On the von Neumann entropy of a bath linearly coupled to a driven quantum system

The change of the von Neumann entropy of a set of harmonic oscillators initially in thermal equilibrium and interacting linearly with an externally driven quantum system is computed by adapting the Feynman-Vernon influence functional formalism. This quantum entropy production has the form of the expectation value of three functionals of the forward and backward paths describing the system history in the Feynman-Vernon theory. In the classical limit of Kramers-Langevin dynamics (Caldeira-Leggett model) these functionals combine to three terms, where the first is the entropy production functional of stochastic thermodynamics, the classical work done by the system on the environment in units of $k_BT$, the second another functional with no analogue in stochastic thermodynamics, and the third is a boundary term.


I. INTRODUCTION
The discovery of fluctuation relations [1][2][3] has transformed classical out-of-equilibrium thermodynamics, giving rise to the new field of stochastic thermodynamics [4][5][6]. A central idea of the field is that classical termodynamics can be extended to single, usually mesoscopic, systems where a surrounding medium takes the role of a heat bath [7]. The fundamental quantity in stochastic thermodynamics is δS env , the entropy production in the environment. Mathematically, this quantity can be defined as the log-ratio of transition probabilities in a forward and a reversed process [8], and fluctuation relations then follow as tautological identities [9,10].
A heat bath is an idealization of a large environment relaxing on a much faster time scale than the system of interest. Hence, the heat bath is arbitrarily close to thermal equilibrium at all times, and the entropy production of the system is nothing but the change of the (thermodynamic) entropy of the bath. By Clausius' formula this gives δS env = βδQ where δQ is the energy (heat) transferred from the system to the bath. Physically, fluctuation relations are non-trivial because it is not obvious that the log-ratio way of defining δS env is the same as βδQ. Indeed, although in retrospect a fairly straightforward fact, for standard physical kinetics without memory (master equations, diffusion equations) this equivalence has only been widely appreciated for a decade and a half [11][12][13].
Potential extensions of statistical thermodynamics and fluctuation relations to the quantum domain have been extensively investigated, and reviewed in [14,15]. However, with the exception of the Jarzynski equality and * eaurell@kth.se † eichhorn@nordita.org Crooks' fluctuation theorem in closed quantum systems [16], the results obtained to date lack the generality and simplicity of fluctuation relations in classical systems. In open quantum systems specific assumptions such as that the dynamics is unital (superoperator preserves the unit operator) [17], or that the quantum jump method [18][19][20] or the Lindblad formalism [21] can be applied, seem to be needed. The task is nontrivial since entropy production, as work and heat, is not a standard quantum operator. Indeed, according to the original proposal for the simplest case of closed quantum system work depends on two quantum measurements [16], while in a recent proposal involving only one measurement [22], a second quantum system is needed to keep track of the work.
The first goal of this contribution is to show that a natural extension to the quantum domain of the thermodynamic version of δS env can be investigated analytically by adapting the method of Feynman and Vernon [23]. This extension, which we will call δS q , is the change of the von Neumann entropy of a heat bath between two measurements on the system. In the context of stochastic thermodynamics a quantity equivalent to δS q was introduced in [24], and more recently investigated in [25].
In the Feynman-Vernon method the bath is taken to consist of harmonic oscillators initially in thermal equilibrium and linearly coupled to the system, which allows for integrating out the bath. As a result δS q can be written as the (quantum) expectation value of three functionals of the system history, similar but not identical to the real and imaginary actions S i and S r in the Feynman-Vernon theory. In the classical limit of a Brownian particle, where the system development is described by a Kramers-Langevin equation [26], these expectation values combine to two averages over the (classical) stochastic process, which will call δS env and δS var , and a boundary term, which we will call ∆S b . δS env is the average over the pro-cess and over a finite time of δS env , the standard entropy production functional in stochastic thermodynamics. By Clausius' formula and for Kramers-Langevin dynamics δS env equals β(δQ f riction +δQ noise ) where δQ f riction and δQ noise are the amounts of energy (heat) transmitted from the system to the bath by respectively the friction force and the random force. δS var is also an average over the same stochastic process, formally the finite part of the square of the random force, and has no analogue in standard stochastic thermodynamics. ∆S b finally depends only on the (classical) transition probability over a finite time interval, and is hence not a functional of the whole (classical) system history. As will be discussed below this quantity has very different properties than one expects for (classical) entropy production.
The paper is organized as follows: in Section II we define δS q , equation (1), and show how to express it as a quantum expectation value, equations (2) and (3). In Section III we introduce the Feynman-Vernon formalism and use it to give an expression for the expectation value, equation (10). In Section IV we evaluate this expression and in parallel give standard results of the Feynman-Vernon theory. The three functionals mentioned above then appear in equations (13) and (14). In Section V we introduce the Caldeira-Leggett limit of the Feynman-Vernon model which leads to classical dynamics with noise and friction; the limit of the three functionals is given in equation (15). In Section VI we analyze the three functionals in this limit and separate out ∆S b , and in Section VII we group the remaining parts into δS env and δS var . For completeness we also give, in Section VI, a derivation of the Caldeira-Leggett result that the Wigner transform of the Feynman-Vernon propagator for the density matrix goes to the transition probability of the (classical) stochastic process. In Section VIII we consider ∆S b in the limit of weak coupling between the system and the bath, and in Section IX we sum up and discuss our results. In Appendices A and B we discuss for completeness the time scales involved and higher-order corrections to the Caldeira-Leggett limit.
We end this Introduction by noting that in the classical limit we are limited to averages of the entropy production over a finite time, which only approximates the entropy production functional of stochastic thermodynamics if the time is short. The question of whether quantum entropy production defined as in this paper can also lead to another definition in terms of forward and reversed (quantum) dynamics is left for future work.

