Excess entropy production in quantum system: Quantum master equation approach
EExcess entropy production in quantum system: Quantum masterequation approach
Satoshi Nakajima ∗ and Yasuhiro Tokura † Graduate School of P ure and Applied Sciences , U niversity of T sukuba , − − , T ennodai , T sukuba , − , J apan
October 31, 2017
Abstract
For open systems described by the quantum master equation (QME), we investigate the excessentropy production under quasistatic operations between nonequilibrium steady states. The averageentropy production is composed of the time integral of the instantaneous steady entropy productionrate and the excess entropy production. We propose to define average entropy production rate usingthe average energy and particle currents, which are calculated by using the full counting statistics withQME. The excess entropy production is given by a line integral in the control parameter space and itsintegrand is called the Berry-Sinitsyn-Nemenman (BSN) vector. In the weakly nonequilibrium regime,we show that BSN vector is described by ln ˘ ρ and ρ where ρ is the instantaneous steady state ofthe QME and ˘ ρ is that of the QME which is given by reversing the sign of the Lamb shift term.If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, the excess entropyproduction approximately reduces to the difference between the von Neumann entropies of the system.Additionally, we point out that the expression of the entropy production obtained in the classicalMarkov jump process is different from our result and show that these are approximately equivalentonly in the weakly nonequilibrium regime. In equilibrium thermodynamics, the central quantity is the entropy S , which describes both the macro-scopic properties of equilibrium systems and the fundamental limits on the possible transitions amongequilibrium states. In equilibrium thermodynamics, the Clausius equality∆ S = βQ, (1.1)tells us how one can determine the entropy by measuring the heat. Here, ∆ S is the change in the entropyof the system during the operation, β is the inverse temperature of the bath contacting with the system,and Q is the heat transferred from the bath to the system during the operation. This equality is universallyvalid for quasistatic transitions between equilibrium states. In the equilibrium classical (quantum) system,the entropy is given by the Shannon entropy of the probability distribution (von Neumann entropy of thedensity matrix) of states.The investigation of thermodynamic structures of nonequilibrium steady states (NESSs) has been atopic of active research in nonequilibrium statistical mechanics [1, 2, 3, 4, 5, 6, 7, 8, 9]. For instance, ∗ Email: [email protected] † Email: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t he extension of the relations in equilibrium thermodynamics, such as the Clausius equality, to NESSshas been one of the central subjects. Recently there has been progress in the extension of the Clausiusequality to NESSs [10, 11, 12] (see also Refs.[13, 14, 15, 16, 17, 18]). In these studies, the excess heat Q b, ex (of the bath b ) [2] has been introduced instead of the total heat Q b from the bath b . The excess heat Q b, ex ,which describes an additional heat induced by a transition between NESSs with time-dependent externalcontrol parameters, is defined by subtracting from Q b the time integral of the instantaneous steady heatcurrent from the bath b . In the weakly nonequilibrium regime, there exists a scalar potential S of thecontrol parameter space which approximately satisfies the extended Clausius equality (cid:88) b β b Q b, ex ≈ ∆ S . (1.2)Here, β b is the inverse temperature of the bath b , ∆ S = S ( α τ ) − S ( α ), α t = ( α t , α t , . . . ) is the valueof the set of the control parameters at time t , and t = 0 and t = τ are the initial and final times of theoperation. We assume multiple baths to maintain the system out of the equilibrium and the symbol (cid:80) b means the summation over the baths. In classical systems, S is the symmetrized Shannon entropy [11].In quantum systems with the time-reversal symmetry, S is the symmetrized von Neumann entropy [12].However, the study of the excess entropy in the quantum system without the time-reversal symmetryseems still lacking. This is the main objective of this paper.In general, the left-hand side (LHS) of (1.2) is replaced by the excess entropy σ ex def = σ − (cid:90) τ dt J σ ss ( α t ) , (1.3)where σ is the average entropy production and J σ ss ( α t ) is the instantaneous steady entropy productionrate at time t [19, 20, 21]. In the quasistatic operation, the excess entropy is given by σ ex = ∆ S + O ( ε δ ) , (1.4)where ε is a measure of degree of nonequilibrium and δ describes the amplitude of the change of thecontrol parameters. Sagawa and Hayakawa [19] studied the full counting statistics (FCS) of the entropyproduction for classical systems described by the Markov jump process and showed that the excess entropyis characterized by the Berry-Sinitsyn-Nemenman (BSN) phase [22].The method of Ref. [19] was generalized to quantum systems