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experiments on mesoscopic quantum systems , and sophisticated numerical techniques for quantum many body theory , promise progress on deep questions about the relationship between microphysics and thermodynamics @xcite . we still require analytical theory , however , to clarify the questions . for example , it should be well known that heat consists microscopically of energy held by degrees of freedom whose evolution is too rapid to perceive or control on macroscopic time scales @xcite . this simple formulation is nonetheless surprising to many physicists , because heat is rarely discussed explicitly except in terms of ensembles . since the ensembles themselves are justified fundamentally as proxies for time averaging over rapid microscopic evolution , the role of time scale separation in defining heat is well established as an implicit assumption ; but it is underdeveloped as an explicit theory . investigating the mesoscopic onset of thermodynamics will require fully developing this theory , because the applicability of ensembles is not obvious in small systems . microscopic energy that remains adiabatically frozen in fast evolution may of course simply be ignored on longer time scales . what the first law of thermodynamics means from a microscopic point of view , however , is that energy can be exchanged slowly between slow and fast degrees of freedom : heat and work may transform into each other . to investigate the mesoscopic onset of thermodynamics , therefore , a simple dynamical model that provides a minimal representation of heat exchange offers a useful paradigm .    in this work we present a simple but non - trivial model for two quantum subsystems in thermal contact , and show how heat exchange between them may be described in familiar quantum mechanical terms . each subsystem is a pair of bosonic modes coupled so as to represent an idealized josephson junction  a two - mode bose - hubbard system . we specify that the subsystem characteristic timescale  the josephson frequency  is to be the shortest one in the entire system , so that observation and control of the system can address only much longer time scales . each subsystem then has , in isolation , exactly two conserved quantum numbers : particle number and energy . for a fixed number of particles , the energy may still be varied independently by exciting the josephson mode ; if the rapid josephson oscillations are averaged over , then states of any energy still count as equilibrium . thus , the internal josephson oscillations of each subsystem are to be regarded as heat . this identification is bold enough that it may need some defense . the 2-mode bose - hubbard model , which has been very extensively studied from many points of view , is certainly a rather minimalist model for a thermal system . it is technically ergodic , inasmuch as the corresponding classical system explores the entire energy surface ( for fixed particle number ) ; but only because that surface is one - dimensional , so that time evolution has no choice but to explore it all . the system is integrable , not chaotic ; and chaos is often invoked as a justification for statistical mechanics . but the precise relevance of chaos to macroscopic behavior , as opposed to time scale separation alone , is far from fully understood . according to the explicit logic normally used to justify ensemble treatments of dynamical systems , and so relate thermodynamics to mechanics , the fact that time averaging reproduces the phase space average over the energy surface should allow us to consider the n - particle excited states of 2-mode bose - hubbard as equilibrium states , as far as any measurements or control operations performed on much longer time scales are concerned .    and if the standard appeal to time averaging allows us to claim our n - particle excited josephson states as equilibria , then it is at least qualitatively reasonable to say that the energy of those excitations is heat . for a macroscopic sample of ordinary gas held in a fixed external potential , with a fixed number of atoms , there is a one - parameter family of equilibrium states , distinguished by total energy ( usually expressed as temperature ) . if we raise this energy without changing the external potential or the particle number , we say that we have added heat to the gas . indeed this is almost the definition of heat , practically speaking . if heat means anything , it must mean this . we therefore argue that the excitations of a two - mode bose - hubbard system can indeed provide a simple model for heat , if they are involved in a dynamical process that occurs slowly compared to their josephson frequency . we arrange for this by placing two such two - mode subsystems in thermal contact , through an additional josephson tunneling between them , with a much smaller rate coefficient than that of intra - subsystem tunneling . the two subsystems can thereby exchange particles , on the slow time scale ; but they can also exchange heat , as the amplitudes of their fast josephson oscillations beat slowly back and forth . our combined four - mode bose - hubbard system is thus a candidate minimalist model for heat exchange . the implications of this model for mesoscopic and quantum thermodynamics depend simply on whether our josephson heat