II. THE FIRST-ORDER CHANGE OF VON NEUMANN ENTROPY
We consider the setting where a quantum system is prepared in an initial pure state |i >< i| at time t i and then attached to a bath with density operator ρ eq B describing a state of thermal equilibrium. Over a time period [t i , t f ] the system and the bath develop in interaction such that the total state at time t f is ρ f T OT . At this point a measurement is made of an operator O which depends on the system variables only, with outcome o f , corresponding to the pure state |f >< f | of the system. This happens with probability P if = Tr B < f |ρ f T OT |f >. By the measurement postulate the total state after the measurement is ρ f,+ T OT = 1 P if |f >< f | ⊕ f |ρ f T OT |f , and we can therefore identify the density operator of the bath, after interacting with the system and after the measurement has been performed on the system, as We assume δρ B = ρ f B − ρ eq B to be small, and the first-order change of the bath entropy is then Equation (1) is our definition of δS q . In a basis of energy eigenstates |n >= |n 1 , n 2 , . . . > the density op- is the energy of the bth degree of freedom of the bath in state n b , and F b is its free energy at inverse temperature β. The expression in (1) can therefore be written and H B is the Hamiltonian of the bath. Note that R if (0) = P if . Equations (2) and (3) are the starting point of our analysis.

III. THE FEYNMAN-VERNON METHOD
The Feynman-Vernon theory is a means to compute P if while we need to compute the slightly more complex quantity R ′ if (0). Let us begin by noting that the time development of an open quantum system is described by a superoperator or quantum map Φ which maps density operators to density operators, and which can always be realized by adding another system or ancilla in state ρ a , acting unitarily on the combined system and ancilla, and then tracing out the ancilla Φρ = Tr a [U (ρ ⊕ ρ a ) U † ] [27][28][29]. The Feynman-Vernon approach consist in writing the two unitary operators U and U † as path integrals while taking the ancilla to be a bath of harmonic oscillators initially in thermal equilibrium, linearly coupled to the system. The total Hamiltonian describing the system and ancilla is thus where C b is the strength of the interaction between the system and bath oscillator b. Integrating out the bath gives Φ in the coordinate representation as (5) and the transition probability as where ψ i and ψ f are the initial and final states in the coordinate representation and K is the Feynman-Vernon propagator of the density operator of the system [23]. The first step in computing K is the path integrals over the bath in U (ρ ⊕ ρ a ) U † with fixed initial and final position of each bath oscillator. The result of these path where the K b 's are propagators of harmonic oscillators with linear terms in the action, X(s)C b x b (s) for the forward path, and Y (s)C b y b (s) for the backward path. The second step is to introduce a coordinate representation of the equilibrium density operator of the bath oscillator, proportional to the propagator in imaginary time. Integrating out the four positions q i , q f , q ′ f and q ′ i then gives the Feynman-Vernon influence functional where Z b = 2 sinh ω b ǫ 2 −1 is the partition function of b at inverse temperature β, and the delta function is the coordinate representation of the trace over the final state of b. The Feynman-Vernon propagator can then be written as double path integral over the system variables A comparison of (3) and (7) shows that to compute the first-order change of the von Neumann entropy of the bath we need in terms of which

