Quantum-classical correspondence principle for heat distribution in quantum Brownian motion

Quantum Brownian motion, described by the Caldeira-Leggett model, brings insights to understand phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum-classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat, and its long-time limit reflects complete thermalization of the system. Our research brings justification for the definition of the quantum fluctuating heat via two-point measurements.


I. INTRODUCTION
In the past few decades, the discovery of fluctuation theorems [1][2][3][4] and the establishment of the framework of stochastic thermodynamics [5][6][7] deepened our understanding about the fluctuating nature of thermodynamic quantities (such as work, heat and entropy production) in microscopic systems [8][9][10][11][12][13]. Among various fluctuation theorems, the nonequilibrium work relation [2] sharpens our understanding of the second law of thermodynamics by presenting an elegant and precise equality associating the free energy change with the fluctuating work. Such a relation was later extended to the quantum realm based on the two-point measurement definition of the quantum fluctuating work [14,15], soon after its discovery in the classical regime. The work statistics has been widely studied in various microscopic classical and quantum systems [16][17][18][19][20][21][22][23][24][25][26]. Historically, the quantum-classical correspondence principle played an essential role in the development of the theory of quantum mechanics and the interpretation of the transition from quantum to classical world [27,28].
In Refs. [19,22,24], it is demonstrated that the existence of the quantum-classical correspondence principle for work distribution brings justification for the definition of quantum fluctuating work via two-point measurements.
Compared to work statistics, heat statistics relevant to thermal transport associated with a nonequilibrium stationary state has been extensively studied [29][30][31][32][33][34][35][36][37][38], but the heat statistics in a finite-time quantum thermodynamic process [39][40][41] and its quantum-classical correspondence have been less explored. A challenge is that the precise description of the bath dynamics requires handling a huge number of degrees of freedom of the heat bath.
Different approaches have been proposed to calculate the quantum fluctuating heat and its statistics, such as the non-equilibrium Green's function approach to quantum thermal transport [29,32,36,[42][43][44] and the path-integral approach to quantum thermodynamics [45][46][47][48][49]. However, very few analytical results about the heat statistics have been obtained for the relaxation processes in open quantum systems. These analytical results are limited to either the relaxation dynamics described by the Lindblad master equation [39,40] or the long-time limit independent of the relaxation dynamics [50]. On the other hand, some results about the heat statistics in the classical Brownian motion model have been reported [51][52][53][54][55][56][57][58][59].
How the quantum and the classical heat statistics (especially associated with the relaxation dynamics in finite time) are related to each other has not been explored so far, probably due to the difficulty in studying the heat statistics in open quantum systems [60][61][62].
In this article, we study the heat statistics of a quantum Brownian motion model described by the Caldeira-Leggett Hamiltonian [48,[63][64][65][66][67][68], where the heat bath is modeled as a collection of harmonic oscillators. Although it is well known that the dynamics of such an open quantum system can approach that of the classical Brownian motion in the classical limit → 0 [64], less is known about the heat statistics of this model during the finitetime relaxation process. We here focus on the relaxation process without external driving (the Hamiltonian of the system is time-independent), and the quantum fluctuating heat can thus be defined as the difference of the system energy between the initial and the final measurements [69]. Under the Ohmic spectral density, the dynamics of the composite system is exactly solvable in the continuum limit of the bath oscillators [70]. By employing the phasespace formulation approach [71][72][73], we obtain analytical results of the characteristic function of heat for the Caldeira-Leggett model at any relaxation time τ with an arbitrary friction coefficient κ. Previously, such an approach was employed to study the quantum corrections to work [74][75][76] and entropy [77,78]. Analytical results of the heat statistics bring important insights to understand the fluctuating property of heat. By taking the classical limit → 0, the heat statistics of the Caldeira-Leggett model approaches that of the classical Brownian motion model. Thus, our results verify the quantum-classical correspondence principle for heat distribution, and provide justification for the definition of the quantum fluctuating heat via two-point measurements. We also verify from the analytical results that the heat statistics satisfies the exchange fluctuation theorem of heat [4].
The rest of this article is organized as follows. In Sec. II, we introduce the Caldeira-Leggett model and define the quantum fluctuating heat. In Sec. III, the analytical results of the characteristic function of heat are obtained by employing the phase-space formulation approach. We show the quantum-classical correspondence of the heat distribution, and discuss the heat distribution in the long-time limit or with the extremely weak or strong coupling strength. The conclusion is given in Sec. IV.

