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+{"text":"\\section{Introduction}\n\\label{s1}\nThe quiet Sun covers most of the solar surface, in particular at activity minimum, but \nalso plays an important role even during the active phase of the solar cycle. The magnetic \nfield in the quiet Sun is composed of the network \\citep{1967SoPh....1..171S}, \ninternetwork \\citep[IN,][]{1971IAUS...43...51L, 1975BAAS....7..346L}, and the ephemeral \nregions \\citep{1973SoPh...32..389H}. For an overview of the small-scale magnetic \nfeatures, see \\citet{1993SSRv...63....1S, 2009SSRv..144..275D, 2014masu.book.....P, \n2014A&ARv..22...78W}.\n\nThe IN features are observed within the supergranular cells and carry hecto-Gauss fields  \n\\citep[][]{1996A&A...310L..33S, 2003A&A...408.1115K, 2008A&A...477..953M} although \nkilo-Gauss fields have also been observed in the IN  \\citep{2010ApJ...723L.164L, \nLagg16}. They evolve as unipolar and bipolar features with typical lifetimes of less than \n10 minutes \\citep{2010SoPh..267...63Z, 2013apj...774..127l,lsa}, i.e., they continuously \nbring new flux to the solar surface, either flux that has been either freshly generated, \nor recycled. They carry fluxes $\\le 10^{18}$ Mx, with the lower limit on the smallest \nflux decreasing with the increasing spatial resolution and polarimetric sensitivity of \n the observing instruments, although the identification technique also plays an important \nrole. \n\nEphemeral regions are bipolar magnetic features appearing within the supergranular cells \ncarrying fluxes $\\approx 10^{19}$ Mx \\citep{2001ApJ...548..497C, 2003ApJ...584.1107H} and \nare much longer-lived compared to the IN features, with lifetimes of 3 -- 4.4 hours \n\\citep{2000RSPTA.358..657T, 2001ApJ...555..448H}. The ephemeral regions also bring new \nmagnetic flux to the solar surface. \n\nThe network is more stable, with typical lifetimes of its structure of a few \nhours \nto a day, although the individual kG magnetic elements within the network live for a much \nshorter time, as the entire flux within the network is exchanged within a period of 8--24 \nhr \\citep{2003ApJ...584.1107H, 2014apj...797...49g}. The flux in the network is fed by \nephemeral regions \\citep{0004-637x-487-1-424, 2001ApJ...555..448H} and IN features \n\\citep[][]{2014apj...797...49g}. The network features are found along the supergranular \nboundaries and carry fields of kG strength with a typical flux of $10^{18}$ Mx \n\\citep{1995soph..160..277w}. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{f1}\n\\caption{Two sample magnetograms recorded at $t$ = 00:47 UT and 00:58 UT on 2009 June 9 \nwith the \\textsc{Sunrise}\/IMaX instrument during its first science flight.}\n\\label{magnetogram}\n\\end{figure*}\n\nThe magnetic flux is produced by a dynamo, the location of which is currently the subject \nof debate, as is whether there is only a single dynamo acting in the Sun \n\\citep[e.g.,][]{2012A&A...547A..93S} or whether there is a small-scale dynamo acting in \naddition to a global dynamo \\citep{1993A&A...274..543P, 1999ApJ...515L..39C, \n2001A&G....42c..18C, 2007A&A...465L..43V, 2008A&A...481L...5S, 2010A&A...513A...1D, \n2013A&A...555A..33B, 2015ApJ...803...42H, 2016ApJ...816...28K}. In addition, it is unclear \nif all the magnetic flux appearing on the Sun is actually new flux produced by a dynamo, \nor possibly recycled flux transported under the surface to a new location, where it \nappears again \\citep[e.g.,][]{2001ASPC..236..363P}. This may be particularly important at \nthe smallest scales. \n\nAn important parameter constraining the production of magnetic flux is the amount of \nmagnetic flux appearing at the solar surface. In particular, the emergence of magnetic \nflux at very small scales in the quiet Sun provides a probe for a possible small-scale \ndynamo acting at or not very far below the solar surface. The deep minimum between solar \ncycles 23 and 24 offered a particularly good chance to study such flux emergence, as the \nlong absence of almost any activity would suggest that most of the emerging flux is newly \nproduced one and is not flux transported from decaying active regions to the quiet Sun \n(although the recycling of some flux from ephemeral regions cannot be ruled out).\n\nThe IN quiet Sun displays by far the largest magnetic flux emergence rate (FER). Already, \n\\citet{1987SoPh..110..101Z} pointed out that two orders of magnitude more flux appears in \nephemeral regions than in active regions, while the FER in the IN is another two orders \nof \nmagnitude larger. This result is supported by more recent studies \n\\citep[e.g.,][]{2002ApJ...565.1323S, 2009SSRv..144..275D, 2009ApJ...698...75P, \n2011SoPh..269...13T}.  Given the huge emergence rate of the magnetic flux in the IN, it \nis of prime importance to measure the amount of flux that is brought to the surface \nby these features.\n \nThe current estimates of the FER in the IN vary over a wide range, which include: \n$10^{24}\\rm{\\,Mx\\, day^{-1}}$ \\citep[][]{1987SoPh..110..101Z}, $3.7\\times \n10^{24}\\rm{\\,Mx\\,day^{-1}}$ \n\\citep[$120\\rm{\\,Mx\\,cm^{-2}\\,day^{-1}}$,][]{2016ApJ...820...35G} and \n$3.8\\times10^{26}\\rm{\\,Mx\\,day^{-1}}$ \\citep[][]{2013SoPh..283..273Z}.\nBy considering all the magnetic features (small-scale features and active regions), \n\\citet{2011SoPh..269...13T} measure a global FER of $3\\times 10^{25}\\rm{\\,Mx\\,day^{-1}}$ \n($450 \\rm{\\,Mx \\,cm^{-2} \\,day^{-1}}$), while \\citet{th-thesis} measures $3.9 \\times \n10^{24} \\rm{\\,Mx \\,day^{-1}}$ ($64\\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$), whereby almost all of \nthis flux emerged in the form of small IN magnetic features. The FER depends on the \nobservations and the method used to measure it. A detailed comparison of the FERs from \ndifferent works is presented Section~\\ref{s6}.\n\nTo estimate the FER, \\citet{1987SoPh..110..101Z} and \\citet{2011SoPh..269...13T}  \nconsidered features with fluxes $\\ge 10^{16}$ Mx, while \\citet{2013SoPh..283..273Z} and \n\\citet{2016ApJ...820...35G} included features with fluxes as low as $6\\times 10^{15}$ \nMx and $6.5\\times10^{15}$ Mx (M. Go\\v{s}i\\'{c}, priv. comm.), respectively. However, with \nthe launch of the balloon-borne \\textsc{Sunrise}{} observatory in 2009 \\citep{2010ApJ...723L.127S, \n2011SoPh..268....1B, 2011SoPh..268..103B, 2011SoPh..268...35G} carrying the Imaging \nMagnetograph eXperiment \\citep[IMaX,][]{2011SoPh..268...57M}, it has now become possible \nto estimate the FER including the contribution of IN features with fluxes as low as \n$9\\times10^{14}$ Mx \\citep[][hereafter referred to as LSA17]{lsa}. The IMaX instrument \nhas \nprovided unprecedented high-resolution magnetograms of the quiet Sun observed at 5250\\, \n\\AA. The high resolution is the main reason for the lower limiting flux. A detailed \nstatistical analysis of the IN features observed in Stokes $V$ recorded by \\textsc{Sunrise}\/IMaX \nis carried out in LSA17. In the present paper we estimate the FER in the IN region using \nthe same data.\n\nIn Section~\\ref{s2} we briefly describe the employed \\textsc{Sunrise}{} data. The IN features \nbringing flux to the solar surface that are considered in the estimation of FER are \noutlined in Section~\\ref{s3}. The FER from \\textsc{Sunrise}{} are presented, discussed and \ncompared with previously obtained results in Section~\\ref{s4} while our conclusions \nare presented in Section~\\ref{s8}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{f2}\n\\caption{Panels a--d: Schematic representation of the different paths by \nwhich magnetic flux is brought to the solar surface and of its subsequent evolution.  The \nquantities $f_i$ and $f_m$ are the instantaneous and maximum fluxes of a feature, \nrespectively, where instantaneous means just after it appears. {Panel e}: Typical \nvariation in the flux of a feature, born by  {unipolar and bipolar} appearances \n(top) and born by splitting\/merging (bottom), over time. The flux gained by them after \nbirth is $\\Delta f=f_m - f_i$. The features born by splitting carry fluxes $f_{i1,2}$ at \nbirth and reach $f_{m1,2}$ in the course of their lifetime, gaining flux $\\Delta f_{1,2}$ \nafter birth. $f_{i1}+f_{i2}$ is equal to the flux of the parent feature at the time of its \nsplitting. The feature born by merging carries flux equal to the sum of the fluxes of the \nparent features $f_{i1}+f_{i2}$. The blue line indicates the merging of the two features.}\n\\label{flux}\n\\end{figure*}\n\n\\section{Data}\n\\label{s2}\nThe data used here were obtained during the first science flight of \\textsc{Sunrise}{} described \nby \\citet{2010ApJ...723L.127S}. We consider 42 maps of the line-of-sight (LOS) magnetic \nfield, $B_{\\rm LOS}$, obtained from sets of images in the four Stokes parameters  \nrecorded with the IMaX instrument between 00:36 and 00:59 UT on 2009 June 9 at the solar \ndisk center, with a cadence of 33\\,s,  {a spatial resolution of \n$0.^{\\prime\\prime}15-0.^{\\prime\\prime}18$ (plate scale is $0.^{\\prime\\prime}054$ per \npixel), and an effective field of view (FOV) of $43^{\\prime\\prime} \\times \n43^{\\prime\\prime}$ after phase diversity reconstruction}.  {The data were \nreconstructed with a point spread function determined by in-flight phase diversity \nmeasurements to correct for the low-order aberrations of the telescope \\citep[defocus, \ncoma, astigmatism, etc., see][]{2011SoPh..268...57M}. The instrumental noise of the \nreconstructed data was $3\\times10^{-3}$ in units of continuum intensity. For \nidentification of the features, spectral averaging was done which further reduced the \nnoise to $\\sigma=1.5\\times10^{-3}$. All features with signals above a $2\\sigma$ \nthreshold, which corresponds to 12\\,G,  were used \\citep[][]{2011SoPh..268...57M}.}\n\nTo measure the flux, we use $B_{\\rm LOS}$ determined with the center-of-gravity (COG) \nmethod \n\\citep[][LSA17]{1979A&A....74....1R, 2010A&A...518A...2O}.  {The inclination of the \nIN fields has been under debate, with several studies dedicated to measuring their \nangular distribution. By analysing the data from \\textit{Hinode}, \n\\citet{2007ApJ...670L..61O, 2008ApJ...672.1237L} concluded the IN fields to be \npredominantly horizontal. However, using the same dataset, \\citet{2011ApJ...735...74I} \nfound some of the IN fields to be vertical. \\citet{2014A&A...569A.105J} arrived at similar \nconclusions (vertical inclination) by analysing the magnetic bright points observed from \nthe first flight of \\textsc{Sunrise}{}. Variations in the inclination of the IN fields with \nheliocentric angle ($\\mu$) have been reported by \\citet{2012ApJ...751....2O, \n2013A&A...550A..98B, 2013A&A...555A.132S}. Isotropic and quasi-isotropic distribution of \nthe IN field inclinations is favoured by \\citet{2008A&A...479..229M}, using the Fe\\,{\\sc \ni} $1.56\\,\\mu$m infrared lines, and by \\citet{2009ApJ...701.1032A,2014A&A...572A..98A} \nusing \\textit{Hinode} data. More recently, \\citet{2016A&A...593A..93D} found the \ndistribution of IN field inclination to be quasi-isotropic by applying 2D inversions on \n\\textit{Hinode} data and comparing them with 3D magnetohydrodynamic simulations. For a \ndetailed review on this, see \\citet{2015SSRv..tmp..113B}.}\n\n {In the determination of the FER, we use $B_{\\rm LOS}$ for consistency and for \neasier comparison with earlier studies on the FER. Also, the determination of the exact \namount of flux carried in horizontal field features is non-trivial and requires estimates \nof the vertical thickness of these features and the variation of their field strength with \nheight. In addition, if they are loop-like structures \\citep[as is suggested by \nlocal-dynamo simulations, e.g.,][]{2007A&A...465L..43V}, then there is the danger of \ncounting the flux multiple times if one or both of their footpoints happen to be resolved \nby the \\textsc{Sunrise} data. We avoid this by considering only the vertical component of the \nmagnetic field. It is likely that we miss the flux carried by unresolved magnetic loops by \nconcentrating on Stokes $V$, but this problem is suffered by all previous studies of FER \nand should decrease as the spatial resolution of the observations is increased. With the \n\\textsc{Sunrise}{} I data analyzed here having the highest resolution, we expect them to see more \nof the flux in the footpoints of the very small-scale loops that appear as horizontal \nfields in \\textit{Hinode} and \\textsc{Sunrise}{} data \\citep{2010A&A...513A...1D}.}\n\nThe small-scale features were identified and tracked using the feature tracking code \ndeveloped in LSA17.  {For the sake of completeness we summarize the most \nrelevant results from LSA17 as follows. All the features covering at least 5 pixels \nwere considered with Stokes $V$ being larger than $2\\sigma$ in each pixel. To determine \nthe flux per feature,  the $B_{\\rm LOS}$ averaged over the feature, denoted as \n\\textlangle$B_{\\rm LOS}$\\textrangle{} was used.} \\textlangle$B_{\\rm LOS}$\\textrangle{} \nhad values up to 200\\,G, even when the maximum field strength in the core of the feature \nreached kG values.  {A total of 50,255 features of both polarities were identified. \nThe sizes of the features varied from 5-1,585 pixels, corresponding to an area range of \n$\\approx 8\\times 10^{-3}-2.5$\\,Mm$^2$. The tracked features had lifetimes ranging from \n0.55 to 13.2 minutes.} The smallest detected flux of a feature was $9\\times10^{14}$ Mx \nand \nthe largest $2.5\\times10^{18}$ Mx.\n\n \nAt the time of the flight of \\textsc{Sunrise}{} in 2009 the Sun was extremely quiet, with no signs \nof activity on the solar disk. Two sample magnetograms at  00:47 UT and 00:58 UT are shown \nin Figure~\\ref{magnetogram}. Most of the features in these maps are part of the IN and in \nthis paper, we determine the rate at which they bring flux to the solar surface. \n\n\\section{Processes increasing magnetic flux at the solar surface}\n\\label{s3}\nThe different processes increasing the magnetic flux at the solar surface are \nschematically represented in Figure~\\ref{flux}a -- \\ref{flux}d. In the figure, $f_i$ \nrefers to the flux of the feature at its birth and $f_m$ is the maximum flux that a \nfeature attains over its lifetime. A typical evolution of the flux of a feature born by\n {unipolar or bipolar} appearances, and by splitting\/merging is shown in \nFigure~\\ref{flux}e (top and bottom, respectively). The gain in the flux of a feature \nafter its birth is $f_m-f_i$. Magnetic flux at the surface increases through the \nfollowing processes:\n\n    \\begin{table*}\n          \\centering\n          \\caption{The instantaneous and maximum fluxes of the features, \n  {integrated over all features and time frames}, in different \n processes measured in LSA17}\n            \\begin{tabular}{|c|c|c|c|c|}\n            \\hline\n Process & Instantaneous flux & Maximum flux & Flux gain & Factor of increase\\\\\n & ($f_i$ in Mx) & ($f_m$ in Mx) & ($\\Delta f= f_m-f_i$ in Mx) & ($f_m\/f_i$)\\\\\n \\hline\n   {Unipolar appearance} & $4.69\\times10^{19}$ & $9.69\\times10^{19}$ &  \n$4.99\\times10^{19}$ &2.06\\\\\n  Splitting & $1.76\\times10^{20}$& $2.12\\times10^{20}$&$3.60\\times10^{19}$&1.20\\\\\n  Merging & $1.64\\times 10^{20}$& $1.85\\times 10^{20}$&$2.20\\times10^{19}$&1.31\\\\\n   {Bipolar appearance} & $3.85\\times10^{18}$ & $9.53\\times10^{18}$& \n$5.67\\times10^{18}$&2.47\\\\\n  \\hline\n            \\end{tabular}\n            \\label{table1}\n        \\end{table*}\n\n\n\\begin{enumerate}\n \\item{ {Unipolar appearance}}: The birth of an isolated feature with no spatial \noverlap with  any of the existing features in the current and\/or previous time frame \n(Figure~\\ref{flux}a). \n \\item { {Bipolar appearance}}: Birth of bipolar features, with the two polarities \nclosely spaced, and either appearing simultaneously or separated by a couple of time \nframes ( {referred to as} time symmetric and asymmetric emergence in LSA17; \nsee also Figure~\\ref{flux}b). \n\\item { {Flux gained by features in the course of their \nlifetime}}: The gain in the \nflux of a feature in the course of its lifetime, i.e. the increase in flux between its \nbirth and the time it reaches its maximum flux, before dying in one way or another, either \nby interacting with another feature, or by disappearing.\n\n This gain can take place in features born in different ways, be it by growth, or \nthrough the merging or splitting of pre-existing features (Figures~\\ref{flux}c, \n\\ref{flux}d and \\ref{flux}e).\n\\end{enumerate}\n\n {Note that the bipolar appearance of magnetic flux is often referred to as \n`emergence' in earlier papers including LSA17. However, the term `emergence' in FER \ndescribes the appearance of new flux at the solar surface from all the three processes \ndescribed above. To avoid confusion, we refer to the emergence of  bipolar features as \nbipolar appearance.}  {Of all the newly born features over the entire time series, \n19,056 features were unique (for area ratio 10:1, see Section~\\ref{s4}). Among them 48\\% \n(8728 features) were unipolar and 2\\% (365 features) were part of bipolar appearances. \nFeatures born by splitting constituted 38\\% (6718 features), and 12\\% (2226 features) \nwere born by merging. The remaining 1019 features correspond to those alive in the \nfirst frame. A comparison of the rates of birth and death of the features by various \nprocesses for different area ratio criteria is given in Table~2 of LSA17.}\n\nIn the FER estimations, the flux brought by the features born by \n {unipolar and bipolar} appearances is the maximum flux that they attain \n($f_m$) over their lifetime. In the case of features born by splitting or merging, the \nflux gained after birth is taken as the flux brought by them to the surface. This gain is \nthe difference between their flux at birth $f_i$ and the maximum flux they attain $f_m$, \ni.e. $f_m - f_i$. \n\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.3\\textwidth]{f3}\n \\caption{Schematic representation of multiple peaks in the flux of a feature occurring \nin the course of its lifetime. In the FER estimations, we consider only the largest gain, \ni.e., flux increase during first peak in this example.}\n \\label{fg}\n\\end{figure}\n\nOur approach is conservative in the sense that if a feature reaches multiple peaks of flux \nin the course of its lifetime, as in the example shown in Figure~\\ref{fg}, then we \nconsider only \nthe largest one (the flux gained during the first peak in Figure~\\ref{fg}), and neglect \nincreases in flux contributing to smaller peaks such as the second and third peaks in \nFigure~\\ref{fg}.  {Multiple peaks in the flux of a feature (shown in \nFigure~\\ref{fg}) are rarely seen, as most features do not live long enough to display \nthem (see LSA17). }\n\n {Changes in the flux of a feature in the course of its lifetime can cause \nit to seemingly appear and disappear with time if its total flux is close to the \nthreshold set in the study (given by a signal level twice the noise in at least five\ncontiguous pixels). If it disappears and reappears again, then it will be counted twice. \nThis introduces uncertainties in the measurement of FER. Uncertainties are discussed in \nSection~\\ref{s7}.}\n\n\\section{Results and Discussions}\n\\label{s4}\n\n\\subsection{Flux emergence rate (FER)}\n\\label{s5}\nIn this work, we consider the results from the area ratio criterion 10:1 of LSA17. In \nthat paper, the authors devise area ratio criteria (10:1, 5:1, 3:1 and 2:1) to avoid that \na feature dies each and every time that a tiny feature breaks off, or merges with it. \nFor example in a splitting event, the largest of the features formed by splitting \nmust have an area less than $n$ times the area of the second largest, under the $n:1$ area \nratio criterion.  {We have verified that the choice of the area ratio criterion does \nnot drastically alter the estimated FER, with variations being less than 10\\% for area \nratios varying between 10:1 and 2:1.}\n\nThe  {instantaneous} and maximum fluxes of the features in different processes are \ngiven in Tables~$1-5$ of LSA17. A summary is repeated in Table~\\ref{table1} for \nconvenience {, where fluxes are given for features born by the four processes listed \nin the first column. The instantaneous flux, in the second column, refers to the flux of \na feature at its birth ($f_i$ in Figure~\\ref{flux}). In the third column is the maximum \nflux of a feature during its lifetime ($f_m$ in Figure~\\ref{flux}). The flux gain in the \nfourth column is the difference of the second and third columns ($\\Delta f=f_m-f_i$). \nIn the fifth column is the factor by which the flux increases from its birth to its peak \n($f_m\/f_i$). The fluxes given in this table are the sum total over all features in the \nentire time series, for each process.