailed review, building up our generalized Jaynes-Cummings-Rabi model while making connections to prior work. We then survey the mean-field steady states of the model in Sec.~\\ref{sec:mean-field} and show how a common critical drive strength links the dissipative extension of the Dicke-model quantum phase transition to the breakdown of photon blockade.  Finally, in Sec.~\\ref{sec:quantum_fluctuations}, we turn from the mean-field treatment to full quantum mechanical calculations for the special case of one two-state system. We recover the critical drive strength from the quasi-energy spectrum of the model Hamiltonian and show how mean-field predictions can  still provide a guide to the physics in the presence of quantum fluctualtions. Conclusions are presented in Sec.~\\ref{sec:conclusions}\n\n\\section{Background}\n\\label{sec:background}\n\\subsection{The Dicke quantum phase transition in the rotating-wave approximation}\n\\label{sec:Dicke_rotating_wave}\nIn their original paper \\cite{hepp&lieb_1973a} ``On the Superradiant Phase Transition for Molecules in a Quantized Radiation Field: the Dicke Maser Model,'' Hepp and Lieb first introduce an ``interesting caricature$\\ldots$invented by Dicke'' \\cite{dicke_1954} of the interaction between quantized radiation in a box and a system of $N$ molecules. The caricature assumes single-mode radiation, two-state molecules, and the rotating-wave approximation; it generalizes the Tavis-Cummings model \\cite{tavis&cummings_1968} to non-zero detuning, and, adopting natural units with $\\hbar=1$, is defined by the Hamiltonian\n\\begin{equation}\nH_0=\\omega a^\\dagger a+\\omega_0J_z+\\frac{\\lambda}{\\sqrt N}(aJ_++a^\\dagger J_-),\n\\label{eqn:hamiltonian_rotating_wave}\n\\end{equation}\nwhere $\\omega$ is the frequency of the field, $\\omega_0$ the resonance frequency of the two-state molecules, and $\\lambda$ is a coupling strength; annihilation and creation operators for the field mode obey the boson commutation relation, $[a,a^\\dagger]=1$, and the collective operators for $N$ two-state systems obey angular momentum commutation relations, $[J_-,J_+]=-2J_z$, $[J_\\mp,J_z]=\\pm J_\\mp$. Hepp and Lieb exactly compute thermodynamic functions in the limit $N\\to\\infty$ and find a critical temperature, $T_c>0$, for any coupling strength above\n\\begin{equation}\n\\lambda_0=\\sqrt{\\omega\\omega_0}.\n\\label{critical_point_eta=0}\n\\end{equation}\nConsidering zero temperature, as we do in this paper, $\\lambda_0$ has the significance of a critical coupling strength, where for $\\lambda\\le\\lambda_0$ the photon number is zero in the ground state, while it follows the formula\n\\begin{equation}\n\\frac{\\langle a^\\dagger a\\rangle_0}N=\\frac{\\omega_0}{4\\omega}\\frac{\\lambda^4-\\lambda_0^4}{\\lambda^2\\lambda_0^2}\n\\label{eqn:photon_number_eta=0}\n\\end{equation}\nwhen $\\lambda>\\lambda_0$. Soon after the rigorous calculation of Hepp and Lieb, the same result was derived by Wang and Hioe \\cite{wang&hioe_1973} using a simpler method [see their Eq.~(40)].\n\\subsection{Counter-rotating terms}\n\\label{sec:Dicke_counter_rotating}\nThe method of Wang and Hioe readily generalizes to an interaction without the rotating-wave approximation: $aJ_++a^\\dagger J_-\\to (a+a^\\dagger)(J_-+J_+)$. The calculation, made by Hepp and Lieb \\cite{hepp&lieb_1973b} and Carmichael \\emph{et al.} \\cite{carmichael_etal_1973}, retains the phase transition and the form of Eq.~(\\ref{eqn:photon_number_eta=0}), but unlike in the rotating-wave approximation, the state of nonzero photon number now assigns a definite phase to the field, and the critical coupling is changed to $\\sqrt{\\omega\\omega_0}\/2$. In fact Hepp and Lieb \\cite{hepp&lieb_1973b} consider a Hamiltonian generalized in the form\n\\begin{eqnarray}\nH_\\eta&=&\\omega a^\\dagger a+\\omega_0J_z\\notag\\\\\n&&+\\frac{\\lambda}{\\sqrt N}(aJ_++a^\\dagger J_-)+\\eta\\frac{\\lambda}{\\sqrt N}(a^\\dagger J_++aJ_-),\n\\label{eqn:hamiltonian_counter_rotating}\n\\end{eqnarray}\nwith $\\eta$ a parameter. We let $\\eta$ vary from 0 to 1 and show (Sec.~\\ref{sec:epsilon=0}) that there are actually two critical coupling strengths marking transitions to states of definite phase:\n\\begin{equation}\n\\lambda_\\eta^\\pm=\\frac1{1\\pm\\eta}\\sqrt{\\omega\\omega_0}.\n\\label{eqn:critical_points_kappa=0}\n\\end{equation}\nMoreover, photon numbers for solutions bifurcating from both critical points, $\\lambda_\\eta^+$  and $\\lambda_\\eta^-$, follow the same form, that of Eq.~(\\ref{eqn:photon_number_eta=0}):\n\\begin{equation}\n\\frac{\\langle a^\\dagger a\\rangle_\\eta^\\pm}N=\\frac{\\omega_0}{4\\omega}\\frac{\\lambda^4-(\\lambda_\\eta^\\pm)^4}{\\lambda^2(\\lambda_\\eta^\\pm)^2}.\n\\label{eqn:photon_number_kappa=0}\n\\end{equation}\nThe transition at $\\lambda_\\eta^+$ corresponds to the extension of the Dicke phase transition of Ref.~\\cite{hepp&lieb_1973a} discussed in Refs.~\\cite{hepp&lieb_1973b} and \\cite{carmichael_etal_1973}: the zero photon state becomes unstable and is replaced by a stable state of nonzero photon number. The transition at $\\lambda_\\eta^-$, not identified before to our knowledge, marks a restabilization of the zero photon state and the birth of an unstable state of nonzero photon number. It provides the fulcrum upon which the unification of the coherently driven extension of the Dicke phase transition and the breakdown of photon blockade turns.\n\\subsection{Dissipative realization}\n\\label{sec:dissipative_realization}\nWhile Dicke's paper \\cite{dicke_1954} generated enormous interest in superradiance as a transient, away-from-equilibrium process \\cite{gross&haroche_1982}, the Dicke quantum phase transition of Hepp and Lieb was, for many years, largely seen as academic---beyond the reach of experiments due to a needed coupling strength on the order of the transition frequency, and, on the theory side, suspect because of approximations used in the Dicke model \\cite{rzazewski_etal_1975,bialynicki-birula&rzazewski_1979,gawedzki&rzazewski_1981,rzazewski&wodkiewicz_1991}.  Dissipative realizations of the Dicke Hamiltonian as an effective Hamiltonian overcome these obstacles by replacing a transition from a ground to an excited state by one between a pair of ground states. Specifically, we have the scheme introduced by Dimer \\emph{et al.} \\cite{dimer_etal_2007,baden_etal_2014} in mind; although there are essentially parallel setups, where internal states are replaced by momentum states of a Bose-Einstein condensate \\cite{baumann_etal_2010,baumann_etal_2011}.\n\nWe consider a pair of Raman transitions between states $|1\\rangle$ and $|2\\rangle$---the two-state system---as sketched in Fig.~\\ref{fig:fig1}, where one leg of each transition is driven by a laser field, with amplitudes and frequencies $\\Omega_{1,2}$ and $\\omega_{1,2}$, and the other creates and annihilates cavity photons of frequency $\\omega$, with coupling strength to the cavity mode $g$. Adopting this configuration, with the excited states (not shown) adiabatically eliminated, and in an interaction picture---free Hamiltonian $\\omega_+a^\\dagger a+\\omega_-J_z$, $\\omega_{\\pm}=(\\omega_1\\pm\\omega_2)\/2$---an effective Hamiltonian is realized in the form of Eq.~(\\ref{eqn:hamiltonian_counter_rotating}):\n\\begin{eqnarray}\nH_\\eta^\\prime&=&\\Delta a^\\dagger a+\\Delta_0J_z\\notag\\\\\n&&+\\frac{\\lambda}{\\sqrt N}(aJ_++a^\\dagger J_-)+\\eta\\frac{\\lambda}{\\sqrt N}(a^\\dagger J_++aJ_-),\n\\label{eqn:hamiltonian_raman_model}\n\\end{eqnarray}\nwith effective frequencies\n\\begin{eqnarray}\n\\Delta&=&\\omega-\\omega_+=\\frac{\\delta_1+\\delta_2}2,\\\\\n\\label{eqn:delta}\n\\Delta_0&=&\\omega_0-\\omega_-=\\frac{\\delta_1-\\delta_2}2,\n\\label{eqn:delta_zero}\n\\end{eqnarray}\nwhere $\\delta_1$ and $\\delta_2$ are Raman detunings (Fig.~\\ref{fig:fig1}), and the coupling constants $\\lambda$ and $\\eta\\lambda$ follow from the strength of the Raman coupling (see Ref.~\\cite{dimer_etal_2007}). We consider an initial state $|0\\rangle|1\\rangle$, with $|0\\rangle$ the cavity mode vacuum, in which case the Raman driving is a source of photons through the counter-rotating interaction, an external drive that is off-set by the cavity loss; thus, the dissipative realization of the generalized Dicke Hamiltonian, Eq.~(\\ref{eqn:hamiltonian_counter_rotating}), is modeled by the master equation\n\\begin{equation}\n\\frac{d\\rho}{dt}=-i[H_\\eta^\\prime,\\rho]+\\kappa{\\mathcal L}[a]\\rho,\n\\label{eqn:master_equation}\n\\end{equation}\nwhere $\\kappa$ is the loss rate and ${\\mathcal L}[\\xi]\\,\\cdot=2\\xi \\cdot \\xi^\\dagger-\\xi^\\dagger \\xi\\cdot-\\cdot \\xi^\\dagger\\xi$. We show (Sec.~\\ref{sec:epsilon=0}) that in the presence of dissipation, for $\\eta<\\eta_\\kappa$,\n\\begin{equation}\n\\eta_\\kappa\\equiv\\frac{\\kappa}{|\\Delta|}\\left[1+\\sqrt{1+\\frac{\\kappa^2}{\\Delta^2}}\\mkern3mu\\right]^{-1},\n\\label{eqn:eta_critical}\n\\end{equation}\nthere is no critical coupling strength, while for $\\eta\\geq\\eta_\\kappa$, there are two that for $\\kappa\\to0$ reduce to Eq.~(\\ref{eqn:critical_points_kappa=0}):\n\\begin{equation}\n\\lambda_\\eta^\\pm\\equiv\\frac{\\sqrt{|\\Delta\\Delta_0|}}{1-\\eta^2}\\left[1+\\eta^2\\mp2\\eta\\sqrt{1-\\frac{(1-\\eta^2)^2}{4\\eta^2}\\frac{\\kappa^2}{\\Delta^2}}\\mkern3mu\\right]^{1\/2}.\n\\label{eqn:lambda_critical}\n\\end{equation}\nPhoton numbers generalizing Eq.~(\\ref{eqn:photon_number_kappa=0}) are recovered from the mean-field steady state in Sec.~\\ref{sec:epsilon=0} [Eq.~(\\ref{eqn:photon_number_epsilon=0})].\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=2.75in]{figure1.pdf}\n\\end{center}\n\\caption{Schematic of the open system realization of the model Hamiltonian, Eq.~(\\ref{eqn:hamiltonian_raman_model}). A pair of ground states, denoted $|1\\rangle$ and $|2\\rangle$, are coupled to an optical cavity mode, frequency $\\omega$, via far-from-resonance Raman transitions, where bold dashed arrows represent external laser drives while the transfer of photons to and from the cavity mode, coupling strength $g$, is represented by bold solid arrows; $\\Omega_{1,2}$ and $\\omega_{1,2}$ are drive amplitudes and frequencies, and $\\delta_{1,2}$ are detunings; excited states are assumed far from resonance and not shown.}\n\\label{fig:fig1}\n\\end{figure}\n\n\\subsection{Extended model with coherent drive}\nEquations~(\\ref{eqn:hamiltonian_raman_model}) and (\\ref{eqn:master_equation}) set out a driven and dissipative model where the driving of the field mode is mediated by externally driven Raman transitions; the dissipative realization of the effective rotating and counter-rotating interactions amounts to a \\emph{nonlinear} driving of the field mode. In studies of the so-called breakdown of photon blockade  \\cite{alsing&carmichael_1991,carmichael_2015,armen_etal_2009,fink_etal_2017}, the mode is subject to a coherent drive, i.e., \\emph{linear} driving by an external field. We now extend our model by adding a coherent drive of amplitude $\\sqrt N\\epsilon$ and frequency $\\omega_d$---a detuning $\\omega_d-\\omega_+$ in the interaction picture of Eq.~(\\ref{eqn:hamiltonian_raman_model}). Choosing $\\omega_1$ and $\\omega_2$ so that $\\omega_+=\\omega_d$, the master equation then becomes\n\\begin{equation}\n\\frac{d\\rho}{dt}=-i[H_\\eta^\\prime,\\rho]-i\\sqrt N\\epsilon[a^\\dagger+a,\\rho]+\\kappa{\\mathcal L}[a]\\rho,\n\\label{eqn:master_equation_drive}\n\\end{equation}\nwhere, from Eq.~(\\ref{eqn:delta}), $\\Delta=\\omega-\\omega_d$ is now the detuning of the field mode from the drive.\n\nThe next section explores the parameter dependence of the mean-field steady states of Eq.~(\\ref{eqn:master_equation_drive}). In particular, we  connect the breakdown of photon blockade, realized for $\\eta=0$, to the coherently driven extension of the Dicke quantum phase transition. We show that an $\\eta$-dependent critical point organizes behavior as a function of drive strength; we then establish a link through the previously unreported phase of the generalized model presented in Ref.~\\cite{hepp&lieb_1973b}, i.e., the second critical coupling strength $\\lambda_\\eta^-$.\n\n\\section{Mean-Field Steady States}\n\\label{sec:mean-field}\nThe mean-field Maxwell-Bloch equations derived from the master equation, Eq.~(\\ref{eqn:master_equation_drive}), are:\n\\begin{eqnarray}\n\\frac{d\\alpha}{dt}&=&-(\\kappa+i\\Delta)\\alpha-i\\frac{\\lambda}{\\sqrt N}\\frac12(\\beta+\\eta\\beta^*)-i\\sqrt N\\epsilon,\n\\label{eqn:mean-field_alpha}\\\\\n\\frac{d\\beta}{dt}&=&-i\\Delta_0\\beta+2i\\frac{\\lambda}{\\sqrt N}(\\alpha+\\eta\\alpha^*)\\zeta,\n\\label{eqn:mean-field_beta}\\\\\n\\frac{d\\zeta}{dt}&=&-i\\frac{\\lambda}{\\sqrt N}\\left[(\\alpha\\beta^*-\\alpha^*\\beta)-\\eta(\\alpha\\beta-\\alpha^*\\beta^*)\\right],\n\\label{eqn:mean-field_zeta}\n\\end{eqnarray}\nwith $\\alpha\\equiv\\langle a\\rangle$, $\\beta\\equiv2\\langle J_-\\rangle$, and $\\zeta\\equiv2\\langle J_z\\rangle$. We first outline a general approach to their steady state solution, where, introducing intensive variables\n\\begin{equation}\n\\bar\\alpha\\equiv\\alpha\/\\sqrt N,\\qquad\\bar\\beta\\equiv\\beta\/N,\\qquad\\bar\\zeta\\equiv\\zeta\/N,\n\\label{eqn:scaling}\n\\end{equation}\nEqs.~(\\ref{eqn:mean-field_alpha}) and (\\ref{eqn:mean-field_beta}) require\n\\begin{eqnarray}\n\\bar\\beta_x&=&2\\lambda\\frac{1+\\eta}{\\Delta_0}\\bar\\alpha_x\\bar\\zeta,\n\\label{eqn:steady-state_beta1}\\\\\n\\bar\\beta_y&=&2\\lambda\\frac{1-\\eta}{\\Delta_0}\\bar\\alpha_y\\bar\\zeta,\n\\label{eqn:steady-state_beta2}\n\\end{eqnarray}\nwith $\\bar\\alpha_x$ and $\\bar\\alpha_y$ satisfying the simultaneous equations:\n\\begin{eqnarray}\n\\kappa\\bar\\alpha_x-\\left[\\Delta+\\lambda^2\\frac{(1-\\eta)^2}{\\Delta_0}\\bar\\zeta\\right]\\bar\\alpha_y&=&0,\n\\label{eqn:steady-state_alpha1}\\\\\n\\kappa\\bar\\alpha_y+\\left[\\Delta+\\lambda^2\\frac{(1+\\eta)^2}{\\Delta_0}\\bar\\zeta\\right]\\bar\\alpha_x&=&-\\epsilon.\n\\label{eqn:steady-state_alpha2}\n\\end{eqnarray}\nWe may then solve Eqs.~(\\ref{eqn:steady-state_beta1})--(\\ref{eqn:steady-state_alpha2}) for $|\\bar\\beta|^2$ in terms of $\\bar\\zeta$ and impose the conservation law $\\bar\\zeta^2+|\\bar\\beta|^2=1$; hence we find an autonomous equation satisfied by $\\bar\\zeta$,\n\\begin{equation}\n(1-\\bar\\zeta^2)[P(\\bar\\zeta)]^2=\\frac{4\\epsilon^2}{\\lambda^2(1+\\eta)^2}\\bar\\zeta^2Q(\\bar\\zeta),\n\\label{eqn:6th-order_polynomial}\n\\end{equation}\nwith $P(\\bar\\zeta)$ and $Q(\\bar\\zeta)$ both quadratic:\n\\begin{equation}\nP(\\bar\\zeta)=\\bar\\zeta^2+2\\frac{\\Delta\\Delta_0(1+\\eta^2)}{\\lambda^2(1-\\eta^2)^2}\\bar\\zeta+\\frac{\\Delta_0^2(\\kappa^2+\\Delta^2)}{\\lambda^4(1-\\eta^2)^2},\n\\label{eqn:p_quadratic}\n\\end{equation}\nand\n\\begin{equation}\nQ(\\bar\\zeta)=\\bar\\zeta^2+2\\frac{\\Delta\\Delta_0}{\\lambda^2(1-\\eta)^2}\\bar\\zeta+\\frac{\\Delta_0^2\\kappa^2}{\\lambda^4(1-\\eta^2)^2}+\\frac{\\Delta^2\\Delta_0^2}{\\lambda^4(1-\\eta)^4}.\n\\label{eqn:q_quadratic}\n\\end{equation}\nSteady-state solutions for $\\bar\\zeta$ are seen to be roots of a 6th-order polynomial, with a possible six distinct solutions for any setting of the parameters: $\\eta$, $\\Delta$, $\\Delta_0$, $\\lambda$, $\\epsilon$, and $\\kappa$. In the following, for the most part, we set $\\Delta_0=\\Delta$ and keep $\\kappa\/\\lambda$ fixed; we then explore the parameter dependence in the $(\\Delta\/\\lambda,\\epsilon\/\\lambda)$-plane for different choices of $\\eta$. To start, we recover the results summarized in Secs.~\\ref{sec:Dicke_rotating_wave} and \\ref{sec:Dicke_counter_rotating} from our general solution scheme.\n\\subsection{Zero drive: $\\epsilon=0$}\n\\label{sec:epsilon=0}\nIn the absence of a coherent drive, the right-hand side of Eq.~(\\ref{eqn:6th-order_polynomial}) is zero, and the 6th-order polynomial satisfied by $\\bar\\zeta$ reduces to\n\\begin{equation}\n(1-\\bar\\zeta^2)[P(\\bar\\zeta)]^2=0.\n\\label{eqn:6th-order_polynomial_epsilon=0}\n\\end{equation}\nEquations~(\\ref{eqn:steady-state_alpha1}) and (\\ref{eqn:steady-state_alpha2}) are replaced by the homogeneous system\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\Delta_0\\kappa&-\\Delta\\Delta_0-\\lambda^2(1-\\eta)^2\\bar\\zeta\\\\\n\\noalign{\\vskip4pt}\n\\Delta\\Delta_0+\\lambda^2(1+\\eta)^2\\bar\\zeta&\\Delta_0\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip4pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=0.\n\\label{eqn:steady-state_alpha_epsilon=0}\n\\end{equation}\nNoting then that the determinant of this homogeneous system is $\\lambda^4(1-\\lambda^2)^2P(\\bar\\zeta)$, the condition for nontrivial solutions for $\\bar\\alpha$ is $P(\\bar\\zeta)=0$. Thus, the roots $\\bar\\zeta=\\pm1$ of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}) correspond to the trivial solution, $\\bar\\alpha=0$, while the roots of $P(\\bar\\zeta)=0$,\n\\begin{equation}\n\\bar\\zeta_\\pm=-\\frac{\\Delta\\Delta_0}{\\lambda^2(1-\\eta^2)^2}\\mkern-2mu\\left[1+\\eta^2\\mp2\\eta\\sqrt{1-\\frac{(1-\\eta^2)^2}{4\\eta^2}\n\\frac{\\kappa^2}{\\Delta^2}}\\mkern3mu\\right],\n\\label{eqn:nontrivial_zeta_epsilon=0}\n\\end{equation}\nyield nontrivial solutions for $\\bar\\alpha$. The latter are physically acceptable if $\\bar\\zeta_\\pm$ are real and $|\\bar\\zeta_\\pm|\\leq1$; the first condition is satisfied if $\\eta\\geq\\eta_\\kappa$, $\\eta_\\kappa$ defined in Eq.~(\\ref{eqn:eta_critical}), and the second gives the critical coupling strengths, $\\lambda_\\eta^\\pm$, defined in Eq.~(\\ref{eqn:lambda_critical}); for $\\eta\\geq\\eta_\\kappa$ and $\\lambda_\\eta^+\\leq\\lambda\\leq\\lambda_\\eta^-$, $\\bar\\zeta_+$ is the only acceptable root, while $\\bar\\zeta_+$ and $\\bar\\zeta_-$ are both acceptable if $\\lambda\\geq\\lambda_\\eta^-$.\n\nNote that $\\Delta$ and $\\Delta_0$ are detunings and therefore two cases arise, one with $\\Delta\\Delta_0$ positive and $\\bar\\zeta_\\pm<0$, and the other with $\\Delta\\Delta_0$ negative and $\\bar\\zeta_\\pm>0$. Considering steady states only, there is no physical difference between the cases as a quick inspection of Eqs.~(\\ref{eqn:mean-field_alpha})-(\\ref{eqn:mean-field_zeta}) shows---simply reverse the signs of $\\Delta_0$ and $\\bar\\zeta$ in Eq.~(\\ref{eqn:mean-field_beta}); steady state stability can change, though. We always illustrate results with $\\Delta_0=\\Delta$, whence $\\Delta\\Delta_0$ is positive.\n\nBy eliminating $\\Delta_0\\kappa$ from the homogeneous system, Eq.~(\\ref{eqn:steady-state_alpha_epsilon=0}), we may solve for\n\\begin{eqnarray}\n(\\bar\\alpha_x^\\pm)^2&=&-|\\bar\\alpha_\\pm|^2\\frac{\\Delta\\Delta_0+\\lambda^2(1-\\eta)^2\\bar\\zeta_\\pm}{4\\lambda^2\\eta\\bar\\zeta_\\pm},\n\\label{eqn:nontrivial_alphax_epsilon=0}\\\\\n(\\bar\\alpha_y^\\pm)^2&=&+|\\bar\\alpha_\\pm|^2\\frac{\\Delta\\Delta_0+\\lambda^2(1+\\eta)^2\\bar\\zeta_\\pm}{4\\lambda^2\\eta\\bar\\zeta_\\pm},\n\\label{eqn:nontrivial_alphay_epsilon=0}\n\\end{eqnarray}\nand hence, using Eqs.~(\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}), and the conservation law $\\bar\\zeta^2+|\\bar\\beta|^2=1$, find\n\\begin{equation}\n|\\bar\\alpha_\\pm|^2=-\\frac{\\Delta_0}{4\\Delta}\\frac{1-\\bar\\zeta_\\pm^2}{\\bar\\zeta_\\pm}.\n\\label{eqn:photon_number_epsilon=0}\n\\end{equation}\nThis result gives back Eq.~(\\ref{eqn:photon_number_kappa=0}), with $\\omega\\to\\Delta$ and $\\omega_0\\to\\Delta_0$, when $\\kappa=0$.\n\nFigure \\ref{fig:fig2} displays four cross-sections of the parameter space for $\\epsilon=0$ and $\\Delta_0=\\Delta$, each subdivided according to the number of distinct steady-state solutions. Frames (a) and (c) apply to the non-dissipative model ($\\kappa=0$), while frames (b) and (d) include cavity mode loss. Two complementary perspectives are provided: first, in frames (a) and (b), where the cut is the ($\\lambda\/\\Delta$,$\\eta$)-plane, and then, in frames (c) and (d), where the ($\\Delta\/\\lambda$,$\\eta$)-plane is shown. The first view envisages the coupling strength $\\lambda$, at fixed detuning $\\Delta$, as the control parameter, the historical view suggested by Refs.~\\cite{hepp&lieb_1973a,wang&hioe_1973,hepp&lieb_1973b,carmichael_etal_1973}; the second envisages $\\Delta$ as the control parameter, with $\\lambda$ fixed, which is more natural for experiments in optics and the perspective carried through the remainder of the paper. To connect with Secs.~\\ref{sec:Dicke_rotating_wave} and \\ref{sec:Dicke_counter_rotating}, we note the following points:\n\\begin{enumerate}[(i)]\n\\item\nThe Dicke quantum phase transition in the rotating-wave approximation, originally proposed by Hepp and Lieb \\cite{hepp&lieb_1973a}, maps to the line $\\eta=0$ in frames (a) and (c). The critical point $\\lambda\/\\Delta=\\Delta\/\\lambda=1$ marks a transition from the trivial solution to one with photon number $|\\alpha_\\pm|^2=(\\Delta^4-\\lambda^4)\/4\\lambda^2\\Delta^2$ [Eqs.~(\\ref{eqn:photon_number_eta=0}) and (\\ref{eqn:photon_number_epsilon=0})], where $\\bar\\zeta_\\pm=-\\Delta^2\/\\lambda^2$ is a double root of $P(\\bar\\zeta)=0$;  $\\bar\\beta\/\\bar\\alpha=-2\\Delta\/\\lambda$, but there is no preferred phase for $\\bar\\beta$, since Eqs.~(\\ref{eqn:nontrivial_alphax_epsilon=0}) and (\\ref{eqn:nontrivial_alphay_epsilon=0}) reduce to the tautology $0=0$.\n\\item\nThe $\\eta=0$ transition does not occur in the presence of dissipation, as in frames (b) and (d) the $\\eta=0$ axis bounds only the $R_2$ region.\n\\item\nThe critical point on the line $\\eta=0$ [frames (a) and (c)] splits into a pair of critical points when $\\eta>0$, subdividing the plane into regions of two, three, and four distinct solutions (two, four, and six solutions when double roots of $[P(\\bar\\zeta)]^2=0$ are considered). The transition at $\\lambda_{\\eta=1}^+=\\Delta\/2$ from region $R_2$ to $R_3$ recovers the renormalized critical point \\cite{carmichael_etal_1973} when the rotating-wave approximation is lifted---the $R_2\/R_3$ boundary carries that renormalization through as a function of $\\eta$. To our knowledge, the critical point defining the $R_3\/R_4$ boundary has not been reported before, although Hepp and Lieb do discuss a model that embraces our inclusion of the parameter $\\eta$ \\cite{hepp&lieb_1973b}. The transition between regions $R_3$ and $R_4$ is central to the unification we present with a coherent drive included (Sec.~\\ref{sec:coherent_drive_intermediate_eta}).\n\\item\nContrasting the situation in (i), nontrivial solutions in regions $R_3$ and $R_4$ assign $\\bar\\beta$ and $\\bar\\alpha$ a definite phase, through Eqs.~(\\ref{eqn:steady-state_beta1}), (\\ref{eqn:steady-state_beta2}), (\\ref{eqn:nontrivial_alphax_epsilon=0}), and (\\ref{eqn:nontrivial_alphay_epsilon=0}).\n\\item\nWhile the map from frame (b) to frame (d) appears straightforward, the map from frame (c) to frame (d) is not: a diagram with two boundaries at fixed $\\eta$ now acquires three, as the $R_2\/R_4$ boundary bends up to meet $\\eta=1$. This follows from the term $\\kappa^2\/\\Delta^2$ under the square root in Eq.~(\\ref{eqn:nontrivial_zeta_epsilon=0}): when $\\kappa\\neq0$, $\\bar\\zeta_\\pm$ are complex for $\\eta>\\eta_\\kappa$, a $\\Delta$-dependent condition at fixed $\\kappa$ [Eq.~(\\ref{eqn:eta_critical})].\n\\end{enumerate}\n\nFigure \\ref{fig:fig3} further illustrates the parameter dependence of the mean-field steady states in the absence of a drive. The symmetrical presentation of the phase diagram in frame (a) is modelled after Ref.~\\cite{carmichael_2015} (Figs.~1 and 2) and carried through in  Figs.~\\ref{fig:fig4}, \\ref{fig:fig5}, and \\ref{fig:fig7}. Frames (b)-(e) show steady states and their stability as a function of detuning for $\\eta=0.2$ and $\\eta=0.6$; they illustrate how the regions in frame (a) interconnect as solutions track smoothly with the changing detuning and bifurcate at boundaries:\n\\begin{description}\n\\item[Region $R_2$]\nSolutions $\\bar\\zeta=\\pm1$ only; the solution $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\item[Region $R_3$]\nSolutions $\\bar\\zeta=\\pm1$ and the root $\\bar\\zeta_+$ of $P(\\bar\\zeta)=0$; the solutions $\\bar\\zeta=\\pm1$ are both unstable and $\\bar\\zeta^+$ is stable. Three solutions in total.\n\\item[Region $R_4$]\nSolutions $\\bar\\zeta=\\pm1$ and the  roots $\\bar\\zeta_+$ and $\\bar\\zeta_-$ of $P(\\bar\\zeta)=0$; the solutions $\\bar\\zeta=-1$ (+1) and $\\bar\\zeta_+$ ($\\bar\\zeta_-$) are stable (unstable). Four solutions in total.\n\\end{description}\n\\noindent\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.7in]{figure2a.pdf}\\includegraphics[width=1.7in]{figure2b.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.7in]{figure2c.pdf}\\includegraphics[width=1.7in]{figure2d.pdf}\n\\end{center}\n\\caption{Mean-field phase diagram for zero drive and $\\Delta_0=\\Delta$: (a) $\\kappa\/\\Delta=0$, (b) $\\kappa\/\\Delta=0.7$, (c) $\\kappa\/\\lambda=0$, and (d) $\\kappa\/\\lambda=0.1$. The cut through parameter space is the $(\\eta,\\lambda\/\\Delta)$-plane in (a) and (b), and the $(\\eta,\\Delta\/\\lambda)$-plane in (c) and (d).}\n\\label{fig:fig2}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure3a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure3b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure3c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure3d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure3e.pdf}\n\\end{center}\n\\caption{Mean-field steady states for zero drive and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.1$ and $\\eta=0.2$ [(b),(c)] and $\\eta=0.6$ [(d),(e)]. The two sweeps through the phase diagram are indicated by dashed lines in (a); solid red (dashed blue) lines indicate locally stable (unstable) steady states in (b)-(e).}\n\\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Critical drive strength: $\\Delta_0=0$}\n\\label{sec:critical_drive}\nWe turn now to the dependence on the coherent drive strength, where we begin by identifying the critical point that organizes behavior as function of $\\epsilon$. To this end, we must first give special consideration to $\\Delta_0=0$, a limit not readily recovered from our general solution scheme, due to the $\\Delta_0$ in the denominator of Eqs.~(\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}); we essentially review an analysis presented by Alsing and Carmichael \\cite{alsing&carmichael_1991}, but extended here to arbitrary $\\eta$.\n\nFrom Eqs.~(\\ref{eqn:p_quadratic}) and (\\ref{eqn:q_quadratic}), when $\\Delta_0=0$, $P(\\bar\\zeta)=Q(\\bar\\zeta)=\\bar\\zeta^2$, and the 6th-order polynomial satisfied by $\\bar\\zeta$ becomes\n\\begin{equation}\n(1-\\bar\\zeta^2)\\bar\\zeta^4=\\left(\\epsilon\/\\epsilon_{\\rm crit}\\right)^2\\bar\\zeta^4,\n\\label{eqn:6th-order_polynomial_Delta_0=0}\n\\end{equation}\nwith\n\\begin{equation}\n\\epsilon_{\\rm crit}\\equiv\\frac12\\lambda(1+\\eta),\n\\label{eqn:critical_drive}\n\\end{equation}\nwhere the significance of $\\epsilon_{\\rm crit}$ as a critical drive strength is elaborated below.\nEquations (\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}) carry over in the form\n\\begin{equation}\n\\bar\\alpha_x\\bar\\zeta=\\bar\\alpha_y\\bar\\zeta=0,\n\\label{eqn:alpha_zeta_Delta_0=0}\n\\end{equation}\nand Eqs.~(\\ref{eqn:steady-state_alpha1}) and (\\ref{eqn:steady-state_alpha2}) as\n\\begin{eqnarray}\n\\kappa\\bar\\alpha_x-\\Delta\\bar\\alpha_y-\\lambda\\frac12(1-\\eta)\\bar\\beta_y&=&0,\n\\label{eqn:steady-state_alpha1_Delta_0=0}\\\\\n\\kappa\\bar\\alpha_y+\\Delta\\bar\\alpha_x+\\lambda\\frac12(1+\\eta)\\bar\\beta_x&=&-\\epsilon.\n\\label{eqn:steady-state_alpha2_Delta_0=0}\n\\end{eqnarray}\nWorking then from Eq.~(\\ref{eqn:alpha_zeta_Delta_0=0}), we can identify two distinct classes of solutions, one holding below $\\epsilon_{\\rm crit}$ and the other above.\n\n\\subsubsection{Solutions with $\\bar\\alpha_x=\\bar\\alpha_y=0$ ($\\epsilon\\leq\\epsilon_{\\rm crit}$)}\nEquation (\\ref{eqn:alpha_zeta_Delta_0=0}) may be satisfied with $\\bar\\alpha_x=\\bar\\alpha_y=0$, which, from Eqs.~(\\ref{eqn:steady-state_alpha1_Delta_0=0}) and (\\ref{eqn:steady-state_alpha2_Delta_0=0}), requires\n\\begin{equation}\n\\bar\\beta_x=-\\epsilon\/\\epsilon_{\\rm crit},\\qquad\\bar\\beta_y=0,\n\\end{equation}\nand hence, from the conservation law $\\bar\\zeta^2+|\\bar\\beta|^2=1$,\n\\begin{equation}\n\\bar\\zeta=\\pm\\sqrt{1-\\left(\\epsilon\/\\epsilon_{\\rm crit}\\right)^2}.\n\\label{eqn:zeta_below_Delta_0=0}\n\\end{equation}\nThe same result follows directly from Eq.~(\\ref{eqn:6th-order_polynomial_Delta_0=0}) under the assumption $\\bar\\zeta\\neq0$. This solution is physically acceptable for $\\epsilon\\leq\\epsilon_{\\rm crit}$, though larger drives require Eq.~(\\ref{eqn:alpha_zeta_Delta_0=0}) to be satisfied in another way.\n\n\\subsubsection{Solutions with $\\bar\\zeta=0$ ($\\epsilon\\geq\\epsilon_{\\rm crit}$)}\nEquation (\\ref{eqn:alpha_zeta_Delta_0=0}) may also be satisfied with $\\bar\\zeta=0$, which leaves only the phase of $\\bar\\beta$ to be determined:\n\\begin{equation}\n\\bar\\beta=e^{i\\phi}.\n\\end{equation}\nFrom Eq.~(\\ref{eqn:mean-field_zeta}), the phase of $\\bar\\alpha$ must satisfy\n\\begin{equation}\n{\\rm Im}\\big[\\bar\\alpha(e^{-i\\phi}-\\eta e^{i\\phi})\\big]=0,\n\\end{equation}\nand also, from Eq.~(\\ref{eqn:mean-field_alpha}),\n\\begin{equation}\n\\bar\\alpha=-i\\frac{\\epsilon+\\epsilon_{\\rm crit}(e^{i\\phi}+\\eta e^{-i\\phi})\/(1+\\eta)}{\\kappa+i\\Delta}.\n\\end{equation}\nThe phase $\\phi$ is therefore a solution of the transcendental equation\n\\begin{equation}\n\\epsilon\\cos\\phi+\\epsilon_{\\rm crit}=\\frac{\\Delta\\sin\\phi}{\\kappa(1-\\eta^2)}[\\epsilon(1+\\eta)^2+\\epsilon_{\\rm crit}4\\eta\\cos\\phi].\n\\end{equation}\nIf we then take $\\Delta=0$ as well as $\\Delta_0=0$ (and $\\eta\\neq1$), we arrive at the much simpler equation\n\\begin{equation}\n\\phi=\\cos^{-1}(-\\epsilon_{\\rm crit}\/\\epsilon),\n\\end{equation}\nwith solution $\\phi=\\pi$ for $\\epsilon=\\epsilon_{\\rm crit}$ and two solutions for the phase of $\\bar\\beta$ above $\\epsilon_{\\rm crit}$. This prediction of a bistability in phase above $\\epsilon_{\\rm crit}$ recovers the so-called Spontaneous Dressed-State Polarization of Alsing and Carmichael \\cite{alsing&carmichael_1991} (see also \\cite{kilin&krinitskaya_1991}) but generalized to $\\eta\\neq0$.\n\n\\subsection{Rotating-wave approximation with coherent drive: $\\eta=0$}\n\\label{sec:rotating_wave_coherent_drive_eta=0}\nWe now begin to lay out the connection between the breakdown of photon blockade and the coherently driven extension of the Dicke quantum phase transition. In this section, we introduce the breakdown of photon blockade  as the coherently driven extension of Sec.~\\ref{sec:epsilon=0} in the limit $\\eta=0$. In so doing, we introduce a completely new region of nontrivial steady states, one disconnected and distinct from regions $R_3$ and $R_4$ of Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}. What follows recovers results from Ref.~\\cite{carmichael_2015}.\n\nReturning to the 6th-order polynomial satisfied by $\\bar\\zeta$, Eq.~(\\ref{eqn:6th-order_polynomial}), with $\\eta$ zero, $Q(\\bar\\zeta)=P(\\bar\\zeta)$, and the polynomial takes the simpler form\n\\begin{equation}\n(1-\\bar\\zeta^2)[P(\\bar\\zeta)]^2=\\bar\\epsilon^2\\bar\\zeta^2P(\\bar\\zeta),\n\\label{eqn:6th-order_polynomial_eta=0}\n\\end{equation}\nwith\n\\begin{equation}\nP(\\bar\\zeta)=(\\bar\\Delta_0\\bar\\kappa)^2+(\\bar\\Delta_0\\bar\\Delta+\\bar\\zeta)^2,\n\\label{eqn:p_quadratic_eta=0}\n\\end{equation}\nwhere we have introduced parameters scaled by $\\epsilon_{\\rm crit}$:\n\\begin{equation}\n\\bar\\epsilon\\equiv\\epsilon\/\\epsilon_{\\rm crit},\\qquad (\\bar\\kappa,\\bar\\Delta,\\bar\\Delta_0)\\equiv(\\kappa,\\Delta,\\Delta_0)\/2\\epsilon_{\\rm crit}.\n\\label{eqn:scaled_parameters}\n\\end{equation}\nThe roots of $P(\\bar\\zeta)=0$ are nonphysical (complex) when $\\eta=0$ [Eq.~(\\ref{eqn:nontrivial_zeta_epsilon=0})] and therefore $P(\\bar\\zeta)$ may be cancelled on both sides of Eq.~(\\ref{eqn:6th-order_polynomial_eta=0}), which means there are at most four distinct solutions.\n\nTurning then to the field, the homogeneous system, Eq.~(\\ref{eqn:steady-state_alpha_epsilon=0}), is replaced by\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\bar\\kappa&-\\bar\\Delta-\\bar\\Delta_0^{-1}\\bar\\zeta\\\\\n\\noalign{\\vskip2pt}\n\\bar\\Delta+\\bar\\Delta_0^{-1}\\bar\\zeta&\\bar\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip2pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=\\left(\n\\begin{matrix}\n0\\\\\n\\noalign{\\vskip2pt}\n-\\bar\\epsilon\/2\n\\end{matrix}\n\\right)\n\\label{eqn:inhomogeneous_system_eta=0}\n\\end{equation}\nwith solution for the field amplitude ($\\bar\\Delta_0\\neq0$)\n\\begin{equation}\n\\bar\\alpha=-i\\frac{\\bar\\epsilon\/2}{\\bar\\kappa+i\\left(\\bar\\Delta+\\bar\\Delta_0^{-1}\\bar\\zeta\\right)}.\n\\label{eqn:steady-state_alpha3_eta=0}\n\\end{equation}\nThus, the field mode responds to coherent driving as a resonator in the presence of a nonlinear dispersion, where the dispersion is defined by solutions to Eq.~(\\ref{eqn:6th-order_polynomial_eta=0}). If we then note that $P(\\bar\\zeta)=\\bar\\Delta_0^2\\bar\\epsilon^2\/4|\\bar\\alpha|^2$ [Eqs.~(\\ref{eqn:p_quadratic_eta=0}) and (\\ref{eqn:steady-state_alpha3_eta=0})], whence, from Eq.