A. The Caldeira-Leggett model
The quantum Brownian motion is generally described by the Caldeira-Leggett model [64,65], where the system is modeled as a single particle moving in a specific potential, and the heat bath is a collection of harmonic oscillators. For simplicity, we choose the harmonic potential for the system [66,[79][80][81], where the dynamics of such an open quantum system can be solved analytically. The system will relax to the equilibrium state at the temperature of the heat bath. We study the heat distribution of such a quantum relaxation process, and obtain analytically the characteristic function of heat and its classical correspondence based on the phase-space formulation of quantum mechanics.
The total Hamiltonian of the composite system is H tot = H S + H B + H SB with each term where m 0 , ω 0 ,q 0 ,p 0 (m n , ω n ,q n ,p n with n = 1, 2, 3, ..., N ) are the mass, frequency, position and momentum of the system (the n-th bath harmonic oscillator), and C n is the coupling strength between the system and the n-th bath harmonic oscillator. The counter-term n [C 2 n /(2m n ω 2 n )]q 2 0 is included in the interaction Hamiltonian H SB to cancel the frequency shift of the system.
The spectral density is defined as J(ω) := n [C 2 n /(2m n ω n )]δ(ω − ω n ). We adopt an Ohmic spectral density with the Lorentz-Drude cutoff [67] where κ is the friction coefficient. A sufficiently large cutoff frequency Ω 0 (Ω 0 ω 0 ) is applied to ensure a finite counter-term, and the dynamics with the timescale exceeding 1/Ω 0 is Markovian. Under such a spectral density, the dissipation dynamics of the Caldeira-Leggett model with a weak coupling strength κ ω 0 reproduces that of the classical underdamped Brownian motion when taking the classical limit → 0 [64].
We assume the initial state to be a product state of the system and the heat bath which makes it possible to define the quantum fluctuating heat via two-point measurements.
Here, ρ S (0) is the initial state of the system, and ρ G B = exp(−βH B )/Z B (β) is the Gibbs distribution of the heat bath with the inverse temperature β and the partition function

B. The quantum fluctuating heat in the relaxation process
We study the heat distribution of the relaxation process based on the two-point measurement definition of the quantum fluctuating heat. When no external driving is applied to the system, the Hamiltonian of the system is time-independent. Since no work is performed during the relaxation process, the quantum fluctuating heat can be defined as where E S l (E S l ) is the eigenenergy of the system corresponding to the outcome l (l ) at the initial (final) time t = 0 (t = τ ). The two-point measurements over the heat bath can be hardly realized due to a huge number of degrees of freedom of the heat bath [20], while the measurements over the small quantum system are much easier in principle. The positive sign corresponds to the energy flowing from the heat bath to the system.
For the system prepared in an equilibrium state, no coherence exists in the initial state, and the initial density matrix of the system commutes with the Hamiltonian of the system, with the conditional transition probability γ τ,l l = Tr (P S l ⊗ I B )U tot (τ )(P S l ⊗ ρ G B )U † tot (τ ) and the initial probability p l = Tr[ρ(0)P S l ]. Here,P S l = |l l| is the projection operator corresponding to the outcome l. The heat distribution is defined as The characteristic function of heat χ τ (ν) is defined as the Fourier transform of the heat distribution χ τ (ν) := l ,l exp[iν(E S l − E S l )]p τ,l l , which can be rewritten explicitly as where U tot (τ ) = exp(−iH tot τ / ) is the unitary time-evolution operator of the composite system.
Our goal is to analytically calculate the characteristic function χ τ (ν). Previously, the quantum-classical correspondence principle for heat statistics has been analyzed with the path-integral approach to quantum thermodynamics [48], yet the explicit result of the characteristic function (or generating function) of heat has not been obtained so far. We employ the phase-space formulation approach to solve this problem, and rewrite the characteristic function Eq. (9) into where the system Hamiltonian in the Heisenberg picture is and the density-matrix-like operator η(0) is We express Eq. (10) with the phase-space formulation of quantum mechanics [71][72][73][74][75][76] where z represents a point z = [q, p] = [q 0 , ..., q N , p 0 , ..., p N ] in the phase space of the composite system, and the integral is performed over the whole phase space. The subscript "w" indicates the Weyl symbol of the corresponding operator, and P (z) is the Weyl symbol of the operator η(0), which is explicitly defined as [71] P (z) := dy q − y 2 In the following, we will calculate the heat statistics Eq. (13) by employing the phase-space formulation approach.