} \n\nTo compute the FER, we add the fluxes from the \nvarious processes described in the previous section. For features born by appearance \n {(unipolar and bipolar), we take their maximum flux of the features ($f_m$)} to be \nthe fresh flux emerging at the surface. For the features formed by merging or splitting \nonly the flux increase after the birth  {($\\Delta f$)} is considered. From the  \nfirst two processes alone, the total flux brought to the surface is $1.1\\times10^{20}$ Mx \nover an FOV of $43^{\\prime\\prime} \\times 43^{\\prime\\prime}$ in 22.5 minutes. This gives \nan \nFER of $700 \\rm{\\, \\, Mx\\, cm^{-2}\\,day^{-1}}$. Including the flux gained by split\/merged \nfeatures increases the FER to $1100 \\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$. Figure~\\ref{flux1} \nshows the contribution from each process to the total FER. The isolated features appearing \non the solar surface contribute the largest, nearly 60\\%. Given that the emerging bipoles \ncontain only 2\\% of the total observed flux (Table~5 in LSA17), they contribute only \nabout \n5.7\\% to the FER. \n\nHowever, the flux brought to the solar surface by features born by splitting or merging, \nafter their birth is $5\\times10^{19}$ Mx, which is quite significant and contributes \n$\\approx35\\%$ to the FER. The contribution to solar surface flux by this process is \ncomparable to the flux brought to the surface by features born by  {unipolar} \nappearance ($9.7\\times10^{19}$ Mx) and nearly an order of magnitude higher than that flux \nfrom features born by  {bipolar appearance} ($9.5\\times10^{18}$ Mx). \n\nOver their lifetimes, the features born by splitting and by merging gain 1.2 times their \ninitial flux (i.e., $f_m = 1.2 \\times f_i$). The fluxes gained by features born by \nappearance relative to their flux at birth is slightly higher ($\\approx2$ times, i.e., \n$f_m = 2\\times f_i$). This is because the initial magnetic flux of the features born by \nappearance is quite low.  The flux at birth of split or merged features is already quite \nhigh because the parent features which undergo splitting or merging are at later stages in \ntheir lives (see Figure~\\ref{flux}e). This is also evident from the fact that the average \ninitial flux per feature of the features born by splitting or merging ($2.9\\times10^{16}$ \nMx and $7.4\\times10^{16}$ Mx, respectively) is an order of magnitude higher than the \naverage initial flux per feature of the appeared  {unipolar or bipolar} features \n($5.4\\times10^{15}$ Mx, see Table~2 of LSA17).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{f4}\n\n \\caption{The percentage of contribution to the flux emergence rate (FER) from different \nprocesses bringing flux to the solar surface. In the case of  {unipolar and bipolar} \nappearances, the maximum flux of the feature is used to determine the FER. For the \nfeatures born by splitting\/merging, the flux gained by them after birth is considered. \nThis gain is the difference $f_m - f_i$ in Figures~\\ref{flux}c, \\ref{flux}d and \n\\ref{flux}e.}\n \\label{flux1}\n\\end{figure}\n\nThe fact that the small-scale magnetic features are the dominant source of fresh flux in \nthe quiet photosphere is discussed in several publications \\citep{2002ApJ...565.1323S, \n2009SSRv..144..275D, 2009ApJ...698...75P, 2011SoPh..269...13T}. Our results extend these \nearlier findings to lower flux per feature values. As shown in Figure~\\ref{hist}, over the \nrange $10^{15}-10^{18}$ Mx, nearly $65\\%$ of the detected features carry a flux $ \\le \n10^{16}$ Mx (left panel). They are also the dominant contributors to the FER (right \npanel). In this figure, only the features that are born by  {unipolar and bipolar} \nappearances are considered. Below $2\\times10^{15}$ Mx, we see a drop as we approach the \nsensitivity limit of the instrument. \n\n\\subsubsection{Flux loss rate}\nFlux is lost from the solar surface by disappearance, cancellation of opposite \npolarity features, and decrease in the flux of the features in the course of their \nevolution (i.e. the opposite process to the ``flux gain'' described earlier in \nSection~\\ref{s3}). As seen from Tables~3 and 4 of LSA17, the increase in flux at the \nsolar surface balances the loss of flux, as it obviously must if the total amount of flux \nis to remain unchanged. To compute the flux loss rate, we take the maximum flux of the \nfeatures that die by cancellation and by disappearance to be the flux lost by them in the \ncourse of their lifetime and during disappearance or cancellation. For the \nfeatures that die by splitting\/merging we take the difference between the maximum flux of \nthe features and the flux at their death as a measure of the flux lost during their \nlifetimes. By repeating the analyses for the 10:1 area ratio criterion, we find that the \nflux is lost from the solar surface at a rate of 1150$\\rm{\\, \\, Mx\\, cm^{-2}\\,day^{-1}}$ \nwhich corresponds within 4.5\\% to the obtained FER. This agreement serves as a \nconsistency check of the FER value that we find.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{f5}\n \\caption{{Left panel:} histogram of the number of features born by \n {unipolar and bipolar} appearances, carrying fluxes in the range $10^{15}-10^{18}$ \nMx. {Right panel:} the flux emergence rate from the features born by \n {unipolar and bipolar} appearances as a function of their flux.}\n \\label{hist}\n\\end{figure*}\n\n\n\\subsection{Uncertainties}\n\\label{s7}\n {Although most of the uncertainties and ambiguities that arise during feature \ntracking have been carefully taken care of, as discussed in LSA17, some additional \nones which can affect the estimated FER are addressed below.}\n\n {In our computation of the FER, the features born before the time series began and \nthe features still alive at the end are not considered. According to LSA17, the first and \nthe last frames of the time series had 1019 and 1277 features, respectively.  To estimate \ntheir contribution, we assume that the features still living at the end have a similar \nlifetime, size, flux distribution and formation mechanism as the total number of features \nstudied. We attribute the appropriate  average flux at birth and the average flux gain \nfor \nfeatures born by splitting, merging, unipolar and bipolar appearance. After including \nthese additional fluxes, we get an FER of $\\approx 1150 \\rm{\\, \\, \nMx\\,cm^{-2}\\,day^{-1}}$, \ncorresponding to a $4-5\\%$ increase. With this method, we are associating the features \nwith flux gain than they might actually contribute (as many of them are likely to reach \ntheir maximum flux only after the end of the time series). This will be balanced out by \nnot considering the features that are already alive at the beginning (also, it is \nimpossible to determine the birth mechanism of these features).}\n\n {Furthermore, in the analysis of LSA17, the features touching the spatial \nboundaries were not counted. An estimate of their contribution, in ways similar to the \nabove, leads to a further increase of the FER by $5-6\\%$. Thus combining the features in \nthe first and last frames and the features touching spatial boundaries together increase \nthe FER by $\\approx 10\\%$. } \n\nMeanwhile, as discussed in Section~\\ref{s3}, in the case of flux gained after birth by \nfeatures born from splitting or merging, we consider only the gain to reach the maximum \nflux in the feature and not the smaller gains required to reach secondary maximum of flux \nin the feature, if any (see Figure~\\ref{fg}).  {These instances are quite rare. \nTo estimate their contribution, we consider all features living for at least four minutes \n(eight time steps) so as to distinguish changes in flux from noise fluctuations. They \nconstitute a small fraction of $\\approx 4\\%$. If all these features are assumed to show \ntwo maxima of equal strength, then they increase the FER by $\\approx \n1.5\\%$. This is a generous estimate and both these assumptions are unlikely to be met. \nHowever this is balanced out by not considering the features that have more than two \nmaxima. Thus the increase in FER is quite minor.} \n\nAdditionally, some of the features identified in a given time frame could disappear, i.e. \ndrop below the noise level, for the next couple of frames, only to reappear after that. \n {This is unlikely to happen due to the thermal or mechanical changes for the  \n\\textsc{Sunrise}{} observatory, flying in a highly stable environment at float altitude, and with \nactive thermal control of critical elements in the IMaX instrument. As mentioned in \nSection~\\ref{s3}, the appearance and disappearance of features could also occur due to \nthe \napplied threshold on the signal levels. In our analyses, the reappeared features are \ntreated as newly appeared. This leads to a higher estimation of the FER. \n\\citet{2016ApJ...820...35G} have estimated that accounting for reappeared features \ndecreases the FER by nearly $10\\%$. If we assume the same amount of decrease in the FER \nfrom the reappeared features in our dataset, then we finally obtain an FER of 1100$ \n\\rm{\\, \n\\, Mx\\, cm^{-2}\\,day^{-1}}$.}\n\n\n\\subsection{Comparison with previous studies}\n\\label{s6}\nBelow, we compare our results from \\textsc{Sunrise}{} data with those from the \\textit{Hinode} \nobservations analysed in three recent publications. Although all these papers use \nobservations from the same instrument, they reach very different estimates of FERs. The \nimportant distinguishing factor between them is the method that is used to identify the \nmagnetic features and to calculate the FER. For comparison we summarize the main \nresult that we have obtained here. We find that in the quiet Sun (composed dominantly of \nthe IN) the FER is $1100 \\rm{\\,Mx\\,cm^{-2}\\,day^{-1}}$. This corresponds \nto $6.6 \\times 10^{25} \\rm{\\,Mx\\,day^{-1}}$ under the assumption that the whole Sun is as \nquiet as the  {very tranquil} \\textsc{Sunrise}\/IMaX FOV. \n\nAccording to \\cite{2016ApJ...820...35G}, the flux appearance or emergence rate in \nthe IN region is $120 \\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$, which corresponds to $3.7 \n\\times 10^{24} \\rm{\\,\\,Mx\\,day^{-1}}$ over the whole surface and the contribution from \nthe \nIN is assumed to be $\\approx50\\%$. The authors track individual features and measure \ntheir \nfluxes, which is similar to the method used in LSA17. Their estimate is an order of \nmagnitude lower than the FER obtained in the present paper. This difference can be \nexplained by the higher spatial resolution of \\textsc{Sunrise}{} compared to \\textit{Hinode}. The \nisolated magnetic feature with the smallest flux detected in \\textsc{Sunrise}\/IMaX data is \n$9\\times10^{14}$ Mx  {(see LSA17)}, which is nearly an order of magnitude smaller \nthan the limit of $6.5\\times10^{15}$ Mx (M. Go\\v{s}i\\'{c}, priv. comm.), underlying the \nanalysis of \\citet{2016ApJ...820...35G}. Additionally, the IMaX data are recorded with \n33\\,s cadence, while the two data sets analysed by the above authors have cadences of 60 \nand 90\\,s each. A higher cadence helps in better tracking of the evolution of features \nand \ntheir fluxes.  {Also, a significant number of the very short-lived features that we \nfind may have been missed by \\citet{2016ApJ...820...35G}.}\n\n\\citet{2011SoPh..269...13T}, also using \\textit{Hinode} observations, estimate the FER \nby fitting a power law to the distribution of frequency of emergence ($\\rm{Mx^{-1} \n\\,cm^{-2} \\,day^{-1}}$) over a wide range of fluxes ($10^{16}-10^{22}$ Mx, which covers \nboth, small-scale features as well as active regions). It is shown that a single power \nlaw index of -2.7 can fit the entire range. Depending on the different emergence \ndetection methods used and described by these authors, such as Bipole Comparison (BC), \nTracked Bipolar (TB) and Tracked Cluster (TC), the authors find a wide range of FERs from \n32 to $470 \\rm{\\,Mx \\,cm^{-2}\\,day^{-1}}$ which correspond to 2.0 to $28.7 \\times 10^{24} \n\\rm{\\,Mx \\,day^{-1}}$ over the whole solar surface \\citep[Table 2 \nof][]{2011SoPh..269...13T,th-thesis}.  To match their results from \\textit{Hinode} with \nother studies, the authors choose an FER of $450\\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$, from the \nhigher end of the range (C. Parnell, priv. comm.). This is nearly four times higher than \nthe value quoted in \\cite{2016ApJ...820...35G}, who also used the \\textit{Hinode} \nobservations and a smaller minimum flux per feature, so that they should in principle \nhave caught more emerging features. However \\citet{th-thesis}, using a power law \ndistribution similar to \\citet{2011SoPh..269...13T} and a slightly different index of\n-2.5, estimates an FER of $64 \\rm{\\,Mx \\,cm^{-2}\\,day^{-1}}$.  {A possible reason \nfor this difference, as briefly discussed in both these studies, could be the different \nfeature tracking and identification methods used. In \\citet{2011SoPh..269...13T}, all the \nfeatures identified by BC, TB and TC methods are considered in determining the FER. \nAccording to the authors, the BC method counts the same feature multiple times and \nover-estimates the rate of flux emergence. However in \\citet{th-thesis}, only the \nfeatures \ntracked by TB method are used. The large differences in the FERs from the three detection \nmethods quoted in Table~2 of \\citet{2011SoPh..269...13T}, support this line of \nreasoning. \n The FER in \\citet{th-thesis} is roughly half that found by \\citet{2016ApJ...820...35G} \nand hence is at least in qualitative agreement.} The FER estimated by us is 2.5 times \nhigher than  {the largest value obtained by} \\citet{2011SoPh..269...13T} and 17 \ntimes higher than that of \\citet{th-thesis}.\n\nAnother recent estimate of the FER is by \\citet{2013SoPh..283..273Z}. Using \n\\textit{Hinode} observations, they estimate that the IN fields contribute up to \n$3.8\\times10^{26} \\rm{\\,Mx\\,day^{-1}}$ to the solar surface. This is an order of \nmagnitude higher than the global FER of $3 \\times 10^{25}\\rm{\\,Mx\\,day^{-1}}$ published \nby \\citet{2011SoPh..269...13T} and is two orders of magnitude higher than the \n$3.7\\times10^{24}\\rm{\\,Mx\\,day^{-1}}$ obtained by \\citet{2016ApJ...820...35G}.  In \n\\citet{2013SoPh..283..273Z}, it is assumed that every three minutes, the IN features \nreplenish the flux at the solar surface  with an average flux density of \n12.4\\,G ($\\,\\rm{\\,Mx\\,cm^{-2}})$. Here, three minutes is taken as the average lifetime of \nthe IN features \\citep{2010SoPh..267...63Z}.  {Their FER is nearly six times higher \nthan our estimate, although the lowest flux per feature to which \\textit{Hinode}\/SOT is \nsensitive is significantly larger than for \\textsc{Sunrise}\/IMaX (due to the lower spatial \nresolution of \nthe former). To understand this difference, we applied the method of \n\\citet{2013SoPh..283..273Z} to the \\textsc{Sunrise}\/IMaX observations. From the entire time \nseries, the total sum of the flux in all features with flux $> 9\\times10^{14}$ Mx is \n$1.1\\times10^{21}$\\,Mx over an area of $3.9\\times10^{20}$\\,cm$^{2}$. This gives us an \naverage flux density of 2.8\\,G, which is  4.5 times smaller than 12.4\\,G of \n\\citet{2013SoPh..283..273Z}. If the IN features are assumed to have an average lifetime \nof three minutes, similar to \\citet{2013SoPh..283..273Z}, then FER over the whole solar \nsurface is $8.2\\times10^{25}$\\,Mx\\,day$^{-1}$. This is 1.2 times higher than our original \nestimate from the feature tracking method. If instead, we take the average lifetime of \nthe features in our dataset from first appearance to final disappearance at the surface \nof \n$\\approx1.8$ minutes, we get an FER of $1.38\\times10^{26}$\\,Mx\\,day$^{-1}$, nearly 1.9 \ntimes higher than our original estimate and still 2.8 times smaller than \n\\citet{2013SoPh..283..273Z}. This is longer than 1.1 minute quoted in LSA17, which \nincludes death of a feature by splitting or merging (see LSA17), i.e. processes that do \nnot remove flux from the solar surface.}\n\n {To be sure that the problem does not lie in the COG technique employed here, we \nalso estimated the average flux density by considering the $B_{\\rm LOS}$ from the \nrecently \navailable inversions of the \\textsc{Sunrise}{} data \\citep{fatima}. The $B_{\\rm LOS}$ values \nreturned by the inversions differ from those given by the COG technique by about 5\\% on \naverage (individual pixels show much larger differences, of course), so that this cannot \nexplain the difference  to the value adopted by \\citet{2013SoPh..283..273Z}. If all the \npixels, including noise, are considered then the average flux density is 10.7\\,G. This is \nan absolute upper limit of the average flux density as a large part of it is due to noise \n and it is still lower than the IN signal of  12.4\\,G, estimated by \n\\citet{2013SoPh..283..273Z}.}\n {Thus the high value of FER from \\citet{2013SoPh..283..273Z} is at least partly \ndue to their possibly too high value of average flux density. The observations analysed \nby these authors clearly show network and enhanced network features. If some of these \nare misidentified, then this would result in a higher average flux density. If this is \nindeed the case, then the estimate of the lifetime of 3 minutes may also be too short \n\\citep[the technique of][neglects any possible correlation between magnetic flux and \nlifetime of a feature]{2013SoPh..283..273Z}. Although the amount of flux in IN \nfields is not expected to change significantly with time or place \n\\citep[see][]{2013A&A...555A..33B}, this is not true for the amount of \nflux in the network, which changes significantly. For example another time series taken \nby \\textsc{Sunrise}{} during its first flight, having slightly more network in the FOV, is found \nto \nhave an average $B_{\\rm LOS}$ of around 16\\,G (including noise), which is higher than the \n12.4\\,G used by \\citet{2013SoPh..283..273Z}. However, this is just a qualitative \nassessment and the very large FER found by \\citet{2013SoPh..283..273Z} needs to be probed \nquantitatively in a future study.}\n\n\\section{Conclusions}\n\\label{s8}\nIn this paper, we have estimated the FER in the quiet Sun from the IN features \nusing the observations from \\textsc{Sunrise}\/IMaX recorded during its first science flight in \n2009. We have included the contribution from features with fluxes in the range \n$9\\times10^{14} - 2.5\\times10^{18}$ Mx, whose evolution was followed directly. By \naccounting for the three important processes that bring flux to the solar surface: \n {unipolar and bipolar} appearances, and flux gained after birth by features born by \nsplitting or merging over their lifetime, we estimate an FER of $1100 \n\\rm{\\,Mx\\,cm^{-2}\\,day^{-1}}$. The third process is found to contribute significantly to \nthe FER. The smaller features with fluxes $\\le10^{16}$ Mx bring most of the flux to the \nsurface. Since our studies include fluxes nearly an order of magnitude smaller than the \nsmallest flux measured from the \\textit{Hinode} data, our FER is also an order of \nmagnitude higher when compared to studies using a similar technique \n\\citep[i.e.,][]{2016ApJ...820...35G}.  {We compare also with other estimates of the \nFER in the literature. \\citet{2011SoPh..269...13T} obtained a range of values. Those near \nthe lower end of the range \\citep[also quoted by ][]{th-thesis}, which are possibly the \nmore reliable ones, are roughly consistent with the results of \n\\citet{2016ApJ...820...35G}. The high FER of $3.8\\times10^{26}\\,{\\rm Mx\\, day^{-1}}$ \nfound by \\citet{2013SoPh..283..273Z} is, however, difficult to reconcile with any other \nstudy. It is likely so high partly due to the excessively large $B_{\\rm LOS}$ of \nIN fields of 12.4\\,G used by these authors, which is more than a factor of 4 \ntimes larger than the averaged $B_{\\rm LOS}$ of 2.8 G that we find. Even the absolute \nupper limit of the spatially averaged $B_{\\rm LOS}$ in our data (including noise) is \nbelow the value used by \\citet{2013SoPh..283..273Z}. We therefore \nexpect that they have overestimated the FER.}\n\n {There is clearly a need for further investigation, not only to quantify  the \nreasons for the different results obtained by different techniques. There are also still \nmultiple open questions. What is the cause of the increase and decrease of the flux of a \nfeature during its lifetime? Is this due to interaction with ``hidden'' flux? Is this \nhidden flux not visible because it is weak and thus below the noise threshold, or because \nit is structured at very small scales, i.e. it is below the spatial resolution? How \nstrongly does the ``hidden'' or missed flux change with changing spatial resolution? The \nmost promising approach to answering these and related questions is to study the flux \nevolution in an magnetohydrodynamic simulation that includes a working small-scale \nturbulent dynamo.