and applied to studies of the quantumpump [23, 24, 25]. Here we briefly explain the studies of the quantum pump. At t = 0 and t = τ ,we perform projection measurements of a time-independent observable O of the baths and obtain theoutcomes o (0) and o ( τ ). The generating function of ∆ o = o ( τ ) − o (0) is Z τ ( χ ) = (cid:90) d ∆ o P τ (∆ o ) e iχ ∆ o (1.5)where P τ (∆ o ) is the probability density distribution of ∆ o and χ is called the counting field. To calculatethe generating function, the method using the quantum master equation (QME) with the counting field(FCS-QME) [26] had been proposed. The solution of the FCS-QME, ρ χ ( t ), provides the generatingfunction as Z τ ( χ ) = Tr S [ ρ χ ( τ )], where Tr S denotes the trace of the system. The Berry phase [27] of theFCS-QME is the BSN phase. The average of the difference of the outcomes is given by (cid:104) ∆ o (cid:105) = (cid:90) τ dt i O ( t ) , (1.6)2here i O ( t ) is the current of an operator O . If the state of the system at t = 0 is the instantaneous steadystate and the modulation of the control parameters is slow, the following relation holds: (cid:104) ∆ o (cid:105) = (cid:90) τ dt i O ss ( α t ) + (cid:90) C dα n A On ( α ) , (1.7)where i O ss ( α t ) is the instantaneous steady current of O . C is the trajectory from α to α τ . and A On ( α ) isthe BSN vector derived from the BSN phase. α n is n -th component of the control parameters and thesummation symbol for n is omitted. The derived formula of the BSN vector depends on the approximationsused for the QME. The Born-Markov approximation with or without the rotating wave approximation(RWA) [28] is frequently used. The QME in the Born-Markov approximation without RWA sometimesviolates the non-negativity of the system reduced density operator. The QME of the RWA or the coarse-graining approximation (CGA) [29, 30] is the Lindblad type which guarantees the non-negativity [28]. If O is the total particle number of a bath b , there are several methods to calculate A On ( α ) of (1.7) .In this paper, we propose to identify˙ σ ( t ) def = (cid:88) b β b ( t ) (cid:2) − i H b ( t ) − µ b ( t ) (cid:8) − i N b ( t ) (cid:9)(cid:3) (1.8)with the average entropy production rate , where µ b is the chemical potential of the bath b , and i H b ( t ) and i N b ( t ) are energy and particle currents from the system to the bath b , respectively. H b and N b representthe Hamiltonian and the total particle number of the bath b , respectively. This is a straightforwardextention of the entropy production rate argued for an NESS [36] to a time-dependent system. Now, theexcess entropy is obtained by σ ex = (cid:90) τ dt [ ˙ σ ( t ) − J σ ss ( α t )] = (cid:90) C dα n A σn ( α ) , (1.9)where we used (1.7) in the second equation with A σn ( α ) def = (cid:88) b β b (cid:2) − A H b n ( α ) − µ b (cid:8) − A N b n ( α ) (cid:9)(cid:3) . (1.10)Here, A H b n ( α ) and A N b n ( α ) are the BSN vectors of H b and N b . It should be noted that β b and µ b couldalso be the elements of the set of the control parameters, α . The following expression is the main resultof this manuscript, A σn ( α ) = − Tr S (cid:20) ln ˘ ρ ( α ) ∂ρ ( α ) ∂α n (cid:21) + O ( ε ) , (1.11)without any assumption on the time-reversal symmetry. ρ ( α ) is the instantaneous steady state of theQME and ˘ ρ ( α ) is that of the QME which is given by reversing the sign of the Lamb shift term. If thesystem Hamiltonian is non-degenerate or the Lamb shift term is negligible, we obtain σ ex = S vN ( ρ ( α τ )) − S vN ( ρ ( α )) + O ( ε δ ) , (1.12) For non-interacting system, A On ( α ) is calculated from the Brouwer formula[31] using the scattering matrix. Recently,the quantum pump in interacting systems has been actively researched. There are three theoretical approaches. The firstis the Green’s function approach, [32]. The second is the generalized master equation approach [34, 35]. The third is theFCS-QME approach. Reference[25] showed the equivalence between the second and the third approaches for all orders ofpumping frequency (see also [33]). S vN ( ρ ) def = − Tr S [ ρ ln ρ ] is the von Neumann entropy.The structure of the paper is as follows. First, we explain the FCS-QME (2.1) and the formula forthe excess entropy. Then we introduce the generalized QME in 2.2. In 2.3, we explain the RWA andafter this section we focus on the RWA except for 3.3. In 3.1, the BSN vector A σn in the equilibrium isdiscussed. In 3.2, the main result of this manuscript, (1.11), is derived. Next we mention the results inthe Born-Markov approximation (3.3). In 4, we compare the preceding study on the entropy productionin the classical Markov jump process [21, 37] with ours. In 5, we consider the time-reversal operation.At last (6), we summarize this paper. In the Appendix A, we derive the formula of the derivative of thevon Neumann entropy and in the Appendix B, we show the details of the derivation of the relation in aweakly nonequilibrium regime. In the Appendix C, we explain the definition of entropy production of theMarkov jump process and the result of Ref. [21]. We