Quantum corrections to the entropy in a driven quantum Brownian motion model
本站小编 Free考研考试/2022-01-02
Tian Qiu1, Hai-Tao Quan,1,2,3,∗1School of Physics, Peking University, Beijing 100871, China 2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 3Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing, 100871, China
First author contact:∗Author to whom any correspondence should be addressed. Received:2021-04-29Revised:2021-06-3Accepted:2021-06-4Online:2021-07-16
Abstract The quantum Brownian motion model is a typical model in the study of nonequilibrium quantum thermodynamics. Entropy is one of the most fundamental physical concepts in thermodynamics. In this work, by solving the quantum Langevin equation, we study the von Neumann entropy of a particle undergoing quantum Brownian motion. We obtain the analytical expression of the time evolution of the Wigner function in terms of the initial Wigner function. The result is applied to the thermodynamic equilibrium initial state, which reproduces its classical counterpart in the high temperature limit. Based on these results, for those initial states having well-defined classical counterparts, we obtain the explicit expression of the quantum corrections to the entropy in the weak coupling limit. Moreover, we find that for the thermodynamic equilibrium initial state, all terms odd in ℏare exactly zero. Our results bring important insights to the understanding of entropy in open quantum systems. Keywords：quantum Brownian motion;time-dependent evolution;quantum entropy
PDF (367KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Tian Qiu, Hai-Tao Quan. Quantum corrections to the entropy in a driven quantum Brownian motion model. Communications in Theoretical Physics, 2021, 73(9): 095602- doi:10.1088/1572-9494/ac0813
1. Introduction
Quantum thermodynamics [1–5] is an emerging field studying the nonequilibrium statistical mechanics of the quantum dissipative systems [6–9]. Quantum work [10–14], quantum heat [15–20], and quantum entropy production [21–25] are among the most basic concepts, which play important roles in the study of work extraction and heat transfer in quantum devices, such as the quantum heat engines and refrigerators [26–41].
A typical exactly solvable model used for addressing these problems is the quantum Brownian motion model proposed by Caldeira and Leggett [9, 42]. It consists of a system described by the Hamiltonian ${\hat{H}}_{S}$ (often a harmonic oscillator [43, 44]), a heat bath of harmonic oscillators with the Hamiltonian ${\hat{H}}_{B}$, and the interaction Hamiltonian ${\hat{H}}_{\mathrm{SB}}$. One can analytically integrate out the degrees of freedom of the heat bath, which brings important insights to the understanding of the thermodynamics of open quantum systems. In the studies about entropy production and heat transfer, previous efforts have been focused mainly on the entropy production in the heat bath. For example, in [21], by adapting the Feynman–Vernon influence functional formalism, the change of the von Neumann entropy of the heat bath is computed. In [17, 19, 20], the energy exchange, and thus the entropy exchange, between the system and the heat bath is calculated. However, the entropy production in the system of interest (the particle undergoing quantum Brownian motion, especially when the Hamiltonian of the system is time-dependent) has largely been unexplored so far (but see [23]).
Usually it is a very challenging task to calculate analytically the time evolution of the reduced density matrix, and hence the time evolution of the von Neumann entropy of the quantum Brownian motion model, especially when the Hamiltonian of the system is time-dependent. In this article, we study the time evolution of the quantum entropy and its corrections to the classical entropy in an open quantum system. Specifically, we calculate the von Neumann entropy in the quantum Brownian motion model subject to a driving force. The von Neumann entropy of a quantum system described by the density matrix $\hat{\rho }$ is given by [45, 46]$\begin{eqnarray}{S}_{q}=-\mathrm{Tr}\left[\hat{\rho }\mathrm{ln}\hat{\rho }\right].\end{eqnarray}$Here, we have set the Boltzmann’s constant to be 1, and thus the entropy becomes dimensionless. It is difficult to calculate the von Neumann entropy through its definition equation (1), because one has to diagonalize an infinite-dimensional matrix in order to compute the trace of a function. But by reformulating the problem in the phase space the calculation can be significantly simplified [47]. (In [47], we discuss the quantum corrections to the Gibbs entropy for specific states, and it has nothing to do with the time evolution of the reduced density matrix of a quantum Brownian motion model.) By solving the quantum Langevin equation exactly, we obtain the analytical expression of the Wigner function at an arbitrary time t in terms of the initial Wigner function [48]. Then by adapting the method in [47], for initial states which have well-defined classical counterparts, we find that if we expand the von Neumann entropy in powers of ℏ, the zeroth-order term reproduces the classical Gibbs entropy. We can also obtain the quantum correction to the entropy and its time evolution in the weak coupling limit. Our results bring important insights to the understanding of the entropy in an open quantum system.
This article is organized as follows. We begin in section 2 with a description of the model and the quantum Langevin equation. In section 3 we derive the general solution to the quantum Langevin equation of the driven quantum Brownian motion model. In section 4 we obtain the expression of the reduced Wigner function of a driven quantum Brownian motion model. In section 5, by using the method developed in [47], we calculate the quantum corrections to the entropy and its dynamical evolution. Finally, in section 6 we make some remarks and summarize our results.
2. The model and the derivation of the quantum Langevin equation
We consider the quantum Brownian motion described by the Caldeira–Leggett model [42, 43]. The system that we consider is a harmonic oscillator subject to a time-dependent driving force $\hat{f}(t)$ [44]. The system is linearly coupled to a heat bath consisting of a set of harmonic oscillators. The Hamiltonian of the composite system is given by ${\hat{H}}_{\mathrm{tot}}(t)={\hat{H}}_{S}(t)+{\hat{H}}_{B}+{\hat{H}}_{\mathrm{SB}}$, with$\begin{eqnarray}{\hat{H}}_{S}(t)=\displaystyle \frac{{\hat{{p}}}^{2}(t)}{2{m}_{0}}+\displaystyle \frac{1}{2}{m}_{0}{\omega }_{0}^{2}{\left(\hat{{q}}(t)-\displaystyle \frac{\hat{f}(t)}{{m}_{0}{\omega }_{0}^{2}}\right)}^{2},\end{eqnarray}$$\begin{eqnarray}{\hat{H}}_{B}=\sum _{j=1}^{N}\left(\displaystyle \frac{{\hat{p}}_{j}^{2}(t)}{2{m}_{j}}+\displaystyle \frac{1}{2}{m}_{j}{\omega }_{j}^{2}{\hat{q}}_{j}^{2}(t)\right),\end{eqnarray}$$\begin{eqnarray}{\hat{H}}_{\mathrm{SB}}=-\hat{{q}}(t)\sum _{j=1}^{N}{C}_{j}{\hat{q}}_{j}(t)+\sum _{j=1}^{N}\displaystyle \frac{{C}_{j}^{2}}{2{m}_{j}{\omega }_{j}^{2}}{\hat{{q}}}^{2}(t),\end{eqnarray}$where m0, ω0, $\hat{{q}}$, $\hat{{p}}$ and mj, ωj, ${\hat{q}}_{j}$, ${\hat{p}}_{j}$ are the mass, angular frequencies, coordinates and momenta of the system and the jth harmonic oscillator of the heat bath, respectively, and Cj (j=1, 2, 3, …) are the coupling constants. Here, we have included the counterterm ${\sum }_{j}({C}_{j}^{2}/2{m}_{j}{\omega }_{j}^{2}){\hat{{q}}}^{2}(t)$ in the interaction Hamiltonian to cancel the negative frequency shift of the potential [49].