~(\\ref{eqn:6th-order_polynomial_eta=0}),\n\\begin{equation}\n\\bar\\zeta=\\pm\\frac{|\\bar\\Delta_0|}{\\left(\\bar\\Delta_0^2+4|\\bar\\alpha|^2\\right)^{1\/2}},\n\\end{equation}\nwe recover the autonomous equation of state for the field mode \\cite{carmichael_2015}:\n\\begin{equation}\n\\bar\\alpha=-i\\frac{\\bar\\epsilon\/2}{\\bar\\kappa+i\\left[\\bar\\Delta\\pm\\hbox{sgn}\\left(\\bar\\Delta_0)(\\bar\\Delta_0^2+4|\\bar\\alpha|^2\\right)^{-1\/2}\\right]}.\n\\label{eqn:state_equation_eta=0}\n\\end{equation}\n\nFigure \\ref{fig:fig4} illustrates the results for mean-field steady states obtained from Eqs.~(\\ref{eqn:6th-order_polynomial_eta=0}) and (\\ref{eqn:state_equation_eta=0}) when $\\Delta_0=\\Delta$. The phenomenology follows that mapped out in Fig.~4 of Ref.~\\cite{carmichael_2015}, where regions of two and four distinct solutions [frame (a)] interconnect through the frequency pulling of vacuuum Rabi resonances located  at $\\Delta\/2\\epsilon_{\\rm crit}=\\pm1$ for $\\epsilon\/\\epsilon_{\\rm crit}\\to0$:\n\\begin{description}\n\\item[Region $R_2^a$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\item[Region $R_4$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two additional solutions that arise from the bistable folding of the solution that approaches $\\bar\\zeta=-1$; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable), and the two additional solutions are stable and unstable. Four solutions in total.\n\\item[Region $R_2^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of large detuning; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\end{description}\nWe emphasize that regions $R_2^a$ and $R_2^b$ comprise a single connected region of two distinct solutions in frame (a) of Fig.~\\ref{fig:fig4}; region $R_4$ does not touch the $\\Delta\/2\\epsilon_{\\rm crit}$ axis, although it comes close when $\\kappa\/\\lambda$ is small. We note also that regions $R_4$ of Fig.~\\ref{fig:fig3} and $R_4$ of Fig.~\\ref{fig:fig4} are distinct and do not share a common boundary; their interface occurs away from $\\eta=0$ and is discussed in Sec.~\\ref{sec:coherent_drive_intermediate_eta}.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure4a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure4b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure4c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure4d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure4e.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure4f.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure4g.pdf}\n\\end{center}\n\\caption{Mean-field steady states for $\\eta=0$ and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.02$ and $\\epsilon\/\\epsilon_{\\rm crit}=0.6$ [(b),(c)], $\\epsilon\/\\epsilon_{\\rm crit}=1.0$ [(d),(e)], and $\\epsilon\/\\epsilon_{\\rm crit}=1.2$ [(f),(g)]. The three sweeps through the phase diagram are indicated by dashed lines in (a); solid red (dashed blue) lines indicate stable (unstable) steady states in (b)-(g); dashed black lines demark the range of bistability in (c).}\n\\label{fig:fig4}\n\\end{figure}\n\n\\subsection{The quantum Rabi Hamiltonian with coherent drive: $\\eta=1$}\n\\label{sec:Dicke_coherent_drive_eta=1}\nTaking now the opposite limit, $\\eta=1$, we meet with a region of nontrivial steady states that is contiguous with $R_3$ of Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}. The new region supports four distinct solutions, while $R_3$ supports only three. Nonetheless, the boundary forms a continuous interface since one solution in $R_3$ corresponds to a double root of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0})---a root of $[P(\\bar\\zeta)]^2=0$; the coherent drive lifts this degeneracy and splits one distinct solution into two.\n\nIn order to avoid the divergence of $P(\\bar\\zeta)$ and $Q(\\bar\\zeta)$ as $\\eta\\to1$, we take Eqs.~(\\ref{eqn:p_quadratic}) and (\\ref{eqn:q_quadratic}) over in the form\n\\begin{equation}\n(1-\\eta)^2P(\\bar\\zeta)=4\\bar\\Delta\\bar\\Delta_0\\bar\\zeta+4\\bar\\Delta_0^2(\\bar\\kappa^2+\\bar\\Delta^2),\n\\end{equation}\nand\n\\begin{equation}\n(1-\\eta)^4Q(\\bar\\zeta)=16\\bar\\Delta^2\\Delta_0^2,\n\\end{equation}\nin which case the 6th-order polynomial in $\\bar\\zeta$, Eq.~(\\ref{eqn:6th-order_polynomial}), simplifies as\n\\begin{equation}\n(1-\\bar\\zeta^2)\\mkern-2mu\\left[\\bar\\zeta+\\frac{\\bar\\Delta_0}{\\bar\\Delta}(\\bar\\kappa^2+\\bar\\Delta^2)\\right]^2=\\bar\\epsilon^2\\bar\\zeta^2,\n\\label{eqn:6th-order_polynomial_eta=1}\n\\end{equation}\nagain a 4th-order polynomial with two or four physically acceptable solutions. In the $\\bar\\epsilon\\to0$ limit, the range of four solutions is confined by the inequality\n\\begin{equation}\n\\frac{|\\bar\\Delta_0|}{|\\bar\\Delta|}(\\bar\\kappa^2+\\bar\\Delta^2)\\leq1,\n\\end{equation}\nwhich recovers the $\\lambda_{\\eta\\to1}^+$ threshold of Eq.~(\\ref{eqn:lambda_critical}). Note also that, as advertised, the root  $\\bar\\zeta=-(\\bar\\Delta_0\/\\bar\\Delta)(\\bar\\kappa^2+\\bar\\Delta^2)$ on the $\\bar\\epsilon=0$ boundary is a double root; thus the region $R_4$ of Fig.~\\ref{fig:fig5}---four distinct roots in the interior---interfaces continuously with the three distinct roots of region $R_3$ in Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}.\n\nTurning to the field, from Eqs.~(\\ref{eqn:steady-state_alpha1}) and (\\ref{eqn:steady-state_alpha2}), Eq.~(\\ref{eqn:inhomogeneous_system_eta=0}) ($\\eta=0$) is replaced by\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\bar\\kappa&-\\bar\\Delta\\\\\n\\noalign{\\vskip2pt}\n\\bar\\Delta+\\bar\\Delta_0^{-1}\\bar\\zeta&\\bar\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip2pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=\\left(\n\\begin{matrix}\n0\\\\\n\\noalign{\\vskip2pt}\n-\\bar\\epsilon\/2\n\\end{matrix}\n\\right),\n\\label{eqn:inhomogeneous_system_eta=1}\n\\end{equation}\nwhere the coupling through $\\bar\\zeta$ is no longer symmetrical in the off-diagonals of the matrix on the left-hand side, and is therefore not serving the function of a nonlinear dispersion. Indeed, the physical interpretation for $\\eta=1$ says the coupling through $\\bar\\zeta$ belongs on the right-hand side of Eq.~(\\ref{eqn:inhomogeneous_system_eta=1}) where it acts as a nonlinear drive. The interpretation is made particularly clear if we write\n\\begin{equation}\n\\bar\\beta=\\bar\\Delta_0^{-1}2\\bar\\alpha_x\\bar\\zeta,\n\\end{equation}\nEqs.~(\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}), and then, from $\\bar\\zeta^2+|\\bar\\beta|^2=1$,\n\\begin{equation}\n\\bar\\zeta=\\pm|\\bar\\Delta_0|(\\bar\\Delta_0^2+4\\bar\\alpha_x^2)^{-1\/2}.\n\\end{equation}\nNow, moving the term  $\\Delta_0^{-1}\\bar\\alpha_x\\bar\\zeta$ to the right-hand side of Eq.~(\\ref{eqn:inhomogeneous_system_eta=1}), the equation is rewritten as\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\bar\\kappa&-\\bar\\Delta\\\\\n\\noalign{\\vskip2pt}\n\\bar\\Delta&\\bar\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip2pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=\\left(\n\\begin{matrix}\n0\\\\\n\\noalign{\\vskip2pt}\n-\\bar\\epsilon\/2\\mp\\bar\\alpha_x(\\bar\\Delta_0^2+4\\bar\\alpha_x^2)^{-1\/2}\n\\end{matrix}\n\\right),\n\\label{eqn:alpha_eta=1}\n\\end{equation}\nwhere, if we can assume $4\\bar\\alpha_x^2\\gg\\bar\\Delta_0^2$, we find two solutions with the amplitude of the coherent drive simply changed from $\\bar\\epsilon$ to $\\bar\\epsilon\\pm1$:\n\\begin{equation}\n\\bar\\alpha=-i\\frac{(\\bar\\epsilon\\pm1)\/2}{\\bar\\kappa+i\\bar\\Delta},\n\\label{eqn:alpha_eta=1_approx}\n\\end{equation}\nand $\\bar\\zeta=\\pm|\\bar\\Delta_0|\/|\\bar\\alpha_x|$, $\\bar\\beta=\\pm{\\rm sgn}(\\bar\\Delta_0){\\rm sgn}(\\bar\\alpha_x)$.\n\nMore generally, Fig.~\\ref{fig:fig5} shows the dependence of mean-field steady states on drive amplitude and detuning for $\\eta=1$ and $\\bar\\Delta_0=\\bar\\Delta$; frames (b)-(g) illustrate results for three sweeps through a parameter space that divides into just two separate regions [frame (a)]:\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure5a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure5b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure5c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure5d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure5e.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure5f.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure5g.pdf}\n\\end{center}\n\\caption{Mean-field steady states for $\\eta=1$ and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.02$  and $\\epsilon\/\\epsilon_{\\rm crit}=0.6$ [(b),(c)], $\\epsilon\/\\epsilon_{\\rm crit}=1.0$ [(d),(e)], and $\\epsilon\/\\epsilon_{\\rm crit}=1.2$ [(f),(g)]. The three sweeps through the phase diagram are indicated by dashed lines in (a); solid red (dashed blue) lines indicate stable (unstable) steady states in (b)-(g); dashed black lines demark the range of bistability in (c).}\n\\label{fig:fig5}\n\\end{figure}\n\\begin{description}\n\\item[Region $R_4$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two that approach the root $\\bar\\zeta_+=-\\bar\\kappa^2-\\bar\\Delta^2$ of $[P(\\bar\\zeta)]^2=0$; the solutions that approach $\\bar\\zeta=\\bar\\zeta_+$ ($\\pm1$) are stable (unstable); the solution that approaches $\\bar\\zeta=-1$ links in a closed loop to one of the solutions approaching $\\bar\\zeta_+$. Four solutions in total.\n\\item[Region $R_2^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of large detuning; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\end{description}\nWe note the following additional points:\n\\begin{enumerate}[(i)]\n\\item\nTwo of the four solutions in region $R_4$ are consistent with the assumption adopted above Eq.~(\\ref{eqn:alpha_eta=1_approx}) [frame (c) of Fig.~\\ref{fig:fig5}] so long as $\\bar\\kappa\\ll1$; the remaining two solutions satisfy Eq.~(\\ref{eqn:alpha_eta=1}) but do not admit the approximation leading to Eq.~(\\ref{eqn:alpha_eta=1_approx}).\n\\item\nThe boundary between regions $R_4$ and $R_2^b$ in frame (a) of Fig.~\\ref{fig:fig5} follows the curve\n\\begin{equation}\n\\bar\\epsilon=\\left\\{1-\\left[\\frac{|\\bar\\Delta_0|}{|\\bar\\Delta|}(\\bar\\kappa^2+\\bar\\Delta^2)\\right]^{2\/3}\\right\\}^{3\/2}.\n\\end{equation}\nThe boundary is a line of double roots of Eq.~(\\ref{eqn:6th-order_polynomial_eta=1}), and the curve may be found by equating derivatives on the left- and right-hand sides of this equation.\n\\item\nThe critical point $\\epsilon_{\\rm crit}$ [Eq.~(\\ref{eqn:critical_drive})] organizes behavior as a function of drive strength and detuning in much the same way as it does for  $\\eta=0$.\n\\item\nThe closed loop in frame (b) of Fig.~\\ref{fig:fig5} is similar to the loop in frame (b) of Fig.~\\ref{fig:fig4}; both shrink with increasing drive strength to eventually vanish at the critical point---frames (d) of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5}. Note, though, that the stabilities are interchanged; this change is clearly reflected in the accompanying plots of $|\\bar\\alpha|^2$ [frames (c) of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5}].\n\\item\nThe stable solutions displayed in frames (c), (e), and (g) of Fig.~\\ref{fig:fig5} are all single nearly Lorentzian peaks; the splitting in the corresponding frames of Fig.~\\ref{fig:fig4} does not occur.\n\\end{enumerate}\n\n\\subsection{Intermediate regime: $0<\\eta<1$}\n\\label{sec:coherent_drive_intermediate_eta}\nSummarizing what we have learned: with no counter-rotating interaction, the dissipative Dicke system shows no phase transition as a function of coupling strength [$\\eta=0$ in frames (b) and (d) of Fig.~\\ref{fig:fig1}], although the breakdown of photon blockade takes place in the presence of a coherent drive (Fig.~\\ref{fig:fig4}); the dissipative system does, however, show the standard phase transition when $\\eta=1$, where it is deformed by a coherent drive and vanishes with increasing drive strength at a renormalized photon-blockade-breakdown critical point (Fig.~\\ref{fig:fig5}).\n\nIn this section we unify these limiting cases by letting $\\eta$ vary continuously between 0 and 1. We show how the previously unreported phase of the Dicke system, i.e., region $R_4$ of Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}, underlies this unification.\n\nWe begin with the interface between frame (a) of Fig.~\\ref{fig:fig3} and frame (a) of Fig.~\\ref{fig:fig5}, where regions of three and four distinct solutions connect on the boundary $\\bar\\epsilon=0$, $\\eta=1$: moving off the boundary with a perturbation $\\bar\\epsilon\\to\\delta\\bar\\epsilon$ lifts the degeneracy of a double root of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}), and thus provides the link between regions. Something similar is encountered on the $\\bar\\epsilon=0$ boundary with $\\eta_\\kappa<\\eta<1$ (e.g., along the lines $\\eta=0.6$ and $\\eta=0.2$ in Fig.~\\ref{fig:fig3}); however, now two regions, $R_3$ and $R_4$, link to contiguous regions under the perturbation $\\bar\\epsilon\\to\\delta\\bar\\epsilon$. Since $R_4$ accommodates \\emph{two} double roots of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}), we predict its linkage to a contiguous region of six distinct solutions in the presence of a coherent drive.\n\nWe illustrate this situation in Fig.~\\ref{fig:fig6} where we plot the function $\\sqrt{1-\\bar\\zeta^2}P(\\bar\\zeta)$---the square root of the left-hand side of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0})---for four detunings along the $\\eta=0.2$ sweep of Fig.~\\ref{fig:fig3}: frames (a), (b), (c), (d) refer, in sequence, to points in regions $R_2$, $R_3$, $R_4$, $R_2$ along the sweep---moving inwards from either end; they show examples of two, three, four, and again two distinct roots. The trivial roots, $\\bar\\zeta=\\pm1$, appear in every plot, while the additional roots [frames (b) and (c)] are double roots of $[P(\\bar\\zeta)]^2=0$. The perturbation $\\bar\\epsilon\\to\\delta\\bar\\epsilon$ replaces each dashed line in the figure by a pair of curves $\\pm\\delta\\bar\\epsilon|\\bar\\zeta|\\sqrt{Q(\\bar\\zeta)}$, and thus lifts the degeneracy of each double root. [It is readily shown that $Q(\\bar\\zeta)>0$.]\n\nFigure~\\ref{fig:fig7} shows how the results displayed in Figs.~\\ref{fig:fig3}-\\ref{fig:fig5} are generalized for $\\eta=0.2$ and $\\bar\\Delta_0=\\bar\\Delta$. The parameter space is now divided into a larger number of regions, integrating those already met in the three limiting cases:\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.7in]{figure6a.pdf}\\includegraphics[width=1.7in]{figure6b.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.7in]{figure6c.pdf}\\includegraphics[width=1.7in]{figure6d.pdf}\n\\end{center}\n\\caption{The square root of the left-hand side of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}) as a function of $\\bar\\zeta$ for $\\eta=0.2$ and $\\Delta_0=\\Delta$: $\\kappa\/2\\epsilon_{\\rm crit}=1\/12$ and $|\\Delta|\/2\\epsilon_{\\rm crit}=1.0$, $0.7$, $0.5$, $0.15$ [(a)-(d)]. Zeros of this function (crossings of the black dashed lines) are roots of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}).}\n\\label{fig:fig6}\n\\end{figure}\n\n\\begin{description}\n\\item[Region $R_2^a$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\item[Region $R_6$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and four additional solutions---two that approach each of the double roots, $\\bar\\zeta_\\pm$, of $[P(\\bar\\zeta)]^2=0$. The solutions approaching $\\bar\\zeta=-1$ and $\\bar\\zeta_+$ ($+1$ and $\\bar\\zeta_-$) are stable (unstable). Six solutions in total.\n\\item[Region $R_4^a$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two that approach the double root $\\bar\\zeta_+$ of $[P(\\bar\\zeta)]^2=0$; the solutions approaching $\\bar\\zeta_+$ ($\\pm1$) are stable (unstable). Four solutions in total.\n\\item[Region $R_4^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two additional solutions that arise from the bistable folding of the solution that approaches $\\bar\\zeta=-1$; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable), and the two additional solutions are one stable\/unstable. Four solutions in total.\n\\item[Region $R_2^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of large detuning; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\end{description}\n\nFrames (b)-(e) of Fig.~\\ref{fig:fig7} show how the corresponding plots in Fig.~\\ref{fig:fig3} change when the degeneracy of the double roots ($\\bar\\epsilon=0$) is lifted ($\\bar\\epsilon\\neq0$) to link regions $R_3$ and $R_4$ of Fig.~\\ref{fig:fig3} to regions $R_4^a$ and $R_6$, respectively, of Fig.~\\ref{fig:fig7}. (Note, however, that $\\kappa\/\\lambda$ takes different values in the figures, so region boundaries do not line up.) The change is clearly seen, for example, comparing frame (b) of Fig.~\\ref{fig:fig3} with frames (b) and (d) of Fig.~\\ref{fig:fig7}: a single stable upper branch---Fig.~\\ref{fig:fig3}---is split into two stable upper branches---Fig.~\\ref{fig:fig7}; and a single unstable branch connecting upper and lower stable branches in Fig.~\\ref{fig:fig3}  splits into two unstable branches in Fig.~\\ref{fig:fig7} [near overlapping dashed lines in frame (d)]. In this way features met separately in the limiting cases of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5} are linked together.\n\nFinally, for larger amplitudes of the drive---e.g., adding sweeps at $\\bar\\epsilon=0.6$, 1.0, and 1.2 in frame (a) of Fig.~\\ref{fig:fig7}---mean-field steady states follow the breakdown of photon blockade, as in frames (b)-(g) of Fig.~\\ref{fig:fig4}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure7a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure7b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure7c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure7d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure7e.pdf}\n\\end{center}\n\\caption{Mean-field steady states for $\\eta=0.2$ and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.02$ and $\\epsilon\/\\epsilon_{\\rm crit}=0.2$; [(d),(e)] expands the view in [(b),(c)]. The sweep through the phase diagram is indicated by the dashed line in (a); solid red (dashed blue and magenta) lines indicate stable (unstable) steady states in (b)-(e); dashed black lines demark the range of bistability or tristability in (c).}\n\\label{fig:fig7}\n\\end{figure}\n\n\\section{Quantum Fluctuations: One Two-State System}\n\\label{sec:quantum_fluctuations}\nWhile the mean field analysis may be highly suggestive of what to expect from an experimental realization of our generalized Jaynes-Cummings-Rabi model, an account in these terms is incomplete---fluctuations are neglected. We encounter coexisting steady states, for example, and although both are stable under small perturbations when Maxwell-Bloch equations are solved, what of the stability once quantum fluctuations are introduced?\n\nIt is beyond the scope of this work to address questions like this in any detail. We limit ourselves here to a few observations about the full quantum treatment for the case $N=1$, where a number of calculations are feasible, some analytical and some numerical, to parallel results for the breakdown of photon blockade \\cite{carmichael_2015}. While it may seem that $N=1$ takes us very far from a many-particle limit where contact with mean-field results may be made, this is not generally the case: it is shown in Ref.~\\cite{carmichael_2015} that the many-photon limit is a strong-coupling limit, and many of the figures from Sec.~\\ref{sec:mean-field} have photon numbers ranging in the hundreds for $N=1$---after the scaling of Eq.~(\\ref{eqn:scaling}) is undone.\n\nIn this section, we show that the $\\eta$-dependence of the critical drive strength (Sec.~\\ref{sec:critical_drive}) follows from the quasi-energy spectrum, extending the previous calculation of the spectrum for $\\eta=0$~\\cite{alsing_etal_1992} to the general case. We then address the role of multi-photon resonances in the limit of small $\\eta$, where we uncover behavior similar to multi-photon blockade \\cite{shamailov_etal_2010} under weak coherent driving, but only for even numbers of photons absorbed. Finally, we use quantum trajectories to explore the accessibility of co-existing mean-field steady states in the presence of fluctuations.\n\n\\subsection{Quasienergies for $\\Delta_0=\\Delta=0$}\n\nEver since the seminal work of Jaynes and Cummings \\cite{jaynes&cummings_1963} (see also \\cite{paul_1963})), the energy spectrum of a single two-state system interacting with one mode of the radiation field has been a fundamental element of quantum optics models and physical understanding. The level scheme is remarkably simple when compared with extensions to the quantum Rabi model \\cite{braak_2011} and generalizations to include a counter-rotating interaction after the manner of Sec.~\\ref{sec:Dicke_counter_rotating} \\cite{tomka_etal_2014}. Alsing \\emph{et al}. \\cite{alsing_etal_1992} showed that the simplicity carries over to the driven Jaynes-Cummings Hamiltonian when the two-state system and radiation mode are resonant with the drive. In this case, a Bogoliubov transformation diagonalizes the interaction picture Hamiltonian, so that quasienergies are recovered. The critical drive $\\epsilon_{\\rm crit}$ is then the point at which all quasienergies collapse to zero. In this section we show that the method employed by Alsing \\emph{et al}. carries through for arbitrary $\\eta$, and the collapse to zero reproduces Eq.~(\\ref{eqn:critical_drive}).\n\nWe consider the Hamiltonian $H_\\eta^{\\prime\\prime}=H_\\eta^\\prime+\\sqrt N\\epsilon(a^\\dagger+a)$, where $H_\\eta^\\prime$ is given by Eq.~(\\ref{eqn:hamiltonian_raman_model}). Taking the coherent drive on resonance and considering just one two-state system, the Hamiltonian is\n\\begin{equation}\nH_\\eta^{\\prime\\prime}=\\lambda(a\\sigma_++a^\\dagger\\sigma_-)+\\eta\\lambda(a^\\dagger\\sigma_++a\\sigma_-)+\\epsilon(a^\\dagger+a).\n\\label{eqn:hamiltonian_zero_detuning_one_atom}\n\\end{equation}\nWe seek solutions to the eigenvalue problem $H_\\eta^{\\prime\\prime}|\\psi_E\\rangle=E|\\psi_E\\rangle$, where $E$ is a quasienergy and\n\\begin{equation}\n|\\psi_E\\rangle=|\\psi^{(1)}_E\\rangle|1\\rangle+|\\psi^{(2)}_E\\rangle|2\\rangle,\n\\label{eqn:eigenket}\n\\end{equation}\nwith the kets $|\\psi^{(1,2)}_E\\rangle$ expanded over the Fock states, $|n\\rangle$, $n=1,2,\\ldots$, of the field mode; we must find allowed values of $E$ and the corresponding field kets.\n\nIt is straightforward to show that the field kets satisfy the homogeneous system of equations\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\epsilon(a^\\dagger+a)-E&\\lambda(a^\\dagger+\\eta a)\\\\\n\\noalign{\\vskip4pt}\n\\lambda(\\eta a^\\dagger+a)&\\epsilon(a^\\dagger+a)-E\n\\end{matrix}\n\\mkern2mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n|\\psi^{(1)}_E\\rangle\\\\\n\\noalign{\\vskip4pt}\n|\\psi^{(2)}_E\\rangle\n\\end{matrix}\n\\right)=0,\n\\label{eqn:quasienergy_homogeneous_system}\n\\end{equation}\nwhence multiplication on the left by ${\\rm diag}(\\eta a^\\dagger+a,a^\\dagger+\\eta a)$ takes us to the coupled equations:\n\\begin{eqnarray}\n-\\epsilon(1-\\eta)|\\psi^{(1)}_E\\rangle&=&[\\epsilon(a^\\dagger+a)-E](\\eta a^\\dagger+a)|\\psi^{(1)}_E\\rangle\\notag\\\\\n\\noalign{\\vskip2pt}\n&&+\\lambda[aa^\\dagger+\\eta(a^{\\dagger 2}+a^2)+\\eta^2a^\\dagger a]|\\psi^{(2)}_E\\rangle,\\notag\\\\\n&&\\label{eqn:eigenket_again1}\\\\\n\\epsilon(1-\\eta)|\\psi^{(2)}_E\\rangle&=&[\\epsilon(a^\\dagger+a)-E](a^\\dagger+\\eta a)|\\psi^{(2)}_E\\rangle\\notag\\\\\n\\noalign{\\vskip2pt}\n&&+\\lambda[a^\\dagger a+\\eta(a^{\\dagger 2}+a^2)+\\eta^2aa^\\dagger]|\\psi^{(1)}_E\\rangle.\\notag\\\\\n&&\\label{eqn:eigenket_again2}\n\\end{eqnarray}\nWe then use Eq.~(\\ref{eqn:quasienergy_homogeneous_system}) to substitute for $(\\eta a^\\dagger+a)|\\psi^{(1)}_E\\rangle$ and $(a^\\dagger+\\eta a)|\\psi^{(2)}_E\\rangle$, respectively, on the right-hand sides of Eqs.~(\\ref{eqn:eigenket_again1}) and (\\ref{eqn:eigenket_again2}), and thus obtain the more compact form:\n\\begin{eqnarray}\n\\left(O(E)-\\lambda^2\\frac{1-\\eta^2}2\\right)|\\psi^{(1)}_E\\rangle-\\epsilon\\lambda(1-\\eta)|\\psi^{(2)}_E\\rangle&=&0,\\mkern20mu\n\\label{eqn:eigenket_yet_again1}\\\\\n\\noalign{\\vskip2pt}\n\\left(O(E)+\\lambda^2\\frac{1-\\eta^2}2\\right)|\\psi^{(2)}_E\\rangle+\\epsilon\\lambda(1-\\eta)|\\psi^{(1)}_E\\rangle&=&0,\n\\label{eqn:eigenket_yet_again2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nO(E)&=&\\lambda^2(1+\\eta^2)\\frac{a^\\dagger a+aa^\\dagger}2+\\lambda^2\\eta\\left(a^{\\dagger 2}+a^2\\right)\\notag\\\\\n&&-\\left[\\epsilon\\left(a^\\dagger+a\\right)-E\\right]^2.\n\\label{eqn:o_operator}\n\\end{eqnarray}\nSince the coefficients of the second terms on the left-hand side are constants, Eqs.~(\\ref{eqn:eigenket_yet_again1}) and (\\ref{eqn:eigenket_yet_again2}) can now be readily uncoupled, and yield the autonomous equation\n\\begin{equation}\nO_+(E)O_-(E)|\\psi^{(1,2)}_E\\rangle=0,\n\\label{eqn:oplus_by_ominus_equation}\n\\end{equation}\nwhere\n\\begin{equation}\nO_{\\pm}(E)=O(E)\\pm\\lambda^2\\frac{1-\\eta^2}2\\Lambda^{1\/2},\n\\label{eqn:oplusminus_operator}\n\\end{equation}\nwith\n\\begin{equation}\n\\Lambda=1-\\frac1{(1+\\eta)^2}\\frac{4\\epsilon^2}{\\lambda^2}.\n\\label{eqn:Lambda_definition}\n\\end{equation}\n\nNote now that $O_+(E)$ and $O_-(E)$ commute, and so the general solution to Eq.~(\\ref{eqn:oplus_by_ominus_equation}) expands as\n\\begin{equation}\n|\\psi_{E}^{(1,2)}\\rangle=c_+^{(1,2)}|\\psi_E^{(+)}\\rangle+c_-^{(1,2)}|\\psi_E^{(-)}\\rangle,\n\\label{eqn:oplus_by_ominus_solution}\n\\end{equation}\nwhere $|\\psi_E^{(+)}\\rangle$ and $|\\psi_E^{(-)}\\rangle$ satisfy\n\\begin{equation}\nO_{\\pm}(E)|\\psi_E^{(\\pm)}\\rangle=0.\n\\label{eqn:oplusminus_equation}\n\\end{equation}\nMoreover, the operators $O_{\\pm}(E)$ are quadratic in creation and annihilation operators, so the diagonalization may be completed by a Bogoliubov transformation: introduce parameters $\\nu$, $\\xi$, $\\alpha(E)$, and $\\mu_\\pm(E)$, such that\n\\begin{equation}\nO_\\pm(E)=\\nu U^\\dagger[\\xi,\\alpha(E)]\\frac{a^\\dagger a+aa^\\dagger}2U[\\xi,\\alpha(E)]+\\mu_\\pm(E),\n\\label{eqn:oplusminus_unitary}\n\\end{equation}\nwhere the unitary $U[\\xi,\\alpha(E)]\\equiv D[\\alpha(E)]S(\\xi)$ executes a displacement and then a squeeze,\n\\begin{equation}\na\\buildrel U\\over\\to[a+\\alpha(E)]\\cosh\\xi+[a^\\dagger+\\alpha(E)]\\sinh\\xi,\n\\label{eqn:a_transform}\n\\end{equation}\nwhence, from Eq.~(\\ref{eqn:oplusminus_equation}),\n\\begin{equation}\n\\left(\\frac{a^\\dagger a+aa^\\dagger}2+\\frac{\\mu_\\pm(E)}\\nu\\right)\\!\\left\\{U[\\xi,\\alpha(E)]|\\psi_E^{(\\pm)}\\rangle\\right\\}=0.\n\\end{equation}\nThe number operator now acts on the left-hand side, and $|\\psi_E^{(+)}\\rangle$ and $|\\psi_E^{(-)}\\rangle$ are displaced and squeezed Fock states:\n\\begin{equation}\n|\\psi_{E_{n_\\pm}}^{(\\pm)}\\rangle=U^\\dagger[\\xi,\\alpha(E_{n_\\pm})]|n_\\pm\\rangle,\n\\end{equation}\n$n_\\pm=0,1,2,\\ldots$, where allowed quasienergies are indexed by the integers $n_\\pm$ and must satisfy\n\\begin{equation}\nn_\\pm+\\frac12+\\frac{\\mu_\\pm(E_{n_\\pm})}\\nu=0.\n\\label{eqn:quasienergy_constraint}\n\\end{equation}\nIt remains to equate terms on both sides of Eq.~(\\ref{eqn:oplusminus_unitary}) to fix the parameters of the Bogoliubov transformation, which yields\n\\begin{equation}\n\\nu=\\lambda^2(1-\\eta^2)\\Lambda^{1\/2},\\qquad\n\\xi=\\frac12\\ln\\left(\\frac{1+\\eta}{1-\\eta}\\Lambda^{1\/2}\\right),\n\\label{eqn:paramters1}\n\\end{equation}\nand\n\\begin{eqnarray}\n\\alpha(E)&=&\\frac{2\\epsilon E}{\\lambda^2(1+\\eta)^2}\\Lambda^{-1},\\\\\n\\noalign{\\vskip4pt}\n\\mu_\\pm(E)&=&\\pm\\frac\\nu2-E^2\\Lambda^{-1},\n\\label{eqn:parameters2}\n\\end{eqnarray}\nand thus the allowed quasienergies follow from\n\\begin{equation}\nn_\\pm+\\frac12\\pm\\frac12-E_{n_{\\pm}}^2\\frac1{\\lambda^2(1-\\eta^2)}\\Lambda^{-3\/2}=0.\n\\label{eqn:energies1}\n\\end{equation}\n\nEquation (\\ref{eqn:energies1}) is the targeted result, which reveals the generalized critical drive strength. It is helpful, however, for clarity, to recognize that $n_+$ and $n_-$ provide a double coverage of the nonnegative integers---traced to the \\emph{two} components on the right-hand side of Eq.~(\\ref{eqn:oplus_by_ominus_solution})---and to replace $n_\\pm$ by a single index $n$: first, associate $n=0$ with $n_-=0$, from which Eq.~(\\ref{eqn:energies1}) yields the quasienergy\n\\begin{equation}\nE_0=0,\n\\label{eqn:energy_zero}\n\\end{equation}\nwith corresponding ket\n\\begin{equation}\n|\\psi_{E_0}^{(-)}\\rangle=U^\\dagger[\\xi,\\alpha(E_0)]|0\\rangle;\n\\end{equation}\nand second, associate $n=1,2,\\dots$ with both $n_+=n-1$ and $n_-=n$, both of which, when substituted in Eq.~(\\ref{eqn:energies1}), yield the quasienergy doublet\n\\begin{equation}\nE_{n,\\pm}=\\pm\\lambda\\sqrt n\\sqrt{1-\\eta^2}\\Lambda^{3\/4},\n\\label{eqn:quasienergy_doublet}\n\\end{equation}\nalthough with distinct corresponding kets:\n\\begin{eqnarray}\n|\\psi_{E_{n,\\pm}}^{(+)}\\rangle&=&U^\\dagger[\\xi,\\alpha(E_{n,\\pm})]|n-1\\rangle,\\\\\n\\noalign{\\vskip2pt}\n|\\psi_{E_{n,\\pm}}^{(-)}\\rangle&=&U^\\dagger[\\xi,\\alpha(E_{n,\\pm})]|n\\rangle.\n\\end{eqnarray}\n\nIt is clear from Eq.~(\\ref{eqn:quasienergy_doublet}) that all quasienergies collapse to zero for $n$ finite and $\\Lambda=0$, a condition that returns, from Eq.~(\\ref{eqn:Lambda_definition}), the critical drive strength $\\epsilon_{\\rm crit}$ [Eq.~(\\ref{eqn:critical_drive})]. From this fully quantum mechanical point of view, $\\epsilon_{\\rm crit}$ marks a transition from a discrete quasienergy spectrum to a continuous one; the continuous side is recovered from the limit $\\Lambda\\to0$, $n\\to\\infty$, $\\sqrt n\\Lambda^{3\/4}$ constant. Note that a continuous spectrum is also recovered in the limit $\\eta\\to1$, $n\\to\\infty$, $\\sqrt n\\sqrt{1-\\eta^2}$ constant. A continuous spectrum is expected for $\\eta=1$, since if we set $\\eta=1$ in Eq.~(\\ref{eqn:quasienergy_homogeneous_system}), $E$ is an eigenvalue of the quadrature operator $a^\\dagger+a$.\n\nThe coefficients $c_\\pm^{(1,2)}$ [Eq.~(\\ref{eqn:oplus_by_ominus_solution})] follow from Eqs.~(\\ref{eqn:quasienergy_homogeneous_system}) and (\\ref{eqn:eigenket_yet_again1}), and normalization (see Ref.~\\cite{alsing_etal_1992}).\n\n\\subsection{Multi-photon resonance}\nWith the focus on just one two-state system, Figs.~\\ref{fig:fig4}, \\ref{fig:fig5}, and \\ref{fig:fig7} show photon numbers ranging from zero to a few thousand, and although numbers are smaller in Fig.~\\ref{fig:fig3}, the range is similar when $\\kappa\/\\lambda$ is set to $0.02$ instead of $0.1$. While we might expect mean-field theory to be broadly reliable for thousands, even hundreds of photons, it will surely miss important features when photon numbers are small. Indeed, photon blockade is a photon by photon effect, underpinned, not by a mean-field nonlinearity, but by a strongly anharmonic ladder of few-photon excited states; it breaks down through multi-photon absorption, where, in Fig.~4 of Ref.~\\cite{carmichael_2015}, for example,  multi-photon resonances dominate the response to weak driving and the mean-field story of dispersive bistability is not picked up until $\\epsilon\/\\epsilon_{\\rm crit}\\sim0.4$.\n\nRecall now that in its dissipate realization (Sec.~\\ref{sec:dissipative_realization}) our generalized model involves not one, but two external\ndrives---a linear drive of strength $\\epsilon$, and a second, nonlinear drive of strength $\\eta$. We show now that the multi-photon response to weak driving carries over, with minor modification, from linear to nonlinear driving.\n\nReinstating detuning and setting $\\Delta_0=\\Delta$, we consider the Hamiltonian $H^{\\prime\\prime\\prime}_\\eta=\\Delta a^\\dagger a+\\Delta\\sigma_z+H^{\\prime\\prime}_\\eta$, where $H^{\\prime\\prime}_\\eta$ is given by Eq.~(\\ref{eqn:hamiltonian_zero_detuning_one_atom}). It is convenient for clarity, however, to adopt an interaction picture, where we define\n\\begin{eqnarray}\nH_\\eta^{\\prime\\prime}(t)&\\equiv&U_0^\\dagger(t)H^{\\prime\\prime}_\\eta U_0(t)\\notag\\\\\n\\noalign{\\vskip2pt}\n&=&H_{\\rm JC}+H_{\\epsilon}(t)+H_\\eta(t),\n\\end{eqnarray}\n$U_0(t)\\equiv\\exp[-i\\Delta(a^\\dagger a+\\sigma_z)t]$, and thus isolate the Jaynes-Cummings interaction,\n\\begin{equation}\nH_{\\rm JC}=\\lambda(a\\sigma_++a^\\dagger\\sigma_-),\n\\end{equation}\nwhich is perturbed by the linear drive\n\\begin{equation}\nH_\\epsilon(t)=\\epsilon(a^\\dagger e^{i\\Delta t}+ae^{-i\\Delta t}),\n\\end{equation}\nand the nonlinear drive\n\\begin{equation}\nH_\\eta(t)=\\eta\\lambda(a^\\dagger\\sigma_+e^{2i\\Delta t}+a\\sigma_-e^{-2i\\Delta t}).\n\\end{equation}\nWe also recall the eigenvalues and eigenkets of $H_{\\rm JC}$:\n\\begin{equation}\nE_0^{\\rm JC}=0,\\qquad E_{n,\\pm}^{\\rm JC}=\\pm\\lambda\\sqrt n,\n\\end{equation}\n$n=1,2,\\ldots$, and\n\\begin{eqnarray}\n|E_0^{\\rm JC}\\rangle&=&|0\\rangle|1\\rangle,\\\\\n|E_{n,\\pm}^{\\rm JC}\\rangle&=&\\frac1{\\sqrt2}\\left(|n\\rangle|1\\rangle\\pm|n-1\\rangle|2\\rangle\\right),\n\\end{eqnarray}\nwhere the first (second) ket refers to the field mode (two-state system) in each product on the right-hand side.