consider system S weakly coupled to several baths (although we used the same symbol S as theentropy in 1, S only means the ‘system’ in the following discussions). In order to maintain the systemout of equilibrium and in NESS, the system needs to be coupled with more than one bath. The totalHamiltonian is given by H tot ( t ) = H S ( α S ( t )) + (cid:88) b [ H b + H Sb ( α Sb ( t ))] . (2.1) H S ( α S ) is the system Hamiltonian and α S denotes a set of control parameters of the system. H b is theHamiltonian of the bath b . H Sb ( α Sb ) is the coupling Hamiltonian between S and the bath b , and α Sb iscorresponding set of control parameters. We denote the inverse temperature and the chemical potentialof the bath b by β b and µ b which can be the control parameters, and α b denotes the set of β b and β b µ b .We symbolize the set of all control parameters ( α S , { α Sb } b , α B def = { α b } b ) by α . While α S and α Sb are dynamical parameters like energy levels, tunnel coupling strengths or a magnetic field, α B are thethermodynamical parameters. We denote the set of all the linear operators of S by B.Consider slow modulation of the control parameters during 0 ≤ t ≤ τ . At t = 0 and t = τ , we performprojection measurements of time-independent observables { O µ } of the baths which commute with eachother. The index µ distinguishes the time-independent observables of the baths. { o ( τ ) µ } ( { o (0) µ } ) denotesthe set of outcomes at t = τ ( t = 0). The generating function Z τ ( { χ O µ } ) = (cid:90) (cid:89) µ d ∆ o µ P τ ( { ∆ o µ } ) e i (cid:80) µ χ Oµ ∆ o µ (2.2)is the Fourier transform of the joint probability density distribution P τ ( { ∆ o µ } ) where ∆ o µ def = o ( τ ) µ − o (0) µ .Here, χ O µ is the counting field for O µ . The generating function is given by Z τ ( { χ O µ } ) = Tr tot [ ρ χ tot ( t = τ )] (2.3)using an operator of the total system ρ χ tot ( t ) obeying the modified von Neumann equation [26] ddt ρ χ tot ( t ) = − i [ H tot ( t ) , ρ χ tot ( t )] χ . (2.4)4n this paper, we set (cid:126) = 1. Here, for two operators A and B , [ A, B ] χ def = A χ B − BA − χ and A χ def = e i (cid:80) µ χ Oµ O µ / Ae − i (cid:80) µ χ Oµ O µ / . (2.5) χ denotes the set of the counting fields { χ O µ } . The initial condition of ρ χ tot ( t ) is given by ρ χ tot (0) = (cid:80) { o ν } P { o ν } ρ tot (0) P { o ν } [26]. Here, ρ tot (0) is the initial state of the total system, { o ν } denotes eigenvaluesof { O ν } and P { o ν } is a projection operator defined by O µ P { o ν } = o µ P { o ν } , P { o ν } P { o (cid:48) ν } = P { o ν } (cid:81) µ δ o µ ,o (cid:48) µ ,and P †{ o ν } = P { o ν } . We suppose ρ tot (0) = ρ (0) ⊗ ρ B ( α B (0)) , (2.6)where ρ (0) is the initial state of the system and ρ B ( α B (0)) def = (cid:79) b b ( α b (0)) e − β b (0)[ H b − µ b (0) N b ] (2.7)with Ξ b ( α b ) def = Tr b [ e − β b [ H b − µ b N b ] ]. Tr b denotes the trace of the bath b , and Tr B denotes the trace over allbaths’ degrees of freedom. Then, we have ρ χ tot (0) = ρ (0) ⊗ (cid:88) { o ν } P { o ν } ρ B ( α B (0)) P { o ν } . (2.8)We suppose [ H b , N b ] = 0. If all O µ commute with H b and N b , P { o ν } commutes with ρ B ( α B (0)) and ρ χ tot (0) = ρ (0) ⊗ ρ B ( α B (0)) holds because (cid:80) { o ν } P { o ν } = 1.We defined ρ χ ( t ) def = Tr B [ ρ χ tot ( t )] and the generating function is calculated with Z τ ( { χ O µ } ) = Tr S [ ρ χ ( t = τ )] . (2.9)We assume ρ χ tot ( t ) ≈ ρ χ ( t ) ⊗ ρ B ( α B ( t )) (0 < t ≤ τ ) (2.10)where ρ B ( α B ( t )) def = (cid:79) b b ( α b ( t )) e − β b ( t )[ H b − µ b ( t ) N b ] . (2.11)The FCS-QME [26] is dρ χ ( t ) dt = ˆ K χ ( α t ) ρ χ ( t ) , (2.12)and the initial condition is ρ χ (0) = ρ (0). Here ˆ K χ ( α t ) is the Liouvillian modified by χ . The Liouvillianis given by ˆ K χ ( α ) • = − i [ H S ( α S ) , • ] + (cid:88) b L χb ( α ) • . (2.13)Here and in the following, • ∈ B. L χb ( α ) describes the coupling effects between S and the bath b anddepends on used approximations, for instance, the Born-Markov approximation without or within theRWA and the CGA. Generally, L χb ( α ) has the form: L χb ( α ) • = (cid:88) a c χba ( α ) A a • B a , (2.14)5here A a and B a belong to B and depend on α S , and c χba ( α ) is a complex number which depends on α S , α Sb and α b . If and only if A a , B a (cid:54) = 1, c χba ( α ) depends on χ . After 2.3 we choose the Born-Markovapproximation within RWA; however, in this subsection we assume only Markov property (i.e., ˆ K χ justdepends on α t ). Explicit expression of (2.14) will be given in 2.3 . At χ = 0, the FCS-QME becomes thequantum master equation (QME) dρ ( t ) dt = ˆ K ( α t ) ρ ( t ) . (2.15)ˆ K ( α t ) equals ˆ K χ ( α t ) at χ = 0. In the following, a symbol X without χ denotes X χ | χ =0 .In the Liouville space [25, 26], the left and right eigenvalue equations of the Liouvillian areˆ K χ ( α ) | ρ χn ( α ) (cid:105)(cid:105) = λ χn ( α ) | ρ χn ( α ) (cid:105)(cid:105) , (2.16) (cid:104)(cid:104) l χn ( α ) | ˆ K χ ( α ) = λ χn ( α ) (cid:104)(cid:104) l χn ( α ) | . (2.17)In the Liouville