}\n\n\\begin{acknowledgements}\n {We thank F. Kahil for helping with the comparison of COG and inversion results.} \nH.N.S. and  {L.S.A.} acknowledge the financial support from the Alexander von \nHumboldt foundation. The German contribution to \\textsc{Sunrise}{} and its reflight was funded by \nthe Max Planck Foundation, the Strategic Innovations Fund of the President of the Max \nPlanck Society (MPG), DLR, and private donations by supporting members of the Max Planck \nSociety, which is gratefully acknowledged. The Spanish contribution was funded by the \nMinisterio de Econom\\'i\u00ada y Competitividad under Projects ESP2013-47349-C6 and \nESP2014-56169-C6, partially using European FEDER funds. The HAO contribution was partly \nfunded through NASA grant number NNX13AE95G. This work was partly supported by the BK21 \nplus program through the National Research Foundation (NRF) funded by the Ministry of \nEducation of Korea.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nWillis materials \\cite{Willis_WM_1981, Milton_NJP_2006, Srivastava_IJSNM_2015, Koo_NatComms_2016, Nassar_JMPS_2017, Sieck_PRB_2017, Muhlestein_NatComms_2017, DiazRubio_PRB_2017, Ponge_EML_2017, Li_NatComms_2018, Quan_PRL_2018} are complex media in which the unusual coupling between stress and particle velocity on one hand and linear momentum and strain on the other hand enable unique properties that only recently began to be explored experimentally. These include unsurpassed control over the propagation of elastic waves \\cite{Milton_NJP_2006}, independent engineering of transmitted and reflected wave fronts \\cite{Koo_NatComms_2016, Muhlestein_NatComms_2017, Li_NatComms_2018}, and broadband sound isolation in thin structures \\cite{Popa_NatComms_2018}.\n\nWe uncover in this article a previously unexplored property of Willis materials, namely the ability to produce linear and broadband non-reciprocal sound transport through remarkably thin metamaterial layers, a feature unachievable through other methods. Acoustic non-reciprocity has recently become an appealing concept because it enables improved sound control, acoustic imaging, and signal processing in acoustic devices \\cite{Haberman_Fleury_review}. The initial work involved non-linear structures in which the behavior of higher harmonics was non-reciprocal while the frequency of the impinging excitation remained reciprocal \\cite{liu2000locally, liang2009pass, liang2010pass, boechler2011bifurcation, Popa_NatComms_2014}. However, truly unidirectional devices require an asymmetric material response at the fundamental frequency in linear media. This has been a more difficult task. Nevertheless, linear non-reciprocal devices based on acoustic circulator-like structures have been demonstrated \\cite{fleury2014circulator} but they are narrow band and bulky. Topological insulators have been shown to break sound propagation reciprocity \\cite{yang2015topological, ni2015topologically} but these techniques provide narrow band solutions which are not applicable to three-dimensional waves. Materials with time modulated properties support non-reciprocal broadband sound, however, the modulation must occur on a significant length scale, therefore these materials necessarily occupy a large volume which becomes an issue especially at low frequencies \\cite{Trainiti_NJP_2016,attarzadeh_AA_2018,Nanda_JASA_2018}.\n\nHere we show that active metamaterials supporting decoupled Willis material parameters overcome the limitations of previous approaches. Specifically, this article brings three major contributions. First, essentially all theoretical analysis on Willis materials assume unbiased passive media in which the two Willis coupling tensors are related (coupled) and satisfy a strict set of constraints \\cite{Quan_PRL_2018}. We demonstrate experimentally that breaking the constraints of passivity leads to highly non-reciprocal and broadband linear media whose deeply subwavelength dimensions are unachievable through other methods. Second, we present a method to extract the effective material properties of a metamaterial in which the Willis coupling terms are allowed to be decoupled. We employ this method to show that our metamaterial is characterized by one very large and one negligible Willis coupling term in a broad band of frequencies, a feature not possible in passive media. Third, we show experimentally for the first time non-reciprocal sound transport in a multi-dimensional space. Interestingly, our metamaterial replicates in acoustics a mechanism for highly non-reciprocal electromagnetic wave transport based on electromagnetic bianisotropy \\cite{Popa_PRB_2012}, which is a solid experimental justification of the term acoustic bianisotropy recently given to Willis materials \\cite{Sieck_PRB_2017}. However, unlike the electromagnetics case, the acoustic metamaterial is very broadband having an operation bandwidth of at least one octave. This suggests that broadband non-reciprocal bianisotropic metamaterials in electromagnetics are within reach.\n\n\\section{Willis Coupling \\& Non-reciprocity}\\label{media}\nThe constitutive relations in Willis acoustic media can be written in the following form using standard index notation \\cite{nemat2011overall}\n\\begin{equation} \\label{eq1}\n\\begin{split}\n    -p &= Bu_{k,k}+S_kv_k, \\\\                               \n    \\mu_{i} &= D_{i}u_{k,k}+ \\rho_{ij}v_j, \\\\\n\\end{split}\n\\end{equation}\nwhere the acoustic pressure $p$ is related to the volumetric strain $u_{k,k}$ via the bulk modulus $B$ and to the particle velocity vector $v_k$ via the Willis coupling vector $S_{k}$. The momentum density vector $\\mu_{i}$ is related to the particle velocity vector $v_j$ via the second order mass density tensor $\\rho_{ij}$ and the volumetric strain $u_{k,k}$ via the Willis coupling  vector $D_{i}$. \n\nThe coupling between adjacent unit cells inside a metamaterial means that the dynamics of the unit cells inside the middle of the metamaterial are slightly different from the dynamics of edge unit cells when the unit cell size is larger than roughly a tenth of the operating wavelength \\cite{Sieck_PRB_2017}. Therefore, the description of the metamaterial acoustic behavior in terms of macroscopic material parameters becomes insufficient. A better description of the metamaterial dynamics is given in terms of microscopic polarizabilities \\cite{Sieck_PRB_2017, Popa_NatComms_2018}. According to this description, acoustic Willis metamaterials are modeled as collections of highly subwavelength sources that sense the monopole (i.e., local pressure field $p_{loc}$) or dipole moments (i.e., local velocity $v_{loc}$) in the surrounding acoustic field and create local monopole (i.e., pressure) or dipole moments (i.e., particle velocity) in response according to the following equations\n\\begin{equation}\n\\begin{aligned}\np_d&=0, &(v_d)_i&=(\\alpha_d)_{ij}(v_{loc})_j,\\\\\np_m&=\\alpha_mp_{loc}, &(v_m)_i&=0,\\\\\np_{dm}&=(\\alpha_{dm})_i Z_0 (v_{loc})_i, &(v_{dm})_i&=0,\\\\\np_{md}&=0, &(v_{md})_i&=(\\alpha_{md})_i Z_0^{-1} p_{loc},\n\\end{aligned}\n\\label{p&v}\n\\end{equation}\nwhere the subscripts $d$ and $m$ refer to the conventional dipole-to-dipole and monopole-to-monopole sources and $dm$ and $md$ refer to the Willis dipole-to-monopole and monopole-to-dipole sources. The local pressure generated by these sources are $p_d$, $p_m$, $p_{dm}$ and $p_{md}$ and the generated local particle velocities are $v_d$, $v_m$, $v_{dm}$ and $v_{md}$. The characteristic acoustic impedance of air is $Z_0$. Finally, the unitless polarizabilities $\\alpha_d$, $\\alpha_m$, $\\alpha_{dm}$ and $\\alpha_{md}$ quantify the linear relationship between the generated pressure and particle velocity and the sensed local fields.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3.3truein]{figures\/configml.eps}\n  \\caption{Metasurface represented by collocated polarizability sheets having the acoustic polarizabilities $\\alpha_d$, $\\alpha_m$, $\\alpha_{md}$, and $\\alpha_{dm}$. The pressure and velocity fields excited by the incident field $p_i$ contribute to the local pressure $p_{loc}$ and velocity $v_{loc}$ at the position of the sheets.}\n  \\label{microconfig}\n\\end{figure}\n\nPassive materials undergoing no external biasing require correlated Willis coupling polarizabilities $\\alpha_{dm}$ and $\\alpha_{md}$ \\cite{muhlestein2016macro}. We show here that we can break this correlation in active metamaterials. Namely, we generate strong non-reciprocal sound transport in metamaterials in which we control $\\alpha_{dm}$ and $\\alpha_{md}$ independently.\n\nTo simplify the analysis, we consider a one-dimensional scenario in which a plane wave $p_i$ is normally incident on a one-unit-cell-thick metasurface modeled as four co-located polarizability sheets (see Fig. \\ref{microconfig}). In this scenario the vector and tensor quantities in Eqs. (\\ref{p&v}) become scalars. Consequently, the local pressure ($p_{loc}$) and particle velocity ($v_{loc}$) at the location of the polarizability sheets are the superposition of the incident and generated fields. Their expressions are given by the following equations.\n\\begin{equation}\n\\begin{aligned}\np_{loc}&= p_i+\\alpha_{m}p_{loc}+\\alpha_{dm}v_{loc}Z_0,\\\\\nv_{loc}&= s Z_0^{-1} p_i+\\alpha_dv_{loc}+\\alpha_{md} Z_0^{-1} p_{loc},\n\\end{aligned}\n\\label{pi}\n\\end{equation}\nwhere $s$ is a parameter that quantifies the direction of the incident wave, i.e., $s=1$ if the incident wave propagates in $+x$ direction and $s=-1$ if the incident wave propagates in $-x$ direction.\n\nEquations (\\ref{p&v}) and (\\ref{pi}) lead to the following relationships between the local pressure, local particle velocity, and the incident acoustic pressure at the location of the polarizability sheets\n\\begin{equation}\n\\begin{aligned}\np_{loc}&= \\frac{s\\alpha_{dm}+1-\\alpha_{d}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}p_i,\\\\\nv_{loc}&= \\frac{s(1-\\alpha_{m})+\\alpha_{md}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}Z_0^{-1}p_i.\n\\end{aligned}\n\\label{local}\n\\end{equation}\n\nAccording to Fig. \\ref{microconfig}, the relationship between the incident pressure $p_i$, reflected pressure $p_r$, and transmitted pressure $p_t$ can be described as follows.\n\\begin{equation}\n\\begin{aligned}\np_t&=p_i+p_m+p_{dm}+s(v_{md}+v_d)Z_0,\\\\\np_r&=p_m+p_{dm}-s(v_{md}+v_d)Z_0.\\\\\n\\end{aligned} \n\\label{Ps}\n\\end{equation}\n\nEquations (\\ref{p&v}), (\\ref{local}), and (\\ref{Ps}) lead to the following scattering parameters calculated in terms of polarizabilities:\n\\begin{equation}\n\\begin{aligned}\nS_{21}&\\equiv\\frac{p_t}{p_i}|_{s=+1}=\\frac{(1+\\alpha_{md})(1+\\alpha_{dm})-\\alpha_m\\alpha_d}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \\\\\nS_{11}&\\equiv\\frac{p_r}{p_i}|_{s=+1}=\\frac{\\alpha_m-\\alpha_d+\\alpha_{dm}-\\alpha_{md}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \\\\\nS_{22}&\\equiv\\frac{p_r}{p_i}|_{s=-1}=\\frac{\\alpha_m-\\alpha_d+\\alpha_{md}-\\alpha_{dm}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \\\\\nS_{12}&\\equiv\\frac{p_t}{p_i}|_{s=-1}=\\frac{(1-\\alpha_{md})(1-\\alpha_{dm})-\\alpha_m\\alpha_d}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \n\\end{aligned}\n\\label{S-p}\n\\end{equation}\nwhere $S_{ij}$ ($i,j=\\overline{1,2}$) are the usual $S$-parameters representing the pressure reflection and transmission coefficients for propagation in the $+x$ and $-x$ directions.\n\nAll passive materials under no external biasing satisfy $\\alpha_{dm}=-\\alpha_{md}$ and the material is reciprocal, i.e., $S_{12}=S_{21}$. However, if $\\alpha_{dm}$ and $\\alpha_{md}$ could be controlled independently, the reciprocity could be broken. For instance, letting $\\alpha_{dm}=0$ and $\\alpha_{md}\\neq 0$ results in very different transmission coefficients $S_{21}$ and $S_{12}$, which means that the polarizability sheets behave as a highly non-reciprocal medium. We demonstrate this point experimentally by constructing a non-reciprocal active acoustic metamaterial in which $\\alpha_{dm}\\approx 0$ and $\\alpha_{md}\\neq 0$. The value of $\\alpha_{md}$ will be chosen to maximize the amplitude of the non-reciprocity factor, defined as $S_{21}\/S_{12}$.\n\n\\section{Active metamaterial design and testing}\\label{method}\nBianisotropic (Willis) metamaterials in which the Willis coupling terms are independently controlled have been shown to provide excellent sound isolation capabilities \\cite{Popa_NatComms_2018}. We reconfigure this metamaterial platform to generate highly non-reciprocal responses. Specifically, the unit cell shown in Fig. \\ref{fi3-1}a consists of a 3.5 cm by 3.5 cm printed circuit board (PCB) designed to implement a monopole-to-dipole source. An omnidirection micro-electro-mechanical (MEMS) transducer senses the local pressure, i.e., the monopole moment in the field, and drives a dipole source composed of two back-to-back flat transducers working $180^\\circ$ out-of-phase via an amplifier circuit of impulse response $g$. The sensing transducer is placed in the plane of symmetry of the dipole in order to ensure that the generated local particle velocity does not contribute to the local pressure $p_{loc}$ sensed by the MEMS transducer. As a result, the unit cell's monopole-to-dipole polarizability is proportional to $g$ and thus it is controlled by the electronics. At the same time the one-directional nature of the amplifier enforces $\\alpha_{dm}=0$.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3.3truein]{figures\/unitcell_waveguide.eps}\n  \\caption{Metamaterial fabrication and measurements. (a) Unit cell schematic; (b) Constructed 5-unit-cell metasurface; (c) The experimental setup shows the metasurface placed inside a two-dimensional acoustic waveguide. The pressure fields are measured in the highlighted areas.}\n  \\label{fi3-1}\n\\end{figure}\n\nA Willis metasurface consisting of five identical unit cells is constructed to demonstrate its non-reciprocal nature (see Fig. \\ref{fi3-1}b). The thickness of the metasurface with all components installed is approximately 9 mm. The metasurface is tested inside a two-dimensional waveguide of dimensions 1.2 m by 1.2 m. A schematic of the experimental setup is presented in Fig. \\ref{fi3-1}c. An external source speaker is placed at the edge of the waveguide. The metasurface is placed 58 cm away from the speaker in a window cut in the middle of a wall dividing the waveguide into two regions. The metasurface is designed so that its effective polarizabilities $\\alpha_m$ and $\\alpha_d$ match those of the wall. The external speaker produces short Gaussian pulses centered on 3000 Hz and having a width of 5 periods at the center frequency.\n\nWe scan the pressure fields in the vicinity of the metasurface in the region highlighted in Fig. \\ref{fi3-1}c in two scenarios. In the first scenario, the external speaker faces the side of the metasurface that is opaque to sound, which we call the reverse orientation in analogy to the reverse polarization of an electronic diode. In the second scenario, the speaker faces the transparent side (forward orientation). \n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=7truein]{figures\/mega.eps}\n  \\caption{\n    Measured transmitted sound pressure distribution. (a) Set up of the waveguide indicating the region measured for transmitted sound pressures; (b) Transmitted sound pressure distribution at one time point, left: pressure distribution for reverse orientation, right: pressure distribution for forward orientation; (c) Transmitted sound pressure distribution at 3000 Hz, left: pressure distribution for reverse orientation, right: pressure distribution forforward orientation; the regions surrounded by the dotted lines are the areas affected by the metasurface (i.e.,, effective area).\n  }\n    \\label{merit}\n\\end{figure*}\n\\section{Experimental Results}\\label{result}\nThe metasurface non-reciprocal behavior is measured directly by comparing the transmitted fields obtained in these two orientations (see Fig. \\ref{merit}a). Figure \\ref{merit}b shows the spatial distribution of the transmitted acoustic waves for the reverse and forward orientations and measured at time instances that better show the transmitted pulses. The spatial region influenced by the metasurface is marked by the dotted lines obtained by connecting the position of the external speaker and the edges of the metasurface. These measurements confirm qualitatively that the metasurface is transparent in the forward orientation and opaque in the reverse orientation. In contrast, outside the dotted lines the measured pressure is the same for both orientations which confirms the reciprocal behavior of the conventional wall.\n\nTo quantify the performance of the metasurface as a broadband non-reciprocal medium, we compute the spectra of the transmitted waves from the time domain measurements. Figure \\ref{merit}c shows the spectra obtained for the reverse (left) and forward orientations (right) at the incident pulse center frequency of 3000 Hz, which further confirms the strongly asymmetric transmission characteristics. Furthermore, we define the non-reciprocity factor as the ratio between the frequency domain pressure measured in the reverse orientation ($p_t^{rev}$) to the pressure measured in the forward orientation ($p_t^{fwd}$) 4 cm behind the metasurface (point A in Fig. \\ref{merit}c), namely $p_t^{rev}\/p_t^{fwd}$, which is a quantity equal to $S_{21}\/S_{12}$.\n\n\\begin{figure}\n  \\includegraphics[width=3.2truein]{figures\/nonr.eps}\n  \\caption{%\n   Amplitude and phase of the non-reciprocity factor retrieved from acoustic pressure field measurement.\n  }\n  \\label{nonr}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3.4truein]{figures\/polarizabilities.eps}\n  \\caption{%\n    Acoustic polarizabilities calculated from measurements.\n  }\n  \\label{alpha}\n\\end{figure}\n Figure \\ref{nonr} shows the amplitude and phase of the non-reciprocity factor in the 2000 Hz to 4000 Hz octave excited by the external source. The figure confirms that the transmitted pressure is significantly different in the entire octave (i.e.,, $p_t^{rev}\/p_t^{fwd}\\neq 1$). The measured acoustic pressure amplitudes are significantly different at most frequencies. For instance, the amplitudes are more than 5 dB apart in the entire 2600 Hz to 3800 Hz band. Interestingly, whenever the $p_t^{fwd}$ and $p_t^{rev}$ amplitudes become comparable, their phases are almost $180^\\circ$ out-of-phase. This is explained by a transmitted field dominated by the dipole moment created by the active metasurface in response to the sensed monopole moment. The fields produced in the transmission directions have the same magnitude but different signs for two opposite directions of propagation.\n \nA key advantage of our active Willis material is its ability to control the Willis polarizabilities $\\alpha_{md}$ and $\\alpha_{dm}$ independently, which  is not possible in passive media \\cite{Sieck_PRB_2017, Quan_PRL_2018}. We compute the acoustic polarizabilities by inverting Eqs. (\\ref{S-p}) and obtain\n\n\\begin{equation}\n\\begin{aligned}\n\\alpha_m&=\\frac{S_{12}S_{21}-(1-S_{11})(1-S_{22})}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\\\\\n\\alpha_d&=\\frac{S_{12}S_{21}-(1+S_{11})(1+S_{22})}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\\\\\n\\alpha_{md}&=\\frac{S_{21}-S_{12} + S_{22}-S_{11}}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\\\\\n\\alpha_{dm}&=\\frac{S_{21}-S_{12} + S_{11}-S_{22}}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\n\\end{aligned} \n\\label{alpha-S}\n\\end{equation}\nwhere the numerical values of the $S$-parameters are evaluated from the fields measured behind and in front of the metasurface using a standard approach \\cite{Popa_PRB_2013, Popa_JASA_2016, Muhlestein_NatComms_2017}. Specifically, we measure the reflection ($S_{11}$) and transmission ($S_{21}$) coefficients in the the reverse orientation by exciting the metasurface with a speaker placed on the $x<0$ side of the metasurface (see Fig. \\ref{merit}a). The reflection and transmission coefficients measured in the forward orientation ($S_{22}$ and $S_{12}$) are obtained by placing the excitation speaker on the other side of the metasurface in the $x>0$ region. The speaker is positioned several wavelengths away from the metasurface to assure that the wavefront curvature at the metasurface is relatively small. The small curvature  justifies our one-dimensional analysis as explained in past work describing the process of measuring reflection and transmission coefficients from field measurements \\cite{Popa_JASA_2016}.