\n\nNote now that the perturbation $H_\\epsilon(t)$ has non-zero matrix elements between neighboring pairs of kets in the $n$-step sequence\n\\begin{equation}\n|E_0^{\\rm JC}\\rangle\\rightarrow|E_{1,\\pm}^{\\rm JC}\\rangle\\rightarrow\\cdots\\rightarrow|E_{n-1,\\pm}^{\\rm JC}\\rangle\\rightarrow|E_{n,\\pm}^{\\rm JC}\\rangle,\n\\end{equation}\n$n=1,2,\\ldots$, while $H_\\eta(t)$ has non-zero matrix elements between pairs of kets in the $n\/2$-step sequence\n\\begin{equation}\n|E_0^{\\rm JC}\\rangle\\rightarrow|E_{2,\\pm}^{\\rm JC}\\rangle\\rightarrow\\cdots\\rightarrow|E_{n-2,\\pm}^{\\rm JC}\\rangle\\rightarrow|E_{n,\\pm}^{\\rm JC}\\rangle,\n\\end{equation}\n$n=2,4,\\ldots$. There are thus matrix elements to mediate multi-photon transitions from $|E_0^{\\rm JC}\\rangle$ to $|E_{n,\\pm}^{\\rm JC}\\rangle$ driven by either perturbation, but with the qualification that $H_\\eta(t)$ can only drive those with even $n$; resonance is achieved under the condition\n\\begin{equation}\n\\Delta=\\pm\\lambda\/\\sqrt n,\n\\end{equation}\nwhich is met either by $n$ steps of $\\Delta$ off-setting $\\pm\\lambda\\sqrt n$, or $n\/2$ steps of $2\\Delta$.\n\nFrame (a) of Fig.~\\ref{fig:fig8} illustrates the breakdown of photon blockade from a fully quantum mechanical point of view; we identify up to six multi-photon resonances before they begin to merge and wash out due to power broadening at higher drives. This figure displays quantum corrections, for $N=1$, to the mean-field results of Fig.~\\ref{fig:fig4}, where at high drives---$\\epsilon\/\\epsilon_{\\rm crit}=0.40$ and $0.48$---the layout of frame (a) of Fig.~\\ref{fig:fig4} begins to appear with the photon number averaged over fluctuation-driven switching between the pair of coexisting mean-field steady states.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure8a.pdf}\n\\vskip0.1in\n\\includegraphics[width=3.4in]{figure8b.pdf}\n\\end{center}\n\\caption{Steady-state photon number expectation computed from the master equation, Eq.~(\\ref{eqn:master_equation_drive}), for $N=1$, $\\Delta_0=\\Delta$, and $\\kappa\/\\lambda=0.02$: (a) $\\eta=0$ and $\\epsilon\/\\epsilon_{\\rm crit}=0.08$, $0.16$, $0.24$, $0.32$, $0.40$, $0.48$ (lower to upper) and (b) $\\epsilon\/\\epsilon_{\\rm crit}=0$ and $\\eta=0.04$, $0.08$, $0.12$, $0.16$, $0.20$, $0.24$ (lower to upper); successive curves  are displace upwards by $0.2$ and $0.3$ in (a) and (b), respectively.}\n\\label{fig:fig8}\n\\end{figure}\n\nFrame (b) of Fig.~\\ref{fig:fig8} shows the similar figure for driving through the nonlinear perturbation $H_\\eta(t)$. Once again multi-photon resonances are seen, but only three of the previous six---those corresponding to the absorption of two, four, and six photons. The figure in this case adds quantum corrections to the mean-field results of Fig.~\\ref{fig:fig3} (but note that $\\kappa\/\\lambda$ is $0.02$ in Fig~\\ref{fig:fig8} and $0.1$ in Fig.~\\ref{fig:fig3}).\n\n\\subsection{Quantum induced switching between mean-field steady states}\nWhile multi-photon resonances are completely beyond the scope of mean-field results, Fig.~\\ref{fig:fig8} does provide a hint of  mean-field predictions once photon numbers rise above two or three, where, in the vicinity of zero detuning, we see clear evidence of regions $R_2^a$ in Fig.~\\ref{fig:fig4} and $R_2$ in Fig.~\\ref{fig:fig3}. In this section, we use quantum trajectory simulations to further trace connections between the mean-field theory and a full quantum treatment.\n\nNote, first, that unlike the common situation for phase transitions of light, where the many-photon limit is a weak-coupling limit (Secs.~IVA and IVC of Ref.~\\cite{carmichael_2015}), the photon number for our generalized Jaynes-Cummings Rabi model scales with $N(\\lambda\/\\kappa)^2$---i.e., the many-photon limit is a strong-coupling limit; this is seen, for example, from Eq.~(\\ref{eqn:alpha_eta=1_approx}), which, undoing the scaling of Eqs.~(\\ref{eqn:scaling}) and (\\ref{eqn:scaled_parameters}), reads\n\\begin{equation}\n|\\alpha|^2=N\\left(\\frac{\\lambda}{\\kappa}\\right)^2\\frac{(1+\\eta)^2}4\\frac{(\\bar\\epsilon\\pm 1)^2}{1+(\\Delta\/\\kappa)^2}.\n\\label{eqn:photon_number_eta=1_unscaled}\n\\end{equation}\nThe scaling is also apparent from a comparison between frames (c) and (e) of Fig.~3, and frames (c), (e), and (f) of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5}, and frame (c) of Fig.~\\ref{fig:fig7}: with $\\lambda\/\\kappa=10$ in Fig.~\\ref{fig:fig3}, photon numbers range from 4 to 40, while with five times larger coupling in Figs.~\\ref{fig:fig4}, \\ref{fig:fig5}, and \\ref{fig:fig7} they range in the hundreds and thousands; indeed,  frames (c), (e), and (f) of Fig.~\\ref{fig:fig5} rise to reach photon numbers of $6.4\\times10^3$, $10^4$, and $1.21\\times10^4$, respectively, at zero detuning [Eq.~({\\ref{eqn:photon_number_eta=1_unscaled}}]. Such high numbers can be reached with just one two-state system, since, when the coupling is strong, there is no need for a large value of $N$ to offset a weak nonlinearity per photon.\n\nAmongst the many effects of quantum fluctuations, in the following we target just two: first, mean-field steady states that are stable under Maxwell-Bloch equations are expected to be metastable in the presence of quantum fluctuations; and, second, isolated stable steady states---e.g., the lower state in frames (b) and (c) of Fig.~\\ref{fig:fig5} [the minus sign in Eq.~(\\ref{eqn:photon_number_eta=1_unscaled})]---might be accessed via quantum fluctuations. These effects are illustrated in Figs.~\\ref{fig:fig9} and \\ref{fig:fig10}, where we plot quantum trajectories of the photon number expectation while the detuning is slowly swept, from negative to positive. The coupling $\\lambda\/\\kappa=10$ is used in Fig.~\\ref{fig:fig9} in order to keep the maximum photon number relatively low, while the larger value in Fig.~\\ref{fig:fig10} maps to the mean-field results of Fig.~\\ref{fig:fig7}.\n\n\\begin{figure}[htpb!]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure9a.pdf}\n\\vskip0.2in\n\\includegraphics[width=3.4in]{figure9b.pdf}\n\\vskip0.2in\n\\includegraphics[width=3.4in]{figure9c.pdf}\n\\end{center}\n\\caption{Sample quantum trajectories as a function of scanned detuning and steady-state $Q$ functions for $N=1$, $\\Delta_0=\\Delta$, $\\epsilon\/\\epsilon_{\\rm crit}=0.2$, $\\kappa\/\\lambda=0.1$, and $\\eta=1$, $0.8$, $0.6$ (top to bottom); in all frames the detuning is scanned from $\\Delta\/\\lambda=-1$ to $+1$ in a time $\\kappa T=6\\times10^4$. Two sample scans are plotted in each frame (solid yellow and cyan lines) against the background of mean-field steady states (solid red and dashed blue curves). The inset $Q$ functions are for detunings $\\Delta\/2\\epsilon_{\\rm crit}=0$ (left) and $\\Delta\/2\\epsilon_{\\rm crit}=0.04$, $0.015$, $0.04$ (top to bottom) (right).}\n\\label{fig:fig9}\n\\end{figure}\n\nFigure \\ref{fig:fig9} presents a sequence of plots illustrating the role of quantum fluctuations as we move away from the limit of the coherently driven extension of the Dicke phase transition of Sec.~\\ref{sec:Dicke_coherent_drive_eta=1} into the intermediate regime of Sec.~\\ref{sec:coherent_drive_intermediate_eta}. Beginning with $\\eta=1$, the upper frame shows quantum trajectories tracking the two mean-field curves plotted from Eq.~(\\ref{eqn:photon_number_eta=1_unscaled}). Both trajectories (yellow and cyan lines) start on the left by following the higher mean-field branch, but quantum fluctuations allow the isolated [see frames (b) and (c) of Fig.~\\ref{fig:fig5}] lower branch to be accessed too. The two branches correspond to fields that are $\\pi$ out of phase in the imaginary direction at zero detuning---inset $Q$ function to the left---and rotate to eventually align with the real axis as the detuning is changed---inset $Q$ function to the right.\n\nSimilar results are plotted for $\\eta=0.8$ and $\\eta=0.6$ in the middle and bottom frames, respectively. Once again, mean-field curves are faithfully followed over segments of the path, but the switching between branches is more common. The most prominent feature, however, is the dramatic loss of stability around zero detuning: although the mean-field analysis finds a stable steady state at zero photon number [region $R_2^a$ in frame (a) of Fig.~\\ref{fig:fig5}], the full quantum treatment yields fluctuations spanning the two previously stable coherent states; the fluctuations are particularly apparent from the inset $Q$ functions in the middle frame of Fig~{\\ref{fig:fig9}}. The spikes that accompany switches between branches are not numerical artifacts; they are decaying oscillations---evidence of a spiraling trajectory for the field amplitude in the approach to the new locally stable state.\n\nFigure \\ref{fig:fig10} presents the results of two detuning scans for $\\lambda\/\\kappa=50$ and $\\eta=0.2$, corresponding to the parameters of Fig.~\\ref{fig:fig7}. In one scan the quantum trajectory follows the highest branch of stable mean-field solutions all the way up to its maximum. Much more commonly, though, the trajectory switches between this branch and the vacuum state in the region of $\\Delta\/2\\epsilon_{\\rm crit}=\\pm 0.1$, as illustrated by the second scan. In this region the quantum fluctuations show clear evidence of the three coexisting stable mean-field steady states illustrated in frame (e) of Fig.~\\ref{fig:fig7} (region $R_6$)---inset $Q$ function to the right.\n\\begin{figure}[htpb!]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure10.pdf}\n\\end{center}\n\\caption{As in Fig.~{\\ref{fig:fig9}} but for $\\kappa\/\\lambda=0.02$ and $\\eta=0.2$, and with the detuning scanned from $\\Delta\/\\lambda=-0.6$ to $+0.6$ in a time $\\kappa T=6\\times10^4$. The inset $Q$ functions are for detunings $\\Delta\/2\\epsilon_{\\rm crit}=0$ (left) and $\\Delta\/2\\epsilon_{\\rm crit}=0.12$ (right).}\n\\label{fig:fig10}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\\noindent\nWe have generalized the dissipative extension \\cite{dimer_etal_2007} of the Dicke model \\cite{dicke_1954} of light interacting with matter in two directions, thus linking the superradiant phase transition of Hepp and Lieb \\cite{hepp&lieb_1973a,hepp&lieb_1973b} to the breakdown of blockade \\cite{carmichael_2015,fink_etal_2017}. Although the former was originally approached through exact calculations in the thermodynamic limit for $N$ two-state systems in thermal equilibrium, and the latter as a phenomenon of single systems, both might be engineered in many- and one-two-state-system versions, with the same underlying mean-field phenomenology and where the central issue of photon number in the presence of dissipation is governed not by the number of two-state systems only, but also the ratio of coupling strength to photon loss \\cite{carmichael_2015}---even one two-state system can control many photons in cavity and circuit QED \\cite{armen_etal_2009,fink_etal_2017}.\n\nWe adopted a generalization introduced by Hepp and Lieb \\cite{hepp&lieb_1973b}, and taken up in a number of recent publications \\cite{stepanov_etal_2008,schiro_etal_2012,tomka_etal_2014,xie_etal_2014,tomka_etal_2015,wang_etal_2016,moroz_2016,kirton_etal_2018}, where the interaction Hamiltonian is made from a sum of rotating and counter-rotating terms of variable relative strength; in this way we span the continuum from the Jaynes-Cummings to the quantum Rabi interaction. We also added direct driving of the field mode, since that, not the counter-rotating interaction, creates photons in the breakdown of photon blockade. We analyzed mean-field steady states as a function of adjustable parameters for this extended model and found that a common critical drive strength, $\\epsilon_{\\rm crit}=\\lambda(1+\\eta)\/2$, links the superradiant phase transition  to the breakdown of photon blockade---$\\lambda$ is the coupling strength and $\\eta$ the relative strength of counter-rotating to rotating interactions.  More generally, we found that the extended phase diagram moves from a region of pure superradiant character into the region of broken blockade, passing through a phase that although present in the generalized model of Hepp and Lieb \\cite{hepp&lieb_1973b} is not identified in that work.\n\nWe then carried our analysis beyond mean-field steady states to a fully quantum treatment for the limiting case of one two-state system: we extended a prior calculation of quasi-energies \\cite{alsing_etal_1992} to the generalized Hamiltonian---resonant driving of the field mode and no dissipation---and obtained numerical results with both detuning and photon loss  included. The quasi-energy spectrum for one two-state system was shown to be singular at $\\epsilon_{\\rm crit}$, where it undergoes a transition from discrete to continuous, and numerical simulations broadly support mean-field results, though expanding the view from earlier work \\cite{carmichael_2015,shamailov_etal_2010} of multi-photon resonances at weak drive and exhibiting quantum-fluctuation-induced switching amongst locally stable mean-field steady states.\n\nThe aim of this study has been to uncover connections between different dissipative quantum phase transition for light and we have left many directions untouched; for example, a broader investigation of a very rich parameter space and the fully quantum treatment. We expect future work on the theoretical side will fill the gaps and hope that experiments in the spirit of Refs.~\\cite{baumann_etal_2010,baumann_etal_2011,baden_etal_2014,fink_etal_2017,armen_etal_2009} will prove feasible.\n\n\\section*{Acknowledgments}\nThis work was supported by the Marsden fund of the RSNZ. Quantum trajectory simulations were carried out on the NeSI Pan Cluster at the University of Auckland, supported by the Center for eResearch, University of Auckland.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec-Intro}\nA number of different high-energy, $\\geq$ 1GeV, neutrino sources have been proposed in literature, that include active galactic nuclei (AGNs)\\citep{ste91,sza94,nell93, ato01,alv04}, gamma-ray bursts (GRBs)\\citep{wax97,der03,raz04b,mur06,gup07}, supernova remnants \\citep{alv02,cos05} and core collapse supernovae \\citep{wax01, wan07}, although   long duration GRBs have been found to be tightly connected with core-collapse supernovae \\citep{hjo03,sta03}.  Properties of neutrino fluxes, energy range, shape of the energy spectra and flavor content depend on physical conditions in the sources.  \nNeutrinos are useful for studying sources, especially when photons cannot escape directly. They could be the only prompt signatures of the \"hidden\" sources.  These have been associated to core collapse of massive stars leading to supernovae (SNe) of type Ib,c and II with mildly relativistic jets emitted by a central engine, a black hole or a highly magnetized neutron star.  Depending on the initial density and metallicity, the pre-supernova star could have different radii. Type Ic supernovae are believed to be He stars with radius $R_\\star\\approx $ 10$^{11}$ cm and Supernovae of type II and Ib are thought to have a radius of $R_\\star\\approx$ 3$\\times$10$^{12}$ cm. \\\\\nRecently, IceCube reported the detection of two neutrino-induced events with energies between 1- 10 PeV \\citep{aar13}.  These events have been discussed as having an extragalactic origin, for instance; GRBs\\citep{cho12}  and low-luminosity GRBs \\citep{liu13}. On the other hand, high-energy neutrinos are produced in the decay of charged pions and muons when energetic protons in the jet interact with synchrotron thermalized photons or nucleons\/mesons (pp, pn)\/($\\pi$, K) in the shocks.  For internal shocks, synchrotron radiation and the number density of particles could be calculated with enough accuracy if we know the distribution of the magnetic field and the particle momentum in the shocked region.  These quantities are calculated using the energy equipartition hypothesis through the equipartition parameters; electron equipartition ($\\epsilon_e=U_e\/U$) and magnetic equipartition $\\epsilon_B=U_B\/U$\\citep{mes98}. Many authors \\citep{barn12,fra12,sac12,kum10,she10} have estimated these parameters to be  $\\epsilon_e\\simeq$ 0.1, and  0.1$\\leq \\epsilon_B\\leq 10^{-4}$, to obtain a good description of more than a dozen of GRBs.\\\\\nOn the other hand, the neutrino flavor ratio is expected to be, at the source,  $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$=1 : 2 : 0 and on Earth (due to neutrino oscillations between the source and Earth) $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$=1 : 1 : 1 and  $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$=1 : 1.8 : 1.8  for neutrino energies less and greater than 100 TeV, respectively, for gamma ray bursts ($\\phi^0_{\\nu_l}$ is the sum of $\\nu_l$ and $\\bar{\\nu}_l$) \\citep{kas05}.   Also it has been pointed out that measurements of the deviation of this standard flavor ratio of astrophysical high-energy neutrinos may probe new physics \\citep{lea95,ath00, kas05}. \nAs it is known, neutrino properties get modified when it propagates in a medium.  Depending on their flavor, neutrinos interact via neutral and\/or charged currents, for instance,   $\\nu_e$ interacts with electrons via both neutral and charged currents, whereas $\\nu_\\nu(\\nu_\\tau)$ interacts only via the neutral current. This induces a coherent effect in which maximal conversion of $\\nu_e$ into $\\nu_\\mu (\\nu_\\tau)$ takes place. The resonant conversion of neutrino from one flavor to another due to the medium effect is well known as the Mikheyev-Smirnov-Wolfenstein effect  \\citep{wol78}.  \nResonance condition of high-energy neutrinos in  hidden jets has been studied in the literature \\citep{men07,raz10, sah10}.   Recently,  \\citet{2013arXiv1304.4906O} studied the three-flavor neutrino oscillations on the surface of the star  for neutrino energy  in the range  (0.1 - 100) TeV.  They found that  those neutrinos generated on the surface  with  energies of less than 10 TeV could oscillate.    Unlike previous studies, we show that these sources  are capable of generating  PeV neutrinos pointing them out as possible progenitors of the first observation of PeV-energy neutrinos with IceCube \\citep{aar13}. Besides,  we  do a full analysis of resonance conditions (two- and three-flavors) for neutrinos produced at different places in the star,  estimating the flavor ratios on Earth.\\\\\nIn this paper we both show that PeV neutrinos can be produced in hidden jets and  estimate the flavor ratio of high-energy neutrinos expected on Earth.  Firstly, we compute the energy range of neutrinos produced by cooling down of hadrons and mesons accelerated in a mildly relativistic jet. After that we take different matter density profiles to show that neutrinos may oscillate resonantly depending on the neutrino energy and  mixing neutrino parameters.    Finally, we discuss our results in the fail jet framework.\n\\section{Jet dynamics}\nFor the internal shocks, we consider a mildly relativistic shock propagating with bulk Lorentz factor $\\Gamma_b=10^{0.5}\\Gamma_{b,0.5}$.  Behind the shock, the comoving number density of particles   and density of energy are  $n'_e=n'_p=1\/(8\\,\\pi\\,m_p\\,c^5)\\,\\Gamma_b^{-4}\\,E_j\\,t^{-2}_{\\nu,s}\\,t^{-1}_j=3.1\\times10^{18}$ cm$^{-3}\\,\\,t^{-2}_{\\nu,s}$ and $n'_p m_p c^2$, respectively, where we have taken the set of typical values for which the jet drills but hardly breaks through the stellar envelope:  the jet kinetic energy $E_j=10^{51.5} E_{j,51.5}$ erg, the variability time scale of the central object $t_\\nu=t_{\\nu, {\\rm s}}\\,{\\rm s}$  with $t_{\\nu,{\\rm s}}$= 0.1 and 0.01, and  the jet duration $t_j=10\\,t_{j,1}$ s \\citep{raz05,and05,2013MNRAS.432..857M}.   We assume that electrons and protons  are accelerated in the internal shocks to a power-law distribution $N(\\gamma_j) d\\gamma_j\\propto \\gamma_j^{-p} d\\gamma_j$.   The internal shocks due to shell collisions take place at a radium $r_j=2\\Gamma_b^2\\,c\\,t_\\nu= 6 \\times   10^{11}\\,\\rm{cm}\\,\\Gamma^2_{0.5}\\,t_{v,s}$.  Electrons, with minimum energy $E_{e,m}=\\frac{p-2}{p-1} \\epsilon_e\\,m_p c^2 \\Gamma_b$   and maximum energy limited by the dynamic time scale $t'_{dyn}\\simeq t_\\nu\\Gamma_b$, cool down rapidly by synchrotron radiation in the presence of the magnetic field given by\n\\begin{eqnarray}\nB'&=&\\biggl(\\frac{\\epsilon_B}{c^3}\\,\\Gamma_b^{-4}\\,E_j\\,t^{-2}_\\nu\\,t^{-1}_j  \\biggr)^{1\/2}\\cr\n&=& 3.43\\times10^8\\,{\\rm G}\\, \\Gamma_{b,0.5}^{-2}\\,E^{1\/2}_{j,51.5}\\,t^{-1\/2}_{j,1}\\,\\epsilon_B^{1\/2}\\,t^{-1}_{\\nu,s}\\,,\n\\label{mfield}\n\\end{eqnarray}\nwhere here and further on  the magnetic equipartition parameter and $t_{\\nu,{\\rm s}}$  lie in the range  0.1$\\leq \\epsilon_B\\leq 10^{-4}$ and 0.1 $\\leq t_{\\nu,{\\rm s}} \\leq $  0.01, respectively. The radiated  photon energies by electron synchrotron emission  with energy $E_e$ is $E_{syn,\\gamma}=eB'\/(\\hbar  m_e^3c^5) E^2_e$, and  also  the opacity to Thomson scattering by these photons is\n\\begin{eqnarray}\n\\tau_{th}'&=&\\frac{\\sigma_T}{4\\pi\\,m_p\\,c^4} \\Gamma_b^{-3}\\,E_j\\,t^{-1}_\\nu\\,t^{-1}_j\\cr\n&=&3.9\\times 10^5\\, \\Gamma_{b,0.5}^{-3}\\,E_{j,51.5}\\,t^{-1}_{j,1}\\,t^{-1}_{\\nu,s}\\,.\n\\label{opde}\n\\end{eqnarray}\nDue to the large Thomson optical depth, synchrotron photons will thermalize to a black body temperature, therefore the peak energy is given by\n\\begin{eqnarray}\nE'_{\\gamma}\\sim k_B\\,T_{\\gamma}&=&\\biggl(\\frac{15(\\hbar\\,c)^3}{8\\pi^4\\,c^3}\\biggr)^{1\/4}\\,\\epsilon_e^{1\/4}\\, E_j^{1\/4}\\,\\Gamma^{-1}_b\\,t_v^{-1\/2}\\,t^{-1\/4}_j\\cr\n&=&1.36\\,{\\rm keV}\\, E_{j,51.5}^{1\/4}\\,\\Gamma^{-1}_{b,0.5}\\,t^{-1\/4}_{j,1}\\,\\epsilon^{1\/4}_{e,-1}\\,t_{\\nu,s}^{-1\/2}\\,,\n\\label{enph}\n\\end{eqnarray}\nand the number density of thermalized photons is  \n\\begin{eqnarray}\n\\eta'_\\gamma&=&\\frac{2\\,\\zeta(3)}{\\pi^2\\,(c\\,\\hbar)^3}\\,\\biggl(\\frac{15\\,\\hbar\\,\\epsilon_e\\, E_j}{8\\pi^4\\,\\Gamma^{4}_b\\,t_v^{2}\\,t_j}  \\biggr)^{3\/4}\\cr\n&=&2.86 \\times 10^{23} {\\rm cm^{-3}}\\, E^{3\/4}_{j,51.5}\\,\\Gamma^{-3}_{b,0.5}\\,t^{-3\/4}_{j,1}\\,\\epsilon^{3\/4}_{e,-1}\\,t_{\\nu,s}^{-3\/2}\\,.\n\\label{denph}\n\\end{eqnarray}\nAlthough keV photons can hardly escape due to the high optical depth,  they are  able to interact with relativistic protons accelerated in the jet, producing high-energy neutrinos via  charged pion decay. The pion energies  depend on the proton energy and  characteristics of the jet. \n\\section{Hadronic model}\nProtons accelerated in internal shocks, on the one hand,  radiate photons by  synchrotron radiation and  also scatter  the internal photons by  inverse Compton (IC) scattering, and on the other hand, interact with  thermal keV  photons and  hadrons by  p$\\gamma$ and  p-hadron interactions.  The  optical depths for p$\\gamma$  and  p-hadron  interactions are\n\\begin{eqnarray}\n\\tau'_{p\\gamma}&=&\\frac{4\\,\\zeta(3)\\sigma_{p\\gamma}}{\\pi^2\\,(c\\,\\hbar)^3}\\,\\biggl(\\frac{15\\,\\hbar\\,\\epsilon_e\\, E_j}{8\\pi^4\\,\\Gamma^{8\/3}_b\\,t_v^{2\/3}\\,t_j}  \\biggr)^{3\/4}\\cr\n&=&3.19\\times 10^6\\, E^{3\/4}_{j,51.5}\\,\\Gamma^{-2}_{b,0.5}\\,t^{-3\/4}_{j,-1}\\,\\epsilon^{3\/4}_{e,-1}\\,t_{\\nu,s}^{-1\/2}\\,,\n\\label{optpg}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\tau'_{pp}&=&\\frac{\\sigma_{pp}}{4\\,\\pi\\,m_p\\,c^5}\\,  E_j\\, \\Gamma^{-3}_b\\,t_v^{-1}\\,t_j^{-1}  \\cr\n&=&1.77\\times 10^4\\, E_{j,51.5}\\, \\Gamma^{-3}_{b,0.5}\\,t_{j,1}^{-1}\\,t_{\\nu,s}^{-1}\\,,\n\\label{optpp}\n\\end{eqnarray}\nrespectively. Due to the optical depths  for p$\\gamma$ and p-hadron interactions are very high,  p$\\gamma$ and p-hadron are effective, although p-hadron interactions are more effective at lower energy than p$\\gamma$ interactions \\citep{raz04b}.\n\\subsection{Cooling time scales}\nThe shock acceleration time for an energy  proton, $E'_p$, is\n\\begin{eqnarray}\nt'_{acc}&=&\\frac{2\\pi\\xi}{c} r_L =\\frac{2\\pi\\xi\\,c^{1\/2}\\,B'_{c,p}}{m_p^2}\\,E'_p\\,\\epsilon^{-1\/2}_B\\, E^{-1\/2}_j\\,\\Gamma^{2}_b\\,t_v\\,t^{1\/2}_j\\cr\n&=&2.04\\times 10^{-12}{\\rm s}\\,E'_p\\,\\xi\\, E^{-1\/2}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t^{1\/2}_{j,1}\\,\\epsilon^{-1\/2}_B\\,t_{\\nu,s}\\,,\n\\label{tacc}\n\\end{eqnarray}\nwhere $r_L$ is the Larmor's radius and $\\xi$ is a factor of equality. The acceleration time, $t'_{acc}$, gives an account of the maximum  proton energy achieved, when it is  compared with the maximum cooling time scales.  In the following subsections we are going to calculate the cooling time scales for protons and mesons.\n\n\n\\subsubsection{Proton cooling time scales}\n\n\nThe cooling time scale for proton synchrotron radiation is\n\\begin{eqnarray}\nt'_{p,syn}&=&\\frac{E'_p}{(dE'_p\/dt)_{syn}}=\\frac{6\\pi\\,m_p^4\\,c^6}{\\sigma_T\\,\\beta^2\\,m_e^2\\,E'_p}\\,\\epsilon^{-1}_B\\, E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=& 38.3\\, {\\rm s}\\,E'^{-1}_{p,9}\\, E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_B\\,t^2_{\\nu,s}\\,.\n\\label{tsyn}\n\\end{eqnarray}\nProtons  in the shock region  can upscatter the thermal keV photons $E'_{IC,\\gamma}\\sim\\gamma^2_p\\,E'_\\gamma$  with peak energy and density given in eqs.(\\ref{enph}) and (\\ref{denph}).  The IC cooling time scale in the Thomson regimen is\n\\begin{eqnarray}\nt'^{th}_{p,ic}&=&\\frac{E'_p}{(dE'_p\/dt)^{th}_{ic}}=\\frac{m_p^4\\,c^4\\,\\pi^6(c\\,\\hbar)^2}{5\\,\\sigma_T\\,\\beta^2\\,m_e^2\\,\\zeta(3)\\,E'_p}\\,\\epsilon^{-1}_e\\, E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=& 383.1\\, {\\rm s}\\,E'^{-1}_{p,9}\\, E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_{e,-1}\\,t^2_{\\nu,s}\\, .\n\\label{tic}\n\\end{eqnarray}\nAlso, the IC cooling time scale in the Klein-Nishina (KN) regimen,   $E'_pE'_\\gamma\/m_p^2c^4=\\Gamma_{KN}$ with ($\\Gamma_{KN}=1$),  is\n\\begin{eqnarray}\nt'^{KN}_{p,ic}&=&\\frac{E'_p}{(dE'_p\/dt)^{KN}_{ic}}=\\frac{3\\pi^4(c\\,\\hbar)^3 \\,E'_p\\,\\epsilon^{-1\/2}_e\\, E^{-1\/2}_j\\,\\Gamma^{2}_b\\,t_v\\,t^{1\/2}_j }{2\\sqrt{30\\hbar}\\,\\sigma_T\\,\\beta^2\\,m_e^2\\,c^5\\,\\zeta(3)}\\cr\n&=&5.15 \\times 10^{-10} \\, {\\rm s}\\,E'_{p,9}\\,E^{-1\/2}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t^{1\/2}_{j,1}\\,\\epsilon^{-1\/2}_{e,-1}\\,t_{\\nu,s}\\,.\n\\end{eqnarray}\nOn the other hand,  protons could upscatter thermal photons according  to Bethe-Heitler (BH)  process.  The proton energy loss is taken away by the pairs produced in this process.   The cooling time scale for the BH scattering is \n\\begin{eqnarray}\nt'_{BH}&=&\\frac{E'_p}{(dE'_p\/dt)_{BH}}=\\frac{E'_p}{n'\\,c\\sigma_{BH}\\Delta E'_p}\\cr\n&=&\\frac{E'_p\\,(m^2_p\\,c^4+2E'_pE'_\\gamma)^{1\/2}}{2n'_\\gamma\\,m_e\\,c^3\\sigma_{BH}(E'_p+E'_\\gamma)}\\,,\n\\end{eqnarray}\nwhere $\\sigma_{BH}=\\alpha r^2_e\\, ((28\/9) \\,ln[2E'_p\\,E'_\\gamma\/(m_pm_ec^4)]-106\/9)$.\nThe energy loss rate due to pion production for p$\\gamma$ interactions is \\citep{ste68,bec09}\n\\begin{eqnarray}\nt'_{p\\gamma}&=&\\frac{\\pi^2\\,(c\\,\\hbar)^3}{0.3\\,c\\,\\sigma_{p\\gamma}\\,\\zeta(3)}\\,\\biggl(\\frac{8\\pi^4\\,\\Gamma^{4}_b\\,t_v^{2}\\,t_j}{15\\,\\hbar\\,\\epsilon_e\\, E_j}  \\biggr)^{3\/4}\\cr\n&=&1.32\\times10^{-5}\\,\\,{\\rm s}\\, E^{-3\/4}_{j,51.5}\\Gamma^{3}_{b,0.5}\\,t^{3\/4}_{j,1} \\,\\epsilon^{-3\/4}_{e,-1}\\,t_{\\nu,s}^{3\/2}\\,,\n\\end{eqnarray}\nand for p-hadron interactions  is  \\citep{der03,der09}\n\\begin{eqnarray}\nt'_{pp}&=&\\frac{10\\,\\pi\\,m_p\\,c^4}{\\sigma_{pp}}\\,  E_j^{-1}\\, \\Gamma^{4}_b\\,t_v^{2}\\,t_j \\cr \n&=&4.47\\times 10^{-4}\\,{\\rm s}\\,E_{j,51.5}^{-1}\\, \\Gamma^{4}_{b,0.5}\\,t_{j,1} \\,t_{\\nu,s}^{2}\\,.\n\\end{eqnarray}\nIn figs \\ref{ptime_r1}  and \\ref{ptime_r2}  we have plotted the proton cooling time scales  when the magnetic field is distributed in order to 0.1$\\leq \\epsilon_B\\leq 10^{-4}$ and internal shocks take place at  r=$6\\times 10^{9}$ cm and   r=$6\\times 10^{10}$ cm, respectively.\n\\subsubsection{Meson cooling time scales}\nHigh-energy charged pions and kaons produced by p-hadron and p$\\gamma$  interactions    ($p+\\gamma\/p \\to X+\\pi^{\\pm}\/K^{\\pm}$)  radiate  in the presence of the magnetic field  (eq. \\ref{mfield}). Therefore,  their cooling time scales  are  \n\\begin{eqnarray}\nt'_{\\pi^+, syn}&=&  \\frac{E'_{\\pi^+}}{(dE'_{\\pi^+}\/dt)} \\simeq  \\frac{6\\pi c^6 m^4_{\\pi^+}}{\\sigma_T\\,\\beta^2\\,m_e^2}\\,\\epsilon^{-1}_B\\, E^{-1}_j\\,\\Gamma^{2}_b\\,t^2_v\\,t_j \\,E'^{-1}_{\\pi^+}\\cr\n&=&1.9\\times 10^{-2} \\,{\\rm s}\\, E'^{-1}_{\\pi^+,9}\\,E^{-1}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_B\\,t^2_{\\nu,s}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nt'_{K^+, syn}&=&  \\frac{E'_{k^+}}{(dE'_{k^+}\/dt)} \\simeq  \\frac{6\\pi c^6 m^4_{k^+}}{\\sigma_T\\,\\beta^2\\,m_e^2}\\,\\epsilon^{-1}_B\\, E^{-1}_j\\,\\Gamma^{2}_b\\,t^2_v\\,t_j  \\,E'^{-1}_{k^+}\\cr\n&=&2.94 \\,{\\rm s}\\,E'^{-1}_{k^+,9}\\, E^{-1}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_B\\,t^2_{\\nu,s}\\,.\n\\end{eqnarray}\nAs protons can collide with secondary pions and kaons ($\\pi^+ p$ and $K^+p$), then its respective cooling time scale is given by\n\\begin{eqnarray}\nt'_{had}&=&\\frac{10\\,\\pi\\,m_p\\,c^4}{\\sigma_{(pK\/p\\pi^+)}}\\,  E_j^{-1}\\, \\Gamma^{4}_b\\,t_v^{2}\\,t_j \\cr \n&=& 4.47\\times 10^{-9} \\,{\\rm s}\\,   E_{j,51.5}^{-1}\\, \\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,t_{\\nu,s}^{2}\\,.\n\\end{eqnarray}\nHere we have used the cross-section   $\\sigma_{(pK^+\/p\\pi^+)} \\approx 3\\times 10^{-26}$ cm$^2$.  Because the mean lifetime of these mesons may be  comparable with the synchrotron  and hadron time scales in some energy range, it is necessary to consider  the cooling time  scales related to their mean lifetime which are given by\n\\begin{eqnarray}\nt'_{\\pi^+,dec}&=&\\frac{E'_{\\pi^+}}{m_{\\pi^+}c^2}\\,\\tau_{\\pi^+}\\cr\n&=&=1.87\\times 10^{-7} \\,{\\rm s}\\,E'_{\\pi^+,9}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nt'_{K^+,dec}&=&\\frac{E'_{K^+}}{m_{K^+}c^2}\\tau_{K^+}\\cr\n&=&2.51\\times 10^{-8} \\,{\\rm s}\\,E'_{K^+,9}\\,,\n\\end{eqnarray}\nwhere $\\tau_{\\pi^+\/K^+}$is the mean lifetime for $\\pi^+\/K^+$ and E$_{\\pi^+\/K^+,9}$= 10$^9$ E$_{\\pi^+\/K^+}$ eV.\\\\\nIn figs \\ref{mtime_r1}  and \\ref{mtime_r2}  we have plotted the meson cooling time scales  when internal shocks happen at  r=$6\\times 10^{9}$ cm and   r=$6\\times 10^{10}$ cm and the magnetic equipartition parameter is in the range  0.1$\\leq \\epsilon_B\\leq 10^{-4}$.\n\\subsection{Neutrino production}\nThe single-pion production channels are $p+\\gamma\\to n+\\pi^+$ and $p+\\gamma\\to p+ \\pi^0$, where the relevant pion decay chains are $\\pi^0\\to 2\\gamma$, $\\pi^+\\to \\mu^++\\nu_\\mu\\to e^++\\nu_e+\\bar{\\nu}_\\mu+\\nu_\\mu$ and $\\pi^-\\to \\mu^-+\\bar{\\nu}_\\mu\\to e^-+\\bar{\\nu}_e+\\nu_\\mu+\\bar{\\nu}_\\mu$ \\citep{der03}, then the threshold neutrino energy from p$\\gamma$ interaction is  \n\\begin{eqnarray}\nE'_{\\nu,\\pi}&=&2.5\\times 10^{-2}\\biggl(\\frac{8\\pi^4}{15\\hbar}\\biggr)^{1\/4}\\,\\,\\frac{(m^2_\\Delta-m_p^2)}{(1-cos\\theta)}\\cr\n&&\\hspace{3.4cm}\\times\\, \\epsilon^{-1\/4}_e\\, E^{-1\/4}_j\\,\\Gamma_b\\,t^{1\/2}_v\\,t^{1\/4}_j\\cr\n&=& 9.72 \\,{\\rm TeV}\\, E^{-1\/4}_{j,51.5}\\,\\Gamma_{b,0.5}\\,t^{1\/4}_{j,1}\\,   \\epsilon^{-1\/4}_{e,-1}\\,t^{1\/2}_{\\nu,s}\\,.\n\\end{eqnarray}\nComparing the time cooling scales  we can estimate the neutrino break energy for each process. Equaling  $t_{acc}\\simeq t'_{p,syn}$, we can  approximately estimate the maximum proton energy\n\\begin{eqnarray}\nE'_{p,max}&=&\\biggl(\\frac{3\\,e\\,m_p^4\\,c^{11\/2}}{\\sigma_T\\,\\xi\\,\\beta^2\\,m_e^2}\\biggr)^{1\/2}\\,\\epsilon^{-1\/4}_B\\, E^{-1\/4}_j\\,\\Gamma_b\\,t^{1\/2}_v\\,t^{1\/4}_j\\cr\n&=&4.3\\times 10^{3} \\,{\\rm GeV}\\, E^{-1\/4}_{j,51.5}\\,\\Gamma_{b,0.5}\\,t^{1\/4}_{j,1}\\,\\epsilon^{-1\/4}_B\\,t^{1\/2}_{\\nu,s}\\,.\n\\end{eqnarray}\nFrom the condition of the synchrotron cooling time scales  for mesons ($t'_{\\pi^+,syn}=t'_{had}$ and $t'_{K^+,syn}=t'_{had}$), one may roughly define the neutrino break energies  as\n\\begin{eqnarray}\nE'_{\\nu,\\pi^+syn}&=&0.15\\times \\frac{m^4_{\\pi^+}\\,c^2\\,\\sigma_{pp}}{m_p\\,\\sigma_T\\,\\beta^2\\,m_e^2}\\epsilon^{-1}_B\\cr\n&=&10.5 \\,{\\rm GeV}\\,  \\epsilon^{-1}_{B}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nE'_{\\nu,k^+syn}&=&0.3\\times \\frac{m^4_{k^+}\\,c^2\\,\\sigma_{pp}}{m_p\\,\\sigma_T\\,\\beta^2\\,m_e^2}\\epsilon^{-1}_B\\cr\n&=&3.28 \\,{\\rm TeV}\\,\\epsilon^{-1}_{B}\\,.\n\\end{eqnarray}\nFrom the lifetime condition of  cooling time scale  ($t'_{\\pi^+,dec}=t'_{had}$ and  $t'_{K^+,dec}=t'_{had}$),  one again we can obtain  the neutrino break energies, which for these cases are \n\\begin{eqnarray}\nE'_{\\nu, \\pi^+lt}&=&2.5\\frac{\\pi\\,m_p\\,m_{\\pi^+}\\,c^6}{\\sigma_{pp}}\\,\\tau^{-1}_{\\pi^+}  \\,E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=&0.6 \\,{\\rm TeV}\\,E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,t^2_{\\nu,s}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nE'_{\\nu, k^+lt}&=&5\\frac{\\pi\\,m_p\\,m_{k^+}\\,c^6}{\\sigma_{pp}}\\,\\tau^{-1}_{k^+}  \\,E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=&8.92 \\,{\\rm TeV}\\,E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,t^2_{\\nu,s}\\,.\n\\end{eqnarray}\nIt is important to say that muons may be suppressed by electromagnetic energy losses and in that case would not contribute  much to high-energy neutrino production. The ratio $\\frac{t'_{\\pi^+\/K^+,cool}}{t'_{\\pi^+\/K^+,dec}}$, where $t'_{\\pi^+\/K^+,cool}=\\frac{t'_{\\pi^+\/K^+,em}\\cdot \\,\\,\\,t'_{\\pi^+\/K^+,had}} {t'_{\\pi^+\/K^+,em}\\,\\,+\\, \\,\\,t'_{\\pi^+\/K^+,had} }$, determines the suppression of mesons before they decay to neutrinos \\citep{raz05}. \\\\\nIn fig. \\ref{prod_neu} we have plotted the neutrino energy created by distinct interaction processes at  different distances,   $6\\times 10^{9}$ cm (above) and $6\\times 10^{10}$ cm (below),  as a function of the magnetic equipartition parameter.