space, A ∈ B is described by | A (cid:105)(cid:105) . The inner produce is defined by (cid:104)(cid:104) A | B (cid:105)(cid:105) = Tr S ( A † B )( A, B ∈ B). In particular, (cid:104)(cid:104) | A (cid:105)(cid:105) = Tr S A holds. A superoperator which operates to a liner operator of thesystem becomes an operator of the Liouville space. The left eigenvectors l χn ( α ) and the right eigenvectors ρ χm ( α ) satisfy (cid:104)(cid:104) l χn ( α ) | ρ χm ( α ) (cid:105)(cid:105) = δ nm . The mode which has the eigenvalue λ χn ( α ) with the maximum realpart is assigned by the label n = 0. Because the conservation of the probability, (cid:104)(cid:104) | ρ ( t ) (cid:105)(cid:105) = 1, and using(2.15), the relation ddt (cid:104)(cid:104) | ρ ( t ) (cid:105)(cid:105) = (cid:104)(cid:104) | ˆ K ( α t ) | ρ ( t ) (cid:105)(cid:105) = 0 (2.18)leads (cid:104)(cid:104) | ˆ K ( α ) = 0, in the limit χ → λ χ ( α ) becomes 0 and (cid:104)(cid:104) l χ ( α ) | becomes (cid:104)(cid:104) | (i.e., l ( α ) is anidentity operator). In addition, the special state | ρ ( α ) (cid:105)(cid:105) determined by ˆ K ( α ) | ρ ( α ) (cid:105)(cid:105) = 0 represents the instantaneous steady state .The formal solution of the FCS-QME (2.12) is | ρ χ ( t ) (cid:105)(cid:105) = T exp (cid:20)(cid:90) t ds ˆ K χ ( α s ) (cid:21) | ρ (0) (cid:105)(cid:105) , (2.19)where T denotes the time-ordering operation. Using this, we obtain the averages of ∆ o µ at time t (cid:104) ∆ o µ (cid:105) t = ∂∂ ( iχ O µ ) (cid:104)(cid:104) | ρ χ ( t ) (cid:105)(cid:105) (cid:12)(cid:12)(cid:12) χ =0 = (cid:90) t du (cid:104)(cid:104) | ˆ K O µ ( α u ) | ρ ( u ) (cid:105)(cid:105) ≡ (cid:90) t du i O µ ( u ) , (2.20)where X O µ ( α ) def = ∂X χ ( α ) ∂ ( iχ O µ ) (cid:12)(cid:12) χ =0 , (2.21)when X is an (super)operator or a c-number and i O µ ( u ) is the current of O µ at time u . Here, we used (cid:104)(cid:104) | ˆ K ( α ) = 0. Moreover, using (cid:104)(cid:104) l ( α ) | = (cid:104)(cid:104) | , λ ( α ) = 0 and (2.17), we obtain (cid:104)(cid:104) | ˆ K O µ ( α ) = λ O µ ( α ) (cid:104)(cid:104) | − (cid:104)(cid:104) l O µ ( α ) | ˆ K ( α ) . (2.22)6ere, (cid:104)(cid:104) l O µ ( α ) | is defined by ∂ (cid:104)(cid:104) l χ ( α ) | ∂ ( iχ Oµ ) (cid:12)(cid:12) χ =0 , then l O µ = − ∂l χ ( α ) ∂ ( iχ Oµ ) (cid:12)(cid:12) χ =0 holds. Now, the current i O µ ( t ) is givenby [24] i O µ ( t ) = (cid:104)(cid:104) | ˆ K O µ ( α t ) | ρ ( t ) (cid:105)(cid:105) = λ O µ ( α t ) (cid:104)(cid:104) | ρ ( t ) (cid:105)(cid:105) − (cid:104)(cid:104) l O µ ( α t ) | ˆ K ( α t ) | ρ ( t ) (cid:105)(cid:105) = λ O µ ( α t ) − (cid:104)(cid:104) l O µ ( α t ) | ddt | ρ ( t ) (cid:105)(cid:105) . (2.23)The current can also be written as i O µ ( t ) = (cid:104)(cid:104) | W O µ ( α t ) | ρ ( t ) (cid:105)(cid:105) , (2.24)where W O µ ( α ) is the current operator defined by (cid:104)(cid:104) | W O µ ( α ) = (cid:104)(cid:104) | ˆ K O µ ( α ) , (2.25)i.e., Tr S [ W O µ ( α ) • ] = Tr S [ ˆ K O µ ( α ) • ] for any • ∈ B. Therefore, using (2.14), the current operator is givenby W O µ ( α ) = (cid:88) b,a c O µ ba ( α ) B a A a . (2.26)Using (2.22), the instantaneous steady current is given by (cid:104)(cid:104) | W O µ ( α ) | ρ ( α ) (cid:105)(cid:105) = λ O µ ( α ) ≡ i O µ ss ( α ) . (2.27)In the following, we suppose that the state of the system at t = 0 is the instantaneous steady state, ρ (0) = ρ ( α ). Then, ρ ( t ) = ρ ( α t ) + O ( ω/ Γ) holds [25] where ω = 2 π/τ and Γ = min n (cid:54) =0 {− Re( λ n ) } . In ω (cid:28) Γ limit, we obtain i O µ ( t ) = i O µ ss ( α t ) − (cid:104)(cid:104) l O µ ( α t ) | ddt | ρ ( α t ) (cid:105)(cid:105) + O (cid:0) ω Γ (cid:1) , (2.28)which leads to (cid:104) ∆ o µ (cid:105) τ = (cid:90) τ dt i O µ ss ( α t ) + (cid:90) C dα n A O µ n ( α ) + O (cid:0) ω Γ (cid:1) , (2.29)where in the second term, the summation symbol (cid:80) n is omitted. Here, α n is the n -th component of thecontrol parameters, C is the trajectory from α to α τ , and A O µ n ( α ) def = −(cid:104)(cid:104) l O µ ( α ) | ∂∂α n | ρ ( α ) (cid:105)(cid:105) , (2.30)is the BSN vector. The BSN vector is also given by [25] A O µ n ( α ) = (cid:104)(cid:104) | W O µ ( α ) R ( α ) ∂∂α n | ρ ( α ) (cid:105)(cid:105) , (2.31)where R ( α ) is the pseudoinverse of the Liouvillian defined by R ( α ) ˆ K ( α ) = 1 − | ρ ( α ) (cid:105)(cid:105)(cid:104)(cid:104) | . (2.32)7he expression of (2.29) was originally derived like the following. The formal solution of the FCS-QMEis expanded as | ρ χ ( t ) (cid:105)(cid:105) = (cid:88) n c χn ( t ) e (cid:82) t ds λ χn ( α s ) | ρ χn ( α t ) (cid:105)(cid:105) . (2.33)Because e (cid:82) t ds λ χn ( α s ) ( n (cid:54) = 0) exponentially damps as a function of time, only n = 0 term remains ifΓ τ (cid:29)