The equation of motion of the time-dependent operator can be obtained by using the Heisenberg equation$\begin{eqnarray}{\rm{i}}{\hslash }\dot{\hat{O}}=[\hat{O},\hat{H}],\end{eqnarray}$which gives the time derivative (denoted by the superposed dot) of an arbitrary operator $\hat{O}$. Then we have [50]$\begin{eqnarray}\left\{\begin{array}{c}\dot{\hat{{q}}}=\displaystyle \frac{1}{{m}_{0}}\hat{{p}},\\ \dot{\hat{{p}}}=-{m}_{0}{\omega }_{0}^{2}\hat{{q}}+\hat{f}(t)+\sum _{j=1}^{N}{C}_{n}\left({\hat{q}}_{j}-\displaystyle \frac{{C}_{j}}{{m}_{j}{\omega }_{j}^{2}}\hat{{q}}\right),\end{array}\right.\end{eqnarray}$for the system, and$\begin{eqnarray}\left\{\begin{array}{c}{\dot{\hat{q}}}_{j}=\displaystyle \frac{1}{{m}_{j}}{\hat{p}}_{j},\\ {\dot{\hat{p}}}_{j}=-{m}_{j}{\omega }_{j}^{2}\left({\hat{q}}_{j}-\displaystyle \frac{{C}_{j}}{{m}_{j}{\omega }_{j}^{2}}\hat{{q}}\right),\end{array}\right.\end{eqnarray}$for the nth harmonic oscillator of the heat bath. Solving equation (5) and substituting it into equation (4), one can obtain the quantum Langevin equation of the driven Brownian particle [50]$\begin{eqnarray}\begin{array}{l}{m}_{0}\ddot{\hat{{q}}}(t)+{\int }_{0}^{t}{\rm{d}}{t}^{{\prime} }\mu (t-{t}^{{\prime} })\dot{\hat{{q}}}({t}^{{\prime} })+{m}_{0}{\omega }_{0}^{2}\hat{{q}}(t)+\mu (t)\hat{{q}}(0)\\ \ \ =\ \hat{F}(t)+\hat{f}(t),\end{array}\end{eqnarray}$where μ(t) is the memory function and it is given by$\begin{eqnarray}\mu (t)=\sum _{j}\displaystyle \frac{{C}_{j}^{2}}{{m}_{j}{\omega }_{j}^{2}}\cos ({\omega }_{j}t),\end{eqnarray}$and $\hat{F}(t)$ is the fluctuating force operator and it can be expressed in terms of the initial bath variables$\begin{eqnarray}\hat{F}(t)=\sum _{j}{C}_{j}\left[{\hat{q}}_{j}(0)\cos ({\omega }_{j}t)+{\hat{p}}_{j}(0)\displaystyle \frac{\sin ({\omega }_{j}t)}{{m}_{j}{\omega }_{j}}\right].\end{eqnarray}$It is straightforward to show that the correlation and the commutator can be expressed as [48, 50]$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2}\langle \hat{F}(t)\hat{F}({t}^{{\prime} })+\hat{F}({t}^{{\prime} })\hat{F}(t)\rangle =\displaystyle \frac{1}{\pi }{\displaystyle \int }_{0}^{\infty }{\rm{d}}\omega \\ \ \times \ \mathrm{Re}\{\tilde{\mu }(\omega +i{0}^{+})\}{\hslash }\omega \coth \left(\displaystyle \frac{{\hslash }\omega }{2{kT}}\right)\cos \left[\omega (t-{t}^{{\prime} })\right],\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}\left[\hat{F}(t),\hat{F}({t}^{{\prime} })\right] & = & \displaystyle \frac{2{\hslash }}{{\rm{i}}\pi }{\int }_{0}^{\infty }{\rm{d}}\omega \\ & & \times \mathrm{Re}\{\tilde{\mu }(\omega +{\rm{i}}{0}^{+})\}\omega \sin [\omega ({\text{}}t-{t}^{{\prime} })],\end{array}\end{eqnarray}$where the bracket $\langle ...\rangle $ depicts the quantum expectation value, and $\tilde{\mu }(z)$ is the Fourier transform of the memory function:$\begin{eqnarray}\tilde{\mu }(z)={\int }_{0}^{\infty }{\rm{d}}t\ \mu (t){{\rm{e}}}^{{\rm{i}}{zt}},\ \mathrm{Im}\ z\gt 0.\end{eqnarray}$In the following, we will try to solve equation (6) by using the Green function approach.
3. General solution to the quantum Langevin equation (6)
For the stationary process, the system is held fixed at the origin in the distant past [50]. From equation (6), we obtain the quantum Langevin equation for the stationary process$\begin{eqnarray}\begin{array}{l}{m}_{0}{\ddot{\hat{{q}}}}^{(s)}(t)+{\int }_{-\infty }^{t}{\rm{d}}{t}^{{\prime} }\mu (t-{t}^{{\prime} }){\dot{\hat{{q}}}}^{(s)}({t}^{{\prime} })+{m}_{0}{\omega }_{0}^{2}{\hat{{q}}}^{(s)}(t)\\ \ \ =\ \hat{F}(t)+\hat{f}(t),\end{array}\end{eqnarray}$where $\hat{f}(t)$ is switched on at t=0, and the solution can be written as [48]$\begin{eqnarray}{\hat{{q}}}^{(s)}(t)={\hat{{q}}}^{(F)}(t)+{\hat{{q}}}^{(f)}(t),\end{eqnarray}$where$\begin{eqnarray}{\hat{{q}}}^{(F)}(t)={\int }_{-\infty }^{t}{\rm{d}}{t}^{{\prime} }\ G(t-{t}^{{\prime} })\hat{F}({t}^{{\prime} }),\end{eqnarray}$$\begin{eqnarray}{\hat{{q}}}^{(f)}(t)={\int }_{-\infty }^{t}{\rm{d}}{t}^{{\prime} }\ G(t-{t}^{{\prime} })\hat{f}({t}^{{\prime} }),\end{eqnarray}$and the Green function G(t) is given by$\begin{eqnarray}G(t)=\displaystyle \frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\rm{d}}\omega \,\alpha (\omega +i{0}^{+})\,{{\rm{e}}}^{-{\rm{i}}\omega t},\end{eqnarray}$with the response function$\begin{eqnarray}\alpha (z)=\displaystyle \frac{1}{-{m}_{0}{z}^{2}-{\rm{i}}z\tilde{\mu }(z)+{m}_{0}{\omega }_{0}^{2}}.\end{eqnarray}$From equations (9) and (14a), we obtain the correlation$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2}\left\langle {\hat{{q}}}^{(F)}(t){\hat{{q}}}^{(F)}({t}^{{\prime} })+{\hat{{q}}}^{(F)}({t}^{{\prime} }){\hat{{q}}}^{(F)}(t)\right\rangle \\ \ =\ \displaystyle \frac{{\hslash }}{\pi }{\displaystyle \int }_{0}^{\infty }{\rm{d}}\omega \ \mathrm{Im}\{\alpha (\omega +{\rm{i}}{0}^{+})\}\coth \left(\displaystyle \frac{{\hslash }\omega }{2{kT}}\right)\cos [\omega (t-{t}^{{\prime} })].\end{array}\end{eqnarray}$
One can easily find that for negative times the Green function (15) vanishes, and for positive times equation (15) is a solution to the following equation$\begin{eqnarray}\begin{array}{l}{m}_{0}\ddot{G}(t)+{\int }_{0}^{t}{\rm{d}}{t}^{{\prime} }\mu (t-{t}^{{\prime} })\dot{G}({t}^{{\prime} })+{m}_{0}{\omega }_{0}^{2}G(t)\\ \ \ +\ \mu (t)G(0)=0,\end{array}\end{eqnarray}$with the initial conditions G(0)=0 and $\dot{G}(0)=1/{m}_{0}$. Then we obtain the general solutions to the quantum Langevin equation (6) [48]$\begin{eqnarray}\hat{{q}}(t)={m}_{0}\dot{G}(t)\hat{{q}}(0)+G(t)\hat{{p}}(0)+\hat{X}(t)+\hat{Y}(t),\end{eqnarray}$$\begin{eqnarray}\dot{\hat{{q}}}(t)={m}_{0}\ddot{G}(t)\hat{{q}}(0)+\dot{G}(t)\hat{{p}}(0)+\dot{\hat{X}}(t)+\dot{\hat{Y}}(t),\end{eqnarray}$where $\hat{X}(t)$ is the position operator relevant to the fluctuating force $\hat{F}(t)$ [48]$\begin{eqnarray}\hat{X}(t)={\int }_{0}^{t}{\rm{d}}{t}^{{\prime} }\ G(t-{t}^{{\prime} })\hat{F}({t}^{{\prime} }),\end{eqnarray}$and $\hat{Y}(t)$ is the position operator relevant to the driven force $\hat{f}(t)$$\begin{eqnarray}\hat{Y}(t)={\int }_{0}^{t}{\rm{d}}{t}^{{\prime} }\ G(t-{t}^{{\prime} })\hat{f}({t}^{{\prime} }).\end{eqnarray}$In the following, based on equations (19a) and (19b), we try to obtain the analytical expression of the reduced Wigner function of the system.