III. RESULTS OF THE CHARACTERISTIC FUNCTION OF HEAT
We show a sketch of the derivation of the heat statistics χ τ (ν) with the details left in Appendix A. We specifically consider the system is initially prepared at an equilibrium state with the inverse temperature β and the partition function The heat bath is at the inverse temperature β, which is different from β . In Eq. (13), the two Weyl symbols e iνH H and where the explicit expressions of the matricesΛ νz (τ ) and Λ βz are given in Eqs. (A10) and (A36), respectively.
Substituting Eqs. (15) and (16) into Eq. (13), the characteristic function of heat at any relaxation time τ with an arbitrary friction coefficient κ is finally obtained as where the quantities Ξ and Θ are Induced by the friction, the frequency of the system harmonic oscillator is shifted toω 0 = and the variance Var with Similarly, one can calculate the higher cumulants from the analytical results of the heat statistics. In the following, we will examine the properties of the heat statistics of the quantum Brownian motion.
A. Quantum-classical correspondence principle for heat statics and the exchange fluctuation theorem of heat We further take the classical limit → 0, or more rigorously β ω 0 → 0. The two quantities approach Ξ → ν/β and Θ → β /β, and the characteristic function of heat [Eq.
(17)] becomes which is consistent with the results obtained from the classical Brownian motion described by the Kramers equation (see Ref. [58] or Appendix C). The average heat is with From Eq. (17) [or the classical counterpart Eq. (25)], one can see the characteristic function of heat exhibits the following symmetry which shows the heat distribution satisfies the exchange fluctuation theorem of heat in the [4]. By setting ν = 0, we obtain the

B. Long-time limit
In the long-time limit τ → ∞, the characteristic functions of heat [Eqs. (17) and (25)] become and Such results, independent of the relaxation dynamics, are in the form reflecting complete thermalization of the system [53]. For example, the relaxation of a harmonic oscillator governed by the quantum-optical master equation gives the identical characteristic function of heat in the long-time limit [39]. In Appendix D, we demonstrate that the characteristic function of heat for any relaxation processes with complete thermalization is always in the form of Eq. (34). With the simple expressions (32) and (33) of the characteristic functions, the heat distributions are obtained from the inverse Fourier transform as and which are exactly the same as the long-time results obtained in Ref. [39].