\n\nThe polarizabilities computed with Eqs. (\\ref{alpha-S}) are shown in Fig. \\ref{alpha}. The results demonstrate our ability to control the Willis polarizabilities independently. Specifically, $\\alpha_{dm}\\approx 0$ in the selected frequency range while $\\alpha_{md}$ is significantly different from $\\alpha_{dm}$, a feature not possible in passive structures. Remarkably, $\\alpha_{md}$ is the second highest polarizabilitiy generated in a very compact metasurface. The largest polarizability is $\\alpha_{d}$ which confirms that the unpowered metasurface behaves like a membrane that produces significant dipole moment but small monopole moment, i.e., $\\alpha_{m}$ is small. The measurements also show that the effective polarizabilities are relatively constant in frequency, which demonstrates the broadband nature of the metamaterial. The measured variation with frequency is caused by numerical artifacts typically appearing in the process of extracting the effective material properties of opaque and acoustically thin structures \\cite{lee2010composite, Zigonenau_JAP_2011}.\n\n\\section{Conclusion}\nTo conclude, we demonstrated experimentally that the independent control of the two acoustic Willis coupling terms enables extremely non-reciprocal linear media composed of highly subwavelength and broadband unit cells, which are features unachievable in passive structures. Interestingly, our approach mirrors a method to obtain large non-reciprocal electromagnetic wave transport in media having asymmetric magneto-electric coupling terms, which is a strong experimental confirmation of the connection between electromagnetic bianisotropy and Willis media. Furthermore, we showed how the four basic acoustic polarizabilities of acoustic media can be retrieved from sound field measurements performed around the metamaterial. The extracted polarizabilities confirm our ability to design broadband and non-resonant Willis materials having strong bianisotropic responses. We believe that the range of effective acoustic properties enabled by active Willis materials will afford an unparalleled level of control over the sound propagation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe goal of quantum computing is to implement a programmable quantum information processor. Such a processor requires access to a universal gate set from which any quantum algorithm can be constructed. Universal gate sets can be formed from single-qubit gates supplemented by a two-qubit entangling gate\\cite{divincenzo1995universalqc}.\nFurthermore, fault-tolerance is necessary in order to perform arbitrarily long and precise computations, which, for the most lenient error correcting surface codes, puts a lower bound of around 0.99 on the required gate fidelities\\cite{raussendorf2007, corcoles2015, obrien2017, fowler2012}.\nExtensible high-fidelity entangling two-qubit gates are thus key elements in any multi-purpose quantum information processor.\n\nSingle-qubit gate operations are routinely performed with fidelities above 0.99\\cite{buluta2011review, gustavsson2013, reagor2018rigetti, rol2017, sheldon2016, chen2016, barends2014cz, wright2019, harty2014, itoh2014, zajac2018}, but pushing two-qubit gate fidelities above 0.99 still proves a daunting task.\nDespite the challenges in realizing a low loss environment while at the same time having high control of two-qubit operations, several two-qubit gates have been reported to do so.\nThe first group to accomplish this was Benhelm \\textit{et al.}, who in 2008 demonstrated a M\u00f8lmer-S\u00f8rensen-type entangling gate\\cite{sorensen1999, sorensen2000} with a fidelity of 0.993 using laser-controlled trapped calcium ions\\cite{benhelm2008}.\nSince then, similar ion trap experiments have realized high-fidelity two-qubit gates\\cite{gaebler2016, ballance2016, erhard2019, harty2016, ballance2015}.\nAnother promising qubit architecture is silicon-based quantum dots\\cite{koiller2001, friesen2003, itoh2014, zajac2018}, where controlled-rotation gates were recently benchmarked with a fidelity of 0.98\\cite{huang2019}.\n\n\nIn superconducting qubits the controlled-phase (\\textsf{CZ}) gate~\\cite{barends2014cz, kelly2014cz, Yu2014, rol2019, kjaergaard2019} and the cross-resonance (\\textsf{CR}) gate\\cite{Sheldon2016crgate} have been shown to exceed a fidelity of 0.99.\nOther two-qubit gates, like the $i\\textsf{SWAP}$ and $\\sqrt{i\\textsf{SWAP}}$ gates\\cite{mckay2016iswap, Dewes2012sqrtiswap, salathe2015iswap, reagor2018rigetti, caldwell2017}, $b\\textsf{SWAP}$ gate\\cite{poletto2012bswap}, the resonator induced phase (\\textsf{RIP}) gate\\cite{paik2016ripgate}, and a parametric \\textsf{CZ} gate\\cite{reagor2018rigetti, caldwell2017}, have been demonstrated with fidelities in the 0.9's.\nThese quantum gates are typically performed with transmons\\cite{transmon_original, you2007pretransmon, Barends2013xmon, Yu2014}, coupled directly to each other or via a separate coupling element, e.g. a transmission line resonator or a tunable coupler.\n\n\n\n\nIn this work, we propose the implementation of controlled two-qubit operations utilizing quantum interference patterns in a network of four qubits.\nAs a specific architecture, where this four-qubit gate can be implemented natively, we consider superconducting transmon qubits placed in a diamond-shaped geometry.\nThe qubits are coupled only through simple capacitive couplings. A similar 2D array of transmons was considered in Refs.~\\cite{lloyd2016, ciani2019}, but with different couplings and purpose.\nThe system comprises a four-qubit quantum gate (`the diamond gate'), where the state of two qubits control a two-qubit gate operation on the remaining two qubits.\nSince the diamond gate natively implements multiple unitaries, it is a useful addition to the gate set used for quantum simulation and quantum compilation.\nDue to its ability to perform (controlled) two-qubit entangling operations, supplementing the diamond gate with single-qubit operations allows for universal quantum computing on the target qubits.\n\nIn Sec.~\\ref{sec:system} we discuss the operation of the diamond gate, and in Sec.~\\ref{sec:extensibility} how it can constitute a building block in an extensible quantum computer.\nIn Sec.~\\ref{sec:simulations} we simulate the transmon implementation of the gate, using parameters from state-of-the-art superconducting qubits, in a Lindblad master equation simulation.\nWe find that the gate generally operates with fidelity around 0.99 in less than 100 ns.\nFinally, in Sec.~\\ref{sec:qutrits} we consider the effects of couplings to higher-energy states in the transmon spectrum, leading to undesired leakage across the control. We show how this behavior can be counteracted by engineering a cross-coupling to cancel the effects.\nThis is a passive scheme, in contrast to the microwave pulse-based scheme recently shown to reduce leakage in the \\textsf{CZ} gate\\cite{rol2019}.\n\nThroughout this paper, we use units where $\\hbar = 1$.\n\n\n\n\n\n\\section{Results}\n\n\\subsection{Four-qubit diamond gate}\n\\label{sec:system}\n\nConsider the four-qubit Hamiltonian being a sum of the non-interacting part\n\\begin{equation}\n   H_0 = -\\frac{1}{2} (\\Omega + \\Delta) (\\sigma_z^\\text{T1} + \\sigma_z^\\text{T2})\n   - \\frac{1}{2} \\Omega (\\sigma_z^\\text{C1} + \\sigma_z^\\text{C2}) \\; ,\n   \\label{eq:H0}\n\\end{equation}\nwhere $\\Omega + \\Delta$ ($\\Omega$) is the fixed frequency of the target (control) qubits, and the interaction terms\n\\begin{equation}\n   H_\\text{int} = J_\\text{C} \\, \\sigma_y^\\text{C1}\\sigma_y^\\text{C2}\n   + J (\\sigma_y^{\\text{T}1} + \\sigma_y^{\\text{T}2}) (\\sigma_y^{\\text{C}1} + \\sigma_y^{\\text{C}2}) \\; .\n   \\label{eq:Hint}\n\\end{equation}\nHere $\\sigma_z^j = \\dyad{0}{0}_j - \\dyad{1}{1}_j$ and $\\sigma_y^j = i\\dyad{1}{0}_j - i\\dyad{1}{0}_j$ are Pauli operators on qubit $j$, and the qubit frequencies are assumed positive such that $\\ket{0}_j$ is the non-interacting qubit ground state. For simplicity we have assumed that the two target (control) qubits are on resonance, and that all the couplings between the target and control qubits have the same strength $J$, although, as we will show later, this contraint is not needed for high performance of the gate. The four-qubit system is sketched in Figure~\\ref{fig:superconducting_circuit}a.\nAs we will discuss in the following, the system implements a four-qubit gate, which we will refer to as `the diamond gate' due to the geometry of the system.\n\nSuperconducting circuits offer a natural platform for implementing this type of Hamiltonian\\cite{krantz2019}. Specifically, by truncating the Hilbert space for each degree of freedom to qubits, the circuit of four capacitively coupled transmon qubits in Figure~\\ref{fig:superconducting_circuit}b implements the Hamiltonian.\nLater, we analyze the model including the second excited state of the transmon qubits.\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{figure1.pdf}\n  \\caption{\n  \\textbf{(a)} The diamond gate: Four-qubit system consisting of two target qubits (T1 and T2) and control qubits (C1 and C2) coupled through exchange interactions (dashed lines) with the indicated strengths.\n  \\textbf{(b)} Lumped element superconducting circuit diagram of four capacitively coupled transmons, where each colored subcircuit corresponds to the same-colored qubit in \\textbf{(a)}.\n  \\textbf{(c)--(d)} Example transformations implemented by the diamond gate, $U$, of Eq.~\\eqref{eq:U}.}\n  \\label{fig:superconducting_circuit}\n\\end{figure*}\n\nWe now consider the interaction Hamiltonian, $H_\\text{int}$, in the frame rotating with $H_0$ and simplify the expression by assuming $\\lvert 2\\Omega \\rvert \\gg \\lvert J \\rvert$ (rotating wave approximation), which allows us to ignore the most rapidly oscillating terms. The system Hamiltonian is then\n\\begin{equation}\n   H = J_\\text{C} \\, \\sigma_+^\\text{C1}\\sigma_-^\\text{C2}\n   + J \\, e^{i\\Delta t}\n   (\\sigma_+^{\\text{T}1} + \\sigma_+^{\\text{T}2}) (\\sigma_-^{\\text{C}1} + \\sigma_-^{\\text{C}2})\n   + \\text{H.c.} \\; ,\n   \\label{eq:H}\n\\end{equation}\nwith $\\sigma_+^j = \\dyad{1}{0}_j$ and $\\sigma_-^j = \\dyad{0}{1}_j$ on qubit $j$. This Hamiltonian governs the dynamics resulting from the interactions in the model. We show in Appendix~\\ref{sec:unitary_gate_analysis} that the effective unitary time-evolution of $H$ gives rise to a four-qubit gate operating by means of controlled quantum interference (the diamond gate).\nThe analysis is based on a Magnus expansion of $H$ within Floquet theory, which assumes $\\lvert \\Delta \\rvert \\gg \\lvert J \\rvert, \\lvert J_\\text{C} \\rvert$, i.e. a qubit detuning much larger than the coupling strengths.\n\nThe diamond gate is a four-way controlled two-qubit gate operation on the target qubits T1 and T2.\nConsider the following gates in the target qubit computational basis, $\\{ \\ket{00}_\\text{T}, \\ket{01}_\\text{T}, \\ket{10}_\\text{T}, \\ket{11}_\\text{T} \\}$, where the superscripts refer to the control setting (discussed below):\n\\begin{align}\n    \\label{eq:UT00}\n    U^{00}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$-1$} & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$-1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$-1$}\n    \\end{pmatrix}\n    = \\textsf{ZZ} \\cdot \\textsf{CZ} \\cdot \\textsf{SWAP}\n    \\; , \\\\\n    \\label{eq:UT11}\n    U^{11}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$-1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$-1$} & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$-1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$1$}\n    \\end{pmatrix}\n    = - \\textsf{CZ} \\cdot \\textsf{SWAP}\n    \\; , \\\\\n    \\label{eq:UTpl}\n    U^{\\Psi^+}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$-1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$1$} & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$-1$}\n    \\end{pmatrix}\n    e^{-it_g J_\\text{C}}\n    = - \\textsf{ZZ} \\, e^{-it_g J_\\text{C}}\n    \\; , \\\\\n    \\label{eq:UTmi}\n    U^{\\Psi^-}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$1$} & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$1$}\n    \\end{pmatrix}\n    e^{+it_g J_\\text{C}}\n    = \\textsf{II} \\, e^{+it_g J_\\text{C}}\n    \\; .\n\\end{align}\nHere $t_g$ is the gate time given by\n\\begin{equation}\n   t_g = \\frac{\\pi\\lvert \\Delta\\rvert}{4J^2} \\; .\n   \\label{eq:tg}\n\\end{equation}\nEqs.~\\eqref{eq:UT00}--\\eqref{eq:UTmi} show the two-qubit operations in terms of well-known gates from the literature, see e.g. Ref.~\\cite{nielsen_chuang_2010}. Here $\\textsf{ZZ}$ is understood as a $\\textsf{Z}$ gate on each target qubit. Thus we see that $U^{00}_\\text{T}$ and $U^{11}_\\text{T}$ are two different combined swap and phase operations.\nAccess to just one of these entangling gates will facilitate universal quantum computing.\nThe third gate, $U^{\\Psi^-}_\\text{T}$, is a phase operation distinguishing target states with different parity (addition of T1 and T2's bit value modulo 2) by application of a relative sign.\nThe final gate, $U_\\text{T}^{\\Psi^-}$, which just adds a global phase, is the identity gate. We can therefore regard the preceding three gates as actual computational gates, while $U_\\text{T}^{\\Psi^-}$ is the idle position of the device.\n\nThe above two-qubit gates are controlled by the state of the control qubits, which we describe in the following orthonormal basis: $\\{ \\ket{00}_\\text{C}, \\ket{11}_\\text{C}, \\ket{\\Psi^+}_\\text{C}, \\ket{\\Psi^-}_\\text{C} \\}$.\nWe refer to this basis, which mixes computational basis states and the Bell states $\\ket{\\Psi^\\pm}_\\text{C} = (\\ket{01}_\\text{C} \\pm \\ket{10}_\\text{C})\/\\sqrt 2$, as the control basis. The full four-qubit unitary operation of the diamond gate is\n\\begin{equation}\n    \\label{eq:U}\n    \\begin{aligned}\n        U ={} &\\dyad{00}_\\text{C} U^{00}_\\text{T} + \\dyad{11}_\\text{C} U^{11}_\\text{T} \\\\\n              &+ \\dyad{\\Psi^+}_\\text{C} U^{\\Psi^+}_\\text{T} + \\dyad{\\Psi^-}_\\text{C} U^{\\Psi^-}_\\text{T} \\; .\n    \\end{aligned}\n\\end{equation}\nCast this way, it is evident that $U$ describes a four-way controlled operation on the target qubits. If the control qubits are initialized in one of the control basis states, only the corresponding gate among \\eqref{eq:UT00}--\\eqref{eq:UTmi} is performed. The control state is unchanged after the gate operation.\nFigure~\\ref{fig:superconducting_circuit}c--d illustrate the gate operation on the target state $\\ket{01}_\\text{T}$ in the cases where the control is $\\ket{00}_\\text{C}$ and $\\ket{\\Psi^+}_\\text{C}$, respectively. However, these gate diagrams only show the gate operation for these two control states, and in general the diamond gate performs a unitary operation on any initial four-qubit state.\nA more sophisticated decomposition of the full unitary $U$ is given i Figure~\\ref{fig:gate_circuit} in Appendix~\\ref{sec:unitary_gate_analysis}, where we note that the complexity in terms of number of $\\textsf{CNOT}$ gates is 42.\nHave access to four controlled two-qubit operations natively is useful for quantum simulation and may ease quantum gate compilation significantly.\n\n\n\nAs shown in Appendix~\\ref{sec:unitary_gate_analysis}, the unitary time-evolution under the Hamiltonian of Eq.~\\eqref{eq:H} approximately gives rise to $U$. Within the first order Magnus expansion, the approximation is exact when $J_\\text{C} = 0$, however a non-zero coupling between the control qubits is needed in order to initialze the control Bell states.\nSuch a coupling allows the triplet states $\\{ \\ket{00}_\\text{C} , \\ket{11}_\\text{C}, \\ket{\\Psi^+}_\\text{C} \\}$ to mix slightly during the gate operation, in which case the separation of control states in Eq.~\\eqref{eq:U} is no longer exact.\nThis leads to small gate infidelities of the order $(2J\/\\Delta)^2 = \\pi\/(t_g\\Delta)$ when then control qubits are initialized in $\\ket{00}_\\text{C}$ or $\\ket{11}_\\text{C}$, and twice as large when the control is in $\\ket{\\Psi^+}_\\text{C}$.\nFor typical superconducting circuit parameter values, like the ones used in the following section, these infidelities are on the order $10^{-3}$ to $10^{-2}$.\nNotice that the infidelity scales inversely with the gate time, leading to a trade-off between a fast gate and high-fidelity coherent operations.\nSince the singlet state $\\ket{\\Psi^-}_\\text{C}$ does not mix with the triplet states, the idle gate operation is not affected by the coupling $J_\\text{C}$, and the gate fidelity is only limited by other factors, e.g. qubit decoherence.\n\nAs mentioned above, the performance of the gate is increased if $J_\\text{C} = 0$, however a non-zero direct coupling between the control qubits is necessary if we wish to preparate the entangled Bell states.\nIn the following, we will assume a fixed value of $J_\\text{C}$, although ideally a tunable coupler\\cite{yan2018} can be used to turn on the coupling only during control state preparation.\nIf the control qubits are detuned from the target qubits, $\\lvert \\Delta \\rvert \\gg \\lvert J \\rvert$, we can initialize the control state without affecting the target qubits.\nThis detuning can be achieved by flux tunable devices, or by fabricating single-junction qubits with different frequencies.\nThus, ignoring the oscillating terms of Eq.~\\ref{eq:H}, we have effectively decoupled the control and target qubits.\nWe note that the effective Hamiltonian of the control qubits in the rotating frame, $J_\\text{C} (\\sigma_+^\\text{C1}\\sigma_-^\\text{C2} + \\sigma_-^\\text{C1}\\sigma_+^\\text{C2})$, has a zero-energy subspace spanned by $\\ket{00}_\\text{C}$ and $\\ket{11}_\\text{C}$, and eigenstates $\\ket{\\Psi^\\pm}_\\text{C}$ of energy $\\pm J_\\text{C}$.\nAn energy separation of $J_\\text{C}\/2\\pi \\sim \\SI{20}{\\mega\\hertz}$ allows us to initialize the control in $\\ket{\\Psi^\\pm}_\\text{C}$ by driving energy transitions\\cite{poletto2012bswap, Sheldon2016crgate}.\nTo initialize the control in $\\ket{00}_\\text{C}$ or $\\ket{11}_\\text{C}$, we can induce Rabi oscillations between these two states by driving the control qubits similarly to the procedure analyzed in Ref.~\\cite{didier2018}.\n\n\n\\subsection{Extensible quantum computer}\n\\label{sec:extensibility}\n\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{figure2.pdf}\n    \\caption{Proposed architecture for an extensible quantum computer. \\textbf{(a)} Four connected copies of the four-qubit diamond gate device. Detuning the qubits on the plaquettes A from the qubits on the plaquettes B allows each four-qubit device to run the diamond gate independently, while tuning the connecting qubits into resonance allows swap operations between plaquettes A and B. \\textbf{(b)} A sequence of diamond gates $U$ of Eq.~\\eqref{eq:U} in each plaquette and two-qubit swaps between the plaquettes running on the 16-qubit quantum computer.}\n    \\label{fig:extensibility}\n\\end{figure}\n\nThe four-qubit quantum interference device can constitute a building block in an extensible quantum computer by connecting several copies.\nOne possible architecture is illustrated in Figure~\\ref{fig:extensibility}a, where a 16-qubit quantum computer is constructed by connecting four copies of the four-qubit device, for instance through capacitive couplings.\nOn the plaquettes labelled A the control qubits are oriented vertically (1, 2, 13 and 14) and the target qubits horizontally (3, 4, 15 and 16), while the diamond gate devices on the plaquettes B are rotated by ninety degrees, such that control and target qubits from different plaquettes are connected.\nThis design of alternating A and B plaquettes can be extended in a straight-forward manner in one or two dimensions.\n\n\nThe quantum algorithm shown in Figure~\\ref{fig:extensibility}b is a generic algorithm spreading entanglement in the computer. Supplemented with single-qubit rotations, it may serve as a variational quantum eigensolver\\cite{Abhinav2017ibm}.\nThe algorithim can be implemented in the following way. Initially, the plaquette A qubits are far detuned from the plaquette B qubits, allowing each four-qubit diamond gate device to run the unitary gate $U$ of Eq.~\\eqref{eq:U} independently. After the completion of the gates, we can prevent further dynamics within each plaquette by switching the controls to the idle state. Then, by tuning pairs of connected qubits from different plaquettes into resonance, for instance 4 and 5, we can perform swap gates or use a suitable microwave driving to perform other desired two-qubit operations.\nFinally, by tuning the qubits out of resonance, and potentially switching certain controls, we are ready to run the diamond gate again.