IV. EVALUATING THE ENTROPY PRODUCTION FUNCTIONAL
The propagator of a harmonic oscillator is the exponential of terms constant, linear and quadratic in the initial and final position. After integrating out these initial and final positions the modified influence functional is are the normalizations of the propagators, the constant term B is given by and the matrix M (ǫ) and the vector u are respectively given by In above B, C, ω and m are as for oscillator b. The pre-factors of the exponential in (11) combine This cancels with the term U b in (2), and the quantity sought is thus Equation (12) is the first result of this paper giving the entropy production in the bath between two measurements on the system as the expected value of a functional of the forward and backward system paths, divided by the transition probability. To proceed further we note that the product b F b , which is defining the real and imaginary parts of the Feynman-Vernon influence action [23,26]. These are quadratic functionals of the paths of the system, where we write X and Y for quantities at time s and X ′ and Y ′ for quantities at the earlier time s ′ , and where the kernels are By a similar analysis as the one leading to S i and S r the functional in (12) can be evaluated to with the kernels Equations (13) and (14) is the second result of this paper. We note that while I (1) equals − 1 β∂ β S r , I (2) and I (3) are new and non-causal terms, i.e. which do not fullfill General Property 5 of influence functionals as discussed on pp 126-127 in [23]. For a physical interpretation we turn to the limit of classical stochastic dynamics.

V. THE CALDEIRA-LEGGETT LIMIT
The classical limit of the Feynman-Vernon theory was computed in [26]. The spectrum of the bath oscillators is then assumed continuous with density f (ω) such that f (ω)C 2 ω /m(ω) equals 2ηω 2 π up to some upper cut-off Ω. The parameter η has the dimension M L/T of a classical friction coefficient, and the first kernel in the Feynman-Vernon theory then tends to k i ≈ −η d d(s−s ′ ) δ(s − s ′ ). The corresponding action S i is a potential renormalization plus a term − η 2 (X − Y )(Ẋ +Ẏ )ds. As a stochastic integral this finite part of S i has to be interpreted in the post-point (anti-Itô) prescription since the time derivatives stem from the integral over s ′ up to s. The other kernel k r has only small contributions for |ω| > 1 β and hence describes, if Ω is large enough, a memory kernel of width β , independently of Ω. If all times of interest in the system are longer than β then S r ≈ η β (X −Y ) 2 ds. The different time scales involved are briefly discussed in Appendix A and possible higher-order corrections in Appendix B.
In the same limit as above the three terms in (13) tend to The integral for I (2) in (13) can be extended over the whole domaini.e. t f t f (· · · ) ds ′ ds -and expanded around the diagonal (s = s ′ ). Contrary to the case of S i there is therefore not any potential renormalization term from I (2) in the Caldeira-Leggett limit. Furthermore, the limit of I (2) given in (15) does not depend on the discretization scheme since the Itô contributions cancel. Therefore, we can alternatively write this term as and and with the mid-point prescription for both terms. Consquently, the term ∆S b is to be interpreted as which is a pure boundary contribution. Of the three terms in (15) it is the average of I (3) which has the most physical immediate meaning, as it must tend to βη v 2 dt whenẊ andẎ are approximated by a classical velocity v. To a friction force −ηv corresponds a reaction force ηv from the system on the bath, and the work done by this force is η v 2 dt. The entropy production from I (3) is therefore βδQ f riction , where δQ f riction is the energy (heat) transferred from the system to the bath, as announced in Introduction.