\n\\section{Density profile of the source}\nAnalytical and numerical models of density distribution in a  pre-supernova have shown a decreasing dependence  on radius  $\\rho\\propto r^{-n}$, with n=3\/2 - 3 above the core, being 3\/2 and 3  convective and radiative envelopes respectively \\citep{woo93, shi90, arn91}.  In particular,  distributions with $\\rho \\propto r^{-3}$ and $\\rho \\propto r^{-17\/7}$ have been proposed  to describe simple blast wave distributions \\citep{bet90, che89}.   Following \\cite{men07}, we use three models of  density profile; Model [A], Model [B] and Model [C].\\\\ Model [A] ,\n\\begin{equation}\n\\hspace{0.7cm}{\\rm [A]} ~~\\rho(r) = 4.0\\times 10^{-6} \\left( \\frac{R_\\star}{r} -1\\right)^3 ~{\\rm g~cm}^{-3}\\,,\n\\label{dens-pro-A} \\\\\n\\end{equation}\ncorresponds to a polytropic hydrogen envelope with $\\rho(r)\\propto r^{-3}$, scaling valid in the range $r_{jet}\\geq r \\geq R_\\star$.    Model [B], \n\\begin{eqnarray}\n&&{\\rm [B]}~~\\rho(r) = 3.4\\times 10^{-5} ~{\\rm g~cm}^{-3}\\cr\n&&\\times\n\\cases{ \n(R_\\star\/r)^{17\/7}\\,; \\hspace{1cm}10^{10.8} ~{\\rm cm}<r<r_b = 10^{12} ~{\\rm cm}   &\\nonumber \\cr\n(R_\\star\/r_b)^{17\/7} (r-R_\\star)^5\/(r_b - R_\\star)^5\\,;\\hspace{0.3cm} r>r_b\\,,& \\cr\n} \n\\label{dens-pro-B} \\\\\n\\end{eqnarray}\nis a power-law fit with an effective polytropic index $n_{eff}=17\/7$ as done for SN 1987A \\citep{che89}.  Here $r_j \\sim  10^{10.8} ~{\\rm cm} $ is the radius of inner border of the envelope, where the density $\\rho=0.4$ g cm$^{-3}$. Associating the number of electron per nucleon $Y_e=$0.5, we obtained the number density of electrons as $N_e=N_a\\,\\rho(r)\\, Y_e=$1.2$\\times$10$^{23}$ cm$^{-3}$ and    Model [C], \n \\begin{eqnarray}\n{\\rm [C]} ~~\\rho(r) = 6.3\\times 10^{-6} {\\it A}\n\\left( \\frac{R_\\star}{r} -1 \\right)^{n_{\\rm eff}}  \n~{\\rm g~cm}^{-3} \\cr\n(n_{\\rm eff}, {\\it A})=\n\\cases{ \n(2.1,20) ~;~ &$10^{10.8} ~{\\rm cm}< r < 10^{11} ~{\\rm cm}$ \\cr\n(2.5,1) ~;~ &$r > 10^{11} ~{\\rm cm}$\\,, \\cr\n}\n\\label{dens-pro-C}\n\\end{eqnarray}\nincludes a sharp drop in density at the edge of the helium core \\citep{mat99}. \n\\section{Neutrino Mixing}\nIn the following subsections we are going to describe the neutrino oscillations in the matter (along the jet for three density profiles given in section 4 ) and  in vacuum (its path up to  Earth). We will be using  the best fit parameters for two-neutrino mixing (solar, atmospheric and accelerator neutrino experiments) and three-neutrino mixing.\nThe best fit value of   solar, atmospheric and accelerator neutrino experiments  are given as follow.\\\\\n\\textbf{Solar Neutrinos} are electron neutrinos produced in the thermonuclear reactions which generate the solar energy.  The Sudbury Neutrino Observatory (SNO) was designed to measure the flux of neutrinos produced by $^8$B decays in the sun, so-called $^8$B neutrinos, and to study neutrino oscillations, as proposed by \\cite{che85}.  A two-flavor neutrino oscillation analysis gave the following parameters:  $\\delta m^2=(5.6^{+1.9}_{-1.4})\\times 10^{-5}\\,{\\rm eV^2}$ and $\\tan^2\\theta=0.427^{+0.033}_{-0.029}$\\citep{aha11}.\\\\\n\\textbf{Atmospheric Neutrinos} are electron neutrinos  $\\nu_e$  produced mainly from the decay chain $\\pi\\to \\mu+\\nu_\\mu$ followed by $\\mu\\to e+\\nu_\\mu+\\nu_e$.  Super-Kamiokande (SK) observatory  observes interactions between neutrinos with electrons or with nuclei or water   via the water Cherenkov method. Under a two-flavor disappearance model with separate mixing parameters between neutrinos and antineutrinos there were found the following parameters for the SK-I + II + III data: $\\delta m^2=(2.1^{+0.9}_{-0.4})\\times 10^{-3}\\,{\\rm eV^2}$ and $\\sin^22\\theta=1.0^{+0.00}_{-0.07}$ \\citep{abe11a}.\\\\\n \\textbf{Reactor Neutrinos} are  produced in Nuclear reactors.  Kamioka Liquid scintillator Anti-Neutrino Detector (KamLAND) was initially designed to detect reactor neutrinos and so later it was prepared to measure $^7$Be solar neutrinos. A two neutrino oscillation analysis gives  $\\delta m^2=(7.9^{+0.6}_{-0.5})\\times 10^{-5}\\,{\\rm eV^2}$ and $\\tan^2\\theta=0.4^{+0.10}_{-0.07}$\\citep{ara05,shi07,mit11}.\\footnote{this value was obtained using a global analysis of data from KamLAND and solar-neutrino experiments}.\\\\\n\\textbf{Accelerator Neutrinos} are mostly produced by $\\pi$ decays (and some K decays), with the pions produced by the scattering of the accelerated protons on a fixed target. The beam can contain both $\\mu$- and e-neutrinos and antineutrinos.  There are two categories: Long and short baselines.\\\\\nLong-baseline experiments with accelerator beams run  with a baseline of about a hundred of kilometers.  K2K experiment was designed to measure neutrino oscillations using a man-made beam with well controlled systematics, complementing and confirming the measurement made with atmospheric neutrinos.  $\\delta m^2=(2.8^{+0.7}_{-0.9})\\times 10^{-3}\\,{\\rm eV^2}$ and $\\sin^22\\theta=1.0$\\citep{ahn06}.\\\\\nShort-baseline experiments with accelerator beams run  with a baseline of about hundreds of meters. Liquid Scintillator Neutrino Detector (LSND) was designed to search for $\\nu_\\mu\\to\\nu_e$ oscillations using $\\nu_\\mu$ from $\\pi^+$ decay in flight \\citep{ath96, ath98}.  The region of parameter space has been partly tested by Karlsruhe Rutherford medium energy neutrino KARMEN \\citep{arm02} and MiniBooNe experiments. \\cite{chu02} found two well-defined regions of oscillation parameters with either $\\delta m^2  \\approx  7\\, {\\rm eV^2}$ or $\\delta m^2 < 1\\, {\\rm eV^2} $ compatible with both LAND and KARMEN experiments, for the complementary confidence. The MiniBooNE experiment was specially designed to verify the LSND's neutrino data. It  is currently running at Fermilab and is searching for  $\\nu_e (\\bar{\\nu}_e)$ appearance in a $\\nu_\\mu(\\bar{\\nu}_\\mu)$ beam. Although MiniBooNE found no evidence for an excess of $\\nu_e$ candidate events above 475 MeV in the $\\nu_\\mu\\to\\nu_e$ study, there was observed a 3.0$\\sigma$ excess of electron-like events below 475 MeV\\citep{agu09,agu10,agu07}.  In addition, in the $\\bar{\\nu}_\\mu\\to\\bar{\\nu}_e$ study, MiniBooNE found evidence of oscillations in the 0.1 to 1.0 eV$^2$, which are consistent with LSND results \\citep{ath96, ath98}.\\\\\nCombining solar, atmospheric, reactor and accelerator parameters, the best fit values of the three neutrino mixing  are\n\n for $\\sin^2_{13} < 0.053$: \\citep{aha11}\n \\begin{equation}\n \\Delta m_{21}^2= (7.41^{+0.21}_{-0.19})\\times 10^{-5}\\,{\\rm eV^2};   \\hspace{0.1cm} \\tan^2\\theta_{12}=0.446^{+0.030}_{-0.029}\\,,\n \\end{equation}\nand  for $\\sin^2_{13} < 0.04$: \\citep{wen10}\n\\begin{equation}\n\\Delta m_{23}^2=(2.1^{+0.5}_{-0.2})\\times 10^{-3}\\,{\\rm eV^2};    \\hspace{0.1cm} \\sin^2\\theta_{23}=0.50^{+0.083}_{-0.093}\\,\n\\label{3parosc}\n\\end{equation}\n\\subsection{Neutrino oscillation inside the jet}\nWhen  neutrino oscillations take place in the matter, a resonance could occur that dramatically enhances the flavor mixing and can lead to maximal conversion from one neutrino flavor to another.  This resonance depends on the effective potential,  density profile of the medium,  and oscillation parameters. As  $\\nu_e$ is the one that can  interact  via CC,   the effective potential can be obtained  calculating the difference between the potential due to CC and  NC contributions \\citep{kuo89}.\n\\subsubsection{Two-Neutrino Mixing}\nIn this subsection, we will consider the neutrino oscillation process $\\nu_e\\leftrightarrow \\nu_{\\mu, \\tau}$. The evolution equation for the propagation of neutrinos in the above medium is given by\n\\begin{equation}\ni\n{\\pmatrix {\\dot{\\nu}_{e} \\cr \\dot{\\nu}_{\\mu}\\cr}}\n={\\pmatrix\n{V_{eff}-\\Delta \\cos 2\\theta & \\frac{\\Delta}{2}\\sin 2\\theta \\cr\n\\frac{\\Delta}{2}\\sin 2\\theta  & 0\\cr}}\n{\\pmatrix\n{\\nu_{e} \\cr \\nu_{\\mu}\\cr}},\n\\end{equation}\nwhere $\\Delta=\\delta m^2\/2E_{\\nu}$, $V_{eff}=\\sqrt 2G_F\\, N_e$ is the effective potential,     $E_{\\nu}$ is the neutrino energy,   and $\\theta$ is the neutrino mixing angle. For anti-neutrinos one has to replace $N_e$ by $-N_e$. The conversion probability for a given time $t$ is\n\\begin{equation}\nP_{\\nu_e\\rightarrow {\\nu_{\\mu}{(\\nu_\\tau)}}}(t) = \n\\frac{\\Delta^2 \\sin^2 2\\theta}{\\omega^2}\\sin^2\\left (\\frac{\\omega t}{2}\\right\n),\n\\label{prob}\n\\end{equation}\nwith\n\\begin{equation}\n\\omega=\\sqrt{(V_{eff}-\\Delta \\cos 2\\theta)^2+\\Delta^2 \\sin^2\n    2\\theta}.\n\\end{equation}\nThe oscillation length for the neutrino is given by\n\\begin{equation}\nL_{osc}=\\frac{L_v}{\\sqrt{\\cos^2 2\\theta (1-\\frac{V_{eff}}{\\Delta \\cos 2\\theta})^2+\\sin^2 2\\theta}},\n\\label{osclength}\n\\end{equation}\nwhere $L_v=2\\pi\/\\Delta$ is the vacuum oscillation length. If the density of the medium is such that the condition $\\sqrt2 G_F\\,N_e=\\Delta\\,\\cos2\\theta$ is satisfied,   the resonant condition, \n\n\\begin{equation}\nV_{eff,R}=\\Delta \\cos 2\\theta\\,,\n\\label{reso2d}\n\\end{equation}\ncan come about, therefore the resonance length can be written as\n\\begin{equation}\nL_{res}=\\frac{L_v}{\\sin 2\\theta}.\n\\label{oscres}\n\\end{equation}\nCombining eqs (\\ref{oscres}) and (\\ref{reso2d}) we can obtain the resonance density as a function of resonance length\n{\\scriptsize \n\\begin{equation}\\label{p1}\n\\textbf{$\\rho_R$}=\n\\cases{\n\\frac{3.69\\times 10^{-4}}{E_{\\nu,TeV}}\t\\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{4.4 \\times 10^{12} \\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2}{\\rm gr\/cm^3}       &   {\\rm sol.} \\,, \\nonumber\\cr\n\\frac{1.39\\times 10^{-2}}{E_{\\nu,TeV}}\t\\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{1.18 \\times 10^{11}\\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2} {\\rm gr\/cm^3}     &   {\\rm atmosp.}\\,,\\nonumber\\cr\n\\frac{3.29}{E_{\\nu,TeV}}\t                             \\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{4.9 \\times 10^8\\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2} {\\rm gr\/cm^3}           &   {\\rm accel.}\\,, \\cr\n}\n\\end{equation}\n}\n\\noindent where sol, atmosp. and accel. correspond to solar, atmospheric and accelerator  parameters.\\\\\nIn addition of the resonance condition, the dynamics of this transition must be  determined by adiabatic conversion through the adiabaticity  parameter \n \\begin{equation}\n\\gamma\\equiv \\frac{\\delta m^2}{2E}\\sin2 \\theta\\,\\tan2 \\theta\\frac{1}{\\mid \\frac1\\rho\\, \\frac {d\\rho}{dr}\\mid}_R\\,,\n\\end{equation}\nwith  $\\gamma\\gg$ 1  or the flip probability  given by\n\\begin{equation}\n P_f= e^{-\\pi\/2\\,\\gamma}\\,,\n \\label{flip}\n \\end{equation}\n where $\\rho$ is given by eqs. (\\ref{dens-pro-A}), (\\ref{dens-pro-B})  and (\\ref{dens-pro-C}).\n\n\n\\subsubsection{Three-neutrino Mixing}\nTo determine the neutrino oscillation probabilities we have to solve the evolution equation of the neutrino system in the matter. In a three-flavor framework, this equation is given by\n\\begin{equation}\ni\\frac{d\\vec{\\nu}}{dt}=H\\vec{\\nu},\n\\end{equation}\nand the state vector in the flavor basis is defined as\n\\begin{equation}\n\\vec{\\nu}\\equiv(\\nu_e,\\nu_\\mu,\\nu_\\tau)^T.\n\\end{equation}\nThe effective Hamiltonian is\n\\begin{equation}\nH=U\\cdot H^d_0\\cdot U^\\dagger+diag(V_{eff},0,0),\n\\end{equation}\nwith\n\\begin{equation}\nH^d_0=\\frac{1}{2E_\\nu}diag(-\\Delta m^2_{21},0,\\Delta^2_{32}).\n\\end{equation}\nwith the same potential $V_{eff}$ given for two-neutrino mixing subsection and $U$  the three\nneutrino  mixing matrix given by \\cite{gon03,akh04,gon08,gon11}\n\\begin{equation}\nU =\n{\\pmatrix\n{\nc_{13}c_{12}                    & s_{12}c_{13}                    & s_{13}\\cr\n-s_{12}c_{23}-s_{23}s_{13}c_{12} & c_{23}c_{12}-s_{23}s_{13}s_{12}   & s_{23}c_{13}\\cr\ns_{23}s_{12}-s_{13}c_{23}c_{12}  &-s_{23}c_{12}-s_{13}s_{12}c_{23}   &  c_{23}c_{13}\\cr\n}},\n\\end{equation}\nwhere $s_{ij}=\\sin\\theta_{ij}$ and  $c_{ij}=\\cos\\theta_{ij}$ and we have taken the Dirac phase $\\delta=0$. For anti-neutrinos one has to replace $U$ by $U^*$.  The different neutrino probabilities are given as\n{\\scriptsize \n\\begin{eqnarray}\nP_{ee}&=&1-4s^2_{13,m}c^2_{13,m}S_{31}\\,,\\nonumber\\\\\nP_{\\mu\\mu}&=&1-4s^2_{13,m}c^2_{13,m}s^4_{23}S_{31}-4s^2_{13,m}s^2_{23}c^2_{23}S_{21}-4\nc^2_{13,m}s^2_{23}c^2_{23}S_{32}\\,,\\nonumber\\\\\nP_{\\tau\\tau}&=&1-4s^2_{13,m}c^2_{13,m}c^4_{23}S_{31}-4s^2_{13,m}s^2_{23}c^2_{23}S_{21}-4\nc^2_{13,m}s^2_{23}c^2_{23}S_{32}\\,,\\nonumber\\\\\nP_{e\\mu}&=&4s^2_{13,m}c^2_{13,m}s^2_{23}S_{31}\\,,\\nonumber\\\\\nP_{e\\tau}&=&4s^2_{13,m}c^2_{13,m}c^2_{23}S_{31}\\,,\\nonumber\\\\\nP_{\\mu\\tau}&=&-4s^2_{13,m}c^2_{13,m}s^2_{23}c^2_{23}S_{31}+4s^2_{13,m}s^2_{23}c^2_{23}S_{21}+4\nc^2_{13,m}s^2_{23}c^2_{23}S_{32}\\,,\\nonumber\\\\\n\\end{eqnarray}\n}\nwhere\n\\begin{equation}\n\\sin\n2\\theta_{13,m}=\\frac{\\sin2\\theta_{13}}{\\sqrt{(\\cos2\\theta_{13}-2E_{\\nu}V_e\/\\delta\n    m^2_{32})^2+(\\sin2\\theta_{13})^2}},\n\\end{equation}\nand\n\\begin{equation}\nS_{ij}=\\sin^2\\biggl(\\frac{\\Delta\\mu^2_{ij}}{4E_{\\nu}}L\\biggr).\n\\end{equation}\nHere $\\Delta\\mu^2_{ij}$ are given by \n\\begin{eqnarray}\n\\Delta\\mu^2_{21}&=&\\frac{\\Delta\n  m^2_{32}}{2}\\biggl(\\frac{\\sin2\\theta_{13}}{\\sin2\\theta_{13,m}}-1\\biggr)-E_{\\nu}V_e\\,,\\nonumber\\\\\n\\Delta\\mu^2_{32}&=&\\frac{\\Delta\n  m^2_{32}}{2}\\biggl(\\frac{\\sin2\\theta_{13}}{\\sin2\\theta_{13,m}}+1\\biggr)+E_{\\nu}V_e\\,,\\nonumber\\\\\n\\Delta\\mu^2_{31}&=&\\Delta m^2_{32} \\frac{\\sin2\\theta_{13}}{\\sin2\\theta_{13,m}}\\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\sin^2\\theta_{13,m}&=&\\frac12\\biggl(1-\\sqrt{1-\\sin^22\\theta_{13,m}}\\biggr)\\,,\\nonumber\\\\\n\\cos^2\\theta_{13,m}&=&\\frac12\\biggl(1+\\sqrt{1-\\sin^22\\theta_{13,m}}\\biggr)\\,.\n\\end{eqnarray}\nThe oscillation length for the neutrino is given by\n\\begin{equation}\nL_{osc}=\\frac{L_v}{\\sqrt{\\cos^2 2\\theta_{13} (1-\\frac{2 E_{\\nu} V_e}{\\delta m^2_{32} \\cos 2\\theta_{13}} )^2+\\sin^2 2\\theta_{13}}},\n\\label{osclength}\n\\end{equation}\nwhere $L_v=4\\pi E_{\\nu}\/\\delta m^2_{32}$ is the vacuum oscillation length. From the resonance condition,  $\\sqrt2 G_F\\,N_e=\\Delta \\cos2\\theta_{13}$, the resonance length and density  are related as\n\\begin{equation} \n\\rho_R=\\frac{1.9\\times 10^{-2}}{E_{\\nu,TeV}}\t\\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{8.2 \\times 10^{10} \\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2}{\\rm gr\/cm^3}\\,.\n\\label{p2}\n\\end{equation}\nOn the other hand, generalizing the adiabaticity  parameter, $\\gamma$, to three-mixing neutrinos, it can be written as\n \\begin{equation}\n\\gamma\\equiv \\frac{\\delta m_{32}^2}{2E}\\sin2 \\theta_{13}\\,\\tan2 \\theta_{13}\\frac{1}{\\mid \\frac1\\rho\\, \\frac {d\\rho}{dr}\\mid}_R\\,,\n\\end{equation}\nwith the flip probability given by eq.  (\\ref{flip}).\n\\subsection{Neutrino Oscillation from  Source to Earth}\nBetween the surface of the star and the Earth the flavor ratio $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$   is affected by the full three description  flavor mixing, which is calculated as follow. The probability for a neutrino to oscillate from a flavor estate $\\alpha$ to a flavor state $\\beta$ in a time starting from the emission of neutrino at star t=0, is given as\n\\begin{eqnarray}\nP_{\\nu_\\alpha\\to\\nu_\\beta} &=&\\mid <  \\nu_\\beta(t) | \\nu_\\alpha(t=0) >  \\mid\\cr\n&=&\\delta_{\\alpha\\beta}-4 \\sum_{j>i}\\,U_{\\alpha i}U_{\\beta i}U_{\\alpha j}U_{\\beta i}\\,\\sin^2\\biggl(\\frac{\\delta m^2_{ij} L}{4\\, E_\\nu}   \\biggr)\\,.\n\\end{eqnarray}\nUsing the set of parameters give in eq. (\\ref{3parosc}), we can write the mixing matrix\n\\begin{equation}\nU =\n{\\pmatrix\n{\n0.816669\t   &  0.544650     &     0.190809\\cr\n -0.504583  & 0.513419      &\t 0.694115\\cr\n 0.280085   &  -0.663141    &     0.694115\\cr\n}}\\,.\n\\end{equation}\nAveraging the sin term in the probability to $\\sim 0.5$ for larger distances L \\citep{lea95}, the probability matrix for a neutrino flavor vector of ($\\nu_e$, $\\nu_\\mu$, $\\nu_\\tau$)$_{source}$ changing to a flavor vector  ($\\nu_e$, $\\nu_\\mu$, $\\nu_\\tau$)$_{Earth}$ is given as\n\\begin{equation}\n{\\pmatrix\n{\n\\nu_e   \\cr\n\\nu_\\mu   \\cr\n\\nu_\\tau   \\cr\n}_{Earth}}\n=\n{\\pmatrix\n{\n0.534143\t  & 0.265544\t  & 0.200313\\cr\n 0.265544\t  & 0.366436\t  &  0.368020\\cr\n 0.200313\t  & 0.368020\t  & 0.431667\\cr\n}}\n{\\pmatrix\n{\n\\nu_e   \\cr\n\\nu_\\mu   \\cr\n\\nu_\\tau   \\cr\n}_{source}}\n\\label{matrixosc}\n\\end{equation}\nfor distances longer than the solar system. \n\\section{Results and Discussions}\nWe have considered a core collapse of massive stars leading to supernovae (SNe) of type Ib,c and II with mildly relativistic jets.  Although this mildly relativistic jet may not be able to break through the stellar envelope,  electrons and protons are expected to be accelerated in the internal shocks,   and then to be cooled down by synchrotron radiation, inverse Compton and hadronic processes (p$\\gamma$ and p-hadron\/meson).   Photons from  electron synchrotron radiation thermalized  to a some keV-peak  energy serve  as  cooling mechanism for accelerated protons by means of  p$\\gamma$ interactions. Another cooling mechanism of protons considered here are the  p-p interactions, due to  the high number density of protons  (3.1 $\\times 10^{20}$ cm$^{-3} \\leq n'_p \\leq $ 3.1 $\\times 10^{22}$ cm$^{-3}$ ) \\citep{raz05}.   In p$\\gamma$ and p-p interactions, high-energy pions and kaons are created  which  in turn interact with protons by $\\pi$-p and $K$-p interactions, producing  another hadronic\/meson  cooling mechanism.   To illustrate the degree and  energy region of efficiency of each cooling process, we have plotted the proton (figures \\ref{ptime_r1} and \\ref{ptime_r2}) and meson (figures \\ref{mtime_r1} and \\ref{mtime_r2}) time scales when   internal shocks take place at   $6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm  and, the magnetic field lies in the range  3.4$\\times 10^7$ G  $\\leq B' \\leq$ 1.1$\\times 10^{10}$ G.  Comparing  the time scales in figures \\ref{ptime_r1} and \\ref{ptime_r2}, one can observe that the maximum proton energy is when the acceleration and synchrotron time scales are equal; it happens  when proton energy is in the range $10^{15} eV \\leq E'_p \\leq 10^{16}$  eV which corresponds to  internal shocks at  $6\\times 10^9$ cm  with $B' = 1.1\\times 10^{10}$ G   and  $6\\times 10^{10}$ cm   with $B'=3.4\\times 10^7$ G, respectively.  In figs. \\ref{mtime_r1} and \\ref{mtime_r2},   one can see that hadronic time scales are equal to  other  time scales at different energies.  For instance,  internal shocks at $6\\times 10^{10}$ cm and $B'=1.1\\times 10^{9}$ G,  the time scales of pion synchrotron emission and  hadronic are equal  for  pion energy  $\\sim 5\\times 10^{11}$ eV. Computing the break meson energies for which time scales are equal to each other, we  can estimate the break neutrino energies.   From the equality  of  kaon\/pion lifetime and  synchrotron  cooling time scales we obtain the break neutrino energies  $\\sim$(24\/179) GeV and  $\\sim$428 GeV\/69 TeV, respectively.  Also, considering   p$\\gamma$ interactions  the  threshold neutrino energy $\\sim$ 3 TeV is obtained.  Taking into account the distances of  internal shocks  ($6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm)\nwe have plotted  the neutrino energy  as a function of the magnetic equipartition parameter in the range  0.1$\\leq \\epsilon_B\\leq 10^{-4}$  (3.4$\\times 10^7$ G  $\\leq B' \\leq$ 1.1$\\times 10^{10}$ G).  As shown in the fig.  \\ref{prod_neu},  neutrino energy between 1 - 10 PeV can be generated for  $\\epsilon_B$  between 3.5$\\times 10^{-3}$  and 4.1$\\times 10^{-4}$, that corresponds to a magnetic field in the range  2.02$\\times 10^8$ (2.02$\\times 10^9$) G - 6.9$\\times 10^7$ (6.9$\\times 10^8$) G at $6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm from the central engine, respectively. Under this scenario, chocked jets are bright in high-energy neutrinos and dark in gamma rays.\\\\\nOn the other hand, taking into account the range of neutrino energy  (24 GeV$\\leq E_\\nu\\leq$ 69 TeV),  internal shocks  at a distance of  6$\\times 10^{10}$ cm,  strength of  magnetic field of 1.1$\\times 10^{10}$ G   and considering  three models of density profile  (see section III. eqs. \\ref{dens-pro-A}, \\ref{dens-pro-B} and \\ref{dens-pro-C}) of a pre-supernova star,  we  present a full description of two- and three-flavor neutrino oscillations.   Based on these models of density profiles we calculate the effective potential, the resonance condition and,   the resonance length and density.    From the resonance  condition, we obtain the  resonance density ($\\rho_R$)  as a function of resonance length ($l_R$) for two (eq. \\ref{p1}) and three flavors (eq. \\ref{p2}). We overlap the plots of the density profiles as a function of distance with  the resonance conditions (resonance density as a function of resonance length). They are shown  in Fig \\ref{twoflavor} (two flavors) and in Fig.  \\ref{threeflavor} (three flavors).    For two flavors,   we have taken into account solar (top), atmospheric (middle) and accelerator (bottom) parameters of neutrino experiments.   Using solar parameters, the resonance length is in the range  $\\sim (10^{11} -  10^{14.2})$ cm  and resonance density in $\\sim  (10^{-2} - 10^{-4})$g\/cm$^3$. As can be seen,  neutrinos with energy 24 GeV  are the only ones that meet the resonance condition for all models of density profiles while neutrinos of energy  178 GeV meet marginally the resonance condition just for the model [B].   Neutrinos with other energy cannot meet the resonance condition. Using atmospheric parameters, the resonance length lies in the range $\\sim (10^{9.1} -  10^{13.3})$ cm and the resonance density in $\\sim  (10^{1} - 10^{-4})$g\/cm$^3$.   As shown,  neutrinos in the energy range of 178 GeV - 3 TeV can oscillate many times before leaving the source. Although the resonance length of neutrino with energy 24 GeV is smaller than star radius, the resonance density is greater than other  models.   Using accelerator parameters, the resonance length is less than $\\sim 10^{10.2}$ cm and  the resonance density lies in the range  $\\sim  (10^{2} - 10^{2})$g\/cm$^3$. Although the resonance length is smaller than the star radius for two flavors, the one that meets the resonance density  is the neutrino energy  69 TeV.   For three flavors,  the range of resonance length  is $\\sim (10^{9} -  10^{12.5})$ cm  and resonance density is  $\\sim  (0.9 - 10^{-4})$g\/cm$^3$, presenting a similar behavior to that described by means of atmospheric parameters.   \nAs the dynamics of resonant transitions  is not only determined by the resonance condition, but also by adiabatic conversion,  we plot the flip probability as a  function of neutrino energy for two  (fig \\ref{twoflip}) and three flavors (fig \\ref{threeflip}). Dividing the plots of flip probabilities in three regions of less than 0.2,  between 0.2 and 0.8 and greater than 0.8, we have that  in the first case (P$_\\gamma \\leq$ 0.2), a pure adiabatic conversion  occurs, the last case (P$_\\gamma \\geq$ 0.8) is a strong violation of adiabaticity  and the intermediate region 0.2 $<$ P$_\\gamma$ $<$ 0.8 represents the transition region \\citep{dig00}. In Fig. \\ref{twoflip},  the top, middle and bottom  plots are obtained  using solar, atmospheric and  accelerator parameters of neutrino oscillations, respectively.    As shown in top figure,  the pure adiabatic conversion occurs when neutrino energy is  less than $5\\times 10^{11}$ eV for model [A] and [C] and,   $\\sim 10^{12}$ eV for model [B] and, the strong violation of adiabaticity is given for neutrino energy greater than $6\\times 10^{12}$ eV  in the three profiles.    In the middle figure, one can see that independently of the profile, neutrinos with energy of less than E$_\\nu$=10$^{14}$ eV can have pure adiabatic conversions.    In the bottom figure,  the three models of  density profiles have the same behavior for the whole energy range.   Neutrinos with   energy less than $\\sim 10^{11.3}$ eV and  greater than  $\\sim 10^{13.2}$ eV present conversion adiabatically pure and strong violation, respectively.   In fig. \\ref{threeflip},   the flip probability for three flavors are plotted.   The energy range for each region of P$_\\gamma$ changes marginally   according to the model of  density profile.  Neutrinos with E$\\sim 10^{12}$ eV are capable of having  pure adiabatic conversion in [B] but not in [A] or [C].  The strong violation of adiabaticity begins when the neutrino energies are E$\\sim 10^{13}$ eV and E$\\sim 10^{13.8}$ eV, for [A] and [C], respectively. \\\\\nOn the other hand,   we have also plotted (fig. \\ref{proen}) the oscillation probabilities for three flavors  as a function of energy when neutrinos keep moving at a distance of   r=$10^{11}$ cm (above) and r=$10^{12}$ cm (below) from the core.   In the top figure, the survival probability of electron neutrino, P$_{ee}$, is close to one regardless of  neutrino energy, therefore  the conversion probabilities  P$_{\\mu e}$ and P$_{\\tau e}$ are close to zero, as shown.  Depending on the neutrino energy, the survival probability of muon and tau neutrino, P$_{\\mu \\mu}$ and P$_{\\tau \\tau}$, oscillates between zero and one. For example,   for E$\\sim 430$ GeV,  the conversion probability of muon  P$_{\\mu \\tau}$ is close to zero while the survival probability of muon and tau neutrino, P$_{\\mu \\mu}$ and P$_{\\tau \\tau}$,  are close to one, and for E$\\sim 1$ TeV  probabilities change dramatically, being P$_{\\mu \\tau}\\sim 1$ and P$_{\\tau \\tau}$=P$_{\\mu \\mu}\\sim$ 0. In the bottom figure, neutrinos are moving along the jet  at r=10$^{12}$ cm and  although the survival and conversion probabilities  have  similar behaviors to those moving to  r=10$^{11}$ cm,  they  are changing faster.  To have a better understanding, we have separated all probabilities and plotted them in fig.  \\ref{prosep}.\nFrom up to down,  the probabilities of  electron neutrino and   survival probability of muon neutrino are shown in the first and second graph, respectively,   and  the conversion and survival probability of tau  neutrino are plotted in the third and four graph, respectively. Moreover, we have plotted in figs. \\ref{prob_dist} and \\ref{prob_dist2} the oscillation probabilities  as a function of distance, when  neutrinos are produced at a radius  $6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm, respectively,  and continue to propagate along the jet. We take into account  four neutrino energies $E_\\nu$=178 GeV, $E_\\nu$=428 GeV, $E_\\nu$=3 TeV and $E_\\nu$=69 TeV.  \nAs shown, as neutrino energy  increases, the probabilities oscillate less. For instance,  when an electron neutrino with energy $E_\\nu$=178 GeV  propagates along the jet,  the survival probability of electron changes from one at  $\\sim 8\\times 10^{10}$ cm to zero at $\\sim 9.5\\times 10^{10}$ cm.   For  $E_\\nu$=428 GeV(3 TeV), the survival probabilities  change from one at  $9.1\\times 10^{10}$ ($6.0\\times 10^{10}$) cm  to zero at $1.8\\times 10^{11}$($3.5\\times 10^{11}$) cm  and for $E_\\nu$=69 TeV, the probability  is constant in this range (greater than $\\sim 10^{12}$ cm).  In the last case, neutrino does not oscillate to another flavor during its propagation.\nFinally, considering a flux ratio for $\\pi$, K and $\\mu$ decay  of 1:  2:  0, the density profile [A] and  oscillation probabilities at three distances (10$^{11}$ cm,  10$^{11.5}$ cm and 10$^{12}$ cm), we show in table 1 the flavor ratio on the surface of star. Also, computing the vacuum oscillation effects between the source and Earth (Eq. \\ref{matrixosc}),  we estimate and show in table \\ref{flaratio}  the flavor ratio expected on Earth when neutrinos emerge  from the star at L=(10$^{11}$,  10$^{11.5}$  and 10$^{12}$) cm . \n\n\n\\begin{table}\n\\begin{center}\\renewcommand{\\tabcolsep}{0.2cm}\n\\renewcommand{\\arraystretch}{0.89}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n$E_{\\nu}$  &$\\phi_{\\nu_e}:\\phi_{\\nu_\\mu}:\\phi_{\\nu_\\tau}$ &$\\phi_{\\nu_e}:\\phi_{\\nu_\\mu}:\\phi_{\\nu_\\tau}$&$\\phi_{\\nu_e}:\\phi_{\\nu_\\mu}:\\phi_{\\nu_\\tau}$  \\\\\n(TeV)&(L=10$^{11}$ cm)&(L=10$^{11.5}$ cm)&(L=10$^{12}$ cm)\\\\ \\hline\n\n0.024     &  0.946:1.949:0.115  &  0.697:1.405:0.899   &  0.881:1.578:0.541  \\\\\\hline\n\n0.178     & 0.510:1.814:0.676  &  0.987:1.386:0.627  &  0.507:1.807:0.686 \\\\\\hline\n\n0.428     & 0.983:1.589:0.428  &  0.659:1.871:0.524   &  0.538:1.721:0.741\\\\\\hline\n\n3            & 0.896:1.212:0.892   &  0.502:1.753:0.744   &  0.501:1.762:0.737 \\\\\\hline\n\n68.5     &  0.999:1.997:0.003   &  0.998:1.972:0.030   &  0.979:1.746:0.275 \\\\\\hline\n\n\\end{tabular}\n\\label{tatm}\n\\end{center}\n\\caption{\\small\\sf The flavor ratio on the surface of source for five neutrino energies  (E$_{\\nu}$=24 GeV, 178 GeV, 428 GeV, 3 TeV and 68.5 TeV), leaving the star to three distances L=10$^{11}$ cm, 10$^{11.5}$ cm, and 10$^{12}$ cm.  }\n\\label{flaratio}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\\renewcommand{\\tabcolsep}{0.2cm}\n\\renewcommand{\\arraystretch}{0.89}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n$E_{\\nu}$  &$\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$ &$\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$&$\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$  \\\\\n(TeV)&(L=10$^{11}$ cm)&(L=10$^{11.5}$ cm)&(L=10$^{12}$ cm)\\\\ \\hline\n\n0.024     &  1.046:1.008:0.956   &   0.925:1.031:1.045 &  0.998:1.011:0.991 \\\\\\hline\n\n0.178     &   0.889:1.049:1.062  &   1.021:1.000:0.978 &  0.888:1.049:1.063 \\\\\\hline\n\n0.428     &   1.033:1.000:0.966  &  0.954:1.053:1.047  &  0.893:1.046:1.061 \\\\\\hline\n\n3             &   0.979:1.010:1.011  &   0.883:1.049:1.067  &  0.883:1.050:1.067 \\\\\\hline\n\n68.5       &   1.065:0.998:0.936  &   1.063:0.999:0.939 &   1.042:1.001:0.957\\\\\\hline\n\n\\end{tabular}\n\\label{tatm}\n\\end{center}\n\\caption{\\small\\sf The flavor ratio expected on Earth for five neutrino energies  (E$_{\\nu}$=24 GeV, 178 GeV, 428 GeV, 3 TeV and 68.5 TeV), leaving the star to three distances L=10$^{11}$ cm, 10$^{11.5}$ cm, and 10$^{12}$ cm.}\n\\label{flaratio}\n\\end{table}\n\\section{Summary and conclusions}\nWe have done a wide description of  production channels  of high-energy neutrinos in a middle relativistic hidden jet and also shown that neutrinos with energies between  1 - 10 PeV can be generated. Taking  into account a particular range of neutrino energies  generated  in the internal shocks at a distance of  6$\\times 10^{10}$ cm and with a distribution of  magnetic field  1.1$\\times 10^{10}$ G,  we have shown  their oscillations between flavors along the jet for  three models of density profiles.  For two neutrinos mixing,  we have used the fit values of neutrino oscillation parameters from solar, atmospheric, and accelerator experiments and  analyzing the resonance condition we found that  the resonance lengths are the largest  and  resonance densities are the smallest for solar parameters and  using  accelerator parameters we have obtained the opposite situation,  the resonance lengths are the smallest  and  resonance densities are the largest. The most favorable condition for high-energy neutrinos  to oscillate resonantly before going out of the source is given through atmospheric parameters and   these conversions would be pure adiabatic.\nFor three neutrino mixing,  we have calculated the ratio flavor on the surface of the source as well as that expected on Earth. Our analysis shows that deviations  from 1:1:1 are obtained  at different energies and places along the jet, which is given in table 2.     From analysis of flip probability we also show that neutrinos may oscillate depending on their energy and the parameters of neutrino experiments.    As a particular case, when the three-flavor  parameters are considered (fig. \\ref{threeflip}), we obtain that neutrino energies above  $\\geq$ 10 TeV can hardly oscillate, obtaining the same result given by \\citet{2013arXiv1304.4906O}.\\\\  As shown,  depending on the flavor ratio obtained on Earth we could differentiate  the progenitor,  its density profile at different depths in the source,  as well as understand similar features between lGRBs and core collapse supernovae.  Distinct times of arrival of neutrino flavor ratio will provide constraints on density profiles at different places in the star  \\citep{bar12}. \nThese observations in detectors such as IceCube, Antares and KM3Net would be a compelling evidence that chocked jets  are bright in neutrinos \\citep{abb12, abb13, pra10,lei12}.\nThe number of sources with hidden jets  may be much larger than the exhibited one, limited only by the ratio of type Ib\/c and type II SNe to GRB rates. Within 10 Mpc, the rate of core-collapse supernovae is $\\sim$1 - 3 yr$^{-1}$, with a large contribution of galaxies around  3 - 4 Mpc.   