4. Analytical expression of the reduced Wigner function
In this section, we calculate the reduced Wigner function of the system. By tracing out the degrees of freedom of the heat bath, we get the time-dependent reduced Wigner function of the system [51]$\begin{eqnarray}W({q},{p};t)=\int {\rm{d}}{\boldsymbol{q}}\int {\rm{d}}{\boldsymbol{p}}\ {W}_{\mathrm{tot}}({q},{p};{\boldsymbol{q}},{\boldsymbol{p}};t).\end{eqnarray}$Here Wtot is the Wigner function of the composite system, with ${\boldsymbol{q}}=({q}_{1},{q}_{2},\ldots ,{q}_{N})$ and ${\boldsymbol{p}}=({p}_{1},{p}_{2},\ldots ,{p}_{N})$ the coordinates and the momenta of the heat bath, respectively. Because the evolution of the composite system satisfies the Liouville–von Neumann equation, we have [48, 51]$\begin{eqnarray}{W}_{\mathrm{tot}}({q},{p};{\boldsymbol{q}},{\boldsymbol{p}};t)={W}_{\mathrm{tot}}({q}(0),{p}(0);{\boldsymbol{q}}(0),{\boldsymbol{p}}(0);0),\end{eqnarray}$where ${q}(0)$, ${p}(0)$, ${\boldsymbol{q}}(0)$, ${\boldsymbol{p}}(0)$ are the initial values of the coordinates and the momenta, and ${q}={q}(t)$, ${p}={p}(t)$, ${\boldsymbol{q}}={\boldsymbol{q}}(t)$, ${\boldsymbol{p}}={\boldsymbol{p}}(t)$ are the solutions of the equation of motion (4) and (5). We assume that the initial state of the composite system is in a factorized form, i.e. a direct product of the density operator of the system and that of the heat bath, and the heat bath is in a thermal equilibrium state at the inverse temperature β. Then we obtain [48, 51]$\begin{eqnarray}\begin{array}{rcl}{W}_{\mathrm{tot}}\left({q}(0),{p}(0);{\boldsymbol{q}}(0),{\boldsymbol{p}}(0);0\right) & = & W\left({q}(0),{p}(0);0\right)\\ & & \times \ \prod _{j=1}^{N}{w}_{j}({q}_{j}(0),{p}_{j}(0)),\end{array}\end{eqnarray}$where wj(qj(0), pj(0)) is the Wigner function of the jth oscillator of the heat bath with the mass mj and the frequency ωj,$\begin{eqnarray}\begin{array}{rcl}{w}_{j}({q}_{j}(0),{p}_{j}(0)) & = & 2\tanh \displaystyle \frac{\beta {\hslash }{\omega }_{j}}{2}\\ & & \times \exp \left[-\displaystyle \frac{{p}_{j}^{2}(0)+{m}_{j}^{2}{\omega }_{j}^{2}{q}_{j}^{2}(0)}{{m}_{j}{\hslash }{\omega }_{j}\coth (\beta {\hslash }{\omega }_{j}/2)}\right].\end{array}\end{eqnarray}$Substituting equations (23)–(25) into equation (22), one obtains$\begin{eqnarray}\begin{array}{rcl}W({q},{p};t) & = & \int {\rm{d}}{\boldsymbol{q}}(t)\int {\rm{d}}{\boldsymbol{p}}(t){\text{}}W({q}(0),{p}(0);0)\\ & & \times \prod _{j=1}^{N}{w}_{j}\left({q}_{j}(0),{p}_{j}(0)\right).\end{array}\end{eqnarray}$Using the general solutions of the quantum Langevin equation (19), we transform the integration variables from the final coordinates of the heat bath $({\boldsymbol{q}}(t),{\boldsymbol{p}}(t))$ to the initial coordinates of the heat bath $({\boldsymbol{q}}(0),{\boldsymbol{p}}(0))$, while holding ${q}(0)$ and ${p}(0)$ fixed [48]$\begin{eqnarray}{\rm{d}}{\boldsymbol{q}}(t){\rm{d}}{\boldsymbol{p}}(t)=\displaystyle \frac{1}{{m}_{0}^{2}\left({\dot{G}}^{2}-G\ddot{G}\right)}{\rm{d}}{\boldsymbol{q}}(0){\rm{d}}{\boldsymbol{p}}(0).\end{eqnarray}$Substituting equation (27) into equation (26), we obtain$\begin{eqnarray}W({q},{p};t)=\displaystyle \frac{\langle W({q}(0),{p}(0);0)\rangle }{{m}_{0}^{2}\left({\dot{G}}^{2}-G\ddot{G}\right)},\end{eqnarray}$where the bracket represents the average over the initial equilibrium distribution of the heat bath, and ${q}(0)$ and ${p}(0)$ in the integrand can be obtained by inverting equations (19a) and (19b)$\begin{eqnarray}{q}(0)=\displaystyle \frac{{m}_{0}\dot{G}({q}-X-Y)-G\left({p}-{m}_{0}\dot{X}-{m}_{0}\dot{Y}\right)}{{m}_{0}^{2}\left({\dot{G}}^{2}-G\ddot{G}\right)},\end{eqnarray}$$\begin{eqnarray}{p}(0)=\displaystyle \frac{-{m}_{0}^{2}\ddot{G}({q}-X-Y)+{m}_{0}\dot{G}\left({p}-{m}_{0}\dot{X}-{m}_{0}\dot{Y}\right)}{{m}_{0}^{2}\left({\dot{G}}^{2}-G\ddot{G}\right)}.\end{eqnarray}$We would like to emphasize that in obtaining equations (29a) and (29b), we have performed the Weyl–Wigner transform [1] over equations (19a) and (19b). That is, the operators $\hat{{p}}$, $\hat{{q}}$, $\hat{{p}}(0)$, $\hat{{q}}(0)$, $\hat{X}$, $\hat{Y}$ in equations (19a) and (19b) have been replaced by variables ${p}$, ${q}$, ${p}(0)$, ${q}(0)$, X, Y. We can calculate the average in equation (28) by taking the Fourier transform of the initial reduced Wigner function [48]$\begin{eqnarray}\begin{array}{rcl}W({q}(0),{p}(0);0) & = & \displaystyle \frac{1}{{\left(2\pi {\hslash }\right)}^{2}}\int {\rm{d}}Q\\ & & \times \int {\rm{d}}P\tilde{W}(Q,P;0){{\rm{e}}}^{\tfrac{{\rm{i}}}{{\hslash }}(P{q}+Q{p})}.\end{array}\end{eqnarray}$Inserting this into equation (28), after some simplifications, one can obtain [48]$\begin{eqnarray}\begin{array}{l}W\left({q},{p};t\right)=\displaystyle \frac{1}{{\left(2\pi {\hslash }\right)}^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}r\\ \ \times \ {\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}s\tilde{W}\left({m}_{0}\dot{G}r+{Gs},{m}_{0}^{2}\ddot{G}r+\dot{G}s;0\right)\\ \ \times \ {{\rm{e}}}^{\tfrac{{\rm{i}}}{{\hslash }}\left(r{p}+s{q}-{m}_{0}\dot{Y}r-{Ys}\right)}\\ \ \times \ {{\rm{e}}}^{-\tfrac{1}{2{{\hslash }}^{2}}\left({m}_{0}^{2}\langle {\dot{X}}^{2}\rangle {r}^{2}+{m}_{0}\langle X\dot{X}+\dot{X}X\rangle {rs}+\langle {X}^{2}\rangle {s}^{2}\right)}.\end{array}\end{eqnarray}$Here, we have transformed the integration variables Q and P into r and s by$\begin{eqnarray}Q={m}_{0}\dot{G}r+{Gs},P={m}_{0}^{2}\ddot{G}r+\dot{G}s,\end{eqnarray}$and we, due to the Gaussian property of X(t), have used [48]$\begin{eqnarray}\langle {{\rm{e}}}^{-\tfrac{{\rm{i}}}{{\hslash }}({m}_{0}\dot{X}r+{Xs})}\rangle ={{\rm{e}}}^{-\tfrac{1}{2{{\hslash }}^{2}}\left({m}_{0}^{2}\langle {\dot{X}}^{2}\rangle {r}^{2}+{m}_{0}\langle X\dot{X}+\dot{X}X\rangle {rs}+\langle {X}^{2}\rangle {s}^{2}\right)}.\end{eqnarray}$We would like to emphasize that equation (31) is the one of the main results of our paper, i.e. the analytical expression of the time evolution of the reduced Wigner function for a quantum Brownian particle in a driven harmonic potential. In this evaluation, the Green function G(t) is given by equation (15), X(t) is given by equation (20) and its correlations are evaluated using equation (9), and Y(t) is given by equation (21).