really behaves as heat . if it does not , because our integrable system is in some way too trivial , then this will be a useful insight into the precise role of chaos . even a reductio ad absurdum is still a reductio . but in fact we will show that the beating josephson excitations do behave very similarly to heat in a recognized , though unusual , form : second sound , which is normally described as a temperature oscillation . we identify a nonlinear collective effect that makes josephson oscillations of heat second josephson oscillations  the slowest collective excitation of the 4-mode system about an excited equilibrium state . our analytical derivation of this effect , from a simple quantum mechanical hamiltonian , indicates how thermodynamics may emerge from quantum dynamics in small systems . our complete system is thus a rather symmetric four - mode bose - hubbard model , with two sharply differing tunneling rates . such systems may be realized to a good approximation in current cold atom laboratories @xcite , either using four closely neighboring microtrap potentials , or with a double well trap holding atoms with two internal states coupled by a rabi laser @xcite . our system can probably also be realized at least classically with superconducting josephson junctions , but with the cold atom context in mind , we will refer to the particles as atoms .        by exploiting the small ratio between the slow and fast tunneling rates , we are able to analyze collective excitations in this model beyond the linear bogoliubov approximation , and show that the low - frequency behavior of the joint system consists of two josephson oscillations : the elementary bogoliubov excitation whereby the two subsystems exchange atoms with each other , and a nonlinear beat mode by which they exchange heat . insofar as a josephson oscillation represents a two - mode analog of bogoliubov zero sound in a dilute bose gas , we identify our second josephson mode as an analog of second sound ( see for example @xcite ) . we offer it as a simple example of a typical phenomenon of mesoscopic thermodynamics . our system s four - mode bose - hubbard ( bh ) hamiltonian with onsite interaction @xmath0 and index @xmath1 is @xmath2 where @xmath3 is the high tunneling rate within each subsystem pair , and @xmath4 the low rate between the two subsystems . @xmath5 conserves the total atom number @xmath6 , and in the present paper we assume for simplicity an eigenstate of @xmath7 with large eigenvalue @xmath8 . the global phase zero mode corresponding to @xmath7 is therefore eliminated , and we are left with the three nontrivial tunneling modes illustrated in fig .  2 .        in the linear bogoliubov approximation the frequencies of these elementary excitations above the @xmath8-atom ground state are @xmath9 these three modes are josephson oscillations , and @xmath10 , @xmath11 , @xmath12 are the three josephson frequencies for the three independent josephson oscillations shown in fig . as is well known , these collective mode frequencies are shifted away from the pure tunneling ( or for internal transitions , rabi ) frequencies @xmath4 , @xmath3 by the nonlinear interactions . repulsive interactions ( @xmath13 ) raise the josephson frequencies , while attractive interactions ( @xmath14 ) lower them .    in the @xmath15 case we consider , only one of these three elementary modes is slow , and the others are fast . since excitations of the bogoliubov modes provide a complete basis for the @xmath8-particle many - body hilbert space , it may seem puzzling that there are in fact ( as we shall see ) two distinct collective modes in this system with low frequencies ( of order @xmath4 rather than @xmath3 ) . this situation is actually common in nonlinear quantum many - body systems , however . the ` type ii ' excitations in the one - dimensional lieb - lieniger gas @xcite , second sound in superfluid helium @xcite , and even ordinary sound in air , are all examples of low - frequency collective excitations that are not to be found among a system s low - frequency elementary excitations , despite the fact that these are a complete set . the resolution of this paradox is that the frequency of beats between our system s two high frequency modes , @xmath16 is of order @xmath4 , and it is the interference beat between the two nearly degenerate high - frequency modes that provides the extra low - frequency collective mode . what we now show , by calculating to leading order in the small frequency ratio @xmath17 , but nonlinearly in excitation amplitude , is that within the low - frequency regime this beat mode is an independent excitation in its own right , whose frequency is shifted away from @xmath18 by nonlinear effects . to go accurately beyond the linear approximation within the standard symmetry - breaking bogoliubov theory requires careful treatment of zero modes , but it is straightforward if we follow a number - conserving approach @xcite . applying the holstein - primakoff transformation ( hpt ) @xcite , we re - write @xmath19 in terms of the total atom number on each side @xmath20 , and atom - moving operators @xmath21 that commute with @xmath22 : @xmath23 it is easy to show that @xmath24=1 $ ] and @xmath25=0 $ ] . in these terms the hamiltonians of the subsystems read @xmath26    we can then implement the bogoliubov formalism