diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzppta" "b/data_all_eng_slimpj/shuffled/split2/finalzzppta" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzppta" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{Intro}\n\nLight from distant galaxies and quasars is gravitationally lensed by mass along the line of sight. For a flux-limited survey of quasars, lensing magnification biases the observed number density \\cite{Turner:1984ch,Fugmann:1988,Narayan:1989}. For example, a positive mass fluctuation along the line of sight can increase the apparent sky area and the observed flux. The geometrical area increase decreases the quasar number density, while the flux increase promotes intrinsically faint objects above the magnitude threshold increasing the source density. Together these effects introduce a correction to the quasar number density called magnification bias. Detections of the magnification of distant quasars by low-redshift galaxies confirm the presence of this effect (e.g. \\cite{Gaztanaga:2002qk,Scranton:2005ci,Menard:2009yb}). \n\nMany authors have studied the effect of lensing magnification on observations of the galaxy correlation function and power spectrum. Another way of inferring the large-scale mass distribution is through measurements of the neutral Hydrogen density. As light from distant quasars passes through clouds of neutral Hydrogen, photons with rest-frame frequency at the Lyman-$\\alpha$ transition ($1216 \\AA$) are absorbed. The observed quasar spectrum contains troughs corresponding to absorption by neutral Hydrogen at redshift $z=\\nu_\\alpha\/(\\nu_{\\rm obs}(1+ v_{\\rm pec})) -1$, where $\\nu_{\\rm obs}$ is the observed frequency of the trough,\n$v_{\\rm pec}$ is the peculiar velocity at $z$ (speed of light is set to unity), and $\\nu_{\\alpha}$ is the Lyman-$\\alpha$ frequency.\\footnote{Here, the redshift $z$ refers to the intrinsic or cosmological redshift, i.e. the redshift if the absorbing materials were comoving. The observed redshift and the intrinsic redshift are related by $1 + z_{\\rm obs} = (1 + z)(1 + v_{\\rm pec})$.} This Lyman-$\\alpha$ forest (of absorption features in the quasar spectrum) is a cosmological tool that can be used to probe the neutral Hydrogen along the line of sight (see for example \\cite{Meiksin:2009} and references therein). Here we ask two questions: what is the effect of gravitational lensing on measurements of the Lyman-$\\alpha$ forest? And how can we extract the gravitational signal from such measurements?\n\nLet us first discuss qualitatively what we expect to happen. Consider a single line of sight towards a quasar at comoving distance $\\chi_Q$. Lensing magnification changes the observed quasar flux $f$, but in a frequency ($\\nu$)\nindependent way \n \\begin{equation}\n\\label{Eq:fmag}\n f(\\nu)\\rightarrow f(\\nu)\n\\cdot \\mu (\\chi_Q) \n\\end{equation} \nwhere ${\\mu}(\\chi_Q)$ is the magnification which depends on the density fluctuations along the line of sight. Since all frequencies are treated in the same way, fluctuations in the flux as a function of frequency are \\emph{unaffected} --- just as if we were looking at an intrinsically brighter or dimmer quasar. Measurements of for example, the flux power spectrum, from an \\emph{individual} line of sight are unchanged by magnification. On the other hand, magnification does change \\emph{which} lines of sight are observed. More precisely, a selection bias is introduced: more (fewer) quasars are observed along lines of sight that lead to a positive (negative) magnification correction to quasar number counts. Let ${n}(\\chi, \\mbox{\\boldmath$\\theta$})$ be the number density of quasars at comoving distance $\\chi$ in direction $\\mbox{\\boldmath$\\theta$}$. Under the effect of lensing magnification,\n\\begin{equation} \n\\label{nlensed}\n{n}(\\chi, \\mbox{\\boldmath$\\theta$}) \\rightarrow {n}(\\chi, \\mbox{\\boldmath$\\theta$}) {\\mu}(\\chi)^{2.5s-1}, \\qquad s= {1\\over {\\rm ln\\,} 10} {\\int dm \\, \\epsilon(m) (dn^0\/dm) \\over \\int dm \\, \\epsilon(m) n^0(m)} \\, .\n\\end{equation}\nHere, $n^0(m)$ is the luminosity function of quasars at magnitude $m$ (i.e. $dm \\, n^0(m)$ is the number density of quasars at magnitude $m \\pm dm\/2$), and $\\epsilon(m)$ quantifies the sample definition -- the simplest example is a step function which cuts off all quasars fainter than some limiting magnitude $m_{\\rm lim.}$, in which case $s$ reduces to the more familiar expression $s = d\\log_{10} N(< m_{\\rm lim.}) \/ dm_{\\rm lim.}$, where $N(< m_{\\rm lim.})$ is the total number of quasars brighter than $m_{\\rm lim.}$. This is a well known result, but a derivation is summarized in Appendix \\ref{derive}. With lensing magnification, the lines of sight we observe do not constitute a fair sample of the density field. Therefore, while lensing magnification has \\emph{no} effect on measurements of the flux fluctuation from a \\emph{single} quasar line of sight, measurements dependent upon the density field \\emph{averaged over multiple quasars} \\emph{will} be biased. This occurs since the observable (e.g. the neutral hydrogen density) is correlated with the magnification which depends on the gravitational potential along the line of sight.\n\nWhat is perhaps the more interesting question is how we could extract the lensing signal from observations of the forest. The discussions above make clear gravitational lensing\naffects both the observed brightness and number density of quasars. One could imagine cross-correlating these quantities with the Lyman-alpha forest observables. We will consider several possibilities and identify the ones with an interesting signal-to-noise.\n\nThe rest of the paper is organized as follows. In \\S\\ref{LensBiasDeriv} we develop a simple description of the effect of lensing magnification on measurements of the Lyman-$\\alpha$ forest, the key result is Eq. (\\ref{Eq:MagBiasFinal}). In \\S \\ref{Sims}--\\S\\ref{FluxPDF} we present the effect of lensing bias on the flux power spectrum, the effective optical depth and the flux probability distribution (PDF) determined by applying a biased weighting scheme to mock absorption spectra from simulations and compare these results with analytical estimates in terms of the flux-mass polyspectra. In \\S \\ref{MagCorrelation} we discuss the exciting possibility of observing the lensing-induced correlation between quasar magnitudes and the flux fluctuations, the flux power spectrum or the mean flux. While this work is dedicated to discussing magnification bias, dust along the line of sight would have a similar (though generally opposite signed) effect, this is discussed in \\S \\ref{dust}. We present concluding remarks and discuss the implications of this work for existing and future Lyman-$\\alpha$ measurements in \\S \\ref{Conclusions}. Appendix \\ref{derive} contains a unified derivation of the magnitude-forest correlations and the lensing bias discussed in the paper. In Appendix \\ref{Extrap} we discuss some issues regarding large-scale flux-mass correlations measured from our simulations and the accuracy of an analytic description. Appendix \\ref{estimator} derives an estimator for the flux-magnitude correlation and its associated error. \n\nBefore proceeding we should mention some related literature. The magnification bias to the statistics of metal absorption lines in quasar spectra was investigated in \\cite{Thomas:1990aj} and a method for detecting statistical lensing by absorbers was proposed in \\cite{Menard:2005}. Lensing effects on the statistics of damped Lyman-$\\alpha$ systems and the inferred density of neutral Hydrogen was considered in \\cite{Bartelmann:1996}. More recently, \\cite{Menard:2003vf} studied magnification bias due to intervening absorbers in the 2dF quasar survey and \\cite{Menard:2008} in SDSS. Recent work by \\cite{Vallinotto:2009wa,Vallinotto:2009jx} proposed correlating lensing in the cosmic microwave background with fluctuations in the forest to extract the flux-mass correlation.\n\n\\section{Lensing as Noise\/Bias}\n\\label{lensingnoise}\n\n\\subsection{Formalism}\n\\label{LensBiasDeriv}\n\nWe are interested in some Lyman-alpha forest observable $\\mathcal{O}$, which can represent the flux power spectrum, the flux transmission\/decrement,\nthe flux fluctuation (around its mean), and so on. Let us use $O_I$ to denote the observable measured from a quasar labeled by $I$. We typically form an estimator by averaging over quasars:\n\\begin{equation} \n\\label{Eq:Osum}\n{\\mathcal{O}}_{\\rm obs}=\\frac{\\sum_I w_I {\\mathcal{O}}_I}{\\sum_I w_I} \n\\end{equation} \nwhere $w_I$ denotes weights, the simplest example of which is $w_I = 1$.\n\nTwo important points. First $\\mathcal{O}_I$, the observable on a quasar by quasar basis, is generally {\\it not} affected by gravitational lensing. This is because gravitational lensing affects all wavelengths equally. For instance, $\\mathcal{O}_I$ could represent the flux transmission which is $f\/f_C$ (where $f$ and $f_C$ are the flux and continuum respectively as a function of frequency), or the flux fluctuation $\\delta_f = (f - \\bar f)\/\\bar f$ (where $\\bar f$ is the flux averaged for the particular line of sight in question). Since gravitational lensing brightens or dims $f$, $\\bar f$ and $f_C$ all equally independent of wavelength, there is no effect on $\\mathcal{O}_I$.\n\nThe other important point is that $\\mathcal{O}_{\\rm obs}$ {\\it is} generally affected by lensing. The crucial observation is that $w_I$ in Eq. (\\ref{Eq:Osum}) reveals only part of the weighting that is going on; in any given dataset, we inevitably give zero weights to quasars which are too faint to observe. This means that even if one chooses $w_I = 1$, one is merely performing a straight average over one's sample, as opposed to an average over all possible quasars. To account for this fact, it is helpful to pixelize the sky, and rewrite ${\\mathcal{O}}_{\\rm obs}$ as \n\\begin{eqnarray}\n\\label{Eq:Osum2}\n{\\mathcal{O}}_{\\rm obs} = {\\sum_i w_i n_i \\mathcal{O}_i \\over \\sum_i w_i n_i}\n\\end{eqnarray}\nwhere $i$ is the pixel label. One could conceptually think of each pixel as sufficiently small that the number of quasars in it $n_i$ is either $1$ or $0$; we will more generally think of $n_i$ as simply the number density of quasars in pixel $i$. This way of rewriting is useful because it makes explicit the fact that $\\mathcal{O}_{\\rm obs}$ is a weighted average -- it is weighted by the number density of quasars, on top of whatever additional weighting $w_i$ one might wish to apply. In other words, since our Lyman-alpha forest observable can be measured only if there exists a background quasar in the sky location of interest, the observable is always implicitly weighted by the abundance of quasars.\n\n\nEq. (\\ref{nlensed}) tells us that\n\\begin{eqnarray} \nn_i = n_i^{\\rm intrinsic} (1 + \\delta^\\mu_i)\n\\end{eqnarray}\nwhere $n_i^{\\rm intrinsic}$ is the intrinsic pre-lensed quasar number density, and $\\delta^\\mu_i$ is the lensing modification given by\n\\begin{eqnarray}\n\\label{Eq:KappaDef}\n\\delta^\\mu_i \\equiv (5s - 2) \\kappa_i \\quad , \\quad {\\kappa}_i =\\int_0^{\\chi_Q} d\\chi'\\frac{\\chi_Q-\\chi'}{\\chi_Q}\\chi' \\nabla_\\perp^2\\phi(\\chi', i)\n\\end{eqnarray}\nassuming fluctuations are small. The expression for the convergence $\\kappa_i$ assumes spatial flatness, but can be easily generalized to non-flat universes ($\\chi_Q$ is the comoving distance to quasar). Here, $1 + 2\\kappa$ is the weak lensing approximation to the lensing magnification $\\mu$ of Eqs. (\\ref{Eq:fmag}) and (\\ref{nlensed}) (the full nonlinear expression is given in footnote \\ref{nonlinfootnote}). Using $w_i = 1$ in Eq. (\\ref{Eq:Osum2}), assuming there is no correlation between the forest observable and the {\\rm intrinsic} quasar number fluctuation, and ignoring corrections of the integral constraint type (see Appendix \\ref{derive} and for example, \\cite{Hui:1998ix}), we obtain the following ensemble average, to the lowest order in fluctuations:\n\\begin{eqnarray}\n\\langle \\mathcal{O}_{\\rm obs} \\rangle = \\mathcal{O}_{\\rm true} + \\langle \\mathcal{O}_i \\delta^\\mu_i \\rangle\n\\end{eqnarray}\nwhere $\\langle \\mathcal{O}_i \\delta^\\mu_i \\rangle$ is the correlation between the observable and the magnification fluctuation at the same $i$ (i.e. zero lag). Dropping the $i$ label, the lensing induced measurement bias on an observable $\\mathcal{O}$ is therefore\n\\begin{eqnarray}\n\\label{Eq:MagBiasFinal}\n\\langle \\mathcal{O}_{\\rm obs} \\rangle - \\mathcal{O}_{\\rm true} = \\langle \\mathcal{O} \\delta^\\mu \\rangle = (5s-2) \\langle \\mathcal{O} \\kappa \\rangle \\, .\n\\end{eqnarray}\nThis is a fundamental expression that we will use repeatedly below. The meaning of each of these symbols is as follows: $\\mathcal{O}$ is the fluctuation which we attempt to measure with the estimator $\\mathcal{O}_{\\rm obs}$, which upon ensemble averaging generally differs from the true underlying value $\\mathcal{O}_{\\rm true} = \\langle \\mathcal{O} \\rangle$.\n\nIt is important to emphasize the actual measurement bias could be different. The above estimate assumes that all quasars {\\it within one's sample} are weighted equally (i.e. $w_i = 1$). In practice, one might want to weigh brighter quasars {\\it within one's sample} more strongly. For instance, one could weigh by the net flux of the quasar, which corresponds to inverse variance weighting in the noise dominated regime. This would result in a lensing induced measurement bias of (see Appendix \\ref{derive} for derivation)\n\\begin{eqnarray}\n\\label{Eq:MagBiasFluxWeight}\n\\langle \\mathcal{O}_{\\rm obs} \\rangle - \\mathcal{O}_{\\rm true} = (5s' - 2) \\langle \\mathcal{O} \\kappa \\rangle \\, .\n\\end{eqnarray}\nwhere $s'$ is defined as\n\\begin{eqnarray}\n\\label{sprime}\ns' = {1\\over {\\rm ln\\,} 10} {\\int dm \\, \\epsilon(m) (dn^0\/dm) 10^{-m\/2.5} \\over\\int dm \\, \\epsilon(m) n^0(m) 10^{-m\/2.5}}\n\\end{eqnarray}\nwhich can be contrasted with $s$ defined in Eq. (\\ref{nlensed}). For a step function sample definition $\\epsilon$, $s'$ reduces to $d\\log_{10} N'(< m_{\\rm lim.}) \/ dm_{\\rm lim.} + 0.4$, where $N' = \\int_{-\\infty}^{m_{\\rm lim.}} dm \\, n^0 10^{-m\/2.5}$. This is generally larger than $s$.\n\nThe precise value of $s$ or $s'$ is sample dependent. The relatively low redshift ($1\\chi_\\beta \\sim 4\/5\\chi_Q$ and had set $\\bar{\\chi}\\sim \\chi_\\beta+ (\\chi_Q-\\chi_\\beta)\/2$). It seems that the sparseness of quasars ($\\sim 16\/(\\textrm{deg.})^2$ for SDSS III) does not permit $N_{pairs}(\\theta)$ to be large enough that adding off-axis correlations will drastically improve the signal-to-noise in the near term. \n\n\nAnother possibility is to cross correlate the quasar number density (which is affected by lensing through magnification bias) with the Lyman-alpha forest observable. Such a correlation makes sense only at a non-zero lag (or non-zero smoothing), since the Lyman-alpha forest is observable only if there is a quasar directly behind it. While we do not give explicit estimates here of the signal-to-noise for these cross correlations, they should be useful in disentangling the lensing signal from certain systematic effects, as we will discuss next.\n\n\\subsection{Dust and Other Systematic Effects}\n\\label{dust}\n\nWe discuss here three systematic effects that could complicate the\nmeasurement of the various cross correlations mentioned above.\n\nThe first is the continuum. The continuum presumably is smooth and therefore\nhas fluctuations only on large scales. But since its precise shape is uncertain,\na cross correlation such as the magnitude-flux correlation is susceptible\nto possible contamination from continuum power.\nA fortunate feature of the cross correlation is that it has a definite\nshape predicted by lensing, as well as an amplitude that scales with $\\tilde s$.\nBoth can be exploited to check for such a contamination.\n\nThe second systematic effect we loosely refer to as `background subtraction'.\nRealistic spectra of quasars inevitably contain `background' which can come\nfrom several sources, including the sky and scattered light within the \noptical instrument. While attempts are generally made to subtract these backgrounds\nas accurately as possible, there are inevitably residuals. These residuals\ncould correlate with the quasar magnitude, for instance they could be more\nnoticeable for fainter quasars. This would then produce spurious correlations\nwhen we correlate the quasar magnitude with Lyman-alpha forest observables (deduced\nfrom imperfect data that contain residual backgrounds). \nIn fact, existing flux power spectrum measurements from SDSS \\cite{McDonald:2004eu} \nare known to exhibit an otherwise puzzling correlation: that $P_{ff}$ is systematically higher\nfor fainter quasars, and this correlation is statistically significant\n(the normalized correlation coefficient is $\\sim 5 - 10\\%$ \\cite{Abazajian:2010}). \nSuch a correlation can be explained by this background subtraction effect.\nIt cannot be explained by lensing since it tends to produce a correlation of an opposite\nsign unless $\\tilde s$ has a sign opposite to what is known.\n(It can also be plausibly produced by dust which we will discuss below).\nTo disentangle background subtraction issues from lensing, it would be useful\nto examine magnitude-observable cross correlations at non-zero lag -- it would\nbe quite surprising if background residuals cause correlations between quasar\nmagnitude at one point with Lyman-alpha forest power at another.\n\nThe third systematic effect is dust extinction. Dust modifies flux by\n$f \\rightarrow f e^{-\\tau_{\\rm dust}}$, where $\\tau_{\\rm dust}$ is the optical depth due\nto dust. This modification is frequency dependent (higher optical depth for bluer photons),\nbut the frequency (which translates into scale) dependence is mild on the scales of interest, and therefore effectively acts as a continuum.\nThe net magnification plus dust correction to the quasar number density is\n\\begin{equation}\n{n}\\rightarrow n\\, {\\mu}^{2.5s-1} e^{-2.5 s {\\tau}_{\\rm dust}} \\, .\n\\end{equation}\nIncluding dust extinction, the measurement bias associated with a\nLyman-alpha forest observable (Eq. [\\ref{Eq:MagBiasFinal}]) is changed to \n\\begin{equation} \n\\label{Eq:MagBiasFinaldust}\n\\langle\\mathcal{O}_{\\rm obs} \\rangle\n- \\mathcal{O}_{\\rm true} \n= (5s-2) \\langle \\mathcal{O} \\kappa \\rangle - 2.5 s \\langle \\mathcal{O} \\delta\\tau_{\\rm dust} \\rangle\n\\end{equation}\nwhere ${\\delta\\tau}_{\\rm dust}=\\tau_{\\rm dust}-\\langle \\tau_{\\rm dust}\\rangle$, which is assumed\nto be small, and we have ignored $\\langle \\mathcal{O} \\kappa \\delta\\tau_{\\rm dust} \\rangle$.\n\nAs far as the `signal' part of our discussion is concerned,\nthe magnitude-observable cross correlation (Eq. [\\ref{Eaverage}])\nis modified to\n\\begin{eqnarray}\n\\label{Eaveragedust}\n\\langle \\mathcal{E} \\rangle = 5 \\tilde s\n\\langle \\kappa \\delta\\mathcal{O} \\rangle - 2.5 \\tilde s\n\\langle \\delta\\tau_{\\rm dust} \\delta\\mathcal{O} \\rangle\\, .\n\\end{eqnarray}\n\nA full calculation of the effect of dust is beyond the scope of this paper. \nBut given a model for $\\delta\\tau_{\\rm dust}$ the effect can be calculated in a way very similar to \nlensing -- $\\tau_{\\rm dust}$ after all is another line of sight integral\nover the (dust) density.\nCalculations comparing the amplitudes of magnification and dust corrections to supernova flux have shown $\\langle\\kappa{\\delta\\tau}_{\\rm dust}\\rangle-\\langle{\\delta\\tau}_{\\rm dust} {\\delta\\tau}_{\\rm dust}\\rangle$ to be $\\sim 7-40\\%$ of $\\langle\\kappa \\kappa\\rangle$ at $z_{source}=1.5$ depending on the dust model \\cite{Zhang:2006qm}. At higher redshifts the dust term is expected to be less important, while the lensing effect should grow. On the other hand, recent measurements suggest that at $z=0.3$ dust extinction can cause shifts in source magnitude comparable to those caused by lensing magnification \\cite{Menard:2009yb}. For the Lyman-$\\alpha$ forest, which is typically measured at redshifts $z\\gsim 2$, the dust corrections are likely to be smaller than the lensing corrections but should be included in a full analysis. \n\n\n\\section{Discussion}\n\\label{Conclusions}\nWe have discussed the sampling bias introduced by magnification and dust on measurements of the Lyman-$\\alpha$ forest. In calculating the effect of this bias (summarized in Eq. (\\ref{Eq:MagBiasFinal})) we made the assumption that all quasar spectra are weighted equally. If, on the other hand, forest measurements were weighted by quasar flux, the effect of lensing bias would be larger than what we have found (see for example Eq. (\\ref{Eq:MagBiasFluxWeight})). \n\nIf the quasars are weighted uniformly, we find that magnification bias leads to corrections $\\lsim 1\\%$ to the flux power spectrum (Fig. \\ref{Fig:PffSims}), $\\lsim 0.1\\%$ to the effective optical depth (Fig. \\ref{Fig:TauBias}), $\\lsim 0.1\\%$ to the flux probability distribution function $\\mathcal{P}(f)$, and as large as a few $\\%$ to the probability distribution function for the flux fluctuation $\\mathcal{P}(\\delta_f)$ (see Fig. \\ref{Fig:PDFetauSims} and Fig. \\ref{Fig:PDFdeltaSims}). These estimates assume quasar number count slope $s=1$, probably reasonable for the SDSS quasars used for Lyman-$\\alpha$ forest measurements. The lensing correction varies strongly with redshift with larger corrections present at lower redshift, largely due to non-linear growth of mass fluctuations. The biases induced by magnification to the effective optical depth are significantly smaller than current error bars. For the flux power spectrum, the lensing bias is just within the current error bars, and since it leads to a systematic offset in each data point, may effect measurements of the overall amplitude of the power spectrum. At low redshift the lensing effect on the PDF of $\\delta_f$ can be rather large, reaching several percent at the high $\\delta_f$ end of the PDF at $z=2$, however this occurs only in regions where the PDF itself is very small $\\lsim 0.1$. Lensing magnification is probably unimportant for current measurements of the flux PDF and the quantities derived from it. \n\nOne may wonder whether including nonlinear magnification\\footnote{\\label{nonlinfootnote}To be precise, the true magnification $\\mu=1\/((1-\\kappa)^2-\\gamma_1^2-\\gamma_2^2)$ where $\\gamma_1$ and $\\gamma_2$ are the shear. Throughout this paper we assume $\\kappa \\, , \\gamma_1 \\, ,\\gamma_2$ are small quantities and so we keep only the first order terms in the expression for $\\mu$. Non-linear magnification refers to the higher-order (in $\\gamma$ and $\\kappa$) terms contributing to $\\mu$ \\cite{Menard:2002ia}.} -- which is more important at the small angular separations relevant for the Lyman-$\\alpha$ forest -- could change our results. We have checked that including all $\\kappa$ terms contributing to $\\mu$ in Eq. (\\ref{nlensed}) and Eq. (\\ref{eq:weightdef}), rather than just the linear term as in Eq. (\\ref{Eq:KappaDef}), changes the magnification bias corrections by $\\lsim 2\\%$. The true nonlinear magnification of course depends on the components of the shear as well as the lensing convergence but it would be surprising if they drastically changed our (all-orders in $\\kappa$) estimate of their importance\\footnote{Actually, there is a further caveat here in that we didn't compute the true lensing convergence $\\kappa$ because we have ignored contributions from mass fluctuations at the lower redshifts outside the box. For the non-linear terms, this shouldn't be a good approximation (e.g. $\\langle\\kappa\\kappa\\rangle$ is poorly estimated) nevertheless it would be surprising if this made an orders-of-magnitude difference in the effect of magnification bias. But perhaps this plus the ignored $\\gamma$ terms could increase the importance of non-linear magnification to the $\\sim 20\\%$ reported by \\cite{Menard:2002ia}.}. \n\n\nAn additional caveat to our analysis is that the damped Lyman-$\\alpha$ (DLA) systems and their associated damping wings are not accurately modeled by the simulations and mock spectra. While DLAs are rare, magnification bias could make them more abundant in survey samples. The damping wings of DLAs are known to add spurious power to $P_{ff}(k_{||})$ at the $10-20\\%$ level on large scales \\cite{McDonald:2005ps}. Magnification bias should increase the abundance of DLAs making the power spectrum more biased than our results suggest. Precisely how magnification affects the DLA bias deserves further investigation.\n\n\nPerhaps most interesting is that lensing magnification induces a correlation between Lyman-$\\alpha$ observables and the magnitude of the quasars used to measure them (Fig. \\ref{Fig:PffMag}). Unfortunately even with the large number of quasar spectra the BOSS survey will obtain it will be a challenge to detect correlations between the flux power spectrum, the flux decrement or the mean transmission and quasar magnitude with high signal-to-noise (see Table I). Additionally, it is possible that lines of sight with high magnification also have more metal lines which could further complicate the analysis. Nevertheless these correlations provide a direct measure of how fluctuations in the quasar flux trace the underlying density field, for example they could \\emph{directly} constrain the flux-mass power spectrum and should therefore be targeted. A more thorough analysis of how to detect the flux-magnitude and flux-power correlations is necessary, we leave this to future work. For a recent idea in a similar vein see \\cite{Vallinotto:2009wa,Vallinotto:2009jx} who propose correlating lensing in the cosmic microwave background with fluctuations in the forest to extract flux-mass information. \n\nIt is worth noting that, with a large quasar sample, it may be possible to exploit the dependence on $5s-2$ and\/or $\\chi(z_Q)$ to isolate the magnification correction and to measure the flux-magnitude correlations. The lensing correction depends linearly on $5s-2$, which ranges quite a bit depending on magnitude limit (Fig. \\ref{Fig:slope}). In the weak lensing limit the correction scales as $H_0(\\chi(z_Q)-\\chi(z))\\chi(z)\/\\chi(z_Q)$ which varies from $\\sim \\frac{4}{25}H_0\\chi(z_Q)$ to $0$ as $z$ goes from $z_\\beta$ and $z_Q$. Indeed measuring these distinctive dependencies would help guard against possible instrumental systematics from being confused with the lensing signal. Our estimates of the forest-magnitude correlations in \\S \\ref{magflux}--\\ref{othercorrelations} assumed $z$ was halfway between $z_\\beta$ and $z_Q$. Also, there exist in public data $\\sim 150$ close quasar pairs \\cite{Hennawi:2006xm}, that one might wish to use for this analysis, unfortunately their number is small compared to the $10^5$ expected from SDSS III and will not significantly improve the signal-to-noise. \n\n\nThere is quite a bit of interest in using the Lyman-$\\alpha$ forest to map the 3D density field and use, for example the baryon features in the two-point correlation function or power spectrum to constrain dark energy and the expansion history of the universe \\cite{McDonald:2006qs}. While we have focused on the power spectrum gotten from 1D measurements of the flux fluctuation, the analysis could be extended to include correlations between different quasar spectra (and therefore different sight-lines). Additional corrections due to correlations between flux and convergence across different lines of sight ($\\sim \\langle\\delta_{f}(\\chi\\mbox{\\boldmath$\\theta$})\\delta_{f}(\\chi'\\mbox{\\boldmath$\\theta$}')\\kappa(\\chi\\mbox{\\boldmath$\\theta$})\\rangle$) would arise. However, since lensing is strongest for lenses along the same line of sight as the sources, the terms calculated in this work should be dominant. One would therefore expect the lensing bias to the correlation function around the baryon scale to remain $0.1-1\\%$ \\footnote{We have checked that unlike the case of the 3D galaxy correlation function \\cite{Hui:2007cu,Hui:2007tm} nothing is remarkably different about the lensing bias to the real space flux correlation function as compared to Fourier space.