Furthermore, one can substitute the inverse of the Fourier transform (30) into equation (31) and rewrite the Wigner function in the form of a propagator acting on the initial reduced Wigner function [48]$\begin{eqnarray}\begin{array}{rcl}W({q},{p};t) & = & {\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}{{q}}^{{\prime} }(0){\rm{d}}{{p}}^{{\prime} }(0)}{2\pi {\hslash }}\\ & & \times P\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)W({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0).\end{array}\end{eqnarray}$Here the propagator $P({q},{q};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t)$ can be written as [48]$\begin{eqnarray}P\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)=\displaystyle \frac{{\hslash }}{\sqrt{| {\boldsymbol{A}}(t)| }}{{\rm{e}}}^{-\tfrac{1}{2}{{\boldsymbol{R}}}^{{\rm{T}}}(t){{\boldsymbol{A}}}^{-1}(t){\boldsymbol{R}}(t)},\end{eqnarray}$where $| {\boldsymbol{A}}(t)| $ denotes the determinant of ${\boldsymbol{A}}(t)$, and$\begin{eqnarray}{\boldsymbol{A}}(t)=\left(\begin{array}{cc}{m}_{0}^{2}\langle {\dot{X}}^{2}\rangle & \displaystyle \frac{{m}_{0}}{2}\langle X\dot{X}+\dot{X}X\rangle \\ \displaystyle \frac{{m}_{0}}{2}\langle X\dot{X}+\dot{X}X\rangle & \langle {X}^{2}\rangle \end{array}\right),\end{eqnarray}$$\begin{eqnarray}{\boldsymbol{R}}(t)=\left(\begin{array}{c}{p}-\overline{{p}(t)}\\ {q}-\overline{{q}(t)}\end{array}\right).\end{eqnarray}$Here the overline depicts the average over X(t) (20), and the operators $\overline{{q}(t)}$ and $\overline{{p}(t)}$ correspond to the solutions to equations (19a) and (19b) with initial values ${{q}}^{{\prime} }(0)$ and ${{p}}^{{\prime} }(0)$$\begin{eqnarray}\overline{{q}(t)}={m}_{0}\dot{G}(t)\ {{q}}^{{\prime} }(0)+G(t)\ {{p}}^{{\prime} }(0)+Y(t),\end{eqnarray}$$\begin{eqnarray}\overline{{p}(t)}={m}_{0}\ddot{G}(t)\ {{q}}^{{\prime} }(0)+\dot{G}(t)\ {{p}}^{{\prime} }(0)+{m}_{0}\dot{Y}(t).\end{eqnarray}$
Now we calculate the propagator (35) explicitly. We consider the case of an Ohmic heat bath, the memory function (7) has the form$\begin{eqnarray}\mu (t)=2{\gamma }_{0}\delta (t),\end{eqnarray}$where γ0 is the Newtonian friction constant. Please note that for the Ohmic heat bath, the elements of the matrix ${\boldsymbol{A}}(t)$ will diverge in all cases but for the weak coupling of the system and the heat bath or the classical limit [52]. Therefore, from now on, we restrict our discussions to the weak coupling regime. Substituting equation (39) into equation (16), one can obtain$\begin{eqnarray}\alpha (z)=\displaystyle \frac{1}{-{m}_{0}{z}^{2}-{\rm{i}}z{\gamma }_{0}+{m}_{0}{\omega }_{0}^{2}}.\end{eqnarray}$Substituting equation (40) into equation (15), after the integration, one can obtain the expression of the Green function. It turns out that in the low damping regime (γ0/(m0ω0)<2),$\begin{eqnarray}G(t)=\displaystyle \frac{1}{2{\rm{i}}{m}_{0}{{\rm{\Omega }}}_{0}}({{\rm{e}}}^{-{\lambda }_{2}t}-{{\rm{e}}}^{-{\lambda }_{1}t}),{\lambda }_{\mathrm{1,2}}=\displaystyle \frac{{\gamma }_{0}}{2{m}_{0}}\pm {\rm{i}}{{\rm{\Omega }}}_{0},\end{eqnarray}$where$\begin{eqnarray}{{\rm{\Omega }}}_{0}=\sqrt{\left|\displaystyle \frac{{\gamma }_{0}^{2}}{4{m}_{0}^{2}}-{\omega }_{0}^{2}\right|}.\end{eqnarray}$It is straightforward to prove that in the low damping regimes$\begin{eqnarray}\begin{array}{l}G(-{t}^{{\prime} })-G(t-{t}^{{\prime} })=G(-{t}^{{\prime} })-{m}_{0}G(-{t}^{{\prime} })\\ \times \ \left[\dot{G}(t)+\displaystyle \frac{\xi }{2}G(t)\right]-{m}_{0}G(t)\left[\dot{G}(-{t}^{{\prime} })+\displaystyle \frac{{\gamma }_{0}}{2{m}_{0}}G(-{t}^{{\prime} })\right].\end{array}\end{eqnarray}$From equations (14a) and (20), after performing the Weyl–Wigner transform, we have$\begin{eqnarray}\begin{array}{rcl}X(t) & = & {{q}}^{(F)}(t)-{{q}}^{(F)}(0)\\ & & +{\int }_{-\infty }^{0}{\rm{d}}{t}^{{\prime} }\left[G(-{t}^{{\prime} })-G(t-{t}^{{\prime} })\right]F({t}^{{\prime} }).\end{array}\end{eqnarray}$Substituting equation (43) into equation (44), one obtains$\begin{eqnarray}\begin{array}{rcl}X(t) & = & {{q}}^{(F)}(t)-{m}_{0}\left[\dot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}G(t)\right]{{q}}^{(F)}(0)\\ & & -{m}_{0}G(t){\dot{{q}}}^{(F)}(t).\end{array}\end{eqnarray}$From the Weyl–Wigner transform of the correlations of ${{q}}^{(F)}(t)$, i.e. equation (17), one can obtain the elements of the matrix of ${\boldsymbol{A}}(t)$ [48] as follows$\begin{eqnarray}\begin{array}{rcl}\langle {X}^{2}\rangle & = & \displaystyle \frac{1}{2}\left\{1+{m}_{0}^{2}{\left[\dot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}G(t)\right]}^{2}\right\}s(0)\\ & & -\displaystyle \frac{1}{2}{m}_{0}^{2}{G}^{2}(t)\ddot{s}(0)-{m}_{0}\left[\dot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}G(t)\right]s(t)\\ & & +{m}_{0}G(t)\dot{s}(t),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}\langle {\dot{X}}^{2}\rangle & = & -\displaystyle \frac{1}{2}\left[1+{m}_{0}^{2}{\dot{G}}^{2}(t)\right]\ddot{s}(0)\\ & & +\displaystyle \frac{1}{2}{m}_{0}^{2}{\left[\ddot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}\dot{G}(t)\right]}^{2}s(0)\\ & & -{m}_{0}\left[\ddot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}\dot{G}(t)\right]\dot{s}(t)+{m}_{0}\dot{G}(t)\ddot{s}(t),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}\langle X\dot{X}+\dot{X}X\rangle & = & {m}_{0}^{2}\left[\dot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}G(t)\right]\\ & & \times \ \left[\ddot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}\dot{G}(t)\right]s(0)\\ & & -{m}_{0}^{2}G(t)\dot{G}(t)\ddot{s}(0)-{m}_{0}\left[\ddot{G}(t)+\displaystyle \frac{{\gamma }_{0}}{{m}_{0}}\dot{G}(t)\right]s(t)\\ & & -{\gamma }_{0}G(t)\dot{s}(t)+{m}_{0}G(t)\ddot{s}(t),\end{array}\end{eqnarray}$where$\begin{eqnarray}s(t)=\displaystyle \frac{2{\gamma }_{0}}{\pi }{\int }_{0}^{\infty }{\rm{d}}\omega \displaystyle \frac{{\hslash }\omega }{{m}_{0}^{2}{\left({\omega }_{0}^{2}-{\omega }^{2}\right)}^{2}+{\gamma }_{0}^{2}{\omega }_{0}^{2}}\coth \displaystyle \frac{\beta {\hslash }\omega }{2}\cos \omega t.\end{eqnarray}$Please note that equations (46)–(48) are only valid in the weak coupling regime or the classical limit [52].