1. Solving the time evolution equation of c χ ( t ) in ω (cid:28) Γ limit, we obtain c χ ( τ ) = c χ (0) exp (cid:20) − (cid:90) τ dt (cid:104)(cid:104) l χ ( α t ) | ddt | ρ χ ( α t ) (cid:105)(cid:105) (cid:21) . (2.34)Here, the argument of the exponential function is called the BSN phase. Substituting this expression and c χ (0) = (cid:104)(cid:104) l χ ( α ) | ρ ( α ) (cid:105)(cid:105) into (2.33), we obtain the expression of ρ χ ( τ ) which provides (2.29). However,when we consider only the average of ∆ o µ , the BSN phase is not essential. All informations of the countingfields up to the first order are included in W O µ As discussed in 1, we propose to identify the average entropy production rate with˙ σ ( t ) def = (cid:88) b β b ( t ) (cid:2) − i H b ( t ) − µ b ( t ) (cid:8) − i N b ( t ) (cid:9)(cid:3) . (2.35)This is given by ˙ σ ( t ) = Tr S [ W σ ( α t ) ρ ( t )] with W σ ( α ) def = (cid:88) b β b [ − W H b ( α ) − µ b {− W N b ( α ) } ] . (2.36)The average entropy production is given by σ def = (cid:90) τ dt ˙ σ ( t )= (cid:90) τ dt J σ ss ( α t ) + (cid:90) C dα n A σn ( α ) + O (cid:0) ω Γ (cid:1) , (2.37)where J σ ss ( α ) def = (cid:88) b β b [ − i H b ss ( α ) − µ b {− i N b ss ( α ) } ] (2.38)and A σn ( α ) is defined in (1.10). Here, we used (2.28) for O µ = H b , N b . The excess entropy production isdefined as (1.9) by σ ex def = (cid:90) C dα n A σn ( α ) + O (cid:0) ω Γ (cid:1) . (2.39) In the research of adiabatic pumping, the expression of (2.29) is essential. In Refs.[23, 24, 25], (2.29) with (2.30) wasused to study the quantum pump. On the other hand, in Ref. [35], (2.29) was derived using the generalized master equation[34] and without using the FCS. In Ref. [35], A O µ n ( α ) was described by the quantity corresponding to the current operatorand the pseudoinverse of the Liouvillian, as shown in (2.31). Reference[25] showed the equivalence between the FCS-QMEapproach and the generalized master equation approach for all orders of pumping frequency. .2 Generalized quantum master equation for entropy production We consider a kind of generalized quantum master equation (GQME) ddt ρ λ ( t ) = K λ ( α t ) ρ λ ( t ) , (2.40)with the initial condition ρ λ (0) = ρ (0). Here, λ is a single real parameter. We suppose that the Liouvillianis given by K λ ( α ) • = − i [ H S ( α S ) , • ] + (cid:88) b L λb ( α ) • (2.41)with L λb ( α ) • = (cid:80) a c λba ( α ) A a • B a and c λba ( α ) (cid:12)(cid:12) λ =0 = c ba . While c χba ( α ) of (2.14) depends on χ if and onlyif A a , B a (cid:54) = 1, c λba ( α ) can depend on λ for all a . We suppose that the solution of (2.40) satisfiesTr S [ ρ (cid:48) ( τ )] = σ, (2.42)where X (cid:48) def = ∂X λ ∂ ( iλ ) (cid:12)(cid:12)(cid:12) λ =0 . This condition is equivalent to (cid:104)(cid:104) |K (cid:48) ( α ) = (cid:104)(cid:104) | W σ ( α ) . (2.43)Let’s consider (cid:104)(cid:104) l λ ( α ) |K λ ( α ) = λ λ ( α ) (cid:104)(cid:104) l λ ( α ) | , (2.44)corresponding to (2.17) for n = 0. Similar to (2.27) and (2.30), λ (cid:48) ( α ) = (cid:104)(cid:104) | W σ ( α ) | ρ ( α ) (cid:105)(cid:105) = J σ ss ( α ) , (2.45)and A σn ( α ) = −(cid:104)(cid:104) l (cid:48) ( α ) | ∂∂α n | ρ ( α ) (cid:105)(cid:105) = (cid:104)(cid:104) | W σ ( α ) R ( α ) ∂∂α n | ρ ( α ) (cid:105)(cid:105) , (2.46)hold. Although λ λ ( α ) and l λ ( α ) depend on the choice of K λ ( α ), λ (cid:48) ( α ) and A σn ( α ) do not depend, as canbe seen in the right-hand side (RHS) of the (2.45) and (2.46). The LHS of (2.43) is given by (cid:104)(cid:104) |K (cid:48) ( α ) = (cid:104)(cid:104) | (cid:88) b,a c (cid:48) ba ( α ) B a A a . (2.47)Using this and (2.26), (2.43) becomes (cid:88) b,a c (cid:48) ba ( α ) B a A a = (cid:88) b,a (cid:104) − β b c H b ba ( α ) + β b µ b c N b ba ( α ) (cid:105) B a A a . (2.48)Infinite solutions of this equation exist. One choice of K λ ( α ) satisfying this relation is ˆ K χ ( α ) in the limitof χ H b → − β b λ and χ N b → β b µ b λ .While we can calculate the average of the entropy production as shown in 2.1 and in this subsection,our formalism is not compatible to discuss the higher moments of the entropy production. “Highermoments” ∂ n ∂ ( iλ ) n Tr S [ ρ λ ( τ )] (cid:12)(cid:12) λ =0 ( n = 2 , , · · · ) depend on the choice of K λ ( α ) and currently there seemsno physical guiding principle to determine an adequate K λ ( α ). Although (2.29) is the average of thedifference between outcomes at t = τ and t = 0 of O µ , there is no bath’s operator corresponding to σ if α B are modulated. In contrast, the higher moments of the entropy production could be considered for theclassical Markov jump process. In Appendix C, we review the entropy production of the Markov jumpprocess [21, 37], and in 4, we compare that and (2.37).9 .3 Rotating wave approximation In this subsection, we introduce the FCS-QME within RWA. First, we introduce the CGA. An operatorin the interaction picture corresponding to A ( t ) is defined by A I ( t ) = U † ( t ) A ( t ) U ( t ) with dU ( t ) dt = − i (cid:34) H S ( α S ( t )) + (cid:88) b H b (cid:35) U ( t ) (2.49)and U (0) = 1. The system reduced density operator in the interaction picture is given by ρ I,χ ( t ) =Tr B [ ρ I,χ tot ( t )] where ρ I,χ tot ( t ) = U ( t ) ρ χ tot ( t ) U † ( t ). ρ I,χ tot ( t ) is governed by dρ I,χ tot ( t ) dt = − i [ H I int ( t ) , ρ I,χ tot ( t )] χ , (2.50)with H int = (cid:80) b H Sb . Up to the second order perturbation in H int , we obtain ρ I,χ ( t + τ CG ) = ρ I,χ ( t ) − (cid:90) t + τ CG t du (cid:90) ut ds Tr B (cid:8) [ H I int ( u ) , [ H I int ( s ) , ρ I,χ ( t ) ⊗ ρ B ( α B ( t ))] χ ] χ (cid:9) ≡ ρ I,χ ( t ) + τ CG ˆ L χτ CG ( t ) ρ I,χ ( t ) , (2.51)using the large-reservoir approximation: ρ I,χ tot ( t ) ≈ ρ I,χ ( t ) ⊗ ρ B ( α B ( t )) (2.52)and Tr B [ H I int ( u ) ρ B ( α B ( t ))] = 0. The arbitrary parameter τ CG ( >