}. \n\n\\acknowledgements{We are extremely grateful to Lars Hernquist and Volker Springel for providing us with the simulations data used in our analyses. We thank David Hogg, Guinevere Kauffman, Ian McGreer, Uros Seljak, David Spergel and David Weinberg for discussions. L.H. thanks members of the CCPP (New York), CITA (Toronto) and IAS (Princeton) for their fabulous hospitality. M.L. and L.H. acknowledge support from the DOE under DE-FG02-92ER40699, the\nNASA ATP under 09-ATP09-0049, and the Initiatives in Science and Engineering Program at Columbia University. S.M. is supported by the Cyprus State Scholarship Foundation. M.L. is supported as a Friends of the Institute for Advanced Study Member.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nToday, there is a large amount of data from p-p, Pb-Pb and recently p-Pb collisions at the LHC which need to be interpreted. EPOS, based on 'Parton-based Gribov-Regge theory' \\cite{epo}, aims to reproduce a large range of LHC observables like jets, multiplicity or collective behavior. We will discuss the recent implementation of charm and prompt photons in this event generator. We want hard probes production to be under control for p-p collisions and then, use them for the study of the QGP.\\\\\nFirst, we will quickly show the general features of EPOS. Then the charm production will be detailed and finally our projet on prompt photons will be exposed.\n\n\n\n\n\\section{General presentation}\nSome important features of EPOS are :\n\n\\vspace{2ex}\n\n\\begin{enumerate}\n\\item Being a real event generator\n\\item Multiple interactions based on a quantum formalism\n\\item Perturbative calculation with resummation of collinear corrections at the order $\\left(\\alpha_s(Q^2)\\ln(\\frac{Q^2}{\\mu^2})\\right)^n$\n\\item Core-corona separation\n\\item Hydrodynamics done event by event\n\\item Hadronisation done using a string fragmentation model for the core\n\\end{enumerate}\n\n\\vspace{2ex}\n\nBy ``being a real event generator'', we mean that one event in the LHC $\\simeq$ one event in EPOS. The program will generate pions even if one is only interested in charm. All particles are registered in a table and, at the end, one has to select particles of interest.\\\\\n\nOur model for multiple interactions is based on a marriage of Gribov-Regge theory \\cite{grib,venus} and pQCD. It gives the possibility of a quantum treatment of multiple interactions. By ``based on Gribov-Regge theory'' we mean an assumption about the structure of the $T$ matrix, expressed in terms\nof elementary objects called Pomerons (not the same object in EPOS and Gribov-Regge theory). The total cross section can be expressed as illustrated on figure \\ref{Tm}.\\\\\n\\begin{figure}[h]\n\\includegraphics[width=23pc]{Tmatrix.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{Tm}Pink lines : pomerons. A and B are nuclei. Small horizontal lines are remnants.}\n\\end{minipage}\n\\end{figure}\\\\\nPartial summation provides exclusive cross sections. In Gribov-Regge theory, the elastic amplitude is given by :\n\\begin{equation}\nA_{2\\rightarrow 2}(s,t)=\\sum_n A_n(s,t)\n\\end{equation}\nwith $A_n(s,t)$ corresponding to the amplitude for $n$ pomeron(s) exchange. Following the same idea, we can defined $\\sigma_m$ corresponding to the cross section for $m$ cut pomerons. A cut pomeron, figure \\ref{pom}, is at the origin of particles production. The multiplicity is then, on the average, proportional to :\n\\begin{equation}\nN\\propto m\\sum_m \\sigma_m\n\\end{equation}\nThe treatment is the same for p-p, p-A or A-A collisions.\\\\\n\nThe hydrodynamical evolution is done event by event. Initial conditions are given by the distribution of cut pomerons which correspond to color flux tubes, figure \\ref{flux}. Flux tubes fragment into string pieces which will later constitute particles. These flux tubes will constitute both bulk matter (if the energy density is high enough) and jets. ``Matter'' is defined by the region of high energy density flux tubes (the blue region figure \\ref{jetfluid}). Then, there are 3 possibilities :\n\\begin{enumerate}\n\\item The string piece (the red one) is formed outside the ``matter''. In that case, it simply escapes as a jet.\n\\item The string piece (the pink one) is formed inside the ``matter'' but has not enough energy to escape. It constitutes the ``matter'' and will evolve with the hydrodynamical code.\n\\item The string piece (the blue one) is formed inside the ``matter'' and has enough energy to escape (based on energy loss argument). It escapes as a jet which has interacted with the fluid. \n\\end{enumerate}\nFor more details, see \\cite{hydro,v2}.\n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=12pc]{flux-tube.png}\n\\caption{\\label{flux}Cut pomerons form color flux tubes between the 2 nuclei.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=12pc]{hydro-jet.png}\n\\caption{\\label{jetfluid}Color flux tubes fragment into string piece. The blue region is \"matter\" formed by high energy density flux tubes.}\n\\end{minipage} \n\\end{figure}\\\\\n\n With these prescriptions, we can reproduce the $v_2$ for identified particles or the ridge, even for p-Pb collisions (figure \\ref{v2} and figure \\ref{ridge}).\n\n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{v2.png}\n\\caption{\\label{v2}(Color online) Elliptical flow coefficients v2 for pi-\nons, kaons, and protons. We show ALICE results (squares)\nand EPOS3 simulations (lines). Pions appear red, kaons\ngreen, protons blue.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{ridge.png}\n\\caption{\\label{ridge}(Color online) Associated yield per trigger, pro-\njected onto $\\Delta\\phi$, for $|\\Delta\\eta| >$ 0.8. We show ALICE results\n(black squares) and EPOS3 simulations (red dots).}\n\\end{minipage} \n\\end{figure}\n\n\\newpage\n\n\n\n\n\n\\section{Charm production}\n\nA charm can be produced during the spacelike cascade, the born process, the timelike cascade (partonic shower) and the string fragmentation, see figures \\ref{pom} and \\ref{tim}. \n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{cutladder.png}\n\\caption{\\label{pom} A cut pomeron. The center of the pomeron is a (pQCD) ladder diagram.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{tim-fra.png}\n\\caption{\\label{tim}Partonic shower and hadronization by string fragmentation.}\n\\end{minipage} \n\\end{figure}\nBased on DGLAP formalism, spacelike and timelike cascades resumme collinear divergences.\\\\\n\nDuring the spacelike cascade, a spacelike parton emits timelike particles until he reaches the born process. In the leading log approximation, virtualities\nare strongly ordered :\n\\begin{equation}\n Q_1^2 \\ll Q_2^2 \\ll ... \\ll Q_{born}^2\n\\end{equation}\nWe use the following probability distribution for variables $Q^2$ and $x$ :\n\\begin{equation}\n\\frac{dP(Q^2_0,Q^2,x)}{dxdQ^2}\\propto \\frac{\\alpha_s}{2\\pi}\\frac{p(x)}{Q^2}\\Delta(Q^2_0,Q^2) \\hspace{1cm} Q_0^2Q^2\n\\end{equation}\n$z$ being the splitting variable defined as :\n\\begin{equation}\nz=E_{children}\/E_{parent}\n\\end{equation}\n\n\nFor the implementation of charm production, no parameters have been changed. Our first test is the comparison with FONLL calculations of M. Cacciari \\cite{fonll}, \nfigure \\ref{charm}. At high $p_t$ the shape is the same but our central values are higher.\n\\begin{figure}[h]\n\\includegraphics[width=21pc]{charm-nlo-fonll.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{charm}charm from Cacciari vs EPOS.}\n\\end{minipage}\n\\end{figure}\nNext, we test EPOS for $D+$ and $D0$ mesons, by comparing our results with the Alice experiment \\cite{alice} and FONLL calculation, figure \\ref{D+} and figure \\ref{D0}.\n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{D+.png}\n\\caption{\\label{D+} D+ yield from EPOS compare to FONLL calculation \\cite{fonll} and Alice experiment \\cite{alice}.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=19pc]{D0.png}\n\\caption{\\label{D0}D0 yield from EPOS compare to FONLL calculation \\cite{fonll} and Alice experiment \\cite{alice}.}\n\\end{minipage} \n\\end{figure}\\\\\n\nEPOS uncertainties account only for statistics and there is no charm production during the string fragmentation. Our results are globally in good agreement with Alice and FONLL. \nHowever, charm quarks are missing at very low $p_t$. The explanation could be that charm production during the timelike cascade is too small. In the game of fitting data, we can reproduce Alice results by allowing charm production in string fragmentation (one parameter is changed), figure \\ref{D+st}.\n\\begin{figure}[h]\n\\includegraphics[width=20pc]{D+frag.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{D+st}D+ from EPOS with $c\\bar{c}$ creation during strings fragmentation.}\n\\end{minipage}\n\\end{figure}\n\n\\newpage\n\n\\section{Prompt photons}\nWe want to :\n\\begin{enumerate}\n\\item Study isolation criteria\n\\item Compare EPOS and Jetphox\n\\item Use EPOS for $\\gamma\/jet$ and $\\gamma\/hadron$ correlations\n\\end{enumerate}\nPhotons production in the born process was already implemented, but it was not the case for the spacelike and the timelike cascade. \nIt has been done with the same formalism used for partons i.e based on the probability eq. \\ref{proba1}. \nOne needs to replace $\\alpha_s$ by $\\alpha_{el}$ and use the appropriate splitting function. \nThe splitting of a photon into a pair of particle-antiparticle is neglected.\\\\\n\nLike in experiments, we have an isolation condition on selected photons. Using the table where final particles are registered, \nwe define a cone of radius $R$ around the photon candidate. If the sum of transverse energy of particles in this cone is smaller than a given \nvalue (5 GeV for CMS \\cite{cms}), then the photon is isolated. Here, the fact that EPOS is experiment like is important. In Jetphox, the addition of this isolation criteria gives a non-physical rise of the cross section with $1\/R$ \\cite{jetp}.\\\\\n\nWork on photons is still in progress. Fragmentation photons are strongly suppressed due to isolation requirement, whereas $\\sim 98\\%$ of direct photons \nare isolated, table \\ref{isolation}.\n\\begin{center}\n\\begin{table}[h]\n\\centering\n\\caption{\\label{isolation}Pourcentage of isolated direct photons as a function of transverse mometum for the isolation criteria used by CMS \\cite{cms} ($R=0.4$, $\\sum p_t <5$GeV).} \n\\begin{tabular}{cc}\n\\br\n$p_t$ & $\\#$ of isolated direct photons\/$\\#$ of direct photons\\\\\n\\mr\n11 & 0.972\\\\\n13 & 0.981\\\\\n15 & 0.971\\\\\n17 & 0.973\\\\\n 19 & 0.976\\\\\n 21 & 0.992\\\\\n 23 & 0.947\\\\\n 25 & 0.991\\\\\n 27 & 0.976\\\\\n 29 & 0.966\\\\\n 31 & 0.977\\\\\n 33 & 0.978\\\\\n 35 & 0.978\\\\\n\\br\n\\end{tabular}\n\\end{table}\n\\end{center}\n\nComparison with CMS \\cite{cms}, figure \\ref{phot}, shows that our yield seems to be too low by approximately a factor of $1.5$. Comparison with Jetphox will give precious information which, I hope, will allow us to improve our results for photons.\n\\begin{figure}[h]\n\\includegraphics[width=20pc]{CMS-pho.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{phot}Isolated photons. For more clarity, a +0.5 shift (x axis) is used for Jetphox data.}\n\\end{minipage}\n\\end{figure}\n\n\\newpage\n\n\\section{Conclusion}\nA unified formalism in EPOS is one of its strengths. Heavy quarks and prompt photons production are based on the same equations with the same parameters. Whereas the work on charm is nearly finished, prompt photons physics still need to be improved. We already have very good results for flow \\cite{v2,epos3}, and with the implementation of hard probes, EPOS could be an excellent tool for the study of the QGP.\n\n\\section*{Acknowledgement}\nI would like to thank the \\textit{\\textbf{projet TOGETHER, region Pays de la Loire}} which has financed my thesis.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe physics of Majorana states in condensed matter devices is attracting strong interest since a few years ago\n\\cite{Nayak,Alicea,StanescuREV,Beenakker,Franz,Elliott,Aguado,Lutrev}. \nThe measured zero-bias conductance peaks in hybrid semiconductor\/superconductor nanowires have been attributed to the presence of localized Majorana modes on the two ends of the nanowire\n\\cite{Lutchyn,Oreg,Mourik,HaoZ,Deng,Das}. \nThese peculiar pairs of states may be seen as nonlocal split fermions, protected by an energy gap that separates them from other normal states lying at finite energies. Besides the zero energy of the Majorana state, also the peak height was recently seen to coincide with the expected value $2e^2\/h$ \\cite{HaoZ2}.\n\nMajorana end states in (quasi) 1D nanowires are inherently localized. By contrast, propagating Majorana\nstates can be formed at the edges of 2D-like hybrid structures. We refer, specifically, to the \nhybrid devices of Ref.\\ \\cite{He294}, consisting of a quantum anomalous Hall\ninsulator and a superconductor material. In such systems, chiral Majorana modes propagating along the \nedges in a clockwise or anticlockwise manner, depending on the orientation of a perpendicular magnetic field, are formed \nat the 2D interfaces between the quantum anomalous Hall\nand the superconductor materials\n\\cite{Qi10,Chung,Wang15,Lian16,Kalad}. \nEach chiral Majorana contributes $0.5\\,e^2\/h$ to the linear conductance of the device, \nsuch that by tuning the number of Majoranas the conductance takes values $0.5\\,e^2\/h$ and $1\\,e^2\/h$ \nfor the topological phases with one and two chiral Majoranas, respectively. \n \nIn this work we discuss the connection between chiral Majoranas and optical absorption. We expect that in presence of chiral Majoranas, the optical absorption of circularly polarized light will differ for clockwise and anti-clockwise polarizations. The difference, known as the {\\em circular dichroism} (CD) \\cite{Eisfeld,Longhi}, can thus be seen as a measure of the existence of such chiral states. We want to investigate how this behavior is actually realized by explicit calculations of the optical aborption. In previous works we analyzed the optical absorption of\nlocalized Majoranas in nanowires \\cite{Ruiz,Osca2015}. \nIn those systems the CD vanishes and the presence of the Majorana is signaled by a lower-absorption plateau, starting at mid-gap energy, \nof the $y$-polarized signal with respect ot the $x$-polarized one.\nIt is also worth mentioning that alternative techniques for detecting Majorana fermions, based on \nmicrowave photoassisted tunneling in Majorana nanocircuits have been suggested in Ref.\\ \\cite{Dart}.\n \n\nFor chiral Majoranas in a 2D square or rectangular geometry the CD \nat low energies\nis characterized by a sequence of equally spaced peaks, corresponding to transitions from\nnegative to positive energy Bogoliubov-deGennes quasiparticles.\nIn the usual energy ordering of quasiparticle states ($n=\\pm1, \\pm2, \\dots$),\nthe selection rules are: a) transitions between conjugate states $-n\\to n$\nare forbidden by electron-hole symmetry, b) transitions $-n\\to m$ \nare allowed only when $n$ and $m$ are both even or both odd.\nThe rationale behind rule b) is the constructive interference\nof the corresponding quasiparticle states connected by the excitation operator\non the edges of the system.\nFurthermore, it will be shown below that the\nCD peaks corresponding to those even-even or odd-odd quasiparticle transitions may be either\npositive or negative. In the limit of a long 2D ribbon there is a preferred CD sign, depending on\nthe magnetic field orientation.\nFor a disc geometry the generalized angular momentum $J_z$ becomes a good quantum number.\nThen, the combination of circular and particle-hole symmetries in a disc causes a vanishing absorption \nfor $p_x\\pm ip_y$ fields and, obviously, also a vanishing CD. \n\n\\section{Model}\n\nWe use the model of Ref.\\ \\cite{He294}\nfor a quantum-anomalous Hall (3D) thin film in contact with two different superconductors.\nThis model represents \nthe device as two surfaces with a certain interaction between them, with Majoranas being located at their edges.\nIn a Nambu spinorial representation that groups \nthe field operators in the top $(t)$ and bottom $(b)$ layers,\n$\\left[\n(\n\\Psi^t_{k\\uparrow},\n\\Psi^t_{k\\downarrow},\n\\Psi^{t\\dagger}_{-k\\downarrow},\n-\\Psi^{t\\dagger}_{-k\\uparrow}\n),\n(\n\\Psi^b_{k\\uparrow},\n\\Psi^b_{k\\downarrow},\n\\Psi^{b\\dagger}_{-k\\downarrow},\n-\\Psi^{b\\dagger}_{-k\\uparrow}\n)\\right]^T\n$,\nthe Hamiltonian reads\n\\begin{eqnarray}\n{\\cal H} &=& \n\\left[\\, m_0 + m_1 \\left(p_x^2 +p_y^2\\right)\\, \\right] \\tau_z\\, \\lambda_x \n+ \\Delta_B\\, \\sigma_z - \\mu\\, \\tau_z\\nonumber\\\\\n&-& \\alpha\\, \\left(\\,p_x\\sigma_y-p_y\\sigma_x\\right)\\, \\tau_z\\,\\lambda_z \\nonumber\\\\\n&+& \\Delta_p\\, \\tau_x + \n\\Delta_m\\, \\tau_x\\,\\lambda_z\\, .\n\\end{eqnarray}\nThis Hamiltonian is acting in the combined position-spin-isospin-pseudospin space. Spatial positions are \na treated as a 2D continuum ($xy$) and a discrete two-valued pseudospin ($z$). The two-valued spin, isospin and pseudospin degrees of freedom are represented by $\\sigma$, $\\tau$ and $\\lambda$ Pauli matrices, respectively. The pseudospin ($\\lambda$) is modeling a coupled bilayer system in which quasiparticles move. \nThe set of Hamiltonian parameters is $m_0$, $m_1$, $\\Delta_B$, $\\mu$, $\\alpha$, $\\Delta_p$ \nand $\\Delta_m$. The latter two are given in terms of the pairing interaction in the two layers, $\\Delta_1$ and $\\Delta_2$, by\n\\begin{equation}\n\\Delta_{p,m} = \\frac{\\Delta_1\\pm\\Delta_2}{2}\\; .\n\\end{equation}\n\nBelow we numerically determine the eigenvalues and eigenstates of ${\\cal H}$\nusing a 2D grid for $x$ and $y$. Increasing $\\Delta_B$ the spectrum of low-energy \neigenvalues evolves from a gapped (void) spectrum around zero energy at low $\\Delta_B$'s, to the \nemergence of chiral near-zero-energy modes for sufficiently large values of $\\Delta_B$. When the pairing parameters for each layer are equal ($\\Delta_m=0$) chiral Majoranas appear in pairs ($0-2-\\dots$), while for sufficiently \ndifferent parameters it is $\\Delta_m\\ne 0$ and there may be phases with odd numbers of chiral Majoranas as well.\n\nThe numerical results shown below are given in an effective unit system, characterized by the choice of \n$\\hbar\\equiv 1$, mass $m\\equiv 1\/2m_1 \\equiv 1$ and a chosen length unit\n$L_U$, typically \n$L_U\\approx 1\\, \\mu{\\rm m}$. \nThe corresponding energy unit is then $E_U=\\hbar^2\/mL_U^2$.\n\n\\begin{figure}\n\\includegraphics[width=8.75cm,trim=1.5cm 15.cm 3.cm 2.5cm,clip]{F1}%\n\\caption{Energy eigenvalues closer to zero energy as a function of $\\Delta_B$.\nPanels a) and b) \nare for a square of dimensions $L_x=L_y=10\\, L_U$, while c) and d)\ncorrespond to a rectangle of $L_x=2 L_y=20\\, L_U$. In a) and c) \nthe same pairing energy is assumed in each layer \n$\\Delta_1=\\Delta_2=E_U$ while in b) and d) it is $\\Delta_1=\\Delta_2\/3=E_U$.\nThe framed labels indicate the degeneracy of the near-zero energy states, \nwhich indicates the topological phase. Other parameters:\n$m_0=0$, $\\mu=0$, $\\alpha=E_U L_U$.\n}\n\\label{F1}\n\\end{figure}\n\n\n\n\n\\subsection{Circular dichroism}\n\nWe compute the optical absorption cross section for right ($+$) and left ($-$) circularly-polarized light from \n\\begin{equation}\n{\\cal S}_\\pm(\\omega)= 4m_1^2 \\sum_{k>0, s<0}{ \n\\frac{1}{\\omega_{ks}}\\, \n\\left|\n\\rule{0cm}{0.35cm}\n\\,\\langle k\\vert p_x\\pm ip_y \\vert s\\rangle\\,\\right|^2 \n\\, \n\\delta(\\omega-\\omega_{ks}) }\\; ,\n\\end{equation}\nwhere $\\hbar\\omega_{ks}=\\varepsilon_k-\\varepsilon_s$ is the energy difference between \nparticle (unoccupied) and hole (occupied) states. The prefactor $4m_1^2$ gives the squared \ninverse effective mass ($1\/m_{\\rm eff}^2$) of the Hamiltonian and fixes the dimensions of ${\\cal S}$\nas an area.\nThe circular dichroism at a given frequency ${\\cal S}_{CD}(\\omega)$ is then defined as\nthe difference between the absorptions for the two circular polarizations,\n\\begin{equation}\n{\\cal S}_{CD}(\\omega)={\\cal S}_+(\\omega)-{\\cal S}_-(\\omega)\\; .\n\\end{equation}\nObviously, in absence of any chirality preference ${\\cal S}_{CD}$ exactly vanishes.\n\n\\begin{figure}\n\\includegraphics[width=8.75cm,trim=2cm 15.cm 2.5cm 2.cm,clip]{F2}%\n\\caption{Energy eigenvalues as a function of $\\langle J_z\\rangle$. Panels a) \nand b) correspond to the phases in Fig.\\ \\ref{F1}d with one ($\\Delta_B=2E_U$) and two ($\\Delta_B=4.75 E_U$) Majorana states, respectively. \nThe grey-shaded zones indicate the occupied (hole) states while \nthe arrows in panel a) show the two lowest allowed transitions to the first particle state.\nPanel c) shows the probability density corresponding to the lowest positive-energy state \nin panel a), adding all spin, isospin and pseudospin contributions.}\n\\label{F2}\n\\end{figure}\n\n\n\n\\section{Results and discussion}\n\n\\subsection{Chiral bands}\n\nFigure \\ref{F1} shows the evolution of the eigenvalue \nspectrum as a function of the magnetic field parameter $\\Delta_B$.\nThe results reproduce already known results \\cite{He294}. At vanishing $\\Delta_B$ the spectrum around zero energy is gapped, a gap that tends to close\nwhen increasing $\\Delta_B$\nby the appearance of a quasi-continuum distribution\nof eigenvalues. These low-energy states are indicating the presence of propagating Majoranas, energy discretized due to the finite size of the system.\nWhen $\\Delta_1=\\Delta_2$ (panels a and c) the degeneracy is such that the Majorana branches \nappear in pairs. We also notice that there is no\nqualitative difference in the eigenvalue distribution between a square\nand a rectangle (upper vs lower panels). It is remarkable that when a Majorana phase is well developed\nthe set of low energy states are equally spaced in energy. This is particularly\nclear for $2<\\Delta_B\/E_U<4$ in panels a and c, corresponding to the phases with \n2 Majoranas. It can also be seen in panels b and d for the phases with one Majorana\nwhile, in these panels, the equally spaced distribution is also hinted for the beginning of the phase with 2 Majoranas. \n\nThe chiral character of the gap-closing Majorana states is clearly seen in Fig.\\ \\ref{F2}. The equally spaced states at low energy arrange themselves on a line\n(a chiral band) when plot as a function of the $z$-component of the angular momentum. For positive $\\Delta_B$ the angular momentum decreases with increasing energy, \ncausing empty (particle) states to have negative $\\langle J_z\\rangle$, while occupied (hole) states\nhave positive $\\langle J_z\\rangle$. The results of Fig.\\ \\ref{F2}a,b correspond to the rectangle with different\npairing energies in each layer shown in Fig.\\ \\ref{F1}d. For $\\Delta_B=2 E_U$ (\\ref{F2}a) there is a single \nchiral band, while for $\\Delta_B=4.75 E_U$ (\\ref{F2}b) there are two overlapping bands. Notice that the \noverlap of states in Fig.\\ \\ref{F2}b degrades as the energy deviates from zero, indicating that the second Majorana band is not yet fully settled for this particular $\\Delta_B$. Additionally, Fig. \\ref{F2}c explicitly shows the edge character of the states of a chiral Majorana band. A similar distribution is obtained for all the \nstates in a chiral band.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm,trim=1cm 5.cm 3.5cm 4.5cm,clip]{F3}%\n\\caption{Absorption cross sections ${\\cal S}_+$, ${\\cal S}_-$ and ${\\cal S}_{CD}$ defined in the \nmain text. The shown results correspond to the spectra of Fig.\\ \\ref{F1}d for Zeeman parameters\nof $\\Delta B=0.3 E_U$ (a), $2 E_U$ (b), and $4.75 E_U$ (c).}\n\\label{F3}\n\\end{figure}\n\n\n\n\\subsection{Absorption and CD}\n\nThe absorption cross sections and the CD for the spectra of the rectangle with different pairing energies in the two layers (Fig.\\ \\ref{F1}d) are shown in Fig.\\ \\ref{F3} for selected values of $\\Delta_B$. They correspond to zero (\\ref{F3}a), one (\\ref{F3}b) and two (\\ref{F3}c) chiral bands. As anticipated, in presence of the chiral states the system develops a clear CD. The negative CD peaks dominate, due to the negative slope of the chiral bands (Fig.\\ \\ref{F2}a,b). It is remarkable, however, that a few positive peaks are also present. We attribute them to the fact that in a rectangular geometry $J_z$ is not a good\nquantum number and, therefore, there are states with mixed angular momentum. We have performed calculations in a circular geometry confirming this interpretation. Therefore, quasiparticle scattering by the corners\nplays a nontrivial role on the absorption by chiral edge states. \n\n \n\n\\begin{figure}\n\\includegraphics[width=8.5cm,trim=1cm 11.cm 3.5cm 4.5cm,clip]{F4}%\n\\caption{Absorption cross sections ${\\cal S}_+$, ${\\cal S}_-$ and ${\\cal S}_{CD}$ \n for a square of $L_x=L_y=20\\, L_U$ (a), and for a rectangle of $L_x=6\\, L_y= 60\\, L_U$. In both cases \n we used $\\Delta_B=2\\, E_U$ and $\\Delta_1=\\Delta_2\/3= E_U$.}\n\\label{F4}\n\\end{figure}\n\n\nThe most conspicuous feature of Fig.\\ \\ref{F3}b is the regular energy spacing of the first few CD peaks.\nAnalysing them in terms of energy transitions of the chiral band it is easily noticed\nthat they correspond to jumps of 3, 5, 7, \\dots steps (see arrows in Fig.\\ \\ref{F2}a). We explain this selection rule noticing the following restrictions for transitions from the negative $n$-th state to the positive $m$-th state \n($-n\\to m$):\n\\begin{itemize}\n\\item[a)] transitions between conjugate states $-n\\to n$ are forbidden by particle-hole symmetry \\cite{Ruiz}, \n\\item[b)] $n$ even to $m$ odd transitions (or vice versa) are forbidden because of destructive interference along the nanostructure perimeter with the excitation operator.\n\\end{itemize}\n\nFor a disc, $J_z$ becomes a good symmetry and, by angular momentum conservation with a dipole \noperator only the transition $-1\\to1$ is possible. However, this transition is blocked by rule a)\nand, therefore, no dipole absorption is possible and the CD exactly vanishes. We have also checked this behavior by explicit calculation for a device with circular geometry. For a square and rectangle, quasiparticle scattering by the corners plays a nontrivial role yielding the mentioned deviations with respect to the disc.\n\nThe pattern of equally spaced peaks is fulfilled only when one or several chiral bands are fully developed and they exactly overlap. In Fig.\\ \\ref{F3}c we see that the slight degradation of the two-band overlaps\nof Fig.\\ \\ref{F2}b manifests in a small twofold splitting of the CD peaks. It is also worth stressing that once the chiral bands are fully formed, the energy positions of the first few CD peaks become $\\Delta_B$-independent (cf.\\ Figs.\\ \\ref{F2}b and \\ref{F2}c).\n\n\nFigure \\ref{F4} shows the absorption results for different geometries, a square (\\ref{F4}a) and a \nlong rectangle resembling a 2D ribbon (\\ref{F4}b). For the square, the first CD peaks alternate sign in a\nremarkable way. On the other hand, for the ribbon the alternation is of a longer period, the positive \npeaks having a much lower intensity than the negative ones and there are groups of a few consecutive negative peaks. The 2D ribbon shape thus favors the observation of CD peaks of the same sign. \n\n\\section{Conclusions}\n\nIn this work we have investigated the manifestation of chiral Majorana modes in the CD of the dipole absorption. The chiral bands formed at the edges of a hybrid system made of a quantum-anomalous-Hall-insulator and a superconductor yield equally spaced peaks in the CD signal. We identified the particle-hole \nselection rules responsible for this behavior from the analysis in terms of chiral bands.\nIn a disc there is no CD signal due to the incompatibility of the selection rules with the angular momentum restriction; a square or rectangular geometry (or, more generally, a system with straight edges) is needed.\nThe presence of two chiral bands can be inferred from the small splitting of the CD peaks. Finally, both positive and negative CD peaks can be seen, with a perfect alternation in a square and a favored sign \nin a long 2D ribbon geometry. \n\nOn the whole, our results suggest the use of CD spectroscopy as a valuable \nprobe of chiral Majorana states, complementing the evidences obtained\nwith electrical conductance measurements \\cite{He294}. This may require the use of\nan array of absorbing devices, in order to achieve a combined signal of sufficient intensity.\nAlternatively, techniques such as those developed for single plasmonic nanoparticle sensing \\cite{Olson} \nmight be applied to an isolated chiral-Majorana device. \n\n\n\\begin{acknowledgements}\nThis work was funded by MINECO (Spain), grant FIS2014-52564.