Before proceeding to the next step, let us make a self-consistency check about our results of equations (46)–(48). It is expected that in the high temperature limit, equations (46)–(48) will reproduce their classical counterparts (see section 10.2.1 in [53]). In the high temperature limit, $\coth (\beta {\hslash }\omega /2)\to 2/(\beta {\hslash }\omega )$, then equation (49) becomes$\begin{eqnarray}{s}_{\mathrm{cl}}(t)=\displaystyle \frac{4{\gamma }_{0}}{\beta \pi }{\int }_{0}^{\infty }{\rm{d}}\omega \displaystyle \frac{\cos \omega t}{{m}_{0}^{2}{\left({\omega }_{0}^{2}-{\omega }^{2}\right)}^{2}+{\gamma }_{0}^{2}{\omega }_{0}^{2}}.\end{eqnarray}$After taking the integration, one can obtain$\begin{eqnarray}{s}_{\mathrm{cl}}(t)=\displaystyle \frac{1}{{\rm{i}}\beta {m}_{0}{{\rm{\Omega }}}_{0}{\omega }_{0}^{2}}\left(-{\lambda }_{2}{{\rm{e}}}^{-{\lambda }_{1}t}+{\lambda }_{1}{{\rm{e}}}^{-{\lambda }_{2}t}\right),\displaystyle \frac{{\gamma }_{0}}{{m}_{0}{\omega }_{0}}\lt 2.\end{eqnarray}$Substituting equations (41) and (51) into equations (46)–(48), after some simplification, one can obtain the classical limit of equation (36)$\begin{eqnarray}{{\boldsymbol{A}}}_{\mathrm{cl}}(t)=\left(\begin{array}{cc}{m}_{0}^{2}\langle {\dot{X}}^{2}{\rangle }_{\mathrm{cl}} & \displaystyle \frac{{m}_{0}}{2}\langle X\dot{X}+\dot{X}X{\rangle }_{\mathrm{cl}}\\ \displaystyle \frac{{m}_{0}}{2}\langle X\dot{X}+\dot{X}X{\rangle }_{\mathrm{cl}} & \langle {X}^{2}{\rangle }_{\mathrm{cl}}\end{array}\right),\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}\langle {X}^{2}{\rangle }_{\mathrm{cl}} & = & \displaystyle \frac{{\gamma }_{0}{v}_{\mathrm{th}}^{2}}{{m}_{0}{\left({\lambda }_{1}-{\lambda }_{2}\right)}^{2}}\left[\displaystyle \frac{{\lambda }_{1}+{\lambda }_{2}}{{\lambda }_{1}{\lambda }_{2}}+\displaystyle \frac{4}{{\lambda }_{1}+{\lambda }_{2}}\right.\\ & & \left.\times \left({{\rm{e}}}^{-({\lambda }_{1}+{\lambda }_{2})t}-1\right)-\displaystyle \frac{1}{{\lambda }_{1}}{{\rm{e}}}^{-2{\lambda }_{1}t}-\displaystyle \frac{1}{{\lambda }_{2}}{{\rm{e}}}^{-2{\lambda }_{2}t}\right],\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{m}_{0}^{2}\langle {\dot{X}}^{2}{\rangle }_{\mathrm{cl}} & = & \displaystyle \frac{{m}_{0}^{2}{\gamma }_{0}{v}_{\mathrm{th}}^{2}}{{m}_{0}{\left({\lambda }_{1}-{\lambda }_{2}\right)}^{2}}\left[{\lambda }_{1}+{\lambda }_{2}+\displaystyle \frac{4{\lambda }_{1}{\lambda }_{2}}{{\lambda }_{1}+{\lambda }_{2}}\right.\\ & & \left.\times \left({{\rm{e}}}^{-({\lambda }_{1}+{\lambda }_{2})t}-1\right)-{\lambda }_{1}{{\rm{e}}}^{-2{\lambda }_{1}t}-{\lambda }_{2}{{\rm{e}}}^{-2{\lambda }_{2}t}\right],\end{array}\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{m}_{0}}{2}\langle X\dot{X}+\dot{X}X{\rangle }_{\mathrm{cl}}=\displaystyle \frac{{m}_{0}{\gamma }_{0}{v}_{\mathrm{th}}^{2}}{{\left({\lambda }_{1}-{\lambda }_{2}\right)}^{2}}{\left({{\rm{e}}}^{-{\lambda }_{1}t}-{{\rm{e}}}^{-{\lambda }_{2}t}\right)}^{2},\end{eqnarray}$and ${v}_{\mathrm{th}}^{2}=1/(\beta {m}_{0})$. One can see that equations (53a)–(53c) are exactly the same as equation (10.63) in [53], i.e. our results (46)–(48) reproduce the results of the classical Brownian particle in the high temperature limit (please note that m0 has been set to one and f(t)=0 in equation (10.63) in [53]). Substituting equations (36)–(38b), (50) and (53a)–(53c) into equation (35), one obtains the classical propagator in the phase space, which is the same as equation (10.55) in [53]:$\begin{eqnarray}\begin{array}{l}{P}_{\mathrm{cl}}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)=\displaystyle \frac{{\hslash }}{\sqrt{| {{\boldsymbol{A}}}_{\mathrm{cl}}(t)| }}\\ \ \ \times \ \exp \left[-\displaystyle \frac{1}{2}{\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)\right]}_{{pp}}{\left[{p}-\overline{{p}(t)}\right]}^{2}-{\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)\right]}_{{pq}}\right.\\ \ \ \left.\times \ \left[{p}-\overline{{p}(t)}\right]\left[{q}-\overline{{q}(t)}\right]-\displaystyle \frac{1}{2}{\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)\right]}_{{qq}}{\left[{q}-\overline{{q}(t)}\right]}^{2}\right].\end{array}\end{eqnarray}$Here, the elements of the inverse matrix of ${{\boldsymbol{A}}}_{\mathrm{cl}}(t)$ are given by$\begin{eqnarray}{\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)\right]}_{{pp}}=\langle {X}^{2}{\rangle }_{\mathrm{cl}}/| {{\boldsymbol{A}}}_{\mathrm{cl}}(t)| ,\end{eqnarray}$$\begin{eqnarray}{\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)\right]}_{{pq}}={\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)\right]}_{{qp}}=-{m}_{0}\langle X\dot{X}+\dot{X}X{\rangle }_{\mathrm{cl}}/\left(2| {{\boldsymbol{A}}}_{\mathrm{cl}}(t)| \right),\end{eqnarray}$$\begin{eqnarray}{\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)\right]}_{{qq}}={m}_{0}^{2}\langle {\dot{X}}^{2}{\rangle }_{\mathrm{cl}}/| {{\boldsymbol{A}}}_{\mathrm{cl}}(t)| ,\end{eqnarray}$and$\begin{eqnarray}\left|{{\boldsymbol{A}}}_{\mathrm{cl}}(t)\right|={m}_{0}^{2}\left[\langle {X}^{2}{\rangle }_{\mathrm{cl}}\langle {\dot{X}}^{2}{\rangle }_{\mathrm{cl}}-\langle X\dot{X}+\dot{X}X{\rangle }_{\mathrm{cl}}^{2}/4\right].\end{eqnarray}$