0) is called the coarse-graining time.The CGA [29, 30] is defined by ddt ρ
I,χ ( t ) = ˆ L χτ CG ( t ) ρ I,χ ( t ) . (2.53)At χ = 0, this is Lindblad type. If τ (cid:29) τ CG , the superoperator ˆ L χτ CG ( t ) is described as a function of theset of control parameters at time t . In this paper, we suppose τ (cid:29) τ CG . Moreover, τ CG should be muchshorter than the relaxation time of the system, τ S ∼ . On the other hand, τ S (cid:28) τ should hold for theadiabatic modulation. Hence τ CG (cid:28) (cid:28) τ should hold. By the way, the Born-Markov approximation isgiven by dρ I,χ ( t ) dt = − (cid:90) ∞ ds Tr B (cid:110) [ H I int ( t ) , [ H I int ( t − s ) , ρ I,χ ( t ) ⊗ ρ B ( α B ( t ))] χ ] χ (cid:111) . (2.54)Now we suppose H Sb ( α Sb ) = (cid:88) k,α V bk,α ( α Sb ) a † α c bk + h.c. , (2.55)where a α and c bk are single-particle annihilation operators of the system and of the bath b . Although wehave used indeces α or β to distinguish the system operators, this may not confuse the readers with theset of control parameters or the inverse temperature since they only appear as a subscript of the operator a (or a † ) and the parameters like V bk,α , Φ ± b,αβ , Ψ ± b,αβ or under the summation symbol. The eigenoperatordefined by a α ( ω ) = (cid:88) n,r,m,s δ ω mn ,ω | E n , r (cid:105)(cid:104) E n , r | a α | E m , s (cid:105)(cid:104) E m , s | . (2.56)10s useful to describe the FCS-QME. Here, ω mn def = E m − E n , H S | E n , r (cid:105) = E n | E n , r (cid:105) (2.57)and r denotes the label of the degeneracy. ω is one of the elements of W = { ω mn | (cid:104) E n , r | a α | E m , s (cid:105) (cid:54) = 0 ∃ α } . (2.58) a α ( ω ) and ω depend on α S . (cid:80) ω a α ( ω ) = a α ,[ H S , a α ( ω )] = − ωa α ( ω ) and [ N S , a α ( ω )] = − a α ( ω ) (2.59)hold. Here, N S is total number operator of the system. We suppose [ N S , H S ] = 0. In the CGA orBorn-Markov approximation, the FCS-QME is described by a α ( ω ) and [ a α ( ω (cid:48) )] † ( ω, ω (cid:48) ∈ W ). If H S istime dependent, the generalization of usual RWA [28] with static H S is unclear. In this paper, the RWAis defined as the limit τ CG → ∞ ( τ CG · min ω (cid:54) = ω (cid:48) | ω − ω (cid:48) | (cid:29)
1) of the CGA. If H S is time independent, thisRWA is equivalent to usual RWA.In the following, except for 3.3, we consider the RWA. Then, L χb ( α ) is generally given by (for thedetails of the derivation, please refer [25, 26]) L χb ( α ) • = Π χb ( α ) • − i [ h b ( α ) , • ] , (2.60)where h b ( α ) is a Hermitian operator describing the Lamb shift . h b ( α ) commutes with H S ( α S ) for generalmodel and with N S for the model (2.55). H L ( α ) def = (cid:80) b h b ( α ) is called Lamb shift Hamiltonian . Thesuperoperator Π χb ( α ) represents the dissipation.Here, we suppose the free Hamiltonian of the bath b : H b = (cid:88) k ε bk c † bk c bk , (2.61)and { O µ } = { N b , H b } b . Π χb ( α ) in (2.60) is given byΠ χb ( α ) • = (cid:88) ω (cid:88) α,β (cid:104) Φ − ,χb,αβ ( ω ) a β ( ω ) • [ a α ( ω )] † −
12 Φ − b,αβ ( ω ) • [ a α ( ω )] † a β ( ω ) −
12 Φ − b,αβ ( ω )[ a α ( ω )] † a β ( ω ) • +Φ + ,χb,αβ ( ω )[ a β ( ω )] † • a α ( ω ) −
12 Φ + b,αβ ( ω ) • a α ( ω )[ a β ( ω )] † −
12 Φ + b,αβ ( ω ) a α ( ω )[ a β ( ω )] † • (cid:105) , (2.62)where Φ − ,χb,αβ (Ω) = 2 π (cid:88) k V bk,α V ∗ bk,β F − b ( ε bk ) e iχ Nb e iχ Hb ε bk δ ( ε bk − Ω)= e iχ Nb + iχ Hb Ω Φ − b,αβ (Ω) , (2.63)Φ + ,χb,αβ (Ω) = 2 π (cid:88) k V ∗ bk,α V bk,β F + b ( ε bk ) e − iχ Nb e − iχ Hb ε bk δ ( ε bk − Ω)= e − iχ Nb − iχ Hb Ω Φ + b,αβ (Ω) . (2.64)Here, χ N b and χ H b are the counting fields for N b and H b . If the baths are fermions, F + b ( ε ) = f b ( ε ) def = 1 e β b ( ε − µ b ) + 1 (2.65)11nd F − b ( ε ) = 1 − f b ( ε ). If the baths are bosons, F + b ( ε ) = n b ( ε ) def = 1 e β b ( ε − µ b ) − F − b ( ε ) = 1 + n b ( ε ). The Lamb shift is given by h b ( α ) = (cid:88) ω (cid:88) α,β (cid:16) −