VI. ANALYSIS OF THE CLASSICAL LIMIT
To further analyze the classical limit we use, as in [26], the Markov property of Feynman-Vernon propagator, in this limit. First it is convenient to introduce auxiliary variables q = X+Y , and the Wigner transform of K, written as This P satisfies the Fokker-Planck equation, and is hence the transition probability of a classic stochastic process starting at (q i , p i ) at time t i , and ending up in (q f , p f ) at time t f [26] (see below). It is further convenient to introduce the two overlaps which we assume to be as for coherent states integrated against functions depending sufficiently weakly on phase space, i.e.
The quantum mechanical transition probability P if is then approximately 2π P (q f , p f , q i , p i , t f , t i ).
For a system with Hamiltonian H S = P 2 2M + V (X) the short-time density matrix propagator, forK and from (q, α) at time t i = t f − ∆s to (q ′ , α ′ ) at time t f , is where ∆q = q ′ −q and ∆α = α ′ −α, the arguments on the left hand side are understood and the contribution from S i has been evaluated with the post-point prescription.
The Chapman-Kolmogorov equation forK is, expanding in the increments, The integral over ∆q of a term proportional to (∆q) n in (24) gives, using (23), ( i ∆s M ) n δ (n) (∆α − η M α ′ ∆s). A term proportional to (∆α) m in (24) can be expanded as l m l (∆α − η M α ′ ∆s) l ( η M α ′ ∆s) m−l and when integrated over ∆α this gives zero unless l = n. Using m = l = 0, m = 1 and l = 0 and m = l = 1 equation (24) can hence be written which is the Lindblad equation derived in [26]. For the Wigner transform (25) gives which is, up to terms of order 2 , the Fokker-Planck equation of classical stochastic dynamics with friction coefficient η [26]. We proceed to treat the three functionals in (15) in an analogous manner. Higher-order corrections to Fokker-Planck equation derived here (finite β corrections) are briefly discussed in Appendix B.
A. The I (1) contribution At given initial and final positions of the system, (q i , α i ) and (q f , α f ), the path integrals in (12) give for the I (1) part Combining this with the integrals over initial and final positions and (21) and (22) we can write The term (− 1 2 α 2 ) can be interpreted as ∂ 2 p ′ p ′ acting on e ip ′ α and by integration by parts this gives The classical limit of the contribution of δS q from I (1) is therefore In Section VII below we show how this can be given an interpretation as an average δS var over the stochastic process.
where we have used the mid-point prescription, following the discussion around equation (18). Focusing first on the pre-point term (the term in the inner parenthesis proportional to α), we use, similarly to (24), the shorttime expression (23) and the expansioñ The integral over ∆q gives ( i ∆s M ) n+1 δ (n+1) (∆α − η M α ′ ∆s) for a term in the expansion proportional to (∆q) n . The only contribution of order ∆s is n = 0 and m = l = 1 which gives Combining this with the integrals over initial and final positions as above we have The factor α can be interpreted as − i ∂ p acting on e ipα and the derivative moved then to the last P , while the derivative ∂ α brings down ip ′ / multiplying the first P . Combining these terms gives 1 2π The classical contribution from the first term in 2i S (mid) i to δS q is therefore The second term in (17) (term in inner parenthesis proportional to ∆α) is on the other hand ds dqdαdq ′ dα ′K (q f , α f , q ′ , α ′ , t f , s)( ηi ∆s ) (∆α∆q)K ∆s (q ′ , α ′ , q, α, ·)K(q, α, q i , α i , s, t i ) The only contribution of order ∆s is then n = m = 0 and l = 1 which gives This leads to the very simple classical contribution: By one integration by parts (30) and (31) can be combined to We will show in Section VII that (32) together are nothing but βδQ noise , where δQ noise is the energy (heat) transferred from the system to the bath by the random force.

C. The ∆S b contribution
To compute the classical limit of this term we consider directly the Wigner transform P of the Feynman-Vernon propagator over the whole time interval and interpret i α i multiplyingK as −∂ pi acting on P , and similarly i α f as ∂ p f . This gives As (33) is complete differential (not a proper functional), it is obviously very different from a classical entropy production term, and more akin to a change in state function. In addition, it depends explicitly on initial and final position for which there is no analogy in stochastic thermodynamics. We will return to a discussion of (33) in Section VIII below.

D. The I (3) contribution
For the I (3) part we finally have The integral over ∆q of the term in (∆q) 2 in the inner parenthesis can be evaluated as ( i ∆s M ) n+2 δ (n+2) (∆α − η M α ′ ∆s) which selects n = 0 and m = l = 2. The combination of pre-factors multiplying ∂ 2 ααK is then βη M 2 (− 2 ) and as above we can interpret − 2 ∂ 2 ααK to be p 2 acting on P . This gives a classical contribution to δS q from I (3) as ds dqdpP (q, p, q i , p i , s, t i )βη( p M ) 2 P (q f , p f , q, p, t f , s) P (q f , p f , q i , p i , t f , t i ) (34) As already remarked above, this quantity is βQ f riction .
The remaining term − 1 4 (∆α) 2 in the inner parenthesis above selects l = n = 0 and m = 2 giving βη(− 1 2 )( η M ) 2 α 2 . This is the same contribution as from I (1) , up to the dimensionless factor −( βη 2M ) 2 . Since β is the decorrelation time of the bath and η/M is the (mesoscopic) Langevin relaxation time of the system this factor must be very small in the set-up considered here, and can therefore be ignored.