We now take the thermal equilibrium initial state as an example to calculate the time evolution of the Wigner function of the system (equations (34) and (35)). Please note that the initial state can also be any state other than the thermal equilibrium state. But for simplicity, we use the thermal equilibrium state as an example to demonstrate the effectiveness of our method. We assume that the system is initially prepared in the thermal equilibrium state at the inverse temperature β′, which is different from the temperature of the heat bath β, and the Wigner function of the initial density matrix can be written as$\begin{eqnarray}\begin{array}{l}W\left({{p}}^{{\prime} }(0),{{q}}^{{\prime} }(0);0\right)=2\tanh \displaystyle \frac{{\beta }^{{\prime} }{\hslash }{\omega }_{0}}{2}\\ \ \ \times \ \exp \left[-\displaystyle \frac{1}{2}\left({p}(0){{q}}^{{\prime} }(0)\right){\boldsymbol{B}}\left(\begin{array}{c}{{p}}^{{\prime} }(0)\\ {{q}}^{{\prime} }(0)\end{array}\right)\right],\\ {\boldsymbol{B}}=\left(\begin{array}{cc}2{\left({m}_{0}{\hslash }{\omega }_{0}\coth \tfrac{{\beta }^{{\prime} }{\hslash }{\omega }_{0}}{2}\right)}^{-1} & 0\\ 0 & 2{\left(\tfrac{{\hslash }}{{m}_{0}{\omega }_{0}}\coth \tfrac{{\beta }^{{\prime} }{\hslash }{\omega }_{0}}{2}\right)}^{-1}\end{array}\right).\end{array}\end{eqnarray}$Substituting equations (35) and (57) into equation (34), one can obtain$\begin{eqnarray}\begin{array}{l}W({p},{q};t)=\displaystyle \frac{2\tanh \tfrac{{\beta }^{{\prime} }{\hslash }\omega }{2}}{\sqrt{| {\boldsymbol{A}}(t)| | +{{\boldsymbol{\Lambda }}}^{{\rm{T}}}(t){{\boldsymbol{A}}}^{-1}(t){\boldsymbol{\Lambda }}(t)| }}\\ \ \times \ \exp \left[\displaystyle \frac{1}{2}\left({p}-{m}_{0}\dot{Y}(t)\ \ \ {q}-Y(t)\right)\right.\\ \ \cdot \ \left\{{{\boldsymbol{A}}}^{-1}(t){\boldsymbol{\Lambda }}(t){\left[{\boldsymbol{B}}+{{\boldsymbol{A}}}^{{\rm{T}}}(t){{\boldsymbol{A}}}^{-1}(t){\boldsymbol{\Lambda }}(t)\right]}^{-1}{{\boldsymbol{\Lambda }}}^{{\rm{T}}}(t){{\boldsymbol{A}}}^{-1}(t)\right.\\ \ \left.\left.-{{\boldsymbol{A}}}^{-1}(t)\right\}\cdot \ \left(\begin{array}{c}{p}-{m}_{0}\dot{Y}(t)\\ {q}-Y(t)\end{array}\right)\right],\end{array}\end{eqnarray}$where$\begin{eqnarray}{\boldsymbol{\Lambda }}(t)=\left(\begin{array}{cc}{m}_{0}\dot{G}(t) & {m}_{0}^{2}\ddot{G}(t)\\ G(t) & {m}_{0}\dot{G}(t)\end{array}\right).\end{eqnarray}$From this result, one can easily find that the variances of the Wigner function is independent of the external driving force f(t).
Furthermore, we would like to show how the equilibrium solution arises in the long time limit, i.e. the relaxation process from T′=1/β′ to T=1/β:
First we recall that, so long as the angular frequency of the system ω0 is nonzero, the Green function will vanish as $t\to \infty $ [48], thus ${\boldsymbol{\Lambda }}(t)=0$ when $t\to \infty $.
Next, from equations (46)–(48), we have$\begin{eqnarray}\langle {X}^{2}\rangle =\displaystyle \frac{1}{2}s(0),\langle {\dot{X}}^{2}\rangle =-\displaystyle \frac{1}{2}\ddot{s}(0),\langle X\dot{X}+\dot{X}X\rangle =0.\end{eqnarray}$
In the weak coupling limit$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{{\gamma }_{0}\to 0}\displaystyle \frac{2{\gamma }_{0}}{\pi }\displaystyle \frac{{\hslash }\omega }{{m}_{0}^{2}{\left({\omega }_{0}^{2}-{\omega }^{2}\right)}^{2}+{\gamma }_{0}^{2}{\omega }_{0}^{2}}=\displaystyle \frac{{\hslash }\omega }{{m}_{0}{\omega }_{0}^{2}}\delta (\omega -{\omega }_{0}).\end{eqnarray}$Then we have$\begin{eqnarray}{\boldsymbol{A}}(t)=\left(\begin{array}{cc}\displaystyle \frac{{m}_{0}{\hslash }{\omega }_{0}}{2}\coth \displaystyle \frac{\beta {\hslash }{\omega }_{0}}{2} & 0\\ 0 & \displaystyle \frac{{\hslash }}{2{m}_{0}{\omega }_{0}}\coth \displaystyle \frac{\beta {\hslash }{\omega }_{0}}{2}\end{array}\right).\end{eqnarray}$Substituting equations (59), (60), and (62) into equation (58), one can obtain the asymptotic expression of the Wigner function in the long time limit$\begin{eqnarray}\begin{array}{l}W({q},{p};t)=2\tanh \displaystyle \frac{\beta {\hslash }{\omega }_{0}}{2}\\ \ \ \times \ \exp \left[-\displaystyle \frac{{\left[{p}-{m}_{0}\dot{Y}(t)\right]}^{2}}{{m}_{0}{\hslash }{\omega }_{0}\coth \tfrac{\beta {\hslash }{\omega }_{0}}{2}}-\displaystyle \frac{{\left[{q}-Y(t)\right]}^{2}}{\tfrac{{\hslash }}{{m}_{0}{\omega }_{0}}\coth \tfrac{\beta {\hslash }{\omega }_{0}}{2}}\right].\end{array}\end{eqnarray}$This is the familiar form of the Wigner function of a dragged harmonic oscillator, which is independent of the initial temperature β′. Please note that when the driving force vanishes, i.e. $\hat{f}(t)=0$, our result (63) reproduces the result in [48].