At larger distances,  the expected number of neutrino events in IceCube is still several, and the supernova rate is $\\geq$ 10 yr$^{-1}$ at 20 Mpc \\citep{and05}. Recently,  \\citet{tab10} calculated the events expected in DeepCore and neutrino-induced cascades in km$^3$ detectors for neutrinos energies $\\leq$ 10 GeV and $\\leq$  a few TeV respectively and forecast that $\\sim$ 4 events in DeepCore and $\\sim$ 6 neutrino-induced cascades in IceCube\/KM3Net  would be expected. An extension up to higher energies of this  calculation  should be done to correlate the expected events  in these sources with  the number of PeV-neutrinos  observed with IceCube \\citep{aar13}. \\\\  \nInterference effects in the detector by atmospheric neutrino oscillation are very  small (less than 10 \\%) due to short   path traveled by neutrinos in comparison with cosmological distances \\citep{men07}.\n\n\\section*{Acknowledgements}\n\nWe thank the referee  for a critical reading of the paper and valuable suggestions. We also thank B. Zhang, K. Murase, William H. Lee, Fabio de Colle, Enrique Moreno and Antonio Marinelli for useful discussions.   NF gratefully acknowledges a Luc Binette-Fundaci\\'on UNAM Posdoctoral Fellowship.\n \n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{intro}\n\nWe study transition fronts for one-dimensional reaction-diffusion equations\nwith \\emph{compactly-perturbed ignition-monostable reactions}.\nConsider the evolution PDE\n\\begin{equation}  \\label{eq:main}\nu_t = u_{xx} + f(x,u),\\quad (t,x)\\in \\mathbb{R}\\times \\mathbb{R},\n\\end{equation}\nwhere the nonlinearity $f$ satisfies the following on $\\mathbb{R}\\times [0,1]$:\n\\begin{enumerate}[label=(F\\arabic*)]\n  \\setlength\\itemindent{10pt}\n\\item $f\\ge 0$ is Lipschitz continuous with $\\gamma:=\\mbox{Lip}(f)$, and  $f(x,0)=f(x,1) = 0$ for all $x\\in \\mathbb{R}$;\n  \\label{item:general}\n  \n\\item there exists $L>0$ such that $f(x,u)\\equiv f_0(u)$ for all $|x|\\ge L$, where $f_0$ is an ignition reaction with $f_0\\equiv 0$ on $[0,\\theta_0]\\cup \\{1\\}$, $f_0>0$ on $(\\theta_0 ,1)$, and $f_0$ is non-increasing on $[1-\\theta_1,1]$ for some $\\theta_0,\\theta_1\\in (0,1)$;\n  \\label{item:ignition}\n  \n\\item \\label{item:regularity} the (right hand) derivative $a(x):=f_u(x,0)\\ge 0$ exists, and for all $\\varepsilon>0$, there exists $\\zeta=\\zeta(\\varepsilon)\\in(0,\\theta_0)$ such that\n  \\begin{equation*}\n   \n    (1-\\varepsilon)a(x)u\\leq f(x,u)\\leq (a(x)+\\varepsilon)u\\quad \\mbox{for }(x,u)\\in \\mathbb{R}\\times [0,\\zeta].\n  \\end{equation*}\n\\end{enumerate}\nAs described above, $f$ is obtained by perturbing a homogeneous ignition reaction $f_0$ locally on the interval $[-L,L]$ with an inhomogeneous monostable reaction.\nIn the present work, we are interested in how such perturbation affects the existence of transition fronts.\n\nThe PDE \\eqref{eq:main} and its variations are widely used to model a host of natural processes, including thermal, chemical, and ecological dynamics.\nBy \\ref{item:general}, $u\\equiv 0, 1$ are two equilibrium solutions of \\eqref{eq:main}.\nTherefore, one is usually interested in the transition from the (unstable) state $u\\equiv 0$ to the (stable) state $u\\equiv 1$.\n\\emph{Transition fronts} are a class of solutions that model this phenomenon.\nThey are global-in-time solutions $u:\\mathbb{R}^2 \\to (0,1)$ of \\eqref{eq:main} satisfying\n\\begin{equation}  \\label{item:trans_lims}\n\\lim_{x\\to-\\infty} u(t,x)=1, \\quad\\lim_{x\\to+\\infty} u(t,x)=0\\quad \\mbox{for all }t\\in \\mathbb{R}\n\\end{equation}\nand the bounded front width condition, that is, for all $\\mu \\in (0,\\frac 12 )$,\n\\begin{equation}  \\label{item:trans_width}\n\\sup _{t\\in \\mathbb{R}} L_\\mu(t):=\\sup_{t\\in\\mathbb{R}}\\, \\mbox{diam} \\{x\\in\\mathbb{R}|\\; \\mu\\le u(t,x)\\le 1-\\mu\\}<\\infty.\n\\end{equation}\nThis definition was introduced in \\cite{BH12, Matano, Sen}.\n\nThe study of transition fronts has seen much activity since the seminal works by Fisher \\cite{F37} and Kolmogorov, Petrovskii, and Piskunov \\cite{KPP37}, who first studied \\emph{traveling fronts} for \\eqref{eq:main} with \\emph{homogeneous Fisher-KPP reactions}.\nHere, traveling fronts are transition fronts of the form $u(t,x)=U(x-ct)$ for some speed $c\\in \\mathbb{R}$ and profile $U$ with $\\lim_{y\\to-\\infty}U(y)=1$, $\\lim_{y\\to\\infty}U(y)=0$,\nand Fisher-KPP reactions are those $f$ satisfying \\ref{item:general}, $f'(0)>0$, and $0<f(u)\\le f'(0)u$ for $u\\in(0,1)$.\nIn their work, they found that for each $c\\ge c_*:=2\\sqrt{f'(0)}$, \\eqref{eq:main} admits a unique (modulo translation) traveling front, while no front exists for $c<c_*$.\nBy means of phase plane analysis, it can be shown that the same existence result holds for general homogeneous monostable reactions ($f(u)>0$ on $(0,1)$), although $c_*\\ge 2\\sqrt{f'(0)}$ in general (e.g., see \\cite{AW78}).\nIn contrast, for \\emph{homogeneous ignition} (defined as in \\ref{item:ignition}) and \\emph{bistable reactions} (the same as ignition except $f<0$ on $(0,\\theta_0)$ and $\\int _0^1 f(u)du>0$), there is only one speed $c_*>0$ which gives rise to a unique (up to translation) traveling front. The unique speed $c_*$ will be called \\emph{the spreading speed of }$f$.\n\nOver decades, the study of transition fronts extended to spatially periodic reactions (in which case fronts have time-periodic profiles, and are known as \\emph{pulsating fronts}).\nInstead of surveying the vast literature, let us refer to the review articles by Berestycki \\cite{B03} and Xin \\cite{X00}, and the references therein.\nThe development in general inhomogeneous media is considerably more recent.\nThe first existence result was obtained by Vakulenko and Volpert \\cite{VV11} for small perturbations of homogeneous bistable reactions.\nLater, Mellet, Roquejoffre, and Sire \\cite{MARS10} proved the existence of fronts for ignition reactions of the form $f(x,u)=a(x)f_0(u)$, where $f_0$ is ignition, and $a(x)$ is bounded with $\\inf _{\\mathbb{R}}a(x)>0$, which need not be close to being constant (see also \\cite{NR09} for the case of random media, relying on the notion of generalized random traveling waves developed in \\cite{Sen}). Zlato\\v{s} then extended these results (along with uniqueness and stability) to general inhomogeneous ignition and mixed ignition-bistable media \\cite{Z13, ZPreprint}.\n\n\nTransition fronts has also been investigated in inhomogeneous Fisher-KPP media by several authors. As far as Fisher-KPP reactions are concerned, a strong inhomogeneity in the reaction may prevent existence of transition fronts, while a weak inhomogeneity gives rise to them.\nThis is translated into the following result proved by Nolen, Roquejoffre, Ryzhik and Zlato\\v{s} \\cite{Zlatos} for reactions satisfying $0< f(x,u)\\le a(x)u$ for all $(x,u)\\in \\mathbb{R}\\times (0,1)$, with $a(x):=f_u(x,0)$, $a_-:=\\inf_{x\\in \\mathbb{R}}a(x)>0$, and $a(x)-a_-\\in C_c(\\mathbb{R})$.\nThey found that when the inhomogeneity of $f$ is strong, in the sense that the principal eigenvalue $\\lambda$ of the operator $\\partial_{xx}+a(x)$ satisfies $\\lambda>2a_-$, any non-constant global-in-time solution $u$ of \\eqref{eq:main} is \\emph{bump-like} (i.e. $u(t,x)\\le C_t e^{-c|x|}$), preventing the existence of transition fronts.\nThis in fact is the first known example of a reaction function $f$ such that \\eqref{eq:main} does not admit any transition front.\n\nMoreover, in the same work, they also show that the existence criterion is (almost) sharp.\nIn the case of a weak localized inhomogeneity $\\lambda<2a_-$, for each $c\\in (2\\sqrt{a_-},\\lambda\/\\sqrt{\\lambda-a_-})$ the PDE \\eqref{eq:main} admits a transition front with \\emph{global mean speed} $c$, in the sense that if $X(t):=\\sup \\{x\\in\\mathbb{R}:u(t,x)=\\frac 12\\}$, then\n\\begin{equation}\\label{gms}\n  \\lim_{t-s\\to\\infty}\\frac{X(t)-X(s)}{t-s}=c.\n\\end{equation}\nTo construct a front, they find an appropriate pair of ordered global-in-time super- and sub-solutions $w\\ge v$ that propagate with speed $c$, and recover a front $u$ between them as a locally uniform limit along a subsequence of solutions $(u_n)_{n\\in {\\mathbb{N}}}$ of the Cauchy problem \\eqref{eq:main} with initial data $u_n(-n,\\cdot)=w(-n,\\cdot)$.\nThe same method was deployed and extended by Zlato\\v{s} \\cite{Z12} and by Tao, Zhu, and Zlato\\v{s} \\cite{TZZ13} to prove the existence of fronts for general inhomogeneous KPP and monostable reactions when $a(x)-a_-$ is not compactly supported.\n\nIn the present paper, we modify the approach from \\cite{Zlatos} to establish a similar sharp existence criterion for reactions satisfying Hypothesis (F).\nAs mentioned, such $f$ is obtained by locally perturbing the ignition reaction $f_0$ with a monostable reaction.\nWe therefore show that a strong perturbation in the reaction prevents the existence of fronts, while a weak perturbation admits them.\nThe existence criterion in our case is determined by the spreading speed of the reaction $f_0$ and the supremum of the spectrum of the operator $\\partial_{xx}+a(x)$.\nThe spreading speed of $f_0$ is the unique number $c_0>0$ such that the following ODE admits a unique (up to translation) solution:\n\\begin{equation}\n  \\label{tfeq}\n  U''+c_0U' +f_0(U)=0,\\quad \\lim_{x\\to -\\infty} U(x)=1,\\quad \\lim_{x\\to\\infty} U(x)=0.\n\\end{equation}\nOn the other hand, the supremum of the spectrum of $\\partial_{xx}+a(x)$ is given by\n\\begin{equation*}\n \n  \\lambda := \\sup \\sigma (\\partial_{xx}+a(x))= \\sup_{{\\psi \\in H^1(\\mathbb{R})}:\\,||\\psi||_{L^2}=1} {\\int_\\mathbb{R} (-[\\psi'(x)]^2 + a(x) [\\psi (x)]^2)dx}.\n\\end{equation*}\nSince $a(x)\\ge 0$ is compactly supported by \\ref{item:ignition}, the essential spectrum of $\\partial_{xx}+a(x)$ is $(-\\infty,0]$, which implies $\\lambda\\ge 0$.\nIf $\\lambda>0$ (i.e. $a\\not\\equiv 0$), it is in fact the principal eigenvalue.\nThen a corresponding $L^\\infty$-normalized principal eigenfunction $\\psi$ exists, is unique, and satisfies\n\\begin{equation}\n  \\label{eq:eigen}\n  \\psi '' + a(x)\\psi = \\lambda \\psi,\\quad \\psi>0,\\quad ||\\psi||_{L^\\infty}=1.\n\\end{equation}\n\nThe main results of the present work are stated as follows.\n\n\\begin{theorem}\n\t\\label{thm:nonex}\n\tLet $f$ satisfy \\ref{item:general}--\\ref{item:regularity} for some $f_0$ and $a$, $c_0$ be the spreading speed of $f_0$, $\\lambda$ be the supremum of the spectrum of $\\partial_{xx}+a(x)$,\n\tand assume $\\lambda > c_0^2$.\n\t\n\t\\begin{enumerate}\n\t\t\\item All entire solutions $u$ of \\eqref{eq:main} with $0<u<1$, $\\inf_{(t,x)\\in \\mathbb{R}^2} u=0$ are bump-like. That is, there are $c,C_t>0$ such that $u(t,x)\\le C_te^{-c|x|}$ for all $(t,x)\\in \\mathbb{R}^2$.\n\t\tIn particular, \\eqref{eq:main} does not admit a transition front solution.\n\t\t\n\t\t\\item Assume \\ref{item:regularity} is replaced by the following: there exists $\\zeta  \\in (0,\\theta_0)$ such that $f(x,u) = a(x)u$ for $u\\in [0,\\zeta]$.\n\t\tThen a nonzero bump-like solution of \\eqref{eq:main} exists, is unique (up to a time-shift) among all solutions with $0<u<1$, $\\inf _{(t,x)\\in\\mathbb{R}^2}u=0$,\n\t\tand satisfies $u(t,x)=\\zeta e^{\\lambda t}\\psi(x)$ for all $t\\le 0$ after a time translation, where $\\psi$ is the principal eigenfunction given in \\eqref{eq:eigen}.\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\n\t\\label{thm:ex}\n\tLet $f,$ $c_0,$ and $\\lambda$ be as in Theorem \\ref{thm:nonex}.\n\tIf $\\lambda < c_0^2$, then \\eqref{eq:main} admits a transition front, which is increasing-in-time and has a global mean speed $c_0$.\n\\end{theorem}\n\n\\begin{remark}\nIf \\eqref{eq:main} does not admit a stationary solution other than $u\\equiv 0, 1$ (i.e. $f(\\cdot,\\theta)\\not\\equiv 0$ for all $\\theta \\in (0,1)$), then any entire solution $u\\not\\equiv 1$ to \\eqref{eq:main} must satisfy $\\inf _{(t,x)\\in \\mathbb{R}^2} u=0$ (see the proof of Lemma \\ref{lem:limit}).\nHence in this case, Theorem \\ref{thm:nonex} holds without the infimum assumption.\n\\end{remark}\n\n\\begin{remark}\nIt can been seen in the proof that Theorem \\ref{thm:nonex} still holds if we replace $f(x,u)=f_0(u)$ in \\ref{item:ignition} by $f(x,u)\\le f_0(u)$ for $|x|\\ge L$.\n\\end{remark}\n\n\\begin{remark}\nThe transition front $u$ constructed in Theorem \\ref{thm:ex} satisfies $\\sup _{t\\in \\mathbb{R}} |X\n(t) - c_0t|<\\infty$, which is a stronger condition than \\eqref{gms}. \nIn fact, $u$ can be viewed as a perturbation of a traveling front $U$ of $f_0$.\nBy the asymptotic stability of ignition fronts, one can further show that $u$ converges to $U$ as $t\\to\\pm\\infty$ in the sense that\n\\[ \\lim_{t\\to\\pm \\infty} \\sup_{x\\in\\mathbb{R}} | u(t,x) - U(x-c_0t -x_\\pm) | = 0\\quad \\mbox{for some }x_\\pm\\in \\mathbb{R}. \\]\n\\end{remark}\n\n\\begin{remark}\t\nIn \\cite{Zlatos}, infinitely many transition fronts with different global mean speeds were constructed in the case of Fisher-KPP reactions with weak inhomogeneity.\nIn our case, we only obtain a single transition front with a single global mean speed $c_0$ in Theorem \\ref{thm:ex}. \nIndeed, by a comparison principle argument and stability of front speeds with respect to reactions, one can easily show that any transition front must have a global mean speed $c_0$. \nThough transition fronts in ignition media are expected to be unique \\cite{Z12}, we do not know if the uniqueness still holds in our case of mixed ignition-monostable reactions. \n\n\\end{remark}\n\nOur method is structurally similar to that of \\cite{Zlatos}.\nThe primary difference and technicality in this work is the lack of KPP structure in our reaction.\nIndeed, KPP structure allows one to exploit the intimate connection between solutions of \\eqref{eq:main} and those of the linearized equation $u_t=u_{xx}+a(x)u$ (which are super-solutions to \\eqref{eq:main}).\nTherefore, super- and sub-solutions found in \\cite{Zlatos} are based on (generalized) eigenfunctions of the operator $\\partial_{xx}+a(x)$ and exponential functions.\nIn our case, we clearly lack this convenience.\nHence, to overcome this, we base our super- and sub-solutions on the traveling front of the ignition reaction $f_0$ instead.\nThis modification introduces some difficulties in calculation, but we are still able to obtain a sharp result.\n\nThe body of the paper is organized as follows.\nWe prove the non-existence of transition fronts (Theorem \\ref{thm:nonex}) in the coming section, and present the complimentary existence result (Theorem \\ref{thm:ex}) in Section 3.\n\n\\section*{Acknowledgment}\nAll authors were supported in part by the NSF grant DMS-1056327, and thank Andrej Zlato\\v{s} for directing this research.\nCG gratefuly acknowledges the support of the Fannie and John Hertz Foundation.\nCG, AM, and DW thank the Department of Mathematics at the University of Wisconsin--Madison for its hospitality during the 2013 REU ``Analysis and Differential Equations,'' where this research was mainly conducted.\n\n\\section{Non-existence for $\\lambda >c_0^2$ (proof of Theorem \\ref{thm:nonex})}\n\\label{sec:non}\n\nAs mentioned above, the methods of this section are based on those found in \\cite{Zlatos}.\nIn particular, Theorem \\ref{thm:nonex}, Lemmas \\ref{lem:refined}, \\ref{lem:new}, and their proofs are similar to Theorem 1.2, Lemmas 3.1, 3.2 \\cite{Zlatos}.\nThe primary difference can be found in the proof of Lemma \\ref{lem:new}.\n\nThroughout this section, we assume $f,\\gamma,f_0, \\theta_0,\\theta_1, L, a$ are all from (F), and $\\lambda > c_0^2$.\nFor $\\varepsilon\\in (0,1)$, let $\\lambda_\\varepsilon$ be the principal eigenvalue of the differential operator $\\partial_{xx}+(1-\\varepsilon)a(x)$.\nSince $\\lim_{\\varepsilon\\to 0^+} \\lambda_\\varepsilon=\\lambda>c_0^2$, we may fix $\\varepsilon>0$ such that $\\lambda_\\varepsilon>c_0^2$, and\nlet $\\zeta = \\zeta(\\varepsilon)$ be given in \\ref{item:regularity}.\nFor $M>0$, we let $\\lambda_M = \\lambda_{\\varepsilon,M}$ be the Dirichlet principal eigenvalue of $\\partial_{xx}+(1-\\varepsilon)a(x)$ on $[-M,M]$, and $\\psi_M\\in C^2([-M,M])$ be the corresponding $L^\\infty$-normalized eigenfunction:\n\\begin{gather}\n\t\\label{TS:psi}\n\t\\psi_M'' + (1-\\varepsilon)a(x) \\psi_M = \\lambda_M \\psi_M \\mbox{ on }(-M,M),\\\\\n\t\\psi_M(\\pm M)=0,\\quad ||\\psi_M||_{L^\\infty}=1. \\nonumber\n\\end{gather}\nNote that $\\psi_M>0$ on $(-M,M)$ and $\\lim_{M\\to\\infty} \\lambda_M = \\lambda_\\varepsilon>c_0^2$.\nSo we may again fix $M\\ge L$ large so that $\\lambda_M>c_0^2$. Finally, all constants involved depend on $c_0, M, \\psi_M, \\lambda_M, \\zeta, \\gamma, \\theta_0$.\n\nIn the following, let $u\\in (0,1)$ be an entire solution of \\eqref{eq:main} with $\\inf _{(t,x)\\in \\mathbb{R}}u=0$.\nIn the proofs, we will frequently use the parabolic Harnack inequality for $u$.\nTherefore, for $R,\\sigma>0$, we let $k=k(R,\\sigma)>0$ denote the Harnack constant such that\n\\begin{equation}\n  \\label{h-const}\n\t\\min_{|x-x_0|\\le R} u(t+\\sigma,x) \\ge k \\max _{|x-x_0|\\le R} u(t,x) \\ge k u(t,x_0)\n\\end{equation}\nholds for any $x_0\\in \\mathbb{R}$.\nWe begin with the following simple fact.\n\n\\begin{lemma}\n\t\\label{lem:limit}\n\tThe solution $u$ satisfies $\\lim_{t\\to-\\infty}u(t,x)=0$ locally uniformly.\n\\end{lemma}\n\n\\begin{proof}\n\tBy the Harnack inequality, it suffices to show the limit for $x=0$.\n\tAssume the contrary, so that there exists $\\alpha \\in (0,1)$ and a sequence of times $\\{t_n\\}$ with $t_n\\searrow -\\infty$ such that $u(t_n,0)\\geq \\alpha$.\n\tLet $k=k(M,1)$ be the Harnack constant from \\eqref{h-const},\n\t$\\theta:=\\min\\{k\\alpha, \\zeta\\}$, and extend the eigenfunction $\\psi_M$ from \\eqref{TS:psi} continuously to $\\mathbb{R}$ by setting $\\psi_M\\equiv 0 $ on $[-M,M]^c$.\n\tSince $||\\psi_M||_{L^\\infty}=1$, \\eqref{h-const} (with $(R,\\sigma,x_0,t)=(M,1,0,t_n)$) implies\n\t\\begin{equation}\n\t\t\\label{eq:Harnack_bd}\n\t\tu(t_n+1,x)\\geq \\theta\\psi_M(x)\\quad \\mbox{for all }x\\in {\\mathbb{R}}.\n\t\\end{equation}\n\t\n\tNow let $v:\\mathbb{R}^+\\times \\mathbb{R} \\to [0,1]$ be the solution to the Cauchy problem of \\eqref{eq:main} with initial data $v(0,x) = \\theta\\psi_M(x)$.\n\tWe claim that $v_t\\ge 0$.\n\tBy the comparison principle, it suffices to show $v(s,\\cdot)\\ge \\theta\\psi_M$ for all $s\\ge 0$.\n\tClearly this holds for all $x\\in [-M,M]^c$ because $\\psi_M\\equiv 0$ in this region.\n\tIf $x\\in [-M,M]$ instead, observe that $w(t,x):=\\theta\\psi_M(x)$ is a (stationary) sub-solution of \\eqref{eq:main} by (F3) and \\eqref{TS:psi}. So the comparison principle shows that $v(s,x)\\ge \\theta\\psi_M(x)$ for $(s,x)\\in \\mathbb{R}^+\\times [-M,M]$.\n\tThis implies $v_t\\ge 0$.\n\tLet $v_\\infty(x):= \\lim_{t\\to\\infty}v(t,x)$, which satisfies $v_\\infty '' +f(x,v_\\infty)=0$ on $\\mathbb{R}$ by parabolic regularity.\n\tSince $f\\ge 0$, this forces $v_\\infty \\equiv \\beta$ for some constant $\\beta \\in [\\theta ,1]$.\n\tNow fix $s\\in \\mathbb{R}$.\n\tBy the comparison principle and \\eqref{eq:Harnack_bd}, for all large $n$\n\t\\begin{equation*}\n\t\tu(s,x)\\ge v(s-t_n-1,x)\\quad \\mbox{for all }x\\in \\mathbb{R}.\n\t\\end{equation*}\n\tLetting $n\\to\\infty$, we find that $u(s,\\cdot)\\ge \\beta >0$ for all $s\\in \\mathbb{R}$, contradicting $\\inf_{(t,x)\\in \\mathbb{R}^2} u=0$. Therefore, we must have $\\lim_{t\\to-\\infty}u(t,0)=0$.\n\\end{proof}\n\nWith Lemma \\ref{lem:limit}, after an appropriate time translation we may now assume\n\\begin{gather}\n\t\\label{eq:zetabound}\n\tu(0,0)\\leq \\frac \\zeta 2 \\psi_M(0).\n\\end{gather}\nIn the coming two lemmas, we establish some important bounds on $u$, which play a crucial role in the proof of Theorem \\ref{thm:nonex}.\n\n\\begin{lemma}\n\t\\label{lem:refined}\n\tFor any $c\\in (c_0, \\sqrt{ \\lambda_M} )$, there exists $C_0>0$ (independent of $u$) such that\n\t\\begin{equation}\n\t\t\\label{eq:refined_bound}\n\t\tu(t, x) \\leq C_0 u(0,0) e^{c_0(x+ct)}, \\quad \\text{for }t\\le -1,\\; x\\in [M,\\sqrt{\\lambda_M}(-t-1)-M-1].\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tDenote $u_0 := u(0,0)>0$, $\\psi_0 := \\psi_M(0)>0$, and $D\\subset \\mathbb{R}^2$ the region described in \\eqref{eq:refined_bound}.\n\tTo show \\eqref{eq:refined_bound}, we will prove the following estimate for some $C_0'>0$ (independent of $u)$:\n\t\\begin{equation}\n\t\t\\label{TS:bound2.2}\n\t\tu(t, x) \\leq C'_0 u_0 \\sqrt{|t|}  e^{\\sqrt{\\lambda_M}(x+\\sqrt{\\lambda_M}t)},\\quad \\text{for }(t,x)\\in D.\n\t\\end{equation}\n\tOne can easily show that \\eqref{eq:refined_bound} follows from this with $C_0 := C'_0 \\sup_{t \\leq 0}\\sqrt{|t|}e^{c_0(\\sqrt{\\lambda_M} - c)t}$ (which is finite because $\\sqrt{\\lambda_M}>c$).\n\t\n\tWe prove \\eqref{TS:bound2.2} by contradiction.\n\tLet $k=k(1,1)$ be the Harnack constant from \\eqref{h-const}, and\n\tsuppose there is $(t',x_0)\\in D$ so that \\eqref{TS:bound2.2} does not hold with  $C_0'$ given by\n\t\\[ C_0':=\\frac{\\sqrt {4 \\pi}}{k\\psi_0}e^{\\lambda_M+\\sqrt{\\lambda_M}(M+1)}. \\]\n\tLet $t_0:=t'+1\\le 0$ and \n\t\\begin{gather*}\n\t\t\\beta:=\\frac {x_0+M+1}{2|t_0|\\sqrt{\\lambda_M}},\\quad \n\t\t\\eta:= C'_0ku_0\\sqrt{|t'|}e^{\\sqrt{\\lambda_M}(x_0+\\sqrt \\lambda_Mt') }.\n\t\\end{gather*}\n\tObserve that $\\beta \\in (0,\\frac 12 ]$ as $(t',x_0)\\in D$.\n\tAlso, by \\eqref{h-const} (with $(R,\\sigma,t)=(1,1,t')$) and the opposite of \\eqref{TS:bound2.2} we have $u(t_0,\\cdot)\\ge \\eta \\chi_{[x_0,x_0+1]}$.\n\tApplying the comparison principle ($u$ is a super-solution to the standard heat equation as $f\\ge 0$), for all $x\\in [-M,M]$ we have\n\t\\begin{align}\n\t\tu(t_0+\\beta |t_0|,x) &\\ge \\frac {\\eta}{\\sqrt{4\\pi\\beta |t_0|}} \\int _{x_0}^{x_0+1} e^{-\\frac{(x-y)^2}{4\\beta |t_0|}} dy\\ge \\frac{\\eta}{\\sqrt{4\\pi \\beta |t_0|}} e^{-\\frac{(x_0+M+1)^2}{4\\beta|t_0|}} \\nonumber \\\\\n\t\t&\\ge {2u_0}{\\psi_0}^{-1} e^{\\sqrt{\\lambda_M}(x_0+M+1+\\sqrt{\\lambda_M}t_0)-\\frac{(x_0+M+1)^2}{4\\beta|t_0|}} \\nonumber \\\\\n\t\t& = {2u_0}{\\psi_0}^{-1} e^{\\lambda_M(t_0+\\beta|t_0|)}.\n\t\t\\label{TS:2.2.3}\n\t\\end{align}\n\tHere, the second inequality is due to $-M\\le x \\le y\\le x_0+1$, for then $0\\le y-x\\le M+x_0+1$.\n\tNow let $v(t,x):=2u_0\\psi_0^{-1}e^{\\lambda _M t} \\psi_M(x)$, which by \\eqref{TS:psi} satisfies\n\t\\begin{equation}\\label{pdineq-23}\n\t\tv_t = v_{xx} + (1-\\varepsilon )a(x) v\\quad \\mbox{for all }(t,x)\\in\\mathbb{R}\\times (-M,M),\\quad v(t,\\pm M)=0.\n\t\\end{equation}\n\tFrom \\eqref{TS:2.2.3}, $||\\psi_M||_{L^\\infty}=1$, and \\eqref{eq:zetabound}, we also have\n\t\\begin{equation}\\label{ineq-27}\n\t\t\\begin{split}\n\t\t\tu(t_0+\\beta|t_0|,x)&\\ge v(t_0+\\beta|t_0|,x), \\\\\n\t\t\tv(t,x)\\le v(0,x)&=2u_0\\psi_0^{-1}\\psi_M(x)\\le \\zeta,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere the latter holds for all $t\\le 0$ and $x\\in \\mathbb{R}$.\n\tThe latter with \\eqref{pdineq-23} and \\ref{item:regularity} shows that $v$ is a sub-solution of \\eqref{eq:main} on $\\mathbb{R}^- \\times (-M,M)$.\n\tHence, \\eqref{ineq-27} and the comparison principle (note that $t_0+\\beta |t_0|\\le 0$ as $\\beta\\in (0,\\frac 1 2]$ and $t_0\\le 0$) yield\n\t\\begin{equation*}\n\t\tu(0,x) \\ge v(0 , x)=2u_0\\psi_0^{-1}\\psi_M(x) \\quad  \\mbox{for } x\\in[-M,M].\n\t\\end{equation*}\n\tLetting $x=0$ yields  the contradiction $u_0\\ge 2u_0$ (as $u_0>0$).\n\tTherefore \\eqref{TS:bound2.2} holds.\n\\end{proof}\n\nWith the estimate \\eqref{eq:refined_bound}, we now further refine the bound for $u$ to show that it is bump-like for all large negative time.\n\n\\begin{lemma}\n\t\\label{lem:new}\n\tUnder the same assumptions as Lemma \\ref{lem:refined}, there exist $C>0$ and $\\tau <0$ (both independent of $u$) such that\n\t\\begin{equation}\n\t\t\\label{TS:new}\n\t\tu(t,x)\\le C u(0,0) e^{-c_0|x|+c_0ct},\\quad \\mbox{for } t\\le \\tau,\\, |x|\\ge M.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{remark}\nWe follow the argument in the proof of Theorem 1.2 in \\cite{Zlatos}.\nThe fundamental difference lies in the definition of the super-solution $w=w_1+w_2^{s}$.\nIn \\cite{Zlatos}, $w_2^{s}$ is chosen to be an exponential function (see the definition of $v_{t_0}$ in Section 3 \\cite{Zlatos}).\nIn our case, $w_2^{s}$ is a (leftward-moving) traveling front of a perturbed reaction $f_\\delta$ (defined in the proof).\n\\end{remark}\n\n\\begin{proof}\n\tFirst of all, it suffices to prove \\eqref{TS:new} for the case $x\\ge M$.\n\tThis is because $\\tilde u(t,x):=u(t,-x)$ is still a global solution to \\eqref{eq:main} with $\\tilde f(x,u):=f(-x,u)$ in place of $f(x,u)$.\n\tClearly, $\\tilde f$ and $\\tilde u$ satisfy (F) and \\eqref{eq:zetabound} respectively.\n\tApplying \\eqref{TS:new} to $\\tilde u$ and $x\\ge M$, we find \\eqref{TS:new} for $u$ and $x\\le -M$.\n\t\n\tFix $c\\in (c_0,\\sqrt{\\lambda_M})$.\n\tFor $\\delta>0$ small, consider the perturbation of $f_0$ given by\n\t\\begin{equation*}\n\t\tf_\\delta(u) := \\max_{v\\in [u-\\delta,u+\\delta ]}f_0(v),\n\t\\end{equation*}\n\twhich is a Lipschitz ignition reaction with $\\text{\\rm{supp}}(f_\\delta)= [\\theta_0 -\\delta,1+\\delta]$.\n\tLet $(U_\\delta,c_\\delta)$ be the traveling front solution to the following problem:\n\t\\begin{gather}\n\t\t\\label{eq:tfeq2}\n\t\tU''_\\delta - c_\\delta U' + f_\\delta (U_\\delta ) = 0,\\quad\n\t\t\\lim_{y\\to -\\infty} U_{\\delta}(y) = 0,\\quad  \\lim_{y\\to \\infty} U_{\\delta}(y) = 1+\\delta.\n\t\\end{gather}\n\tNote that $U_\\delta$ is leftward moving, so $U_\\delta'>0$.\n\tBy a simple argument using phase plane analysis and the uniqueness\/stability of solutions to ODEs, one can easily show that $c_\\delta>c_0$ and $\\lim_{\\delta \\searrow 0}c_{\\delta}= c_0$.\n\tThus we may fix $\\delta \\in (0,\\theta_0)$ so that $c_\\delta \\in ( c_0, \\sqrt {c_0c})$.\n\tLet $C_0$ be the constant from Lemma \\ref{lem:refined} and $u_0:=u(0,0)$.\n\tSince $f_\\delta\\equiv 0$ on $[0,\\theta_0-\\delta]$ and \\eqref{eq:tfeq2}, we can specify the translation of $U_\\delta$ so that\n\t\\begin{equation}\\label{Udel-trans}\n\t\tU_\\delta(x)=Ae^{c_\\delta x}\\mbox{ whenever }U_\\delta(x)\\le \\theta_0-\\delta,\\mbox{ where }A:=C_0u_0 e^{(c_0-c_\\delta )M}.\n\t\\end{equation}\n\tFor $s\\in \\mathbb{R}$, define\n\t\\begin{align*}\n\t\tw_1(t,x) := C_0u_0e^{-c_0(x-2M-ct)},\\quad w_2^{s}(t,x) := U_\\delta\\rb {x + c_\\delta t + \\rb{\\frac {c c_0}{c_\\delta} - c_\\delta}s}.\n\t\\end{align*}\n\tLet $\\tau:=\\min\\{T_0,T_1\\}$, where $T_0$, $T_1$ are given by\n\t\\begin{equation*}\n\t\tT_0:= -\\frac {2M+1}{\\sqrt {\\lambda_M}}-1 ,\\qquad C_0 e^{c_0(M+cT_1)}=\\delta.\n\t\\end{equation*}\n\tHere, $T_0$ is defined so that the interval from \\eqref{eq:refined_bound} is non-empty for all $t\\le T_0$.\n\tBy Lemma \\ref{lem:refined},\n\t\\begin{equation}\n\t\t\\label{TS:comp1}\n\t\tu(t,M)\\leq w_1(t,M)\\quad \\mbox{for }t\\le \\tau.\n\t\\end{equation}\n\tWe also claim that for all sufficiently negative $s\\le 0$,\n\t\\begin{equation}\n\t\t\\label{TS:comp2}\n\t\tu(s,x)\\le w_2^{s}(s,x) \\quad \\mbox{for }x\\in [M,\\infty).\n\t\\end{equation}\n\tWe postpone the proof of this claim to first show \\eqref{TS:new}.\n\t\n\tLet $w:=w_1+w_2^{s}$ and $D_{s} := [s,\\tau]\\times [M,\\infty)$.\n\tThen $w$ is a super-solution to \\eqref{eq:main} on $D_{s}$.\n\tAfter all, $w_1(t,x)\\leq w_1(T_1,M)= u_0\\delta\\le \\delta$ for all $(t,x) \\in D_{s}$, so\n\t\\begin{align*}\n\t\tw_t-w_{xx}&=\\partial_t w_1-\\partial_{xx}w_1+\\partial_tw_2^{s}-\\partial_{xx}w_2^{s}\\nonumber \\\\\n\t\t&=c_0(c-c_0)w_1+f_\\delta(w_2^{s})\\geq f_\\delta(w_2^{s})\\nonumber \\\\\n\t\t&\\ge f_0(w)=f(x,w).\n\t\n\t\\end{align*}\n\tThe last inequality follows from $0< w_1\\leq \\delta$ on $D_{s}$ and the definition of $f_\\delta$.\n\tBy \\eqref{TS:comp1}, \\eqref{TS:comp2}, and  the comparison principle, we have $u\\leq w_1+w^{s}_2$ on $D_{s}$, which holds for all large negative $s$.\n\tObserve that the argument of $U_\\delta$ in the definition of $w^{s}_2$ tends to $-\\infty$ as $s\\searrow -\\infty$, since $c_\\delta <\\sqrt {cc_0}$.\n\tHence, $w^{s}_2\\searrow 0$, and $u(t,x)\\le w_1(t,x)$ for all $(t,x)\\in (-\\infty,\\tau]\\times [M,\\infty)$.\n\tThis is \\eqref{TS:new} for $x\\ge M$ if we set $C:=C_0e^{2c_0M}$.\n\t\n\tIt remains to prove \\eqref{TS:comp2}.\n\tLet $\\xi_0 \\in \\mathbb{R}$ satisfy $C_0u_0e^{c_0\\xi_0}=\\theta_0-\\delta$, and define\n\t\\begin{equation*}\n\t\tW(s):= w_2^{s}(s,\\xi_0-cs) = U_\\delta (\\xi_0+c(c_0c_\\delta^{-1}-1)s),\n\t\\end{equation*}\n\twhich is continuous and satisfies $W(-\\infty)=U_\\delta(\\infty)=1+\\delta$ (as $c_0<c_\\delta$).\n\tNow, by this and $c<\\sqrt{\\lambda_M}$, we let $s_0\\le -1$ so that for all $s\\le s_0$ we have\n\t\\begin{equation}\\label{s0-def}\n\t\tW(s)\\ge 1,\\qquad \\xi_0-cs\\le -\\sqrt {\\lambda_M}-M-1-\\sqrt{\\lambda_M}s.\n\t\\end{equation}\n\tNow we show that \\eqref{TS:comp2} holds for all $s\\le s_0$.\n\tIndeed, if $x\\ge \\xi_0-cs$, we have $w_2^{s}(s,x) \\ge w_2^s(s,\\xi_0-cs) =W(s)\\ge 1 > u(s,x)$.\n\tConsider $x\\in [M,\\xi_0-cs)$. By \\eqref{s0-def} and \\eqref{eq:refined_bound}, it suffices to show that $w_2^s(s,x)\\ge C_0u_0e^{c_0(x+cs)}$.\n\tAssume the contrary that it does not hold for some $x_0\\in [M,\\xi_0-cs)$.\n\tIt then follows from the definition of $\\xi_0$ that\n\t\\begin{equation}\\label{cont-05}\n\t\tw_2^s(s,x_0)< C_0u_0e^{c_0(x_0+cs)}\\le C_0u_0e^{c_0\\xi_0}=\\theta_0-\\delta.\n\t\\end{equation}\n\tOn the other hand, by our translation for $U_\\delta$ from \\eqref{Udel-trans}, $c_0<c_\\delta$ and $x_0\\ge M$, it follows\n\t\\begin{align*}\n\t\tw^{s}_2(s,x_0) &= U_\\delta(x_0+cc_0c^{-1}_\\delta s) =  Ae^{c_\\delta x_0+cc_0s}  \\\\\n\t\t&= C_0u_0 e^{c_0M+c_\\delta(x_0-M)+cc_0s}\n\t\t\\ge C_0u_0e^{c_0(x_0+cs)}.\n\t\\end{align*}\n\tThis is a contradiction to the first inequality of \\eqref{cont-05}. Hence, \\eqref{TS:comp2} holds for all $s\\le s_0$.\n\\end{proof}\n\nWe now prove our first main theorem.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:nonex}]\n\t(i) We have already shown in Lemma \\ref{lem:new} that $u$ is bump-like for all $t\\le \\tau$.\n\tIt follows from a standard argument using the comparison principle that $u$ remains bump-like for all time. This clearly implies that transition front does not exist.\n\t\n\t\n\t(ii) Denote $\\varphi(t,x):=\\zeta e^{\\lambda t }\\psi(x)$, which solves the following PDE globally in time:\n\t\\begin{equation}\n\t\t\\label{eq:linearized}\n\t\t\\varphi_t = \\varphi_{xx} + a(x)\\varphi.\n\t\\end{equation}\n\tTo construct a bump-like solution, we let $u(t,x):= \\varphi(t,x)$ for $t\\le 0$, and propagate forward in time by solving the Cauchy problem for \\eqref{eq:main} with initial data $u(0,x) = \\zeta \\psi(x)$ for $t >0$.\n\tNext we prove that this solution is unique up to time translation. Let $\\tilde u\\in (0,1)$ be another solution of \\eqref{eq:main} with $\\inf _{(t,x)\\in \\mathbb{R}^2}\\tilde u=0$.\n\tBy Lemmas \\ref{lem:limit} and  \\ref{lem:new}, $\\tilde u(t,\\cdot)\\to 0$ uniformly as $t\\to-\\infty$.\n\tTherefore, after a time-shift we may assume\n\t\\begin{equation}\\label{sup-trans}\n\t\t\\sup _{(t,x)\\in \\mathbb{R}^-\\times \\mathbb{R}}\\tilde u(t,x)\\le \\frac \\zeta 2 \\psi_M(0).\n\t\\end{equation}\n\tLet $\\tilde \\varphi(t,x):=\\tilde u(t,x)$ for all $t\\le 0$, and propagate forward in time as the solution of \\eqref{eq:linearized} with $\\tilde \\varphi(0,x)=\\tilde u(0,x)$.\n\tSince $f(x,u)=a(x)u$ for $u\\in [0,\\zeta]$ by the assumption, $\\tilde \\varphi$ is an entire solution of \\eqref{eq:linearized}.\n\tBy \\cite[Proposition 2.5]{HP07}, we have $\\tilde \\varphi= q \\varphi$ for some $q>0$, provided that Conditions (A), (H1), and (2.12) from \\cite{HP07} are met (which will be shown shortly).\n\tTherefore, we have $\\tilde \\varphi(t,\\cdot) = \\varphi(t-T,\\cdot)$ with $T:=-\\lambda^{-1}\\log({q}{\\zeta}^{-1})$.\n\tSince $u\\equiv \\varphi$, $\\tilde u \\equiv \\tilde \\varphi$ for all $t\\le 0$, it clearly follows $\\tilde u(t,\\cdot)=u(t-T,\\cdot)$, which shows the uniqueness of solution.\n\t\n\tIt remains to check all the conditions from \\cite{HP07}.\n\tNote that (A) follows from $0\\le a\\le \\gamma\\chi_{[-L,L]}$, and (H1) holds for the PDE \\eqref{eq:linearized} because $\\lambda >0$.\n\tTo show \\cite[(2.12)]{HP07}, we will prove that\n\t\\begin{equation}\n\t\t\\label{TS:b1}\n\t\t\\sup_{x\\in\\mathbb{R}} \\tilde \\varphi(s,x)\\le K \\tilde \\varphi(s,0) \\quad \\mbox{for all }s\\in \\mathbb{R},\n\t\\end{equation}\n\tfor some $K>0$ independent in time.\n\tConsider the above for $s\\le 0$\n\tLet $\\tilde \\varphi^s(t,x):= \\tilde \\varphi(t+s,x)$, which again satisfies \\eqref{eq:zetabound} (by \\eqref{sup-trans}).\n\tFollow from Lemma \\ref{lem:new}, $\\tilde \\varphi^s$ satisfies the estimate \\eqref{TS:new}. With $t=\\tau\\le 0$, we find that\n\t\\begin{equation*}\n\t\t\\sup_{|x|\\ge M}\\tilde \\varphi^s(\\tau,x)\\le C \\tilde\\varphi^s(0,0).\n\t\\end{equation*}\n\tOn the other hand, by the Harnack inequality \\eqref{h-const}, we have $\\max_{|x|\\le M}\\tilde \\varphi^s(\\tau,x)\\le k^{-1}\\tilde \\varphi^s(0,0)$, where $k=k(M,-\\tau)$.\n\tHence, $\\sup_{x\\in \\mathbb{R}}\\tilde \\varphi^s(\\tau,x)\\le A\\varphi^s(0,0)$ with $A:=\\max\\{C,k^{-1}\\}$.