by defining @xmath27 and choosing @xmath28 to diagonalize the quadratic terms in @xmath29 while maintaining @xmath30=1 $ ] . since we are interested in dynamics slow compared to @xmath3 , and in josephson oscillations of finite but not extreme amplitude , we will apply the rotating wave approximation ( rwa ) by dropping from @xmath29 all terms that do not commute with the leading term proportional to @xmath31 . assuming the regime of large occupations for simplicity , we omit corrections of order @xmath32 , and obtain for @xmath33 @xmath34    in the quadratic term of ( [ halpha1 ] ) we note the familiar josephson frequency of the excitations created by @xmath35 , which we have been unable to resist naming ` josons ' . the co - efficient of the term quartic in @xmath36 is in general of the same order as @xmath0 , but opposite sign ; this reflects the well - known fact that the period of nonlinear josephson oscillations lengthens with amplitude @xcite . in quasiparticle terms , we may say that where atoms repel each other , josons attract ; and _ vice versa_.    to simplify the description of the slow tunneling between the @xmath37 subsystems , we now perform a similar hpt for our two atom numbers , while assuming that our system is in an eigenstate of the conserved total @xmath38 , with eigenvalue @xmath8 . this defines a number - conserving operator @xmath39 , which satisfies @xmath40=1 $ ] ; it transfers atoms between the @xmath41 and @xmath42 subsystems . in the large @xmath8 limit that we assume for simplicity , this provides @xmath43    \end{aligned}\ ] ] up to terms of @xmath44 . we can therefore re - write our full hamiltonian ( [ hamiltonian ] ) in terms of @xmath45 , @xmath39 , and @xmath8 .    since @xmath11 is on the order of the problem s high frequency @xmath3 , we see that to leading ( zeroth ) order in @xmath17 , our entire hamiltonian is simply @xmath46 , for @xmath47 in classical terms , the ` total joson number ' @xmath48 is an adiabatic invariant . by means of an rwa we can now compute low frequency dynamics nonlinearly in excitation amplitude , but to leading order in small frequency ratios , by dropping all terms in @xmath5 that do not commute with @xmath48 . since @xmath31 are not separately adiabatic invariant , we must retain terms such as @xmath49 . our result is @xmath50 in the first two lines we can recognize two different forms of the standard single josephson junction hamiltonian . the first line provides josephson oscillations of atoms between @xmath41 and @xmath42 subsystems , with josephson frequency @xmath51 , while the second implies _ josephson oscillations of josons_. the two josephson modes are exactly analogous , except that @xmath52 and @xmath0 are of opposite sign , as anticipated above following ( 5 ) . the third line couples the atom and joson modes , and arises because the frequency of fast josephson oscillations in each @xmath37 subsystem depends on the respective atom number . we can now perform a final hpt to express @xmath36 in terms of @xmath48 and a @xmath53-conserving operator @xmath54 that transfers josons between subsystems . as an adiabatic invariant , @xmath48 is just as good a conserved quantity within the low - frequency regime as @xmath7 itself , and so we may let @xmath55 in the low - frequency theory . to identify collective excitations , we then linearize ( [ hfull ] ) in @xmath56 to obtain for fixed @xmath57 @xmath58 in terms of the atom josephson frequency @xmath59 and the new josephson frequency for josons @xmath60 the decoupled collective mode frequencies in ( [ hfinal ] ) are @xmath61^{1/2}\!\!\!\!\!\!\!\!.\ ] ]    we observe that in the low - amplitude regime @xmath62 , either @xmath63 so that @xmath64 ; or else @xmath65 and thus @xmath66 so that the third line in ( [ hfinal ] ) is much smaller than the first two . thus for @xmath62 the low frequency atom josephson frequency @xmath67 is always essentially unchanged from the standard bogoliubov theory indicated in fig . ( 2 ) ; but it is joined by a second josephson oscillation whose frequency @xmath68 can be significantly lower ( for @xmath13 ) , because @xmath52 is negative . this is our main result . note that the linearization that yields ( [ hfinal ] ) is different from the standard bogoliubov approximation of retaining only quadratic terms in fluctuations of the original @xmath69 about the mean - field ground state . it is a resummed bogoliubov formalism , in which certain dominant nonlinear interactions among atoms have been included in the linearized interactions among josons , which are nonlinearly related to atoms through their definition via the hpt . one may also say that we do pursue the standard ( conserving ) bogoliubov approach , but applied to a two - mode system of interacting conserved atoms and josons . in this sense the extra low - frequency mode is a non - elementary collective mode in the full theory , but an elementary excitation within the adiabatic effective theory for low frequency dynamics . although our derivation is post - bogoliubov in the sense of being nonlinear , it is not essentially non - classical , because we have assumed large @xmath8 , and because all our hpt and rwa steps are exactly paralleled in classical adiabatic theory . we can therefore test our theory of second josephson oscillations numerically by solving the four - mode gross - pitaevskii nonlinear schrdinger equation for the classical limit of our system . we excite intra - subsystem josephson excitations with an oscillating drive , as follows . we relax to an @xmath8-particle mean field ground state , then excite josons by adding a time - dependent potential tilt to each subsystem , which varies periodically with the josephson frequency @xmath11 . we excite the second josephson mode by giving the tilt drives slightly different amplitudes for the @xmath41 and @xmath42 pairs , so that slightly more josons are excited on one side than on the other . the resulting evolution , after the drive is turned off , is shown in fig .  