\n\\end{acknowledgements} \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\chapter{Exciton-Plasmon Interactions in Individual Carbon Nanotubes}\n\\setcounter{page}{0}\n\n\\begin{center}\n{\\huge\\textbf{Exciton-Plasmon Interactions in\\\\[0.25cm] Individual Carbon\nNanotubes\\footnote{To appear in \"Plasmons: Theory and\nApplications\", ed. K.Helsey (Nova Publishers, NY, USA)}}}\\\\[1.0cm]\n\n{\\Large \\textbf{Igor V. Bondarev}}\\\\[0.25cm]\n{\\large {\\it Department of Physics, North Carolina Central University\\\\\nDurham, NC 27707, USA \\\\[0.25cm]{\\tt E-mail:~ibondarev@nccu.edu}}}\\\\[1cm]\n\n{\\Large \\textbf{Lilia M. Woods} ~and~ \\textbf{Adrian Popescu}}\\\\[0.25cm]\n{\\large {\\it Department of Physics, University of South Florida\\\\\nTampa, FL 33620, USA \\\\[1cm]}}\n\n\\begin{minipage}{5in}\n\nWe use the macroscopic quantum electrodynamics approach suitable\nfor absorbing and dispersing media to study the properties and\nrole of collective surface excitations --- excitons and plasmons\n--- in single-wall and double-wall carbon nanotubes. We show that\nthe interactions of excitonic states with surface electromagnetic\nmodes in individual small-diameter ($\\,\\lesssim\\!1$~nm)\nsingle-walled carbon nanotubes can result in strong\nexciton-surface-plasmon coupling. Optical response of individual\nnanotubes exhibits Rabi splitting $\\sim\\!0.1$~eV, both in the\nlinear excitation regime and in the non-linear excitation regime\nwith the photoinduced biexcitonic states formation, as the exciton\nenergy is tuned to the nearest interband surface plasmon resonance\nof the nanotube. An electrostatic field applied perpendicular to\nthe nanotube axis can be used to control the exciton-plasmon\ncoupling. For double-wall carbon nanotubes, we show that at tube\nseparations similar to their equilibrium distances interband\nsurface plasmons have a profound effect on the inter-tube Casimir\nforce. Strong overlapping plasmon resonances from both tubes\nwarrant their stronger attraction. Nanotube chiralities possessing\nsuch collective excitation features will result in forming the\nmost favorable inner-outer tube combination in double-wall carbon\nnanotubes. These results pave the way for the development of new\ngeneration of tunable optoelectronic and nano-electromechanical\ndevice applications with carbon nanotubes.\n\\end{minipage}\n\\end{center}\n\n\\noindent {\\bf Keywords:} Carbon nanotubes, Near-field effects,\nExcitons, Plasmons\n\n\\newpage\n\n\\section{Introduction}\n\nSingle-walled carbon nanotubes (CNs) are quasi-one-dimensional\n(1D) cylindrical wires consisting of graphene sheets rolled-up\ninto cylinders with diameters $\\sim\\!1-10$~nm and lengths\n$\\sim\\!1-10^4\\,~\\mu$m~\\cite{Dresselhaus,Dai,Zheng,Huang}. CNs are\nshown~to~be useful as miniaturized electronic, electromechanical,\nand chemical devices~\\cite{Baughman}, scanning probe\ndevices~\\cite{Popescu}, and nanomaterials for macroscopic\ncomposites~\\cite{Trancik}. The area of their potential\napplications was recently expanded to\nnanophotonics~\\cite{Bondarev10,Bondarev06trends,Bondarev06,Bondarev07,Bondarev07jem,Bondarev07os}\nafter the demonstration of controllable single-atom incapsulation\ninto CNs~\\cite{Shimoda,Jeong,Jeongtsf,Khazaei}, and even to\nquantum cryptography since the experimental evidence was reported\nfor quantum correlations in the photoluminescence spectra of\nindividual nanotubes~\\cite{Imamoglu}.\n\nFor pristine (undoped) single-walled CNs, the numerical\ncalculations predicting large exciton binding energies\n($\\sim\\!0.3\\!-\\!0.6$~eV) in semiconducting\nCNs~\\cite{Pedersen03,Pedersen04,Capaz}~and even in some\nsmall-diameter ($\\sim\\!0.5$~nm) metallic CNs~\\cite{Spataru04},\nfollowed by the results of various exciton photoluminescence\nmeasurements~\\cite{Imamoglu,Wang04,Wang05,Hagen05,Plentz05,Avouris08},\nhave become available.~These works, together with other reports\ninvestigating the role of effects such as intrinsic\ndefects~\\cite{Hagen05,Prezhdo08}, exciton-phonon\ninteractions~\\cite{Plentz05,Prezhdo08,Perebeinos05,Lazzeri05,Piscanec},\nbiexciton formation~\\cite{Pedersen05,Papan},\nexciton-surface-plasmon\ncoupling~\\cite{Qasmi,BondOptCom,BondOptSpectr,BondPRB09}, external\nmagnetic~\\cite{Zaric,Srivastava} and electric\nfields~\\cite{BondPRB09,Perebeinos07}, reveal the variety and\ncomplexity of the intrinsic optical properties of\nCNs~\\cite{Dresselhaus07}.\n\nCarbon nanotubes combine advantages such as electrical\nconductivity, chemical stability, high surface area, and unique\noptoelectronic properties that make them excellent potential\ncandidates for a variety of applications, including efficient\nsolar energy conversion~\\cite{Trancik}, energy\nstorage~\\cite{Shimoda}, optical\nnanobiosensorics~\\cite{Goodsell10}. However, the quantum yield of\nindividual CNs is normally very low. Nanotube composites of CN\nbundles and\/or films could surpass this difficulty, opening up new\npaths for the development of high-yield, high-performance\noptoelectronics applications with CNs~\\cite{Munich,Ferrari}.\nUnderstanding the inter-tube interactions is important in order to\nbe able to tailor properties of CN bundles and films, as well as\nproperties of multi-wall CNs. This is also important for\nexperimental realization of new effects and devices proposed\nrecently, such as trapping of cold\natoms~\\cite{Goodsell10,Fermani2007} and their\nentanglement~\\cite{Bondarev07} near single-walled CNs, surface\nprofiling~\\cite{Popescu} and nanolithography\napplications~\\cite{Popescu2009} with double-wall CNs.\n\nHere, we use the macroscopic Quantum ElectroDynamics (QED)\nformalism developed earlier for absorbing and dispersive\nmedia~\\cite{VogelWelsch,ScheelWelsch,BuhmannWelsch,Bondarev06trends}\nand then successfully employed to study near-field EM effects in\nhybrid CN systems~\\cite{Bondarev06,Bondarev07,Fermani2007}, to\ninvestigate the properties and role of collective surface\nexcitations --- excitons and plasmons --- in single-wall and\ndouble-wall CNs. First, we show that, due to the presence of\nlow-energy ($\\sim\\!0.5\\!-\\!2$~eV) weakly-dispersive interband\nplasmon modes~\\cite{Pichler98} and large exciton excitation\nenergies in the same energy domain~\\cite{Spataru05,Ma}, the\nexcitons can form strongly coupled mixed exciton-plasmon\nexcitations in individual small-diameter ($\\lesssim\\!1$~nm)\nsemiconducting single-walled CNs. The exciton-plasmon coupling\n(and the exciton emission accordingly) can be controlled by an\nexternal electrostatic field applied perpendicular to the CN axis\n(the quantum confined Stark effect). The optical response of\nindividual CNs exhibits the Rabi splitting of $\\sim\\!0.1$~eV, both\nin the linear excitation regime and in the non-linear excitation\nregime with the photoinduced biexcitonic states formation, as the\nexciton energy is tuned to the nearest interband surface plasmon\nresonance of the CN. Previous studies of the exciton-plasmon\ncoupling have been focused on artificially fabricated\n\\emph{hybrid} plasmonic nanostructures, such as dye molecules in\norganic polymers deposited on metallic films~\\cite{Bellessa},\nsemiconductor quantum dots coupled to metallic\nnanoparticles~\\cite{Govorov}, or nanowires~\\cite{Fedutik}, where\nsemiconductor material carries the exciton and metal carries the\nplasmon. Our results are particularly interesting since they\nreveal the fundamental electromagnetic (EM) phenomenon\n--- the strong exciton-plasmon coupling --- in an\n\\emph{individual} quasi-1D nanostructure, a~carbon nanotube, as\nwell as its tunability feature by means of the quantum confined\nStark effect. We expect these results to open up new paths for the\ndevelopment of tunable optoelectronic device applications with\noptically excited carbon nanotubes, including the strong\nexcitation regime with optical non-linearities.\n\nNext, we turn to the double-wall carbon nanotubes to investigate\nthe effect of collective surface excitations on the inter-tube\nCasimir interaction in these systems. The Casimir interaction is a\nparadigm for a force induced by quantum EM fluctuations. The\nfundamental nature of this force has been studied extensively ever\nsince the prediction of the existence of an attraction between\nneutral metallic mirrors in\nvacuum~\\cite{BuhmannWelsch,Klimchitskaya2009}. In recent years,\nthe Casimir effect has acquired a much broader impact due to its\nimportance for nanostructured materials and devices. The\ndevelopment and operation of micro- and nano-electromechanical\nsystems are limited due to unwanted effects, such as stiction,\nfriction, and adhesion, originating from the Casimir\nforce~\\cite{Chan2001}. This interaction is also an important\ncomponent for the stability of nanomaterials. Here, we show that\nat tube separations similar to their equilibrium distances\ninterband surface plasmons have a profound effect on the\ninter-tube Casimir force. Strong overlapping plasmon resonances\nfrom both tubes warrant their stronger attraction. Nanotube\nchiralities possessing such collective excitation features will\nresult in forming the most favorable inner-outer tube combination\nin double-wall carbon nanotubes. This theoretical understanding is\nimportant for the development of nano-electromechanical devices\nwith CNs.\n\nThis Chapter is organized as follows.~Section~\\ref{sec2}\nintroduces the general Hamiltonian of the exciton interaction with\nvacuum-type quantized surface EM modes of a single-walled CN. No\nexternal EM field is assumed to be applied.~The\nvacuum--type--field we consider is created by CN surface EM\nfluctuations.~Section~\\ref{sec3} explains how the interaction\nintroduced results in the coupling of the excitonic states to the\nnanotube's surface plasmon modes.~Here we derive and discuss the\ncharacteristics of the coupled exciton--plasmon excitations, such\nas the dispersion relation, the plasmon density of states (DOS),\nand the optical response functions, for particular semiconducting\nCNs of different diameters. We also analyze how the electrostatic\nfield applied perpendicular to the CN axis affects the CN band\ngap, the exciton binding energy, and the surface plasmon energy,\nto explore the tunability of the exciton-surface-plasmon coupling\nin CNs.~Section~\\ref{sec4} derives and analyzes the Casimir\ninteraction between two concentric cylindrical graphene sheets\ncomprising a double-wall CN. The summary and conclusions of the\nwork are given in Sec.~\\ref{concl}. All the technical details\nabout the construction and diagonalization of the exciton--field\nHamiltonian, the EM field Green tensor derivation, the\nperpendicular electrostatic field effect, are presented in the\nAppendices in order not to interrupt the flow of the arguments and\nresults.\n\n\\section{Exciton-electromagnetic-field interaction\\\\ on the\nnanotube surface}\\label{sec2}\n\nWe consider the vacuum-type EM interaction of an exciton with the\nquantized surface electromagnetic fluctuations of a single-walled\nsemiconducting CN by using our recently developed Green function\nformalism to quantize the EM field in the presence of quasi-1D\nabsorbing\nbodies~\\cite{Bondarev02,Bondarev04,Bondarev04pla,Bondarev04ssc,Bondarev05,Bondarev06trends}.\nNo external EM field is assumed to be applied. The nanotube is\nmodelled by an infinitely thin, infinitely long, anisotropically\nconducting cylinder with its surface conductivity obtained from\nthe realistic band structure of a particular CN. Since the problem\nhas the cylindrical symmetry, the orthonormal cylindrical basis\n$\\{\\mathbf{e}_{r},\\mathbf{e}_{\\varphi},\\mathbf{e}_{z}\\}$ is used\nwith the vector $\\mathbf{e}_{z}$ directed along the nanotube axis\nas shown in Fig.~\\ref{fig1}. Only the axial conductivity,\n$\\sigma_{zz}$, is taken into account, whereas the azimuthal one,\n$\\sigma_{\\varphi\\varphi}$, being strongly suppressed by the\ntransverse depolarization\neffect~\\cite{Benedict,Tasaki,Li,Marinop,Ando,Kozinsky}, is\nneglected.\n\n\\begin{figure}[t]\n\\epsfxsize=9.0cm\\centering{\\epsfbox{fig1.eps}}\\caption{The\ngeometry of the problem.}\\label{fig1}\n\\end{figure}\n\nThe total Hamiltonian of the coupled exciton-photon system on the\nnanotube surface is of the form\n\\begin{equation}\n\\hat{H}=\\hat{H}_F+\\hat{H}_{ex}+\\hat{H}_{int},\\label{Htot}\n\\end{equation}\nwhere the three terms represent the free (medium-assisted) EM\nfield, the free (non-interacting) exciton, and their interaction,\nrespectively.~More explicitly, the second quantized field\nHamiltonian is\n\\begin{equation}\n\\hat{H}_F\\!=\\!\\sum_\\mathbf{n}\\int_0^\\infty\\!\\!\\!d\\omega\\,\\hbar\\omega\n\\hat{f}^\\dag(\\mathbf{n},\\omega)\\hat{f}(\\mathbf{n},\\omega),\n\\label{HF}\n\\end{equation}\nwhere the scalar bosonic field operators\n$\\hat{f}^\\dag(\\mathbf{n},\\omega)$ and $\\hat{f}(\\mathbf{n},\\omega)$\ncreate and annihilate, respectively, the surface EM excitation of\nfrequency $\\omega$ at an arbitrary point\n$\\mathbf{n}\\!=\\!\\mathbf{R}_n\\!=\\!\\{R_{CN},\\varphi_n,z_n\\}$\nassociated with a carbon atom (representing a lattice site --\nFig.~\\ref{fig1}) on the surface of the CN of radius $R_{CN}$. The\nsummation is made over all the carbon atoms, and in the following\nit is replaced by the integration over the entire nanotube surface\naccording to the rule\n\\begin{equation}\n\\sum_\\mathbf{n}\\!\\ldots=\\frac{1}{S_0}\\int\\!d\\mathbf{R}_n\\!\\ldots=\n\\frac{1}{S_0}\\int_0^{2\\pi}\\!\\!\\!\\!d\\varphi_nR_{CN}\\!\\int_{-\\infty}^\\infty\\!\\!\\!\\!dz_n\\!\\ldots,\n\\label{sumrule}\n\\end{equation}\nwhere $S_0\\!=\\!(3\\sqrt{3}\/4)b^2$ is the area of an elementary\nequilateral triangle selected around each carbon atom in a way to\ncover the entire surface of the nanotube, $b\\!=\\!1.42$~\\AA\\space\nis the carbon-carbon interatomic distance.\n\nThe second quantized Hamiltonian of the free exciton (see, e.g.,\nRef.~\\cite{Haken}) on the CN surface is of the form\n\\begin{equation}\n\\hat{H}_{ex}\\!=\\!\\!\\!\\sum_{\\mathbf{n},\\mathbf{m},f}\\!\\!\\!E_f(\\mathbf{n})\nB^\\dag_{\\mathbf{n}+\\mathbf{m},f}B_{\\mathbf{m},f}\\!=\n\\!\\sum_{\\mathbf{k},f}E_f(\\mathbf{k})B^\\dag_{\\mathbf{k},f}B_{\\mathbf{k},f},\n\\label{Hex}\n\\end{equation}\nwhere the operators $B^\\dag_{\\mathbf{n},f}$ and $B_{\\mathbf{n},f}$\ncreate and annihilate, respectively, an exciton with the energy\n$E_f(\\mathbf{n})$ in the lattice site $\\mathbf{n}$ of the CN\nsurface.~The index $f\\,(\\ne\\!0)$ refers to the internal degrees of\nfreedom of the exciton. Alternatively,\n\\begin{equation}\nB^\\dag_{\\mathbf{k},f}=\\frac{1}{\\sqrt{N}}\\sum_{\\mathbf{n}}\\!B^\\dag_{\\mathbf{n},f}\ne^{i\\mathbf{k}\\cdot\\mathbf{n}}~~~\\mbox{and}~~~B_{\\mathbf{k},f}=(B^\\dag_{\\mathbf{k},f})^\\dag\n\\label{Bkf}\n\\end{equation}\ncreate and annihilate the $f$-internal-state exciton with the\nquasi-momentum $\\mathbf{k}\\!=\\!\\{k_{\\varphi},k_{z}\\}$, where the\nazimuthal component is quantized due to the transverse confinement\neffect and the longitudinal one is continuous, $N$ is the total\nnumber of the lattice sites (carbon atoms) on the CN surface. The\nexciton total energy is then written in the form\n\\begin{equation}\nE_f(\\mathbf{k})=E_{exc}^{(f)}(k_{\\varphi})+\\frac{\\hbar^2k_z^2}{2M_{ex}(k_{\\varphi})}\n\\label{Ef}\n\\end{equation}\nHere, the first term represents the excitation energy\n\\begin{equation}\nE_{exc}^{(f)}(k_{\\varphi})=E_g(k_{\\varphi})+E_b^{(f)}(k_{\\varphi})\n\\label{Eexcf}\n\\end{equation}\nof the $f$-internal-state exciton with the (negative) binding\nenergy $E_b^{(f)}$, created via the interband transition with the\nband gap\n\\begin{equation}\nE_g(k_{\\varphi})=\\varepsilon_e(k_{\\varphi})+\\varepsilon_h(k_{\\varphi}),\n\\label{Egkfi}\n\\end{equation}\nwhere $\\varepsilon_{e,h}$ are transversely quantized azimuthal\nelectron-hole subbands (see the schematic in Fig.~\\ref{fig2}).~The\nsecond term in Eq.~(\\ref{Ef}) represents the kinetic energy of the\ntranslational longitudinal movement of the exciton with the\neffective mass $M_{ex}=m_e+m_h$, where $m_e$ and $m_h$ are the\n(subband-dependent) electron and hole effective masses,\nrespectively. The two equivalent free-exciton Hamiltonian\nrepresentations are related to one another via the obvious\northogonality relationships\n\\begin{equation}\n\\frac{1}{N}\\sum_\\mathbf{n}e^{-i(\\mathbf{k}-\\mathbf{k}^\\prime)\\cdot\\mathbf{n}}=\n\\delta_{\\mathbf{k}\\mathbf{k}^\\prime},\\;\\;\n\\frac{1}{N}\\sum_\\mathbf{k}e^{-i(\\mathbf{n}-\\mathbf{m})\\cdot\\mathbf{\\mathbf{k}}}=\n\\delta_{\\mathbf{n}\\mathbf{m}} \\label{orthog}\n\\end{equation}\nwith the $\\mathbf{k}$-summation running over the first Brillouin\nzone of the nanotube.~The bosonic field operators in $\\hat{H}_F$\nare transformed to the $\\mathbf{k}$-representation in the same\nway.\n\n\\begin{figure}[t]\n\\epsfxsize=10.0cm\\centering{\\epsfbox{fig2.eps}}\\caption{Schematic\nof the two transversely quantized azimuthal electron-hole subbands\n(\\emph{left}), and the first-interband ground-internal-state\nexciton energy (\\emph{right}) in a small-diameter semiconducting\ncarbon nanotube. Subbands with indices $j=1$ and 2 are shown,\nalong with the optically allowed (exciton-related) interband\ntransitions~\\cite{Ando}. See text for notations.}\\label{fig2}\n\\end{figure}\n\nThe most general (non-relativistic, electric dipole)\nexciton-photon interaction on the nanotube surface can be written\nin the form (we use the Gaussian system of units and the Coulomb\ngauge; see details in Appendix A)\n\\begin{equation}\n\\hat{H}_{int}=\\sum_{\\mathbf{n},\\mathbf{m},f}\\int_{0}^{\\infty}\\!\\!\\!\\!\\!d\\omega\\,[\\,\n\\mbox{g}_f^{(+)}(\\mathbf{n},\\mathbf{m},\\omega)B^\\dag_{\\mathbf{n},f}-\n\\mbox{g}_f^{(-)}(\\mathbf{n},\\mathbf{m},\\omega)B_{\\mathbf{n},f}\\,]\\,\n\\hat{f}(\\mathbf{m},\\omega)+h.c.,\\label{Hint}\n\\end{equation}\nwhere\n\\begin{equation}\n\\mbox{g}_f^{(\\pm)}(\\mathbf{n},\\mathbf{m},\\omega)=\n\\mbox{g}_f^{\\perp}(\\mathbf{n},\\mathbf{m},\\omega)\\pm\n\\frac{\\omega}{\\omega_f}\\,\\mbox{g}_f^{\\parallel}(\\mathbf{n},\\mathbf{m},\\omega)\n\\label{gpm}\n\\end{equation}\nwith\n\\begin{equation}\n\\mbox{g}_f^{\\perp(\\parallel)}(\\mathbf{n},\\mathbf{m},\\omega)=\n-i\\frac{4\\omega_f}{c^{2}}\\sqrt{\\pi\\hbar\\omega\\,\\mbox{Re}\\,\\sigma_{zz}(R_{CN},\\omega)}\\;\n(\\textbf{d}^f_{\\mathbf{n}})_z\\,^{\\perp(\\parallel)}G_{zz}(\\mathbf{n},\\mathbf{m},\\omega)\n\\label{gperppar}\n\\end{equation}\nbeing the interaction matrix element where the exciton with the\nenergy $E_{exc}^{(f)}=\\hbar\\omega_f$ is excited through the\nelectric dipole transition\n$(\\textbf{d}^f_{\\mathbf{n}})_z\\!=\\!\\langle0|(\\hat{\\mathbf{d}}_\\mathbf{n})_z|f\\rangle$\nin the lattice site $\\textbf{n}$ by the nanotube's transversely\n(longitudinally) polarized surface EM modes. The modes are\nrepresented in the matrix element by the transverse (longitudinal)\npart of the Green tensor $zz$-component\n$G_{zz}(\\mathbf{n},\\mathbf{m},\\omega)$ of the EM subsystem\n(Appendix~B). This is the only Green tensor component we have to\ntake into account.~All the other components can be safely\nneglected as they are greatly suppressed by the strong transverse\ndepolarization effect in\nCNs~\\cite{Benedict,Tasaki,Li,Marinop,Ando,Kozinsky}.~As a\nconsequence, only $\\sigma_{zz}(R_{CN},\\omega)$, the \\emph{axial}\ndynamic surface conductivity per unit length, is present in\nEq.(\\ref{gperppar}).\n\nEquations~(\\ref{Htot})--(\\ref{gperppar}) form the complete set of\nequations describing the exciton-photon coupled system on the CN\nsurface in terms of the EM field Green tensor and the CN surface\naxial conductivity.\n\n\\section{Exciton-surface-plasmon coupling}\\label{sec3}\n\nFor the following it is important to realize that the transversely\npolarized surface EM mode contribution to the\ninteraction~(\\ref{Hint})--(\\ref{gperppar}) is negligible compared\nto the longitudinally polarized surface EM mode contribution. As\na~matter of fact,\n$^{\\perp}G_{zz}(\\mathbf{n},\\mathbf{m},\\omega)\\!\\equiv\\!0$ in the\nmodel of an infinitely thin cylinder we use here (Appendix~B),\nthus yielding\n\\begin{equation}\n\\mbox{g}_f^\\perp(\\mathbf{n},\\mathbf{m},\\omega)\\!\\equiv\\!0,~~~\n\\mbox{g}_f^{(\\pm)}(\\mathbf{n},\\mathbf{m},\\omega)\\!=\\!\n\\pm\\frac{\\omega}{\\omega_f}\\,\\mbox{g}_f^{\\parallel}(\\mathbf{n},\\mathbf{m},\\omega)\n\\label{gfpar}\n\\end{equation}\nin Eqs.~(\\ref{Hint})--(\\ref{gperppar}). The point is that, because\nof the nanotube quasi-one-dimen\\-sionality, the exciton\nquasi-momentum vector and all the relevant vectorial matrix\nelements of the momentum and dipole moment operators are directed\npredominantly along the CN axis (the longitudinal exciton; see,\nhowever, Ref.~\\cite{UryuAndo}). This prevents the exciton from the\nelectric dipole coupling to transversely polarized surface EM\nmodes as they propagate predominantly along the CN axis with their\nelectric vectors orthogonal to the propagation direction.~The\nlongitudinally polarized surface EM modes are generated by the\nelectronic Coulomb potential (see, e.g., Ref.~\\cite{Landau}), and\ntherefore represent the CN surface plasmon excitations.~These have\ntheir electric vectors directed along the propagation direction.\nThey do couple to the longitudinal excitons on the CN surface.\nSuch modes were observed in Ref.~\\cite{Pichler98}. They occur in\nCNs both at high energies (well-known $\\pi$-plasmon~at\n$\\sim\\!6$~eV) and at comparatively low energies of\n$\\sim\\!0.5\\!-\\!2$~eV.~The latter ones are related to the\ntransversely quantized interband (inter-van Hove) electronic\ntransitions. These weakly-dispersive\nmodes~\\cite{Pichler98,Kempa02} are similar to the intersubband\nplasmons in quantum wells~\\cite{Kempa89}. They occur in the same\nenergy range of $\\sim\\!1$~eV where the exciton excitation energies\nare located in small-diameter ($\\lesssim\\!1$~nm) semiconducting\nCNs~\\cite{Spataru05,Ma}. In what follows we focus our\nconsideration on the exciton interactions with these particular\nsurface plasmon modes.\n\n\\subsection{The dispersion relation}\n\nTo obtain the dispersion relation of the coupled exci\\-ton-plasmon\nexcitations, we transfer the total\nHamiltonian~(\\ref{Htot})--(\\ref{Hint}) and (\\ref{gfpar}) to the\n$\\mathbf{k}$-representation using Eqs.~(\\ref{Bkf}) and\n(\\ref{orthog}), and then diagonalize it exactly by means of\nBogoliubov's canonical transformation technique (see, e.g.,\nRef.~\\cite{Davydov}). The details of the procedure are given in\nAppendix~C. The Hamiltonian takes the form\n\\begin{equation}\n\\hat{H}=\\!\\!\\!\\sum_{\\mathbf{k},\\,\\mu=1,2}\\!\\!\\!\\hbar\\omega_\\mu(\\mathbf{k})\\,\n\\hat{\\xi}^\\dag_\\mu(\\mathbf{k})\\hat{\\xi}_\\mu(\\mathbf{k})+E_0\\,.\n\\label{Htotdiag}\n\\end{equation}\nHere, the new operator\n\\begin{eqnarray}\n\\hat{\\xi}_\\mu(\\mathbf{k})=\n\\sum_{f}\\left[u_\\mu^\\ast(\\mathbf{k},\\omega_f)B_{\\mathbf{k},f}\n-v_\\mu(\\mathbf{k},\\omega_f)B_{-\\mathbf{k},f}^\\dag\\right]\\label{xik}\\\\\n+\\int_0^\\infty\\!\\!\\!\\!\\!d\\omega\\left[u_\\mu(\\mathbf{k},\\omega)\\hat{f}(\\mathbf{k},\\omega)-\nv_\\mu^\\ast(\\mathbf{k},\\omega)\\hat{f}^\\dag(-\\mathbf{k},\\omega)\\right]\\nonumber\n\\end{eqnarray}\nannihilates and\n$\\hat{\\xi}^\\dag_\\mu(\\mathbf{k})\\!=\\![\\hat{\\xi}_\\mu(\\mathbf{k})]^\\dag$\ncreates the exciton-plas\\-mon excitation of branch $\\mu$, the\nquantities $u_\\mu$ and $v_\\mu$ are appropriately chosen canonical\ntransformation coefficients. The \"vacuum\" energy $E_0$ represents\nthe state with no exciton-plasmons excited in the system, and\n$\\hbar\\omega_\\mu(\\mathbf{k})$ is the exciton-plasmon energy given\nby the solution of the following (dimensionless) dispersion\nrelation\n\\begin{equation}\nx_\\mu^2-\\varepsilon_f^2-\\varepsilon_f\\frac{2}{\\pi}\\int_0^\\infty\\!\\!\\!\\!\\!dx\\,\n\\frac{x\\,\\bar\\Gamma_0^f(x)\\rho(x)}{x_\\mu^2-x^2}=0\\,.\\label{dispeq}\n\\end{equation}\nHere,\n\\begin{equation}\nx=\\frac{\\hbar\\omega}{2\\gamma_0},~~~\nx_\\mu=\\frac{\\hbar\\omega_\\mu(\\mathbf{k})}{2\\gamma_0},~~~\n\\varepsilon_f=\\frac{E_f(\\mathbf{k})}{2\\gamma_0} \\label{dimless}\n\\end{equation}\nwith $\\gamma_0\\!=\\!2.7$~eV being the carbon nearest neighbor\noverlap integral entering the CN surface axial conductivity\n$\\sigma_{zz}(R_{CN},\\omega)$. The function\n\\begin{equation}\n\\bar\\Gamma_0^f(x)=\\frac{4|d^f_z|^2x^3}{3\\hbar c^3}\n\\left(\\frac{2\\gamma_0}{\\hbar}\\right)^{\\!2} \\label{Gamma0f}\n\\end{equation}\nwith\n$d^f_z\\!=\\!\\sum_{\\mathbf{n}}\\langle0|(\\hat{\\mathbf{d}}_\\mathbf{n})_z|f\\rangle$\nrepresents the (dimensionless) spontaneous decay rate, and\n\\begin{equation}\n\\rho(x)=\\frac{3S_0}{16\\pi\\alpha\nR_{CN}^2}\\;\\mbox{Re}\\frac{1}{\\bar\\sigma_{zz}(x)}\\label{plDOS}\n\\end{equation}\nstands for the surface plasmon density of states (DOS) which is\nresponsible for the exciton decay rate variation due to its\ncoupling to the plasmon modes.~Here, $\\alpha\\!=\\!e^2\/\\hbar\nc\\!=\\!1\/137$ is the fine-structure constant and\n$\\bar\\sigma_{zz}\\!=\\!2\\pi\\hbar\\sigma_{zz}\/e^2$ is the\ndimensionless CN surface axial conductivity per unit length.\n\nNote that the conductivity factor in Eq.~(\\ref{plDOS}) equals\n\\begin{equation}\n\\mbox{Re}\\frac{1}{\\bar\\sigma_{zz}(x)}=-\\frac{4\\alpha\nc}{R_{CN}}\\left(\\frac{\\hbar}{2\\gamma_0x}\\right)\\mbox{Im}\\frac{1}{\\epsilon_{zz}(x)-1}\n\\label{Reoneoversigma}\n\\end{equation}\nin view of Eq.~(\\ref{dimless}) and equation\n\\begin{equation}\n\\sigma_{zz}(x)=-\\frac{i\\omega}{4\\pi\nS\\rho_{T}}\\,[\\epsilon_{zz}(x)-1] \\label{DrudeGauss}\n\\end{equation}\nrepresenting the Drude relation for CNs, where $\\epsilon_{zz}$ is\nthe longitudinal (along the CN axis) dielectric function, $S$ and\n$\\rho_{T}$ are the surface area of the tubule and the number of\ntubules per unit volume,\nrespectively~\\cite{Bondarev04,Bondarev05,Tasaki}.~This relates\nvery closely the surface plasmon DOS function~(\\ref{plDOS}) to the\nloss function $-\\mbox{Im}(1\/\\epsilon)$ measured in Electron Energy\nLoss Spectroscopy (EELS) experiments to determine the properties\nof collective electronic excitations in solids~\\cite{Pichler98}.\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig3.eps}}\\caption{(a),(b)~Calculated\ndimensionless (see text) axial surface conductivities for the\n(11,0) and (10,0) CNs. The dimensionless energy is defined as\n[\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}).}\\label{fig3}\n\\end{figure}\n\nFigure~\\ref{fig3} shows the low-energy behaviors of\n$\\bar\\sigma_{zz}(x)$ and $\\mbox{Re}[1\/\\bar\\sigma_{zz}(x)]$ for the\n(11,0) and (10,0) CNs ($R_{CN}=0.43$~nm and $0.39$~nm,\nrespectively) we study here. We obtained them numerically as\nfollows. First, we adapt the nearest-neighbor non-orthogonal\ntight-binding approach~\\cite{Valentin} to determine the realistic\nband structure of each CN. Then, the room-temperature longitudinal\ndielectric functions $\\epsilon_{zz}$ are calculated within the\nrandom-phase approximation~\\cite{LinShung97,EhrenreichCohen59},\nwhich are then converted into the conductivities $\\bar\\sigma_{zz}$\nby means of the Drude relation. Electronic dissipation processes\nare included in our calculations within the relaxation-time\napproximation (electron scattering length of $130R_{CN}$ was\nused~\\cite{Lazzeri05}). We did not include excitonic many-electron\ncorrelations, however, as they mostly affect the real conductivity\n$\\mbox{Re}(\\bar\\sigma_{zz})$ which is responsible for the CN\noptical absorption~\\cite{Pedersen04,Spataru04,Ando}, whereas we\nare interested here in $\\mbox{Re}(1\/\\bar\\sigma_{zz})$ representing\nthe surface plasmon DOS according to Eq.~(\\ref{plDOS}). This\nfunction is only non-zero when the two conditions,\n$\\mbox{Im}[\\bar\\sigma_{zz}(x)]=0$ and\n$\\mbox{Re}[\\bar\\sigma_{zz}(x)]\\rightarrow0$, are fulfilled\nsimultaneously~\\cite{Kempa02,Kempa89,LinShung97}. These result in\nthe peak structure of the function $\\mbox{Re}(1\/\\bar\\sigma_{zz})$\nas is seen in Fig.~\\ref{fig3}. It~is~also seen from the comparison\nof Fig.~\\ref{fig3}~(b) with Fig.~\\ref{fig3}~(a) that the peaks\nbroaden as the CN diameter decreases. This is consistent with the\nstronger hybridization effects in smaller-diameter\nCNs~\\cite{Blase94}.\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig4.eps}}\\caption{(a),(b)~Surface\nplasmon DOS and conductivities (left panels), and lowest bright\nexciton dispersion when coupled to plasmons (right panels) in\n(11,0) and (10,0) CNs, respectively. The dimensionless energy is\ndefined as [\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}). See text for the dimensionless\nquasi-momentum.