12 Ψ − b,αβ ( ω )[ a α ( ω )] † a β ( ω ) + 12 Ψ + b,αβ ( ω ) a α ( ω )[ a β ( ω )] † (cid:17) , (2.67)where Ψ − b,αβ (Ω) = 2 (cid:88) k V bk,α V ∗ bk,β F − b ( ε bk )P 1 ε bk − Ω , (2.68)Ψ + b,αβ (Ω) = 2 (cid:88) k V ∗ bk,α V bk,β F + b ( ε bk )P 1 ε bk − Ω . (2.69)Here, P denotes the Cauchy principal value. Φ ± b,αβ (Ω) satisfy[Φ ± b,αβ (Ω)] ∗ = Φ ± b,βα (Ω) , (2.70)Φ + b,αβ (Ω) = e − β b (Ω − µ b ) Φ − b,βα (Ω) . (2.71)The latter is the Kubo-Martin-Schwinger (KMS) condition.We introduce projection superoperators P ( α S ) and Q ( α S ) by P ( α S ) | E n , r (cid:105)(cid:104) E m , s | = δ E n ,E m | E n , r (cid:105)(cid:104) E m , s | , (2.72)and Q ( α S ) = 1 − P ( α S ). We define sets of operators B P def = { X ∈ B |P X = X } and B Q def = { X ∈ B |Q X = X } . ˆ K χ P• ∈ B P holds. Then, ˆ K χ Q• ∈ B Q and Q ˆ K χ P = 0 = P ˆ K χ Q , (2.73)hold. This implies that the right eigenvalue equations (2.16) are decomposed into two closed systems ofequations for P ρ χn and for Q ρ χn . Thus, ρ χn is an element of B P or B Q . In particular, ρ χ ∈ B P . Then, thematrix representation of ρ ( α ) by | E n , r (cid:105) is block diagonalized. This implies[ H S ( α S ) , ρ ( α )] = 0 . (2.74)The particle and energy current operators from the system into bath b , w N b ( α ) and w H b ( α ), areusually defined by w X b ( α ) def = − [ L † b ( α ) X S ] † = −L † b ( α ) X S ( X = N, H ) . (2.75)For a superoperator J , J † is defined by (cid:104)(cid:104)J † X | Y (cid:105)(cid:105) = (cid:104)(cid:104) X |J Y (cid:105)(cid:105) ( X, Y ∈ B). L † b ( α ) • = (cid:80) a c ∗ ba ( α ) A † a • B † a holds. w X b ( α ) is a Hermitian operator and is given by w X b ( α ) = − (cid:88) a c ba ( α ) B a X S A a ( X = N, H ) . (2.76)12or the Born-Markov approximation and the CGA, w N b ( α ) = W N b , while w H b ( α ) (cid:54) = W H b ( α ). For RWA, w N b ( α ) = W N b ( α )= (cid:88) ω (cid:88) α,β (cid:110) Φ − b,αβ ( ω )[ a α ( ω )] † a β ( ω ) − Φ + b,αβ ( ω ) a α ( ω )[ a β ( ω )] † (cid:111) , (2.77) w H b ( α ) = W H b ( α )= (cid:88) ω (cid:88) α,β (cid:110) ω Φ − b,αβ ( ω )[ a α ( ω )] † a β ( ω ) − ω Φ + b,αβ ( ω ) a α ( ω )[ a β ( ω )] † (cid:111) , (2.78)hold. Therefore, (2.36) and (2.75) imply that W σ ( α ) is given by W σ ( α ) = (cid:88) b L † b ( α )( β b H S − β b µ b N S ) = (cid:88) b Π † b ( α )( β b H S − β b µ b N S ) . (2.79) In this subsection, we consider equilibrium state β b = β and µ b = µ , and α denotes the set of ( α S , { α Sb } b , β , βµ ). We show that A σn ( α ) is a total derivative of the von Neumann entropy of the instantaneous steadystate. Differentiating (2.44) by iλ and setting λ = 0, we obtain (cid:104)(cid:104) l (cid:48) ( α ) | ˆ K ( α ) + (cid:104)(cid:104) |K (cid:48) ( α ) = λ (cid:48) ( α ) (cid:104)(cid:104) | . (3.1)In the RHS, λ (cid:48) ( α ) = J σ ss ( α ) = 0 holds. The second term of the LHS is (cid:104)(cid:104) | W σ ( α ). (2.79) leads W σ ( α ) = β (cid:88) b L † b ( α )[ H S − µN S ] = β ˆ K † ( α )[ H S − µN S ] , (3.2)i.e., (cid:104)(cid:104) β [ H S − µN S ] | ˆ K ( α ) = (cid:104)(cid:104) | W σ ( α ) . (3.3)Then, (3.1) leads (cid:2) (cid:104)(cid:104) l (cid:48) ( α ) | + (cid:104)(cid:104) β [ H S − µN S ] | (cid:3) ˆ K ( α ) = 0 . (3.4)This implies (cid:104)(cid:104) l (cid:48) ( α ) | = −(cid:104)(cid:104) β [ H S − µN S ] | + c ( α ) (cid:104)(cid:104) | , (3.5)i.e., { l (cid:48) ( α ) } † = − β [ H S − µN S ] + c ( α ) where c ( α ) is an unimportant complex number. The equilibriumstate, ρ ( α ), is given by ρ ( α ) = ρ gc ( α S ; β, βµ ) def = e − β ( H S ( α S ) − µN S ) Ξ( α S ; β, βµ ) , (3.6)with Ξ( α S ; β, βµ ) def = Tr S [ e − β ( H S ( α S ) − µN S ) ]. Then, { l (cid:48) ( α ) } † = ln ρ gc ( α S ; β, βµ ) + c (cid:48) ( α )1 (3.7)with c (cid:48) ( α ) = c ( α ) + ln Ξ( α S ; β, βµ ), holds. Substituting this equation into (2.46), we obtain A σn ( α ) = ∂∂α n S vN ( ρ gc ( α S ; β, βµ )) , (3.8)where we used (A.1) in the Appendix A. 13 .2 Weakly nonequilibrium regime In this subsection, we study the BSN vector and the excess entropy production in a weakly nonequilibriumcondition. We introduce parameters characterizing the degree of nonequilibrium: ε ,b def = β b − β, ε ,b def = β b µ b − βµ, ε def = max b (cid:8) | ε ,b | β , | ε ,b || βµ | (cid:9) , (3.9)where β and βµ are the reference values, which satisfy min b β b ≤ β ≤ max b β b and min b β b µ b ≤ βµ ≤ max b µ b β b . ε is a measure of degree of nonequilibrium. We consider ε (cid:28) K κ ( α ) • def = − i [ H S ( α S ) + κH L ( α ) , • ] + (cid:88) b Π b ( α ) • , (3.10)and corresponding instantaneous steady state ρ ( κ )0 ( α ):ˆ K κ ( α ) ρ ( κ )0 ( α ) = 0 . (3.11)Here, κ is a real parameter satisfying − ≤ κ ≤ (cid:104)(cid:104) | ˆ K κ ( α ) = 0holds. We use the following notations: α ,b def = β b , α ,b def = β b µ b , X def = X (cid:12)(cid:12) α i,b = α i . (3.12)We expand ρ ( κ )0 and l (cid:48) (the derivative of n = 0 left eigenavector for κ = +1) ρ ( κ )0 ( α ) = ρ ( κ )0 + (cid:88) b (cid:16) ε ,b ρ ( κ )1 ,b + ε ,b ρ ( κ )2 ,b (cid:17) + O ( ε ) , (3.13) l (cid:48) ( α ) = l (cid:48) ( α ) + (cid:88) b ( ε ,b k ,b + ε ,b k ,b ) + O ( ε ) , (3.14)with ρ ( κ )0 = ρ gc , l (cid:48) ( α ) = − βH S + βµN S + c ∗ ρ gc + c (cid:48)∗ . (3.15)Here, ρ gc def = ρ gc ( α S ; β, βµ ), c and c (cid:48) are the same with c ( α ) and c (cid:48) ( α ) in 3.1. After some calculations, weobtain following relation ( i = 1 , k i,b = ρ ( − i,b ρ − + c i,b , (3.16)where c i,b is an arbitrary complex number. The details of the derivation are explained in the AppendixB. Using this relation, (3.14) becomes l (cid:48) ( α ) = ln ρ gc ( α S ; β, βµ ) + C ( α )1 + (cid:88) b (cid:88) i =1 ε i,b ρ ( − i,b ρ − + O ( ε )= ln ρ ( − ( α ) + C ( α )1 + O ( ε ) , (3.17)where C ( α ) def = c (cid:48)∗ + (cid:80) b,i