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Numerical simulations}\n\\label{sec:simulations}\n\nAlthough the analytic results suggest a functioning four-qubit diamond gate, we use numerical simulations to quantify the performance of the gates for state-of-the-art superconducting qubit parameters\\cite{wang2018, kounalakis2018, Wendin2016cqed_review}. Decoherence is included via the Lindblad master equation,\n\\begin{equation}\n  \\dot \\rho = - i [H, \\rho]\n  + \\sum_n \\Big[ C_n \\rho C_n^\\dagger\n  - \\frac{1}{2} (\\rho C_n^\\dagger C_n +  C_n^\\dagger C_n \\rho)\n  \\Big] \\; .\n  \\label{eq:master-equation}\n\\end{equation}\nHere $\\rho$ is the density matrix, $H$ is the Hamiltonian of Eq.~\\eqref{eq:H}, and the sum is taken over the following eight collapse operators, $C_n$: $\\sqrt{\\gamma} \\, \\sigma_z^{i}$ inducing pure dephasing and $\\sqrt{\\gamma} \\, \\sigma_-^{i}$ inducing qubit relaxation (photon loss), with $i$ running over all four qubits, denoting by $\\gamma$ the decoherence rate. We solve the master equation numerically using the Python toolbox QuTiP\\cite{qutip}.\n\n\n\n\n\\begin{table}\n    \\begin{tabular}{lll}\n        \\toprule\n        & Parameter set 1 & Parameter set 2 \\\\\n        \\midrule\n        $J_\\text{C} \/ 2\\pi \\, \\si{\\mega\\hertz}$ & $20$ & $20$ \\\\\n        $J \/ 2\\pi \\, \\si{\\mega\\hertz}$          & $65$ & $45$ \\\\\n        $\\Delta \/ 2\\pi \\, \\si{\\giga\\hertz}$     & $2$  & $0.5$ \\\\\n        $\\gamma \/ \\si{\\mega\\hertz}$             & $0.01$ & $0.01$ \\\\\n        \\midrule\n        Predicted $t_g \/ \\si{\\nano\\second}$     & $59.2$ & $30.9$ \\\\\n        Simulated $t_g \/ \\si{\\nano\\second}$     & $59.3$ & $31.5$ \\\\\n        \\midrule\n        $F_{00}(t_g)$          & $0.9943$ & $0.9662$ \\\\\n        $F_{11}(t_g)$          & $0.9931$ & $0.9668$ \\\\\n        $F_{\\Psi^+}(t_g)$      & $0.9881$ & $0.9348$ \\\\\n        $F_{\\Psi^-}(t_g)$      & $0.9968$ & $0.9983$ \\\\\n        $F(t_g)$               & $0.9923$ & $0.9637$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Two sets of model parameters and their corresponding gate times and gate fidelities. The gate fidelities are found at the simulated $t_g$.}\n    \\label{tab:parameters}\n\\end{table}\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{figure3.pdf}\n    \\caption{Fidelities versus time for the individually controlled gates ($F_{00}$, $F_{11}$, $F_{\\Psi^+}$, $F_{\\Psi^-}$) and the total diamond gate ($F$). Insets show zooms around the gate time.\n    The parameters used in \\textbf{(a)} are set 1 from Table~\\ref{tab:parameters}, and in \\textbf{(b)} they are set 2.}\n    \\label{fig:fid_vs_time}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n   \\includegraphics[width=2\\columnwidth]{figure4.pdf}\n   \\caption{Simulations varying the model parameters $J_\\text{C}$, $J$ and $\\Delta$, with qubit decoherence of rate $\\gamma = \\SI{0.01}{\\mega\\hertz}$. While one parameter is varied, the remaining two are fixed at the values marked by the gray vertical lines (parameter set 1 of Table~\\ref{tab:parameters}). \\textbf{(a)--(c)} Gate times, also showing the prediction of Eq.~\\eqref{eq:tg} as the dashed line. \\textbf{(d)--(f)} Gate fidelities, i.e. the fidelities at the simulated gate time.}\n   \\label{fig:simulation_params}\n\\end{figure*}\n\n\n\nAs a quality measure of the gate, we consider the average fidelity\\cite{Nielsen2002avfid} (or simply `fidelity' in the following),\n\\begin{equation}\n    \\label{eq:av_fidelity_def}\n    F(t) \\equiv\n    \\int d\\psi \\matrixel{\\psi}{U_\\text{target}^\\dagger \\mathcal{E}_t(\\dyad{\\psi}) U_\\text{target}}{\\psi} \\; ,\n\\end{equation}\nwhich quantifies how well the quantum map $\\mathcal{E}_t$ approximates the target unitary gate $U_\\text{target}$ over a uniform distribution of input quantum states.\nIf the diamond gate is run with an arbitrary initial state, the integral is taken over all possible four-qubit states, and can be reduced to a sum over a density matrix basis, as shown in Ref.~\\cite{Nielsen2002avfid}.\nPutting $U_\\text{target} = U$ from Eq.~\\eqref{eq:U} and $\\mathcal{E}_t(\\rho(0)) = \\rho(t)$ found from solving Eq.~\\eqref{eq:master-equation}, the computed fidelity quantifies the overall performance of the diamond gate with arbitrary initial states.\nWe denote this fidelity by $F$. Its maximum value (the gate fidelity) defines the gate time, which generally matches the predicted value of Eq.~\\eqref{eq:tg} within a few percent.\nThe sources of gate infidelity are qubit decoherence and state mixing accommodated by a non-zero $J_\\text{C}$.\n\n\n\n\n\n\n\n\n\nIn order to study the performance of the four individual gates of Eqs.~\\eqref{eq:UT00}--\\eqref{eq:UTpl}, we initialize the control qubits in $\\ket{\\phi}_\\text{C} \\in \\{ \\ket{00}_\\text{C}, \\ket{11}_\\text{C}, \\ket{\\Psi^+}_\\text{C}, \\ket{\\Psi^-}_\\text{C} \\}$.\nIn this case the target operation is a single term in Eq.~\\eqref{eq:U}, $U_\\text{target} = \\dyad{\\phi}_\\text{C} U_\\text{T}^\\phi$, and the integral is taken over all states on the form $\\ket{\\phi}_\\text{C}\\ket{\\psi}_\\text{T}$, i.e. only varying the target qubits' state, $\\ket{\\psi}_\\text{T}$.\nThese states span a subspace of the entire four-qubit Hilbert space characterized by the fixed control state, however couplings to other control states leads to leakage out of the subspace, which we take into account with the appropriate modification of the sum formula in Ref.~\\cite{Nielsen2002avfid}. The resulting fidelity is denoted $F_\\phi$, and the value at the gate time is denoted the gate fidelity for the associated gate.\n\n\n\n\nTwo example parameter sets relevant for superconducting qubits are shown in Table~\\ref{tab:parameters}. We use the state-of-the art decoherence rate $\\gamma = \\SI{0.01}{\\mega\\hertz}$, corresponding to a qubit life-time of $\\gamma^{-1} = \\SI{100}{us}$\\cite{wang2018}. Figure~\\ref{fig:fid_vs_time} shows the simulated fidelities as functions of time. As expected, there is a trade-off between a fast gate and high-fidelity operations.\nParameter set 1 operates in $\\SI{59.3}{\\nano\\second}$ with gate fidelities $\\sim 0.99$, which decreases to $\\sim 0.96$ for the very fast $\\SI{31.5}{\\nano\\second}$ gate of parameter set 2.\nThe gate infidelities for each controlled gate follow the expectations discussed in the previous section.\nIn particular, the idle gate fidelity, $F_{\\Psi^-}(t)$, is only limited by qubit decoherence, reducing its value from $1$ to $0.9983$ and $0.9968$, respectively, during the operation time in the two cases.\nFor the remaining three controlled gates, a longer gate time can improve the gate fidelity, with the drawback of increased susceptibility to qubit decoherence.\nUltimately this limits the number of computations the diamond gate device can run successfully.\nFor the purpose of demonstrating the model, we will use parameter set 1 in the following, unless otherwise stated.\n\n\n\n\n\n\nTo probe the sensitivity to the model parameters, we vary each of $\\Delta$, $J$ and $J_\\text{C}$.\nAs is evident from Figure~\\ref{fig:simulation_params}a--c, the simulated gate times follow closely the prediction of Eq.~\\eqref{eq:tg}. Specifically, the gate time is tunable through $\\Delta$ and $J$.\nThe gate fidelities for the individually controlled gates and the total diamond gate are shown in Figure~\\ref{fig:simulation_params}a--f. Except for the phase gate controlled by $\\ket{\\Psi^+}_\\text{C}$, which is affected most strongly by couplings to other control states, the fidelities are above 0.99 over a wide range of parameters.\nDue to the mathematical equivalence between the two swapping gates controlled by $\\ket{00}_\\text{C}$ and $\\ket{11}_\\text{C}$, the gate fidelities for these operations are very similar. We attribute the difference to qubit relaxation, which only affects $\\ket{11}_\\text{C}$ and becomes more pronounced as the gate time increases.\nThe identity gate controlled by $\\ket{\\Psi^-}_\\text{C}$ is only limited by decoherence, and its gate fidelity decreases linearly with the gate time.\n\n\n\\begin{figure}\n   \\includegraphics[width=\\columnwidth]{figure5.pdf}\n   \\caption{Investigating gate stability for the following system infidelities: \\textbf{(a)} Crosstalk coupling between the target qubits. \\textbf{(b)} Random asymmetric noise in the couplings between the target and control qubits. \\textbf{(c)} Control state infidelity. \\textbf{(d)} Qubit decoherence with rate $\\gamma$.}\n   \\label{fig:simulation_noise}\n\\end{figure}\n\n\nWith a superconducting circuit implementation in mind, we consider a variety of system infidelites and their impact on the gate fidelities, see Figure~\\ref{fig:simulation_noise}. Most harmful is a direct capacitive coupling between the target qubits (Figure~\\ref{fig:simulation_noise}a), which allows the target qubits to bypass the control qubits, thereby circumventing the interference condition set by the control qubits.\nThe gate fidelities roughly decrease with the square of the cross-coupling strength $J_\\text{T}$, leading to noticable gate infidelities even for a relatively weak coupling. However, as we will show in the next section, crosstalk should not be suppressed, but rather utilized to combat another effect appearing in superconducting qubits: couplings to higher-energy states in the qubits' spectrum.\n\nFigure~\\ref{fig:simulation_noise}b shows simulation results with random noise on the couplings between the target and control qubits emulating asymmetries present in an actual circuit due to fabrication limits.\nEach data point in the plot corresponds to a simulation with random deviations from the noiseless value, $J$, denoting by $\\delta J$ the maximum deviation over the four couplings.\nThe gate performance is very robust towards this type of noise.\n\nBell state generation, which is required for the control states $\\ket{\\Psi^\\pm}_\\text{C}$, has been shown with a state infidelity of $\\sim 0.005$\\cite{barends2014cz}.\nWe introduce control state infidelity in the following way. For each data point in Figure~\\ref{fig:simulation_noise}c we contruct a random four-by-four Hermitian matrix $M$, from which we construct a unitary matrix $V = e^{i\\epsilon M}$, where $\\epsilon$ is a small real parameter.\nIn the simulations, we apply $V$ to the initial state of the control qubits in order to model imperfect state preparation.\nThe resulting gate fidelity is shown as a function of the maximum infidelity among the four control states.\nThe diamond gate suffers a linear decrease in gate fidelity, but remains high-performing for realistic control state infidelity.\n\nQubit decoherence in the form of relaxation and dephasing is included in the master equation~\\eqref{eq:master-equation} with rate $\\gamma$.\nIn Figure~\\ref{fig:simulation_noise}d we see that the gate fidelity decreases linearly with $\\gamma$.\nEven for qubits with $\\gamma = \\SI{0.05}{\\mega\\hertz}$, corresponding to a lifetime of $\\gamma^{-1} = \\SI{20}{\\micro\\second}$, the gate fidelity is $\\sim 0.98$.\nWe attribute this robustness to the relatively short gate time of $\\SI{59.3}{\\nano\\second}$.\n\n\n\n\n\n\n\n\n\n\\subsection{Higher-energy states}\n\\label{sec:qutrits}\n\nIn the previous section, we treated a model for four coupled qubits. In the superconducting circuit implementation of Figure~\\ref{fig:superconducting_circuit}, these qubits are comprised of the two lowest energy states of the each transmon, $\\ket{0}$ and $\\ket{1}$.\nHowever, in an actual superconducting circuit, the qubits may couple to higher-energy states in the transmon spectrum, which is the spectrum of a slightly anharmonic oscillator\\cite{transmon_original}.\nIn this section, we analyse the effects from including the second excited state, $\\ket{2}$, in the spectrum, thereby turning each qubit into a qutrit.\n\nThe full analysis of the circuit of Figure~\\ref{fig:superconducting_circuit}b is given in Appendix~\\ref{sec:qutrit_analysis}. The resulting four-qutrit Hamiltonian is a sum of the non-interacting part\n\\begin{equation}\n    \\label{eq:qutrit_H0}\n    \\tilde H_0 = -\\frac{1}{2} \\Omega_\\text{T} (\\tilde\\sigma_z^\\text{T1} + \\tilde\\sigma_z^\\text{T2})\n                 -\\frac{1}{2} \\Omega_\\text{C} (\\tilde\\sigma_z^\\text{C1} + \\tilde\\sigma_z^\\text{C2}) \\; ,\n\\end{equation}\nand the interaction terms\n\\begin{equation}\n    \\label{eq:qutrit_Hint}\n    \\tilde H_\\text{int} = J_\\text{T} \\tilde\\sigma_y^\\text{T1} \\tilde\\sigma_y^\\text{T2}\n                        + J_\\text{C} \\tilde\\sigma_y^\\text{C1} \\tilde\\sigma_y^\\text{C2}\n                        + J (\\tilde\\sigma_y^\\text{T1} + \\tilde\\sigma_y^\\text{T2}) (\\tilde\\sigma_y^\\text{C1} + \\tilde\\sigma_y^\\text{C2}) \\; ,\n\\end{equation}\nwhich are analogous to Eqs.~\\eqref{eq:H0}--\\eqref{eq:Hint}. The `Pauli $z$-operator' on qutrit $j$, denoted $\\tilde \\sigma_j$, includes $\\ket{2}_j$ in such a way that it has an energy $\\Omega_j + \\alpha_j$ above $\\ket{1}_j$, with $\\Omega_j$ and $\\alpha_j$ the frequency and anharmonicity, respectively. Typically $\\alpha_j \/ \\Omega_j \\sim -0.05$, yielding a small detuning of the second excited state compared to an equidistant spectrum (i.e. to vanishing anharmonicity). The operator is given as\n\\begin{equation}\n    \\label{eq:qutrit_sigmaz}\n    \\tilde\\sigma_z^j = \\dyad{0}{0}_j - \\dyad{1}{1}_j - \\left( 3 + \\frac{2\\alpha_j}{\\Omega_j} \\right) \\dyad{2}{2}_j \\; ,\n\\end{equation}\nThe `Pauli $y$-operator' on qutrit $j$ is\n\\begin{equation}\n    \\label{eq:qutrit_sigmay}\n    \\tilde\\sigma_y^j = i T_0^j \\dyad{1}{0}_j + i T_2^j \\dyad{2}{1}_j + \\text{H.c.} \\; ,\n\\end{equation}\nwhere $T_0^j \\approx 1$ and $T_2^j \\approx \\sqrt 2$ can be expressed in terms of $\\Omega_j$ and $\\alpha_j$ (see Appendix~\\ref{sec:qutrit_analysis}).\nHence, the coupling between the first and second excited state is as strong as the coupling between the two lowest (qubit) levels.\nDue to the small anharmonicity in transmons, i.e. that the energy separation between the qubit levels almost equals the separation between the first and second excited state, couplings that exchange a single excitation like $\\ket{11} \\rightarrow \\ket{02}$ are not strongly energetically suppressed.\nIn fact, this transition is sometimes used for the \\textsf{CZ} gate\\cite{krantz2019}.\nNotice that this lack of suppression holds for transmons in general, and is not a consequence of the specific model considered here.\n\nThis has two undesired consequences. Firstly, unless $\\abs{J_\\text{C}\/\\alpha_\\text{C}} \\ll 1$, it allows the control state $\\ket{11}_\\text{C}$ to mix with $\\ket{02}_\\text{C}$ and $\\ket{20}_\\text{C}$, leading to a non-conserved control state during the gate operation. This can be resolved by redefining the control state as\n\\begin{equation}\n    \\label{eq:tilde11}\n    \\ket{\\tilde{11}}_\\text{C} = \\cos\\tilde\\theta \\ket{11}_\\text{C} + \\sin\\tilde\\theta \\frac{1}{\\sqrt 2} (\\ket{02}_\\text{C} + \\ket{20}_\\text{C}) \\; ,\n\\end{equation}\nwith the mixing angle $\\tilde\\theta = -\\frac{1}{2} \\arctan (2\\sqrt 2 J_\\text{C} T_1^\\text{C} T_2^\\text{C} \/ \\alpha_\\text{C}) \\sim 0.5$, such that it is an eigenstate of an effective control state Hamiltonian.\nThis introduces a significant component of $(\\ket{02}_\\text{C} + \\ket{20}_\\text{C})\/\\sqrt{2}$, which is avoided if $J_\\text{C} = 0$.\nDetails are found in Appendix~\\ref{sec:qutrit_analysis}.\n\n\\begin{figure}\n   \\includegraphics[width=\\columnwidth]{figure6.pdf}\n   \\caption{Swap rate, found as the inverse of the smallest time $t$ where the swap fidelity (probability) $\\vert \\bra{\\phi}_\\text{C}\\bra{01}_\\text{T} e^{-i (\\tilde H_0 + \\tilde H_\\text{int})t} \\ket{10}_\\text{T} \\ket{\\phi}_\\text{C} \\vert^2$ becomes close to unity, versus crosstalk strength $J_\\text{T}$.\n   Data points are shown with the control state $\\ket{\\phi}_\\text{C}$ set to each of the displayed states.\n   The parameters used in the simulation are\n   $J_\\text{C}\/2\\pi = \\SI{20}{\\mega\\hertz}$, $J\/2\\pi = \\SI{65}{\\mega\\hertz}$, $\\Omega_\\text{C}\/2\\pi = \\SI{7}{\\giga\\hertz}$,\n   $\\Omega_\\text{T}\/2\\pi = \\SI{9}{\\giga\\hertz}$,\n   $\\alpha_\\text{C} = \\SI{-270}{\\mega\\hertz}$ and $\\alpha_\\text{T} = \\SI{-280}{\\mega\\hertz}$.\n   The optimal value of Eq.~\\eqref{eq:JTopt} is marked with a vertical line, $J_\\text{T}^\\text{opt}\/2\\pi = \\SI{-3.66}{\\mega\\hertz}$.}\n   \\label{fig:qutrit_swap_time_vs_JT}\n\\end{figure}\n\n\\begin{figure*}\n   \\includegraphics[width=2\\columnwidth]{figure7.pdf}\n   \\caption{Fidelity for swapping $\\ket{\\psi}_\\text{T} \\leftrightarrow \\ket{\\psi'}_\\text{T}$ for the indicated processes, computed as\n   $\\vert \\bra{\\phi}_\\text{C}\\bra{\\psi'}_\\text{T} e^{-i (\\tilde H_0 + \\tilde H_\\text{int})t} \\ket{\\psi}_\\text{T} \\ket{\\phi}_\\text{C} \\vert^2$, with the control state $\\ket{\\phi}_\\text{C}$ indicated above each column.\n   The parameters used in the simulation are\n   $J_\\text{C}\/2\\pi = \\SI{20}{\\mega\\hertz}$, $J\/2\\pi = \\SI{65}{\\mega\\hertz}$, $\\Omega_\\text{C}\/2\\pi = \\SI{7}{\\giga\\hertz}$,\n   $\\Omega_\\text{T}\/2\\pi = \\SI{9}{\\giga\\hertz}$,\n   $\\alpha_\\text{C} = \\SI{-270}{\\mega\\hertz}$ and $\\alpha_\\text{T} = \\SI{-280}{\\mega\\hertz}$.\n   \\textbf{(a)--(d)} No crosstalk, $J_\\text{T} = 0$. \\textbf{(e)--(h)} Crosstalk is set to its optimal value of Eq.~\\eqref{eq:JTopt}, $J_\\text{T}^\\text{opt}\/2\\pi = \\SI{-3.66}{\\mega\\hertz}$.}\n   \\label{fig:qutrit_swap_fids_vs_time}\n\\end{figure*}\n\nSecondly, excitations to the second excited states allow unwanted processes which bypass the control.\nFor instance, when the diamond gate is desired to be idle, leakage across the control can occur via:\n\\begin{equation}\n    \\label{eq:qutrit_leakage}\n    \\ket{\\Psi^-}_\\text{C} \\ket{10}_\\text{T}\n    \\rightarrow\n    \\frac{1}{\\sqrt 2} ( \\ket{02}_\\text{C} - \\ket{20}_\\text{C} ) \\ket{00}_\\text{T}\n    \\rightarrow\n    \\ket{\\Psi^-}_\\text{C} \\ket{01}_\\text{T} \\; .\n\\end{equation}\nSince this is a second order process in the qutrit model Hamiltonian, it would not pose a threat to the functionality of the diamond gate if it only relied on (generally faster) first order processes. However, the swap operations of Eqs.~\\eqref{eq:UT00}--\\eqref{eq:UT11} are also second order processes, leading to a failure of the idle diamond gate on the same time-scale as the operation of the swap gates. Similarly, the control state $\\ket{\\Psi^+}$ fails to prevent excitation leakage across the control, corrupting the operation of  Eq.~\\eqref{eq:UTpl}.\n\n\n\nHowever, these undesired processes can be mitigated by taking advantage of the effects of crosstalk. The circuit analysis in Appendix~\\ref{sec:qutrit_analysis} reveals a weak unavoidable crosstalk coupling of strength $J_\\text{T}$ in the interaction Hamiltonian~\\eqref{eq:qutrit_Hint}, which by itself has a significant negative impact on the gate fidelities, c.f. Figure~\\ref{fig:simulation_noise}a.\nThis leads directly to leakage across the control through processes of the type\n\\begin{equation}\n    \\label{eq:crosstalk_leakage}\n    \\ket{\\Psi^-}_\\text{C} \\ket{10}_\\text{T}\n    \\rightarrow\n    \\ket{\\Psi^-}_\\text{C} \\ket{01}_\\text{T} \\; .\n\\end{equation}\nThis process has the same unwanted outcome as the one of Eq.~\\ref{eq:qutrit_leakage}. As we show below, we can therefore restore the gate functionality by tuning the value of $J_\\text{T}$ such that these two unwanted leakage processes cancel each other.\nAnalyzing the problem with second order perturbation theory in order to calculate the amplitude of the leaked state (see Appendix~\\ref{sec:qutrit_analysis}), we find destructive interference between these processes when the crosstalk strength takes the optimal value\n\\begin{equation}\n    \\label{eq:JTopt}\n    \\begin{aligned}\n        J_\\text{T}^\\text{opt} ={}\n        &\\frac{(J T_2^\\text{C})^2}{\\Omega_\\text{C} + \\Omega_\\text{T} + \\alpha_\\text{C} + J_\\text{C}(T_1^\\text{C})^2} \\\\\n        &+ \\frac{(J T_2^\\text{C})^2}{\\Omega_\\text{C} - \\Omega_\\text{T} + \\alpha_\\text{C} + J_\\text{C}(T_1^\\text{C})^2} \\; .\n    \\end{aligned}\n\\end{equation}\nThus by tuning the crosstalk strength to $J_\\text{T} = J_\\text{T}^\\text{opt}$, we expect the fidelity for the target qubit swap $\\ket{01}_\\text{T} \\leftrightarrow \\ket{10}_\\text{T}$ to diminish, or equivalently a vanishing swap rate, when the control state is $\\ket{\\Psi^\\pm}_\\text{C}$.\nFigure~\\ref{fig:qutrit_swap_time_vs_JT} shows the swap rate for varying $J_\\text{T}$, with control qubits in each of the four control states.\nWe find two distinct zero-points, one for the data related to the control states $\\ket{00}_\\text{C}$ and $\\ket{\\Psi^\\pm}_\\text{C}$ at the expected value $J_\\text{T}^\\text{opt}$ (vertical line), and one for $\\ket{\\tilde{11}}_\\text{C}$.\nThus, it is possible to prevent the unwanted swap operation for the control states $\\ket{\\Psi^\\pm}_\\text{C}$, but as a consequence also the swap operation controlled by $\\ket{00}_\\text{C}$ is obstructed.