}\\label{fig4}\n\\end{figure}\n\nLeft panels in Figs.~\\ref{fig4}(a) and \\ref{fig4}(b) show the\nlowest-energy plasmon DOS resonances calculated for the (11,0) and\n(10,0) CNs as given by the function $\\rho(x)$ in\nEq.~(\\ref{plDOS}). Also shown there are the corresponding\nfragments of the functions $\\mbox{Re}[\\bar\\sigma_{zz}(x)]$ and\n$\\mbox{Im}[\\bar\\sigma_{zz}(x)]$. In all graphs the lower\ndimensionless energy limits are set up to be equal to the lowest\nbright exciton excitation energy [$E^{(11)}_{exc}=1.21$~eV\n($x=0.224$) and $1.00$~eV ($x=0.185$) for the (11,0) and (10,0)\nCN, respectively, as reported in Ref.\\cite{Spataru05} by directly\nsolving the Bethe-Salpeter equation]. Peaks in $\\rho(x)$ are seen\nto coincide in energy with zeros of\n$\\mbox{Im}[\\bar\\sigma_{zz}(x)]$ \\{or zeros of\n$\\mbox{Re}[\\epsilon_{zz}(x)]$\\}, clearly indicating the plasmonic\nnature of the CN surface excitations under\nconsideration~\\cite{Kempa02,Kempa}. They describe the surface\nplasmon modes associated with the transversely quantized interband\nelectronic transitions in CNs~\\cite{Kempa02}. As is seen in\nFig.~\\ref{fig4} (and in Fig.~\\ref{fig3}), the interband plasmon\nexcitations occur in CNs slightly above the first bright exciton\nexcitation energy~\\cite{Ando}, in the frequency domain where the\nimaginary conductivity (or the real dielectric function) changes\nits sign. This is a unique feature of the complex dielectric\nresponse function, the consequence of the general\nKramers-Kr\\\"{o}nig relation~\\cite{VogelWelsch}.\n\nWe further take advantage of the sharp peak structure of $\\rho(x)$\nand solve the dispersion equation~(\\ref{dispeq}) for $x_\\mu$\nanalytically using the Lorentzian approximation\n\\begin{equation}\n\\rho(x)\\!\\approx\\!\\frac{\\rho(x_p)\\Delta x_{p}^2}\n{(x-x_{p})^2+\\Delta x_{p}^2}\\,. \\label{rhox}\n\\end{equation}\nHere, $x_{p}$ and $\\Delta x_{p}$ are, respectively, the position\nand the half-width-at-half-maximum of the plasmon resonance\nclosest to the lowest bright exciton excitation energy in the same\nnanotube (as shown in the left panels of Fig.~\\ref{fig4}). The\nintegral in Eq.~(\\ref{dispeq}) then simplifies to the form\n\\[\n\\frac{2}{\\pi}\\int_0^\\infty\\!\\!\\!\\!\\!dx\\,\\frac{x\\,\\bar\\Gamma_0^f(x)\\rho(x)}{x_\\mu^2-x^2}\n\\approx\\frac{F(x_p)\\Delta x_{p}^2}{x_\\mu^2-x_p^2}\\!\n\\int_0^\\infty\\!\\!\\!\\!\\!\\frac{dx}{(x-x_{p})^2+\\Delta x_{p}^2}\n\\]\\vspace{-0.25cm}\n\\[\n=\\frac{F(x_p)\\Delta x_{p}}{x_\\mu^2-x_p^2}\n\\left[\\arctan\\!\\left(\\frac{x_p}{\\Delta x_p}\\right)+\n\\frac{\\pi}{2}\\right]\n\\]\nwith $F(x_p)=2x_p\\bar\\Gamma_0^f(x_p)\\rho(x_p)\/\\pi$. This\nexpression is valid for all $x_\\mu$ apart from those located in\nthe narrow interval $(x_p-\\Delta x_p,x_p+\\Delta x_p)$ in the\nvicinity of the plasmon resonance, provided that the resonance is\nsharp enough. Then, the dispersion equation becomes the\nbiquadratic equation for $x_\\mu$ with the following two positive\nsolutions (the dispersion curves) of interest to us\n\n\\begin{equation}\nx_{1,2}=\\sqrt{\\frac{\\varepsilon_f^2+x_{p}^2}{2}\\pm\\frac{1}{2}\n\\sqrt{(\\varepsilon_f^2\\!-x_{p}^2)^2+F_{\\!p}\\,\\varepsilon_f}}\\,.\n\\label{dispsol}\n\\end{equation}\nHere, $F_{\\!p}=4F(x_p)\\Delta x_p(\\pi-\\Delta x_{p}\/x_{p})$ with the\n$\\arctan$-function expanded to linear terms in $\\Delta\nx_p\/x_p\\ll1$.\n\nThe dispersion curves (\\ref{dispsol}) are shown in the right\npanels in Figs.~\\ref{fig4}(a) and \\ref{fig4}(b) as functions of\nthe dimensionless longitudinal quasi-momentum.~In these\ncalculations, we estimated the interband transition matrix element\nin $\\bar\\Gamma_0^f(x_p)$ [Eq.(\\ref{Gamma0f})] from the equation\n$|d_f|^2=3\\hbar\\lambda^3\/4\\tau_{ex}^{rad}$ according to Hanamura's\ngeneral theory of the exciton radiative decay in spatially\nconfined systems~\\cite{Hanamura}, where $\\tau_{ex}^{rad}$ is the\nexciton intrinsic radiative lifetime, and $\\lambda=2\\pi c\\hbar\/E$\nwith $E$ being the exciton total energy given in our case by\nEq.~(\\ref{Ef}).~For zigzag-type CNs considered here, the first\nBrillouin zone of the longitudinal quasi-momentum is given by\n$-2\\pi\\hbar\/3b\\le\\hbar k_z\\le2\\pi\\hbar\/3b$~\\cite{Dresselhaus,Dai}.\nThe total energy of the ground-internal-state exciton can then be\nwritten as $E=E_{exc}+(2\\pi\\hbar\/3b)^2t^2\/2M_{ex}$ with $-1\\le\nt\\le1$ representing the dimensionless longitudinal quasi-momentum.\nIn our calculations we used the lowest bright exciton parameters\n$E^{(11)}_{exc}=1.21$~eV and $1.00$~eV, $\\tau_{ex}^{rad}=14.3$~ps\nand $19.1$~ps, $M_{ex}=0.44m_0$ and $0.19m_0$ ($m_0$ is the\nfree-electron mass) for the (11,0) CN and (10,0) CN, respectively,\nas reported in Ref.\\cite{Spataru05} by directly solving the\nBethe-Salpeter equation.\n\nBoth graphs in the right panels in Fig.~\\ref{fig4} are seen to\ndemonstrate a clear anticrossing behavior with the (Rabi) energy\nsplitting $\\sim\\!0.1$~eV. This indicates the formation of the\nstrongly coupled surface plasmon-exciton excitations in the\nnanotubes under consideration. It is important to realize that\nhere we deal with the strong exciton-plasmon interaction supported\nby an individual quasi-1D nanostructure --- a single-walled\n(small-diameter) semiconducting carbon nanotube, as opposed to the\nartificially fabricated metal-semiconductor nanostructures studied\npreviuosly~\\cite{Bellessa,Govorov,Fedutik} where the metallic\ncomponent normally carries the plasmon and the semiconducting one\ncarries the exciton. It is also important that the effect comes\nnot only from the height but also from the width of the plasmon\nresonance as it is seen from the definition of the $F_p$ factor in\nEq.~(\\ref{dispsol}). In other words, as long as the plasmon\nresonance is sharp enough (which is always the case for interband\nplasmons), so that the Lorentzian approximation (\\ref{rhox})\napplies, the effect is determined by the area under the plasmon\npeak in the DOS function~(\\ref{plDOS}) rather than by the peak\nheight as one would expect.\n\nHowever, the formation of the strongly coupled exci\\-ton-plasmon\nstates is only possible if the exciton total energy is in\nresonance with the energy of a surface plasmon mode. The exciton\nenergy can be tuned to the nearest plasmon resonance in ways used\nfor excitons in semiconductor quantum microcavities\n--- thermally~\\cite{Reithmaier,Yoshie,Peter} (by elevating sample temperature),\nor\/and electrostatically~\\cite{MillerPRL,Miller,Zrenner,Krenner}\n(via the quantum confined Stark effect with an external\nelectrostatic field applied perpendicular to the CN axis). As is\nseen from Eqs.~(\\ref{Ef}) and (\\ref{Eexcf}), the two possibilities\ninfluence the different degrees of freedom of the quasi-1D exciton\n--- the (longitudinal) kinetic energy and the excitation energy,\nrespectively. Below we study the (less trivial) electrostatic\nfield effect on the exciton excitation energy in CNs.\n\n\\subsection{The perpendicular electrostatic field effect}\n\nThe optical properties of semiconducting CNs in an external\nelectrostatic field directed along the nanotube axis were studied\ntheoretically in Ref.~\\cite{Perebeinos07}. Strong oscillations in\nthe band-to-band absorption and the quadratic Stark shift of the\nexciton absorption peaks with the field increase, as well as the\nstrong field dependence of the exciton ionization rate, were\npredicted for CNs of different diameters and chiralities. Here, we\nfocus on the perpendicular electrostatic field orientation.~We\nstudy how the electrostatic field applied perpendicular to the CN\naxis affects the CN band gap, the exciton binding\/excitation\nenergy, and the interband surface plasmon energy, to explore the\ntunability of the strong exciton-plasmon coupling effect predicted\nabove.~The problem is similar to the well-known quantum confined\nStark effect first observed for the excitons in semiconductor\nquantum wells~\\cite{MillerPRL,Miller}. However, the cylindrical\nsurface symmetry of the excitonic states brings new peculiarities\nto the quantum confined Stark effect in CNs. In what follows we\nwill generally be interested only in the lowest internal energy\n(ground) excitonic state, and so the internal state index $f$ in\nEqs.~(\\ref{Ef}) and (\\ref{Eexcf}) will be omitted for brevity.\n\nBecause the nanotube is modelled by a continuous, infinitely thin,\nanisotropically conducting cylinder in our macroscopic QED\napproach, the actual local symmetry of the excitonic wave function\nresulted from the graphene Brillouin zone structure is disregarded\nin our model (see, e.g., reviews~\\cite{Dresselhaus07,Ando}). The\nlocal symmetry is implicitly present in the surface axial\nconductivity though, which we calculate beforehand as described\nabove.\\footnote{In real CNs, the existence of two equivalent\nenergy valleys in the 1st Brillouin zone, the $K$- and\n$K^{\\prime}$-valleys with opposite electron helicities about the\nCN axis, results into dark and bright excitonic states in the\nlowest energy spin-singlet manifold~\\cite{AndoDarkExciton}. Since\nthe electric interaction does not involve spin variables, both\n$K$- and $K^{\\prime}$-valleys are affected equally by the\nelectrostatic field in our case, and the detailed structure of the\nexciton wave function multiplet is not important. This is opposite\nto the non-zero magnetostatic field case where the field affects\nthe $K$- and $K^{\\prime}$-valleys differently either to brighten\nthe dark excitonic states~\\cite{Srivastava}, or to create Landau\nsublevels~\\cite{Ando} for longitudinal and perpendicular\norientation, respectively.}\\label{footnote1}\n\nWe start with the Schr\\\"{o}dinger equation for the electron and\nhole on the CN surface, located at\n$\\textbf{r}_{e}=\\{R_{CN},\\varphi_{e},z_{e}\\}$ and\n$\\textbf{r}_{h}=\\{R_{CN},\\varphi_{h},z_{h}\\}$, respectively. They\ninteract with each other through the Coulomb potential\n$V(\\textbf{r}_e,\\textbf{r}_h)=-e^2\/\\epsilon|\\textbf{r}_e\\!-\\textbf{r}_h|$,\nwhere $\\epsilon=\\epsilon_{zz}(0)$.~The external electrostatic\nfield $\\textbf{F}=\\{F,0,0\\}$ is directed perpendicular to the CN\naxis (along the $x$-axis in Fig.~\\ref{fig1}). The Schr\\\"{o}dinger\nequation is of the form\n\\begin{equation}\n\\left[\\hat{H}_{e}(\\textbf{F})+\\hat{H}_{h}(\\textbf{F})+\nV(\\textbf{r}_e,\\textbf{r}_h)\\right]\\!\n\\Psi(\\textbf{r}_e,\\textbf{r}_h)=E\\Psi(\\textbf{r}_e,\\textbf{r}_h)\n\\label{Schroedinger}\n\\end{equation}\nwith\n\\begin{equation}\n\\hat{H}_{e,h}(\\textbf{F})=-\\frac{\\hbar^2}{2m_{e,h}}\\left(\\!\n\\frac{1}{R_{CN}^2}\\frac{\\partial^2}{\\partial\\varphi^2_{e,h}}+\n\\frac{\\partial^2}{\\partial z^2_{e,h}}\\right)\\!\\mp\ne\\textbf{r}_{e,h}\\!\\cdot\\textbf{F}\\label{Heh}\n\\end{equation}\n\nWe further separate out the translational and relative degrees of\nfreedom of the electron-hole pair by transforming the longitudinal\n(along the CN axis) motion of the pair into its center-of-mass\ncoordinates given by $Z=(m_ez_e+m_hz_h)\/M_{ex}$ and $z=z_e-z_h$.\nThe exciton wave function is approximated as follows\n\\begin{equation}\n\\Psi(\\textbf{r}_e,\\textbf{r}_h)=e^{ik_zZ}\\phi_{ex}(z)\\psi_e(\\varphi_e)\\psi_h(\\varphi_h).\n\\label{wfunceh}\n\\end{equation}\nThe complex exponential describes the exciton center-of-mass\nmotion with the longitudinal quasi-momentum $k_z$ along the CN\naxis. The function $\\phi_{ex}(z)$ represents the longitudinal\nrelative motion of the electron and the hole inside the exciton.\nThe functions $\\psi_e(\\varphi_e)$ and $\\psi_h(\\varphi_h)$ are the\nelectron and hole subband wave functions, respectively, which\nrepresent their confined motion along the circumference of the\ncylindrical nanotube surface.\n\nEach of the functions is assumed to be normalized to unity.\nEquations~(\\ref{Schroedinger}) and (\\ref{Heh}) are then rewritten\nin view of Eqs.~(\\ref{Ef})--(\\ref{Egkfi}) to yield\n\\begin{equation}\n\\left[-\\frac{\\hbar^2}{2m_eR_{CN}^2}\\frac{\\partial^2}{\\partial\\varphi^2_e}-\neR_{CN}F\\cos(\\varphi_e)\\right]\\!\\psi_e(\\varphi_e)=\\varepsilon_{e}\\psi_e(\\varphi_e),\n\\label{wfe}\n\\end{equation}\n\\begin{equation}\n\\left[-\\frac{\\hbar^2}{2m_hR_{CN}^2}\\frac{\\partial^2}{\\partial\\varphi^2_h}+\neR_{CN}F\\cos(\\varphi_h)\\right]\\!\\psi_h(\\varphi_h)\\!=\\!\\varepsilon_h\\psi_h(\\varphi_h)\\!,\n\\label{wfh}\n\\end{equation}\n\\begin{equation}\n\\left[-\\frac{\\hbar^2}{2\\mu}\\frac{\\partial^2}{\\partial\\,\\!z^2}+\nV_{\\mbox{\\small{eff}}}(z)\\right]\\!\\phi_{ex}(z)=E_b\\phi_{ex}(z),\n\\label{wfexc}\n\\end{equation}\nwhere $\\mu=m_em_h\/M_{ex}$ is the exciton reduced mass, and\n$V_{\\mbox{\\small eff}}$ is the effective longitudinal\nelectron-hole Coulomb interaction potential given by\n\\begin{equation}\nV_{\\mbox{\\small\neff}}(z)=\\!-\\frac{e^2}{\\epsilon}\\!\\int_0^{2\\pi}\\!\\!\\!\\!\\!\\!d\\varphi_e\\!\\!\\int_0^{2\\pi}\\!\\!\\!\\!\\!\\!\nd\\varphi_h|\\psi_e(\\varphi_e)|^2|\\psi_h(\\varphi_h)|^2V(\\varphi_e,\\varphi_h,z)\n\\label{Veff}\n\\end{equation}\nwith $V$ being the original electron-hole Coulomb potential\nwritten in the cylindrical coordinates as\n\\begin{equation} V(\\varphi_e,\\varphi_h,z)=\\!\n\\frac{1}{\\{z^2+4R_{CN}^2\\sin^2[(\\varphi_{e}\\!-\\varphi_{h})\/2]\\}^{1\/2}}.\n\\label{V}\n\\end{equation}\nThe exciton problem is now reduced to the 1D equation\n(\\ref{wfexc}), where the exciton binding energy does depend on the\nperpendicular electrostatic field through the electron and hole\nsubband functions $\\psi_{e,h}$ given by the solutions of\nEqs.~(\\ref{wfe}) and (\\ref{wfh}) and entering the effective\nelectron-hole Coulomb interaction potential (\\ref{Veff}).\n\nThe set of Eqs.~(\\ref{wfe})-(\\ref{V}) is analyzed in Appendix~D.\nOne of the main results obtained in there is that the effective\nCoulomb potential (\\ref{Veff}) can be approximated by an\nattractive cusp-type cutoff potential of the form\n\\begin{equation}\nV_{\\mbox{\\small\neff}}(z)\\approx-\\frac{e^2}{\\epsilon[|z|+z_0(j,F)]},\n\\label{Vcutoff}\n\\end{equation}\nwhere the cutoff parameter $z_0$ depends on the perpendicular\nelectrostatic field strength and on the electron-hole azimuthal\ntransverse quantization index $j=1,2,...$ (excitons are created in\ninterband transitions involving valence and conduction subbands\nwith the same quantization index~\\cite{Ando} as shown in\nFig.~\\ref{fig2}). Specifically,\n\\begin{equation}\nz_0(j,F)\\approx2R_{CN}\\frac{\\pi-2\\ln2\\,[1-\\Delta_j(F)]}{\\pi+2\\ln2\\,[1-\\Delta_j(F)]}\n\\label{z0}\n\\end{equation}\nwith $\\Delta_j(F)$ given to the second order approximation in the\nelectric field by\n\\begin{eqnarray}\n\\Delta_j(F)&\\!\\!\\approx\\!\\!&2\\mu M_{ex}\\frac{e^2R_{CN}^6w^2_j}{\\hbar^4}\\,F^2,\\label{DeltaF}\\\\\nw_j&\\!\\!=\\!\\!&\\frac{\\theta(j\\!-\\!2)}{1-2j}+\\frac{1}{1+2j},\\nonumber\n\\end{eqnarray}\nwhere $\\theta(x)$ is the unit step function. Approximation\n(\\ref{Vcutoff}) is formally valid when $z_0(j,F)$ is much less\nthan the exciton Bohr radius $a_B^\\ast$ $(=\\epsilon\\hbar^2\/\\mu\ne^2)$ which is estimated to be $\\sim\\!10R_{CN}$ for the first\n($j\\!=\\!1$ in our notations here) exciton in\nCNs~\\cite{Pedersen03}. As is seen from Eqs.~(\\ref{z0}) and\n(\\ref{DeltaF}), this is always the case for the first exciton for\nthose fields where the perturbation theory applies, i.~e. when\n$\\Delta_1(F)<1$ in Eq.~(\\ref{DeltaF}).\n\nEquation~(\\ref{wfexc}) with the potential (\\ref{Vcutoff}) formally\ncoincides with the one studied by Ogawa and Takagahara in their\ntreatments of excitonic effects in 1D semiconductors with no\nexternal electrostatic field applied~\\cite{Takagahara}.~The only\ndifference in our case is that our cutoff parameter (\\ref{z0}) is\nfield dependent. We therefore follow Ref.~\\cite{Takagahara} and\nfind the ground-state binding energy $E^{(11)}_b$ for the first\nexciton we are interested in here from the transcendental equation\n\\begin{equation}\n\\ln\\!\\left[\\frac{2z_0(1,F)}{\\hbar}\\sqrt{2\\mu|E^{(11)}_b|}\\,\\right]\n+\\frac{1}{2}\\sqrt{\\frac{|E^{(11)}_b|}{Ry^\\ast}}=0. \\label{Ebnum}\n\\end{equation}\nIn doing so, we first find the exciton Rydberg energy,~$Ry^\\ast$\n$(=\\!\\mu e^4\/2\\hbar^2\\epsilon^2)$, from this equation at\n$F\\!=\\!0$. We use the diameter- and chirality-dependent electron\nand hole effective masses from Ref.~\\cite{Jorio}, and the first\nbright exciton binding energy of 0.76~eV for both (11,0) and\n(10,0) CN as reported in Ref.~\\cite{Capaz} from \\emph{ab initio}\ncalculations. We obtain $Ry^\\ast=4.02$~eV and $0.57$~eV for\nthe~(11,0) tube and (10,0) tube, respectively. The difference of\nabout one order of magnitude reflects the fact that these are the\nsemiconducting CNs of different types --- type-I and type-II,\nrespectively, based on $(2n+m)$ families~\\cite{Jorio}. The\nparameters $Ry^\\ast$ thus obtained are then used to find\n$|E^{(11)}_b|$ as functions of $F$ by numerically solving\nEq.~(\\ref{Ebnum}) with $z_0(1,F)$ given by Eqs.~(\\ref{z0}) and\n(\\ref{DeltaF}).\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig5.eps}}\\caption{(a)~Calculated\nbinding energies of the first bright exciton in the (11,0) and\n(10,0) CNs as functions of the perpendicular electrostatic field\napplied.~Solid lines are the numerical solutions to\nEq.~(\\ref{Ebnum}), dashed lines are the quadratic approximations\nas given by Eq.~(\\ref{Ebapprox}). (b)~Field dependence of the\neffective cutoff Coulomb potential (\\ref{Vcutoff}) in the (11,0)\nCN. The dimensionless energy is defined as\n[\\emph{Energy}]\/$2\\gamma_0$, according to Eq.~(\\ref{dimless}).}\n\\label{fig5}\n\\end{figure}\n\nThe calculated (negative) binding energies are shown in\nFig.~\\ref{fig5}(a) by the solid lines. Also shown there by dashed\nlines are the functions\n\\begin{equation}\nE^{(11)}_b(F)\\approx E^{(11)}_b[1-\\Delta_1(F)] \\label{Ebapprox}\n\\end{equation}\nwith $\\Delta_1(F)$ given by Eq.~(\\ref{DeltaF}). They are seen to\nbe fairly good analytical (quadratic in field) approximations to\nthe numerical solutions of Eq.~(\\ref{Ebnum}) in the range of not\ntoo large fields. The exciton binding energy decreases very\nrapidly in its absolute value as the field increases. Fields of\nonly $\\sim\\!0.1-0.2$~V\/$\\mu$m are required to decrease\n$|E^{(11)}_b|$ by a factor of $\\sim\\!2$ for the CNs considered\nhere. The reason is the perpendicular field shifts up the \"bottom\"\nof the effective potential (\\ref{Vcutoff}) as shown in\nFig.~\\ref{fig5}(b) for the (11,0) CN. This makes the potential\nshallower and pushes bound excitonic levels up, thereby decreasing\nthe exciton binding energy in its absolute value. As this takes\nplace, the shape of the potential does not change, and the\nlongitudinal relative electron-hole motion remains finite at all\ntimes.~As a consequence, no tunnel exciton ionization occurs in\nthe perpendicular field, as opposed to the longitudinal\nelectrostatic field (Franz-Keldysh) effect studied in\nRef.~\\cite{Perebeinos07} where the non-zero field creates the\npotential barrier separating out the regions of finite and\ninfinite relative motion and the exciton becomes ionized as the\nelectron tunnels to infinity.\n\nThe binding energy is only the part of the exciton excitation\nenergy (\\ref{Eexcf}). Another part comes from the band gap energy\n(\\ref{Egkfi}), where $\\varepsilon_{e}$ and $\\varepsilon_{h}$ are\ngiven by the solutions of Eqs.~(\\ref{wfe}) and (\\ref{wfh}),\nrespectively. Solving them to the leading (second) order\nperturbation theory approximation in the field (Appendix~D), one\nobtains\n\\begin{equation}\nE^{(jj)}_g(F)\\approx\nE^{(jj)}_g\\!\\left[1-\\frac{m_e\\Delta_j(F)}{2M_{ex}j^2w_j}-\\frac{m_h\\Delta_j(F)}{2M_{ex}j^2w_j}\\right]\\!,\n\\label{EgF}\n\\end{equation}\nwhere the electron and hole subband shifts are written separately.\nThis, in view of Eq.~(\\ref{DeltaF}), yields the first band gap\nfield dependence in the form\n\\begin{equation}\nE^{(11)}_g(F)\\approx\nE^{(11)}_g\\!\\left[1-\\frac{3}{2}\\,\\Delta_1(F)\\right]\\!,\n\\label{EgF11}\n\\end{equation}\nThe bang gap decrease with the field in Eq.~(\\ref{EgF11}) is\nstronger than the opposite effect in the negative exciton binding\nenergy given (to the same order approximation in field) by\nEq.~(\\ref{Ebapprox}). Thus, the first exciton excitation energy\n(\\ref{Eexcf}) will be gradually decreasing as the perpendicular\nfield increases, shifting the exciton absorption peak to the red.\nThis is the basic feature of the quantum confined Stark effect\nobserved previously in semiconductor\nnanomaterials~\\cite{MillerPRL,Miller,Zrenner,Krenner}. The field\ndependences of the higher interband transitions exciton excitation\nenergies are suppressed by the rapidly (quadratically) increasing\nazimuthal quantization numbers in the denominators of\nEqs.~(\\ref{DeltaF}) and (\\ref{EgF}).\n\nLastly, the perpendicular field dependence of the interband\nplasmon resonances can be obtained from the frequency dependence\nof the axial surface conductivity due to excitons (see\nRef.~\\cite{Ando} and refs. therein). One has\n\\begin{equation}\n\\sigma^{ex}_{zz}(\\omega)\\sim\\!\\!\\!\\sum_{j=1,2,...}\\!\\frac{-i\\hbar\\omega\nf_j}{[E^{(jj)}_{exc}]^2\\!-(\\hbar\\omega)^2\\!-2i\\hbar^2\\omega\/\\tau},\n\\label{Ando}\n\\end{equation}\nwhere $f_j$ and $\\tau$ are the exciton oscillator strength and\nrelaxation time, respectively. The plasmon frequencies are those\nat which the function $\\mbox{Re}[1\/\\sigma^{ex}_{zz}(\\omega)]$ has\nmaxima. Testing it for maximum in the domain\n$E^{(11)}_{exc}\\!\\!<\\!\\hbar\\omega<\\!E^{(22)}_{exc}$, one finds the\nfirst interband plasmon resonance energy to be (in the limit\n$\\tau\\!\\rightarrow\\!\\infty$)\n\\begin{equation}\nE^{(11)}_p=\\sqrt{\\frac{[E^{(11)}_{exc}]^2+[E^{(22)}_{exc}]^2}{2}}\\,.\n\\label{Epl}\n\\end{equation}\nUsing the field dependent $E^{(11)}_{exc}$ given by\nEqs.~(\\ref{Eexcf}),~(\\ref{Ebapprox}) and (\\ref{EgF11}), and\nneglecting the field dependence of $E^{(22)}_{exc}$, one obtains\nto the second order approximation in the field\n\\begin{equation}\nE^{(11)}_p(F)\\approx\nE^{(11)}_p\\!\\left[1-\\frac{1+\\!E^{(11)}_{g}\\!\/2E^{(11)}_{exc}}\n{1+\\!E^{(22)}_{exc}\\!\/E^{(11)}_{exc}}\\;\\Delta_1(F)\\right]\\!.\n\\label{EplF}\n\\end{equation}\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig6.eps}}\\vskip-0.3cm\\caption{(a),(b)~Calculated\ndependences of the first bright exciton parameters in the (11,0)\nand (10,0) CNs, respectively, on the electrostatic field applied\nperpendicular to the nanotube axis. The dimensionless energy is\ndefined as [\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}). The energy is measured from the top of the\nfirst unperturbed hole subband.} \\label{fig6}\n\\end{figure}\n\nFigure~\\ref{fig6} shows the results of our calculations of the\nfield dependences for the first bright exciton parameters in the\n(11,0) and (10,0) CNs. The energy is measured from the top of the\nfirst unperturbed hole subband (as shown in Fig.~\\ref{fig2}, right\npanel). The binding energy field dependence was calculated\nnumerically from Eq.~(\\ref{Ebnum}) as described above [shown in\nFig.~\\ref{fig5}~(a)]. The band gap field dependence and the\nplasmon energy field dependence were calculated from\nEqs.~(\\ref{EgF}) and (\\ref{EplF}), respectively. The zero-field\nexcitation energies and zero-field binding energies were taken to\nbe those reported in Ref.~\\cite{Spataru05} and in\nRef.~\\cite{Capaz}, respectively, and we used the diameter- and\nchirality-dependent electron and hole effective masses from\nRef.~\\cite{Jorio}. As is seen in Fig.~\\ref{fig6}~(a)~and (b), the\nexciton excitation energy and the interband plasmon energy\nexperience red shift in both nanotubes as the field increases.\nHowever, the excitation energy red shift is very small (barely\nseen in the figures) due to the negative field dependent\ncontribution from the exciton binding energy.~So,\n$E^{(11)}_{exc}(F)$ and $E^{(11)}_{p}(F)$ approach each other as\nthe field increases, thereby bringing the total exciton energy\n(\\ref{Ef}) in resonance with the surface plasmon mode due to the\nnon-zero longitudinal kinetic energy term at finite\ntemperature.\\footnote{We are based on the zero-exciton-temperature\napproximation in here~\\cite{Suna}, which is well justified because\nof the exciton excitation energies much larger than $k_BT$ in~CNs.\nThe exciton Hamiltonian (\\ref{Hex}) does not require the thermal\naveraging over the exciton degrees of freedom then, yielding the\ntemperature independent total exciton energy (\\ref{Ef}). One has\nto keep in mind, however, that the exciton excitation energy can\nbe affected by the enviromental effect not under consideration in\nhere (see Ref.~\\cite{Miyauchi}).} Thus, the electrostatic field\napplied perpendicular to the CN axis (the quantum confined Stark\neffect) may be used to tune the exciton energy to the nearest\ninterband plasmon resonance, to put the exciton-surface plasmon\ninteraction in small-diameter semiconducting CNs to the\nstrong-coupling regime.\n\n\\subsection{The optical absorption}\n\nHere, we analyze the longitudinal exciton absorption line shape as\nits energy is tuned to the nearest interband surface plasmon\nresonance. Only longitudinal excitons (excited by light polarized\nalong the CN axis) couple to the surface plasmon modes as\ndiscussed at the very beginning of this section (see\nRef.~\\cite{UryuAndo} for the perpendicular light exciton\nabsorption in CNs). We start with the linear (weak) excitation\nregime where only single-exciton states are excited, and follow\nthe optical absorption\/emission lineshape theory developed\nrecently for atomically doped CNs~\\cite{Bondarev06}. (Obviously,\nthe absorption line shape coincides with the emission line shape\nif the monochromatic incident light beam is used in the absorption\nexperiment.) Then, the non-linear (strong) excitation regime is\nconsidered with the photonduced excitation of biexciton states.\n\nWhen the $f$-internal state exciton is excited and the nanotube's\nsurface EM field subsystem is in vacuum state, the time-dependent\nwave function of the whole system \"exciton+field\" is of the\nform\\footnote{See the footnote on page~\\pageref{footnote1} above.}\n\\begin{eqnarray}\n|\\psi(t)\\rangle&=&\\sum_{{\\bf\nk},f}C_{f}(\\textbf{k},t)\\,e^{-i\\tilde{E}_{f}({\\bf k})t\/\\hbar}\n|\\{1_f(\\textbf{k})\\}\\rangle_{ex}|\\{0\\}\\rangle\\label{wfunc}\\\\\n&+&\\sum_{\\bf k}\\int_{0}^{\\infty}\\!\\!\\!\\!\\!\\!d\\omega\\,\nC(\\textbf{k},\\omega,t)\\,e^{-i\\omega t}\n|\\{0\\}\\rangle_{ex}|\\{1(\\mathbf{k},\\omega)\\}\\rangle.\\nonumber\n\\end{eqnarray}\nHere, $|\\{1_f(\\textbf{k})\\}\\rangle_{ex}$ is the excited\nsingle-quantum Fock state with one exciton and\n$|\\{1(\\textbf{k},\\omega)\\}\\rangle$ is that with one surface\nphoton. The vacuum states are $|\\{0\\}\\rangle_{ex}$ and\n$|\\{0\\}\\rangle$ for the exciton subsystem and field subsystem,\nrespectively. The coefficients $C_{f}(\\textbf{k},t)$ and\n$C(\\textbf{k},\\omega,t)$ stand for the population probability\namplitudes of the respective states of the whole system. The\nexciton energy is of the form\n$\\tilde{E}_{f}(\\textbf{k})\\!=\\!E_f(\\textbf{k})\\!-i\\hbar\/\\tau$ with\n$E_f(\\textbf{k})$ given by Eq.~(\\ref{Ef}) and $\\tau$ being the\nphenomenological exciton relaxation time constant [assumed to be\nsuch that $\\hbar\/\\tau\\!\\ll\\!E_{f}(\\textbf{k})$] to account for\nother possible exciton relaxation processes. From the literature\nwe have $\\tau_{ph}\\sim30\\!-\\!100$~fs for the exciton-phonon\nscattering~\\cite{Perebeinos07}, $\\tau_d\\sim50$~ps for the exciton\nscattering by defects~\\cite{Hagen05,Prezhdo08}, and\n$\\tau_{rad}\\sim10\\,\\mbox{ps}-10\\,\\mbox{ns}$ for the radiative\ndecay of excitons~\\cite{Spataru05}. Thus, the scattering by\nphonons is the most likely exciton relaxation mechanism.\n\nUsing Eqs.(\\ref{Bkf}) and (\\ref{orthog}), we transform the total\nHamiltonian~(\\ref{Htot})--(\\ref{Hint}) to the\n$\\mathbf{k}$-representation (see Appendix~A), and apply it to the\nwave function in Eq.~(\\ref{wfunc}). We obtain the following set of\nthe two simultaneous differential equations for the coefficients\n$C_{f}(\\textbf{k},t)$ and $C(\\textbf{k},\\omega,t)$ from the time\ndependent Schr\\\"{o}dinger equation\n\\begin{equation}\n\\mbox{\\it\\.C}_{f}(\\textbf{k},t)\\,e^{-i\\tilde{E}_{f}({\\bf\\!