C. Weak/Strong-coupling limit in finite time
In the weak-coupling limit κ ω 0 , the characteristic function of heat [Eq. (17)] becomes There is only one relaxation timescale associated to κ. Such situation corresponds to the highly underdamped regime of the classical Brownian motion, and a systematic method has been proposed to study the heat distribution [56] as well as the work distribution under an external driving [82,83]. In the strong coupling limit κ ω 0 , the characteristic function of heat [Eq. (17)] becomes The relaxation timescales of the momentum (the first factor) and the coordinate (the second factor) are separated. The long-time limits of both Eqs. (37) and (38) are equal to Eq. (32).
In classical thermodynamics, the usual overdamped approximation neglects the motion of the momentum, hence the heat statistics derived under such an approximation is incomplete [52]. Actually, the momentum degree of freedom also contributes to the heat statistics.   Var Q cl (τ ) = 1/β 2 + 1/β 2 (gray horizontal lines). For κ = 100 (right subfigures), only the momentum degree of freedom is thermalized at this timescale. Thus, the mean value and the variance take half value of their long-time limits. When the coordinate degree of freedom is also thermalized in the long-time limit (τ κ 2 /ω 2 0 = 10 4 ), the mean value and the variance are expected to approach the same values as those in the middle subfigures.

IV. CONCLUSION
Previously, the heat statistics of the relaxation processes has been studied analytically in open quantum systems described by the Lindblad master equation [39,40,50]. However, due to the rotating wave approximation and other approximations. Such quantum systems do not possess a well-defined classical counterpart. Hence, the quantum-classical correspondence principle for heat distribution has not been well established.
In this paper, we study the heat statistics of the quantum Brownian motion model de- We have also discussed the characteristic function of heat in the long-time limit or with the extremely weak/strong coupling strength. In the long-time limit, the form of the characteristic function of heat reflects complete thermalization of the system. In addition, from the analytical expressions of the heat statistics, we can immediately verify the exchange fluctuation theorem of heat. The phase-space formulation can be further utilized to study the joint statistics of work and heat in a driven open quantum system, which will be beneficial to explore the fluctuations of power and efficiency in finite-time quantum heat engines.  17) is obtained from Eq. (13).
1. (z) is [78,81] e iνH H where z(t) gives the trajectory in the phase space determined by the initial point z(0) = z, and Λ νz is a rank-2 diagonal matrix p n = −m n ω 2 n q n + C n q 0 , and the matrix exponential is formally written as We rewrite the quadratic form into z We next carry out every element in Eq. (A9) through the Laplace transforms of Eqs.
Representingq n (s) andp n (s) withq 0 (s) and the initial conditions, we obtain Under the Ohmic spectral density [Eq. (4)], the above equation is simplified to where the summation on the left-hand side of Eq. (A15) can be approximately expressed as with a large cutoff frequency Ω 0 . The inverse Laplace transform gives the differential equation of q 0 (t) as q 0 (t)+κq 0 (t)+ω 2 0 q 0 (t) = −κq 0 (0)δ(t) initial velocity change + n C n m 0 q n (0) sin(ω n t) ω n + q n (0) cos(ω n t) On the right-hand side, the second term presents the stochastic force induced by the heat bath; the first term indicates an abrupt velocity change −κq 0 (0) of the system particle at the initial time t = 0 [63,84,85]. The sudden change of velocity occurs for the system harmonic oscillator when the coupling between the system and the heat bath is switched on. Such an initial slippage is caused by the assumption of the initial product state. To avoid such an initial discontinuous problem, we drop the first term by considering the particle motion as starting at t = 0+ [70]. Under such a modification, the Caldeira-Leggett model can reproduce the complete Langevin equation with an arbitrary friction coefficient κ for both the underdamped and the overdamped regimes, and the heat distribution of the Caldeira-Leggett model approaches that of the classical Brownian motion described by the Kramers equation [86]. In Appendix B, for the classical counterpart of the Caldeira-Leggett model, we show the initial slippage can be naturally eliminated by choosing another initial state.

Calculation of the integral
With the explicit expressions of e iνH H S (τ ) w (z) and P (z), we perform the integral in Eq. (13), and obtain the result of the characteristic function of heat .