\n\tApplying the comparison principle (noting that $w(t,x)=A\\varphi^s(0,0)e^{\\gamma (t-\\tau)}$ is a super-solution to \\eqref{eq:linearized}), we find $\\sup_{x\\in \\mathbb{R}}\\tilde\\varphi^s (0,x)\\le A e^{-\\gamma \\tau} \\tilde\\varphi^s(0,0)$, which is \\eqref{TS:b1} with $K:=Ae^{-\\gamma\\tau}$.\n\t\n\tNow consider \\eqref{TS:b1} with $s>0$.\n\tDecompose $\\tilde \\varphi(0,x)=\\alpha\\psi(x)+\\psi^\\perp(x)$, where $\\psi$, $\\psi^\\perp$ are orthogonal in $L^2(\\mathbb{R})$ (recalling that $\\psi$ is the eigenfunction from \\eqref{eq:eigen}).\n\tLet $\\phi(t,x) := e^{-\\lambda t}\\tilde\\varphi(t,x)$, which by \\eqref{eq:linearized} satisfies\n\t\\begin{equation*}\n\t\n\t\t\\phi_t = (\\partial_{xx}+a(x)-\\lambda)\\phi.\n\t\\end{equation*}\n\tSince the principal eigenvalue $0$ of $\\partial_{xx}+a(x)-\\lambda$ is isolated, it is well-known that $\\phi (t,\\cdot)\\to \\alpha \\psi $ uniformly as time progresses. This clearly implies \\eqref{TS:b1} for $s>0$, as desired.\n\\end{proof}\n\n\\section{Existence for $\\lambda<c_0^2$ (proof of Theorem \\ref{thm:ex})}\n\\label{sec:ex}\n\nIn this section we will prove Theorem \\ref{thm:ex} by finding a transition front solution of \\eqref{eq:main} under the assumption $\\lambda<c_0^2$.\nThe construction is based on the standard solution method mentioned in Section \\ref{intro}---we first construct appropriate pairs of super- and sub-solutions, then obtain an entire solution $u$ sandwiched between them, and use their asymptotic behavior to show that $u$ is a transition front.\n\nThroughout the section, $f,\\gamma,f_0, \\theta_0,\\theta_1, L$, and $a$ are all from Hypothesis (F),\nand we extend $f(x,\\cdot),f_0(\\cdot)$ continuously to $\\mathbb{R}$ by setting $f_0(u)=f(x,u)\\equiv 0$ for $u\\notin [0,1]$.\nFor $\\varepsilon > 0$, let $\\lambda^\\varepsilon>0,\\psi^\\varepsilon> 0$ be the principal eigenvalue and (normalized) eigenfunction of the differential operator $\\partial_{xx} + a(x) + 2\\varepsilon\\chi_{[-L,L]}(x)$, which satisfy\n\\begin{gather*}\n\\partial_{xx}\\psi^\\varepsilon+[a(x)+2\\varepsilon\\chi_{[-L,L]}(x)] \\psi ^\\varepsilon = \\lambda^\\varepsilon \\psi^\\varepsilon \\mbox{ on }\\mathbb{R}\\\\\n\\lim_{|x|\\to\\infty}\\psi^\\eps(x)=0,\\quad ||\\psi^\\eps||_{L^\\infty}=1.\n\\end{gather*}\nSince $a(x)=0$ for $|x|\\ge L$ and $||\\psi^\\varepsilon||_{L^\\infty}=1$, $\\psi^\\varepsilon$ satisfies the exponential bound\n\\begin{equation}\n\\label{TS:psi-exp}\n\\psi^\\eps (x) \\le \\min\\{1,e^{-\\sqrt{\\lambda^\\varepsilon}(|x|-L)}\\}\\quad \\mbox{for all }x\\in \\mathbb{R}.\n\\end{equation}\nNote also that $\\varepsilon\\mapsto \\lambda^\\varepsilon$ is increasing and continuous, with $\\lambda^0 = \\lambda$ and $\\lim_{\\varepsilon \\to \\infty}\\lambda^\\varepsilon \\to \\infty$, so we may fix $\\varepsilon>0$ such that $\\lambda^{\\varepsilon}\\in (\\frac{c_0^2}{16},c_0^2)$, and let $\\zeta = \\zeta(\\varepsilon)\\in (0,\\theta_0)$ be from \\ref{item:regularity}.\nLet $U$ be the unique traveling front of $f_0$ in the sense of \\eqref{tfeq} with $U(0)=\\frac {\\theta_0} 2$.\nGiven $y\\in \\mathbb{R}$, we define\n\\begin{equation}\n\\label{TS:v1-def}\nv^{y}(t,x) := U(x-c_0t + y),\n\\end{equation}\nwhich satisfies \\eqref{eq:main} with $f(x,u)$ replaced by $f_0(u)$.\nFinally, let \\begin{equation}\n\\label{TS:om-eta-def}\n\\omega := \\inf_{x\\in[-L,L]} \\psi^\\varepsilon(x),\\quad \\eta: = c_0\\inf \\left\\{-U'(s);\\;s\\in \\mathbb{R},\\,U(s)\\in \\left[\\frac{\\theta_0}{2},1-\\theta_1\\right]\\right\\}.\n\\end{equation}\nAll the constants involved in this section will depend on $\\gamma, \\theta_0,\\theta_1, L, \\varepsilon, \\zeta, U, c_0, \\omega,$ and $\\eta$.\n\nWe begin with the construction of sub- and super-solutions for $t\\le 0$.\n\\begin{lemma}\n\t\\label{lem:supersub}\n\t\\begin{enumerate}\n\t\t\\item For all $y\\ge L$, $v^y$ given in \\eqref{TS:v1-def} is a sub-solution to \\eqref{eq:main} on $(-\\infty,0)\\times \\mathbb{R}$.\n\t\t\\item  There exists $y_0\\ge L+c_0\\beta(0)$, such that $w$ given as follows is a super-solution to \\eqref{eq:main} on $(-\\infty,0)\\times \\mathbb{R}$\\emph{:}\n\t\t\\begin{gather*}\n\t\tw(t,x) := v^{y_0}(t+\\beta(t),x) + \\phi(t,x),\\\\\n\t\t\\beta(t):= \\frac{\\gamma\\zeta}{4\\sqrt{\\lambda^\\eps }c_0\\eta}e^{2\\sqrt{\\lambda^\\eps }c_0 t},\\quad\n\t\t\\phi(t,x):= \\frac \\zeta 2 e^{\\sqrt{\\lambda^\\eps }c_0 t}\\psi^\\varepsilon(x).\n\t\t\\end{gather*}\n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\t(i) Abbreviate $v=v^y$, recalling that $v$ satisfies \\eqref{eq:main} with $f(x,u)\\equiv f_0(u)$.\n\tThen we must show that\n\t\\begin{equation}\n\t\\label{eq:sub1}\n\tv_t-v_{xx}-f(x,v)=f_0(v)-f(x,v)\\leq 0\\quad \\mbox{for }(t,x)\\in (-\\infty,0)\\times \\mathbb{R}.\n\t\\end{equation}\n\tFor $x\\in [-L,L]^c$, we have $f(x,v)\\equiv f_0(v)$ by \\ref{item:ignition}, and the above holds trivially.\n\tIf $x\\in [-L,L]$ instead, by $U'<0$ we have $v(t,x)=U(x-c_0t+y)\\le U(y-L)\\le U(0)= \\frac {\\theta_0}2$.\n\tTherefore $f_0(v)=0$, and \\eqref{eq:sub1} holds.\n\t\n\t(ii) Let $y_0\\in \\mathbb{R}$ be the unique number such that\n\t\\begin{equation}\\label{y0-def}\n\tU(y_0- c_0\\beta(0)-L) = \\min \\cb{\\frac\\zeta 2, \\frac{\\varepsilon \\omega \\zeta }{2(\\gamma +\\varepsilon )}}.\n\t\\end{equation}\n\tClearly, $y_0\\ge L+c_0\\beta(0)$ because $U(0)=\\frac {\\theta_0}{2} >\\frac{\\zeta}{2}$ and $U'<0$.\n\tAbbreviate $v^{y_0} = v$. Then we must show for all $(t,x)\\in (-\\infty,0)\\times \\mathbb{R}$ that\n\t\\begin{align}\n\tw_t-w_{xx}-f(x,w)\n\t=&f_0(v) + v_t\\,\\beta' + \\sqrt{\\lambda^\\eps }\\left(c_0-\\sqrt{\\lambda^\\eps }\\right)\\phi\\nonumber \\\\\n\t& +[a(x)+2\\varepsilon\\chi_{[-L,L]}(x)] \\phi - f(x,w)\\geq 0,\n\t\\label{eq:super1}\n\t\\end{align}\n\twhere $v$ and $v_t$ are evaluated at $(t+\\beta(t),x)$ in what follows.\n\tConsider first $x\\in [-L,L]$.\n\tSince the first four terms in \\eqref{eq:super1} are nonnegative, it suffices to show that $[a(x)+2\\varepsilon]\\phi\\geq f(x,w)$.\n\tNote that $\\phi(t,x)\\le \\frac{\\zeta}{2}$ as the eigenfunction $\\psi^\\varepsilon$ satisfies $||\\psi^\\varepsilon||_{L^\\infty}=1$.\n\tMoreover, by $U'<0$, $\\beta '>0$ and \\eqref{y0-def}, we have\n\t\\begin{equation*}\n\tv(t+\\beta(t),x)\\le U(y_0-c_0\\beta(0)-L)\\le \\frac{\\zeta}{2}.\n\t\\end{equation*}\n\tTherefore $w=v + \\phi \\le \\zeta$, and $f(x,w)\\leq\\left[a(x)+\\varepsilon\\right]w$ by \\ref{item:regularity}.\n\tOn the other hand, by \\eqref{tfeq} and $f_0\\equiv 0$ on $[0,\\theta_0]$, we have $U(y)=\\frac{\\theta_0}{2}e^{-c_0y}$ for $y\\ge 0$.\n\t\\eqref{y0-def} then implies\n\t\\begin{align*}\n\tv(t+\\beta(t),x)&\\le U(y_0-c_0\\beta(0)-L-c_0t)\\\\\n\t&=U(y_0-c_0\\beta(0)-L)e^{c_0^2 t} \\le \\frac {\\varepsilon \\omega \\zeta}{2(\\gamma+\\varepsilon )} e^{c_0^2t}.\n\t\\end{align*}\n\tCombining this, $\\sqrt{\\lambda^\\varepsilon}<c_0$, $a(x)\\le \\gamma$ and $\\psi^\\varepsilon(x)\\ge \\omega $ on $[-L,L]$ (by \\eqref{TS:om-eta-def}), we compute\n\t\\begin{align*}\n\t[a(x)+2\\varepsilon]\\phi-f(x,w)\n\t&\\geq [a(x)+2\\varepsilon]\\phi-\\left[a(x)+\\varepsilon\\right]w\\\\\n\t&= \\varepsilon\\phi-\\left[a(x)+\\varepsilon\\right]v \n\t\\geq \\frac \\zeta 2 \\left[\\varepsilon \\phi-\\left[a(x)+\\varepsilon\\right]\\left(\\frac{\\varepsilon\\omega}{\\gamma+\\varepsilon }e^{c_0^2t}\\right)\\right]\\\\\n\t&\\geq \\frac {\\varepsilon\\zeta }2 e^{\\sqrt{\\lambda^\\varepsilon}c_0 t} [\\psi^\\varepsilon(x)-\\omega]\\geq 0.\n\t\\end{align*}\n\tHence \\eqref{eq:super1} holds for $(t,x)\\in (-\\infty,0)\\times [-L,L]$.\n\t\n\tNow consider \\eqref{eq:super1} for $x\\in [-L,L]^c$.\n\tSince $f(x,w)=f_0(w)$ and the first four terms of \\eqref{eq:super1} are non-negative, it suffices to show that\n\t\\begin{align}\n\tf_0(v) + v_t\\,\\beta' - f_0(w)\\geq 0.\n\t\\label{eq:super2}\n\t\\end{align}\n\tWe consider three cases for the value of $v$.\n\tWhen $v\\leq \\frac{\\theta_0}{2}$, we have $w=v+\\phi\\leq \\theta_0$ (recalling that $\\phi\\le \\frac{\\zeta}{2}$), so $f_0(w)=0$ and \\eqref{eq:super2} holds.\n\tIf $v\\geq 1-\\theta_1$, $f_0(v)\\geq f_0(w)$ because $f_0$ is non-increasing on $[1-\\theta_1,\\infty)$ by \\ref{item:ignition}; \\eqref{eq:super2} follows.\n\tFinally, suppose $\\frac{\\theta_0}{2} \\leq v\\leq 1-\\theta_1$.\n\tIn this case, by $y_0\\ge c_0\\beta(0)+L$, $U'<0$, and $U(0)=\\frac {\\theta_0}2$, we have $x\\le c_0t-L$.\n\tUsing $\\mbox{Lip}(f)=\\gamma$, \\eqref{TS:psi-exp}, and $v_t\\ge \\eta$ from \\eqref{TS:om-eta-def}, we deduce\n\t\\begin{align*}\n\t|f_0(v+\\phi)-f_0(v)|&\\leq \\gamma \\phi(t,x) \\le \\frac {\\gamma\\zeta } 2 e^{\\sqrt{\\lambda^\\eps }(x+L)+\\sqrt{\\lambda^\\eps }c_0 t}\\le \\frac {\\gamma\\zeta } {2} e^{2\\sqrt{\\lambda^\\eps }c_0t} \\\\\n\t&\\leq v_t\\frac{\\gamma \\zeta }{2\\eta}e^{2\\sqrt{\\lambda^\\eps }c_0 t} =  v_t\\,\\beta'.\n\t\\end{align*}\n\tHence \\eqref{eq:super2} always holds, and $w$ is a super-solution to \\eqref{eq:main} on $(-\\infty,0)\\times\\mathbb{R}$.\n\\end{proof}\n\nNext, we construct super- and sub-solutions for $t\\ge 0$.\n\n\\begin{lemma}\\label{lem:sup-t+}\n\t\\begin{enumerate}\n\t\t\\item There exist $y_1\\le -L$ and $B_1>0$ such that\n\t\t$\\tilde w$ given as follows is a super-solution to \\eqref{eq:main}\n\t\ton $(0,\\infty)\\times \\mathbb{R}$\\emph{:}\n\t\t\\begin{gather*}\n\t\t\\tilde w(t,x):= v^{y_1} (t+\\beta_1(t),x)+\\phi_1 (t,x),\\\\\n\t\t\\beta_1 (t):=B_1(1-e^{-c_0^2t\/8}),\\quad \\phi_1(t,x) := e^{-\\frac{c_0}{4} (x-L -\\frac{c_0}{2} t)}.\n\t\t\\end{gather*}\n\t\t\\item For all $y\\in \\mathbb{R}$, there exists $B_2=B_2(y)>0$ such that\n\t\t$\\tilde v^y$ given as follows is a sub-solution to \\eqref{eq:main}\n\t\ton $(0,\\infty)\\times \\mathbb{R}$\\emph{:}\n\t\t\\begin{gather*}\n\t\t\\tilde v^y(t,x):=  v^{y} (t+\\beta_2(t),x) -\\phi_2 (t,x),\\\\\n\t\t\\beta_2 (t):=B_2 e^{-c_0^2t\/8},\\quad \\phi_2(t,x):= \\frac{16\\gamma }{c_0^2} e^{-\\frac{c_0}{4} (x-L-\\frac{c_0}{2}t)}.\n\t\t\\end{gather*}\n\t\\end{enumerate}\n\n\\end{lemma}\n\n\\begin{proof}\n\t(i) Let $y_1,\\ell,B_1\\in \\mathbb{R}$ be defined as follows:\n\t\\begin{equation}\n\t\\label{TS:3.2-def}\n\t\\phi_1(0,-y_1)=\\frac{\\theta_0}{2},\\quad U(\\ell)=1-\\theta_1,\\quad B_1 := \\frac{8\\gamma }{c_0^2 \\eta }e^{-\\frac{c_0}{4}(\\ell-y_1 -L)}.\n\t\\end{equation}\n\tNote that $y_1\\le -L$ because $\\phi_1(0,L)=1$. As before, we abbreviate $v=v^{y_1}$, and we need to show for all $(t,x)\\in(-\\infty,0)\\times \\mathbb{R}$ that\n\t\\begin{equation}\n\t\\label{eq:tgreater}\n\t\\tilde w_t - \\tilde w_{xx} - f(x,\\tilde w) =  v_t\\,\\beta_1' + \\frac{c_0^2}{16}\\phi_1+f_0(v)- f(x,\\tilde w)\\geq 0,\n\t\\end{equation}\n\twhere $v$ and $v_t$ are evaluated at $(t+\\beta_1(t),x)$ for the remainder of this part.\n\tNote that the first three terms are nonnegative.\n\tIf $x\\le L$, then \\eqref{eq:tgreater} holds by $f(x,\\tilde w) =0$ as $\\tilde w(t,x)\\geq \\phi_1(t,x)\\ge 1$.\n\tNow consider $x>L$, noting that $f(x,\\tilde w) = f_0(\\tilde w)$.\n\tWe again consider three cases for the value of $v$.\n\tWhen $v\\geq 1-\\theta_1$, we have $f_0(v)-f_0(\\tilde w)\\geq 0$ because $f_0$ is non-increasing on $\\left[1-\\theta_1,\\infty\\right)$.\n\t\\eqref{eq:tgreater} follows.\n\tIf $v\\leq \\frac{\\theta_0}{2}$, then \\eqref{TS:v1-def} and $U(0)=\\frac {\\theta_0}{2}$ imply $x-c_0t \\ge -y_1$.\n\tTherefore,\n\t\\begin{equation*}\n\t\\phi_1(t,x)\\le \\phi_1(2t,x)=\\phi_1(0,x-c_0t)\\le \\phi_1(0,-y_1)=\\frac{\\theta_0}{2}.\n\t\\end{equation*}\n\tIt follows that $\\tilde w= v+\\phi_1 \\le \\theta_0$, so $f_0(\\tilde w)=0$, and \\eqref{eq:tgreater} holds again.\n\tFinally, suppose $v\\in [\\frac{\\theta_0}{2}, 1-\\theta_1] $.\n\tFrom \\eqref{TS:3.2-def}, we again have $x-c_0t \\geq \\ell-y_1 $.\n\tUsing the definition of $\\eta$ from \\eqref{TS:om-eta-def},\n\t\\begin{align*}\n\t|f_0(v)-f_0(\\tilde w)|&\\leq \\gamma \\phi_1(t,x) = \\gamma e^{-\\frac{c_0}{4} (x-L -\\frac{c_0}{2} t)}\\\\\n\t&\\leq \\gamma e^{-\\frac{c_0 ^2}{8}t-\\frac{c_0}{4}(\\ell-y_1 -L)} \\le v_t \\frac{\\gamma}{\\eta} e^{-\\frac{c^2_0}{8}t-\\frac{c_0}{4}(\\ell-y_1 -L)} = v_t\\,\\beta_1'.\n\t\\end{align*}\n\t\\eqref{eq:tgreater} again follows, so $\\tilde w$ is a super-solution to \\eqref{eq:main} on $(0,\\infty)\\times \\mathbb{R}$.\n\t\n\t(ii) Define $\\tilde \\theta,\\tilde \\eta,\\tilde \\ell, B_2=B_2(y)$ as follows:\n\t\\begin{align*}\n\t\\tilde \\theta := \\min\\cb{\\frac{\\theta_0}{2},\\frac{\\theta_1}{2},\\frac{c_0^2 \\theta_1}{32\\gamma}},&\\quad \\tilde \\eta:= c_0\\inf\\left\\{-U'(s):s\\in \\mathbb{R},\\,U(s)\\in \\left[\\til \\theta,1-\\til \\theta\\right]\\right\\},\\\\\n\tU({\\til \\ell})= 1-\\til \\theta,&\\quad B_2:= \\frac{2^7 \\gamma^2}{c_0^4\\tilde \\eta }e^{-\\frac{c_0}{4}\\left({\\til \\ell}-y -L\\right)}.\n\t\\end{align*}\n\tAgain abbreviate $v=v^{y}$, $\\tilde v=\\tilde v^y$.\n\tWe will show for all $(t,x)\\in (0,\\infty)\\times \\mathbb{R}$ that\n\t\\begin{align}\n\t\\tilde v_t- \\tilde v_{xx} - f(x, \\tilde v ) = v_t\\,\\be_2' - \\frac{c_0^2}{16}\\phi_2  + f_0(v) - f(x, \\tilde v)\\leq 0,\n\t\\label{eq:subsol}\n\t\\end{align}\n\twhere $v$ and $v_t$ are evaluated at $(t+\\be_2(t),x)$ for the rest of the proof.\n\tNote that $f_0(v)$ is the only term in \\eqref{eq:subsol} which is potentially positive.\n\tConsider $x\\in [-L,L]$, where we have $\\phi_2(t,x)\\geq \\phi_2(0,L) = {16\\gamma}{c_0}^{-2}$.\n\tThen \\eqref{eq:subsol} follows from\n\t\\begin{equation*}\n\t- \\frac{c_0^2}{16}\\phi_2 + f_0(v) \\le -\\gamma+f_0(v) \\leq 0.\n\t\\end{equation*}\n\tNow consider $x\\in[-L,L]^c$, where $f(x,\\tilde v) = f_0(\\tilde v)$.\n\t\\eqref{eq:subsol} holds whenever $v \\leq \\til \\theta(\\le \\theta_0)$, as $f_0(v)=0$.\n\tIf $v\\in[\\til \\theta, 1 - \\til \\theta]$, then $x\\ge c_0 t+{\\til \\ell}-y$ because $U(\\tilde \\ell)=1-\\tilde \\theta$. It then follows from $v_t \\ge \\tilde \\eta>0 $ that\n\t\\begin{align*}\n\t\\abs{f_0(v) - f_0(\\tilde v)} &\\leq \\gamma \\phi_2(t,x) =  \\frac{16\\gamma^2}{c_0^2}e^{-\\frac{c_0}{4}(x-L-\\frac{c_0}{2}t)}\\le \\frac{16\\gamma^2}{c_0^2} e^{-\\frac{c_0^2}{8}t-\\frac{c_0}{4}({\\til \\ell}-y-L)} \\\\\n\t&\\le v_t\\frac{16\\gamma^2}{c_0^2\\tilde \\eta}e^{-\\frac{c_0^2}{8}t-\\frac{c_0}{4}({\\til \\ell}-y-L)} = -v_t\\,\\be_2',\n\t\\end{align*}\n\twhich implies \\eqref{eq:subsol}.\n\tFinally, consider $v \\geq 1 - \\til \\theta(\\ge 1-\\frac{\\theta_1}{2})$.\n\tIf $\\phi_2 \\leq \\frac{\\theta_1}{2}$ then $\\tilde v \\geq  1 - \\theta_1$.\n\tSince $f_0$ is non-increasing on $[1 - \\theta_1,\\infty)$, $f_0( v) \\leq f_0( \\tilde v)$, and \\eqref{eq:subsol} holds again.\n\tIf $\\phi_2\\geq \\frac{\\theta_1}{2}$, then\n\t\\begin{equation*}\n\t\\frac{c_0^2}{16}\\phi_2\\geq \\frac{c_0^2\\theta_1}{32}\\geq \\gamma \\til \\theta \\geq \\gamma \\abs{v - 1} \\geq f_0(v).\n\t\\end{equation*}\n\tHence \\eqref{eq:subsol} holds in all cases, as claimed.\n\\end{proof}\n\nWith the requisite super- and sub-solutions in place, we may now establish Theorem \\ref{thm:ex}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:ex}]\n\tIn what follows, $y_0,y_1, B_1, \\beta, \\phi, w,\\beta_1, \\phi_1,\\tilde w,$ and $\\phi_2$ are constants and functions defined in Lemmas \\ref{lem:supersub}(ii) and \\ref{lem:sup-t+}, and let $v=v^{y_0}$ be given as in \\eqref{TS:v1-def}.\n\t\n\tThe construction of an entire solution $u$ follows the standard procedure.\n\tFor $n\\in \\mathbb N$, let $u_n$ be the solution to \\eqref{eq:main} on $(-n,\\infty)\\times \\mathbb{R}$ with initial data $u_n(-n,x) = v(-n,x)$.\n\tFirst, observe that $u_n$ is increasing in time.\n\tIndeed, since $v$ is a sub-solution of \\eqref{eq:main} on $(-\\infty,0)\\times \\mathbb{R}$ with $v_t>0$ by Lemma \\ref{lem:supersub}, the comparison principle implies $u_n(t,\\cdot)> u_n(0,\\cdot)$ for all $t> 0$.\n\tIt follows thast $\\partial_t u_n> 0$ by the maximum principle.\n\tMoreover, since $v(-n,\\cdot)\\le w(-n,\\cdot)$ by their definition, Lemma \\ref{lem:supersub} and the comparison principle ensure that\n\t\\begin{equation*}\n\tv(t,\\cdot)\\le u_n(t,\\cdot)\\le w(t,\\cdot)\\quad\\mbox{for all }t\\in [-n,0].\n\t\\end{equation*}\n\tBy parabolic regularity, we obtain an increasing in time entire solution $u$ to \\eqref{eq:main} as a locally uniform limit along a subsequence of $(u_n)$ satisfying\n\t\\begin{equation}\n\t\\label{TS:ex-comp}\n\tv(t,\\cdot)\\le u(t,\\cdot)\\le w(t,\\cdot)\\quad \\mbox{for all }t\\le 0.\n\t\\end{equation}\n\t\n\tNext, we check that $u$ fulfills \\eqref{item:trans_lims}.\n\tIt obviously holds for $t\\le 0$ by \\eqref{TS:ex-comp} and the limit behavior of $v,w$ at $\\pm \\infty$.\n\tFor $t>0$, the first limit of \\eqref{item:trans_lims} still holds because $u_t>0$ and $u<1$.\n\tTo prove the second limit condition, we first claim that\n\t\\begin{equation}\n\t\\label{TS:ex-comp1}\n\tu(0,x)\\le \\min\\{ 1, w(0,x) \\} \\le \\tilde w(0,x).\n\t\\end{equation}\n\tFrom this, Lemma \\ref{lem:sup-t+}(i), and the comparison principle,\n\t\\begin{equation}\n\t\\label{TS:ex-comp3}\n\tu(t,\\cdot)\\le \\tilde w(t,\\cdot)\\quad \\mbox{for all }t\\ge 0.\n\t\\end{equation}\n\tThe second limit of \\eqref{item:trans_lims} then follows from $\\lim_{x\\to\\infty}\\tilde w(t,x) =0$ and $u>0$.\n\tNow consider \\eqref{TS:ex-comp1}.\n\tThe first inequality is simply \\eqref{TS:ex-comp} with $t=0$.\n\tFor the second, when $x\\le L$, $\\tilde w(0,x)\\ge \\phi_1(0,x) \\ge 1$.\n\tFor $x>L$, \\eqref{TS:psi-exp}, $\\lambda^\\varepsilon >\\frac{c_0^2}{16}$, $U'<0$, and $y_1\\le y_0-c_0\\beta(0)$ (by Lemmas \\ref{lem:supersub}(ii) and \\ref{lem:sup-t+}(i)) imply\n\t\\begin{align*}\n\tw(0,x) &= U(x+y_0-c_0\\beta(0)) + \\frac{\\zeta}{2}\\psi^\\varepsilon(x)\\le U(x+y_1) + e^{-\\frac{c_0}{4}(x-L)} = \\tilde w(0,x).\n\t\\end{align*}\n\tTherefore \\eqref{TS:ex-comp1} holds.\n\tThis completes the proof of \\eqref{item:trans_lims}.\n\t\n\tIt remains to show the bounded front width condition \\eqref{item:trans_width}.\n\tFix $\\mu \\in (0,\\frac 12 )$ and let\n\t\\begin{equation*}\n\tX^-(t):=\\inf \\{x\\in \\mathbb{R}:u(t,x)\\le 1-\\mu\\},\\quad X^+(t):=\\sup \\{x\\in \\mathbb{R}:u(t,x)\\ge \\mu\\},\n\t\\end{equation*}\n\tfor then we have $L_\\mu(t) = X^+(t)-X^-(t)$.\n\tWe will show that $L_\\mu(t)$ is uniformly bounded in $t\\in \\mathbb{R}$ by considering these three cases: $t\\le 0$, $t> t_\\mu,$ and $t\\in (0,t_\\mu]$, where $t_\\mu$ will be defined shortly.\n\tFor the first case $t<0$, let\n\t\\begin{gather*}\n\t\\rho_-:= U^{-1}(1-\\mu),\\quad \n\t \\rho_+:= \\max\\left\\{ U^{-1}\\left (\\frac \\mu 2\\right), \\frac 1{\\sqrt{\\lambda^\\eps }}\\left |\\log \\frac \\mu 2\\right|+L+y_0-c_0\\beta(0) \\right\\}.\n\t\\end{gather*}\n\tFor all $x< c_0 t + \\rho_--y_0$, by \\eqref{TS:ex-comp} we have\n\t\\begin{equation*}\n\tu(t,x)\\ge v(t,x) = U(x-c_0t +y_0)> U(\\rho_-)= 1-\\mu.\n\t\\end{equation*}\n\tTherefore $X^-(t) \\ge c_0t +\\rho_--y_0$.\n\tNow, suppose $x> c_0t+\\rho_+ -y_0 +c_0\\beta(0)$.\n\tCombining \\eqref{TS:psi-exp}, $||\\psi^\\varepsilon||_{L^\\infty}=1$, \\eqref{TS:ex-comp} and the definition of $\\rho_+$, we compute\n\t\\begin{align*}\n\tu(t,x) &\\le w(t,x) = U(x-c_0(t+\\beta(t))+y_0)+\\frac \\zeta 2 e^{c_0\\sqrt{\\lambda^\\eps }t}\\psi^\\eps(x) \\\\\n\t&\\le  U(x-c_0(t+\\beta(0))+y_0)+ \\frac \\zeta 2e^{-\\sqrt{\\lambda^\\eps }(x-L-c_0t)} \\\\\n\t&< U(\\rho_+)+ e^{-\\sqrt{\\lambda^\\eps }(\\rho_+-L-y_0+c_0\\beta(0))}\\le \\mu .\n\t\\end{align*}\n\tThis implies $X^+(t) \\le c_0t +\\rho_+ - y_0+c_0\\beta(0)$, so\n\t\\begin{equation}\n\t\\label{TS:L-1}\n\tL_\\mu(t) \\le c_0\\beta(0)+\\rho_+ - \\rho_-\\quad \\mbox{for }t\\le 0.\n\t\\end{equation}\n\t\n\tWe now define $t_\\mu$.\n\tGiven $y\\in \\mathbb{R}$, let $\\bar v^y(t,x):= v^{y}(t,x)- \\phi_2(t,x)$, where $v^y,\\phi_2$ are from \\eqref{TS:v1-def} and Lemma \\ref{lem:sup-t+}(ii).\n\tOne can easily verify that $M(y):=\\sup_{x\\in \\mathbb{R}} \\bar v^{y}(0,x)$ is continuous, non-increasing in $y\\in \\mathbb{R}$, $\\lim_{y\\to-\\infty} M(y)=1$, $\\lim_{y\\to\\infty}M(y)=0$, and the supremum is achieved somewhere if $M(y)>0$.\n\tSo we may fix $y_2=y_2(\\mu)$ such that $M(y_2)=1-\\mu$.\n\tFor the remainder of the proof, we abbreviate $\\bar v = \\bar v ^{y_2}$ and let $B_2=B_2(y_2)$, $\\be_2$, and $\\tilde v=\\tilde v^{y_2}$ be from Lemma \\ref{lem:sup-t+}(ii).\n\tLet $x_\\mu\\in \\mathbb{R}$ be a maximizer so that $\\bar v (0,x_\\mu)=1-\\mu$, and define\n\t\\begin{equation}\\label{t-mu-def}\n\tt_\\mu := \\inf \\cb{t>0: u(t,\\cdot)\\ge \\max \\{\\tilde v(0,\\cdot), (1-\\mu)\\chi_{(-\\infty,x_\\mu]}\\}  }\\ge 0.\n\t\\end{equation}\n\tWe claim that $t_\\mu$ is finite.\n\tAfter all, we have $\\tilde v(0,\\cdot)\\le (1-\\mu')\\chi_{I}$ for some bounded interval $I\\subset \\mathbb{R}$ and $\\mu'\\in (0,1)$.\n\tRecall that $u_t>0$ and the limit condition \\eqref{item:trans_lims} holds.\n\tTherefore, we have $u(t,\\cdot)\\nearrow 1$ uniformly on each $(-\\infty,R)$, $R\\in \\mathbb{R}$, which implies $t_\\mu<\\infty$.\n\t\n\tNow consider the front width for $t>t_\\mu$.\n\tCombining \\eqref{t-mu-def}, Lemma \\ref{lem:sup-t+}(ii), the comparison principle, and $\\be_2>0$, we have\n\t\\begin{equation}\n\t\\label{TS:ex-comp2}\n\tu(t_\\mu+t,\\cdot)\\ge \\tilde v(t,\\cdot)\\ge \\bar v(t,\\cdot)\\quad \\mbox{for all }t\\ge 0.\n\t\\end{equation}\n\tRecall $B_1$ from Lemma \\ref{lem:sup-t+}(i) and set\n\t\\begin{equation*}\n\t\\tilde\\rho_+ := \\max \\cb{U^{-1}\\rb{\\frac \\mu 2}, \\frac{4}{c_0} \\abs{\\log\\frac{\\mu}{2}} +y_1+L-c_0B_1}.\n\t\\end{equation*}\n\tThen $X^+(t) \\le c_0(t+B_1) +\\tilde \\rho_+-y_1$.\n\tIndeed, for $x> c_0(t+B_1)+\\tilde\\rho_+-y_1$, \\eqref{TS:ex-comp3} and $\\beta_1< B_1$ imply\n\t\\begin{align*}\n\tu(t,x)&\\le \\tilde w(t,x) = U(x-c_0(t+\\beta_1(t))+y_1) + e^{-\\frac{c_0}{4} (x-L-\\frac{c_0}{2}t)}\\\\\n\t&< U(\\tilde\\rho_+)+ e^{-\\frac{c_0}{4}(c_0B_1+\\tilde \\rho_+-y_1-L)} \\le \\mu.\n\t\\end{align*}\n\tOn the other hand, we claim that $X^-(t)\\ge c_0(t-t_\\mu)+x_\\mu$, implying:\n\t\\begin{equation}\n\t\\label{TS:L-2}\n\tL_\\mu(t) \\le c_0(t_\\mu+ B_1) +\\tilde \\rho_+ -y_1 - x_\\mu,\\quad \\text{for }t\\in (t_\\mu,\\infty).\n\t\\end{equation}\n\tTo prove the claimed bound, it suffices to check that $u(t,x)\\ge 1-\\mu$ for all $x< c_0(t-t_\\mu)+x_\\mu$.\n\tIf $x\\le x_\\mu$, then $u_t>0$ and \\eqref{t-mu-def} show that $u(t,x)> u(t_\\mu ,x)\\ge 1-\\mu$.\n\tNow consider $x\\in (x_\\mu, c_0(t-t_\\mu)+x_\\mu)$.\n\tNote that $v^{y_2}(t,x_\\mu+c_0t)=U(x_\\mu+y_2)$, while $t\\mapsto \\phi_2(t,x_\\mu+c_0t)$ is decreasing.\n\tHence, their difference $\\bar v (t,x_\\mu+c_0t) $ is increasing in $t$, and\n\t\\begin{equation*}\n\t\\bar v(t,x_\\mu+c_0t)> \\bar v(0,x_\\mu)=1-\\mu\\quad  \\mbox{ for all }t>0.\n\t\\end{equation*}\n\tLet $t_*:=c_0^{-1}(x-x_\\mu)\\in (0,t-t_\\mu)$.\n\tThen by $u_t> 0$ and \\eqref{TS:ex-comp2}, it follows that\n\t\\begin{equation*}\n\tu(t,x) > u(t_\\mu+t_*,x)=u(t_\\mu+t_*,x_\\mu+c_0t_*)\\ge \\bar v(t_*,x_\\mu+c_0t_*) > 1-\\mu.\n\t\\end{equation*}\n\tThis proves the claim.\n\t\n\tFinally, consider $t\\in (0,t_\\mu]$.\n\tSince $u_t>0$, the width is bounded by\n\t\\begin{equation*}\n\tL_\\mu(t)\\le X^+(t_\\mu)-X^-(0) \\le c_0 (t_\\mu+B_1) +\\tilde\\rho_+-\\rho_- +y_0-y_1.\n\t\\end{equation*}\n\tWith this, \\eqref{TS:L-1}, and \\eqref{TS:L-2}, $L_\\mu(t)$ is uniformly bounded for all $t\\in \\mathbb{R}$.\n\tThis concludes the proof of \\eqref{item:trans_width}.\n\tTherefore $u$ is an increasing-in-time transition front solution of \\eqref{eq:main}. \n\tIt also obviously holds from the comparisons \\eqref{TS:ex-comp}, \\eqref{TS:ex-comp3} and \\eqref{TS:ex-comp2} that $u$ has a global mean speed $c_0$. \n\tThis completes the proof of Theorem \\ref{thm:ex}. \n\\end{proof}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGeneralized Nash equilibrium problems (GNEPs) have been widely studied in the literature \\cite{Ros65, facchinei2007, facchinei2010} and such a strong interest is motivated by numerous applications ranging from economics to engineering \\cite{pavel2007,kulkarni2012}. In a GNEP, each agent seeks to minimize his own cost function, under local and coupled feasibility constraints. In fact, both the cost function and the constraints depend on the strategies chosen by the other agents. Due to the presence of these shared constraints, the search for generalized Nash equilibria is usually a quite challenging task. \n\nFor the computation of a GNE, various algorithms have been proposed, both distributed \\cite{belgioioso2018,yi2019}, and semi-decentralized \\cite{facchinei2010,belgioioso2017}. When dealing with coupling constraints, a common principle is the focus on a special class of equilibria, which reflect some notion of fairness among the agents. This class is known as \\emph{variational equilibria} (v-GNE) \\cite{Deb52,facchinei2010}. Besides fairness considerations, v-GNE is computationally attractive since it can be formulated in terms of variational inequality, which makes it possible to solve them via operator splitting techniques \\cite{BauCom16,facchinei2007}. A recent breakthrough along these lines is the distributed, preconditioned, forward-backward (FB) algorithm conceived in \\cite{yi2019} for strongly-monotone games. The key lesson from \\cite{yi2019} is that the FB method cannot be directly applied to GNEPs, thus a suitable preconditioning is necessary. From a technical perspective, the FB operator splitting requires that the pseudo-gradient mapping of the game is strongly monotone, an assumption which is not always satisfied\n\nIn this paper we investigate two distributed algorithmic schemes for computing a v-GNE. Motivated by the need to relax the strong monotonicity assumption on the pseudo-gradient of the game, we first investigate a distributed \\textit{forward-backward-forward} (FBF) algorithm \\cite{tseng2000}. We show that a suitably constructed FBF operator splitting guarantees not only fully distributed computation, but also convergence to a v-GNE under the mere assumption of monotonicity of the involved operators. This enables us to drop the strong monotonicity assumption, which is the main advantage with respect to the FB-based splitting methods \\cite{yi2019}.\nAs a second condition, in order to exploit the structure of the monotone inclusion defining the v-GNE problem, we also investigate the \\textit{forward-backward-half-forward} (FBHF) algorithm \\cite{briceno2018}. We would like to point out that both our algorithms are distributed in the sense that each agent needs to know only his local cost function and its local feasible set, and there is no central coordinator that updates and broadcasts the dual variables. The latter is the main difference with semi-decentralized schemes for aggregative games \\cite{Gram17,belgioioso2017}.\n\n\nCompared with the FB and the FBHF algorithms, the FBF requires less restrictive assumptions to guarantee convergence, i.e., plain monotonicity of the pseudo-gradient mapping. On the other hand, the FBF algorithm requires two evaluations of the pseudo-gradient mapping, which means that the agents must communicate at least twice at each iterative step. Confronted with the FBF algorithm, our second proposal, the FBHF algorithm requires only one evaluation of the pseudo-gradient mapping, but needs strong monotonicity to provide theoretical convergence guarantees. Effectively, the FBHF algorithm is guaranteed to converge under the same assumptions as the preconditioned FB \\cite{yi2019}.\n\n\n\n\n\n\n\\paragraph*{Notation}\n${\\mathbb{R}}$ indicates the set of real numbers and $\\bar{\\mathbb{R}}={\\mathbb{R}}\\cup\\{+\\infty\\}$. $\\mathbf{0}_N$ ($\\mathbf{1}_N$) is the vector of N zeros (ones). The Euclidean inner product and norm are indicated with $\\inner{\\cdot,\\cdot}$ and $\\norm{\\cdot}$, respectively.\nLet $\\Phi$ be a symmetric, positive definite matrix, $\\Phi \\succ 0$. The induced inner product is $\\inner{\\cdot,\\cdot}_{\\Phi}:=\\inner{\\Phi\\cdot,\\cdot}$, and the associated norm is $\\norm{\\cdot}_{\\Phi}:=\\inner{\\cdot,\\cdot}_{\\Phi}^{1\/2}$. We call $\\mathcal H_\\Phi$ the Hilbert space with norm $\\norm{\\cdot}_\\Phi$. \nGiven a set $\\mathcal X\\subseteq{\\mathbb{R}}^n$, the normal cone mapping is denoted with $\\NC_{\\mathcal{X}}(x)$. $\\Id$ is the identity mapping. Given a set-valued operator $A$, the graph of $A$ is the set $\\gr(A)=\\{(x,y)\\vert y\\in Ax\\}$ The set of zeros is $\\Zer A=\\{x\\in{\\mathbb{R}}^n \\mid 0\\in Ax\\}$.  \nThe resolvent of a maximally monotone operator $A$ is the map $\\mathrm{J}_{A}=(\\Id+A)^{-1}$. If $g:{\\mathbb{R}}^{n}\\to(-\\infty,\\infty]$ is a proper, lower semi-continuous, convex function, its subdifferential is the maximal monotone operator $\\partial g(x)$. The proximal operator is defined as $\\operatorname{prox}^{\\Phi}_{g}(v)=\\operatorname{J}_{\\Phi\\partial g}(v)$ \\cite{BauCom16}.\nGiven $x_{1}, \\ldots, x_{N} \\in {\\mathbb{R}}^{n}, \\boldsymbol{x} :=\\operatorname{col}\\left(x_{1}, \\dots, x_{N}\\right)=\\left[x_{1}^{\\top}, \\dots, x_{N}^{\\top}\\right]^{\\top}$.\n\n\\section{Mathematical Setup: The Monotone Game and Variational Generalized Nash Equilibria}\n\\label{sec:problem}\nWe consider a game with $N$ agents where each agent chooses an action $x_{i}\\in{\\mathbb{R}}^{n_i}$, $i\\in\\mathcal I=\\{1,\\dots,N\\}$.  \n\nEach agent $i$ has an extended-valued local cost function $J_{i}: {\\mathbb{R}}^n \\to (-\\infty,\\infty]$ of the form \n\\vspace{-.15cm}\\begin{equation}\\label{eq:f}\nJ_{i}(x_{i}, \\boldsymbol x_{-i}):=f_{i}(x_{i},\\boldsymbol x_{-i}) + g_{i}(x_{i}).\n\\vspace{-.15cm}\\end{equation}\nwhere $\\boldsymbol x_{-i}=\\operatorname{col}(\\{x_j\\}_{j\\neq i})$ is the vector of all decision variables except for $x_i$, and $g_{i}:{\\mathbb{R}}^{n_{i}}\\to(-\\infty,\\infty]$ is a local idiosyncratic costs function which is possibly non-smooth. Thus, the function $J_{i}$ in \\eqref{eq:f} has the typical splitting into smooth and non-smooth parts.\n\\begin{standassumption}[Local cost]\nFor each $i\\in\\mathcal I$, the function $g_i$ in \\eqref{eq:f} is lower semicontinuous and convex.\nFor each $i\\in\\mathcal I$, $\\dom(g_{i})=\\Omega_i$ is a closed convex set.\n\\hfill\\small$\\blacksquare$\n\\end{standassumption}\n\n\nExamples for the local cost function are indicator functions to enforce set constraints, or penalty functions that promote sparsity, or other desirable structure. \n\nFor the function $f_i$ in \\eqref{eq:f}, we assume convexity and differentiability, as usual in the GNEP literature \\cite{facchinei2010}.\n\\begin{standassumption}[Local convexity]\n\\label{ass:IC}\nFor each $i \\in \\mathcal{I}$ and for all $\\boldsymbol{x}_{-i} \\in {\\mathbb{R}}^{n-n_i}$, the function $f_{i}(\\cdot, \\boldsymbol{x}_{-i})$ in \\eqref{eq:f} is convex and continuously differentiable. \n\\hfill$\\blacksquare$\n\\end{standassumption}\n\n\n\nWe assume that the game displays joint convexity with affine coupling constraints defining the collective feasible set \n\\vspace{-.15cm}\\begin{equation}\\label{eq:coupling}\n\\boldsymbol{\\mathcal{X}}:=\\left\\{\\boldsymbol x \\in\\boldsymbol\\Omega \\mid A \\boldsymbol{x}-b \\leq {\\boldsymbol{0}}_{m}\\right\\}\n\\vspace{-.15cm}\\end{equation}\nwhere $A:=[A_1,\\dots, A_N]\\in{\\mathbb{R}}^{m\\times n}$ and $b:=\\sum_{i=1}^{N}b_{i}\\in{\\mathbb{R}}^m$. \nEffectively, each matrix $A_i\\in{\\mathbb{R}}^{m\\times n_i}$ defines how agent $i$ is involved in the coupling constraints, thus we consider it to be private information of agent $i$. \nThen, for each $i$, given the strategies of all other agents $\\boldsymbol x_{-i}$, the feasible decision set is\n\\vspace{-.15cm}\\begin{equation}\n\\mathcal{X}_{i}(\\boldsymbol{x}_{-i}) := \\left\\{y_i \\in \\Omega_i \\mid \\, A_i y_i \\leq b-\\textstyle\\sum_{j \\neq i}^{N} A_j x_j\\right\\}.\n\\vspace{-.15cm}\\end{equation}\n\nNext, we assume a constraint qualification condition.\n\n\\begin{standassumption}\\label{ass_X}\n(\\textit{Constraint qualification})\nThe set $\\boldsymbol{\\mathcal{X}}$ in \\eqref{eq:coupling} satisfies Slater's constraint qualification. \n\\hfill\\small$\\blacksquare$\n\\end{standassumption}\nThe aim of each agent is to solve its local optimization problem\n\\vspace{-.15cm}\\begin{equation}\\label{game}\n\\forall i\\in\\mathcal I: \\quad\\left\\{\\begin{array}{cl}\n\\min_{x_i \\in  \\Omega_i} & J_{i}(x_i, \\boldsymbol x_{-i}) \\\\ \n\\text { s.t. } & A_i x_i \\leq b-\\sum_{j \\neq i}^{N} A_j x_j.\n\\end{array}\\right.\n\\vspace{-.15cm}\\end{equation}\n\n\n\n\n\nThus, the solution concept for such a competitive scenario is the generalized Nash equilibrium \\cite{Deb52,facchinei2010}. \n\n\\begin{definition} (\\textit{Generalized Nash equilibrium})\nA collective strategy $\\boldsymbol x^{\\ast}=\\operatorname{col}(x_{1}^{\\ast},\\ldots,x_{N}^{\\ast})\\in \\boldsymbol{\\mathcal{X}}$\nis a generalized Nash equilibrium of the game in \\eqref{game} if, for all $i\\in\\mathcal I$,\n\\vspace{-.15cm}\\begin{equation*}\nJ_i(x^*_i,\\boldsymbol x^*_{-i}) \\leq \\inf\\{ J_i(y,\\boldsymbol x^*_{-i}) \\, \\mid  \\, y\\in\\mathcal X_i(\\boldsymbol x_{-i})\\}.