\\,k})t\/\\hbar}=\n-\\frac{i}{\\hbar}\\sum_{{\\bf\\!\\,k^\\prime}}\\int_{0}^{\\infty}\\!\\!\\!\\!\\!d\\omega\\,\n\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k}^\\prime\\!,\\omega)\\,\nC(\\textbf{k}^\\prime\\!,\\omega,t)\\,e^{-i\\omega\\!\\,t}, \\label{paexc}\n\\end{equation}\n\\begin{equation}\n\\mbox{\\it\\.C\\,}(\\textbf{k}^\\prime\\!,\\omega,t)\\,e^{-i\\omega\\!\\,t}\\delta_{{\\bf\\!\\,k}{\\bf\\!\\,k}^\\prime}=\n-\\frac{i}{\\hbar}\\sum_f\\,[\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k}^\\prime\\!,\\omega)]^\\ast\nC_f(\\textbf{k},t)\\,e^{-i\\tilde{E}_{f}({\\bf\\!\\,k})t\/\\hbar}.\\nonumber\n\\label{papho}\n\\end{equation}\nThe $\\delta$-symbol on the left in Eq.~(\\ref{papho}) ensures that\nthe momentum conservation is fulfilled in the exciton-photon\ntransitions, so that the annihilating exciton creates the surface\nphoton with the same momentum and vice versa. In terms of the\nprobability amplitudes above, the exciton emission intensity\ndistribution is given by the final state probability at very long\ntimes corresponding to the complete decay of all initially excited\nexcitons,\n\\begin{eqnarray}\nI(\\omega)\\!&=&\\!|C(\\mathbf{k},\\omega,t\\!\\rightarrow\\!\\infty)|^{2}=\n\\frac{1}{\\hbar^2}\\sum_f|\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k},\\omega)|^2\\nonumber\\\\\n&\\times&\\!\\left|\\int_0^\\infty\\!\\!\\!\\!\\!dt^\\prime\\,C_f(\\textbf{k},t^\\prime)\\,\ne^{-i[\\tilde{E}_{f}({\\bf\\!\\,k})-\\hbar\\omega]t^\\prime\\!\/\\hbar}\\right|^2\\!.\\label{Iomega}\n\\end{eqnarray}\nHere, the second equation is obtained by the formal integration of\nEq.~(\\ref{papho}) over time under the initial condition\n$C\\,(\\textbf{k},\\omega,0)\\!=\\!0$. The emission intensity\ndistribution is thus related to the exciton population probability\namplitude $C_f(\\mathbf{k},t)$ to be found from Eq.~(\\ref{paexc}).\n\nThe set of simultaneous equations (\\ref{paexc}) and (\\ref{papho})\n[and Eq.~(\\ref{Iomega}), respectively] contains no approximations\nexcept the (commonly used) neglect of many-particle excitations in\nthe wave function (\\ref{wfunc}).~We now apply these equations to\nthe exciton-surface-plasmon system in small-diameter\nsemiconducting CNs. The interaction matrix element~in\nEqs.~(\\ref{paexc}) and (\\ref{papho}) is then given by the\n$\\mathbf{k}$-transform of Eq.~(\\ref{gfpar}), and has the following\nproperty (Appendix~C)\n\\begin{equation}\n\\frac{1}{2\\gamma_0\\hbar}\\,|\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k},\\omega)|^2\n=\\frac{1}{2\\pi}\\,\\bar\\Gamma_0^f(x)\\rho(x) \\label{gpkk}\n\\end{equation}\nwith $\\bar\\Gamma_0^f(x)$ and $\\rho(x)$ given by\nEqs.~(\\ref{Gamma0f}) and (\\ref{plDOS}), respectively. We further\nsubstitute the result of the formal integration of\nEq.~(\\ref{papho}) [with $C\\,(\\textbf{k},\\omega,0)\\!=\\!0$] into\nEq.~(\\ref{paexc}), use Eq.~(\\ref{gpkk}) with $\\rho(x)$\napproximated by the Lorentzian~(\\ref{rhox}), calculate the\nintegral over frequency analytically, and differentiate the result\nover time to obtain the following second order ordinary\ndifferential equation for the exciton probability amplitude\n[dimensionless variables, Eq.~(\\ref{dimless})]\n\\[\n\\mbox{\\it\\\"{C}}_f(\\beta)+[\\Delta\nx_p-\\Delta\\varepsilon_f+i(x_p-\\varepsilon_f)]\\mbox{\\it\\.{C}}_f(\\beta)+(X_f\/2)^{2}C_f(\\beta)\\!=\\!0,\n\\]\nwhere $X_f\\!=\\![2\\Delta x_{p}\\bar\\Gamma_f(x_p)]^{1\/2}$ with\n$\\bar\\Gamma_f(x_{p})\\!=\\!\\bar\\Gamma_0^f(x_{p})\\rho(x_{p})$,\n$\\Delta\\varepsilon_f=\\hbar\/2\\gamma_0\\tau$,\n$\\beta=2\\gamma_0t\/\\hbar$ is the dimensionless time,~and the\n\\textbf{k}-dependence is omitted for brevity. When the total\nexciton energy is close to a plasmon resonance,\n$\\varepsilon_f\\!\\approx\\!x_{p}$, the solution of this equation is\neasily found to be\n\\begin{eqnarray}\nC_f(\\beta)\\!\\!\\!&\\approx&\\!\\!\\!\\frac{1}{2}\\left(\\!1+\\frac{\\delta\nx}{\\sqrt{\\delta x^2-X_f^2}}\\right)e^{-\\left(\\delta\nx-\\sqrt{\\delta x^2-X_f^2}\\right)\\beta\/2}\\hskip0.5cm\\label{Cfapp}\\\\\n&+&\\!\\!\\!\\frac{1}{2}\\left(\\!1-\\frac{\\delta x}{\\sqrt{\\delta\nx^2-X_f^2}}\\right)e^{-\\left(\\delta x+\\sqrt{\\delta\nx^2-X_f^2}\\right)\\beta\/2},\\nonumber\n\\end{eqnarray}\nwhere $\\delta x=\\Delta x_p-\\Delta\\varepsilon_f>0$ and\n$X_f\\!=\\![2\\Delta x_{p}\\bar\\Gamma_f(\\varepsilon_f)]^{1\/2}$. This\nsolution is valid when $\\varepsilon_f\\!\\approx\\!x_p$ regardless\nof~the strength of the exciton-surface-plasmon coupling.~It yields\nthe exponential decay of the excitons into plasmons,\n$|C_f(\\beta)|^2\\!\\approx\\!\\exp[-\\bar\\Gamma_f(\\varepsilon_f)\\beta]$,\nin the weak coupling regime where the coupling parameter\n$(X_f\/\\delta x)^{2}\\!\\ll\\!1$. If, on the other hand, $(X_f\/\\delta\nx)^{2}\\!\\gg\\!1$,~then the strong coupling regime occurs, and the\ndecay of the excitons into plasmons proceeds via damped Rabi\noscillations, $|C_f(\\beta)|^{2}\\!\\approx\\!\\exp(-\\delta\nx\\beta)\\cos^{2}(X_f\\beta\/2)$.~This is very similar to what was\nearlier reported for an excited two-level atom near the nanotube\nsurface~\\cite{Bondarev02,Bondarev04,Bondarev04pla,Bondarev06trends}.~Note,\nhowever, that here we have the exciton-phonon scattering as well,\nwhich facilitates the strong exciton-plasmon coupling by\ndecreasing $\\delta x$ in the coupling parameter.~In other words,\nthe phonon scattering broadens the (longitudinal) exciton momentum\ndistribution~\\cite{Bondarev05Ps}, thus effectively increasing the\nfraction of the excitons with $\\varepsilon_f\\!\\approx\\!x_p$.\n\nIn view of Eqs.~(\\ref{gpkk}) and (\\ref{Cfapp}), the exciton\nemission intensity (\\ref{Iomega}) in the vicinity of the plasmon\nresonance takes the following (dimensionless) form\n\n\\begin{equation}\n\\bar{I}(x)\\approx\\bar{I}_{0}(\\varepsilon_f)\\sum_f\\left|\\int_{0}^{\\infty}\\!\\!\\!\\!\\!\\!d\\beta\\,C_{f}(\\beta)\\,\ne^{i(x-\\varepsilon_f+i\\Delta\\varepsilon_f)\\beta}\\right|^{2},\n\\label{Ix}\n\\end{equation}\nwhere $\\bar I(x)\\!=\\!2\\gamma_0I(\\omega)\/\\hbar$ and $\\bar\nI_{0}\\!=\\!\\bar\\Gamma_f(\\varepsilon_f)\/2\\pi$. After some algebra,\nthis results in\n\\begin{equation}\n\\bar{I}(x)\\approx\\frac{\\bar{I}_{0}(\\varepsilon_f)\\,[(x-\\varepsilon_f)^{2}+\\Delta\nx_p^2]}{[(x-\\varepsilon_f)^{2}-X_f^{2}\/4]^{2}+(x-\\varepsilon_f)^{2}(\\Delta\nx_p^2+\\Delta\\varepsilon_f^2)}\\,, \\label{Ixfin}\n\\end{equation}\nwhere $\\Delta x_p^2>\\Delta\\varepsilon_f^2$. The summation sign\nover the exciton internal states is omitted since only one\ninternal state contributes to the emission intensity in the\nvicinity of the sharp plasmon resonance.\n\nThe line shape in Eq.~(\\ref{Ixfin}) is mainly determined by the\ncoupling parameter $(X_f\/\\Delta x_p)^2$. It is clearly seen to be\nof a symmetric two-peak structure in the strong coupling regime\nwhere $(X_f\/\\Delta x_p)^{2}\\gg1$. Testing it for extremum, we\nobtain the peak frequencies to be\n\\[\nx_{1,2}=\\varepsilon_f\\pm\\frac{X_f}{2}\\sqrt{\\sqrt{1+\n8\\left(\\!\\frac{\\Delta\nx_p}{X_f}\\right)^{2}}\\!\\!-4\\left(\\!\\frac{\\Delta\nx_p}{X_f}\\right)^{2}}\n\\]\n[terms $\\sim\\!(\\Delta x_p)^2(\\Delta\\varepsilon_f)^2\/X_f^4$ are\nneglected], with the Rabi splitting $x_{1}-x_{2}\\!\\approx\\!X_f$.\nIn the weak coupling regime~where $(X_f\/\\Delta x_p)^{2}\\ll1$, the\nfrequencies $x_1$ and $x_2$ become complex, indicating that there\nare no longer peaks at these frequencies. As this takes place,\nEq.~(\\ref{Ixfin}) is approximated with the weak coupling\ncondition, the fact that $x\\!\\sim\\!\\varepsilon_f$, and\n$X_f^2=2\\Delta x_p\\bar\\Gamma_f(\\varepsilon_f)$, to yield the\nLorentzian\n\\[\n\\tilde{I}(x)\\approx\\frac{\\bar{I}_{0}(\\varepsilon_f)\/[1+(\\Delta\\varepsilon_f\/\\Delta\nx_p)^2]}{(x-\\varepsilon_f)^{2}+\\left[\\bar\\Gamma_f(\\varepsilon_f)\/2\\sqrt{1+(\\Delta\\varepsilon_f^{\\!}\/\\Delta\nx_p)^2}\\;\\right]^2}\n\\]\npeaked at $x=\\varepsilon_f$, whose half-width-at-half-maximum~is\nslightly narrower, however, than $\\bar\\Gamma_f(\\varepsilon_f)\/2$\nit should be if the exciton-plasmon relaxation were the only\nrelaxation mechanism in the system.~The reason is the competing\nphonon scattering takes excitons out of resonance with plasmons,\nthus decreasing the exciton-plasmon relaxation rate. We therefore\nconclude that~the phonon scattering does not affect the exciton\nemission\/absorption line shape when the exciton-plasmon coupling\nis strong (it facilitates the strong coupling regime to occur,\nhowever, as was noticed above), and it narrows the (Lorentzian)\nemission\/absorption line when the exciton-plasmon coupling is\nweak.\n\nThe non-linear optical susceptibility is proportional to the\nlinear optical response function under resonant pumping\nconditions~\\cite{Mukamel}. This allows us to use Eq.~(\\ref{Ixfin})\nto investigate the non-linear excitation regime with the\nphotoinduced biexciton formation as the exciton energy is tuned to\nthe nearest interband plasmon resonance. Under these conditions,\nthe third-order longitudinal CN susceptibility takes the\nform~\\cite{Pedersen05,Mukamel}\n\\begin{equation}\n\\chi^{(3)}(x)\\approx\\tilde{I}(x)\\!\\left[\\frac{1}{x-\\varepsilon_f+i(\\Gamma^f\\!\/2+\\Delta\\varepsilon_f)}-\n\\frac{1}{x-(\\varepsilon_f-|\\varepsilon^{X\\!X}_f|)+i(\\Gamma^f\\!\/2+\\Delta\\varepsilon_f)}\n\\right]\\!, \\label{chi3x}\n\\end{equation}\nwhere $\\varepsilon^{X\\!X}_f$ is the (negative) dimensionless\nbinding energy of the biexciton composed of two $f$-internal state\nexcitons, and $\\chi_0$ is the frequency-independent constant. The\nfirst and second terms in the brackets represent bleaching due to\nthe depopulation of the ground state and photoinduced absorption\ndue to exciton-to-biexciton transitions, respectively.\n\n\\begin{figure}[t]\n\\epsfxsize=8.2cm\\centering{\\epsfbox{fig7.eps}}\\caption{(a)~Schematic\n(arbitrary units) of the exchange coupling of two ground-state 1D\nexcitons to form a biexcitonic state. (b)~The coupling occurs in\nthe configuration space of the two independent longitudinal\nrelative electron-hole motion coordinates, $z_1$ and $z_2$, of\neach of the excitons, due to the tunneling of the system through\nthe potential barriers formed by the two single-exciton cusp-type\npotentials [bottom, also in (a)], between equivalent states\nrepresented by the isolated two-exciton wave functions shown on\nthe top.}\\label{fig7}\n\\end{figure}\n\nThe binding energy of the biexciton in a small-diameter\n($\\sim\\!1\\,$nm) CN can be evaluated by the method pioneered by\nLandau~\\cite{LandauQM}, Gor'kov and Pitaevski~\\cite{Pitaevski},\nHolstein and Herring~\\cite{Herring} --- from the analysis of the\nasymptotic exchange coupling by perturbation on the configuration\nspace wave function of the two ground-state one-dimensional (1D)\nexcitons. Separating out circumferential and longitudinal degrees\nof freedom of each of the excitons by means of\nEq.~(\\ref{wfunceh}), one arrives at the biexciton Hamiltonian of\nthe form [see Fig.~\\ref{fig7}~(a)]\n\\begin{equation}\n\\hat{H}(z_1,z_2,\\Delta\nZ)=-\\frac{1}{2}\\left(\\frac{\\partial^2}{\\partial\\,\\!z_1}\n+\\frac{\\partial^2}{\\partial\\,\\!z_2}\\right)\\label{biexcham}\n\\end{equation}\\vspace{-0.5cm}\n\\begin{eqnarray}\n-\\frac{1}{2}\\left[\\frac{1}{|z_{1}|+z_0}+\\frac{1}{|z_{2}+\\Delta\nZ|+z_0}+\\frac{1}{|z_{2}|+z_0}+\\frac{1}{|z_{1}-\\Delta Z|+z_0}\\right]\\nonumber\\\\\n-\\frac{1}{|(z_1+z_2)\/2+\\Delta Z|+z_0}-\\frac{1}{|(z_1+z_2)\/2-\\Delta Z|+z_0}\\hskip0.7cm\\nonumber\\\\\n+\\frac{1}{|(z_1-z_2)\/2+\\Delta Z|+z_0}+\\frac{1}{|(z_1-z_2)\/2-\\Delta\nZ|+z_0}\\,.\\hskip0.5cm\\nonumber\n\\end{eqnarray}\nHere, $z_{1,2}=z_{e1,2}-z_{h1,2}$ is the electron-hole relative\nmotion coordinates of the two 1D excitons, $z_0$ is the cut-off\nparameter of the effective longitudinal electron-hole Coulomb\npotential (\\ref{Vcutoff}), and $\\Delta Z\\!=\\!Z_2-Z_1$ is the\ncenter-of-mass-to-center-of-mass inter-exciton separation\ndistance. Equal electron and hole effective masses $m_{e,h}$ are\nassumed~\\cite{Jorio} and \"atomic units\"\\space are\nused~\\cite{LandauQM,Pitaevski,Herring}, whereby distance and\nenergy are measured in units of the exciton Bohr radius $a^\\ast_B$\nand in units of the doubled ground-state-exciton binding energy\n$2E_b\\!=\\!-2Ry^\\ast\/\\nu_0^2$, respectively. The first two lines in\nEq.~(\\ref{biexcham}) represent two isolated non-interacting 1D\nexcitons [see Fig.~\\ref{fig7}~(a)]. The last two lines are their\nexchange Coulomb interactions --- electron-hole and\nelectron-electron + hole-hole, respectively.\n\nThe Hamiltonian (\\ref{biexcham}) is effectively two dimensional in\nthe configuration space of the two \\emph{independent} relative\nmotion coordinates, $z_1$ and $z_2$. Figure~\\ref{fig7}~(b),\nbottom, shows schematically the potential energy surface of the\ntwo closely spaced non-interacting 1D excitons [line two in\nEq.~(\\ref{biexcham})] in the $(z_1,z_2)$ space. The surface has\nfour symmetrical minima [representing \\emph{equivalent} isolated\ntwo-exciton states shown in Fig.~\\ref{fig7}~(b), top], separated\nby the potential barriers responsible for the tunnel exchange\ncoupling between the two-exciton states in the configuration\nspace. The coordinate transformation $x=(z_1-z_2-\\Delta\nZ)\/\\sqrt{2}$, $y=(z_1+z_2)\/\\sqrt{2}$ places the origin of the new\ncoordinate system into the intersection of the two tunnel channels\nbetween the respective potential minima [Fig.~\\ref{fig7}~(b)],\nwhereby the exchange splitting formula of\nRefs.~\\cite{LandauQM,Pitaevski,Herring} takes the form\n\\begin{equation}\nU_{g,u}(\\Delta Z)-2E_b=\\mp J(\\Delta Z), \\label{Ugu}\n\\end{equation}\nwhere $U_{g,u}$ are the ground and excited state energies,\nrespectively, of the two \\emph{coupled} excitons (the biexciton)\nas functions of their center-of-mass-to-center-of-mass separation,\nand\n\\begin{equation}\nJ(\\Delta Z)=\\frac{2}{3!}\\!\\int_{\\!-\\Delta Z\/\\!\\sqrt{2}}^{\\Delta\nZ\/\\!\\sqrt{2}}\\!dy\\!\\left[\\psi(x,y)\\frac{\\partial\\psi(x,y)}{\\partial\nx}\\right]_{\\!x=0} \\label{J}\n\\end{equation}\nis the tunnel exchange coupling integral, where $\\psi(x,y)$ is the\nsolution to the Schr\\\"{o}\\-dinger equation with the Hamiltonian\n(\\ref{biexcham}) transformed to the $(x,y)$ coordinates. The\nfactor $2\/3!$ comes from the fact that there are two equivalent\ntunnel channels in the problem, mixing three equivalent\nindistinguishable two-exciton states in the configuration space\n[one state is given by the two minima on the $y$-axis, and two\nmore are represented by each of the minima on the $x$-axis ---\ncompare Figs.~\\ref{fig7}~(a) and (b)].\n\nThe function $\\psi(x,y)$ in Eq.~(\\ref{J}) is sought in the form\n\\begin{equation}\n\\psi(x,y)=\\psi_0(x,y)\\exp[-S(x,y)]\\,, \\label{psixy}\n\\end{equation}\nwhere $\\psi_0=\\nu_0^{-1}\\exp[-(|z_1(x,y,\\Delta Z)|+|z_2(x,y,\\Delta\nZ)|)\/\\nu_0]$ is the product of two single-exciton wave\nfunctions\\footnote{This is an approximate solution to the\nShr\\\"{o}dinger equation with the Hamiltonin given by the first two\nlines in Eq.~(\\ref{biexcham}), where the cut-off parameter $z_0$\nis neglected~\\cite{Takagahara}. This approximation greatly\nsimplifies problem solving here, while still remaining adequate as\nonly the long-distance tail of $\\psi_0$ is important for the\ntunnel exchange coupling.} representing the isolated two-exciton\nstate centered at the minimum $z_1\\!=\\!z_2\\!=\\!0$ (or\n$x\\!=\\!-\\Delta Z\/\\sqrt{2}$, $y\\!=\\!0$) of the configuration space\npotential [Fig.~\\ref{fig7}~(b)], and $S(x,y)$ is a slowly varying\nfunction to take into account the deviation of $\\psi$ from\n$\\psi_0$ due to the tunnel exchange coupling to another equivalent\nisolated two-exciton state centered at $z_1\\!=\\Delta Z$,\n$z_2\\!=\\!-\\Delta Z$ (or $x\\!=\\!\\Delta Z\/\\sqrt{2}$, $y\\!=\\!0$).\nSubstituting Eq.~(\\ref{psixy}) into the Schr\\\"{o}dinger equation\nwith the Hamiltonian (\\ref{biexcham}) pre-transformed to the\n$(x,y)$ coordinates, one obtains in the region of interest\n\\[\n\\frac{\\partial S}{\\partial x}=\\nu_0\\left(\\frac{1}{x+3\\Delta\nZ\/\\sqrt{2}}-\\frac{1}{x-\\Delta\nZ\/\\sqrt{2}}+\\frac{1}{y-\\sqrt{2}\\Delta Z}-\\frac{1}{y+\\sqrt{2}\\Delta\nZ}\\right),\n\\]\nup to (negligible) terms of the order of the inter-exciton van der\nWaals energy and up to second derivatives of~$S$. This equation is\nto be solved with the boundary condition $S(-\\Delta\nZ\/\\sqrt{2},y)\\!=\\!0$ originating from the natural requirement\n$\\psi(-\\Delta Z\/\\sqrt{2},y)\\!=\\!\\psi_0(-\\Delta Z\/\\sqrt{2},y)$, to\nresult in\n\\begin{equation}\nS(x,y)=\\nu_0\\!\\left(\\!\\ln\\!\\left|\\frac{x\\!+\\!3\\Delta\nZ\/\\!\\sqrt{2}}{x-\\Delta Z\/\\!\\sqrt{2}}\\right|+\\frac{2\\sqrt{2}\\Delta\nZ(x\\!+\\!\\Delta Z\/\\!\\sqrt{2})}{y^2-2\\Delta Z^2}\\!\\right)\\!.\n\\label{sxy}\n\\end{equation}\n\nAfter plugging Eqs.~(\\ref{sxy}) and (\\ref{psixy}) into\nEq.~(\\ref{J}), and retaining only the leading term of the integral\nseries expansion in powers of $\\nu_0$ subject to $\\Delta Z>1$,\nEq.~(\\ref{Ugu}) becomes\n\\begin{equation}\nU_{g,u}(\\Delta\nZ)\\approx2E_b\\left[1\\pm\\frac{2}{3\\nu_0^2}\\left(\\frac{e}{3}\\right)^{2\\nu_0}\\!\\!\\!\n\\Delta Z\\,e^{-2\\Delta Z\/\\nu_0}\\right]. \\label{Uxx}\n\\end{equation}\nThe ground state energy $U_g$ of two coupled 1D excitons is now\nseen to go through the negative minimum (biexcitonic state) as the\ninter-exciton center-of-mass-to-center-of-mass separation $\\Delta\nZ$ increases (Fig.~\\ref{fig8}). The minimum occurs at $\\Delta\nZ_0=\\nu_0\/2$, whereby the biexciton binding energy is\n$E_{X\\!X}\\!\\approx(2E_b\/9\\nu_0)(e\/3)^{2\\nu_0-1}$, or, expressing\n$\\nu_0$ in terms of $E_b$ and measuring the energy in units of\n$Ry^\\ast$,\n\\begin{equation}\nE_{X\\!X}[\\mbox{in}~Ry^\\ast]\\approx-\\frac{2}{9}\\;|E_b|^{3\/2}\\left(\\frac{e}{3}\\right)^{2\/\\!\\sqrt{|E_b|}\\,-\\,1}\\!\\!\\!.\n\\label{Exx}\n\\end{equation}\n\nThe energy $E_{X\\!X}$ can be affected by the quantum confined\nStark effect since $|E_b|$ decreases quadratically with the\nperpendicular electrostatic field applied as shown in\nFig.~\\ref{fig5}~(a). Since $e\/3\\sim\\!1$, the field dependence in\nEq.~(\\ref{Exx}) mainly comes from the pre-exponential factor. So,\n$|E_{X\\!X}|$ will be decreasing quadratically with the field as\nwell, for not too strong perpendicular fields. At the same time,\nthe equilibrium inter-exciton separation in the biexciton, $\\Delta\nZ_0=\\nu_0\/2\\sim|E_b|^{-1\/2}$, will be slowly increasing with the\nfield consistently with the lowering of $|E_{X\\!X}|$. In the zero\nfield, one has roughly\n$E_{X\\!X}\\!\\sim\\!|E_b|^{3\/2}\\!\\sim\\!R_{CN}^{-0.9}$ for the\nbiexciton binding energy versus the CN radius $R_{CN}$\n($|E_b|\\!\\sim\\!R_{CN}^{-0.6}$ as reported in\nRef.~\\cite{Pedersen03} from variational calculations), pretty\nconsistent with the $R_{CN}^{-1}$ dependence obtained\nnumerically~\\cite{Pedersen05}. Interestingly, as $R_{CN}$ goes\ndown, $|E_{X\\!X}|$ goes up faster than $|E_b|$ does. This is\npartly due to the fact that $\\Delta Z_0$ slowly decreases as\n$R_{CN}$ goes down, --- a theoretical argument in support of\nexperimental evidence for increased exciton-exciton annihilation\nin small diameter CNs~\\cite{THeinz,Valkunas,Kono}.\n\n\\begin{figure}[t]\n\\epsfxsize=11.00cm\\centering{\\epsfbox{fig8.eps}}\\caption{Calculated\nground state energy $U_g$ of the coupled pair of the first bright\nexcitons in the (11,0) CN as a function of the\ncenter-of-mass-to-center-of-mass inter-exciton distance $\\Delta Z$\nand perpendicular electrostatic field applied. Inset shows the\nbiexciton binding energy $E_{X\\!X}$ and inter-exciton separation\n$\\Delta Z_0$ ($y$- and $x$-coordinates, respectively, of the\nminima in the main figure) as functions of the field.}\\label{fig8}\n\\end{figure}\n\nFigure~\\ref{fig8} shows the ground state energy $U_g(\\Delta Z)$ of\nthe coupled pair of the first bright excitons, calculated from\nEq.~(\\ref{Uxx}) for the semiconducting (11,0) CN exposed to\ndifferent perpendicular electrostatic fields. The inset shows the\nfield dependences of $E_{X\\!X}$ [as given by Eq.~(\\ref{Exx})] and\nof $\\Delta Z_0$. All the curves are calculated using the field\ndependence of $E_b$ obtained as described in the previous\nsubsection (Figs.~\\ref{fig5} and~\\ref{fig6}). They exhibit typical\nbehaviors discussed above.\n\n\\begin{figure}[t]\n\\epsfxsize=11.00cm\\centering{\\epsfbox{fig9.eps}}\\vskip-3.5cm\\caption{[(a),\n(b), and (c)] Linear (top) and non-linear (bottom) response\nfunctions as given by Eq.~(\\ref{Ixfin}) and by the imaginary part\nof Eq.~(\\ref{chi3x}), respectively, for the first bright exciton\nin the (11,0) CN as the exciton energy is tuned to the nearest\ninterband plasmon resonance (vertical dashed line). Vertical lines\nmarked as X and XX show the exciton energy and biexciton binding\nenergy, respectively. The dimensionless energy is defined as\n[\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}).}\\label{fig9}\n\\end{figure}\n\nFigure~\\ref{fig9} compares the linear response lineshape\n(\\ref{Ixfin}) with the imaginary part of Eq.~(\\ref{chi3x})\nrepresenting the non-linear optical response function under\nresonant pumping, both calculated for the 1st bright exciton in\nthe (11,0) CN as its energy is tuned (by means of the quantum\nconfined Stark effect) to the nearest plasmon resonance (vertical\ndashed line in the figure). The biexciton binding energy in\nEq.~(\\ref{chi3x}) was taken to be $E_{X\\!X}\\approx52$~meV as\ngiven~by Eq.~(\\ref{Exx}) in the zero field. [Weak field dependence\nof $E_{X\\!X}$ (inset in Fig.~\\ref{fig8}) plays no essential role\nhere as $|E_{X\\!X}|\\ll|E_b|\\approx0.76$~eV regardless of the field\nstrength.] The phonon relaxation time $\\tau_{ph}\\!=\\!30$~fs was\nused as reported in Ref.~\\cite{Perebeinos05}, since this is the\nshortest one out of possible exciton relaxation processes,\nincluding exciton-exciton annihilation\n($\\tau_{ee}\\!\\sim\\!1$~ps~\\cite{THeinz}). Clear line (Rabi)\nsplitting effect $\\sim\\!0.1$~eV is seen both in the linear and in\nnon-linear excitation regime, indicating the strong\nexciton-plasmon coupling both in the single-exciton states and in\nthe biexciton states as the exciton energy is tuned to the\ninterband surface plasmon resonance. The splitting is not masked\nby the exciton-phonon scattering.\n\nThis effect can be used for the development of new tunable\noptoelectronic device applications of optically excited\nsmall-diameter semiconducting CNs in areas such as nanophotonics,\nnanoplasmonics, and cavity quantum electrodynamics, including the\nstrong excitation regime with optical non-linearities. In the\nlatter case, the experimental observation of the non-linear\nabsorption line splitting predicted here would help identify the\npresence and study the properties of biexcitonic states (including\nbiexcitons formed by excitons of different subbands~\\cite{Papan})\nin individual single-walled CNs, due to the fact that when tuned\nclose to a plasmon resonance the exciton relaxes into plasmons at\na rate much greater than\n$\\tau_{ph}^{-1}\\;(\\,\\gg\\!\\tau_{ee}^{-1})$, totally ruling out the\nrole of the competing exciton-exciton annihilation process.\n\n\\section{Casimir Interaction in Double-Wall\\\\ Carbon Nanotubes}\\label{sec4}\n\nHere, we consider the Casimir interaction between two concentric\ncylindrical gra\\-phene sheets comprising a double-wall CN, using\nthe macroscopic QED approach employed above to study the\nexciton-surface-plasmon interactions in single wall\nnanotubes.\\footnote{In this Section only, the International System\nof units is used to make the comparison easier of our theory with\nother authors' results.} The method is fully adequate in this case\nas the Casimir force is known to originate from quantum EM field\nfluctuations. The fundamental nature of this force has been\nstudied for many years since the prediction of the attraction\nforce between two neutral metallic plates in vacuum (see,\nRefs.~\\cite{BuhmannWelsch,Klimchitskaya2009}). After the first\nreport of observation of this spectacular\neffect~\\cite{Sparnaay1958}, new measurements with improved\naccuracy have been done involving different\ngeometries~\\cite{Lamoreaux1997,Mohideen1998,Munday2009}. The\nCasimir force has also been considered theoretically with methods\nprimarily based on the zero-point summation approach and Lifshitz\ntheory~\\cite{Bordag2001,Parsegian2005}.\n\nThe Casimir effect has acquired a much broader impact recently due\nto its importance for nanostructured materials, including graphite\nand graphitic nanostructures~\\cite{Klimchitskaya2009} which can\nexist in different geometries and with various unique electronic\nproperties. Moreover, the efficient development and operation of\nmodern micro- and nano-electromechanical devices are limited due\nto effects such as stiction, friction, and adhesion, originating\nfrom or closely related to the Casimir effect~\\cite{Chan2001}.\n\nThe mechanisms governing the CN interactions still remain elusive.\nIt is known that the system geometry~\\cite{Rajter2007,Popescu2008}\nand dielectric response~\\cite{Fermani2007,Bondarev05} have a\nprofound effect on the interaction, in general, but their specific\nfunctionalities have not been qualitatively and quantitatively\nunderstood. Since CNs of virtually the same radial size can\npossess different electronic properties, investigating their\nCasimir interactions presents a unique opportunity to obtain\ninsight into specific dielectric response features affecting the\nCasimir force between metallic and semiconducting cylindrical\nsurfaces. This can also unveil the role of collective surface\nexcitations in the energetic stability of multi-wall CNs of\nvarious chiral combinations.\n\nSince Lifshitz theory cannot be easily applied to geometries other\nthan parallel plates, researchers have used the Proximity Force\nApproximation (PFA) to calculate the Casimir interaction between\nCNs~\\cite{Rajter2007,Bordag2006} (see also\nRef.~\\cite{Klimchitskaya2009} for the latest review). The method\nis based on approximating the curved surfaces at very close\ndistances by a series of parallel plates and summing their\nenergies using the Lifshitz result. Thus, the PFA is inherently an\nadditive approach, applicable to objects at very close separations\n(still to be greater than objects inter-atomic distances) under\nthe assumption that the CN dielectric response is the same as the\none for the plates. This last assumption is very questionable as\nthe quasi-1D character of the electronic motion in CNTs is known\nto be of principal importance for the correct description of their\nelectronic and optical\nproperties~\\cite{Dresselhaus,Bondarev04,Tasaki}.\n\nWe model the double-wall CN by two infinitely long, infinitely\nthin, continuous concentric cylinders with radii $R_{1,2}$,\nimmersed in vacuum. Each cylinder is characterized by the complex\ndynamic axial dielectric function $\\epsilon _{zz}(R_{1,2},\\omega)$\nwith the \\textit{z}-direction along the CN axis as shown in\nFig.~\\ref{fig10}. The azimuthal and radial components of the\ncomplete CN dielectric tensor are neglected as they are known to\nbe much less than $\\epsilon_{zz}$ for most CNs~\\cite{Tasaki}. The\nQED quantization scheme in the presence of\nCNs~\\cite{BuhmannWelsch,Bondarev05} generates the second-quantized\nHamiltonian\n\\[\n\\hat{H}=\\sum_{i=1,2}\\int_{0}^{\\infty}\\!\\!\\!d\\omega\\hbar\\omega\\int\nd\\mathbf{R}_{i}\\hat{f}^{\\dag}(\\mathbf{R}_{i},\\omega)\\hat{f}(\\mathbf{R}_{i},\\omega)\n\\]\nof the vacuum-type medium assisted EM field, with the bosonic\noperators $\\hat{f}^{\\dag}$ and $\\hat{f}$ creating and\nannihilating, respectively, surface EM excitations of frequency\n\\textit{$\\omega $} at points\n$\\mathbf{R}_{1,2}=\\{R_{1,2},\\varphi_{1,2},z_{1,2}\\}$ of the\ndouble-wall CN system. The Fourier-domain electric field operator\nat an arbitrary point $\\mathbf{r}=(r,\\varphi,z)$ is given by\n\\[\n\\hat{E}(\\mathbf{r},\\omega)=i\\omega\\mu _{0}\\sum _{i=1,2}\\int\nd\\mathbf{R}_{i}\\mathbf{G}(\\mathbf{r},\\mathbf{R}_{i},\\omega)\\cdot\n\\hat{\\mathbf{J}}(\\mathbf{R}_{i},\\omega),\n\\]\nwhere $\\mathbf{G}(\\mathbf{r},\\mathbf{R}_{i},\\omega)$ is the dyadic\nEM field Green's function (GF), and\n\\[\n\\hat{\\mathbf{J}}(\\mathbf{R}_{i},\\omega)=\\frac{\\omega}{\\mu_{0}c^{2}}\n\\sqrt{\\frac{\\hbar\\,\\mbox{Im}\\,\\epsilon_{zz}(R_i,\\omega)}{\\pi\\varepsilon_{0}}}\\,\n\\hat{f}(\\mathbf{R}_{i},\\omega)\\mathbf{e}_{z}\n\\]\nis the surface current density operator selected in such a way as\nto ensure the correct QED equal-time commutation relations for the\nelectric and magnetic field\noperators~\\cite{BuhmannWelsch,Bondarev05}. Here, $\\mathbf{e}_{z}$\nis the unit vector along the CN axis, $\\varepsilon_0$, $\\mu_0$,\nand $c$ are the dielectric constant, magnetic permeability, and\nvacuum speed of light, respectively.