(A41)
We have used the following integral formula where all the eigenvalues of T have positive real parts.
By introducing a diagonal matrix A = Λ βz − iΛ νz , we rewrite Eq. (A41) as SinceΛ νz (τ ) is a rank-2 matrix, we rewrite it as with the vectors v q 0 (τ ) = α 0 , α 1 , ..., α N , v p 0 (τ ) = m 0α0 , m 0α1 , ..., m 0αN ,β 0 , m 0 m 1β 1 , ..., Here, the evolution time t in the terms α n and β n is set to τ . We rewrite the matrix in the determinant [see Eq. (A43)] as with the matrix The determinant in Eq. (A43) can be simplified to The explicit result of M T M is obtained as where the elements are functions of the final time τ , and the functions h 11 (τ ), h 12 (τ ) and with The summations are replaced by the integral with the Ohmic spectral density, and every element in Eq. (A52) is carried out as Then, Eq. (17) is obtained by directly calculating the determinant of a 4 × 4 matrix in Eq. (A49).

Appendix B: Classical Caldeira-Leggett model
We consider the classical Caldeira-Leggett model, where coordinates and momenta commute with each other. To eliminate the initial slippage, the initial state is amended as a coupled state which represents the probability density in the phase space of the composite system. The classical partition function is obtained by performing the integral in the phase space which is independent of the interaction (notice that the partition function of the quantum model relies on the interaction strength [68,87]).
We also define the classical fluctuating heat as the energy difference of the initial and the final system energy. For classical dynamics, the initial and the final states are directly represented by the points in the phase space, and the measurements over the system can be applied without disturbing the composite system. Therefore, the characteristic function of heat is where the energy of the system H S (t) = [p 0 (t)] 2 /(2m 0 ) + m 0 ω 2 0 [q 0 (t)] 2 /2 is determined by p 0 (t) and q 0 (t) associated with the initial point z. We choose a new set of initial variables q 0 , q n := q n − C n q 0 /(m n ω 2 n ), p 0 and p n in the following calculation. We rewrite the evolution of the coordinate q 0 (t) of the system [Eq. (A16)] as with the matrices [88] andΛ The initial vector is now amended to z(0) = (q 0 (0), q 1 , ..., q n , p 0 (0), ..., p N (0)) T .
The initial Hamiltonians H S (0) and H B (0) + H SB (0) are with the matrix According to the integral formula (A42), we carry out the characteristic function Eq.
Appendix C: The characteristic function of heat for the classical Brownian motion We derive the characteristic function of heat for the classical Brownian motion. For an underdamped Brownian particle moving in a potential V (x), the stochastic dynamics is described by the complete Langevin equation The fluctuating force is a Gaussian white noise satisfying the fluctuation-dissipation relation The evolution of the system state is characterized by the probability density function ρ(x, p; t) in the phase space. The stochastic dynamics is then described by the Kramers with the Liouville operator Similarly in the phase space, we calculate the characteristic function of heat for the classical Brownian motion where a probability-density-like function η(x, p; t) also satisfies the dynamic equation (C3) with the initial condition We consider the system potential as a harmonic potential V (x) = mω 2 0 x 2 /2 and the initial system state as an equilibrium state with the inverse temperature β and the classical partition function Z cl S (β ) = 2π/(β ω 0 ). Under such conditions, the probability-density-like function η(x, p; t) is always in a quadratic form, assumed as The Kramers equation (C3) for η(x, p; t) leads to the following ordinary differential equationṡ with the initial conditions a(0) = b(0) = β + iν, c(0) = 0 and Λ(0) = 0. According to the conservation of the probability η(x, p; t)dxdp = const, the coefficient Λ(t) is obtained as Substituting Eq. (C8) into Eq. (C5), we obtain the characteristic function for the classical Brownian motion as To solve the nonlinear differential equations (C10)-(C12), we introduce a new set of variables A = a ab − c 2 , (C15) and obtain the linear differential equations with the initial conditions A(0) = B(0) = 1/(β + iν) and C(0) = 0. The characteristic function Eq. (C14) becomes .