VII. INTERPRETATIONS AS STOCHASTIC FUNCTIONALS
The purpose of this Section if to interpret all the terms derived above except ∆S b as expectation values with respect to the (classical) Kramers-Langevin process. We begin with the term from I (3) in (34), and express it symbolically as The ratio Prob(path)/ q f ,p f qi,pi D(path)Prob(path) is the conditional probability that a given path is chosen among the set that starts at (q i , p i ) and ends up at (q f , p f ). Therefore we may give the interpetation which is βδQ f riction -as already obtained by a simpler argument above. For the term in (32) we want to compare to the energy transferred to the bath from the system by the fluctuating force. To the Fokker-Planck equation corresponds a Kramers-Langevin equatioṅ with a random force F noise = √ 2k B T ηζ where dζ is a standard Wiener increment. To this force there is a reaction force −F noise from the system on the bath and the infintessimal energy transferred to the bath is the work done by this force, − √ 2k B T ηζ • dq, where • indicates the mid-point (Stratonovich) presecription. The average of the (classical) entropy production due to the random force is thus Using alternatively F noise =ṗ − F + η p M we should hence compute We do this by discretizing the time in steps t i = t 0 , t 1 , . . . , t N = t f and using the propagator of the Kramers-Langevin equation N n=1 dq n dp n dq n−1 dp n−1 − p n + p n−1 2M β p n − p n−1 − F n ds + η p n + p n−1 2M ds P (q n−1 , p n−1 , q i , p i )P (q n , p n , q n−1 , p n−1 )P (q f , p f , q n , p n ) The short-time propagator is and we therefore have Integration by parts gives three terms where the derivate ∂ p ′ is moved respectively to δ(·), p ′ +p 2M or P (q f , p f , q ′ , p ′ ). The first term will be overall quadratic in ∆s, and the other two can be compared to (32). Hence we have indeed that (32) is equal to βQ noise . Combining (36) and (39) we have δS env , the average of the entropy production in stochastic thermodynamics, as announced in the Introduction.
The term from I (1) in (27) can also be given a probabilistic interpretation, albeit not a standard one in stochastic thermodynamics. We start by observing that the increments of a standard Wiener process are Gaussian distributed and that for an unconstrained average E[(dζ) 2 ] = ds with no term of order (ds) 2 . If however we average over the paths of the stochastic process that start at (q i , p i ) and end at (q f , p f ) we can have q f ,p f qi,pi D(path)Prob(path) (dp−F ds+η p 2M ds) 2 2kB T η q f ,p f qi,pi D(path)Prob(path) = ds + b[q, p, s|q i , p i , q f , p f ](ds) 2 (39) with a non-trivial coefficient b. We relate the expression in (27) to such a term by observing, in analogy to (39), that p ′ − p − F ∆s + η p ′ +p 2M ∆s 2 2k B T η P ∆s = δ(·) 2k B T η(∆s) 2 ∂ 2 p ′ p ′ f + ∆sf By two integrations by parts we therefore find that (27) is also an average over the stochastic process where b is defined by the ansatz in (39). Formally we could also write (41) as where we mean the finite remainder after a term diverging as (ds) −1 has been subtracted from the random force squared.

VIII. ASYMPTOTIC ANALYSIS OF ∆S b
The most surprising term that have come from the above analysis is the ∆S b given in (33), both because it is like a change in a state function, and also because of its dependence on initial and final position. We will here consider this term in the limit of weak coupling. As this is essentially a classical problem we will adopt the dimension-less units introduced below in Appendix A where the Kramers-Langevin equation readṡ where ζ is standard white noise and where where P is the transition probability of (42). We will only consider the case when t, the duration of the process, is of order one in (42) (t on the order of t osc , the characteristic time of the system, in the original dimensional variables), and γ tending to zero. After the limit in γ is taken we could also allow t to become long. We note that the opposite limit where first t is taken long and then γ is taken to zero is physically more interesting, but also mathematically considerably more complicated [30,31], and outside the scope of this discussion.
Let (q * f , p * f ) be the final position and momentum at time t of the classical conservative system defined by (42) when γ = 0, starting from (q i , p i ). If ∆q f = q f − q * f and ∆p f = p f − p * f are the deviations of the actual final positions from the classical path we assume P (q f , p f , q i , p i ) to be a Gaussian distribution, i.e.