\\vspace{-.4cm}\n\\vspace{-.15cm}\\end{equation*}\n\\hfill\\small$\\blacksquare$\n\\end{definition}\nTo derive optimality conditions characterizing GNE, we define agent $i$'s Lagrangian function as\n$\\mathcal L_{i}(x_{i},\\lambda_i,  \\boldsymbol x_{-i}):=J_{i}(x_i, \\boldsymbol x_{-i})+\\lambda_i^{\\top}(A\\boldsymbol{x}-b)$\nwhere $\\lambda_i\\in{\\mathbb{R}}^m_{\\geq 0}$ is the Lagrange multiplier associated with the coupling constraint $A \\boldsymbol{x} \\leq b$. Thanks to the sum rule of the subgradient for Lipschitz continuous functions \\cite[\\S 1.8]{Cla98}, we can write the subgradient of agent $i$ as \n$ \\partial_{x_{i}}J_{i}(x_{i},\\boldsymbol x_{-i})=\\nabla_{x_{i}} f_{i}(x_{i},\\boldsymbol x_{-i})+\\partial g_{i}(x_{i})$. Therefore, \nUnder Assumption \\ref{ass_X}, the Karush--Kuhn--Tucker (KKT) theorem ensures the existence of a pair $(x^*_{i},\\lambda^*_i)\\in\\Omega_{i}\\times\\mathbb{R}^{m}_{\\geq 0}$, such that \n\\begin{equation}\\label{KKT_game}\n\\forall i\\in\\mathcal I:\\begin{cases}\n\\mathbf{0}_{n_i}\\in \\nabla_{x_{i}} f_{i}(x^*_{i};\\boldsymbol x^*_{-i})+\\partial g_{i}(x^*_{i})+A^{\\top}_{i}\\lambda_i^*\\\\\n\\mathbf{0}_{m}\\in \\NC_{\\mathbb{R}^{m}_{\\geq0}}(\\lambda_i^*)-(A\\boldsymbol{x}^*-b).\n\\end{cases}\n\\vspace{-.15cm}\\end{equation}\n\nWe conclude the section by postulating a standard assumption for GNEP's \\cite{facchinei2010}, and inclusion problems in general \\cite{BauCom16}, concerning the monotonicity and Lipschitz continuity of the mapping that collects the partial gradients $\\nabla_{i} f_{i}$.\n\n\\begin{standassumption}[Monotonicity]\n\\label{ass:GM}\nThe mapping \n\\vspace{-.15cm}\\begin{equation}\\label{eq:F}\nF(\\boldsymbol x):= \\mathrm{col}\\left( \\nabla_{x_1}f_{1}(\\boldsymbol x),\\ldots,\\nabla_{x_N}f_{N}(\\boldsymbol x)\\right)\n\\vspace{-.15cm}\\end{equation}\nis monotone on $\\boldsymbol\\Omega$, i.e., for all $\\boldsymbol x,\\boldsymbol y\\in\\boldsymbol\\Omega$\n$\\langle F(\\boldsymbol x)-F(\\boldsymbol y),\\boldsymbol x-\\boldsymbol y\\rangle\\geq 0.$\nand $\\frac{1}{\\beta}$-Lipschitz continuous, $\\beta > 0$, i.e., for all $\\boldsymbol x,\\boldsymbol y\\in\\boldsymbol\\Omega$,\\vspace{-.15cm}\n$\\norm{F(\\boldsymbol x)-F(\\boldsymbol y)} \\leq \\tfrac{1}{\\beta}\\norm{\\boldsymbol x-\\boldsymbol y}.\n\\hfill\\small$\\blacksquare$\n\\end{standassumption}\n\\vspace{.15cm}\nAmong all possible GNEs of the game, this work focuses on the computation of a \\emph{variational GNE} (v-GNE) \\cite[Def. 3.10]{facchinei2010}, i.e. a GNE in which all players share consensus on the dual variables:\n\\vspace{-.15cm}\\begin{equation}\\label{KKT_VI}\n\\forall i\\in\\mathcal I:\\begin{cases}\n\\mathbf{0}_{n_i}\\in \\nabla_{x_{i}}f_{i}(x^{*}_{i};\\boldsymbol{x}^{*}_{-i})+\\partial g_{i}(x^{*}_{i})+A_{i}^{\\top}\\lambda^{\\ast}\\\\\n\\mathbf{0}_{m}\\in \\NC_{\\mathbb{R}^{m}_{\\geq0}}(\\lambda^*)-(A\\boldsymbol{x}^*-b).\\\\\n\\end{cases}\n\\vspace{-.15cm}\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Distributed Generalized Nash equilibrium seeking via Operator Splitting}\nIn this section, we present the proposed distributed algorithms. We allow each agent to have information on his own local problem data only, i.e., $J_{i},\\Omega_{i}, A_{i}$ and $b_{i}$. We let each agent $i$ control its local decision $x_{i}$, and a local copy $\\lambda_{i}\\in\\mathbb{R}^{m}_{\\geq0}$ of dual variables, as well as a local auxiliary variable $z_{i}\\in\\mathbb{R}^{m}$ used to enforce consensus of the dual variables.\nTo actually reach consensus on the dual variables, we let the agents exchange information via an undirected weighted \\emph{communication graph}, represented by its weighted adjacency matrix $\\boldsymbol W = [w_{i,j}]_{i,j}\\in{\\mathbb{R}}^{N\\times N}$. We assume $w_{ij}>0$ iff $(i,j)$ is an edge in the communication graph. The set of neighbours of agent $i$ in the graph is $\\mathcal{N}_{i}^{\\lambda}=\\{j |w_{i,j}>0\\}$.\n\\vspace{.15cm}\n\\begin{standassumption}[Graph connectivity]\\label{ass:graph}\nThe matrix $\\mathbf{W}$ is symmetric and irreducible.\\hfill\\small$\\blacksquare$\n\\end{standassumption}\nDefine the weighted Laplacian as $\\mathbf{L}:=\\diag\\left\\{(\\mathbf{W}\\mathbf{1}_{N})_{1}, \\dots, (\\mathbf{W}\\mathbf{1}_{N})_{N}\\right\\}-\\mathbf{W}$. It holds that $\\mathbf{L}^{\\top}=\\mathbf{L}$, $\\ker(\\mathbf{L})=\\Span(\\mathbf{1}_{N})$ and that, given Standing Assumption \\ref{ass:graph}, $\\mathbf{L}$ is positive semi-definite with real and distinct eigenvalues $0=s_{1}<s_{2}\\leq \\ldots \\leq s_{N}$.\nMoreover, given the maximum degree of the graph $\\mathcal G^\\lambda$, $\\Delta:=\\max_{i\\in\\mathcal I}(\\mathbf{W}\\mathbf{1}_{N})_{i}$, it holds that $\\Delta \\leq s_{N}\\leq 2\\Delta$. Denoting by $\\kappa:=\\abs{\\mathbf{L}}$, it holds that $\\kappa\\leq 2\\Delta$.  We define the tensorized Laplacian as the matrix $\\bar{\\mathbf{L}}=\\mathbf{L}\\otimes I_{m}$. We set $\\bar{b}=(b_1,\\ldots,b_{N})^{\\top}$, $\\boldsymbol x=\\operatorname{col}(x_{1},\\ldots,x_{N})$ and similarly $\\boldsymbol z$ and $\\boldsymbol \\lambda$.\n\nLet ${\\bf{A}}=\\operatorname{diag}(A_1,\\dots,A_N)$ and define \n\n\\vspace{-.15cm}\\begin{equation}\\label{eq:AB}\n\\begin{aligned}\n\\mathcal{A}(\\boldsymbol{x},\\boldsymbol z,\\boldsymbol \\lambda):=\\operatorname{col}(F({\\bf x}),{\\bf{0}}_{mN},\\bar{\\bf{L}}\\boldsymbol\\lambda+\\bar b),\\\\\n\\mathcal{B}(\\boldsymbol{x},\\boldsymbol z,\\boldsymbol \\lambda):=\\operatorname{col}({\\bf{A}}^{\\top}\\boldsymbol \\lambda, \\bar{\\bf{L}}\\boldsymbol \\lambda,-{\\bf{A}}\\boldsymbol{x}-\\bar{\\bf{L}}\\boldsymbol z)\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nLet us also define the operator $\\mathcal D :=\\mathcal A+\\mathcal B$, and the set-valued operator\n\\vspace{-.15cm}\\begin{equation}\\label{eq:C}\n\\mathcal C(\\boldsymbol{x},\\boldsymbol z,\\boldsymbol \\lambda)=G(\\boldsymbol{x})\\times\\{{\\bf{0}}_{mN}\\}\\times \\NC_{{\\mathbb{R}}_{ \\geq 0}^{mN}}(\\boldsymbol \\lambda)\n\\vspace{-.15cm}\\end{equation}\nwhere $G(\\boldsymbol x)=\\partial g_{1}(x_{1})\\times\\cdots\\times\\partial g_{N}(x_{N})$.\n\nLet us summarize the properties of the operators above.\n\\begin{lemma}\\label{lemma_op}\nThe following statements hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal A$ is maximally monotone and $L_{\\mathcal A}=( \\frac{1}{\\beta}+\\kappa)$-Lipschitz continuous.\n\\item[(ii)] $\\mathcal B$ is maximally monotone and $L_{\\mathcal B}=(2|{\\bf {A}}|+2\\kappa)$-Lipschitz continuous.\n\\item[(iii)] $\\mathcal D$is maximally monotone and $L_{\\mathcal D}=L_{\\mathcal A}+L_{\\mathcal B}$-Lipschitz continuous.\n\\item[(iv)] $\\mathcal C+\\mathcal D$ is maximally monotone.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n(i) The operator $\\mathcal A$ is maximally monotone by \\cite[Prop. 20.23]{BauCom16}.\nFurthermore, given $\\kappa=\\abs{\\mathbf{L}}$ and $\\boldsymbol u:=\\operatorname{col}(\\boldsymbol x,\\boldsymbol z,\\boldsymbol \\lambda)$, it holds that \n\\vspace{-.15cm}\\begin{equation}\\label{A_lip}\n\\begin{aligned}\n\\norm{ \\mathcal A\\boldsymbol u-\\mathcal A\\boldsymbol u'}\\leq& \\norm{F(\\boldsymbol x)-F(\\boldsymbol x')}+\\norm{\\bar{\\mathbf{L}}(\\boldsymbol z-\\boldsymbol z')}\\\\\n\\leq&( \\frac{1}{\\beta}+\\kappa)\\left(\\norm{\\boldsymbol x-\\boldsymbol x'}+\\norm{\\boldsymbol z-\\boldsymbol z'}\\right),\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nshowing that $\\mathcal A$ is $L_{\\mathcal A}:=(\\frac{1}{\\beta}+\\kappa)$-Lipschitz continuous.\\\\\n(ii) The operator $\\mathcal B$ is maximally monotone by \\cite[Cor. 20.28]{BauCom16}.\nBy a computation similar to \\eqref{A_lip}, it can be shown that $\\mathcal B$ is $L_{\\mathcal B}=(2|{\\bf {A}}|+2\\kappa)$-Lipschitz continuous. \\\\\n(iii) The operator $\\mathcal D$ is maximally monotone since $\\dom(\\mathcal B)=\\boldsymbol\\Omega$ \\cite[Prop. 20.23]{BauCom16}. It is $L_{\\mathcal D}=L_{\\mathcal A}+L_{\\mathcal B}$-Lipschitz continuous because sum of Lipschitz operators.\\\\\n(iv) The operator $\\mathcal C$ is maximally monotone by \\cite[Lem. 5]{yi2019} and $\\mathcal C+\\mathcal D$ is maximally monotone by \\cite[Prop. 20.23]{BauCom16}.\n\\end{proof}\n\nThe following result, which immediately follows from the definition of the inner product $\\langle\\cdot,\\cdot\\rangle_\\Phi$, holds for monotone operators and it will be recalled later on.\n\\begin{lemma}\\label{lemma_mono}\nLet $\\Phi\\succ0$ and $\\mathcal T$ be a monotone operator, then $\\Phi^{-1}\\mathcal T$ is monotone in the Hilbert space $\\mathcal H_\\Phi$.\n\\hfill\\small$\\blacksquare$\\end{lemma}\n\n\n\nNow, given the operators $\\mathcal A$, $\\mathcal B$ and $\\mathcal C$ as in \\eqref{eq:AB} and \\eqref{eq:C}, a simple proof shows that the zeros of the sum $\\mathcal A+\\mathcal B+\\mathcal C$ are v-GNE of the game in \\eqref{game}. In fact, this follows verbatim from \\cite[Thm. 2]{yi2019}, so we omit the detailed proof here.\n\\begin{lemma}\nThe set $\\Zer(\\mathcal A+\\mathcal B+\\mathcal C)$ is the set of v-GNE of the game in \\eqref{game} and it is non-empty.\n\\hfill\\small$\\blacksquare$\n\\end{lemma}\n\n\n\n\n\n\\subsection{Forward-backward operator splitting} \n\\label{sec:FB}\nThe aim of this section is to revisit a distributed forward-backward (FB) splitting algorithm for the  distributed computation of a v-GNE, see Algorithm \\ref{FB_algo} \\cite{yi2019}. \n\\begin{algorithm}[h]\n\\caption{Preconditioned Forward Backward}\\label{FB_algo}\nInitialization: $x_i^0 \\in \\Omega_i, \\lambda_i^0 \\in {\\mathbb{R}}_{\\geq0}^{m},$ and $z_i^0 \\in {\\mathbb{R}}^{m} .$\\\\\nIteration $k$: Agent $i$\\\\\n($1$) Receives $x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}, \\lambda_j^k$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$, then updates\n$$\\begin{aligned}\n&x_i^{k+1}=\\operatorname{prox}^{\\rho_i}_{g_{i}}[x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-A_{i}^{T} \\lambda_i^k)]\\\\\n&z_i^{k+1}=z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)\n\\end{aligned}$$\n($2$) Receives $z_{j}^{k+1}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$, then updates\n$$\\begin{aligned}\n \\lambda_i^{k+1}=&\\operatorname{proj}_{{\\mathbb{R}}^m_{\\geq 0}}\\{\\lambda_i^k-\\tau_{i}[A_{i}(2 x_i^{k+1}-x_i^k)-b_{i}\\\\\n&+\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[2(z_i^{k+1}-z_{j, k+1})-(z_i^k-z_j^k)]\\\\\n&+\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)]\\}\\\\\n\\end{aligned}$$\n\\end{algorithm}\nFrom now on, the triplet $\\boldsymbol u:=\\operatorname{col}(\\boldsymbol x,\\boldsymbol z,\\boldsymbol \\lambda)$ defines the state variable of a distributed algorithm. Given the state at iteration $k$, $\\boldsymbol u^k=(\\boldsymbol x^k,\\boldsymbol z^k,\\boldsymbol \\lambda^k)$, the FB algorithm can be written as fixed-point iteration of the form $\\boldsymbol u^{k+1}=T_{\\text{FB}}\\boldsymbol u^k,$ where\n\\vspace{-.15cm}\\begin{equation}\\label{eq:FB}\nT_{\\text{FB}}:= \\mathrm{J}_{\\Phi_{\\text{FB}}^{-1}(\\mathcal C+\\mathcal B)}(\\operatorname{Id}-\\Phi_{\\text{FB}}^{-1}\\mathcal A).\n\\vspace{-.15cm}\\end{equation}\nand $\\Phi_{\\text{FB}}$ is the preconditioning matrix defined as \n\\vspace{-.15cm}\\begin{equation}\\label{eq:phi}\n\\Phi_{\\text{FB}}:=\\left[\\begin{array}{ccc} \n\\rho^{-1} & 0 & -{\\mathbf A}^{\\top}\\\\\n0 & \\sigma^{-1} & -\\bar{\\mathbf{L}}\\\\\n-{\\mathbf A} & -\\bar{\\mathbf{L}} &  \\tau^{-1}\n\\end{array}\\right].\n\\vspace{-.15cm}\\end{equation}\nThe matrices $\\rho=\\diag\\{\\rho_{1}I_{n_{1}},\\ldots,\\rho_{N} I_{n_{N}}\\}$, $\\sigma$ and $\\tau$ (defined analoguosly)\ncollect the step sizes of the primal, the auxiliary and the dual updates, respectively. By choosing the step sizes appropriately, the preconditioning matrix $\\Phi$ can be made positive definite \\cite{belgioioso2018}. The FB algorithm is known to converge to a zero of a monotone inclusion $0 \\in \\mathcal A+\\mathcal B+\\mathcal C$ when the operators are maximally monotone and the single-valued operator $\\Phi_{\\text{FB}}^{-1}\\mathcal A$ is cocoercive \\cite[Thm. 26.14]{BauCom16}. Thus, the pseudo-gradient mapping $F$ in \\eqref{eq:F} should satisfy the following assumption.\n\\begin{assumption}[Strong monotonicity]\n\\label{ass:Hstrong}\nFor all $\\boldsymbol x,\\boldsymbol x'\\in \\boldsymbol{\\Omega}$,\n$\\inner{F(\\boldsymbol x)-F(\\boldsymbol{x}'),\\boldsymbol x-\\boldsymbol x'}\\geq\\eta\\norm{\\boldsymbol{x}-\\boldsymbol{x}'}^{2}$\nfor some $\\eta>0$.\n\\hfill\\small$\\blacksquare$\n\\end{assumption}\n\nTo ensure the cocoercivity condition, we refer to the following result.\n\n\\begin{lemma}\\cite[Lem. 5 and Lem. 7]{yi2019}\\label{lemma_coco}\nLet $\\Phi\\succ0$ and $F$ as in \\eqref{eq:F} satisfy Assumption \\ref{ass:Hstrong}. Then, the following hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal A$ is $\\theta$-cocoercive with $\\theta\\leq\\min\\{1\/2\\Delta,\\eta\\beta^2\\}$.\n\\item[(ii)] $\\Phi^{-1}\\mathcal A$ is $\\alpha\\theta$-cocoercive with $\\alpha=1\/\\abs{\\Phi^{-1}}$. \n\\hfill\\small$\\blacksquare$\n\\end{itemize}\n\\end{lemma}\n\nWe recall that convergence to a v-GNE has been demonstrated in \\cite[Th. 3]{yi2019}, if the step sizes in \\eqref{eq:phi} are chosen small enough \\cite[Lem. 6]{yi2019}. \n\n\n\\subsection{Forward-backward-forward splitting}\n\\label{sec:FBF}\nIn this section, we propose our distributed forward-backward-forward (FBF) scheme, Algorithm \\ref{FBF_algo}.\n\n\\begin{algorithm}\n\\caption{Distributed Forward Backward Forward}\\label{FBF_algo}\nInitialization: $x_i^0 \\in \\Omega_i, \\lambda_i^0 \\in {\\mathbb{R}}_{\\geq0}^{m},$ and $z_i^0 \\in {\\mathbb{R}}^{m} .$\\\\\nIteration $k$: Agent $i$\\\\\n($1$) Receives $x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}$, $ \\lambda_j^k$ and $z_{j,k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&\\tilde x_i^k=\\operatorname{prox}^{\\rho_i}_{g_{i}}[x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-A_{i}^{T} \\lambda_i^k)]\\\\\n&\\tilde z_i^k=z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)\\\\\n&\\tilde\\lambda_i^k=\\operatorname{proj}_{{\\mathbb{R}}^m_{\\geq 0}}\\{\\lambda_i^k-\\tau_{i}(A_{i}x_i^k-b_{i})\\\\\n&\\quad\\quad+\\tau\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_{i}^{k}-z_j^k)-(\\lambda_i^k-\\lambda_j^k)]\\}\n\\end{aligned}$$\n($2$) Receives $\\tilde x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}$, $ \\tilde \\lambda_j^k$and $\\tilde z_{j}^{k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&x_i^{k+1}=\\tilde x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-\\nabla_{\\tilde x_{i}} f_{i}(\\tilde x_i^k,\\tilde {\\boldsymbol{x}}_{-i}^k))\\\\\n&\\quad\\quad\\quad-\\rho_iA_{i}^{T} (\\lambda_i^k-\\tilde \\lambda_{i,k})\\\\\n&z_i^{k+1}=\\tilde z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(\\lambda_i^k-\\lambda_j^k)-(\\tilde\\lambda_i^k-\\tilde\\lambda_j^k)]\\\\\n&\\lambda_i^{k+1}=\\tilde{\\lambda}_i^{k}+\\tau_iA_i(\\tilde{x}_{i}^{k}-x_{i}^{k})\\\\\n&\\quad\\quad\\quad-\\tau_i\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_i^k-z_j^k)-(\\tilde z_i^k-\\tilde z_j^k)]\\\\\n&\\quad\\quad\\quad+\\tau_i\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(\\lambda_{i,k}-\\lambda_j^k)-(\\tilde\\lambda_i^k-\\tilde\\lambda_j^k)]\\\\\n\\end{aligned}$$\n\\end{algorithm}\n\nIn compact form, the FBF algorithm generates two sequences $(\\boldsymbol u^{k},\\boldsymbol v^{k})_{k\\geq 0}$ as follows: \n\\vspace{-.15cm}\\begin{equation}\\label{FBF}\n\\begin{aligned}\n\\boldsymbol u^{k}&=J_{\\Psi^{-1} \\mathcal C}(\\boldsymbol v^{k}-\\Psi^{-1} \\mathcal D \\boldsymbol v^{k})\\\\\n\\boldsymbol v^{k+1}&=\\boldsymbol u^{k}+\\Psi^{-1} (\\mathcal D\\boldsymbol v^{k}-\\mathcal D\\boldsymbol u^{k}).\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\n\nIn \\eqref{FBF}, $\\Psi$ is the block-diagonal matrix of the step sizes: \n\\vspace{-.15cm}\\begin{equation}\\label{Psi}\n\\Psi=\\operatorname{diag}(\\rho^{-1},\\sigma^{-1}, \\tau^{-1}),\n\\vspace{-.15cm}\\end{equation}\n\nWe recall that $\\mathcal D=\\mathcal A+\\mathcal B$ is single-valued, maximally monotone and Lipschitz continuous by Lemma \\ref{lemma_op}. Each iteration differs from the scheme in \\eqref{eq:FB} by one additional forward step and the fact that the resolvent is now defined in terms of the operator $\\mathcal C$ only. Writing the coordinates as $\\boldsymbol u^{k}=(\\tilde{\\boldsymbol{x}}^{k},\\tilde{\\boldsymbol z}^{k},\\tilde{\\boldsymbol \\lambda}^{k})$ and $\\boldsymbol v^{k}=(\\boldsymbol{x}^{k},\\boldsymbol z^{k},\\boldsymbol \\lambda^{k})$, the updates are explicitly given in Algorithm \\ref{FBF_algo}.\n\nFBF operates on the splitting $\\mathcal C+\\mathcal D$ and it can be compactly written as the fixed-point iteration\n$\\boldsymbol v^{k+1}=T_{\\text{FBF}} \\, \\boldsymbol v^{k},$\nwhere the mapping $T_{\\text{FBF}}$ is defined as  \n\\vspace{-.15cm}\\begin{equation}\\label{eq:T_FBF}\nT_{\\text{FBF}}:=\\Psi^{-1}\\mathcal D+(\\Id-\\Psi^{-1}\\mathcal D)\\circ \\operatorname{J}_{\\Psi^{-1}\\mathcal C}\\circ(\\Id-\\Psi^{-1}\\mathcal D). \n\\vspace{-.15cm}\\end{equation}\n\n\nTo ensure convergence of Algorithm \\ref{FBF_algo} to a v-GNE of the game in \\eqref{game}, we need the next assumption.\n\n \n\\begin{assumption}\\label{step_FBF}\n$\\abs{\\Psi^{-1}} < 1\/L_{\\mathcal D}$, with $\\Psi$ as in \\eqref{Psi} and $L_{\\mathcal D}$ being the Lipschitz constant of $\\mathcal D$ as in Lemma \\ref{lemma_op}.\n\\hfill\\small$\\blacksquare$\n\\end{assumption}\n\n\n\n\n\n\n\\begin{theorem}\\label{theo_FBF}\nLet Assumption \\ref{step_FBF} hold. The sequence $(\\boldsymbol x^k,\\boldsymbol \\lambda^k)$ generated by Algorithm \\ref{FBF_algo} converges to \n$\\operatorname{zer}(\\mathcal A+\\mathcal B+\\mathcal C)$, thus the primal variable converges to a v-GNE of the game in \\eqref{game}.\n\\end{theorem}\n\\begin{proof}\nThe fixed-point iteration with $T_{\\text{FBF}}$ as in \\eqref{eq:T_FBF} can be derived from \\eqref{FBF} by substituting $\\boldsymbol u_k$.\nTherefore, the sequence $(\\boldsymbol x^k,\\boldsymbol\\lambda^k)$ generated by Algorithm \\ref{FBF_algo} converges to a v-GNE by \\cite[Th.26.17]{BauCom16} and \\cite[Th.3.4]{tseng2000} since $\\Psi^{-1}\\mathcal A$ is monotone by Lemma \\ref{lemma_mono} and $\\mathcal A+\\mathcal B+\\mathcal C$ is maximally monotone by Lemma \\ref{lemma_op}. See Appendix \\ref{sec:FBF} for details.\n\\end{proof}\n \n\nWe emphasize that Algorithm \\ref{FBF_algo} does not require strong monotonicity (Assumption \\ref{ass:Hstrong}) of the pseudo-gradient mapping $F$ in \\eqref{eq:F}.\nMoreover, we note that the FBF algorithm requires two evaluations of the individual gradients, which requires computing the operator $\\mathcal D$ twice per iteration. At the level of the individual agents, this means that we need two communication rounds per iteration in order to exchange the necessary information. Compared with the FB algorithm, the non-strong monotonicity assumption comes at the price of increased communications at each iteration.\n\n\n\\subsection{Forward-backward-half forward splitting}\n\\label{sec:FBHF}\n\nShould the strong monotonicity condition (Assumption \\ref{ass:Hstrong}) be satisfied, an alternative to the FB is the \\emph{forward-backward-half-forward} (FBHF) operator splitting, developed in \\cite{briceno2018}. Thus, our second GNE seeking algorithm is a distributed FBHF, described in Algorithm \\ref{FBHF_algo}.\n\n\\begin{algorithm}\n\\caption{Distributed Forward Backward Half Forward}\\label{FBHF_algo}\nInitialization: $x_i^0 \\in \\Omega_i, \\lambda_i^0 \\in {\\mathbb{R}}_{\\geq0}^{m},$ and $z_i^0 \\in {\\mathbb{R}}^{m} .$\\\\\nIteration $k$: Agent $i$\\\\\n($1$) Receives $x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}$, $ \\lambda_j^k$ and $z_{j,k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&\\tilde x_i^k=\\operatorname{prox}^{\\rho_i}_{g_{i}}[x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-A_{i}^{T} \\lambda_i^k)]\\\\\n&\\tilde z_i^k=z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)\\\\\n&\\tilde\\lambda_i^k=\\operatorname{proj}_{{\\mathbb{R}}^m_{\\geq 0}}\\{\\lambda_i^k-\\tau_{i}(A_{i}x_i^k-b_{i})\\\\\n&\\quad\\quad+\\tau\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_{i}^{k}-z_j^k)-(\\lambda_i^k-\\lambda_j^k)]\\}\n\\end{aligned}$$\n($2$) Receives $ \\tilde \\lambda_j^k$and $\\tilde z_{j,k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&x_i^{k+1}=\\tilde x_i^k+\\rho_iA_{i}^{T} (\\lambda_i^k-\\tilde \\lambda_{i,k})]\\\\\n&z_i^{k+1}=\\tilde z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(\\lambda_i^k-\\lambda_j^k)-(\\tilde\\lambda_i^k-\\tilde\\lambda_j^k)]\\\\\n&\\lambda_i^{k+1}=\\tilde{\\lambda}_i^{k}+\\tau_iA_i(\\tilde{x}_{i}^{k}-x_{i}^{k})\\\\\n&\\quad\\quad\\quad-\\tau_i\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_i^k-z_j^k)-(\\tilde z_i^k-\\tilde z_j^k)]\\\\\n\\end{aligned}$$\n\\end{algorithm}\nIn compact form, the FBHF algorithm reads as\n\\vspace{-.15cm}\\begin{equation}\\label{FBHF}\n\\begin{aligned}\n\\boldsymbol u^{k} & = \\mathrm{J}_{\\Psi^{-1}\\mathcal C}(\\boldsymbol v^{k}-\\Psi^{-1} (\\mathcal A+\\mathcal B) \\boldsymbol v^{k}) \\\\\n\\boldsymbol v^{k+1} & =\\boldsymbol u^{k}+\\Psi^{-1}(\\mathcal B\\boldsymbol v^k- \\mathcal B\\boldsymbol u^k).\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nWe note that the iterates of FBHF are similar to those of the FBF, but the second forward step requires the operator $\\mathcal B$ only. \nMore simply, we can write the FBHF as the fixed-point iteration\n$\\boldsymbol v^{k+1}=T_{\\text{FBHF}}\\boldsymbol v^{k},$\nwhere \n\\vspace{-.15cm}\\begin{equation}\\label{eq:T_FBHF}\nT_{\\text{FBHF}}=(\\Id-\\Psi^{-1}\\mathcal B)\\circ \\operatorname{J}_{\\Psi^{-1}\\mathcal C}\\circ (\\Id-\\Psi^{-1}\\mathcal D)+\\Psi^{-1}\\mathcal B. \n\\vspace{-.15cm}\\end{equation}\n\nAlso in this case, we have a bound on the step sizes.\n\n \n\\begin{assumption}\\label{step_FBHF}\n$|\\Psi^{-1}| \\leq \\min\\{2\\theta_{\\mathcal A},1\/L_{\\mathcal B}\\}$,\nwith $\\theta_{\\mathcal A}$ as in Lemma \\ref{lemma_coco} and $L_{\\mathcal B}$ as in Lemma \\ref{lemma_op}.\n\\hfill\\small$\\blacksquare$\n\\end{assumption}\n \n\nWe note that in Assumption \\ref{step_FBHF}, the step sizes in $\\Psi$ can be chosen larger compared to those in Assumption \\ref{step_FBF}, since the upper bound is related to the Lipschitz constant of the operator $\\mathcal B$, not of $L_{\\mathcal D}=L_{\\mathcal A}+L_{\\mathcal B}$ as for the FBF (Assumption \\ref{step_FBF}). A similar comparison can be done with respect to the FB algorithm. Intuitively, larger step sizes should be beneficial in term of convergence speed.\n\nWe can now establish our convergence result for the FBHF algorithm.\n\n \n\\begin{theorem}\nLet Assumptions \\ref{ass:Hstrong} and \\ref{step_FBHF} hold. The sequence $(\\boldsymbol x^k,\\boldsymbol \\lambda^k)$ generated by Algorithm \\ref{FBHF_algo} converges to $\\operatorname{zer}(\\mathcal A+\\mathcal B+\\mathcal C)$, thus the primal variable converges to \na v-GNE of the game in \\eqref{game}. \\hfill\\small$\\blacksquare$\n\\end{theorem}\n\n\\begin{proof}\nAlgorithm \\ref{FBHF_algo} is the fixed point iteration in \\eqref{eq:T_FBHF} whose convergence is guaranteed by \\cite[Th. 2.3]{briceno2018} under Assumption \\ref{step_FBHF} because $\\Psi^{-1}\\mathcal A$ is cocoercive by Lemma \\ref{lemma_coco}. See Appendix \\ref{sec:FBHF} for details.\n\\end{proof}\n\n\n\n\n\n\n\\section{Case study and numerical simulations}\nWe consider a networked Cournot game with market capacity constraints \\cite{yi2019}.\nAs a numerical setting, we use a set of 20 companies and 7 markets, similarly to \\cite{yi2019}. Each company $i$ has a local constraint $x_i\\in(0,\\delta_i)$ where each component of $\\delta_i$ is randomly drawn from $[1, 1.5]$. The maximal capacity of each market $j$ is $b_j$, randomly drawn from $[0.5, 1]$. The local cost function of company $i$ is $c_i(x_i) = \\pi_i\\sum_{j=1}^{n_i} ([x_i]_j)^2 + r^\\top x_i$, where $[x_i]_j$ indicates the $j$ component of $x_i$.\nFor all $i\\in\\mathcal I$, $\\pi_i$ is randomly drawn from $[1, 8]$, and the components of $r_i$ are randomly drawn from $[0.1, 0.6]$. Notice that $c_i(x_i)$ is strongly convex with Lipschitz continuous gradient. The price is taken as a linear function $P= \\bar P-DA\\boldsymbol x$ where each component of $\\bar P =\\operatorname{col}(\\bar P_1,\\dots,\\bar P_7)$ is randomly drawn from $[2,4]$ while the entries of $D=\\operatorname{diag}(d_1,\\dots,d_7)$ are randomly drawn from $[0.5,1]$. Recall that the cost function of company $i$ is influenced by the variables of the agents selling in the same market. Such informations can be retrieved from \\cite[Fig. 1]{yi2019}. Since $c_i(x_i)$ is strongly convex with Lipschitz continuous gradient and the prices are linear, the pseudo gradient of $f_i$ is strongly monotone. The communication graph $\\mathcal G^\\lambda$ for the dual variables is a cycle graph with the addition of the edges $(2,15)$ and $(6,13)$. As local cost functions $g_i$ we use the indicator functions. In this way, the proximal step is a projection on the local constraints sets.\n\nThe aim of these simulations is to compare the proposed schemes.\nThe step sizes are taken differently for every algorithm. In particular, we take $\\rho_{\\text{FB}}$, $\\sigma_{\\text{FB}}$ and $\\tau_{\\text{FB}}$ as in \\cite[Lem. 6]{yi2019}, $\\rho_{\\text{FBF}}$, $\\sigma_{\\text{FBF}}$ and $\\tau_{\\text{FBF}}$ such that Assumption \\ref{step_FBF} is satisfied and $\\rho_{\\text{FBHF}}$, $\\sigma_{\\text{FBHF}}$ and $\\tau_{\\text{FBHF}}$ such that Assumption \\ref{step_FBHF} holds. We select them to be the maximum possible.\n\nThe initial points $\\lambda_i^0$ and $z_i^0$ are set to 0 while the local decision variable $x_i^0$ is randomly taken in the feasible sets.\n\nThe plots in Fig. \\ref{distance_sol} show the performance parameter $\\frac{\\norm{\\boldsymbol{x}_{k+1}-\\boldsymbol{x}^{*}}}{\\norm{\\boldsymbol{x}^{*}}}$, that is, the convergence to a solution $\\boldsymbol x^*$, and the CPU time (in seconds) used by each algorithm. We run 10 simulations, changing the parameters of the cost function to show that the result are replicable. The darker line represent the average path towards the solution.\n\n\nThe plot in Fig \\ref{distance_sol} shows that with suitable parameters convergence to a solution is faster with the FBF algorithm which, however, is computationally more expansive than the FB and FBHF algorithms\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=.22]{ave_sol.eps}\\hspace{-.2cm}\n\\includegraphics[scale=.22]{ave_sec.eps}\n\\caption{Relative distance from v-GNE (left) and cumulative CPU time (right).}\\label{distance_sol}\n\\end{figure}\n\\vspace{-.2cm}\n\n\n\n\\section{Conclusion}\n\nThe FBF and the FBHF splitting methods generate distributed equilibrium seeking algorithms for solving generalized Nash equilibrium problems. Compared to the FB, the FBF has the advantage to converge under the non-strong monotonicity assumption. This comes at the price of increased communications between the agents. If strong monotonicity holds, an alternative to the FBF is the FBHF that, in our numerical experience is less computationally expensive than the FBF.\n\n\\section{Appendix}\n\\subsection{Convergence of the forward-backward-forward}\\label{app:FBF}\nWe show the convergence proof for the FBF. From now on, $\\mathsf{H}={\\mathbb{R}}^n\\times{\\mathbb{R}}^{mN}\\times{\\mathbb{R}}^{mN}$ and $\\operatorname{fix}(T)=\\{x\\in\\mathsf{H}:Tx=x\\}$.\n\\begin{proposition}\nIf Assumption \\ref{step_FBF} holds, $\\operatorname{fix}(T_{\\text{FBF}})=\\mathcal Z$. \n\\end{proposition}\n\\begin{proof}\nWe first show that $\\mathcal Z\\subseteq \\operatorname{fix}(T_{\\text{FBF}})$. Let $u^{\\ast}\\in\\mathcal Z$: \n\\vspace{-.15cm}\\begin{equation*}\n\\begin{aligned}\n0\\in \\mathcal Cu^{\\ast} +\\mathcal Du^{\\ast} & \\Leftrightarrow -\\mathcal Du^{\\ast} \\in \\mathcal Cu^{\\ast} \\\\\n&\\Leftrightarrow u^{\\ast}=J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})\\\\\n&\\Leftrightarrow \\Psi^{-1}\\mathcal Du^{\\ast}=\\Psi^{-1}\\mathcal DJ_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})\\\\\n&\\Leftrightarrow u^{\\ast}=T_{\\text{FBF}}u^{\\ast}. \n\\end{aligned}\n\\vspace{-.15cm}\\end{equation*}\nConversely, let $u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}})$. Then \n$u^{\\ast}-J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})=\n\\Psi^{-1}\\mathcal Du^{\\ast}-\\Psi^{-1}\\mathcal DJ_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})$\nans\n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\norm{u^{\\ast}-J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})}\\leq\\\\\n&\\leq \\alpha^{-1}\\norm{\\mathcal Du^{\\ast} -\\mathcal DJ_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})}\\\\\n&\\leq \\tfrac{L}{\\alpha}\\norm{u^{\\ast}-J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nHence,\n$u^{\\ast}=J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})$. \n\\end{proof}\n\n\\begin{proposition}\nFor all $u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}})$ and $v\\in\\mathsf{H}$, there exists $\\varepsilon\\geq 0$ such that \n\\vspace{-.15cm}\\begin{equation}\\label{eq:Fejer}\n\\norm{T_{\\text{FBF}}v-u^{\\ast}}^{2}_{\\Psi}= \\norm{v-u^{\\ast}}^{2}_{\\Psi}-\\left(1-(L\/\\alpha)^{2}\\right)\\norm{u-v}^{2}_{\\Psi}-2\\varepsilon.\n\\vspace{-.15cm}\\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet $u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}})$ and $u=J_{\\Psi^{-1}\\mathcal C}(v-\\Psi^{-1}\\mathcal D v),v^{+}=T_{\\text{FBF}}v$, for $v\\in\\mathsf{H}$ arbitrary. Then, \n\\vspace{-.18cm}\\begin{equation*}\\begin{aligned}\n\\norm{v-u^{\\ast}}_{\\Psi}^{2}&=\\norm{v-u+u-v^{+}+v^{+}-u^{\\ast}}^{2}_{\\Psi}\\\\\n&=\\norm{v-u}^{2}_{\\Psi}+\\norm{u-v^{+}}^{2}_{\\Psi}+\\norm{v^{+}-u^{\\ast}}^{2}_{\\Psi}\\\\\n&+2\\inner{v-u,u-u^{\\ast}}_{\\Psi}+2\\inner{u-v^{+},v^{+}-u^{\\ast}}_{\\Psi}.\n\\end{aligned}\\vspace{-.18cm}\\end{equation*}\nSince, \n$2\\inner{u-v^{+},v^{+}-u^{\\ast}}_{\\Psi}=2\\inner{u-v^{+},v^{+}-u}_{\\Psi}\n+2\\inner{u-v^{+},u-u^{\\ast}}_{\\Psi}=-2\\norm{u-v^{+}}_{\\Psi}^{2}+2\\inner{u-v^{+},u-u^{\\ast}}_{\\Psi}.$\nThis gives \n$\\norm{v-u^{\\ast}}_{\\Psi}^{2}=\\norm{v-u}^{2}_{\\Psi}-\\norm{u-v^{+}}_{\\Psi}^{2}+\\norm{v^{+}-u^{\\ast}}_{\\Psi}^{2}+2\\inner{u-u^{\\ast},v-v^{+}}_{\\Psi}. $\nBy definition of the updates, we have for $\\bar{v}\\equiv Bv,\\bar{u}\\equiv Bu,\\hat{v}\\in Cu$, the identities\n$u+\\Psi^{-1}\\hat{v}=v-\\Psi^{-1}\\bar{v}$ and $v^{+}=u+\\Psi^{-1}(\\bar{v}-\\bar{u}).$\nFurthermore, since $0\\in \\mathcal Du^{\\ast} +\\mathcal Cu^{\\ast} $, there exists $\\hat{v}^{\\ast}\\in \\mathcal Cu^{\\ast} $ and $\\bar{u}^{\\ast}\\equiv \\mathcal Du^{\\ast} $ such that\n$0=\\bar{u}^{\\ast}+\\hat{v}^{\\ast}.$\nIt follows that \n$v-v^{+}=v-u-\\Psi^{-1}(\\bar{v}-\\bar{u})=\\Psi^{-1}(\\hat{v}+\\bar{u}).$\nHence, \n\\vspace{-.18cm}\\begin{equation*}\n\\begin{aligned}\n\\norm{v-u^{\\ast}}_{\\Psi}^{2}=&\\norm{v-u}^{2}_{\\Psi}-\\norm{u-v^{+}}_{\\Psi}^{2}+\\\\\n&+\\norm{v^{+}-u^{\\ast}}_{\\Psi}^{2}+2\\inner{u-u^{\\ast},\\hat{v}+\\bar{u}}\\\\\n=&\\norm{v-u}^{2}_{\\Psi}-\\norm{u-v^{+}}_{\\Psi}^{2}+\\norm{v^{+}-u^{\\ast}}_{\\Psi}^{2}+\\\\\n&+2\\inner{\\hat{v}-\\hat{v}^{\\ast}-\\bar{u}^{\\ast}+\\bar{u},u-u^{\\ast},u-u^{\\ast}}.\n\\end{aligned}\n\\vspace{-.18cm}\\end{equation*}\nSince $(u,\\hat{v}),(u^{\\ast},\\hat{v}^{\\ast})\\in\\gr(C),(u^{\\ast},\\bar{u}^{\\ast}),(u,\\bar{u})\\in\\gr(B)$, it follows from the monotonicity that \n$\\varepsilon:=\\inner{\\hat{v}-\\hat{v}^{\\ast}-\\bar{u}^{\\ast}+\\bar{u},u-u^{\\ast},u-u^{\\ast}}\\geq 0.$\nFinally, observe that $u-v^{+}=\\Psi^{-1}(\\mathcal Du-\\mathcal Dv)$, and that \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\norm{\\Psi^{-1}(\\mathcal Du-\\mathcal Dv)}_{\\Psi}^{2}=\\inner{\\Psi^{-1}(\\mathcal Du-\\mathcal Dv),\\mathcal Du-\\mathcal Dv}\\\\\n&\\leq \\lambda_{\\max}(\\Psi^{-1})\\norm{\\mathcal Du-\\mathcal Dv}^{2}\\leq L^{2}\\lambda_{\\max}(\\Psi^{-1})\\norm{u-v}^{2}\\\\\n&\\leq L^{2}\\tfrac{\\lambda_{\\max}(\\Psi^{-1})}{\\lambda_{\\min}(\\Psi)}\\norm{u-v}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nSince $\\alpha=1\/\\lambda_{\\max}(\\Psi^{-1})=\\lambda_{\\min}(\\Psi)$, it follows from the Lipschitz continuity of the operator $B$ that\n$\\norm{u-v^{+}}_{\\Psi}^{2}\\leq (L\/\\alpha)^{2}\\norm{u-v}^{2}_{\\Psi} $\nand the statement is proven.\n\\end{proof}\n\\begin{corollary}\nIf $L\/\\alpha<1$, the map $T_{\\text{FBF}}:\\mathsf{H}\\to\\mathsf{H}$ is quasinonexpansive in the Hilbert space $(\\mathsf{H},\\inner{\\cdot,\\cdot}_{\\Psi})$, i.e. \n\\vspace{-.15cm}\\begin{equation*}\n\\forall v\\in\\mathsf{H}\\; \\forall u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}}) \\;\\norm{T_{\\text{FBF}}v-u^{\\ast}}_{\\Psi}\\leq\\norm{v-u^{\\ast}}_{\\Psi}.\n\\vspace{-.15cm}\\end{equation*}\n\\end{corollary}\n\\begin{proposition}\nIf Assumption \\ref{step_FBF} holds, the sequence generated by the FBF algorithm, $(v^{k})_{k\\geq 0}$, is bounded in norm, and all its accumulation points are elements in $\\mathcal Z$.\n\\end{proposition}\n\\begin{proof}\nForm \\eqref{eq:Fejer} we deduce that $(v^{k})_{k\\geq 0}$ is Fej\\'{e}r monotone with respect to $\\operatorname{fix}(T_{\\text{FBF}})=\\mathcal Z$. Therefore, it is bounded norm. It remains to show that all accumulation points are in $\\mathcal Z$. By an obvious abuse of notation, let $(v^{k})_{k\\geq 0}$ denote a converging subsequence with limit $u^{\\ast}$. From \\eqref{eq:Fejer} it follows $\\norm{u^{k}-v^{k}}_{\\Psi}\\to 0$, and hence $\\norm{u^{k}-v^{k}}\\to 0$ as $k\\to\\infty$. By continuity, it therefore follows as well $\\norm{\\mathcal Du^k-\\mathcal Dv^k}\\to 0$ as $k\\to\\infty$. Since $u^{k}=J_{\\Psi^{-1}\\mathcal C}(v^{k}-\\Psi^{-1}\\mathcal Dv^{k})$, it follows that $w^{k}:=\\Psi(v^{k}-u^{k})+\\mathcal Du^k-\\mathcal Dv^k\\in \\mathcal Du^k+\\mathcal Cu^{k}.$\nSince $w^{k}\\to 0$ and the operator $\\mathcal C+\\mathcal D$ is maximally monotone by Lemma \\ref{lemma_op} and has a closed graph \\cite[Lem. 3.2]{tseng2000}, we conclude $0\\in \\mathcal Du^{\\ast} +\\mathcal Cu^{\\ast} $. Hence, $u^{\\ast}\\in\\mathcal Z$.