\nOn the other hand, the swap operation controlled by $\\ket{\\tilde{11}}_\\text{C}$ is preserved at $J_\\text{T} = J_\\text{T}^\\text{opt}$, although the gate time is prolonged to around $\\SI{220}{\\nano\\second}$.\nRemarkably, for $J_\\text{T}\/2\\pi \\approx \\SI{-2.5}{\\mega\\hertz}$ the situation is reversed. Here, putting the control in $\\ket{\\tilde{11}}_\\text{C}$ prevents swapping, while the three remaining control states permit it.\nAt each zero-point, the gate time (inverse swap rate) for the swapping gate(s) is prolonged compared to the results in the previous section.\nTo reduce the gate time, one should pick parameters such that the zero-points are further apart, or such that the inclination of the graphs are steeper.\n\nFigure~\\ref{fig:qutrit_swap_fids_vs_time} illustrates in more detail the cancellation of unwanted transfer by crosstalk engineering.\nEach subfigure shows the swap fidelity for different initial target qubit states. The control is initialized in the state indicated above each column.\nFigure~\\ref{fig:qutrit_swap_fids_vs_time}a--d (the top row) show simulations for $J_\\text{T} = 0$, while the crosstalk has been put to its optimal value, $J_\\text{T} = J_\\text{T}^\\text{opt}$, in Figure~\\ref{fig:qutrit_swap_fids_vs_time}e--h (the bottom row).\nAs expected from Figure~\\ref{fig:qutrit_swap_time_vs_JT}, the swap $\\ket{01}_\\text{T} \\leftrightarrow \\ket{10}_\\text{T}$ (dark lines) occurs for any control state when there is no crosstalk, but is controlled uniquely by $\\ket{\\tilde{11}}_\\text{C}$ when the crosstalk is at the optimal value.\nIn the cases of $\\ket{00}_\\text{T}$ and $\\ket{11}_\\text{T}$, we wish to maintain a unit fidelity across all control states, i.e. the states should acquire at most a phase.\nTuning the crosstalk to $J_\\text{T}^\\text{opt}$ also improves the gate operation in this regard.\n\nEngineering crosstalk to mitigate unwanted leakage through higher-excited states is killing two birds with one stone: Each process is harmful to the functionality of the diamond gate, but letting them cancel each other preserves the ability to control the swap operation.\nThe price is the loss of swap functionality in the gate controlled by $\\ket{00}_\\text{C}$, and an increased gate time for the model parameters considered here.\nGenerally, the phases applied to each target state will be modified for all four controlled gates, but we do not pursue an analysis here, as other factors specific to the implementation will contribute to this as well.\nRather, our main goal was to demonstrate a passive method for dealing with undesired leakage processes.\n\n\n\n\n\\section{Discussion}\n\nWe have proposed a quantum interference device by coupling four qubits with exchange interactions.\nBy analyzing the unitary dynamics of the system, we have shown that it realizes the diamond gate: a four-way controlled two-qubit gate, with the ability to run two different entangling swap and phase operations, a (parity) phase operation, an idling gate with no dynamics, or an arbitrary superposition of these.\nWe considered an implementation in superconducting qubits using transmon qubits, and found that it generally operated fast and with high fidelity using state-of-the-art model and noise parameters. When taking second excited states into account, we had to prevent leakage across the control by engineering crosstalk, demonstrating a general method to avoid leakage in superconducting qubit systems.\nThe cost of this was a single redefined control state, one swap gate turning into a phase gate, altered phases on the gates, and a slower gate for the considered parameters.\nHowever, we only consider this analysis a starting point for an actual implementation, which might also include active microwave driving to optimize the operations or to prevent certain transitions.\nIt might also be worthwile to consider other types of superconducting qubits with larger anharmonicity, or entirely different platforms such as lattices of ultracold atoms or ions, where qubit encoded in hyperfine states or vibrational modes are far detuned from the rest of the spectrum.\n\nWe illustrated how the four-qubit diamond gate device can constitute an essential building block in an extensible quantum computer, and proposed a simple scheme where quantum algorithms are run on the computer by parallel processing on each four-qubit module interspersed with two-qubit operations spreading entanglement in the system, and single-qubit operations.\nEvidently, this scheme is adaptable to many different algorithms, and future work will investigate which algorithms are suitable to be implemented in the diamond-plaquette device.\n\n\n\n\n\n\n\\begin{acknowledgments}\n  This research was funded in part by the U.S. Army Research Office Grant No. W911NF-17-S-0008. N.J.S.L., L.B.K., and N.T.Z. acknowledge support from the Carlsberg Foundation and The Danish National Research Council under the Sapere Aude program. M.K. acknowledges support from the Carlsberg Foundation. T.W.L. acknowledges support from Microsoft. N.J.S.L. ackowledge discussions with Al\u00e1n Aspuru-Guzik, Daniel Kyungdeock Park, Kasper Sangild, and Stig Elkj\u00e6r Rasmussen. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements of the US Government.\n\\end{acknowledgments}\n\n\\onecolumngrid\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Flux Qubits}\n\nWe describe our design of the SC flux qubit that prepares an SC island with SC\nphase $\\varepsilon=\\varepsilon^{0}$ or $\\varepsilon^{1}$ depending on the flux\nqubit state $\\left\\vert 0\\right\\rangle _{\\mathrm{flux}}$ or $\\left\\vert\n1\\right\\rangle _{\\mathrm{flux}}$. The phase of the SC island can coherently\ncontrol the coupling between the two Majorana fermions at the end of the STIS\nwire. We would like to have a small phase separation $\\Delta\\varepsilon\n\\equiv\\varepsilon^{1}-\\varepsilon^{0}\\ll\\pi\/2$, so that we can easily switch\noff the coupling $H_{12}^{\\mathrm{MF}}$ when we do not want to couple the\nMajorana fermions (MFs).\n\nThe previous design of flux qubit with three JJs \\cite{Mooij99,Orlando99},\nhowever, is not amenable to achieving a small phase separation $\\Delta\n\\varepsilon\\ll\\pi\/2$, because of the following reason. The three JJs have\nJosephson energy $E_{J,1}=E_{J,2}=E_{J}$ and $E_{J,3}=\\alpha E_{J}$\n\\cite{Mooij99,Orlando99}. The phase difference across the first junction is\n$\\theta=\\pm\\cos^{-1}\\frac{1}{2\\alpha}$. By choosing $\\alpha=\\eta+1\/2$ and\n$\\eta\\ll1$, there is a small phase separation $\\Delta\\varepsilon=2\\left\\vert\n\\theta\\right\\vert \\approx4\\eta^{1\/2}$. However, the energy barrier for the\ntunneling is significantly suppressed $\\Delta U=2\\alpha-\\left(  2-\\frac\n{1}{2\\alpha}\\right)  E_{J}\\approx4\\eta^{2}E_{J}\\sim\\theta^{4}E_{J}$. The\naction associated with the tunneling is $S\\approx\\theta\\sqrt{\\Delta U\/E_{C}%\n}\\approx\\theta^{3}\\sqrt{E_{J}\/E_{C}}\\propto\\theta^{3}E_{J}$, where the last\nstep uses the property that $E_{C}\\propto1\/E_{J}$. In order to maintain a\nsimilar tunneling matrix element between the two potential minima, it requires\nthe unfavorable scaling $E_{J}\\propto1\/\\theta^{3}$. For example, to achieve\n$\\theta=0.1$, we have to increase the area of the Josephson junction by\n$\\sim10^{3}$. Practically, it would also be very challenging to have a precise\nvalue of $\\alpha=\\frac{1}{2}+\\frac{\\theta^{2}}{4}=0.5025$, which fine tunes\nthe phase separation $\\Delta\\varepsilon$. In contrast, the four-junction\ndesign of flux qubit has a favorable scaling of $E_{J,4}=\\beta E_{J}%\n\\propto1\/\\theta$ and there is no fine-tuned parameter. This motivates us to\nredesign the flux qubit with more Josephson junctions.\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=6cm]{fig_FluxQubit.pdf}\n\\end{center}\n\\caption[fig:FluxQubit]{The design of flux qubit consists of a loop of four\nJJs in series that encloses an external magnetic flux $f\\Phi_{0}$. Schematic\nillustration in terms of (a) JJs and (b) SC islands. }%\n\\label{fig:FluxQubit}%\n\\end{figure}\n\nAs shown in Fig.~\\ref{fig:FluxQubit}, the flux qubit consists of a loop of\nfour JJs in series that encloses an external magnetic flux $f\\Phi_{0}$\n($f\\approx1\/2$ and $\\Phi_{0}=h\/2e$ is the SC flux quantum). The first two JJs\nhave equal Josephson coupling energy $E_{J,1}=E_{J,2}=E_{J}$; the third JJ has\ncoupling energy $E_{J,3}=\\alpha E_{J}$, with $0.5<\\alpha<1$; the coupling in\nthe fourth JJ is $E_{J,4}=\\beta E_{J}$, with $\\beta\\gg1$. For JJs with the\nsame thickness but different junction area $\\left\\{  A_{j}\\right\\}  $,\n$E_{J,j}\\propto A_{j}$ and $C_{j}\\propto A_{j}$. The charging energies can be\ndefined as $E_{C,1}=E_{C,2}=E_{C}=\\frac{e^{2}}{2C_{1}}$, $E_{C,3}=\\alpha\n^{-1}E_{C}$ and $E_{C,4}=\\beta^{-1}E_{C}$. Notice that $E_{J,j}E_{C,j}%\n=E_{J}E_{C}$ is independent of $j$. The system may have two stable states with\npersistent circulating current of opposite sign.\n\nHere is a summary of the key results:\n\n1.\\qquad For $\\beta\\rightarrow\\infty$, we may neglect the fourth JJ and reduce\nthe system to the well-studied flux qubit with three JJs\n\\cite{Mooij99,Orlando99}. For $\\beta\\gg1$, there is only a small phase\ndifference across the fourth junction, with $\\theta_{4}=\\pm\\theta_{4}^{\\ast\n}=\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}$ depending on the sign of the\ncirculating current (i.e., the state of flux qubit, with $\\mathbf{Z}%\n_{\\mathrm{flux}}=\\left(  \\left\\vert 0\\right\\rangle \\left\\langle 0\\right\\vert\n-\\left\\vert 1\\right\\rangle \\left\\langle 1\\right\\vert \\right)  _{\\mathrm{flux}%\n}$). We show that the magnitude of the phase difference can be small\n$\\theta_{4}^{\\ast}\\approx\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha\\beta}\\propto\n\\beta^{-1}$. As shown in Fig.~\\ref{fig:FluxQubit}, two SC islands ($c$ and\n$d$) are connected by this junction, if we fix the SC phase $\\phi_{c}$, then\n$\\phi_{d}=\\varepsilon^{0,1}=\\phi_{c}\\pm\\theta_{4}^{\\ast}$ has phase separation\n$\\Delta\\varepsilon=2\\theta_{4}^{\\ast}\\approx\\frac{\\sqrt{4\\alpha^{2}-1}}%\n{\\alpha\\beta}$. The SC island $d$ can be used to coherently control the\ncoupling between the Majorana fermions of the STIS wire.\n\n2.\\qquad There are quantum fluctuations for the phase of the SC island. The\nmagnitude of quantum fluctuations depends on $\\left\\{  E_{C,j}\\right\\}  $ and\n$\\left\\{  E_{J,j}\\right\\}  $. For $\\beta\\gg1$, the dynamics associated with\n$\\theta_{4}$ can be characterized by a harmonic oscillator (HO) Hamiltonian%\n\\[\nH^{\\mathrm{HO}}=\\frac{p_{\\theta_{4}}^{2}}{2M_{4}}+\\frac{E_{J,4}}{2}\\left(\n\\theta_{4}-\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}\\right)  ^{2},\n\\]\nwhere the effective mass is $M_{4}=\\frac{1}{8E_{c_{4}}}$ and the canonical\nmomentum $p_{\\theta_{4}}$ satisfies $\\left[  \\theta_{4},p_{\\theta_{4}}\\right]\n=i$ (with $\\hbar\\equiv1$). We may rewrite $H^{\\mathrm{HO}}=\\left(  a^{\\dag\n}a+1\/2\\right)  \\omega$ and $\\theta_{4}=\\mathbf{Z}_{\\mathrm{flux}}\\theta\n_{4}^{\\ast}+\\zeta\\left(  a^{\\dag}+a\\right)  \/\\sqrt{2}$, where the oscillator\nfrequency is $\\omega=\\sqrt{8E_{J}E_{C}}$ and the magnitude of quantum\nfluctuations is $\\zeta=\\left(  \\frac{8E_{C}}{E_{J}}\\right)  ^{1\/4}\\beta\n^{-1\/2}$. We justify that this simple model agrees very well with the general\nmodel characterizing the quantum fluctuations for the flux qubit with coupled JJs.\n\n3.\\qquad Various parameters characterizing the flux qubit are also calculated,\nincluding the plasma frequencies $\\left\\{  \\omega_{i}\\right\\}  _{i=1,2,3}$,\nbarrier height $\\Delta U$, and the tunneling matrix element $t$. For example,\ngiven parameters $\\alpha=0.8$ and $\\beta=10$, we compute $\\left\\{  \\omega\n_{i}\\right\\}  \\approx\\left(  2.8,2.3,1.8\\right)  \\sqrt{E_{J}E_{C}}\\sim\n\\sqrt{E_{J}E_{C}}$, $\\Delta U\\approx0.26E_{J}$, and $t\\approx1.8\\sqrt\n{E_{J}E_{C}}\\exp\\left[  -0.7\\left(  E_{J}\/E_{C}\\right)  ^{1\/2}\\right]  $. We\nnotice that these parameters for the flux qubit hardly depend on $\\beta$ when\n$\\beta>10$, which verifies the intuition that inserting a large Josephson\njunction to the loop has almost no effect to the properties of the flux qubit.\n\n4.\\qquad We propose two schemes to implement the SC phase-controller, which\ncan fix the phase difference between an SC island and a big SC reservoir.\n\nIn the following, we provide detailed analysis to justify our design of flux\nqubit with four JJs. First, we give the Hamiltonian description for the\nsystem. Then, we calculate the phase separation and quantum fluctuations.\nNext, we numerically obtain various quantities such as plasma frequencies,\nbarrier height, and tunneling matrix element. Our numerical calculation also\nverifies our estimates on phase separation and magnitude of quantum\nfluctuations. After that, we propose two implementations of the SC\nphase-controller. Finally, we derive the energy splitting function $E\\left(\n\\varepsilon\\right)  $ that is highly non-linear in terms of $\\varepsilon$.\n\n\\section{Hamiltonian for Flux Qubit}\n\nThe Hamiltonian for a flux qubit consisting of four JJs in series is%\n\\begin{equation}\nH^{\\mathrm{flux}}=T+U,\n\\end{equation}\nwith the Josephson potential%\n\\begin{equation}\nU=\\sum_{j=1,2,3,4}E_{J,j}\\left(  1-\\cos\\theta_{j}\\right)  ,\n\\end{equation}\nand the capacitive charging energy%\n\\begin{equation}\nT=\\frac{1}{2}\\sum_{j=1,2,3,4}C_{j}V_{j}^{2}.\n\\end{equation}\nHere for the $j$-th Josephson junction, $E_{J,j}$ is the Josephson coupling,\n$\\theta_{j}$ is the gauge-invariant phase difference, $C_{j}$ is the\ncapacitance, and $V_{j}$ is the voltage across the junction. Suppose that all\nthe junctions have the same thickness, we have $E_{J,j}$ $\\propto A_{j}$ and\n$E_{C,j}\\equiv\\frac{e^{2}}{2C_{j}}\\propto A_{j}^{-1}$ where $A_{j}$ is the\narea of the junction and $C_{j}$ is the junction capacitance \\cite{Tinkham96}.\nThe quantity $E_{J,j}E_{C,j}=E_{J}E_{C}$ does not depend on $j$.\n\nThe Josephson potential is constraint by the phase relation\n\\begin{equation}\n\\sum_{j}\\theta_{j}+2f\\pi\\equiv0\\left(  \\operatorname{mod}2\\pi\\right)  ,\n\\label{eq:PhaseConstraint}%\n\\end{equation}\nThis phase relation removes one degree of freedom. We may introduce a 3-vector\n$\\vec{\\theta}=\\left(  \\theta_{1},\\theta_{2},\\theta_{3}\\right)  $ to\ncharacterize the system with four JJs. (Note that we only need to choose three\nindependent phases for 3-vector, e.g., $\\vec{\\theta}=\\left(  \\theta_{1}%\n,\\theta_{2},\\theta_{4}\\right)  $ is also a valid choice.) For $f=1\/2$, the\nJosephson potential is%\n\\begin{align}\nU  &  =-E_{J,1}\\cos\\theta_{1}-E_{J,2}\\cos\\theta_{2}-E_{J,3}\\cos\\theta_{3}\\\\\n&  +E_{J,4}\\cos\\left(  \\theta_{1}+\\theta_{2}+\\theta_{3}\\right)  .\\nonumber\n\\end{align}\nBy appropriately choosing the parameter $\\left\\{  E_{J,j}\\right\\}  $, there\nare only two minima for the Josephson energy at $\\pm\\left\\{  \\theta_{j}^{\\ast\n}\\right\\}  =\\pm\\left\\{  \\theta_{1}^{\\ast},\\theta_{2}^{\\ast},\\theta_{3}^{\\ast\n},\\theta_{4}^{\\ast}\\right\\}  $ (see Fig.~\\ref{fig:Potentialn6}ab). We may\nidentify the two levels of the flux qubit as $\\left\\vert 0\\right\\rangle\n_{\\mathrm{flux}}=\\left\\vert \\left\\{  \\theta_{i}^{\\ast}\\right\\}  \\right\\rangle\n$ and $\\left\\vert 1\\right\\rangle _{\\mathrm{flux}}=\\left\\vert \\left\\{\n-\\theta_{i}^{\\ast}\\right\\}  \\right\\rangle $. Thus, we may denoted the two\npotential minima as $\\left\\{  \\mathbf{Z}_{\\mathrm{flux}}\\theta_{j}^{\\ast\n}\\right\\}  $, with $\\mathbf{Z}_{\\mathrm{flux}}=\\left(  \\left\\vert\n0\\right\\rangle \\left\\langle 0\\right\\vert -\\left\\vert 1\\right\\rangle\n\\left\\langle 1\\right\\vert \\right)  _{\\mathrm{flux}}$.\n\n\\begin{figure*}[ptbh]\n\\begin{center}\n\\includegraphics[width=16cm]{fig_Potential_n6.pdf}\n\\end{center}\n\\caption[fig:Potentialn6]{Contour plots of Josephson energy as a function of\n(a) $\\left\\{  \\theta_{1},\\theta_{2},\\theta_{3}\\right\\}  $ (with $\\theta\n_{4}=\\pi-\\theta_{1}-\\theta_{2}-\\theta_{3}$). (b) $\\left\\{  \\theta_{1}%\n,\\theta_{2},\\theta_{4}\\right\\}  $ (with $\\theta_{3}=\\pi-\\theta_{1}-\\theta\n_{2}-\\theta_{4}$). (c) $\\left\\{  \\theta_{1},\\theta_{4}\\right\\}  $ (with\n$\\theta_{3}=\\pi-\\theta_{1}-\\theta_{2}$ and $\\theta_{4}=0$) (d,e) $\\left\\{\n\\theta_{1},\\theta_{4}\\right\\}  $ (with $\\theta_{1}=\\theta_{2}$ and $\\theta\n_{3}=\\pi-2\\theta_{1}-\\theta_{4}$). (f) Marginal probability distributions of\n$\\theta_{4}$ associated with states $\\left\\vert 0\\right\\rangle _{\\mathrm{flux}%\n}$ (blue solid line) and $\\left\\vert 1\\right\\rangle _{\\mathrm{flux}}$ (red\ndashed line). The parameters are $E_{J}\/E_{C}=80$ and $\\left\\{  E_{J,i}%\n\/E_{J}\\right\\}  _{i=1,2,3,4}=\\left\\{  1,1,\\alpha=0.8,\\beta=10\\right\\}  $.}%\n\\label{fig:Potentialn6}%\n\\end{figure*}\n\nThe capacitive charging energy can be regarded as the kinetic energy\nassociated with the dynamics of $\\vec{\\theta}$. This is because the voltage\nacross the junction is given by the Josephson voltage-phase relation\n$V_{j}=\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  \\dot{\\theta}_{j}$\n\\cite{Tinkham96} and the time derivatives $\\left\\{  \\dot{\\theta}_{j}\\right\\}\n$ obey the constraint $\\dot{\\theta}_{1}+\\dot{\\theta}_{2}+\\dot{\\theta}_{3}%\n+\\dot{\\theta}_{4}=0$ (derived from Eq.(\\ref{eq:PhaseConstraint})). Thus, we\nmay write the capacitive charging energy as%\n\\begin{align}\nT  &  =\\frac{1}{2}\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}\\sum\n_{i,j=1,2,3}C_{ij}\\dot{\\theta}_{i}\\dot{\\theta}_{j}\\\\\n&  =\\frac{1}{2}\\overrightarrow{\\dot{\\theta}}^{T}\\cdot\\mathbf{M}\\cdot\n\\overrightarrow{\\dot{\\theta}}\\\\\n&  =\\frac{1}{2}\\vec{p}^{T}\\cdot\\mathbf{M}^{-1}\\cdot\\vec{p},\n\\end{align}\nwhere the capacitive matrix is%\n\\begin{equation}\nC_{ij}=C_{i}\\delta_{ij}+C_{4},\n\\end{equation}\nthe effective mass tensor is\n\\begin{equation}\n\\mathbf{M}=\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}\\mathbf{C},\n\\end{equation}\nand the canonical momentum is%\n\\begin{equation}\n\\vec{p}=\\mathbf{M}\\cdot\\overrightarrow{\\dot{\\theta}}.\n\\end{equation}\nTherefore, we have reduced the problem to the canonical model of the quantum\nsystem with Hamiltonian\n\\begin{equation}\nH=\\frac{1}{2}\\vec{p}^{T}\\cdot\\mathbf{M}^{-1}\\cdot\\vec{p}+U\\left(  \\vec{\\theta\n}\\right)  , \\label{eq:HamGeneral}%\n\\end{equation}\nwhere the operators satisfy the commutation relation $\\left[  \\theta_{j}%\n,p_{k}\\right]  =i\\delta_{jk}$.\n\nBased on this model, we obtain the phase separation and the magnitude of\nquantum fluctuations in the next two sections.\n\n\\section{Phase Separation between the Potential Minima}\n\nWe now calculate the potential minimum $\\left\\{  \\theta_{j}^{\\ast}\\right\\}  $\nby introducing a Lagrange variable $\\lambda$ associated with the phase\nrelation (Eq.(\\ref{eq:PhaseConstraint})). We study the function%\n\\begin{equation}\nF=-\\sum_{j=1,2,3,4}E_{J,j}\\cos\\theta_{j}-\\lambda\\left(  \\sum_{j}\\theta_{j}%\n-\\pi\\right)  .\n\\end{equation}\nThe first derivatives all vanish at the extreme point:%\n\\begin{equation}\nE_{J,j}\\sin\\theta_{j}^{\\ast}=\\lambda.\n\\end{equation}\nFrom the phase relation $\\sum_{j}\\theta_{j}^{\\ast}=\\sum_{j}\\sin^{-1}%\n\\frac{\\lambda}{E_{J,j}}=\\pi$, we may solve for $\\lambda$.\n\nFor our system with four JJs, we can also calculate $\\theta_{4}^{\\ast}$ by\nseries expansion with respect to the small parameter $\\beta^{-1}$. For\n$\\beta\\rightarrow\\infty$, we have the zeroth order expansion $\\lambda^{\\left(\n0\\right)  }=\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha}$, $\\theta_{1}^{\\left(\n0\\right)  }=\\theta_{2}^{\\left(  0\\right)  }=\\sin^{-1}\\lambda^{\\left(\n0\\right)  }=\\cos^{-1}\\frac{1}{2\\alpha}$, $\\theta_{3}^{\\left(  0\\right)  }%\n=\\pi-2\\cos^{-1}\\frac{1}{2\\alpha}$ and $\\theta_{4}^{\\left(  0\\right)  }=0$.\nThen, to the first order of $\\beta^{-1}$, we have $\\theta_{4}^{\\left(\n1\\right)  }=\\sin^{-1}\\frac{\\lambda^{\\left(  0\\right)  }}{\\beta}\\approx\n\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha\\beta}$. Therefore,\\ the phase difference\nfor the fourth junction is\n\\begin{equation}\n\\theta_{4}^{\\ast}=\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha\\beta}+\\left(  \\beta\n^{-2}\\right)  .\n\\end{equation}\nAs plotted in Fig.~\\ref{fig:BetaDependence2}b, the numerically obtained\nquantity $\\frac{1}{\\Delta\\varepsilon}=\\frac{1}{2\\theta_{4}^{\\ast}}\\propto\n\\beta$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8cm]{fig_BetaDependence_v4.pdf}\n\\end{center}\n\\caption[fig:BetaDependence2]{Parameters for the flux qubit as a function of\n$\\beta$. (a,b,c) Phase separation $\\Delta\\varepsilon$ (dark solid line) and\nthe magnitude of quantum fluctuations $\\zeta$ (purple dashed line) from\nnumerical calculation. The theoretical prediction from\nEq.