\n\n\\begin{figure}[t]\n\\epsfxsize=5.5cm\\centering{\\epsfbox{fig10.eps}}\\caption{Schematic\nof the two concentric CNs in vacuum. The CN radii are $R_1$ and\n$R_2$. The regions between the CN surfaces are denoted as (1),\n(2), and (3).}\\label{fig10}\n\\end{figure}\n\nThe dyadic GF satisfies the wave equation\n\\begin{equation} \\label{GrindEQ1}\n\\nabla\\times\\nabla\\times\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)-\n\\frac{\\omega^{2}}{c^{2}}\\,\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)=\n\\delta(\\mathbf{r}-\\mathbf{r}^\\prime)\\,\\mathbf{I}\n\\end{equation}\nwith $\\mathbf{I}$ being the unit tensor. The GF can further be\ndecomposed as follows\n\\[\n\\mathbf{G}^{(s,f)}=\\mathbf{G}^{(0)}\\delta_{sf}+\\mathbf{G}_{scatt}^{(s,f)}\n\\]\nwhere $\\mathbf{G}^{(0)}$ and $\\mathbf{G}_{scatt}^{(s,f)}$\nrepresent the contributions of the direct and scattered waves,\nrespectively~\\cite{Tai1994,KoreanPaper}, with a point-like field\nsource located in region $s$ and the field registered in region\n$f$ (see Fig.~\\ref{fig10}). The boundary conditions for\nEq.~(\\ref{GrindEQ1}) are obtained from those for the electric and\nmagnetic field components on the CN\nsurfaces~\\cite{Fermani2007,Bondarev04}, which result in\n\\begin{equation}\n\\label{GrindEQ2}\n\\mathbf{e}_{r}\\times\\left[\\left.\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{+}}\n-\\left.\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{-}}\\right]=0,\n\\end{equation}\n\\begin{equation}\n\\label{GrindEQ3}\n{\\mathbf{e}_{r}\\!\\times\\!\\nabla\\!\\times\\!\\left[\\left.\n\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{+}}\\!\\!\\!\n-\\left.\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{-}}\\right]}\\!=\ni\\omega\\mu_{0}\\bm{\\sigma}^{(1,2)}(\\mathbf{r},\\omega)\\!\\cdot\\!\\left.\n\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}}\n\\end{equation}\nwhere $\\mathbf{e}_r$ is the unit vector along the radial\ndirection. The discontinuity in Eq.~(\\ref{GrindEQ3}) results from\nthe full account of the finite absorption and dispersion for both\nCNs by means of their conductivity tensors $\\bm{\\sigma}^{(1,2)}$\napproximated by their largest components\n\\begin{equation}\n\\sigma_{zz}^{(1,2)}(R_{1,2},\\omega)=-\\frac{i\\omega\\varepsilon_{0}}{S\\rho_{T}}\\,\n[\\epsilon_{zz}^{(1,2)}(R_{1,2},\\omega)-1]\\label{DrudeSI}\n\\end{equation}\n[compare with Eq.~(\\ref{DrudeGauss})].\n\nFollowing the procedure described in\nRefs.~\\cite{Tai1994,KoreanPaper}, we expand $\\mathbf{G}^{(0)}$ and\n$\\mathbf{G}_{scatt}^{(s,f)}$ into series of even and odd vector\ncylindrical functions with unknown coefficients to be found from\nEqs.~(\\ref{GrindEQ2}) and (\\ref{GrindEQ3}). This splits the EM\nmodes in the system into TE and TM polarizations, with\nEqs.~(\\ref{GrindEQ2}) and (\\ref{GrindEQ3}) yielding a set of 32\nequations (16 for each polarization) with 32 unknown coefficients.\nThe unknown coefficients are found determining the dyadic GF in\neach region.\\footnote{Due to the lengthy and tedious algebra, this\nderivation will be presented in a separate longer communication.}\n\nUsing the expressions for the electric and magnetic fields, the\nelectromagnetic stress tensor is\nconstructed~\\cite{BuhmannWelsch,Cavero-Pelaez2005}\n\\begin{equation} \\label{GrindEQ4}\n\\mathbf{T}(\\mathbf{r},\\mathbf{r}^\\prime)=\\mathbf{T}_{1}(\\mathbf{r},\\mathbf{r}^\\prime)+\n\\mathbf{T}_{2}(\\mathbf{r},\\mathbf{r}^\\prime)-\\frac{1}{2}\\,\\mathbf{I}\\,Tr\\!\\left[\n\\mathbf{T}_{1}(\\mathbf{r},\\mathbf{r}^\\prime)+\\mathbf{T}_{2}(\\mathbf{r},\\mathbf{r}^\\prime)\\right]\n\\end{equation}\n\\begin{equation} \\label{GrindEQ5}\n\\mathbf{T}_{1}(\\mathbf{r},\\mathbf{r}^\\prime)=\\frac{\\hbar}{\\pi}\\int_{0}^{\\infty}\\!\\!\\!d\\omega\n\\frac{\\omega^{2}}{c^{2}}\\,\\mbox{Im}\\!\\left[\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right]\n\\end{equation}\n\\begin{equation} \\label{GrindEQ6}\n\\mathbf{T}_{2}(\\mathbf{r},\\mathbf{r}^\\prime)=-\\frac{\\hbar}{\\pi}\\int_{0}^{\\infty}\\!\\!\\!d\\omega\n\\,\\mbox{Im}\\!\\left[\\nabla\\times\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\times\n\\stackrel{\\!\\!\\leftarrow}{\\nabla^{\\,\\prime}}\\right]\n\\end{equation}\nWe are interested in the radial component $T_{rr}$ which describes\nthe radiation pressure of the virtual EM field on each CN surface\nin the system. The Casimir force per unit area exerted on the\nsurfaces is then given by~\\cite{BuhmannWelsch}\n\\begin{equation}\nF_{i}=\\lim\\limits_{r\\to\nR_{i}}\\left\\{\\lim\\limits_{\\mathbf{r}^\\prime\\to\\mathbf{r}}\n\\left[T_{rr}^{(i)}(\\mathbf{r},\\mathbf{r}^\\prime)-T_{rr}^{(i+1)}(\\mathbf{r},\\mathbf{r}^\\prime)\n\\right]\\right\\},\\;\\;\\;i=1,2 \\label{GrindEQ7}\n\\end{equation}\n\nThe forces $F_{1,2}$ calculated from Eq.~(\\ref{GrindEQ7}) are of\nequal magnitude and opposite direction, indicating the attraction\nbetween the cylindrical surfaces. The Casimir force thus obtained\naccounts \\textit{simultaneously} for the geometrical curvature\neffects (through the GF tensor) and the finite absorption and\ndissipation of each CN [through their dielectric response\nfunctions (\\ref{DrudeSI})]. The dielectric response functions of\nparticular CNs were calculated from the CN realistic band\nstructure as described above, in Section~\\ref{sec3}. We decomposed\nthem into the Drude contribution and the contribution originating\nfrom (transversely quantized) interband electronic transitions,\n$\\epsilon_{zz}=\\epsilon_{zz}^D+\\epsilon_{zz}^{inter}$, in order to\nbe able to see how much each individual contribution affects the\ninter-tube Casimir attraction.\n\nIt is interesting to consider the case of infinitely conducting\nparallel plates first using Eq.~(\\ref{GrindEQ7}). This is obtaned\nby taking the limits $\\sigma_{zz}^{(1,2)}\\to\\infty$ and\n$R_{1,2}\\to\\infty$ while keeping constant the inter-tube distance,\n$R_{1}\\!-R_{2}=d$. We find\n\\[\nF=-\\frac{\\hbar c}{16\\pi^2R_1^4}\n\\int_0^\\infty\\!\\!\\!dx_1x_1\\sum_{n=0}^\\infty\n\\frac{(2-\\delta_n^0)}{I_n(x_1)K_n(x_2)-I_n(x_2)K_n(x_1)}\n\\]\n\\[\n\\times\\left\\{\\left[x_1^2K^{\\prime\\,2}_n(x_1)+\\left(n^2+x_1^2\\right)K_n^2(x_1)\\right]\n\\left[I_n^2(x_1)K_n(x_2)\/K_n(x_1)-2I_n(x_1)I_n(x_2)\\right]\\right.\n\\]\n\\[\n-\\left[x_1^2I^{\\prime\\,2}_n(x_1)+\\left(n^2+x_1^2\\right)I_n^2(x_1)\\right]K_n(x_1)K_n(x_2)\n\\]\n\\[\n-\\left.2\\left[x_1^2I^\\prime_n(x_1)K^\\prime_n(x_1)+\\left(n^2+x_1^2\\right)I_n(x_1)K_n(x_1)\\right]\nI_n(x_2)K_n(x_1)\\right\\}\n\\]\nwhere $x_{1,2}=xR_{1,2}$, $I_n(x)$ and $K_n(x)$ are the modified\nBessel functions of the first and second kind, respectively. The\nabove expression is obtained by making the transition to imaginary\nfrequencies $\\omega\\rightarrow i\\omega$, and using the Euclidean\nrotation technique as described in\nRefs.~\\cite{Cavero-Pelaez2005,Milton1978}. This can further be\nevaluated by summing up the series over $n$ using the large-order\nBessel function expansions~\\cite{Abramovitz}. This results in\n$F\\!\\sim\\!(-1\/3)(\\hbar c\\pi^2\/240d^4)$ which is $1\/3$ of the\nwell-known result for two parallel\nplates~\\cite{BuhmannWelsch,Klimchitskaya2009}. This deviation\noriginates from $\\epsilon_{zz}\\ne0$ only and the remaining\ndielectric tensor components being zero in our model.\n\n\\begin{figure}[t]\n\\epsfxsize=12.5cm\\centering{\\epsfbox{fig11.eps}}\\vskip-0.2cm\\caption{The\nCasimir force per unit area as a function of the inter-tube\nseparation $d$, for different pairs of CNs. The inset shows force\nfound with the full dielectric function and the Drude contribution\nonly for the same CN pairs indicated in the figure.}\\label{fig11}\n\\end{figure}\n\nFigure~\\ref{fig11} presents results from the numerical\ncalculations of $F$ as a function of the inter-tube\nsurface-to-surface distance for various pairs of CNs with their\nrealistic chirality dependent dielectric responses taken into\naccount. We have chosen the inner CN to be the achiral $(12,12)$\nmetallic nanotube, and to change the outer tubes. As $R_2$ is\nvaried, one can envision double wall CNs consisting of metal\/metal\nor metal\/semiconductor combinations of different chiralities but\nof similar radial dimensions.\n\nFigure~\\ref{fig11} shows that $F$ decreases in strength as the\nsurface-to-surface distance increases. This dependence is\nmonotonic for the zigzag $(m,0)$ and armchair $(n,n)$ outer tubes,\nbut it happens at different rates. The attraction is stronger if\nthe outer CN is an armchair $(n,n)$ one as compared to the\nattraction for the outer $(m,0)$ nanotubes. At the same time, for\nchiral tubes the Casimir force decreases as a function of $d$ in a\nrather irregular fashion. It is seen that for relatively small\n$d$, the interaction force can be quite different. For example,\nthe attraction between $(27,4)@(12,12)$ and $(21,13)@(12,12)$\ndiffer by $\\sim\\!20$~\\% in favor of the second pair, even though\nthe radial difference is only $0.2$~\\AA. The differences between\nthe different CNs become smaller as their separation becomes\nlarger, and they eventually become negligible as the Casimir force\ndiminishes at large distances.\n\nWe also calculate the Casimir force using the\n$\\epsilon_{zz}^D(\\omega)$ contribution alone in each dielectric\nfunction. The inset in Fig.~\\ref{fig11} indicates that the\nattraction is stronger when the interband transitions are\nneglected. The decay of $F$ as a function of $d$ is monotonic.\nIncluding the $\\epsilon_{zz}^{inter}(\\omega)$ term not only\nreduces the force, but also introduces non-linearities due to the\nchirality dependent optical excitations. At large\nsurface-to-surface separations, the discrepancies between the\nforce calculated with the full dielectric response, and those\nobtained with the Drude term only become less significant. We find\nthat for $d\\!\\sim\\!15$~\\AA, this difference is less than $10$~\\%.\n\n\\begin{figure}[t]\n\\epsfxsize=12.5cm\\centering{\\epsfbox{fig12.eps}}\\vskip-0.1cm\\caption{The\nCasimir force per unit area as a function of the inter-tube\nseparation $d$ for selected CN pairs. The insets show the EELS\nspectra for several CNs.}\\label{fig12}\n\\end{figure}\n\nTo investigate further the important functionalities originating\nfrom the cylindrical geometry and the CN dielectric response\nproperties, $F$ is calculated for different achiral inner\/outer\nnanotube pairs. Studying zigzag and armchair CNs allows tracking\ngeneralities from $\\epsilon(\\omega)$ in a more controlled manner.\nThe results are presented in Fig.~\\ref{fig12}. We have chosen\nrepresentatives of three inner CN types -- metallic $(12,12)$,\nsemi-metallic $(21,0)$, and semiconducting $(20,0)$ tubules. They\nare of similar radii, $8.14$~\\AA, $8.22$~\\AA, and $7.83$~\\AA,\nrespectively. We see that depending on the outer nanotube types,\nthe $F$ versus $d$ curves are positioned in three groups. The\nweakest interaction is found when there are two zigzag concentric\nCNs (top two curves). The fact that some of these are\nsemi-metallic and others are semiconducting does not seem to\ninfluence the magnitude and monotonic decrease of the Casimir\nforce.\n\nThe attraction is stronger when there is a combination of an\narmchair and a zigzag CNT as compared to the previous case. The\ncurves for $(m,0)@(12,12)$, $(n,n)@(21,0)$, and $(n,n)@(20,0)$ are\npractically overlapping, meaning that the specific location of the\nzigzag and armchair tubes (inner or outer) is of no significance\nto the force. The small deviations can be attributed to the small\ndifferences in the inner CN radii. Finally, we see that the\nstrongest interaction occurs between two armchair CNs (red curve).\nThese functionalities are not unique just for the considered CNs.\nWe have performed the same calculations for many different achiral\ntubes, and we always find that the strongest interaction occurs\nbetween two armchair CNs and the weakest --- between two zigzag\nCNs (provided that their radial dimensions are similar).\n\nThe results from these calculations are strongly suggestive that\nthe CN collective excitation properties have a strong effect on\ntheir mutual interaction. This is particularly true for the\nrelatively small distances of interest here, for which the\ndominant contribution of plasmonic modes to the Casimir\ninteractions has been realized for planar~\\cite{Kampen1968} and\nlinear~\\cite{Dobson2006} metallic systems. To elucidate this issue\nhere, we calculate the EELS spectra, given by\n$-\\mbox{Im}[1\/\\epsilon(\\omega)]$, and compare them for various\ninner and outer CNs combinations --- Fig.~\\ref{fig12} (inset).\n\nConsidering $F$ as a function of $d$ and the specific form of the\nEELS spectra, it becomes clear from the inset in Fig.~\\ref{fig12}\nthat the low frequency plasmon excitations, given by peaks in\n$-\\mbox{Im}[1\/\\epsilon(\\omega)]$, are key to the strength of the\nCasimir force. We always find that the strongest force is between\nthe tubules with well pronounced overlapping low frequency plasmon\nexcitations. This is consistent with the conclusion of\nRef.~\\cite{Dobson2006} for generic 1D-plasmonic structures.\nHowever, in our case we deal with the interband plasmons\noriginating from the space quantization of the transverse\nelectronic motion, and, therefore, having quite a different\nfrequency-momentum dispersion law (constant) as compared to that\nnormally assumed (linear) for plasmons~\\cite{Pichler98}. A weaker\nforce is obtained if only one of the CNs supports strong low\nfrequency interband plasmon modes. The weakest interaction happens\nwhen neither CN has strong low frequency plasmons. For the cases\nshown in Fig~\\ref{fig12}, one finds well pronounced overlapping\nplasmon transitions in the $(12,12)$ CN at $\\omega_1=2.18$~eV and\n$\\omega_2=3.27$~eV, and at $\\omega_1=1.63$~eV and\n$\\omega_2=2.45$~eV in the $(17,17)$ CN. At the same time, no such\nwell defined strong low frequency excitations in the $(21,0)$ and\n$(30,0)$ CNs are found. Figure~\\ref{fig12} shows that the\nattraction in $(17,17)@(12,12)$ is much stronger than the\nattraction in $(30,0)@(21,0)$, even though the radial sizes of the\ninvolved CNs are approximately the same. One also notes that for\nthe case of $(17,17)@(21,0)$ there is only one such low frequency\nexcitation coming from the armchair tube and, consequently, the\nCasimir force has an intermediate value as compared to the above\ndiscussed two cases.\n\nWe performed calculations of the Casimir force between many CN\npairs and made comparisons between the relevant regions of the\nEELS spectra. It is found that, in general, armchair tubes always\nhave strong, well pronounced interband plasmon excitations in the\nlow frequency range. Zigzag and most chiral CNs have low frequency\ninterband plasmons~\\cite{BondPRB09}, too, but they are not as near\nas well pronounced as those in armchair tubes; their stronger\nplasmon modes are found at higher frequencies.\n\nThese studies are indicative of the significance of the collective\nresponse properties of the involved CNs. Specifically, the\ncollective low energy plasmon excitations and their relative\nlocation can result in nanotube attraction with different\nstrengths. We further investigate this point by considering a\ndouble wall CN with radii $R_1=11.63$~\\AA and $R_2=8.22$~\\AA. The\ndielectric function of each tube is taken to be of the generic\nLorentzian form\n\\begin{equation}\n\\epsilon_{zz}(R_{1,2},\\omega)=1-\\frac{\\Omega^2}{\\omega^2-\\omega_{1,2}^2\n+i\\omega\\Gamma}\\label{epsLorentz}\n\\end{equation}\nwith the typical for nanotubes values $\\Omega=2.7$~eV and\n$\\Gamma=0.03$~eV~\\cite{Fermani2007}. Then, the EELS spectrum has\nonly one plasmon resonance at $\\omega_{1,2}$ for each tube. This\ngeneric form allows us to change the relative position and\nstrength of the plasmon peaks and uncover more characteristic\nfeatures originating from the EELS spectra.\n\n\\begin{figure}[t]\n\\epsfxsize=12.5cm\\centering{\\epsfbox{fig13.eps}}\\caption{The\nCasimir force per unit area as a function of the outer CN plasmon\nfrequency, while the inner CN plasmon peak $\\omega_2$ is constant.\nResults are shown for four values of $\\omega_2$. The dielectric\nfunctions are modeled by a generic Lorentzian as given by\nEq.~(\\ref{epsLorentz}).}\\label{fig13}\n\\end{figure}\n\nIn Fig.~\\ref{fig13}, the force as a function of plasmon frequency\nresonances of the outer CN is shown when the plasmon transition\nfor the inner CNT is kept constant (four values are chosen for\n$\\omega_2$). One sees that the local minima in $F$ versus $\\omega$\noccur when $\\omega_1$ and $\\omega_2$ coincide. In fact, the\nstrongest attraction happens when both CNs have the lowest plasmon\nexcitations at the same frequency $\\omega_1=\\omega_2=0.81$~eV. It\nis evident that the existence of relatively strong low frequency\nEELS spectrum \\textit{and} an overlap between the relevant plasmon\npeaks of the two structures is necessary to achieve a strong\ninteraction.\n\nThis study clearly demonstrates the crucial importance of the\ncollective low energy surface plasmon excitations at relatively\nclose surface-to-surface separations along with the cylindrical\ncircular geometry of the double-wall CN system. The QED approach\nwe used provides the unique opportunity to investigate these\nfeatures together, or separately, and to uncover underlying\nmechanisms of the energetic stability of different double-wall CN\ncombinations. An additional advantage here is that we can\ncalculate the dielectric function explicitly for each chirality.\nThus, we can determine unambiguously how the semiconducting or\nmetallic nature of each CN contributes to their mutual\ninteraction.\n\n\\section{Conclusion}\\label{concl}\n\nWe have shown that the strong exciton-surface-plasmon coupling\neffect with characteristic exciton absorption line (Rabi)\nsplitting $\\sim\\!0.1$~eV exists in small-diameter\n($\\lesssim\\!1$~nm) semiconducting CNs.~The splitting is almost as\nlarge as the typical exciton binding energies in such CNs\n($\\sim\\!0.3-0.8$~eV~\\cite{Pedersen03,Pedersen04,Wang05,Capaz}),\nand of the same order of magnitude as the exciton-plasmon Rabi\nsplitting in organic semiconductors\n($\\sim\\!180$~meV~\\cite{Bellessa}).~It is much larger than the\nexciton-polariton Rabi splitting in semiconductor microcavities\n($\\sim\\!140-400\\,\\mu\\mbox{eV}\\,$\\cite{Reithmaier,Yoshie,Peter}),\nor the exciton-plasmon Rabi splitting in hybrid\nsemiconductor-metal nanoparticle molecules~\\cite{Govorov}.\n\nSince the formation of the strongly coupled mixed exciton-plasmon\nexcitations is only possible if the exciton total energy is in\nresonance with the energy of an interband surface plasmon mode, we\nhave analyzed possible ways to tune the exciton energy to the\nnearest surface plasmon resonance. Specifically, the exciton\nenergy may be tuned to the nearest plasmon resonance in ways used\nfor the excitons in semiconductor quantum microcavities\n--- thermally (by elevating sample\ntemperature)~\\cite{Reithmaier,Yoshie,Peter}, and\/or\nelectrostatically~\\cite{MillerPRL,Miller,Zrenner,Krenner} (via the\nquantum confined Stark effect with an external electrostatic field\napplied perpendicular to the CN axis).~The two possibilities\ninfluence the different degrees of freedom of the quasi-1D exciton\n--- the (longitudinal) kinetic energy and the excitation energy,\nrespectively.\n\nWe have studied how the perpendicular electrostatic field affects\nthe exciton excitation energy and interband plasmon resonance\nenergy (the quantum confined Stark effect). Both of them are shown\nto shift to the red due to the decrease in the CN band gap as the\nfield increases. However, the exciton red shift is much less than\nthe plasmon one because of the decrease in the absolute value of\nthe negative binding energy, which contributes largely to the\nexciton excitation energy. The exciton excitation energy and\ninterband plasmon energy approach as the field increases, thereby\nbringing the total exciton energy in resonance with the plasmon\nmode due to the non-zero longitudinal kinetic energy term at\nfinite temperature.\n\nThe noteworthy point is that the strong exciton-surface-plasmon\ncoupling we predict here occurs in an individual CN as opposed to\nvarious artificially fabricated hybrid plasmonic nanostructures\nmentioned above. We strongly believe this phenomenon, along with\nits tunability feature via the quantum confined Stark effect we\nhave demonstrated, opens up new paths for the development of CN\nbased tunable optoelectronic device applications in areas such as\nnanophotonics, nanoplasmonics, and cavity QED. One straightforward\napplication like this is the CN photoluminescence control by means\nof the exciton-plasmon coupling tuned electrostatically via the\nquantum confined Stark effect.~This complements the microcavity\ncontrolled CN infrared emitter application reported\nrecently\\cite{Avouris08}, offering the advantage of less stringent\nfabrication requirements at the same time since the planar\nphotonic microcavity is no longer required. Electrostatically\ncontrolled coupling of two spatially separated (weakly localized)\nexcitons to the same nanotube's plasmon resonance would result in\ntheir entanglement~\\cite{Bondarev07,Bondarev07jem,Bondarev07os},\nthe phenomenon that paves the way for CN based solid-state quantum\ninformation applications. Moreover, CNs combine advantages such as\nelectrical conductivity, chemical stability, and high surface area\nthat make them excellent potential candidates for a variety of\nmore practical applications, including efficient solar energy\nconversion~\\cite{Trancik}, energy storage~\\cite{Shimoda}, and\noptical nanobiosensorics~\\cite{Goodsell10}. However, the\nphotoluminescence quantum yield of individual CNs is relatively\nlow, and this hinders their uses in the aforementioned\napplications. CN bundles and films are proposed to be used to\nsurpass the poor performance of individual tubes. The theory of\nthe exciton-plasmon coupling we have developed here, being\nextended to include the inter-tube interaction, complements\ncurrently available 'weak-coupling' theories of the\nexciton-plasmon interactions in low-dimensional\nnanostructures~\\cite{Govorov,GovorovPRB} with the very important\ncase of the strong coupling regime. Such an extended theory\n(subject of our future publication) will lay the foundation for\nunderstanding inter-tube energy transfer mechanisms that affect\nthe efficiency of optoelectronic devices made of CN bundles and\nfilms, as well as it will shed more light on the recent\nphotoluminescence experiments with CN\nbundles~\\cite{Munich,Ferrari} and multi-walled CNs~\\cite{Hirori},\nrevealing their potentialities for the development of high-yield,\nhigh-performance optoelectronics applications with CNs.\n\nIn addition, we have first applied the macroscopic QED approach\nsuitable for dispersing and absorbing media to study the Casimir\ninteraction in a double-wall carbon nanotube systems with the\nrealistic dielectric response taken into account. We found that at\ndistances similar to the equilibrium separations between graphitic\nsurfaces ($\\sim\\!3$~\\AA), the attraction is dominated by the low\nenergy (interband) plasmon excitations of both CNs. The key\nattributes of the EELS spectra are the existence of low frequency\nplasmons, their strong and well pronounced nature, and the overlap\nbetween the low frequency plasmon peaks belonging to the two CNTs.\nThus, the chiralities of concentric graphene sheets with similar\nradial sizes exhibiting these features will be responsible for\nforming the most preferred CN pairs. As the inter-tube separation\nincreases, the plasmon effect diminishes and the collective\nexcitations originating from the nanotube metallic or\nsemiconducting nature do not influence the interaction in a\nprofound way.\n\nWe expect our results to pave the way for the development of new\ngeneration of tunable optoelectronic and nano-electromechanical\ndevice applications with single-wall and multi-wall carbon\nnanotubes.\n\n\\begin{center}\n{\\bf Acknowledgements}\n\\end{center}\nI.V.B. is supported by the US National Science Foundation, Army\nResearch Office and NASA (grants ECCS-1045661 \\& HRD-0833184,\nW911NF-10-1-0105, and NNX09AV07A). L.M.W. and A.P. are supported\nby the US Department of Energy contract DE-FG02-06ER46297. Helpful\ndiscussions with Mikhail Braun (St.-Peterburg U., Russia),\nJonathan Finley (WSI, TU Munich, Germany), and Alexander Govorov\n(Ohio U., USA) are gratefully acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\\section{Introduction}\n\nPlease follow the steps outlined below when submitting your manuscript to\nthe IEEE Computer Society Press. This style guide now has several\nimportant modifications (for example, you are no longer warned against the\nuse of sticky tape to attach your artwork to the paper), so all authors\nshould read this new version.\n\n\\subsection{Language}\n\nAll manuscripts must be in English.\n\n\\subsection{Dual submission}\n\nPlease refer to the author guidelines on the WACV 2023 web page\n(\\url{http:\/\/wacv2023.thecvf.com\/submission\/})\nfor a discussion of the policy on dual submissions.\n\n\\subsection{Paper length}\nPapers must be no longer than eight pages, not including references.\nAny pages in excess of eight pages must contain ONLY references ---\nno text, figures, acknowledgements, tables, etc.\n\n{\\bf There will be no extra page charges for WACV 2023.}\n\nOverlength papers will simply not be reviewed. This includes papers\nwhere the margins and formatting are deemed to have been significantly\naltered from those laid down by this style guide. 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(\\LaTeX\\ users may uncomment\nthe \\verb'\\wacvfinalcopy' command in the document preamble.) Reviewers:\nnote that the ruler measurements do not align well with lines in the paper\n--- this turns out to be very difficult to do well when the paper contains\nmany figures and equations, and, when done, looks ugly. Just use fractional\nreferences (e.g.\\ this line is $087.5$), although in most cases one would\nexpect that the approximate location will be adequate.\n\n\\subsection{Mathematics}\n\nPlease number all of your sections and displayed equations. It is\nimportant for readers to be able to refer to any particular equation. Just\nbecause you didn't refer to it in the text doesn't mean some future reader\nmight not need to refer to it. It is cumbersome to have to use\ncircumlocutions like ``the equation second from the top of page 3 column\n1''. (Note that the ruler will not be present in the final copy, so is not\nan alternative to equation numbers). All authors will benefit from reading\nMermin's description of how to write mathematics:\n\\url{http:\/\/www.pamitc.org\/documents\/mermin.pdf}.\n\n\n\\subsection{Blind review}\n\nMany authors misunderstand the concept of anonymizing for blind\nreview. Blind review does not mean that one must remove\ncitations to one's own work---in fact it is often impossible to\nreview a paper unless the previous citations are known and\navailable.\n\nBlind review means that you do not use the words ``my'' or ``our''\nwhen citing previous work. In addition, in means you are extremely careful if you\nshare URLs to source code, github repositories, project websites,\ndatasets, etc. The URL or other information (e.g. your github user ID)\nmay identify you and would violate the anonymization policy.\n\n\nFor example, saying ``this builds on the work of Lucy Smith [1]'' does not say\nthat you are Lucy Smith; it says that you are building on her\nwork. If you are Smith and Jones, do not say ``as we show in\n[7]'', say ``as Smith and Jones show in [7]'' and at the end of the\npaper, include reference 7 as you would any other cited work.\n\nAn example of a bad paper just asking to be rejected:\n\\begin{quote}\n\\begin{center}\n An analysis of the frobnicatable foo filter.\n\\end{center}\n\n In this paper we present a performance analysis of our\n previous paper [1], and show it to be inferior to all\n previously known methods. Why the previous paper was\n accepted without this analysis is beyond me.