3 for a typical case .     and @xmath70 in the mean - field system for the parameters @xmath71 , @xmath72 , @xmath73 , @xmath74 and @xmath75 . red solid line : numerical mean - field @xmath76 . blue solid line : oscillation at second josephson frequency @xmath77 computed in the text , fitted for phase . blue dotted line : oscillation at beat - frequency @xmath78 . black dashed line : numerical @xmath70 . ] the agreement with our linearized theory is excellent , and the nonlinear shifting of the second josephson frequency well below the trivial bogoliubov beat is clear . some atom population oscillation between @xmath41 and @xmath42 subsystems occurs at frequency @xmath79 , this mode having been slightly excited by our driving procedure , but the main atom oscillation is exactly opposite in phase to the joson oscillation . we note that while in incompressible superfluids second sound includes no include significant particle density change , the coupling of density and heat oscillations is expected for second sound in compressible superfluids @xcite . we interpret the coupling between the slow atom and joson oscillations as a simple example of the interconversion of heat and work . the question whether our proposed oscillation of heat carries oscillating entropy is as much about the microscopic definition of entropy , as about the validity of our identification of heat . our josons do carry at least one physically meaningful form of entropy : the von neumann entropy of the single - particle reduced density matrices @xmath80 , projected onto either half of our system . these are simply two - by - two matrices , diagonal in the basis of even and odd modes excited by @xmath81 : @xmath82\!\ ! , \label{palpha}\end{aligned}\ ] ] for @xmath83 . the local single - particle entropies @xmath84 on each side , @xmath85\end{aligned}\ ] ] are genuine entropies inasmuch as they characterize the first order coherence , or equivalently the condensate depletion , within the two - mode bose - hubbard subsystems . they are direct analogs , for their respective two - mode subsystems , of the single - particle entropy which has been used since boltzmann to represent thermodynamic entropy in statistical mechanics . since @xmath86 is a simple function of @xmath87 , it is clear that our josons carry entropy at least in this single - particle sense . one could even consider letting @xmath88 and @xmath89 in ( [ palpha ] ) and ( [ entropy ] ) , in order to define hermitian entropy operators @xmath90 which would be ordinary quantum mechanical observables . one of the basic questions in quantum thermodynamics is whether quantum entropy should be an observable operator just like quantum energy , with state - independent eigenvalues that appear as measurements with state - dependent probabilities , or whether it should remain a fundamentally statistical quantity like the full von neumann entropy @xmath91 , which is a nonlinear function of the density operator @xmath92 that represents the quantum state of an open system . our simple model thus addresses this question by suggesting a form of entropy operator , whose performance in the role of thermodynamic entropy may be assessed experimentally .    the general implication of our work for mesoscopic quantum thermodynamics would seem to be that it is rather easy to identify first law phenomenology , whereby parametric oscillations among high - frequency internal degrees of freedom provide those extra contributions to the low - frequency energy budget which in thermodynamics are called heat . inasmuch as this heat oscillates in our case , rather than diffusing , we must recognize that second law phenomenology , with irreversible spontaneous processes , may be more elusive in small quantum systems . on the other hand , the full range of quantum dynamics in even the four - mode bose - hubbard model , in parameter ranges not considered here , is rich beyond the scope of this paper . it includes quantum chaos @xcite , whose role in mesoscopic irreversibility amply deserves further study . support from the deutsche forschungsgemeinschaft via the graduiertenkolleg `` nichtlineare optik und ultrakurzzeitphysik '' is gratefully acknowledged .
as a model for mesoscopic quantum systems in thermal contact , we consider a four - mode bose - hubbard model with two greatly differing tunneling rates . by a series of holstein - primakoff transformations we show that the low - frequency dynamics of this system consists in general of two slow josephson oscillations , rather than the single slow mode predicted by linear bogoliubov theory . 
 we identify the second slow josephson oscillation as a heat exchange mode analogous to second sound .