~(\\ref{eq:QuantumFluctuations}) (gray solid line in (c)) agrees very well\nwith the numerical calculation. (d,e,f) The two ground states of the flux\nqubit can be coupled by intra-cell tunneling (red solid lines) and inter-cell\ntunneling (blue dashed lines) \\cite{Mooij99,Orlando99}, characterized by the\naction $S$, potential barrier $\\Delta U$, and tunneling rate $t$. Panels (a)\nand (f) assume $E_{J}\/E_{C}=80$.}%\n\\label{fig:BetaDependence2}%\n\\end{figure}\n\n\\section{Models for Quantum Fluctuations of the SC Phase}\n\nIn this section, we consider two models for quantum fluctuations across the\nJJs. The quantum fluctuations across the JJs are due to the finite mass matrix\nfor the Hamiltonian (Eq.(\\ref{eq:HamGeneral})), which is proportional to the\ncapacitance matrix. In the following, we first provide a simple, intuitive\nmodel to characterize the quantum fluctuations associated with $\\theta_{4}$\nfor $\\beta\\gg1$. Then, we consider the general model of multiple coupled\nharmonic oscillators, and obtain the formula for the magnitude of quantum\nfluctuations associated with a projected degree of freedom. We find very good\nagreement between the two models when $\\beta\\gg1$, which justifies the simple model.\n\n\\subsection{Simple Model -- One Harmonic Oscillator}\n\nGiven very large $\\beta$, we may neglect the higher order couplings to other\nJJs and consider the reduced Hamiltonian for the fourth JJ%\n\\begin{equation}\nH_{J,4}=\\frac{1}{2}C_{4}V_{4}^{2}+E_{J,4}\\left(  1-\\cos\\left(  \\theta\n_{4}-\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}\\right)  \\right)  ,\n\\end{equation}\nwith displaced potential minimum at $\\mathbf{Z}_{\\mathrm{flux}}\\theta\n_{4}^{\\ast}$. We then use the harmonic approximation and obtain%\n\\begin{equation}\nH^{\\mathrm{HO}}=\\frac{p_{\\theta_{4}}^{2}}{2M_{4}}+\\frac{E_{J,4}}{2}\\left(\n\\theta_{4}-\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}\\right)  ^{2},\n\\label{eq:Osc2}%\n\\end{equation}\nwhere $M_{4}=\\frac{1}{8E_{C,4}}$ and $E_{C,4}=\\frac{e^{2}}{2C_{4}}$. The\noscillator has frequency $\\omega=\\sqrt{8E_{J,4}E_{C,4}}=\\sqrt{8E_{J}E_{C}}$\nand characteristic length\n\\begin{equation}\n\\zeta=\\left(  \\frac{8E_{C,4}}{E_{J,4}}\\right)  ^{1\/4}=\\left(  \\frac{8E_{C}%\n}{E_{J}}\\right)  ^{1\/4}\\beta^{-1\/2}. \\label{eq:QuantumFluctuations}%\n\\end{equation}\nHere $\\zeta$ is also the magnitude of quantum fluctuations.\n\n\\subsection{General Model -- Coupled Harmonic Oscillators}\n\nWe now consider the general model of multiple coupled harmonic oscillators,\nand obtain the formula for the magnitude of quantum fluctuations associated\nwith a projected degree of freedom.\n\nClose to the potential minimum $\\left\\{  \\mathbf{Z}_{\\mathrm{flux}}\\theta\n_{j}^{\\ast}\\right\\}  $, we may expand the potential function to the second\norder of $\\vec{x}\\equiv\\vec{\\theta}-\\mathbf{Z}_{\\mathrm{flux}}\\vec{\\theta\n}^{\\ast}$%\n\\begin{equation}\nU=U_{\\min}+\\frac{1}{2}\\sum_{i,j=1,2,3}K_{ij}x_{i}x_{j}+O\\left(  x^{3}\\right)\n\\end{equation}\nwith%\n\\begin{equation}\nK_{ij}=\\left.  \\frac{d^{2}U}{d\\theta_{i}d\\theta_{j}}\\right\\vert _{\\vec{\\theta\n}=\\vec{\\theta}^{\\ast}}.\n\\end{equation}\nThe effective Hamiltonian around the minimum describes a system of coupled\nHarmonic oscillators:%\n\\begin{align}\nH_{\\mathrm{oscillator}}  &  =\\frac{1}{2}\\sum_{i,j=1,2,3}M_{ij}\\dot{x}_{i}%\n\\dot{x}_{j}+\\frac{1}{2}\\sum_{i,j=1,2,3}K_{ij}x_{i}x_{j}\\\\\n&  =\\frac{1}{2}\\vec{p}^{T}\\cdot\\mathbf{M}^{-1}\\cdot\\vec{p}+\\frac{1}{2}\\vec\n{x}^{T}\\cdot\\mathbf{K}\\cdot\\vec{x}%\n\\end{align}\nwhere we have used the definition $\\vec{p}=\\mathbf{M}\\cdot\\overrightarrow\n{\\dot{\\theta}}=\\mathbf{M}\\cdot\\overrightarrow{\\dot{x}}$.\n\nTo solve for the system of coupled oscillators, we perform the following\ntransformation to make the mass matrix\/tensor isotropic. For real and\nsymmetric mass matrix ($M_{ij}=M_{ji}$), there is an orthogonal transformation\n$V_{1}$ (with $V_{1}^{-1}=V_{1}^{T}$) that diagonalizes the mass matrix%\n\\begin{equation}\nV_{1}\\mathbf{M}V_{1}^{T}=\\mathbf{\\Lambda}%\n\\end{equation}\nwhere $\\Lambda_{ij}=\\lambda_{i}\\delta_{ij}$ is the diagonal matrix. The\neigenvalue $\\lambda_{i}$ is effective mass along the $i$-th principle axis. By\ninverting both sides, we have $V_{1}\\mathbf{M}^{-1}V_{1}^{T}=\\mathbf{\\Lambda\n}^{-1}$. For diagonal matrix, we may also define $\\left(  \\Lambda^{\\pm\n1\/2}\\right)  _{ij}=\\lambda_{i}^{\\pm1\/2}\\delta_{ij}$. Introducing the\ntransformation%\n\\begin{align}\n\\vec{x}^{\\prime}  &  =\\Lambda^{1\/2}V_{1}\\cdot\\vec{x}\\\\\n\\vec{p}^{\\prime}  &  =\\Lambda^{-1\/2}V_{1}\\cdot\\vec{p}%\n\\end{align}\nwe have%\n\\begin{equation}\nH_{\\mathrm{oscillator}}=\\frac{1}{2}\\vec{p}^{\\prime T}\\cdot\\vec{p}+\\frac{1}%\n{2}\\vec{x}^{\\prime T}\\cdot\\mathbf{\\tilde{K}}\\cdot\\vec{x}%\n\\end{equation}\nwhere%\n\\begin{equation}\n\\mathbf{\\tilde{K}=}\\Lambda^{-1\/2}\\left(  V_{1}KV_{1}^{T}\\right)\n\\Lambda^{-1\/2}.\n\\end{equation}\nFinally, we diagonalize the real symmetric matrix $\\mathbf{\\tilde{K}}$ via\northogonal transformation $V_{2}$%\n\\begin{equation}\nV_{2}\\mathbf{\\tilde{K}}V_{2}^{T}=\\mathbf{\\Omega}%\n\\end{equation}\nwhere $\\Omega_{ij}=\\omega_{i}^{2}\\delta_{ij}$ is the diagonal matrix. Overall,\nthe position and momentum transform as\n\\begin{subequations}\n\\label{eq:Transformation}%\n\\begin{align}\n\\vec{y}  &  =V_{2}\\vec{x}^{\\prime}=V_{2}\\Lambda^{1\/2}V_{1}\\cdot\\vec{x}\\\\\n\\vec{q}  &  =V_{2}\\vec{p}^{\\prime}=V_{2}\\Lambda^{-1\/2}V_{1}\\cdot\\vec{p}%\n\\end{align}\nThe eigenvalue $\\omega_{i}^{2}$ is the square of the $i$-th oscillator\nfrequency (also called plasma frequency). Given parameters $\\alpha=0.8$ and\n$\\beta=10$, we calculate the plasma frequencies $\\left\\{  \\omega_{i}\\right\\}\n_{i=1,2,3}=\\left\\{  2.8,2.3,1.8\\right\\}  \\sqrt{E_{J}E_{C}}$. We may also vary\nthe parameter $\\beta$, and observe that the plasma frequencies only depend\nvery weakly for $\\beta\\gg1$.\n\nNote that position and momentum have different transformations $V_{2}%\n\\Lambda^{1\/2}V_{1}$ and $V_{2}\\Lambda^{-1\/2}V_{1}$. Furthermore, these\ntransformations are not orthogonal transformations. However, as long as the\ntransformations preserve the commutation relation $\\left[  \\hat{y}_{j},\\hat\n{q}_{k}\\right]  =\\left[  \\hat{x}_{j},\\hat{p}_{k}\\right]  =i\\delta_{jk}$, we\ncan still perform quantization over the transformed coordinate.\n\nFollowing this procedure, we can numerically compute the magnitude of quantum\nfluctuations of $\\theta_{4}$ as detailed below.\n\n\\subsection{Quantum Fluctuations in SC Phase}\n\nWe now quantize the phase difference across the fourth junction $\\theta_{4}$.\nNear the potential minimum at $\\left\\{  \\theta_{i}^{\\ast}\\right\\}  $, we\nperform the transformation of Eq.\\ (\\ref{eq:Transformation}) along with the\nquantization $\\hat{y}_{i}=\\frac{\\hat{a}_{i}^{\\dag}+\\hat{a}_{i}}{\\sqrt{2}}$ and\n$\\hat{q}_{i}=\\frac{\\hat{a}_{i}^{\\dag}-\\hat{a}_{i}}{\\sqrt{2}i}$, we obtain the\nHamiltonian for three uncoupled harmonic oscillators%\n\\end{subequations}\n\\begin{equation}\n\\tilde{H}_{\\mathrm{oscillator}}=\\sum_{i=1,2,3}\\hbar\\omega_{i}\\left(  \\hat\n{a}_{i}^{\\dag}\\hat{a}_{i}+\\frac{1}{2}\\right)  ,\n\\end{equation}\nwith eigenfrequencies of $\\left\\{  \\omega_{i}\\right\\}  _{i=1,2,3}$. Each\noscillatory mode may induce quantum fluctuations in $\\theta_{4}$, with\ncharacteristic length scale $\\zeta_{i}$. The operator form of $\\theta_{4}$ can\nbe written as%\n\\begin{equation}\n\\theta_{4}=\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}+\\sum_{i=1,2,3}\\zeta\n_{i}\\frac{\\hat{a}_{i}^{\\dag}+\\hat{a}_{i}}{\\sqrt{2}}%\n\\end{equation}\n\n\nWe calculate the values of $\\zeta_{i}$ as the following. In the $y$%\n-coordinate, the characteristic displacement vector for the $i$-th mode is\n$\\vec{l}_{i}^{\\left(  y\\right)  }=\\sqrt{\\frac{\\hbar}{\\omega_{i}}}\\vec{e}%\n_{i}^{\\left(  y\\right)  }$, with unit vector $\\vec{e}_{i}$ along the $i$-th\ndirection. We may transform this back to the $x$-coordinate, $\\vec{l}%\n_{i}^{\\left(  x\\right)  }=\\left(  V_{2}\\Lambda^{1\/2}V_{1}\\right)  ^{-1}%\n\\cdot\\sqrt{\\frac{\\hbar}{\\omega_{i}}}\\vec{e}_{i}^{\\left(  y\\right)  }$. Note\nthat $\\vec{l}_{i}^{\\left(  x\\right)  }$ is no longer orthogonal. From $\\vec\n{l}_{i}^{\\left(  x\\right)  }$, we can obtain the characteristic fluctuating\nscale of $\\zeta_{i}=\\left\\vert \\sum_{k=1,2,3}\\left(  \\vec{l}_{i}^{\\left(\nx\\right)  }\\right)  _{k}\\right\\vert $. The magnitude of quantum fluctuations\nof $\\theta_{4}$ can be computed as\n\\begin{equation}\n\\zeta=\\left(  \\sum_{i=1,2,3}\\left\\vert \\zeta_{i}^{2}\\right\\vert \\right)\n^{1\/2} \\label{eq:QuantumFluctuations2}%\n\\end{equation}\nfor independent fluctuations from the three uncoupled harmonic oscillators.\n\nAs plotted in Fig.~\\ref{fig:BetaDependence2}c, the numerically obtained value\nfor $\\zeta$ (using Eq.~(\\ref{eq:QuantumFluctuations2})) agrees very well with\nthe prediction from the simple model (using Eq.~(\\ref{eq:QuantumFluctuations}%\n)), which scales as $\\beta^{-1\/2}$. Therefore, the simple model of\nEq.~(\\ref{eq:QuantumFluctuations}) provides a reliable description to\ncharacterize the dynamics associated with $\\theta_{4}$.\n\n\\section{Tunneling Matrix Element}\n\n\\subsection{WKB method}\n\nWe use the WKB method to estimate the tunneling matrix element\n\\cite{Orlando99}. The action associated with the pathway $\\vec{\\theta}\\left(\nr\\right)  $ from $\\vec{\\theta}\\left(  0\\right)  =\\vec{\\theta}_{a}$ to\n$\\vec{\\theta}\\left(  1\\right)  =\\vec{\\theta}_{b}$ is%\n\\begin{equation}\nS=\\int_{\\vec{\\theta}_{a}}^{\\vec{\\theta}_{b}}\\sqrt{2\\left(  U-E\\right)  }%\n\\sqrt{d\\vec{\\theta}^{T}\\cdot\\mathbf{M}\\cdot d\\vec{\\theta}},\n\\end{equation}\nand the tunneling matrix element can be estimated as\n\\begin{equation}\nt\\approx\\frac{\\hbar\\omega}{2\\pi}~e^{-S\/\\hbar}.\n\\end{equation}\nThe phase space $\\vec{\\theta}$ has period $2\\pi~$for all three directions. We\nmay introduce the unit cell with volume $\\left(  2\\pi\\right)  ^{3}$ and three\nbasis vectors $\\vec{a}_{1}=2\\pi\\left(  1,0,0\\right)  $, $\\vec{a}_{2}%\n=2\\pi\\left(  0,1,0\\right)  $, and $\\vec{a}_{3}=2\\pi\\left(  0,0,1\\right)  $.\n(Note that we may choose the shape of the unit cell for our convenience.)\nRegarding the tunneling pathway, we may choose the initial point $\\vec{\\theta\n}_{a}^{\\ast}=-\\vec{\\theta}^{\\ast}$, while the choice for final point is not\nunique as $\\vec{\\theta}_{b}=\\vec{\\theta}^{\\ast}+\\sum_{i}n_{i}\\vec{a}_{i}$ for\nintegers $\\left\\{  n_{i}\\right\\}  $. However, we may require that the final\npoint, $\\vec{\\theta}_{b}^{\\ast}$, be the one that has the minimum action from\nthe initial point, and we choose the unit cell so that it includes the minimum\naction pathway that connects $\\vec{\\theta}_{a}^{\\ast}$ and $\\vec{\\theta}%\n_{b}^{\\ast}$. After this procedure, the intra-cell tunneling has the minimum\naction, compared to all inter-cell tunneling pathways.\n\n\\subsection{Pathway with minimum action}\n\nWe should use the pathway $\\vec{\\theta}\\left(  r\\right)  $ with extreme action\nfor the WKB method, i.e.,%\n\\[\n\\frac{\\delta S}{\\delta\\vec{\\theta}\\left(  r\\right)  }=0.\n\\]\nWe obtain these extreme pathways using the following approach. First, we\ndiscretize the integration%\n\\begin{equation}\nS=\\sum_{i=1}^{N}\\left\\vert \\vec{x}_{i}-\\vec{x}_{i-1}\\right\\vert ~f\\left(\n\\frac{\\vec{x}_{i}+\\vec{x}_{i-1}}{2}\\right)  ,\n\\end{equation}\nwith $\\vec{x}_{0}=\\vec{\\theta}_{a}$ and $\\vec{x}_{N}=\\vec{\\theta}_{b}$. Then\nwe calculate the derivatives with respect to $\\vec{r}_{i}$%\n\\begin{align}\n\\frac{\\delta S}{\\delta\\vec{r}_{i}} &  =\\frac{\\Delta\\vec{x}_{i}}{\\left\\vert\n\\Delta\\vec{x}_{i}\\right\\vert }f\\left(  \\frac{\\vec{x}_{i}+\\vec{x}_{i-1}}%\n{2}\\right)  -\\frac{\\Delta\\vec{x}_{i+1}}{\\left\\vert \\Delta\\vec{x}%\n_{i+1}\\right\\vert }f\\left(  \\frac{\\vec{x}_{i+1}+\\vec{x}_{i}}{2}\\right)\n\\nonumber\\\\\n&  +\\left\\vert \\Delta\\vec{x}_{i}\\right\\vert \\bigtriangledown f\\left(\n\\frac{\\vec{x}_{i}+\\vec{x}_{i-1}}{2}\\right)  +\\left\\vert \\Delta\\vec{x}%\n_{i+1}\\right\\vert \\bigtriangledown f\\left(  \\frac{\\vec{x}_{i+1}+\\vec{x}_{i}%\n}{2}\\right)  .\n\\end{align}\nFor extreme pathway, these derivatives should vanish. If we start with an\nnon-extreme pathway with non-vanishing derivatives, we can update the pathway\nso that the updated pathway becomes closer to the extreme pathway. By\nrepeating the update procedure many times, we will obtain a pathway that is\nvery close to the extreme pathway. Since we know in advance that we are\nlooking for the pathway that gives the minimum action, we apply the following\nupdate rules for the $k$th update:%\n\\begin{equation}\n\\vec{r}_{i}^{\\left(  k+1\\right)  }=\\vec{r}_{i}^{\\left(  k\\right)  }%\n-\\epsilon\\frac{\\delta S}{\\delta\\vec{r}_{i}}%\n\\end{equation}\nwhere $\\epsilon$ determines the evolution rate and $i=1,2,\\cdots,N-1$. For\nsufficiently small evolution rate%\n\\begin{equation}\nS\\left[  \\left\\{  \\vec{r}_{i}^{\\left(  k+1\\right)  }\\right\\}  \\right]\n-S\\left[  \\left\\{  \\vec{r}_{i}^{\\left(  k\\right)  }\\right\\}  \\right]\n=-\\epsilon\\left(  \\frac{\\delta S}{\\delta\\vec{r}_{i}}\\right)  ^{2}+O\\left(\n\\epsilon^{2}\\right)  \\leq0,\n\\end{equation}\nwhich ensures continuous reduction of the action.\n\n\\subsection{Tunneling rates and $\\beta$ Dependence}\n\nThis algorithm gives us the correct pathway that locally minimize the action\nbetween neighboring potential wells. We find that the pathways are essentially\nthe straight lines connecting the different minima, with no significant\ndifference in terms of the action values. For example, given parameters\n$\\alpha=0.8$ and $\\beta=10$, the intra-cell action is $S_{in}\\approx\n0.7\\hbar\\sqrt{E_{J}\/E_{C}}$ and the smallest inter-cell action is\n$S_{out}\\approx1.4\\hbar\\sqrt{E_{J}\/E_{C}}$. For $E_{J}\/E_{C}\\approx80$, we\nwill have $t_{2}\/t_{1}\\approx\\exp\\left[  \\left(  S_{1}-S_{2}\\right)\n\/\\hbar\\right]  \\sim10^{-3}\\ll1$. The barrier height for intra-cell tunneling\nis $\\Delta U=0.25E_{J}$.\n\nWhen $\\beta\\rightarrow\\infty$, all quantities (potential minimum position,\nplasma frequencies, action for tunneling, energy barrier, and tunneling matrix\nelement) reduce to the case with three JJs. As illustrated in\nFig.~\\ref{fig:BetaDependence2}, the deviation scales as $\\beta^{-1}$. For\n$\\beta\\geq10$, the perturbation from $\\beta$ is very small.\n\nFor practical parameters of mesoscopic aluminum junctions with critical\ncurrent density $500$ A\/cm$^{2}$ \\cite{Mooij99,Orlando99}, a junction with an\narea of $A=0.2\\times0.4$ $\\mu m^{2}$ can achieve $E_{J}\\approx200$ GHz and\n$E_{J}\/E_{C}\\approx80$, which corresponds to the first two junctions\n$E_{J,1}=E_{J,2}=E_{J}$. Since $E_{J,j}\\propto A_{j}$, the fourth junction\n$E_{J,4}=\\beta E_{J}$ should have an area of approximately $1\\times1$ $\\mu\nm^{2}$ to achieve $\\beta\\approx10.$\n\n\\section{Phase-Controllers}\n\nFor the STIS quantum wire, we would like to fix the phase difference between\ntwo disconnected SC islands. For example, we would like have $\\phi_{l}%\n-\\phi_{u}=\\pi\/2$, $\\phi_{r}-\\phi_{u}=\\pi\/2$, $\\phi_{c}-\\phi_{u}=\\phi_{c}+\\pi$,\nas shown in Fig.~\\ref{fig:TopoFluxQubits}ac. The idea is to connect the two SC\nislands via a phase-controller. The phase-controller has two large SC islands\nwith a controllable phase difference $\\gamma$. Using large SC islands for\nphase-controllers reduces quantum fluctuations in the SC phase.\n\nIn this section, we consider two approaches to building a phase-controller\nusing Josephson junctions (JJs). The phase difference $\\gamma$ between the two\nlarge SC islands can be controlled by either the external magnetic flux\n$\\gamma=\\gamma\\left(  \\Phi_{x}\\right)  $ or the electric current\n$\\gamma=\\gamma\\left(  I\\right)  $, as detailed below.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=8.7cm]{fig_PhaseController.pdf}\n\\end{center}\n\\caption[fig:PhaseController]{The phase-controllers can establish desired\nphase difference $\\gamma$ between SC islands $a$ and $b$. For example, (a) the\nflux phase-controller with external magnetic flux $\\Phi_{x}$, and (b) the\ncurrent phase-controller with external electric current $I$.}%\n\\label{fig:PhaseController}%\n\\end{figure}\n\n\\subsection{Flux phase-controller}\n\nThe \\emph{flux phase-controller} uses an rf SQUID loop that is interrupted by\na single Josephson junction (Fig.~\\ref{fig:PhaseController}a). We may change\nthe external magnetic flux enclosed by the loop, to induce the desired phase\ndifference between the two SC\\ islands, $a$ and $b$. The rf SQUID loop has\ninductance $L$, and the JJ has Josephson coupling energy $E_{J}$ and\ncapacitive charging energy $E_{C}$ ($=e^{2}\/2C$). The Hamiltonian for the flux\nphase-controller is%\n\\begin{equation}\nH=U+T,\n\\end{equation}\nwhere the potential energy is%\n\\begin{equation}\nU=\\frac{1}{2L}\\left(  \\Phi_{x}-\\frac{\\Phi_{0}}{2\\pi}\\gamma\\right)  ^{2}%\n+E_{J}\\left(  1-\\cos\\gamma\\right)\n\\end{equation}\nwith $\\Phi_{x}$ for the external flux enclosed by the loop and $\\gamma$ for\nthe gauge invariant phase difference across the junction, and the capacitive\ncharging energy is%\n\\begin{equation}\nT=\\frac{1}{2}CV^{2}%\n\\end{equation}\nwith the voltage $V=\\frac{\\Phi_{0}}{2\\pi}\\frac{d\\gamma}{dt}$ and the effective\nmass $m_{eff}=C\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}.$\n\nWe find that the potential minimum satisfies the condition%\n\\begin{equation}\n\\Phi_{x}=\\frac{\\Phi_{0}}{2\\pi}\\gamma+E_{J}L\\frac{2\\pi}{\\Phi_{0}}\\sin\\gamma,\n\\end{equation}\nThis expression can be used to determine $\\gamma$ as a function of $\\Phi_{x}$.\nFor $L\\rightarrow0$, we have $\\gamma=-2\\pi\\frac{\\Phi_{x}}{\\Phi_{0}}$. For\nsufficiently small $L$ (satisfying $L<\\frac{\\Phi_{0}^{2}}{4\\pi^{2}E_{J}}$),\nthis relation is still single valued for all $\\gamma$. Therefore, we can\ndeterministically control the gauge invariant phase difference $\\gamma$ by\napplying an appropriate $\\Phi_{x}$.\n\nWe then consider the perturbation around the potential minimum and obtain the\nplasma frequency $\\omega_{p}\\approx\\left(  LC\\right)  ^{-1\/2}$ for $L\\ll\n\\frac{\\Phi_{0}^{2}}{4\\pi^{2}E_{J}}$. The quantum fluctuations of $\\gamma$\naround the potential minimum has characteristic scale\n\\begin{equation}\n\\zeta_{\\mathrm{flux}}=\\sqrt{\\frac{\\hbar}{m_{eff}\\omega_{p}}}\\approx2\\sqrt{\\pi\n}\\left(  \\frac{E_{c}}{E_{L}}\\right)  ^{1\/4}.\n\\end{equation}\n\n\n\\subsection{Current phase-controller}\n\nAlternatively, we may also control the phase differences via the external\ncurrent $I$ through a Josephson junction (Fig.~\\ref{fig:PhaseController}a).\nThe Hamiltonian for such a \\emph{current phase-controller} is%\n\\begin{equation}\nH=U+T,\n\\end{equation}\nwhere the potential energy is%\n\\begin{equation}\nU=-I\\frac{\\Phi_{0}}{2\\pi}\\gamma+E_{J}\\left(  1-\\cos\\gamma\\right)\n\\end{equation}\nwith $\\gamma$ for the gauge invariant phase difference across the junction,\nand the capacitive charging energy is%\n\\begin{equation}\nT=\\frac{1}{2}CV^{2}%\n\\end{equation}\nwith the voltage $V=\\frac{\\Phi_{0}}{2\\pi}\\frac{d\\gamma}{dt}$ and the effective\nmass $m_{eff}=C\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}$.\n\nAs long as $I<I_{c}\\equiv\\frac{2\\pi}{\\Phi_{0}}E_{J}$, we have the potential\nminimum at\n\\begin{equation}\n\\gamma=\\sin^{-1}I\/I_{c}.\n\\end{equation}\nWe then consider the perturbation around the potential minimum and obtain the\nplasma frequency $\\omega_{p}\\approx\\left(  \\frac{2eI_{c}\\cos\\gamma^{\\ast}%\n}{\\hbar C}\\right)  ^{1\/2}$. The quantum fluctuations of $\\gamma$ around the\npotential minimum has characteristic scale\n\\begin{equation}\n\\zeta_{\\mathrm{current}}=\\sqrt{\\frac{\\hbar}{m_{eff}\\omega_{p}}}=\\left(\n\\frac{8}{\\cos\\gamma^{\\ast}}\\frac{E_{c}}{E_{J}}\\right)  ^{1\/4}.\n\\end{equation}\n\n\n\\subsection{Comparison between flux and current phase-controllers}\n\nBoth flux and current phase-controllers enable us to reliably induce phase\ndifference between two SC islands. We compare the two phase-controllers in the\nfollowing aspects:\n\n(1) Tunable phase rage: A single flux phase-controller can create all desired\nphase differences in the range $\\left[  -\\pi,\\pi\\right]  $. In contrast, a\nsingle current phase-controller can only create phase difference in the range\n$\\left(  -\\pi\/2,\\pi\/2\\right)  $. However, by using two or more current\nphase-controllers in series, it is possible to create arbitrary phase differences.