5. Quantum corrections to the entropy
Based on the above results, we now calculate the quantum corrections to the entropy of a dragged harmonic oscillator which is undergoing quantum Brownian motion. First, we assume that the initial state of the system has a well-defined classical counterpart [47], i.e. when we expand the initial Wigner function in powers of ℏ, there are no terms in negative powers of ℏ$\begin{eqnarray}W\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)={W}_{\mathrm{cl}}+({\rm{i}}{\hslash }){W}^{(1)}+{\left({\rm{i}}{\hslash }\right)}^{2}{W}^{(2)}+o({{\hslash }}^{2}),\end{eqnarray}$where ${W}_{\mathrm{cl}}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)$ is the corresponding classical probability distribution in the phase space. Next, we expand the propagator (35) in powers of ℏ. Because$\begin{eqnarray}{\hslash }\coth \displaystyle \frac{\beta {\hslash }\omega }{2}=\displaystyle \frac{2}{\beta \omega }-{\left({\rm{i}}{\hslash }\right)}^{2}\displaystyle \frac{\beta \omega }{6}+o({{\hslash }}^{2}),\end{eqnarray}$in the weak coupling limit, from equation (49), we have$\begin{eqnarray}s(t)={s}_{\mathrm{cl}}(t)+{\left({\rm{i}}{\hslash }\right)}^{2}{s}^{(2)}(t)+o({{\hslash }}^{2}),\end{eqnarray}$where s(2)(t) is given by$\begin{eqnarray}{s}^{(2)}(t)=-\displaystyle \frac{\beta }{6{m}_{0}}\cos ({\omega }_{0}t).\end{eqnarray}$From equations (46)–(48), we know that$\begin{eqnarray}\langle {X}^{2}\rangle =\langle {X}^{2}{\rangle }_{\mathrm{cl}}+{\left({\rm{i}}{\hslash }\right)}^{2}\langle {X}^{2}{\rangle }^{(2)}+o({{\hslash }}^{2}),\end{eqnarray}$$\begin{eqnarray}\langle {\dot{X}}^{2}\rangle =\langle {\dot{X}}^{2}{\rangle }_{\mathrm{cl}}+{\left({\rm{i}}{\hslash }\right)}^{2}\langle {\dot{X}}^{2}{\rangle }^{(2)}+o({{\hslash }}^{2}),\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}\langle X\dot{X}+\dot{X}X\rangle & = & \langle X\dot{X}+\dot{X}X{\rangle }_{\mathrm{cl}}\\ & & +{\left({\rm{i}}{\hslash }\right)}^{2}\langle X\dot{X}+\dot{X}X{\rangle }^{(2)}+o({{\hslash }}^{2}),\end{array}\end{eqnarray}$where the expressions of $\langle {X}^{2}{\rangle }^{(2)}$, $\langle {\dot{X}}^{2}{\rangle }^{(2)}$ and $\langle X\dot{X}+\dot{X}X{\rangle }^{(2)}$ can be obtained by substituting equation (67) into equations (46)–(48). Then the matrix ${\boldsymbol{A}}(t)$ can be expanded in powers of ℏas follows$\begin{eqnarray}{\boldsymbol{A}}(t)={{\boldsymbol{A}}}_{\mathrm{cl}}(t)+{\left({\rm{i}}{\hslash }\right)}^{2}{{\boldsymbol{A}}}^{(2)}(t)+o({{\hslash }}^{2}),\end{eqnarray}$where ${{\boldsymbol{A}}}^{(2)}(t)$ is given by$\begin{eqnarray}{{\boldsymbol{A}}}^{(2)}(t)=\left(\begin{array}{cc}{m}_{0}^{2}\langle {\dot{X}}^{2}{\rangle }^{(2)} & \displaystyle \frac{{m}_{0}}{2}\langle X\dot{X}+\dot{X}X{\rangle }^{(2)}\\ \displaystyle \frac{{m}_{0}}{2}\langle X\dot{X}+\dot{X}X{\rangle }^{(2)} & \langle {X}^{2}{\rangle }^{(2)}\end{array}\right).\end{eqnarray}$Then we can similarly expand $| {\boldsymbol{A}}(t)| $ and ${{\boldsymbol{A}}}^{-1}(t)$ in powers of ℏ$\begin{eqnarray}| {\boldsymbol{A}}(t)| =| {{\boldsymbol{A}}}_{\mathrm{cl}}(t)| +{\left({\rm{i}}{\hslash }\right)}^{2}| {{\boldsymbol{A}}}_{\mathrm{cl}}(t)| \mathrm{Tr}\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t){{\boldsymbol{A}}}^{(2)}(t)\right]+o({{\hslash }}^{2}),\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{A}}}^{-1}(t)={{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)-{\left({\rm{i}}{\hslash }\right)}^{2}{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t){{\boldsymbol{A}}}^{(2)}(t){{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t)+o({{\hslash }}^{2}).\end{eqnarray}$Substituting equations (73a) and (73b) into equation (35), we obtain the expression of the propagator in powers of ℏas follows$\begin{eqnarray}\begin{array}{l}P\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)={P}_{\mathrm{cl}}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)\\ \ \ \ +\ {\left({\rm{i}}{\hslash }\right)}^{2}{P}^{(2)}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)+o({{\hslash }}^{2}),\end{array}\end{eqnarray}$where ${P}_{\mathrm{cl}}({q},{p};{{q}}^{{\prime} }(0),{p}(0);t)$ is given by equation (54), and$\begin{eqnarray}\begin{array}{l}{P}^{(2)}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)=\displaystyle \frac{1}{2}{P}_{\mathrm{cl}}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)\\ \ \ \times \ \left({{\boldsymbol{R}}}^{{\rm{T}}}(t){{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t){{\boldsymbol{A}}}^{(2)}(t){{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t){\boldsymbol{R}}(t)-\mathrm{Tr}\left[{{\boldsymbol{A}}}_{\mathrm{cl}}^{-1}(t){{\boldsymbol{A}}}^{(2)}(t)\right]\right).\end{array}\end{eqnarray}$Substituting equations (64) and (74) into equations (34), we obtain the time evolution of the Wigner function of the system in powers of ℏ$\begin{eqnarray}\begin{array}{rcl}W({q},{p};t) & = & {W}_{\mathrm{cl}}({q},{p};t)+({\rm{i}}{\hslash }){W}^{(1)}({q},{p};t)\\ & & +{\left({\rm{i}}{\hslash }\right)}^{2}{W}^{(2)}({q},{p};t)+o({{\hslash }}^{2}),\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{W}_{\mathrm{cl}}({q},{p};t) & = & {\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}{{q}}^{{\prime} }(0){\rm{d}}{{p}}^{{\prime} }(0)}{2\pi {\hslash }}{P}_{\mathrm{cl}}\left({q},{p};{{q}}^{{\prime} }(0),{p}(0);t\right)\\ & & \times {W}_{\mathrm{cl}}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{W}^{(1)}({q},{p};t) & = & {\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}{{q}}^{{\prime} }(0){\rm{d}}{{p}}^{{\prime} }(0)}{2\pi {\hslash }}{P}_{\mathrm{cl}}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)\\ & & \times {W}^{(1)}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{W}^{(2)}({q},{p};t)={\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}{{q}}^{{\prime} }(0){\rm{d}}{{p}}^{{\prime} }(0)}{2\pi {\hslash }}\left[{P}_{\mathrm{cl}}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)\right.\\ \ \ \times \ {W}^{(2)}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)+{P}^{(2)}\left({q},{p};{{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);t\right)\\ \ \ \left.\times \ {W}_{\mathrm{cl}}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)\right].\end{array}\end{eqnarray}$One can see that ${W}_{\mathrm{cl}}({q},{p};t)$ is the corresponding classical probability distribution at time t in the phase space, while ${W}^{(1)}({q},{p};t)$ and ${W}^{(2)}({q},{p};t)$ are the first- and the second-order quantum corrections at time t to the classical probability distribution, respectively. It is worth mentioning that this result is a dynamical extension of the one in [1].