\n\\end{proof}\n\n\\subsection{Convergence of the forward-backward-half-forward}\n\\label{sec:FBHF}\nWe here provide the convergence proof for the FBHF.\n\\begin{proposition}\nIf Assumption \\ref{step_FBHF} holds, the sequence generated by the FBHF algorithm converges to $\\mathcal Z$.\n\\end{proposition}\nSince, $w-u\\in\\Psi^{-1}\\mathcal C u$, it follows that $(u,w-u)\\in\\gr(\\Psi^{-1}\\mathcal C)$. Additionally, $0\\in \\mathcal Du^{\\ast} +\\mathcal Cu^{\\ast} $, implying that $(u^{\\ast},-\\Psi^{-1}\\mathcal Du^{\\ast})\\in \\gr(\\Psi^{-1}\\mathcal C)$. Monotonicity of the involved operators, implies that \n$\\inner{u-u^{\\ast},w-u-\\Psi^{-1}\\mathcal Du^{\\ast}}_{\\Psi}\\leq 0,\\text{ and }\n\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bu^{\\ast} -\\mathcal Bu)}_{\\Psi}\\leq 0, $\nUsing these two inequalities, we see \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\inner{u-u^{\\ast},u-w-\\Psi^{-1}\\mathcal Bu}_{\\Psi}=\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Au^{\\ast}}_{\\Psi}\\\\\n&+\\inner{u-u^{\\ast},u-w-\\Psi^{-1}\\mathcal Du^{\\ast}}_{\\Psi}\\\\\n&+\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Du^{\\ast} -\\mathcal Bu)}_{\\Psi}\\leq \\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Au^{\\ast}}_{\\Psi}\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nTherefore, \n\\vspace{-.15cm}\\begin{equation}\\label{step}\n\\begin{aligned}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}=\\\\\n&2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Bv+w-u}_{\\Psi}+2\\inner{u-u^{\\ast},u-w-\\Psi^{-1}\\mathcal Bu}_{\\Psi}\\\\\n&\\leq 2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Bv+w-u}_{\\Psi}+2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Au^{\\ast}}_{\\Psi}\\\\\n&=2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Dv+w-u}_{\\Psi}\\\\\n&+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}\\\\\n&=2\\inner{u-u^{\\ast},v-u}_{\\Psi}+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi},\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nwhere in the last equality we have used the identity $w=v-\\Psi^{-1}\\mathcal Dv$. Using the cosine formula, \n\\eqref{step} becomes \n\\vspace{-.15cm}\\begin{equation}\n\\begin{aligned}\\label{eq:step}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}\\leq \\norm{v-u^{\\ast}}_{\\Psi}^{2}-\\norm{u-u^{\\ast}}_{\\Psi}^{2}\\\\\n&-\\norm{v-u}_{\\Psi}^{2}+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}.\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nThe cocoercivity of $\\Psi^{-1}\\mathcal A$ in $(\\mathsf{H},\\inner{\\cdot,\\cdot}_{\\Psi})$ gives for all $\\varepsilon>0$\n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}=\\\\\n&2\\inner{v-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}+2\\inner{u-v,\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}\\\\\n&\\leq -2\\alpha\\theta\\norm{\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}^{2}+2\\inner{u-v,\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}\\\\\n&=-2\\alpha\\theta\\norm{\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}^{2}+\\tfrac{1}{\\varepsilon}\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&+\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}\\\\\n&=\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\left(2\\alpha\\theta-\\tfrac{1}{\\varepsilon}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nCombining this estimate with (\\ref{eq:step}), we see \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}\\leq \\norm{v-u^{\\ast}}_{\\Psi}^{2}-\\norm{u-u^{\\ast}}_{\\Psi}^{2}\\\\\n&-\\norm{v-u}_{\\Psi}^{2}+\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\left(2\\alpha\\theta-\\tfrac{1}{\\varepsilon}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nTherefore, \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\norm{v^{+}-u^{\\ast}}^{2}_{\\Psi\n=\\norm{u+\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)-u^{\\ast}}^{2}_{\\Psi}\\\\\n&=\\norm{u-u^{\\ast}}^{2}_{\\Psi}+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}\\\\\n&+\\norm{\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}^{2}\\\\\n&\\leq \\norm{u-u^{\\ast}}^{2}_{\\Psi}+\\norm{\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}^{2}-\\norm{u-u^{\\ast}}_{\\Psi}^{2}\\\\\n&-\\hspace{-.1cm}\\left(2\\alpha\\theta-\\tfrac{1}{\\varepsilon}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}+\\norm{v-u^{\\ast}}_{\\Psi}^{2}\\hspace{-.1cm}-\\hspace{-.1cm}\\norm{v-u}_{\\Psi}^{2}\\\\\n&+\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nSince, \n$\\norm{\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}^{2}_{\\Psi}\\leq (L\/\\alpha)^{2}\\norm{v-u}^{2}_{\\Psi},$\nthe above reads as\n\\vspace{-.15cm}\\begin{equation*}\n\\begin{aligned}\n\\norm{T_{\\text{FBHF}}v-u^{\\ast}}_{\\Psi}^{2}\\leq& \\norm{v-u^{\\ast}}^{2}_{\\Psi}-L^{2}\\left(\\tfrac{1-\\varepsilon}{L^{2}}-\\tfrac{1}{\\alpha^{2}}\\right)\\norm{v-u}^{2}_{\\Psi}\\\\\n&-\\tfrac{1}{\\alpha\\varepsilon}\\left(2\\theta\\varepsilon-\\tfrac{1}{\\alpha}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation*}\nIn order to choose the largest interval for $1\/\\alpha$ ensuring that the second and third terms are negative, we set\n$\\chi\\leq\\min\\{2\\theta,1\/L\\}$.\nThen,\n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n\\norm{T_{\\text{FBHF}}v-u^{\\ast}}_{\\Psi}^{2}\\leq& \\norm{v-u^{\\ast}}^{2}_{\\Psi}-L^{2}\\left(\\chi^{2}-\\tfrac{1}{\\alpha^{2}}\\right)\\norm{v-u}^{2}_{\\Psi}\\\\\n&-\\tfrac{2\\theta}{\\alpha\\chi}\\left(\\chi-\\tfrac{1}{\\alpha}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\tfrac{\\chi}{2\\theta}\\norm{v-u-\\tfrac{2\\theta}{\\chi}(\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast}))}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nFrom here, we obtain convergence of the sequence $(v^{k})_{k\\geq 0}$ as a consequence of \\cite[Thm. 2.3]{briceno2018} for $1\/\\alpha\\in(0,\\chi)$.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe 2014 Nobel Prize in Chemistry was awarded for development of methods of super-resolution optical imaging, which, in particular, relied on single optical emitters imaging and localization \\cite{Betzig2006science,Dickson1997nature}. Despite a tremendous progress of nanoscopic fluorescence-based imaging, which has made possible through those pioneering work, identification of single emitters remains to be a challenge \\cite{Mortensen2010natmeth} and often relies on ultra-bright emission, which is not always affordable for biological systems, and some prior knowledge of the system, which is often not available. Thus, it would be highly desirable to be able to characterize individual emitters and quantify their presence in any given imaging volume. This can be generalized to a broader fundamental problem of counting the number of emitters in a sample from optically collected data, which has significant implications beyond the commonly used fluorescence imaging. For example,\nis it possible to determine the number of emitters contributing to a fluorescence or Raman signal, on the basis of imaging data alone?  Furthermore, if it is possible, what is the limit to which we may determine the number of emitters?  In particular we want to be able to determine the number of molecules, which might be in the range of 10, 100, or 1,000.  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{schematic.pdf}\n    \\caption{Schematic showing the concept to determine the  number of emitters $M$ in one sample with unknown detection probability $P$. A pulsed laser excites the sample and emitted photons are measured using a the photon-number-resolving detector, PNRD.  a histogram of the number of photons in each collected pulse in analysed using a Maximum Likelihood Estimator, which enables the determination of both $M$ and $P$.  Such discrimination is not possible using only classical intensity measurements.}\n    \\label{fig:schematic}\n\\end{figure}\n\nThe problem with determination of the number of emitters from intensity is fundamental.  For a classical fluorescence measurement, we observe an intensity, which we may express as $I = M I_0$, where $I$ is the total intensity, $M$ the number of emitters and $I_0$ is the intensity of each emitter, where we assume each emitter has the same emission intensity, for simplicity.  However, without \\emph{a priori} knowledge of $I_0$ or $M$, it is not possible to discriminate between cases where there are more dimmer emitters, or fewer brighter emitters.  Additional practical confouding issues include uncertainty in the probability of collecting light emitted from the emitters, which may vary due to experimental conditions.\n\nQuantum mechanically, however, there is the prospect for distinguishing different emitter configurations.  It is known that photon anti-bunching signals (Hanbury Brown and Twiss experiment) can be used to distinguish the number of emitters \\cite{Monticone2014prl}.  This technique is typically used when the number of emitters is few (e.g. to determine the difference between 1, 2 or 3 emitters) \\cite{Worboys2020pra, Davin2021arxiv, Chen2017nphoton, Schwartz2013nanoletter}. The Hanbury Brown and Twiss (HBT) experiment, in its simplest form, uses two SPDs that sample the same optical field of view via a beamsplitter \\cite{Stevens2013book}.  By observing the signal in coincidence, some information about the number of emitters can be obtained, which leads to the well-known result for the background-free, equal brightness HBT signal at coincidence:\n\\begin{align}\n    g^{(2)}(0) = 1 - \\frac{1}{M}.\n\\end{align}\nOne way to improve determination of the number of emitters is to increase the number of SPDs. This approach leads to considerable complexity due to the increased number of beamsplitters and coincidence electronics that is required \\cite{Steven2014oe}.  \n\nMeasurement of photon number has traditionally been a difficult task.  Originally, this task was performed using single photon detectors (SPD) such as photomultiplier tubes, and later avalanche photodiodes. Such devices permit a binary measurement of the number of photons: they measure either 0 photons, or more than 0 photons, but a single device typically does not allow for a  more sophisticated determination of the number of photons. Alternatively, new generations of photon number resolving (PNR) detectors are becoming available. PNR detectors have the ability to perform a direct projective measurement of the number of photons in a pulse of light. Compared with non-PNR detection, PNR detection can provide more information about noise and receiver imperfections \\cite{Becerra2015natphotonics}. Several techniques have been applied in realising photon number resolving \\cite{Provaznik2020oe, Thekkadath2020thesis}, including multiplexed APD \\cite{Kardynal2008nphoton}, CMOS image sensors \\cite{Ma2017optica}, superconductor nanowire \\cite{Cahall2017optica}, and superconducting transition-edge sensor (TES) \\cite{Schmidt2018LowTemPhys}. Additionally, there are  multipixel photon counters (MPPC) that have the ability to distinguish from one up to 10 photons \\cite{Kalashnikov2011oe}. Superconducting transition-edge sensors have recently been reported to resolve photon numbers up to 16 with the efficiency of over 90\\% \\cite{Morais2020arXiv}. A study has reported a 24-pixel PNR detector based on superconducting nanowires that achieves the detection of $n=0-24$ photons \\cite{Mattioli2016oe}. Given the increase in technology it is is expected that this upper limit will soon be exceeded, and the availability of such detectors will become more widespread.  It is therefore timely to see the effects that such detectors will have on the determination of the number of emitters in an unknown sample.  \n\nHere we show that photon number resolving measurements enable the determination of emitter number more generally. The schematic is shown in Fig.\\ref{fig:schematic}.  We theoretically determine the photon number probability distribution for $M$ emitters, with photon detection probability $p$.  On the basis of this, we show maximum likelihood estimation and the Cramer-Rao lower bound for the simultaneous determination of both the number of emitters and the probability of detection.  This analysis enables us to provide scaling laws for the number of experiments required to distinguish between different configurations.  \n\nThis paper is organised as follows: We first discuss the photon statistics from an ensemble of $M$ classically identical emitters (ie emitters with the same emission probability in the same field of view with the same emission properties such as polarisation and wavelength, although we stress that the emitters are assumed to be not quantum indistinguishable).  We then show the maximum likelihood determination of the number of emitters and photon detection probability for particular cases, as a function of the number of experiments.  Lastly we present the Cramer-Rao lower bound for the scaling.\n\n\n\\section{Photon number resolving detection probabilities}\n\nWe are concerned with the problem of simultaneously determining the number of emitters, and the collection probability for a number of emitters.  We consider an experimental configuration where $M$ (unknown) emitters are excited by a short pulse laser, and the fluorescence signal collected confocally.  Each emitter is assumed to emit no more than one photon per excitation pulse, and we assume that the probability of detecting a photon from each emitter in that pulse is $p$.  The photon resolving detector performs a projective measurement in the photon number basis, and we may write down the binomially distributed probability of detecting $N$ photons from the $M$ emitters as \n\n\\begin{align}\n\\mathcal{P}(N|M,p) = \\frac{M!}{\\left(M-N\\right)! N!} p^N \\left(1 - p\\right)^{M-N}. \\label{eq:P(N)}\n\\end{align}\n\n\nWe can explore Eq.~\\ref{eq:P(N)} in various limits, however to address the original problem, the clearest case to consider is where we have a known (measured) brightness, but where the actual number of emitters and their probability of emission is unknown.  Therefore,  we set $\\lambda = M p$, so that $\\lambda$ is the expected number of photons emitted per experiment, where each experiment is a Bernouilli trial.  Note that although $N$ is quantised, $\\lambda$ is not.  \n\nEq.~\\ref{eq:P(N)} in terms of $\\lambda$ becomes\n\\begin{align}\n    \\mathcal{P}(N|\\lambda,M) = \\frac{M!}{\\left(M-N\\right)! N!} \\left(\\frac{\\lambda}{M}\\right)^N \\left(1 - \\frac{\\lambda}{M}\\right)^{M-N}.\n\\end{align}\nThis result should be compared with the standard Poisson distribution, which is expected in the limit $M\\rightarrow \\infty$\n\\begin{align}\n    \\lim_{M\\rightarrow\\infty}\\mathcal{P}(N|M,p) = \\frac{\\lambda^N e^{-\\lambda}}{N!}.\n\\end{align}\nAnalytical results for this are shown in Fig.~\\ref{fig:poissonDistri}. Fig.~\\ref{fig:poissonDistri}(a) shows the probability of obtaining $N$ photons for different $M$.  As shown in Fig.~\\ref{fig:poissonDistri}(a), the greatest change in $\\mathcal{P}(N)$ occurs at $N\\approx \\lambda$, although it is clear that the \\emph{entire} distribution provides information about $M$.  Hence it is important that any photon resolving detector should be at least able to detect $\\lambda$ photons for maximum ability to determine $M$.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[trim=0cm 5cm 0cm 5cm, width=\\textwidth]{Poisson.pdf}\n\\caption{(a) Series of curves showing the distribution of probability distribution function for the number of photons, $N$, for different $M$ and $p$ such that $\\lambda = Mp = 20$. The dotted line shows the limit for a Poisson distribution.  As $M$ increases the peak broadens, and approaches the Poisson limit. By measuring the distribution, the number of emitters should be distinguishable, however it is important to stress that the differences are small, and noise will make sure determination difficult. (b) The peak of the probability distribution, at $N = \n\\lambda$, is the point that shows the largest dependence on number of emitters however as this curve shows, even variation of $M$ over three orders of magnitude only leads to a change in the probability of $N=20$ photon events from  $\\mathcal{P}(N = 20| M = 30, \\lambda = 20) = 15.3\\%$ to $\\mathcal{P}(N = 20| M = 10^5, \\lambda = 20) = 8.88\\%$}\n\\label{fig:poissonDistri}\n\\end{figure}\n\n\nTo explore the determination of both $M$ and $p$, we begin by generating synthetic data obtained by sampling Eq.~\\ref{eq:P(N)} for a finite number of numerical experiments.  This yields a histogram of events, such as that shown in Fig.~\\ref{fig:BarFig}.  This data was generated on the basis of $\\nu = 100$ experiments, with $M=40$ atoms and probability of detection $p=0.2$.  Also shown is the probability distribution function, $\\mathcal{P}(N)$ under the same circumstances. As the number of experiments increases, the synthetic data and probability distribution function should converge.  \n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{BarFig.pdf}\n    \\caption{Plot showing relative frequency data (bars) and expected (ideal) probability of detecting a number of photons given $M=40$ emitters, each with a detection probability of $p=0.2$.  The synthetic data was generated on the basis of 100 experiments and assumes no noise other than random fluctuations provided by Eq.~\\ref{eq:P(N)}.\n    }\n    \\label{fig:BarFig} \n\\end{figure}\n\nTo address the issue of how well we can determine the model parameters using the data, we turn to maximum likelihood estimation.  For a given set of data $\\bm{N} = \\left(N_1, N_2, ...N_{\\nu}\\right)$, where the $N_i$ correspond to the number of photons resolved in experiment $i$, which are independent and identically distributed with probability mass function $\\mathcal{P}(N_i|\\bm{\\theta})$, where $\\bm{\\theta} = \\left(M,p\\right)\\in\\mathbb{Z}^+\\bigtimes[0,1]$ is the vector of parameters to be estimated. Furthermore, let $\\bm{\\theta}_0$ be the ground truth and $L(\\bm{\\theta}|N_i)$ the associated likelihood function of $\\bm{\\theta}$ given data $N_i$. \n\nThen we can write the joint log likelihood function as follows:\n\\begin{align}\n\\ell\\left(\\bm{\\theta}\\big|\\mathbf{N}\\right) = \\log L\\left(\\bm{\\theta}\\big|\\mathbf{N}\\right)=\\sum_{i = 1}^{\\nu}\\log L\\left(\\bm{\\theta}\\big|{N}_i\\right),\n\\end{align}\nwhere $\\nu$ is the number of experiments.  \n\nAccordingly, the MLE of $\\bm{\\theta}$ is given by\n\\begin{align}\n\\hat{\\bm{\\theta}}=\\mathop{\\arg \\max}\\limits_{\\bm{\\theta}\\in\\mathbb{Z}^+\\bigtimes [0,1]} \\ell\\left(\\bm{\\theta}\\big|\\mathbf{N}\\right).\n\\end{align}\nAs $\\nu$ increases, from the consistency of MLE \\cite{rao1973linear}, we expect that the $\\hat{\\bm{\\theta}}$ approaches $\\bm{\\theta}_0$. \n\nIt is easier to determine sample brightness than the number of emitters.  This accords with our classical intuition, namely that on the basis of intensity-only measurements it should be \\emph{only} possible to determine the mean brightness, and \\emph{impossible} to determine the number of emitters (ie few bright emitters should be indistinguishable from many dim emitters).  It is therefore useful to transform our parameters from $\\bm{\\theta}=(M,p)$ to $\\bm{\\beta}=(\\lambda, \\xi)$ where $\\xi = M\/p$.  With this parameterisation, the probability distribution function becomes\n\\begin{align}\n\\mathcal{P}(N|\\bm{\\beta})=\\mathcal{P}(N|\\lambda,\\xi)=\\frac{(\\sqrt{\\lambda\\xi})!}{(\\sqrt{\\lambda\\xi}-N)!N!} \\left(\\frac{\\lambda}{\\xi}\\right)^{N}\\left[1-\\left(\\frac{\\lambda}{\\xi}\\right)\\right]^{\\sqrt{\\lambda\\xi}-N}\\label{pdf2}\n\\end{align}\n\n\n\n\\section{Uncertainty of the estimates: Cramer Rao lower bound}\n\nTo obtain the scaling laws for estimating $M$, and $p$, we now proceed to calculate the Cramer-Rao lower bound (CRLB). The Cramer-Rao Lower Bound (CRLB) gives a lower estimate for the variance of an unbiased estimator. The Fisher Information Matrix (FIM)\\cite{Nishiyama2019arxiv,Ly2017jmp} is required to calculate the CRLB.  To do this, we need to find the derivative of (\\ref{pdf2}) w.r.t $\\lambda$ and $\\xi$. However the likelihood function $L(\\bm{\\beta}|N)$ is not differentiable since $\\sqrt{\\lambda\\xi}\\in\\mathbb{Z}^+$. To implement the derivative we use the $x!=x\\Gamma(x)$ to transfer $(\\sqrt{\\lambda\\xi})!$ into a continuous function with respect to $\\lambda$ and $\\xi$. Additionally, we have $\\left[x\\Gamma(x)\\right]'=\\Gamma(x)+x\\Gamma(x)\\psi(x),$\nwhere $\\psi(\\cdot)$ is digamma function.\n\nLet $\\bar{L}(\\bm{\\beta}|N)$ be the approximated likelihood function (after replacing the factorial term associated to $\\lambda$ and $\\xi$ by the interpolation function), then \n\\begin{align}\n &\\frac{\\partial \\bar{L}(\\bm{\\beta}|N)}{\\partial\\lambda}\\notag\\\\ \n=&\\sqrt{\\frac{\\xi}{\\lambda  }}\\alpha_1\\left\\{\\xi  \\lambda ^2+N^2 \\sqrt{\\xi  \\lambda }-N \\left[\\left(\\sqrt{\\xi  \\lambda }-1+\\lambda \\right) \\sqrt{\\xi  \\lambda }+\\lambda  \\right]+\\left(\\lambda -\\sqrt{\\xi  \\lambda }\\right)\\alpha_2\\right\\}\\\\\n&\\frac{\\partial \\bar{L}(\\bm{\\beta}|N)}{\\partial\\xi}\\notag\\\\ \n=&\\lambda\\alpha_1\\left[-\\lambda  \\sqrt{\\xi  \\lambda }-N^2+N \\left(-\\sqrt{\\frac{\\lambda }{\\xi }}+\\sqrt{\\xi  \\lambda }+\\lambda +1\\right)+\\left(\\sqrt{\\frac{\\lambda }{\\xi }}-1\\right)\\alpha_2\\right]\n\\end{align}\nwhere\n\\begin{align}\n    \\alpha_1&=\\frac{\\Gamma \\left(\\sqrt{\\xi  \\lambda }\\right) \\left(\\frac{\\lambda }{\\xi }\\right)^{N\/2} \\left(1-\\sqrt{\\frac{\\lambda }{\\xi }}\\right)^{\\sqrt{\\xi  \\lambda }-N-1}}{2 N! \\sqrt{\\xi  \\lambda } \\left(\\sqrt{\\xi  \\lambda }-N\\right)^2 \\Gamma \\left(\\sqrt{\\xi  \\lambda }-N\\right)}\\\\\n    \\alpha_2&= \\left(\\xi  \\lambda -N \\sqrt{\\xi  \\lambda }\\right) \\left[\\log \\left(1-\\sqrt{\\frac{\\lambda }{\\xi }}\\right)+\\psi\\left(\\sqrt{\\xi  \\lambda }\\right)-\\psi \\left(\\sqrt{\\xi  \\lambda }-N\\right)\\right]\n\\end{align}\nThen the $(i,j)$-th element, $\\forall i,j=1,2$, in FIM, i.e. $\\mathbf{I}_N(\\bm{\\beta})_{i,j}$, is\n\\begin{align}\n\\mathbf{I}_N(\\bm{\\beta})_{1,1}&=\\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\lambda}\\right]^2\\frac{1}{f(\\lambda,\\xi|N)}\\right\\}\\label{I11_2}\\\\\n\\mathbf{I}_N(\\bm{\\beta})_{2,2}&=\\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\xi}\\right]^2\\frac{1}{f(\\lambda,\\xi|N)}\\right\\}\\label{I22_2}\\\\\n\\mathbf{I}_N(\\bm{\\beta})_{1,2}=\\mathbf{I}_N(\\bm{\\beta})_{2,1}&=\\sum_{N=0}^n\\left[\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\lambda}\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\xi}\\frac{1}{f(\\lambda,\\xi|N)}\\right]\\label{I21_2}\n\\end{align}\n\n\nEquivalently, the FIM for $\\bm{\\theta}=(M,p)$ is\n\\begin{align}\n    \\mathbf{I}_N(\\bm{\\theta})_{1,1}\n    \\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial L(\\bm{\\theta}|N)}{ \\partial M}\\right]^2\\frac{1}{L(\\bm{\\theta}|N)}\\right\\},\\notag\n   \n    \\label{eq:I11}\n\\end{align}\nwhere $L(\\bm{\\theta}|N)$ is not differentiable again since it is discrete in $M$. We can find a approximated $f(M,p|N)$, i.e.  $\\bar{f}(M,p|N)$, using the similar method in obtaining $\\bar{f}(\\lambda,\\xi|N)$. Then we have\n\\begin{align}\n&\\frac{ \\partial \\bar{f}(M,p|N)}{ \\partial M}\\notag\\\\\n=&\\frac{\\Gamma (M) p^N (1-p)^{M-N} }{N! (M-N)^2 \\Gamma (M-N)}\\left\\{M (N-M) \\left[\\psi(M-N)-\\psi(M)-\\log (1-p)\\right]-N\\right\\}\n\\end{align}\n\n\nSimilarly, we have \n\\begin{align}\n\\mathbf{I}_N(\\bm{\\theta})_{2,2}=\\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial f(M,p|N)}{ \\partial p}\\right]^2\\frac{1}{f(M,p|N)}\\right\\},\\label{I22}\n\\end{align}\nand\n\\begin{align}\n\\mathbf{I}_N(\\bm{\\theta})_{2,1}&=\\mathbf{I}_N(\\bm{\\theta})_{1,2}\\notag\\\\\n&\\approx\\sum_{N=0}^n\\left\\{\\frac{ \\partial \\bar{f}(M,p|N)}{ \\partial M}\\frac{ \\partial f(M,p|N)}{ \\partial p}\\frac{1}{f(M,p|N)}\\right).\\label{I12}\n\\end{align}\nwhere \n\\begin{align}\n\\frac{ \\partial f(M,p|N)}{ \\partial p}=-\\frac{M! p^{N-1} (1-p)^{M-N-1} (M p-N)}{N! (M-N)!}\n\\end{align}\nThe CRLB is given by the inverse of the FIM,\n\\begin{align}\n\\mathbf{C} = \\mathbf{I}_N(\\bm{\\theta})^{-1}\\big|_{M=M_0,p=p_0}.\n\\end{align}\nGiven that there are $\\nu$ i.i.d. experiments,  so the underlying Cramer Rao lower bound  is\n\\begin{align}\n\\mathbf{C}_{\\nu} = \\frac{\\mathbf{C}}{\\nu} =  \\frac{1}{\\nu}\\mathbf{I}_N(\\bm{\\theta})^{-1}\\big|_{M=M_0,p=p_0}\\label{eq:CRLB}\n\\end{align}\n\nWe proceeded to compare our maximum likelihood simulations with the CRLB. The parameters $(M,p)$ are estimated using increasing number of experiments, i.e. $\\nu$. For each $\\nu$, we performed $500$ independent Monte Carlo simulations.  By performing an ensemble of numerical experiments, we could compare the estimated values in the $(M,p)$ space with a 2D confidence region, i.e. the CRLB 95\\% error ellipse (within which the probability that the random estimated value $\\bm{\\theta}=(M,p)$ will fall inside the ellipse is 95\\%).\nThe simulation results are shown in Fig.~\\ref{fig:CRLB} with ground truth $p_0=0.2$ and $M_0=40$. \n\nFig.~\\ref{fig:CRLB} shows a series of maximum likelihood determinations of the number of emitters and probability of detection per emitter, for ground truth $\\bm{\\theta_0} = (M,p) = (40,0.2)$.  We show the results in $\\left(\\lambda,\\xi\\right)$ and the data converted back into $\\left(M,p\\right)$ space, for increasing experiments $\\nu$.  Each point represents the maximum likelihood determination and the solid curve is the 95\\% confidence interval.  Observe that in $\\left(\\lambda,\\xi\\right)$ space we obtain a standard error ellipse in Fig.\\ref{fig:CRLB} (b), whereas in $\\left(M,p\\right)$ space in Fig.\\ref{fig:CRLB} (a), the ellipse is converted according to reciprocal functions: $p = \\sqrt{\\lambda\/\\xi}$ and $M = \\sqrt{\\lambda \\xi}$. The shape of the error region in $\\left(M,p\\right)$ space is a consequence of the classical ambiguity between more dim emitters and fewer brighter emitters.  Nevertheless, as can be seen, by applying quantum measurements, some bounding on the number of emitters can be obtained, with increasing certainty as the number of experiments increases.  For simplicity, we have not enforced the requirement that CRLBs of $\\left(M,p\\right)$ and $\\left(\\lambda,\\xi\\right)$ are positive, although these values are strictly positive in the simulation data, hence the maximum likelihood and CRLB values do not agree for small $\\nu$. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.8\\textwidth]{CRLB_MP_LX.pdf}\n    \\caption{ The Monte-Carlo simulation results (blue dots) using maximum likelihood and the Cramer-Rao lower bound (red) showing as 95\\% confidence interval ellipse in both (a) ($M,p$) space and (b) ($\\lambda,\\xi$) space. $p_0=0.2$ and $M_0=40$.}\n    \\label{fig:CRLB}\n\\end{figure}\n\n\nTo justify the performance of the maximum likelihood model, we compare the variances of predicted $M$ and $p$ results with CRLB. Fig.\\ref{fig:scalinglaw} presents two configurations, $\\bm{\\theta_0} = \\left(40,0.2\\right)$ and $\\bm{\\theta_0} = \\left(100,0.1\\right)$ with CRLB in $(\\lambda,\\xi)$ space and $(M,P)$ space. Both showing asymptotic trend to CRLB. A log-log scaling law is observed here, i.e the $\\log(\\text{Variance})$ scales with $-\\log(\\nu)$. Ideally the variance of estimated data cannot be lower than CRLB, but for small $\\nu$ the estimator is biased when there are too few measurement data, and CRLB only holds when the estimator is unbiased.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.8\\textwidth]{ScalingLaw.pdf}\n    \\caption{The scaling of Cramer-Rao lower bound (red) and the Monte-Carlo simulation (blue) using maximum likelihood. Ground truth (a) $p_0=0.2$, $M_0=40$ (b) $p_0=0.1$, $M_0=100$. (a)i-ii, (b)i-ii present variances in (M,P) space and (a)iii-iv, (b)iii-iv are variances in ($\\lambda,\\xi$) space. It can be observed that the sampled variance of $p$ is slightly smaller than the CRLB when the number of experiments is small. This is because the MLE is biased on those points due to the small size of data \\cite{WANG201579}.}\n    \\label{fig:scalinglaw}\n\\end{figure}\n\n\nFig.\\ref{fig:NexpMap} (a) shows the number of experiments that required to meet the CRLB criterion, here we define the CRLB criterion as the relative variance of $M$: $\\mathrm{Var}[M]\/M=1\\%$. Here the lower bound of variance of $M$ is the corresponding CRLB element in the matrix of Eq.~\\ref{eq:CRLB}. The contours show the resolved photon number with max likelihood $\\lambda=MP$. In the top half of the map where the probability $p$ roughly larger than 0.5, the number of experiments to achieve CRLB is relatively small ($<10^6$), even for large emitter numbers $10^3$. In the bottom half of the map where $p<0.5$, with the decrease of the probability $p$ the number of experiments to achieve CRLB increases dramatically. With such low brightness or detected probability when it reaches to large emitter numbers $M$, measurements that required to determine $M$ is several orders higher than high brightness scenario.\n \n Along one contour with a fixed $\\lambda$, $\\nu$ increases with the increase of $M$ and decrease of $p$, which means given a detected photon number distribution with the peak occurrence locating at $\\lambda$ (similar to Fig.\\ref{fig:BarFig}), more measurements are required to resolve many low brightness emitters than few high brighter emitters. Fig.\\ref{fig:NexpMap} (b) shows the relationships of $\\nu$ and $M$ along one contour with fixed $\\lambda$, from $\\lambda=5$ to 50. The curves show an elbow shape when approaching large amount of emitters. To the left of the elbow shape, measurements $\\nu$ increases dramatically with emitter numbers. To the right of elbow shape, e.g.$M=200$, the small $\\lambda$ curves stay on top of the large $\\lambda$ ones. This is because the small $\\lambda$ indicates a small probability of detecting photons from each emitter, and results in more measurements being required to determine the number of emitters.\n \nWe now consider the example of quantitative fluoresence. If we consider a sample of 1,000 fluorophores, in a field of view with probability of photon collection of $1\\%$ from each emitter, then the number of measurements required to achieve a determination of the number of emitters with a relative variance of $1\\%$ is around $1.96 \\times 10^9$.  Photon number resolving measurements can be performed at of order microsecond timescales \\cite{Morais2020arXiv}.  This means that the length of time required to achieve to determine the number of identical (but unknown) fluorescent emitters is if order $\\sim$30min.  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[trim=0cm 5cm 0cm 5cm, width=\\textwidth]{NuMap.pdf}\n    \\caption{(a) Number of experiments to achieve CRLB with the relative variance of M: {$\\mathrm{Var}[M]\/M=1\\%$}. Here the lower bound of variance of M is the CRLB(M) element in the matrix of Eq.\\ref{eq:CRLB}. The contours show the $\\lambda=MP$. (b) The relationships of $\\nu$ and $M$ along contours with fixed $\\lambda$ value, from $\\lambda=5$ to 50, generated by extracting the $\\nu$ values along contours in (a). }\n    \\label{fig:NexpMap}\n\\end{figure}\n\n\n\\section{Conclusions}\nWe have shown that photon number resolving measurements can help to identify the number of emitters in a field of view, even without \\emph{a priori} knowledge of the brightness of the emitters.    Our results enable the prediction of the number of experiments required for a particular variance to be achieved.  As the number of emitters increases, the photon distribution approaches Poissonian, and in this limit, resolution of the number of emitters becomes increasingly difficult.\n\nOur results show the idealised case, of equal brightness emitters.  Naturally, variations in the emission probability (for example by particles locate in different parts of the optical point spread function, or with different local environments) will lead to increased number of experiments.  Nevertheless, our analysis is likely to guide future experiments in quantitative tests of biological pathways.  Practical systems will also have to contend with variations in photo-bleaching of emitters that may limit the practically achievable number of experiments.  As such, our results  provide an opportunity to bound the expected sample variance, and hence to give limits on the number of emitters that might be contributing to a signal - bounds that are not possible to impose given the current limits of classical fluorescence based imaging.  \n\n\n\\section*{Acknowledgements}\nThis work was funded by the Air Force Office of Scientific Research (FA9550-20-1-0276).  ADG also acknowledges funding from the Australian Research Council (CE140100003 and FT160100357).\nVVY acknowledges partial support from the National Science Foundation (NSF) (DBI-1455671, ECCS-1509268, CMMI-1826078), the Air Force Office of Scientific Research (AFOSR) (FA9550-15-1-0517, FA9550-20-1-0366, FA9550-20-1-0367), Army Medical Research Grant (W81XWH2010777), the National Institutes of Health (NIH) (1R01GM127696-01, 1R21GM142107-01), the Cancer Prevention and Research Institute of Texas (CPRIT) (RP180588).\n\n\n\\section*{Reference}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}