\n\n(2) Quantum fluctuations: Both controllers have similar scaling for the\nquantum fluctuations, which scales as $\\left(  E_{c}\/E_{L,J}\\right)  ^{1\/4}$.\nIn order to minimize the quantum fluctuations, we may use the JJ loops with a\nsmall inductance (i.e., $E_{L}>E_{J}$) for flux phase-controller, while we may\nuse the JJ loops with large Josephson coupling energy $E_{J}$ for current\nphase-controller. Note that for current phase-controller the quantum\nfluctuations becomes unfavorably large when $\\gamma\\approx\\pi\/2$, which can be\novercome by using two or more controllers in series to reduce the fluctuations.\n\n\\subsection{Parameters}\n\nWe may estimate the quantum fluctuations for practical devices. According to\nthe experimental parameters of the large Josephson junctions \\cite{Martinis02}%\n: the charging energy $E_{c}\\equiv e^{2}\/2C=0.15$ mK and the Josephson\ncoupling energy $E_{J}=I_{c}\\Phi_{0}\/2\\pi=500$ K, which gives us\n$\\zeta_{\\mathrm{current}}\\approx(8\\ast\\frac{0.00015}{500})^{1\/4}=0.04$. It is\nalso possible to build a SQUID loop with very small inductive energy\n$E_{L}\\equiv\\Phi_{0}^{2}\/2L=645$ K \\cite{Friedman00}, and we can obtain\n$\\zeta_{\\mathrm{flux}}\\approx2\\sqrt{\\pi}\\ast(\\frac{0.00015}{645})^{1\/4}%\n\\approx0.08$. By further increasing the junction area, we may further decrease\n$E_{c}$ and increase $E_{J}$, which should give us more reduced quantum\nfluctuations from the phase-controller.\n\n\\section{Brief Derivation for Energy Splitting $E\\left(  \\varepsilon\\right)\n$}\n\nWe now briefly derive the energy splitting function $E\\left(  \\varepsilon\n\\right)  $, which is a highly non-linear function of $\\varepsilon$. The\nderivation mostly follows Ref.~\\cite{FuL08}.\n\nWe start with the effective\\ Hamiltonian \\cite{FuL08}%\n\\begin{equation}\nH^{\\mathrm{STIS}}=-iv_{F}\\tau^{x}\\partial_{x}+\\delta_{\\varepsilon}\\tau\n^{z}\\mathrm{,\\ }%\n\\end{equation}\nwith $\\delta_{\\varepsilon}=-\\Delta_{0}\\sin\\varepsilon\/2$ (differed from\n\\cite{FuL08} by a minus sign, due to a slightly different assignment of SC\nphases). The Hamiltonian can be written as%\n\\begin{equation}\nH^{\\mathrm{STIS}}\\left(  k\\right)  =v_{F}k\\tau^{x}+\\delta_{\\varepsilon}%\n\\tau^{z}=\\left(\n\\begin{array}\n[c]{cc}%\n\\delta_{\\varepsilon} & v_{F}k\\\\\nv_{F}k & -\\delta_{\\varepsilon}%\n\\end{array}\n\\right)  ,\n\\end{equation}\nwhere $k$ is the wave vector in the quantum wire. The eigen-energies for\n$H^{\\mathrm{STIS}}\\left(  k\\right)  $ are $E^{\\pm}\\left(  \\delta_{\\varepsilon\n},k\\right)  =\\pm\\sqrt{\\delta_{\\varepsilon}^{2}+v_{F}^{2}k^{2}}$. For a finite\nSTIS quantum wire with length $L$, it can only support a discretized set of\nwave vectors satisfying the boundary condition \\cite{FuL08}%\n\\begin{equation}\n\\tan kL=-\\frac{v_{F}k}{\\delta_{\\varepsilon}}=kL\/\\Lambda_{\\varepsilon},\n\\label{eq:BC}%\n\\end{equation}\nwith a dimensionless parameter\n\\begin{equation}\n\\Lambda_{\\varepsilon}\\equiv-\\delta_{\\varepsilon}L\/v_{F}.\n\\end{equation}\nFor given $\\Lambda_{\\varepsilon}$, we may solve Eq. (\\ref{eq:BC}) and obtain a\nset of solutions $kL=f_{n}\\left(  \\Lambda_{\\varepsilon}\\right)  $ with index\n$n=0,1,2,\\cdots$ for different bands. The function $f_{n}\\left(  y\\right)  $\nis just the inverse function of $y=x\/\\tan\\left(  x\\right)  $ associated with\nthe $n$th invertible domain.\n\nFor the lowest band, we have\n\\begin{equation}\nE\\left(  \\varepsilon\\right)  \\equiv E^{+}\\left(  \\delta_{\\varepsilon}%\n,k_{0}\\right)  =\\Delta E\\sqrt{\\Lambda_{\\varepsilon}^{2}+f_{0}^{2}\\left(\n\\Lambda_{\\varepsilon}\\right)  },\n\\end{equation}\nwith $\\Delta E=v_{F}\/L$. Note that $kL=f_{0}\\left(  \\Lambda_{\\varepsilon\n}\\right)  $ is purely imaginary for $\\Lambda_{\\varepsilon}\\in\\left(\n1,\\infty\\right)  $, and it is real for $\\Lambda_{\\varepsilon}\\in\\left(\n-\\infty,1\\right)  $. Physically, imaginary $kL$ corresponds to localized MFs\nat the ends of the quantum wire, and real $kL$ indicates delocalized MFs.\nThose higher bands (with $n\\geq1$) are associated with the excitation modes of\nthe quantum wire, with excitation energy at least $\\Delta E$ \\cite{FuL08}.\n\nIn summary, we have%\n\\begin{equation}\n\\frac{E\\left(  \\varepsilon\\right)  }{\\Delta E}=\\sqrt{\\Lambda_{\\varepsilon}%\n^{2}+f_{0}^{2}\\left(  \\Lambda_{\\varepsilon}\\right)  },\n\\end{equation}\nwhich is plotted in Fig~\\ref{fig:Potential}a.\n\n\n\\end{document} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\n\nIn this work, we introduce a unified 3D pipeline for OAR localization-segmentation rather than novel localization or segmentation architectures. The proposed localization framework and the  unified two-step approach for multi-organ segmentation work directly in 3D volumetric data to exploit 3D context information. We have validated our method in a public benchmark data set, which shows consistent improvement in Dice score compared to the baseline, especially for the small structures. Future work includes directly predicting the extension of the bounding boxes along the centroid prediction to discard the need for population statistics.\n\n\n\n\n\\section{Experiments and Discussion}\\label{chapter:experiments}\n\nTo validate the performance of the proposed approach, we perform the segmentation of 16 anatomical structures considered as OAR: lungs, kidneys, liver, spleen, pancreas, aorta, urinary bladder, sternum, thyroid, vertebra, right-, and left-psoas and rectus abdominal muscles.\n\n\n\n\\textbf{Data set:} The data set used in the experiments consists of CT scans from the gold corpus, and silver corpus in the VISCERAL dataset \\cite{Jimenez2016}. 74 CT scans from the silver corpus data are used for training and hyper-parameter tuning. The annotations in this set are automatically labeled by fusing the results of multiple algorithms. 23 CT scans from the gold corpus, which contains manually annotated labels, are used for testing. \n\n\n\n\nAll volumes are resampled to $3mm^3$ and $1mm^3$ for the localization and segmentation stages, respectively. We normalized each volume to zero mean and unit variance.\n\n\nTo evaluate the overall performance of the proposed two-stage approach for multi-organ segmentation, we report the average dice score between the ground truth and the predicted segmentation map. In addition, organ-wise dice scores are reported using box plots.\n\n\\textbf{Optimization:} We train the networks using Adam optimizer with a decaying learning rate initialized at $1e^{-3}$ and $1e^{-5}$ for the multi-variate regression model and segmentation model, respectively. We use a Mini-batch size of 1 in all experiments. The training was continued till validation loss converged. All the experiments are conducted on an NVIDIA Titan Xp GPU with 12GB vRAM.\n\n\n\\textbf{Gaussian Variance:} For the generation of ground-truth Gaussian heat maps, several approaches were tested. Experimental results showed that an isotropic variance of \\(150mm^2\\) gives the best performance. However, larger variances resulted in less precise predictions, while smaller variances tended to cause localization failures, where no strong centroid prediction was generated. Organ-size-specific heat-maps had the tendency to suffer from the same pitfalls, being either too large or too small.\n\n\n\\textbf{Threshold for Centroid Prediction:} We found an empirical value of $\\tau = 0.1$ to obtain the organ centroid prediction using the validation set. Furthermore, we found that heat maps' values  $<0.1$ translate to the absence of organ or detection failure.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\textbf{Quantitative Evaluation:} For the quantitative evaluation, we report our results and compare them with two approaches. Here is the definition of the experiments:\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{$\\bullet$}\n  \\item \\textbf{Baseline (3D U-Net)}: segmentation directly in 3D, this is a 3D version of the original U-Net with weighted cross-entropy for class imbalance.\n  \\item \\textbf{2D+3D Two-step}: 2D localization proposed in \\cite{DeVos2017} followed by 3D U-Net segmentation.\n  \\item \\textbf{3D+3D Two-step}: Proposed 3D localization followed by 3D U-Net segmentation.\n\\end{itemize}\n\n\n\nIn \\textbf{Table} \\ref{tab:total_dice} we report the overall Dice score for every method. We can observe that our proposed approach results in an improvement of $1.4\\%$ compared to the baseline. Similarly, the 2D+3D Two-step approach results in a similar performance to our proposed approach. Nevertheless, notice that the 2D localization proposed by DeVos et al. \\cite{DeVos2017} requires the training of three independent 2D networks in axial, coronal, and sagittal views to generate the final bounding box, while our approach only requires one network.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ll}\n\\textbf{Method}                              & \\textbf{Dice Score} \\\\\n\\noalign{\\hrule height 0.8pt}\nBaseline (3D U-Net)                           & 0.9120 \\(\\pm\\) 0.026                        \\\\ \\hline\n2D+3D Two-step            & \\textbf{0.9274 \\(\\pm\\) 0.016 }                       \\\\ \\hline\n3D+3D Two-step & \\textbf{0.9260 \\(\\pm\\)  0.018}                       \\\\ \\hline\n\\end{tabular}\n\\caption{ \\small{Quantitative results: the table describes the mean and standard deviation of the global Dice scores for all evaluated models.}}\n    \\label{tab:total_dice}\n\\end{center}\n\\end{table}\nIn \\textbf{Fig.} \\ref{fig:dice_scores} we can observe the organ-wise performance across different evaluated methods using box plots. We can notice that Dice scores for big structures, for instance, lungs and liver, do not improve significantly compared to the baseline. In contrast, the proposed two-step approach provides significant improvement for small structures such as the spleen, urinary bladder, aorta, trachea, thyroid gland, first lumbar vertebra, and muscles. We attribute this to the fact that training a network to segment all structures at once creates class imbalance and small structures are underrepresented. The proposed two-step approach resolves this problem, delivering better performance for these structures.\n\n\\textbf{Qualitative Evaluation:} In \\textbf{Fig} \\ref{fig:segimg}  we show a qualitative comparison between the baseline, the two-step approach with 2D localization, and the proposed approach. We highlight ROIs with green boxes to show regions where our approach improves the segmentation and red boxes to indicate regions where baseline produces false positives. We can observe the correlation of Dice score improvement in the quantitative results with the visual assessment, particularly for the small organs. For instance, the highlighted red boxes show examples of false positives for the trachea and lungs. Moreover, observing the last two columns of \\textbf{Fig} \\ref{fig:segimg}, we can visually recognize that the proposed approach results in more smooth and continuous segmentation compared to both the baselines and 2D+3D approach. This is clearly observed on the boundaries of the lungs' segmentations indicated with yellow arrows in \\textbf{Fig.} \\ref{fig:segimg}.\n\n\n\n\n\n\n\n\n\\begin{figure}[h]\n  \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/img_comparison.png}\n   \n    \\caption{\\small{Qualitative results: ROIs in green indicate regions where the proposed approach improves the segmentation. ROIs in red indicate regions where the baseline produces false positives. Yellow arrows point to regions where the proposed approach presents smooth and continuous segmentations.}}\n  \\label{fig:segimg}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nLocalization and segmentation of OAR are two necessary pre-processing steps for many automatic medical image analysis tasks, such as radiation therapy planning \\cite{lipkova2019personalized,ezhov2020real} and lesion detection \\cite{bilic2019liver}. Expert radiologists perform these steps manually, making the process time-consuming, costly, and dependent on the level of expertise. These problems call for computer-aided systems for automatic analysis. \n\n\\begin{figure}[ht!]\n  \\centering\n    \\includegraphics[width=1\\linewidth]{figures\/method_overview_two_rows.png}\n    \\caption{\\small{Overview of the proposed approach. The input is a 3D CT scan. In the localization step, our proposed multi-variate regression network together with population statistics predicts organs' bounding boxes. In the segmentation step, segmentation is performed on organ-wise ROIs. The output of the network is the merged segmentation map after aggregation.}}\n  \\label{fig:overview}\n\\end{figure}\nIn recent years, deep convolutional neural networks have been widely adopted in medical imaging applications for both localization \\cite{DeVos2017,ren2015faster,xu2019efficient,navarro2020deep,waldmannstetter2020reinforced} and segmentation \\cite{ronneberger2015u,cciccek20163d,Navarro2019,qasim2020red}. These methods focus either on localization or segmentation. While the detection task prioritizes coarse-level organ representation to estimate relative position, the segmentation task requires fine feature representation to determine organ boundaries. We argue that in a holistic pipeline, prior knowledge about organ location obtained from the localization task is useful for the segmentation model to specialize in organ-specific feature learning solely. In this vein, few approaches aim at jointly solving the two tasks \\cite{he2017mask,zhao2019knowledge, dabiri2020deep,hussain2021cascaded}.\nNevertheless, the model complexity of Mask-RCNN \\cite{he2017mask} makes it unsuitable for 3D volumetric analysis. Likewise, methods similar to the proposed by Zhao et al. \\cite{zhao2019knowledge} require a registration step resulting in a time-consuming approach. On the other hand, \\cite{hussain2021cascaded, dabiri2020deep}  use 2D networks for localization using axial, coronal, and sagittal views without exploiting the 3D information. Other approaches like in \\cite{vesal2020fully} segments the organ of interest twice: one in a coarse resolution using the gradcam for localization and later used a crop around the attention map to segment the organ. In \\cite{liu2019automatic} patches generated from superpixels are classified as candidate regions of the organ of interest in the localization step followed by 2.5D segmentation networks. In contrast, our approach uses one network for centroids prediction and one for organs-specific segmentation.\n\nThe aforementioned two-step approaches are either too computational expensive for medical imaging or lack ability to fully exploit 3D information. In this work, we focus on bringing together 3D localization and segmentation pipelines to fully exploit 3D context information, rather than proposing novel localization or segmentation architectures. With this intention, we propose a holistic two-step approach to efficiently exploit the organ-size invariant relative positional information in a 3D context. Firstly, inspired by \\cite{payer2020coarse}, we formulate the localization problem as a 3D multi-variate regression problem to predict the organs' centroids and hence the bounding boxes. Secondly, we leverage 3D segmentation networks to obtain the final segmentation map around the localized region obtained from the first step. Lastly, we aggregate each organ segmentation, resulting in a multi-organ segmentation map.\n\n\n\n\n\n\n\n\n\n\n\n\\section{METHODOLOGY}\n\\label{chapter:methodology}\nOur proposed holistic approach consists of two stages: the localization and the segmentation stage. We obtain the final segmentation map by aggregating the individual segmentation predictions. Fig \\ref{fig:overview}. shows an overview of the proposed approach. In the rest of the section, we describe the details of each component.\n\n\n\\subsection{Localization}\n\n\\subsubsection{\\textbf{Multi-variate Gaussian Regression}}\nWe formulate the organ localization task as a multi-variate Gaussian regression problem for organs' centroids. These predictions in combination with average organs' bounding boxes enables the exploitation of 3D context information. Formally, we define the input 3D medical image as $\\mathbf{X} \\in \\mathbb{R}^{(h \\times w \\times d)}$. For each organ, we define the ground truth as a Gaussian heat map of the form \n$\\mathbf{y_i}= -e^{{\\parallel x- \\mu_{i} \\parallel}^{2} \/\\ 2 \\sigma^2}$, where $\\mu_{i}$ is the centroid of the $i^{th}$ organ and $\\sigma^2$ controls the variance of the Gaussian. The ground truth annotation results in $\\mathbf{Y} \\in \\mathbb{R}^{(h \\times w \\times d \\times n)}$, where $n$ is the number of organs to be localized.\n\nThe proposed network architecture is inspired by the seminal works in Sekuboyina et al. \\cite{Sekuboyina2018,sekuboyina2020labeling,payer2020coarse} for vertebrae localization and segmentation. We created a 3D version of the authors' approach \\cite{Sekuboyina2018,sekuboyina2020labeling} to work directly in 3D volumetric data, instead of 2D projections, depicted in \\textbf{ Fig.} \\ref{fig:architecture}. In contrast to \\cite{payer2020coarse}, we tailored the 3D localization proposed for vertebrae and adapted it to organs. This architecture allows the network to exploit the 3D context information and efficiently use computational resources for different CT organs. The loss function used for optimization is the $\\mathcal{L}_{2}$ loss:\n\\begin{equation}\n\\mathcal{L}_{2} = \\frac{1}{h\\times w\\times d \\times n} \\sum_{} {\\parallel \\mathbf{Y} - \\tilde{\\mathbf{Y}} \\parallel}^2\n\\end{equation}\nwhere $\\mathbf{\\tilde{Y}}$ stands for the predicted Gaussian heat map. \n\n\n\n\\begin{figure}[h!]\n  \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/network.png}\n    \\caption{\\small{Proposed 3D Multi-variate Gaussian regression network. Each output channel corresponds to one organ heat map.}}\n  \\label{fig:architecture}\n\\end{figure}\n\n\\subsubsection{\\textbf{Bounding Box Generation}}\nTo obtain bounding boxes from the predictions of the multi-variate Gaussian regression model $\\mathbf{\\tilde{Y}} \\in \\mathbb{R}^{(h \\times w \\times d \\times n)}$, the following steps are executed:\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{$\\bullet$}\n  \\item Voxels with Gaussian heat map's values above a threshold $\\tau$ form a region potentially containing the organ centroid. This method is preferred over taking the highest heat-map probability as it is sensitive to outliers and false positives.\n  \\item We define the mid-point of this region as the organ centroid prediction.\n  \\item For each organ, an average bounding box size is computed from the training set.\n  \\item Finally, this bounding box is centered around the aforementioned centroid prediction plus $v$ voxels in each direction resulting in the final box. Adding this margin ensures the organ is contained inside the box.\n\\end{itemize}\n\n\n\n\n\n\n\n\\subsection{Organ-wise 3D Segmentation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFrom the predicted organ's bounding boxes, we crop organ-wise ROIs from the CT scans. Then, we use these ROIs to train organ-specific 3D segmentation networks. This model segregation enables us to customize the networks to learn organ-specific feature representations. Furthermore, the localization stage endows the networks to process high-resolution 3D information efficiently. In this work, we adopted 3D U-Nets \\cite{cciccek20163d} to generate the organ segmentation maps. \n\n\n\nWe use a combination of binary cross-entropy and Dice as loss function:\n\\begin{equation}\n\\mathcal{L} = \\underbrace{- \\sum_{x}^{} {g_{}{(x)} \\log{p_{}{(x)}}}}_\\text{Cross-Entropy Loss} - \\underbrace{{\\frac{2 \\sum_{x}^{} {p_{}{(x)}} g_{}{(x)}} {\\sum_{x}^{} {p_{}^{2}{(x)}} + \\sum_{x}^{} {g_{}^{2}{(x)}} } }}_\\text{Dice Loss}\n\\end{equation}\nwhere $p(x)$ is the probability of a pixel in location $x$ to belong to the foreground class and $g(x)$ is the ground truth for the corresponding pixel.\n\n\\begin{figure*}[h]\n  \\centering\n    \\includegraphics[width=0.7\\linewidth]{figures\/dice_scores.png}\n    \\caption{\\small{Organ-wise box plots comparison across all tested methods. A constant improvement of  Dice score is obtained with the proposed approach compared to the baseline.}}\n  \\label{fig:dice_scores}\n\\end{figure*}\nWe aggregate the segmented crops from every organ to generate the final segmentation map, bringing the crops back to the original image space. During this aggregation process, there may exist overlap between organs that are close to each other. Those overlapping voxels, for which segmentation networks disagree, are set to the predictions with the highest probability value.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}