Finally, using the methods developed in [47], we obtain the quantum corrections to the classical Gibbs entropy of a dragged harmonic oscillator which is undergoing quantum Brownian motion$\begin{eqnarray}{S}_{q}(t)={S}_{\mathrm{cl}}(t)+({\rm{i}}{\hslash }){S}^{(1)}(t)+{\left({\rm{i}}{\hslash }\right)}^{2}{S}^{(2)}(t)+o({{\hslash }}^{2}),\end{eqnarray}$where$\begin{eqnarray}{S}_{\mathrm{cl}}(t)=-\int \displaystyle \frac{{\rm{d}}{q}{\rm{d}}{p}}{2\pi {\hslash }}\ {W}_{\mathrm{cl}}\mathrm{ln}{W}_{\mathrm{cl}},\end{eqnarray}$$\begin{eqnarray}{S}^{(1)}(t)=-\int \displaystyle \frac{{\rm{d}}{q}{\rm{d}}{p}}{2\pi {\hslash }}\ \left[{W}^{(1)}\mathrm{ln}{W}_{\mathrm{cl}}+{W}^{(1)}\right],\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{S}^{(2)}(t) & = & -\displaystyle \int \displaystyle \frac{{\rm{d}}{q}{\rm{d}}{p}}{2\pi {\hslash }}\ \left[{W}^{(2)}\mathrm{ln}{W}_{\mathrm{cl}}+{W}^{(2)}+\displaystyle \frac{{\left({W}^{(1)}\right)}^{2}}{2{W}_{\mathrm{cl}}}\right.\\ & & \left.-\displaystyle \frac{{W}_{\mathrm{cl}}{\left(\leftarrow {\partial }_{{p}}\to {\partial }_{{q}}-\leftarrow {\partial }_{{q}}\to {\partial }_{{p}}\right)}^{2}{W}_{\mathrm{cl}}}{16{W}_{\mathrm{cl}}}+\displaystyle \frac{G({W}_{\mathrm{cl}})}{12{W}_{\mathrm{cl}}^{2}}\right].\end{array}\end{eqnarray}$Here$\begin{eqnarray}\begin{array}{rcl}G({W}_{\mathrm{cl}}) & = & \left({\partial }_{{p}}^{2}{W}_{\mathrm{cl}}\right){\left({\partial }_{{q}}{W}_{\mathrm{cl}}\right)}^{2}+\left({\partial }_{{q}}^{2}{W}_{\mathrm{cl}}\right){\left({\partial }_{{p}}{W}_{\mathrm{cl}}\right)}^{2}\\ & & -2\left({\partial }_{{q}}{W}_{\mathrm{cl}}\right)\left({\partial }_{{p}}{W}_{\mathrm{cl}}\right)\left({\partial }_{{q}{p}}{W}_{\mathrm{cl}}\right).\end{array}\end{eqnarray}$One can find that Scl(t) is exactly the corresponding classical Gibbs entropy, while S(1)(t) and S(2)(t) are the first- and the second-order quantum corrections to the entropy, respectively.
As a demonstration, we take the thermal equilibrium initial state as an example to illustrate our results (80)–(84). We assume that the system is prepared initially in the thermal equilibrium state at the inverse temperature β′. The initial Wigner function is given by equation (57), which can be expanded in the form of equation (64) and$\begin{eqnarray}{W}_{\mathrm{cl}}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)={\beta }^{{\prime} }{\hslash }{\omega }_{0}{{\rm{e}}}^{-{\beta }^{{\prime} }\epsilon \left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0)\right)},\end{eqnarray}$$\begin{eqnarray}{W}^{(1)}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)=0,\end{eqnarray}$$\begin{eqnarray}{W}^{(2)}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)={W}_{\mathrm{cl}}\displaystyle \frac{{\beta }^{{\prime} 2}{\omega }_{0}^{2}}{12}\left[1-{\beta }^{{\prime} }\epsilon \left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0)\right)\right],\end{eqnarray}$where$\begin{eqnarray}\epsilon \left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0)\right)=\displaystyle \frac{{{p}}^{{\prime} 2}(0)}{2{m}_{0}}+\displaystyle \frac{1}{2}{m}_{0}{\omega }_{0}^{2}{{q}}^{{\prime} 2}(0)\end{eqnarray}$is the Hamiltonian of a single harmonic oscillator. Substituting equations (64), (74), (75), and (85a)–(85c) into equation (76), one can obtain the expansion of the reduced Wigner function at an arbitrary time t. Finally, we obtain the the quantum corrections to the entropy by substituting equations (77)–(79) into equations (81)–(83). One can find that for the thermodynamic equilibrium initial state, all terms odd in ℏare exactly zero due to ${W}^{(1)}\left({{q}}^{{\prime} }(0),{{p}}^{{\prime} }(0);0\right)=0$. The evolution of the classical Wigner function ${W}_{\mathrm{cl}}({q},{p};t)$ is given by equation (77), and Scl(t) reproduces the classical Gibbs entropy for a dragged Brownian harmonic oscillator. And the lowest order quantum correction to the entropy is given by equation (83).
6. Discussion and summary
Before concluding this paper, we would like to give the following remarks.(I)In calculating the von Neumann entropy, we trace out the degrees of freedom of the heat bath, and ignore completely the entanglement between the system and the heat bath. It can be seen that when neglecting the entanglement, the von Neumann entropy of the system reproduces its classical counterpart in the classical limit (81). However, it is unclear to us if it is proper to neglect the entanglement in the study of quantum information related problems, e.g. the Landauer’s principle [54, 55]. How the entanglement between the system and the heat bath will influence the Landauer’s principle in an open quantum system is still an open question. (II)Exactly solvable models can bring important insights. The dynamical evolution of the quantum Brownian motion model under a time-dependent Hamiltonian is of great importance in the study of nonequilibrium quantum thermodynamics, e.g. finite time quantum heat engines, quantum Landauer’s principle, and quantum fluctuation theorems. But it is usually extremely difficult to solve exactly due to the huge number of degrees of freedom of the heat bath. Luckily, for this specific model, we obtain the analytical results of the time evolution of the Wigner function and the von Neumann entropy. The exact solutions of the quantum corrections to the entropy will be helpful for analyzing the interplay between quantum mechanics and thermodynamics at extremely low temperature. (III)We also notice that in [56–58], the author presented a method to calculate the von Neumann entropy of quantum states whose Wigner function is in a Gaussian form. So this method can also be applied to calculate the entropy of equation (58). However, their method is not applicable when the Wigner function is non-Gaussian. Nevertheless, the method for calculating the von Neumann entropy presented in equations (80)–(84) is valid for whatever states as long as they have well-defined classical counterparts.
In summary, in this paper, we study the time evolution of the von Neumann entropy of a quantum Brownian particle under a driving force. By solving the quantum Langevin equation, we obtain the analytical expression of the Wigner function at an arbitrary time t. As an example, we obtain the evolution of the Wigner function explicitly when the system is initially prepared in a thermal equilibrium state, and it reproduces the classical probability distribution in the high temperature and the weak coupling limit. Based on the above results and the results of the ℏexpansion of the von Neumann entropy in the phase space, we prove that the zeroth-order term reproduces the Gibbs entropy, and we obtain the explicit expression of the time evolution of the quantum corrections to the Gibbs entropy. Moreover, we find that for the thermodynamic equilibrium initial state, all terms odd in ℏare exactly zero.
In the classical stochastic thermodynamics, fluctuating work (heat) is defined along individual stochastic trajectory in the phase space [59]. Nevertheless, it is elusive to define a trajectory-dependent work (heat) in open quantum systems, because there is no well-defined trajectory in the Hilbert space due to the Heisenberg uncertainty principle. We plan to extend our current investigation to these problems and we believe that further studies along this line will advance our understanding about the relationship between the quantum and the classical work and heat and may bring important insights to some fundamental problems in quantum thermodynamics.
Acknowledgments
H T Quan acknowledges support from the National Science Foundation of China under Grants Nos. 11775001, 11534002, and 11825001.
,1,2,3,∗1School of Physics, 