\n\n [1] Removed for blind review\n\\end{quote}\n\n\nAn example of an acceptable paper:\n\n\\begin{quote}\n\\begin{center}\n An analysis of the frobnicatable foo filter.\n\\end{center}\n\n In this paper we present a performance analysis of the\n paper of Smith \\etal [1], and show it to be inferior to\n all previously known methods. Why the previous paper\n was accepted without this analysis is beyond me.\n\n [1] Smith, L and Jones, C. ``The frobnicatable foo\n filter, a fundamental contribution to human knowledge''.\n Nature 381(12), 1-213.\n\\end{quote}\n\nIf you are making a submission to another conference at the same time,\nwhich covers similar or overlapping material, you may need to refer to that\nsubmission in order to explain the differences, just as you would if you\nhad previously published related work. In such cases, include the\nanonymized parallel submission~\\cite{Authors20} as additional material and\ncite it as\n\\begin{quote}\n[1] Authors. ``The frobnicatable foo filter'', F\\&G 2020 Submission ID 324,\nSupplied as additional material {\\tt fg324.pdf}.\n\\end{quote}\n\nFinally, you may feel you need to tell the reader that more details can be\nfound elsewhere, and refer them to a technical report. For conference\nsubmissions, the paper must stand on its own, and not {\\em require} the\nreviewer to go to a techreport for further details. Thus, you may say in\nthe body of the paper ``further details may be found\nin~\\cite{Authors20b}''. Then submit the techreport as additional material.\nAgain, you may not assume the reviewers will read this material.\n\nSometimes your paper is about a problem which you tested using a tool which\nis widely known to be restricted to a single institution. For example,\nlet's say it's 1969, you have solved a key problem on the Apollo lander,\nand you believe that the WACV 70 audience would like to hear about your\nsolution. The work is a development of your celebrated 1968 paper entitled\n``Zero-g frobnication: How being the only people in the world with access to\nthe Apollo lander source code makes us a wow at parties'', by Zeus \\etal.\n\nYou can handle this paper like any other. Don't write ``We show how to\nimprove our previous work [Anonymous, 1968]. This time we tested the\nalgorithm on a lunar lander [name of lander removed for blind review]''.\nThat would be silly, and would immediately identify the authors. Instead\nwrite the following:\n\\begin{quotation}\n\\noindent\n We describe a system for zero-g frobnication. This\n system is new because it handles the following cases:\n A, B. Previous systems [Zeus et al. 1968] didn't\n handle case B properly. Ours handles it by including\n a foo term in the bar integral.\n\n ...\n\n The proposed system was integrated with the Apollo\n lunar lander, and went all the way to the moon, don't\n you know. It displayed the following behaviours\n which show how well we solved cases A and B: ...\n\\end{quotation}\nAs you can see, the above text follows standard scientific convention,\nreads better than the first version, and does not explicitly name you as\nthe authors. A reviewer might think it likely that the new paper was\nwritten by Zeus \\etal, but cannot make any decision based on that guess.\nHe or she would have to be sure that no other authors could have been\ncontracted to solve problem B.\n\\medskip\n\n\\noindent\nFAQ\\medskip\\\\\n{\\bf Q:} Are acknowledgements OK?\\\\\n{\\bf A:} No. Leave them for the final copy.\\medskip\\\\\n{\\bf Q:} How do I cite my results reported in open challenges?\n{\\bf A:} To conform with the double blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\\medskip\\\\\n\n\\begin{figure}[t]\n\\begin{center}\n\\fbox{\\rule{0pt}{2in} \\rule{0.9\\linewidth}{0pt}}\n \n\\end{center}\n \\caption{Example of caption. It is set in Roman so that mathematics\n (always set in Roman: $B \\sin A = A \\sin B$) may be included without an\n ugly clash.}\n\\label{fig:long}\n\\label{fig:onecol}\n\\end{figure}\n\n\\subsection{Miscellaneous}\n\n\\noindent\nCompare the following:\\\\\n\\begin{tabular}{ll}\n \\verb'$conf_a$' & $conf_a$ \\\\\n \\verb'$\\mathit{conf}_a$' & $\\mathit{conf}_a$\n\\end{tabular}\\\\\nSee The \\TeX book, p165.\n\nThe space after \\eg, meaning ``for example'', should not be a\nsentence-ending space. So \\eg is correct, {\\em e.g.} is not. The provided\n\\verb'\\eg' macro takes care of this.\n\nWhen citing a multi-author paper, you may save space by using ``et alia'',\nshortened to ``\\etal'' (not ``{\\em et.\\ al.}'' as ``{\\em et}'' is a complete word.)\nHowever, use it only when there are three or more authors. Thus, the\nfollowing is correct: ``Frobnication has been trendy lately.\n It was introduced by Alpher~\\cite{Alpher02}, and subsequently developed by\n Alpher and Fotheringham-Smythe~\\cite{Alpher03}, and Alpher \\etal~\\cite{Alpher04}.''\n\nThis is incorrect: ``... subsequently developed by Alpher \\etal~\\cite{Alpher03} ...''\nbecause reference~\\cite{Alpher03} has just two authors. If you use the\n\\verb'\\etal' macro provided, then you need not worry about double periods\nwhen used at the end of a sentence as in Alpher \\etal.\n\nFor this citation style, keep multiple citations in numerical (not\nchronological) order, so prefer \\cite{Alpher03,Alpher02,Authors20} to\n\\cite{Alpher02,Alpher03,Authors20}.\n\n\n\\begin{figure*}\n\\begin{center}\n\\fbox{\\rule{0pt}{2in} \\rule{.9\\linewidth}{0pt}}\n\\end{center}\n \\caption{Example of a short caption, which should be centered.}\n\\label{fig:short}\n\\end{figure*}\n\n\\section{Formatting your paper}\n\nAll text must be in a two-column format. The total allowable width of the\ntext area is $6\\frac78$ inches (17.5 cm) wide by $8\\frac78$ inches (22.54\ncm) high. Columns are to be $3\\frac14$ inches (8.25 cm) wide, with a\n$\\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the\nfirst page) should begin 1.0 inch (2.54 cm) from the top edge of the\npage. The second and following pages should begin 1.0 inch (2.54 cm) from\nthe top edge. On all pages, the bottom margin should be 1-1\/8 inches (2.86\ncm) from the bottom edge of the page for $8.5 \\times 11$-inch paper; for A4\npaper, approximately 1-5\/8 inches (4.13 cm) from the bottom edge of the\npage.\n\n\\subsection{Margins and page numbering}\n\nAll printed material, including text, illustrations, and charts, must be kept\nwithin a print area 6-7\/8 inches (17.5 cm) wide by 8-7\/8 inches (22.54 cm)\nhigh.\n\nPage numbers should be in footer with page numbers, centered and .75\ninches from the bottom of the page and make it start at the correct page\nnumber rather than the 9876 in the example. To do this find the secounter\nline (around line 33 in this file) and update the page number as\n\\begin{verbatim}\n\\setcounter{page}{123}\n\\end{verbatim}\nwhere the number 123 is your assigned starting page.\n\n\\subsection{Type-style and fonts}\n\nWherever Times is specified, Times Roman may also be used. If neither is\navailable on your word processor, please use the font closest in\nappearance to Times to which you have access.\n\nMAIN TITLE. Center the title 1-3\/8 inches (3.49 cm) from the top edge of\nthe first page. The title should be in Times 14-point, boldface type.\nCapitalize the first letter of nouns, pronouns, verbs, adjectives, and\nadverbs; do not capitalize articles, coordinate conjunctions, or\nprepositions (unless the title begins with such a word). Leave two blank\nlines after the title.\n\nAUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title\nand printed in Times 12-point, non-boldface type. This information is to\nbe followed by two blank lines.\n\nThe ABSTRACT and MAIN TEXT are to be in a two-column format.\n\nMAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use\ndouble-spacing. All paragraphs should be indented 1 pica (approx. 1\/6\ninch or 0.422 cm). Make sure your text is fully justified---that is,\nflush left and flush right. Please do not place any additional blank\nlines between paragraphs.\n\nFigure and table captions should be 9-point Roman type as in\nFigures~\\ref{fig:onecol} and~\\ref{fig:short}. Short captions should be centred.\n\n\\noindent Callouts should be 9-point Helvetica, non-boldface type.\nInitially capitalize only the first word of section titles and first-,\nsecond-, and third-order headings.\n\nFIRST-ORDER HEADINGS. (For example, {\\large \\bf 1. Introduction})\nshould be Times 12-point boldface, initially capitalized, flush left,\nwith one blank line before, and one blank line after.\n\nSECOND-ORDER HEADINGS. (For example, { \\bf 1.1. Database elements})\nshould be Times 11-point boldface, initially capitalized, flush left,\nwith one blank line before, and one after. If you require a third-order\nheading (we discourage it), use 10-point Times, boldface, initially\ncapitalized, flush left, preceded by one blank line, followed by a period\nand your text on the same line.\n\n\\subsection{Footnotes}\n\nPlease use footnotes\\footnote {This is what a footnote looks like. It\noften distracts the reader from the main flow of the argument.} sparingly.\nIndeed, try to avoid footnotes altogether and include necessary peripheral\nobservations in\nthe text (within parentheses, if you prefer, as in this sentence). If you\nwish to use a footnote, place it at the bottom of the column on the page on\nwhich it is referenced. Use Times 8-point type, single-spaced.\n\n\n\\subsection{References}\n\nList and number all bibliographical references in 9-point Times,\nsingle-spaced, at the end of your paper. When referenced in the text,\nenclose the citation number in square brackets, for\nexample~\\cite{Authors20}. Where appropriate, include the name(s) of\neditors of referenced books.\n\n\\begin{table}\n \\begin{center}\n {\\small{\n\\begin{tabular}{llr}\n\\toprule\nMethod & Frobnability & Accuracy \\\\\n\\midrule\nTheirs & Frumpy & 30.2\\%\\\\\nYours & Frobbly & 45.2\\%\\\\\nOurs & Makes one's heart Frob & 99.9\\%\\\\\n\\bottomrule\n\\end{tabular}\n}}\n\\end{center}\n\\caption{Results. Ours are better. If you prefer, you can put table\ncaptions on top of the tables instead of on the bottom.}\n\\end{table}\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered. Please ensure that any point you wish to\nmake is resolvable in a printed copy of the paper. Resize fonts in figures\nto match the font in the body text, and choose line widths which render\neffectively in print. Many readers (and reviewers), even of an electronic\ncopy, will choose to print your paper in order to read it. You cannot\ninsist that they do otherwise, and therefore must not assume that they can\nzoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it's almost always best to use\n\\verb+\\includegraphics+, and to specify the figure width as a multiple of\nthe line width as in the example below\n{\\small\\begin{verbatim}\n \\usepackage[dvips]{graphicx} ...\n \\includegraphics[width=0.8\\linewidth]\n {myfile.eps}\n\\end{verbatim}\n}\n\n\n\\subsection{Color}\n\nPlease refer to the author guidelines on the WACV 2023 web page \n(\\url{http:\/\/wacv2023.thecvf.com\/submission\/})\nfor a discussion of the use of color in your document.\n\n\\section{Final copy}\n\nYou must include your signed IEEE copyright release form when you submit\nyour finished paper. We MUST have this form before your paper can be\npublished in the proceedings.\n\nPlease direct any questions to the production editor in charge of these\nproceedings at the IEEE Computer Society Press: \n\\url{https:\/\/www.computer.org\/about\/contact}.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Introduction}\n\nPlease use this template for two separate purposes. (1) If you are\nsubmitting a paper to the second round for a paper that received a\nRevise decision in the first round, please use this template to\naddress the reviewer concerns and highlight the changes made since the\noriginal submission. (2) If you submit a paper to the second round,\nafter you receive the reviews, you may optionally submit a rebuttal to\naddress factual errors or to supply additional information requested by\nthe reviewers (this rebuttal will be seen by Area Chairs but not by\nthe reviewers). In either case, you are limited to a {\\bf one page}\nPDF file. Please follow the steps and style guidelines outlined below\nfor submitting your author response.\n\n\nThe rebuttal must adhere to the same blind-submission as the original submission and must comply with this rebuttal-formatted template.\n\n\n\\subsection{Response length}\nAuthor responses must be no longer than 1 page in length including any references and figures. Overlength responses will simply not be reviewed. This includes responses where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide. Note that this \\LaTeX\\ guide already sets figure captions and references in a smaller font.\n\n\\section{Formatting your Response}\n\n{\\bf Make sure to update the paper title and paper ID in the appropriate place in the tex file.}\n\nAll text must be in a two-column format. The total allowable width of the text\narea is $6\\frac78$ inches (17.5 cm) wide by $8\\frac78$ inches (22.54 cm) high.\nColumns are to be $3\\frac14$ inches (8.25 cm) wide, with a $\\frac{5}{16}$ inch\n(0.8 cm) space between them. The top margin should begin\n1.0 inch (2.54 cm) from the top edge of the page. The bottom margin should be\n1-1\/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \\times\n11$-inch paper; for A4 paper, approximately 1-5\/8 inches (4.13 cm) from the\nbottom edge of the page.\n\nPlease number all of your sections and any displayed equations. It is important\nfor readers to be able to refer to any particular equation.\n\nWherever Times is specified, Times Roman may also be used. Main text should be\nin 10-point Times, single-spaced. Section headings should be in 10 or 12 point\nTimes. All paragraphs should be indented 1 pica (approx. 1\/6 inch or 0.422\ncm). Figure and table captions should be 9-point Roman type as in\nFigure~\\ref{fig:onecol}.\n\n\nList and number all bibliographical references in 9-point Times, single-spaced,\nat the end of your response. When referenced in the text, enclose the citation\nnumber in square brackets, for example~\\cite{Alpher02}. Where appropriate,\ninclude the name(s) of editors of referenced books.\n\n\\begin{figure}[t]\n\\begin{center}\n\\fbox{\\rule{0pt}{1in} \\rule{0.9\\linewidth}{0pt}}\n \n\\end{center}\n \\caption{Example of caption. It is set in Roman so that mathematics\n (always set in Roman: $B \\sin A = A \\sin B$) may be included without an\n ugly clash.}\n\\label{fig:long}\n\\label{fig:onecol}\n\\end{figure}\n\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered. Please ensure that any point you wish to make is resolvable in a printed copy of the response. Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print. Many readers (and reviewers), even of an electronic copy, will choose to print your response in order to read it. You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it's almost always best to use \\verb+\\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below\n{\\small\\begin{verbatim}\n \\usepackage[dvips]{graphicx} ...\n \\includegraphics[width=0.8\\linewidth]\n {myfile.eps}\n\\end{verbatim}\n}\n\n\n{\\small\n\\bibliographystyle{ieee}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we propose a novel approach for learning low bit-width DNNs models by distilling knowledge from multiple quantized teachers. We introduce the idea of collaborative learning that allows teachers to form importance-aware shared knowledge, which will be used to guide the student. The proposed framework also leverages the idea of mutual learning that allows both teachers and student to adjust their parameters to achieve an overall object function. The proposed framework allows end-to-end training in which not only network parameters but also the importance factors indicating the contributions of teachers to the shared knowledge are updated simultaneously. The experimental results on CIFAR-100 and ImageNet datasets with AlexNet and ResNet18 architectures demonstrate that the low bit-width models trained with the proposed approach achieve competitive results compared to the state-of-the-art methods.\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\subsection{Experimental setup}\n\\paragraph{\\textbf{Datasets.}}\nWe conduct experiments on {CIFAR-100} \\cite{cifar} and {ImageNet} (ILSVRC-2012) \\cite{imagenet} datasets. CIFAR-100 dataset consists of $100$ classes with the total of $60,000$ images, where $50,000$ and $10,000$ images are used for training and testing set, respectively. ImageNet is a large scale dataset with $1,000$ classes in total. This dataset contains $1.2$ million images for training and $50,000$ images for validation, which is used as test set in our experiments.\n\n\\paragraph{\\textbf{Implementation details.}} \nWe evaluate our proposed method on two common deep neural networks AlexNet \\cite{alexnet} and ResNet18 \\cite{Resnet}. Regarding AlexNet, batch normalization layers are added after each convolutional layer and each fully connected layer, which are similar to works done by \\cite{HWGQ, dorefa}. In all experiments, similar to previous works \\cite{LQNets,guidedQuantize}, in the training, we use the basic augmentation including horizontal flips, resizing\nand randomly cropping that crops images to $227\\times 227$ and $224\\times 224$ pixels for ResNet18 and AlexNet, respectively. \nWe use Stochastic Gradient Descents with a momentum of $0.9$ and a mini-batch size of $256$. The learning rate $lr$ for network models is set to $0.1$ and $0.01$ for ResNet and AlexNet, respectively.\nThe learning rate for the importance factors ($\\pi$) in teacher models is set to $lr\/10$\n\n\n\nWhen training ResNet18 model on CIFAR-100, we train the model with 120 epochs. The learning rate is decreased by a factor of 10 after 50 and 100 epochs. When training ResNet18 model on ImageNet, we train the model with 100 epochs. The learning rate is decreased by a factor of 10 after $30$, $60$, and $90$ epochs. When training AlexNet model, for both CIFAR-100 and ImageNet, we train the model with $100$ epochs and we adopt $cosine$ learning rate decay.\n\nWe set the weight decay to 25e-6 for the 1 or 2-bit precision and set it to 1e-4 for higher precisions. Regarding hyper-parameters of the total loss (\\ref{eq:final_loss}), {we empirically set $\\alpha=1$, $\\beta=0.5$}. \nIn our experiments, the shared knowledge is formed at certain layers of teachers and is distilled to the correspondence layers of student. \nSpecifically, the shared knowledge is formed at the last convolutional layers of each convolution block, i.e., layers $2, 5$, and $7$ of AlexNet teachers and from layers $5, 9, 13$, and $17$ of ResNet18 teachers.\nMeanwhile, we set $\\gamma$ to $100$ or $1$ when attention loss or FitNet loss is used in $\\mathcal{L}_{feat}$. We do not quantize the first and last layers. \n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Ablation studies}\n\\label{subsec:ablation}\n\n\n\nWe conduct several ablation studies on CIFAR-100 with ResNet18 and AlexNet to demonstrate the effectiveness of our proposed method.\nFor ablation studies, we use HWGQ quantizer \\cite{HWGQ} for the proposed CMT-KD. In addition, for the intermediate features-based distillation ($\\mathcal{L}_{feat}$) in the final loss (eq. (\\ref{eq:final_loss})), the attention loss is used. We consider the following settings. \n\n\\begin{table}[!t]\n\\def1.05{1.0}\n \\centering\n \\begin{tabular}{c|c|c|c}\n \\hline\n Models & Bit-width & Top 1 & Top 5 \\\\\n\n \\hline\n \\multirow{5}{*}{Single model} \n & FP & 72.4 & 91.3 \\\\\n & 8 bits& 70.9 & 90.9 \\\\\n & 6 bits& 70.8 & 90.8 \\\\\n & 4 bits& 70.7 & 90.8 \\\\\n & 2 bits& 69.4 & 90.5 \\\\\n \\hline\n KD (from FP teacher) & \\multirow{5}{*}{2 bits} &\n {71.3} & {91.6} \\\\\n Average teacher & & 71.0 & 91.6 \\\\\n \n CMT-KD (w\/o Att) & & 71.8 & 91.8 \\\\\n CMT-KD (w\/o ML) & & 70.9 & 91.3 \\\\\n CMT-KD & & 72.1 & 91.9 \\\\\n \\hline\n Combined teacher &4, 6, 8 bits & 72.3 & 91.7 \\\\\n \n \n \\hline\n \\end{tabular}\n \\caption{Ablation studies on the CIFAR-100 dataset with AlexNet. The descriptions of settings are presented in Section \\ref{subsec:ablation}.}\n \\label{tab:ablationAlexNet} \n \n\\end{table}\n\n\\begin{table}[!t]\n\\def1.05{1}\n \\centering\n \\begin{tabular}{c|c|c|c}\n \n \\hline\n Models & Bit-width & Top 1 & Top 5 \\\\\n \n \\hline\n \\multirow{5}{*}{Single model} \n &FP& 75.3 & 93.1 \\\\\n &8 bits& 75.2 & 92.9 \\\\\n &6 bits& 74.9 & 92.7 \\\\\n &4 bits& 74.9 & 92.5 \\\\\n &2 bits& 72.9 & 91.9 \\\\\n \\hline\n KD (from FP teacher) & \\multirow{5}{*}{2 bits} & {75.1} & {92.8} \\\\\n Average teacher & & 76.0 & 93.8 \\\\\n \n CMT-KD (w\/o Att) & & 76.5 & 93.9 \\\\\n CMT-KD (w\/o ML) & & 75.0 & 93.2 \\\\\n CMT-KD & & {78.3} & {94.4} \\\\\n \\hline\n Combined teacher &4, 6, 8 bits & 79.5 & 94.9 \\\\\n \n \\hline\n \\end{tabular}\n \\caption{Ablation studies on the CIFAR-100 dataset with ResNet18. The descriptions of settings are presented in Section \\ref{subsec:ablation}.}\n \\label{tab:ablationResNet18} \n \\vspace{-0.5em}\n\\end{table}\n\n\n \n\\paragraph{\\textbf{Single model.}} We evaluate single models with different precisions (i.e., full precision, 8-bit, 6-bit, 4-bit, and 2-bit precisions) without any distillation methods. The results for AlexNet and ResNet18 architectures are presented in Table \\ref{tab:ablationAlexNet} and Table \\ref{tab:ablationResNet18}, respectively. With the AlexNet architecture, the results show that the 4-bit, 6-bit, 8-bit models achieve comparable results. There are considerable large gaps between these models and the full precision model. There are also large gaps between those models and the 2-bit model. With the ResNet18 architecture, the 8-bit model achieves comparable results to the full precision model. There is a small gap between the 6-bit, 4-bit models and the 8-bit model. Similar to the observation on the AlexNet, there is a large gap between the 2-bit model and the full precision model. \n\\paragraph{\\textbf{Knowledge distillation from the full precision teacher.}} In this setting, we train a 2-bit student model with the knowledge distillation from the full precision teacher, i.e., the KD (from FP teacher) setting in Table \\ref{tab:ablationAlexNet} and Table \\ref{tab:ablationResNet18}. For this setting, we follow \\cite{KD_Hinton}, i.e., when training the student, in addition to the cross-entropy loss, the softmax outputs from the teacher will be distilled to the student. When AlexNet is used, this setting achieves better performance than quantized single models. When ResNet18 is used, this setting achieves comparable results to the 8-bit single quantized model. \n\n\\paragraph{\\textbf{Knowledge distillation from an ensemble of multiple quantized teachers.}}\nIn this setting, we separately train three teachers with different precisions, i.e., 4-bit, 6-bit, and 8-bit teachers. The averaged softmax outputs of teachers are distilled to the 2-bit student. This setting is noted as ``Average teacher\" in Table \\ref{tab:ablationAlexNet} and Table~\\ref{tab:ablationResNet18}. It is worth noting that this setting is also used in the previous work \\cite{DBLP:conf\/kdd\/YouX0T17}.\n When AlexNet is used, at the top-1 accuracy, this setting produces the 2-bit student that achieves comparable performance with the quantized single teachers.\n However, its performance ($71.0\\%$) is still lower than the full precision model ($72.4\\%$). When ResNet18 is used, this setting improves the performance over the full precision model, i.e., the gain is $0.7\\%$ for both top-1 and top-5 accuracy. \n\n\\paragraph{\\textbf{Effectiveness of collaborative and mutual learning.}} We consider different settings of the proposed framework. \nFor the results in Table \\ref{tab:ablationAlexNet} and Table \\ref{tab:ablationResNet18}, when training CMT-KD models, we use 3 teachers, i.e., 4-bit, 6-bit, and 8-bit teachers. In those tables, CMT-KD means that the models are trained with the total loss (\\ref{eq:final_loss}). CMT-KD (w\/o Att) means that the models are trained with the loss (\\ref{eq:final_loss}) but the intermediate features-based component $\\mathcal{L}_{feat}$ is excluded. CMT-KD (w\/o ML) means that the models are trained with the loss (\\ref{eq:final_loss}) but the mutual learning component $(\\mathcal{L}_{KL}^S + \\mathcal{L}_{KL}^T)$ is excluded. The results show that the mutual learning loss is more effective than the intermediate features-based loss. However, both components are necessary to achieve the best results, i.e., CMT-KD. \n\nWhen AlexNet is used, the full precision (FP) model slightly outperforms the proposed CMT-KD at top-1 accuracy.\nWhen ResNet18 is used, CMT-KD outperforms the FP model at both top-1 and top-5 accuracy. A significant gain is achieved, i.e., $3\\%$, at top-1 accuracy. It is worth noting that when ResNet18 is used, the CMT-KD significantly outperforms the 2-bit single model, the 2-bit model when using the average teacher, and the 2-bit model when distilling from the FP model. Those results confirm the effectiveness of the proposed method. \n\n\\paragraph{\\textbf{Combined teacher.}} We also evaluate the performance of the combined teacher in the collaborative learning in our proposed method, i.e., the predictions which are made by the classifier corresponding to the $\\mathcal{L}^T_{CE}$ loss in Figure \\ref{fig:1}. Overall, this setting produces the best results, except for top-1 accuracy with AlexNet architecture. It achieves better performance than the ``Average teacher'' setting. \nWith ResNet18, this setting significantly outperforms the full precision model. Those results confirm the effectiveness of the proposed collaborative learning between teachers. \n\\begin{table}[]\n\\vspace{-1em}\n\\vspace{0.5em}\n\\def1.05{1.1}\n\\centering\n \\begin{tabular}{c|ccc}\n \\hline Setting & & AlexNet & ResNet18 \\\\\n \\hline \\multirow{2}{*}{ (a) } & Top-1 & {72.1} & {78.3} \\\\\n & Top-5 & 91.9 & 94.4 \\\\\n \\hline \\multirow{2}{*}{ (b) } & Top-1 & 71.1 & 78.1 \\\\\n & Top-5 & 91.2 & 94.3 \\\\\n \\hline\n \\end{tabular}\n\\vspace{-0.5em}\n\\caption{Impact of the number of teachers on the CMT-KD 2-bit students. The results are on the CIFAR-100 dataset. (a) Using 4-bit, 6-bit, and 8-bit teachers. (b) Using 4-bit and 8-bit teachers.}\n\\label{tab:numteachers}\n\\end{table}\n\n\\paragraph{\\textbf{Impact of the number of teachers.}} The results in Table \\ref{tab:numteachers} show the impact of the number of teachers on the performance of the 2-bit CMT-KD student models. The results show that using 3 teachers (4-bit, 6-bit, and 8-bit) slightly improves the performance when using 2 teachers (4-bit and 8-bit).\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Comparison with the state of the art}\nIn this section, we compare our proposed method CMT-KD against state-of-the-art network quantization methods, including LQ-Net \\cite{LQNets}, LQW+CAQ \\cite{LQW_CAQ}, HWGQ \\cite{HWGQ}, and DoReFa-Net \\cite{dorefa}. We also make a comparison between our approach and methods that apply both distillation and quantization consisting of PQ+TS+Guided \\cite{guidedQuantize}, QKD \\cite{QKD}, SPEQ \\cite{SPEQ}. \nFor CMT-KD, we use three teachers (4-bit, 6-bit, and 8-bit teachers) to guide the learning of compact quantized 2-bit weights ($K_w = 2$) and 2-bit activations ($K_a = 2$) students. Meanwhile, we use 2-bit, 4-bit, and 8-bit teachers to guide the learning of compact quantized 1-bit weights ($K_w = 1$) and 2-bit activations ($K_a = 2$) students.\nWe do not consider 1-bit activations because the previous \\cite{HWGQ,NIPS2016_d8330f85,xnor_net,dorefa} show that 1-bit quantization for activations is not sufficient for good performance. \n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[!t] \n\\def1.05{1.05}\n\\centering\n\\vspace{0.5em}\n\\begin{tabular}{l|c|c|c|c}\n\\hline \n\\multirow{2}{*}{ Method } & \\multicolumn{2}{|c|}{ AlexNet } & \\multicolumn{2}{c}{ ResNet18 } \\\\\n\\cline{2-5} & Top-1 & Top-5 & Top-1 & Top-5 \\\\\n\\hline \n\\hline\n\\multicolumn{1}{l|}{Full precision} & 72.4 & 91.3 & 75.3 & 93.1 \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=1$, $K_a=2$ } \\\\\n\\hline\nLQ-Nets \\cite{LQNets} & 68.7 & 90.5 & 70.4 & 91.2 \\\\\nLQW + CAQ \\cite{LQW_CAQ} & 69.3 & 91.2 & 72.1 & 91.6 \\\\\nHWGQ \\cite{HWGQ} & 68.6 & 90.8 & 71.0 & 90.8 \\\\\n{CMT-KD-FitNet} & 69.9 & \\textbf{91.3} & \\textbf{76.1} & \\textbf{93.7} \\\\\n {CMT-KD-Att} & \\textbf{70.4} & {91.1} & {75.6} & {93.5} \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=2$, $K_a=2$ } \\\\\n\\hline\nPQ+TS+Guided \\cite{guidedQuantize} & 64.6 & 87.8 & - & - \\\\\nLQ-Net \\cite{LQNets} & 69.2 & 91.2 & 70.8 & 91.3 \\\\\nLQW + CAQ \\cite{LQW_CAQ} & 69.9 & {91.3} & 72.1 & 91.6 \\\\\nHWGQ \\cite{HWGQ} & 69.4 & 90.5 & 72.9 & 91.9 \\\\\n{CMT-KD-FitNet} & {70.0} & {90.7} & \\textbf{78.7} & \\textbf{94.6} \\\\\n{CMT-KD-Att} &\\textbf{72.1} & \\textbf{91.9} & 78.3 & 94.4 \\\\\n\\hline\n\\end{tabular}\n\\caption{The comparative results on the CIFAR-100 dataset. We report the results of CMT-KD when FitNet loss or attention loss (Att) is used for the intermediate features-based distillation. CMT-KD uses HWGQ quantizer to quantize teachers and student. The results of HWGQ are reported by using the official released code.}\n\\label{tab:sota_cifar10} \n\\end{table}\n\\paragraph{\\textbf{Comparative results on CIFAR-100.}} Table \\ref{tab:sota_cifar10} presents the top-1 and top-5 classification accuracy on CIFAR-100 dataset of different network quantization methods for AlexNet and ResNet18. {The results of competitors are cited from \\cite{LQW_CAQ,guidedQuantize}}. \nWe report the results of our CMT-KD when FitNet loss or attention loss is used for $\\mathcal{L}_{feat}$ in Eq. (\\ref{eq:final_loss}). Those models are denoted as CMT-KD-FitNet or CMT-KD-Att. The quantizer HWGQ \\cite{HWGQ} is used to quantize teachers and student networks when training CMT-KD models. \nOverall, the best CMT-KD models outperform most of the competitor quantization methods. When AlexNet is used,\nthe CMT-KD (for both FitNet and Att) models outperform the compared quantization methods at top-1 accuracy. However, the proposed models achieve lower performance than the FP model at top-1 accuracy. This may be due to the limit in the capacity of the AlexNet model, which consists of only 5 convolutional layers. \n\nWhen ResNet18 is used, our CMT-KD models outperform the full precision model. Especially when using 2-bit weights and 2-bit activations, the improvements of the CMT-KD-FitNet over the FP model are $3.4\\%$ and $1.5\\%$ for top-1 and top-5 accuracy, respectively.\n{It is also worth noting that the proposed models significantly improve over the HWGQ method~\\cite{HWGQ} which uses HWGQ quantizer to quantize FP models, i.e., with $K_w=2, K_a = 2$, CMT-KD-FitNet outperforms HWGQ \\cite{HWGQ} $5.8\\%$ at top-1 accuracy.} \n\n\\begin{table}[!t]\n\\def1.05{1.05}\n\\centering\n\\vspace{0.5em}\n\\begin{tabular}{l|c|c|c|c}\n\\hline \n\\multirow{2}{*}{ Method } & \\multicolumn{2}{|c|}{ AlexNet } & \\multicolumn{2}{c}{ ResNet18 } \\\\\n\\cline{2-5} & Top-1 & Top-5 & Top-1 & Top-5 \\\\\n\\hline \n\\hline\n\\multicolumn{1}{l|}{ Full precision} & 61.8 & 83.5 & 70.3 & 89.5 \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=1$, $K_a=2$ } \\\\\n\\hline\nDoReFa-Net \\cite{dorefa} & 49.8 & - & 53.4 & - \\\\\nLQ-Nets \\cite{LQNets} & 55.7 & 78.8 & \\textbf{62.6} & \\textbf{84.3} \\\\\nHWGQ \\cite{HWGQ} & 52.7 & 76.3 & 59.6 & 82.2 \\\\\n{CMT-KD (HWGQ)} &\\textbf{56.2} &\\textbf{79.1} & 60.6 & 83.5 \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=2$, $K_a=2$ } \\\\\n\\hline \nDoReFa-Net \\cite{dorefa} & 48.3 & 71.6 & 57.6 & 80.8 \\\\\nQKD \\cite{QKD} & - & - & 67.4 & 87.5 \\\\\nPQ + TQ Guided \\cite{guidedQuantize} & 52.5 & 77.3 & - & - \\\\\nLQ-Net \\cite{LQNets} & 57.4 & 80.1 & 64.9 & 85.9 \\\\\nSPEQ \\cite{SPEQ} & 59.3 & - & 67.4 & - \\\\\nHWGQ \\cite{HWGQ,HWGQ_Rethinking} & 58.6 & 80.9 & 65.1 & 86.2 \\\\\nCMT-KD (HWGQ) &59.2 &81.3 & {65.6} & {86.5} \\\\\nLSQ (with distill) \\cite{LSQ} & - & - & \\textbf{67.9} & \\textbf{88.1} \\\\\nLSQ* (w\/o distill) & - & - & {66.7} & {87.1} \\\\\nCMT-KD (LSQ) &\\textbf{59.3} &\\textbf{81.5} & {67.8} & {87.8} \\\\\n\\hline\n\\end{tabular}\n\\caption{The comparative results on the ImageNet dataset. CMT-KD uses the attention loss for the intermediate features-based distillation. CMT-KD (HWGQ) and CMT-KD (LSQ) denote models when HWGQ \\cite{HWGQ} and LSQ \\cite{LSQ} quantizers are used in our framework, respectively. We report experimental results for LSQ (please refer to footnote 1) without distillation in LSQ* row.} \n\\label{tab:sota_imagenet} \n\\end{table}\n\\vspace{-0.5em}\n\\paragraph{\\textbf{Comparative results on ImageNet.}}\n\n\n\n\\begin{figure*}[!t]\n \\centering\n \\begin{subfigure}[b]{0.37\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{img\/layer11.pdf}\n \\caption{Layer 5}\n \\label{fig:layer 1}\n \\end{subfigure}\n \n \\hspace{0.8cm}\n \\begin{subfigure}[b]{0.37\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{img\/layer22.pdf}\n \\caption{Layer 9}\n \\label{fig:layer 2}\n \\end{subfigure}\n \n \\begin{subfigure}[b]{0.37\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{img\/layer33.pdf}\n \\caption{Layer 13}\n \\label{fig:layer 3}\n \\end{subfigure}\n \n \\hspace{0.8cm}\n \\begin{subfigure}[b]{0.37\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{img\/layer44.pdf}\n \\caption{Layer 17}\n \\label{fig:layer 4}\n \\end{subfigure}\n \\caption{The importance factors of the three ResNet18 teachers (4-bit, 6-bit, 8-bit) on the CIFAR-100 dataset during the training process.}\n \\label{fig:importancefactor}\n \\vspace{-1em}\n\\end{figure*}\n\nTable \\ref{tab:sota_imagenet} presents the top-1 and top-5 classification accuracy on ImageNet dataset of different network quantization methods for AlexNet and ResNet18. The results of competitors are cited from the corresponding papers.\nAt $K_w = 1$ and $K_a =2$, when using AlexNet, the proposed CMT-KD significantly outperforms HWGQ \\cite{HWGQ}. The gain is $4.5\\%$ and $2.8\\%$ for top-1 and top-5 accuracy, respectively. For ResNet18, we also achieved an improvement of $1\\%$ compared to HWGQ at top-1 accuracy. \nWith $K_w = 2$ and $K_a =2$, the CMT-KD (HWGQ) outperforms the HWGQ \\cite{HWGQ} method by $0.6\\%$, and $0.5\\%$ on top-1 accuracy for AlexNet and ResNet18, respectively. \n\nAs the proposed framework is flexible to quantizers, we also report in Table \\ref{tab:sota_imagenet} the results when LSQ~\\cite{LSQ} quantizer is used to quantize teacher and student networks in our framework, i.e., CMT-KD (LSQ). LSQ is a quantization method in which the step size is learned during training.\n When ResNet18 is used, given the same LSQ quantizer implementation\\footnote{\\label{note1}The official source code of LSQ is not available. We adopt the LSQ quantizer from an un-official implementation from \\url{https:\/\/github.com\/hustzxd\/LSQuantization} for our experiments.}, our method CMT-KD (LSQ) can boost the top-1 accuracy of LSQ* by $1.1\\%$. \nHowever, we note that the results reported in LSQ \\cite{LSQ} slightly outperforms CMT-KD (LSQ). It is worth noting that in order to achieve the reported results ($67.9\\%$ top-1 and $88.1\\%$ top-5), LSQ \\cite{LSQ} also uses knowledge distillation to distill knowledge from the full precision model to the quantized model\\footnote{\nWe are unsuccessful in reproducing results reported in LSQ \\cite{LSQ}. For example, without distillation for LSQ, we only achieve a result of $66.7\\%$ for top-1 accuracy when using ResNet18 with $K_w=2, K_a=2$, while in \\cite{LSQ}, the authors reported $67.6\\%$ at the same setting.}. \nOur best method CMT-KD (LSQ) compares favorably to the recent method SPEQ~\\cite{SPEQ} for both AlexNet and ResNet18 models.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{\\textbf{Visualization of the importance factors ($\\pi$) of teachers.}}\nFigure \\ref{fig:importancefactor} visualizes the importance factors of three teachers (4-bit, 6-bit, 8-bit) when using ResNet18 architecture for the teachers and student. The experiment is conducted on CIFAR-100 dataset. The visualized importance factors are at the last convolutional layers in each block of ResNet18 during training, i.e., layers 5, 9, 13, 17. They are also the layers in which the shared knowledge is formed. \nFigure \\ref{fig:layer 2} shows that the highest precision teacher (8-bit) does not always give the highest contributions to the shared knowledge at all layers. For example, at layer 9, the importance factor of the 6-bit teacher is higher than the importance factor of the 8-bit teacher. At the last convolutional layer (i.e., layer 17), the 8-bit teacher dominates other teachers and gives most of the contributions to the shared knowledge. In addition, the importance factors are mainly updated at the early training stage of the framework. \n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\nDeep Convolutional Neural Networks (CNNs) have achieved tremendous successes in a variety of computer vision tasks, including image classification, object detection, segmentation, and image retrieval. However, CNNs generally require excessive memory and expensive computational resources, limiting their usage in many applications. Therefore, a great number of researches have been devoted to {making} CNNs lightweight and to {improving} inference efficiency for practical applications.\n\nAn effective approach is to use low bit-width weights and\/or low bit-width activations. This approach not only can reduce the memory footprint but also achieves a significant gain in speed, as the most computationally expensive convolutions can be done by bitwise operations \\cite{xnor_net}. Although existing quantization-based methods \\cite{HWGQ,lossaware,xnor_net,tbn,LQNets,dorefa,TTQ} achieved improvements, there are still noticeable accuracy gaps between the quantized CNNs and their full precision counterparts, especially in the challenging cases of 1 or 2 bit-width weights and activations. \n\nModel compression using knowledge distillation is another attractive approach to reduce the computational cost of DNNs \\cite{overhaul_fea_distill,KD_Hinton,QKD,apprentice,FitNets}. In knowledge distillation, a smaller student network is trained to mimic the behaviour of a cumbersome teacher network. \nTo further enhance the performance of {the} student network, some works~\\cite{adaptive_multi_teacher,DBLP:conf\/ecai\/ParkK20,DBLP:conf\/kdd\/YouX0T17} propose to distill knowledge from multiple teachers. However, in those works, teacher models are separately pretrained, which would limit the collaborative learning between teachers. It also limits the mutual learning between student network and teacher networks. \n\nTo improve the compact low bit-width student network, in this paper, we propose a novel framework -- Collaborative Multi-Teacher Knowledge Distillation (CMT-KD), which encourages the {collaborative learning} between teachers and the mutual learning between teachers and student. \n\nIn the collaborative learning process, knowledge at corresponding layers from teachers will be combined to form an importance-aware shared knowledge which will subsequently be used as input for the next layers of teachers.\nThe collaborative learning between teachers is expected to form a valuable shared knowledge to be distilled to the corresponding layers in the student network. To our best knowledge, this paper is the first one that proposes this kind of collaborative learning for knowledge distillation.\n\nIt is worth noting that our novel framework design allows end-to-end training, in which not only the teachers and student networks but also\nthe contributions (i.e., importance factors) of teachers to the shared knowledge are also learnable during the learning process. It is also worth noting that the proposed framework is flexible -- different quantization functions and different knowledge distillation methods can be used in our framework.\n\nTo evaluate the effectiveness of the proposed framework, we conduct experiments on CIFAR-100 and ImageNet datasets with AlexNet and ResNet18 architectures. The results show that the compact student models trained with our framework achieve competitive results compared to previous works.\n\n\n\n\n\n\n\n\n\n\n\\section{Proposed method}\n\\label{sec:proposed}\n\n\\subsection{The proposed framework}\n\\begin{figure*}[!h]\n\\centering\n \\includegraphics[scale=1.8, width=0.9\\linewidth]{img\/KD_quantization-WACV2023_final.pdf}\n \n \\caption{The framework of our proposed collaborative multi-teacher knowledge distillation for low bit-width DNNs. \n \n The collaborative learning among a set of quantized teachers forms useful shared knowledge at certain layers through importance parameters $(\\pi)$. \n $\\mathcal{L}^T_{KL}$ and $\\mathcal{L}^S_{KL}$ are for mutual learning between teachers and student\n via ensemble logits $\\overline{\\mathbf{z}}$, where $\\overline{\\mathbf{z}}$ is calculated from teachers logits $\\mathbf{z}^T$ and student logits $\\mathbf{z}^S$. \n CE means Cross Entropy, and KL means Kullback-Leibler divergence. $\\mathcal{D}(.)$ denotes the loss for intermediate features-based distillation that could be attention loss or hint loss (FitNet). }\n \n\\label{fig:1} \n\\vspace{-0.5em}\n\\end{figure*}\n\nWe propose a novel framework -- Collaborative Multi-Teacher Knowledge Distillation (CMT-KD) as illustrated in Figure~\\ref{fig:1}, which encourages collaborative learning between teachers and mutual learning between teachers and student. \n\nFirst, we propose a novel collaborative learning of multiple teachers. During the training process, the collaborative learning among a set of quantized teachers forms useful importance-aware shared knowledge at certain layers,\nwhich is distilled to the corresponding layers in the student network.\nSecond, the learning process between the student and the teachers is performed in a mutual-learning manner \\cite{DML} via ensemble logits $\\overline{\\mathbf{z}}$ from teachers logits $\\mathbf{z}^T$ and student logits $\\mathbf{z}^S$. \n\n\nFurthermore, our framework design allows end-to-end training, in which not only the teachers and student networks but also the contributions (i.e., importance factors) of teachers to the shared knowledge are also learnable during the learning process. It is also worth noting that the proposed framework is flexible -- different quantization functions \\cite{HWGQ,HWGQ_Rethinking,LSQ,LQW_CAQ,QIL,LQNets} and different knowledge distillation methods \\cite{mimic,ABdistill,overhaul_fea_distill,FitNets,attention_transfer} can be used in our framework.\n\n\n\n\\subsection{Collaborative learning of multiple teachers}\nTeacher model selection is important in knowledge distillation. In~\\cite{QKD}, the authors show that if there is a large gap in the capacity between teacher and student, the knowledge from teacher may not be well transferred to student. To control the power of teacher, in our work, we consider teacher and student models that have the same architecture. However, teachers are quantized with higher different bit-widths. \nThe knowledge at corresponding layers from teachers will be fused to form a shared knowledge which will subsequently be used as input for the next layers of teachers. This forms collaborative learning between teachers. It is expected that different teachers have different capacities, and therefore, they should have different contributions to the shared knowledge. To this end, for each teacher, an importance factor that controls how much knowledge the teacher will contribute to the shared knowledge will also be learned. It is worth noting that the learning of importance factors will encourage the collaborative learning between teachers to produce a suitable shared knowledge that the student can effectively mimic.\n\n\nFormally, given a quantization function $Q(x, b)$, \nthe $i^{th}$ teacher is quantized using $Q(W_i, b_i)$ and $Q(A_i, b_i)$, in which $W_i$ and $A_i$ represent for weights and activations, respectively; $b_i$ is the bit-width. \nThe shared knowledge $F_k$ of corresponding layers of $n$ teachers at $k^{th}$ layer index is formulated as follows\n\n\\begin{equation}\n\\begin{gathered}\nF_k = \\sum_{i=1}^{n}\\pi^{k}_i * Q(A_i^k, b_i) \\\\\n \\text { s.t. } \\sum_i \\pi^{k}_i =1, \\pi^{k}_i \\in[0,1] \\text {,}\n\\end{gathered}\n\\label{eq:Fk}\n\\end{equation}\nwhere\n${\\pi^{k}_i}$ represents the importance of teacher $i^{th}$. To handle the constraint over $\\pi$ in (\\ref{eq:Fk}), in the implementation, a softmax function is applied to $\\pi^{k}_i$ values before they are used to compute $F_k$.\nThe importance parameters $\\pi$ and the model weights of teachers and student are optimized simultaneously via end-to-end training.\n\\subsection{Other components}\n\n\\subsubsection{Quantization function}\nQuantization function maps a value $x \\in R$ to a quantized value $\\overline{x} \\in \\{q_1, q_2, ..., q_n\\}$ using a quantization function Q with a precision bit-width $b$. The quantized value is defined as\n\\begin{equation}\n \\overline{x}=Q(x, b).\n\\end{equation}\n\n\n\nDifferent quantization methods have been proposed \\cite{HWGQ,LSQ,QIL,xnor_net,LQNets,dorefa}. In this paper, we consider the half-wave Gaussian quantization (HWGQ) \\cite{HWGQ} as a quantizer to be used in our framework, which is an effective and simple uniform quantization approach. \nTo quantize weights and activations, they first pre-compute the optimal value $q_i$ by using uniform quantization {for unit Gaussian distribution}. \n{Depending on variance $\\sigma$ of weights and activations}, the quantized value of $x$ is expressed as\n\\begin{equation}\n \\overline{x} = \\sigma*q_i.\n\\end{equation}\n\n\n\n\\subsubsection{Intermediate features-based distillation}\n\nThe shared knowledge from teachers will be used to guide the learning of student. \nLet $F^T_k$ and $F^S_k$ be the shared feature map of teachers and the feature map of the student at $k^{th}$ layers of models, respectively. Let $\\mathcal{I}$ be the selected layer indices for intermediate features-based distillation.\nThe intermediate features-based knowledge distillation loss is defined as follows\n\n\n\\begin{equation}\n \\mathcal{L}_{feat} = \\sum_{k \\in \\mathcal{I}} \\mathcal{D} \\left(F_{k}^{T}, F_{k}^{S}\\right),\n \\label{eq:fea_loss}\n\\end{equation}\nwhere $\\mathcal{D}$ is the distance loss measuring the similarity between features $F^T_k$ and $F^S_k$.\nDifferent forms of $\\mathcal{D}$ can be applied in our framework. In this work, we consider two widely used distance losses, i.e., the attention loss~\\cite{attention_transfer} and the FitNet loss\\cite{FitNets}. \n\n\n\n\nThe attention loss~\\cite{attention_transfer} is defined as follows\n\\begin{equation}\n \\mathcal{D}_{A T}=\\sum_{k \\in \\mathcal{I}}\\left\\|\\frac{Q_{k}^{T}}{\\left\\|Q_{k}^{T}\\right\\|_{2}}-\\frac{Q_{k}^{S}}{\\left\\|Q_{k}^{S}\\right\\|_{2}}\\right\\|_{p},\n\\end{equation}\nwhere $Q_k^S$ and $Q_k^S$ are the attention maps of features $F_k^S$ and $F_k^S$, respectively. $\\|.\\|_p$ is the $l_p$ norm function that could be a $l_1$ or $l_2$ normalization. The attention map for a feature map $F \\in R^{c \\times h \\times w}$ that has $c$ channels is defined as $Q=\\sum_{j=1}^{c}|A_j|^p$, where $A_j = F(j, :,:)$ is the $j^{th}$ channel of $F$; $|.|$ is the element-wise absolute function. In our implementation we use $p=2$. \nThe FitNet loss \\cite{FitNets} is defined as follows\n\\begin{equation}\n \\mathcal{D}_{HT}=\\sum_{k \\in \\mathcal{I}}\\left\\|F_{k}^{T} - r(F_{k}^{S})\\right\\|_{p},\n\\end{equation}\nwhere $r(.)$ is a convolutional layer that adapts the student feature map $F_k^{S}$ before comparing it to the shared knowledge teacher feature map $F_k^{T}$. In our method, we follow existing works \\cite{overhaul_fea_distill,FitNets} in which $r(.)$ a convolutional layer with a kernel size of $1\\times1$.\n\\subsubsection{Mutual learning between teachers and student}\nIn addition to the intermediate features-based distillation, we also leverage the mutual learning \\cite{DML} defined on the logits from networks for learning. The mutual learning allows student to give feedback about its learning to teachers. This learning mechanism encourages both teachers and student to simultaneously adjust their parameters to achieve an overall learning objective, e.g., minimizing a total loss function.\n\nThe mutual learning \\cite{DML} applies $KL$ losses on the softmax outputs of networks.\nHowever, due to the diversity of the output logits from different networks, this method may hurt the performance of models. \nTo overcome this issue, we adopt KDCL-MinLogit \\cite{KDCL}, which is a simple and effective method to ensemble logits from teachers and student. In particular, this method selects the minimum logit value of each category. \n\nLet $\\mathbf{z}^T$ and $\\mathbf{z}^S$ be the logit outputs of the combined\nteacher model $T$ and a student model $S$, $z^{T,c}$ and $z^{S,c}$ be the elements of $\\mathbf{z}^T$ and $\\mathbf{z}^S$ corresponding to the target class $c$, $\\mathbf{1}$ be the vector with all $1s$ elements, we denote $\\mathbf{z}^{T, c} = \\mathbf{z}^T - z^{T,c}\\mathbf{1}$ and $\\mathbf{z}^{S,c} = \\mathbf{z}^S - z^{S,c}\\mathbf{1}$. \nThe element $\\overline{z_i}$ of the ensemble logits $\\overline{\\mathbf{z}}$ is computed as follows\n\\begin{equation}\n \\overline{z_i} = min\\{z_i^{T,c}, z_i^{S,c}\\}, i=1, 2, ..., m\n \n\\end{equation}\nwhere $z_i^{T,c}$, $z_i^{S,c}$ are the $i^{th}$ elements of $\\mathbf{z}^{T, c}$ and $\\mathbf{z}^{S, c}$, and $m$ is the number of classes.\n\n\nThe mutual learning is defined as follows\n\n\\begin{equation}\n \\mathcal{L}_{KL}^{S}= \\mathcal{T}^{2} \\times KL(\\overline{\\mathbf{p}}|| \\mathbf{p}^{S}),\n\\end{equation}\n\\begin{equation}\n \\mathcal{L}_{KL}^{T}= \\mathcal{T}^{2} \\times KL(\\overline{\\mathbf{p}}|| \\mathbf{p}^{T}),\n\\end{equation}\nwhere $KL$ means the Kullback-Leibler divergence and $\\mathcal{T}$ is the temperature parameter. $\\overline{\\mathbf{p}}$, $\\mathbf{p}^{S}$, and $\\mathbf{p}^{T}$ are the soft logits of $\\overline{\\mathbf{z}}$, $\\mathbf{z}^{S}$, and $\\mathbf{z}^{T}$, respectively. The soft logit $\\mathbf{p}$ is defined as $\\mathbf{p}=softmax(\\frac{\\mathbf{z}}{\\mathcal{T}})$.\n\n\n\n\n\n\nFinally, the overall loss of our proposed collaborative and mutual learning in classification task is defined as\n\n\\begin{equation}\n\\small\n\\label{eq:final_loss}\n \\begin{aligned}\n \\mathcal{L} = \\alpha \\times (\\mathcal{L}_{CE}^{S}+ \\mathcal{L}_{CE}^{T}) \n + \\beta \\times (\\mathcal{L}_{KL}^S + \\mathcal{L}_{KL}^T) + \\gamma \\times \\mathcal{L}_{feat},\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha$, $\\beta$, and $\\gamma$ are the hyper-parameters of total loss for optimization and $\\mathcal{L}_{CE}$ is the standard cross-entropy loss calculated using the corresponding soft logits and the ground-truth labels. During the training process,\nboth the model weights of teachers, student and the importance factors of teachers, i.e., $\\pi_i^k ~\\forall i,k$, will be updated by minimizing $\\mathcal{L}$, using gradient descent. \n\n\\section{Related work}\n\\label{sec:related_work}\nOur work is closely related to two main research topics in the literature: network quantization and knowledge distillation (KD).\n\n\\textbf{Network quantization.}\nEarlier works in network quantization have applied the basic form of weight quantization to directly constrain the weight values into the binary\/ternary space without or with a scaling factor, i.e., $\\{-1, 1\\}$ \\cite{BinaryConnect}, $\\{-\\alpha, \\alpha\\}$ \\cite{xnor_net}, or $\\{-\\alpha, 0, \\alpha\\}$ \\cite{TWN}. \nSince quantization of activations can substantially reduce complexity further \\cite{xnor_net,BinaryNet,LQNets,dorefa}, this research topic attracts more and more attention \\cite{xnor_net,LQNets,dorefa}. \nIn~\\cite{xnor_net,BinaryNet}, the authors propose to binarize both weights and activations to $\\{-1,1\\}$. However, there are considerable accuracy drops compared to full precision networks. \n{To address this problem, the generalized low bit-width quantization \\cite{log_data_represent,dorefa} is\nstudied.\n}\n{In half-wave Gaussian quantization (HWGQ) \\cite{HWGQ}, the authors propose a practical and simple uniform quantization method that exploits the statistics of network activations and batch normalization.}\nIn LQ-Nets \\cite{LQNets}, the authors propose to train a quantized CNN and its associated non-uniform quantizers jointly.\nThe approach in Learned Step Size Quantization (LSQ) \\cite{LSQ} learns uniform quantizers using trainable interval values. In quantization-interval-learning (QIL) \\cite{QIL}, the authors introduce a trainable quantizer that additionally performs both pruning and clipping.\n\n\\textbf{Knowledge Distillation (KD)} is a common method in training smaller networks by distilling knowledge from a large teacher model \\cite{KD_Hinton}. The rationale behind this is to use extra supervision in the forms of classification probabilities \\cite{KD_Hinton}, intermediate feature representations~\\cite{mimic,ABdistill,overhaul_fea_distill,FitNets}, attention maps \\cite{attention_transfer}.\nKnowledge distillation approaches transfer the knowledge of a teacher network to a student network in two different settings: offline and online. In the offline learning, KD uses a fixed pre-trained teacher network to transfer the knowledge to the student network. Deep mutual learning \\cite{DML} mitigates this limitation by conducting online distillation in one-phase training between two peer student models.\n\n\\textbf{Multi-Teacher Knowledge Distillation.}\nThe approach in \\cite{MultiLang_NMT} applies multi-teacher learning into multi-task learning where each teacher corresponds to a task. Similarly, the approach in \\cite{UHC} trains a classifier in each source and unifies their classifications on an integrated label space.\nThe approach in \\cite{DBLP:conf\/kdd\/YouX0T17} considers knowledge from multiple teachers equally by averaging the soft targets from different pretrained teacher networks. In\n\\cite{adaptive_multi_teacher}, the authors propose to learn a weighted combination of pretrained teacher representations. \n{Different from \\cite{DBLP:conf\/kdd\/YouX0T17,adaptive_multi_teacher}, in this work, we propose a novel online distillation method that captures importance-aware knowledge from different teachers\nbefore distilling the captured knowledge to the student network.} {\nIn \\cite{DBLP:conf\/ecai\/ParkK20}, the last feature map of student network is fed through different non-linear layers; each non-linear layer is for each teacher. The student network and the non-linear transformations are trained such that the output of those non-linear transformations mimic the last feature maps of the corresponding teacher networks.}\nThe previous works \\cite{adaptive_multi_teacher,DBLP:conf\/ecai\/ParkK20, DBLP:conf\/kdd\/YouX0T17} mainly learn full precision student models from a set of full precision pretrained teacher models, while we focus on learning quantized models. Specifically, we aim to learn a quantized student model with guidance from a set of quantized teacher models with different precisions. \nIn addition, different from previous works \\cite{DBLP:conf\/kdd\/YouX0T17,adaptive_multi_teacher,DBLP:conf\/ecai\/ParkK20} in which teachers are fixed when training student, our method simultaneously trains student and teachers using the collaborative and mutual learning. \n\n\n\n\\textbf{Quantization + Knowledge distillation}. Some works have tried to adopt knowledge distillation methods to assist the training process of low precision networks \\cite{SPEQ,QKD,apprentice,distill_quant,guidedQuantize}. \nIn Apprentice (AP) \\cite{apprentice}, the teacher and student networks are initialized with the corresponding pre-trained full precision networks. After lowering the precision of the student network, the student is then fine-tuned using distillation.\nDue to AP's initialization of the student, AP might get stuck in a local minimum in the case of very low bit-width quantized student networks. \nBecause of the inherent differences between the feature distributions of the full-precision teacher and low-precision student network, using a fixed teacher as in \\cite{apprentice} can limit the knowledge transfer. QKD \\cite{QKD} and Guided-Quantization \\cite{guidedQuantize} mitigate the issue of AP by jointly training the teacher and student models, which makes a teacher adaptable to the student model.\nIn our work, to further mitigate the problem of different {feature distributions} between teacher and student models, instead of using a full-precision teacher model, we propose to use a set of quantized teacher models. Using quantized models would help the teachers obtain more suitable knowledge for a quantized student model to mimic.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}