c i,b ε i,b . Substituting this equation into (2.46), we obtain A σn ( α ) = − Tr S (cid:104) ln ρ ( − ( α ) ∂ρ (1)0 ( α ) ∂α n (cid:105) + O ( ε ) , (3.18)14here the notation ρ ( α ) = ρ (1)0 ( α ) and ˘ ρ ( α ) = ρ ( − ( α ) is used in 1 for clarity. We supposed [ ρ gc , ρ ( − i,b ] =0, which leads ln ρ ( − ( α ) = ln ρ gc + (cid:88) i,b ε i,b ρ ( − i,b ρ − + O ( ε ) . (3.19)This supposition is satisfied if [ N S , ρ ( − ( α )] = O ( ε ) (which leads [ N S , ρ ( − i,b ] = 0) or βµ = 0 holds. If H S is non-degenerate, [ N S , ρ ( − ( α )] = 0 holds, then [ N S , ρ ( − i,b ] = 0, [ ρ gc , ρ ( − i,b ] = 0 and (3.17) hold.If the states of the baths are in the canonical distributions ( µ b → ρ gc is replaced by the canonicaldistribution and (3.17) holds without any assumption.If [ H L ( α ) , ρ ( κ )0 ( α )] = 0 , (3.20)holds, ρ ( κ )0 ( α ) is independent of κ ( ρ ( κ )0 ( α ) = ρ ( α )), then (3.18) becomes A σn ( α ) = ∂∂α n S vN ( ρ ( α )) + O ( ε ) , (3.21)using (A.1). (3.20) holds if H S is non-degenerate. (3.21) can be shown from [ H L , ρ (1) i,b ] = 0, which is weakerassumption than (3.20) and is derived from (3.20) for κ = 1. If we neglect the Lamb shift Hamiltonian,namely we consider the QME for ˆ K ( α ), (3.21) holds (with a replacement ρ → ρ (0)0 ). From (3.21), weobtain σ ex = S vN ( ρ ( α τ )) − S vN ( ρ ( α )) + O ( ε δ ) , (3.22)with δ = max n,α ∈ C | α n − α n || ¯ α n | , (3.23)where ¯ α n is typical value of the n -th control parameter.Yuge et al. [20] applied the FCS-QME approach to the excess entropy production of the quantumsystem. They introduced a time-dependent observable A ( t ) = − (cid:80) b β b ( t )[ H b − µ b ( t ) N b ] and consideredthe outputs at t = 0 and t = τ as a (0) and a ( τ ). Then, they identified the average σ (cid:48) def = (cid:104) a ( τ ) − a (0) (cid:105) asthe average entropy production. However, σ (cid:48) seems not the average entropy production σ . The average σ (cid:48) can be rewritten as σ (cid:48) ≈ Tr tot [ A ( τ ) ρ tot ( τ )] − Tr tot [ A (0) ρ tot (0)] = (cid:90) τ dt (cid:26) ddt Tr tot [ A ( t ) ρ tot ( t )] (cid:27) ≈ − (cid:90) τ dt (cid:88) b (cid:104) dβ b ( t ) dt (cid:104) H b (cid:105) t − d [ β b ( t ) µ b ( t )] dt (cid:104) N b (cid:105) t (cid:105) + (cid:90) τ dt (cid:88) b (cid:104) β b ( t ) {− ddt (cid:104) H b (cid:105) t } − β b ( t ) µ b ( t ) {− ddt (cid:104) N b (cid:105) t } (cid:105) . (3.24)Here, (cid:104)•(cid:105) t def = Tr tot [ • ρ tot ( t )], ρ tot ( t ) is the total system state and Tr tot denotes the trace of the totalsystem. The integrand of the second term of the last expression of (3.24) roughly equals to ˙ σ The Here, we supposed ddt (cid:104) O (cid:105) t ≈ i O ( t ) for O = H b , N b . However, because the thermodynamic parameters β b and µ b aremodulated, ddt (cid:104) H b (cid:105) t and ddt (cid:104) N b (cid:105) t also include the currents from the outside of the total system to the bath b . time-independent observable although A ( t ) is time-dependent.These two issues are the problems of Ref. [20]. Nevertheless, the obtained Liouvillian (of which theLamb shift Hamiltonian is neglected) incidentally satisfies (2.43). Using that Liouvillian, for the systemwith time-reversal symmetry, Yuge et al. studied the relation between A σn ( α ) and the symmetrized vonNeumann entropy. In contrast, we do not suppose the time-reversal symmetry to derive (3.18). In 5, weconsider the time-reversal symmetric system. We denote the BSN vector for the entropy production and instantaneous steady state of the Born-Markovapproximation by A σ, BM n ( α ) and ρ BM0 ( α ). Then, A σ, BM n ( α ) = A σn ( α ) + O ( v ) , (3.25) S vN ( ρ BM0 ( α )) = S vN ( ρ ( α )) + O ( v ) , (3.26)hold [20]. Here, v = u and u ( (cid:28)
1) describes the order of H Sb . The above two equations and (3.21) lead A σ, BM n ( α ) = ∂∂α n S vN ( ρ BM0 ( α )) + O ( ε ) + O ( v ) . (3.27) In this section, we compare the preceding study on the entropy production in the classical Markov jumpprocess [21, 37] with ours. We consider the Markov jump process among the states n = 1 , , · · · , N , wherethe definitions are explained in Appendix C. The probability to find the system in a state n is p n ( t ) andit obeys the master equation: dp n ( t ) dt = N (cid:88) m =1 K nm ( α t ) p m ( t ) . (4.1)The Liouvillian is given by K nm ( α ) = (cid:88) b K ( b ) nm ( α ) (4.2)where K ( b ) nm originates the couping between the system and the bath b . (cid:80) n K ( b ) nm ( α ) = 0 holds. Wesuppose that K ( b ) mn ( α ) (cid:54) = 0(= 0) holds if K ( b ) nm ( α ) (cid:54) = 0(= 0) for all n (cid:54) = m . The definition of the entropyproduction for each Markov jump process (C.1) is (C.4). The average entropy production σ C is given by(see (C.10)) σ C = (cid:90) τ dt (cid:88) n,m σ C nm ( α t ) p m ( t ) , (4.3)where σ C nm ( α ) = − K nm ( α ) ln K nm ( α ) K mn ( α ) . (4.4)16e denote the solution of the QME with RWA by ρ ( t ). We suppose p n ( t ) def = (cid:104) n | ρ ( t ) | n (cid:105) is governedby (4.1) with K ( b ) nm ( α ) = (Π b ( α )) nn,mm . Here, | n (cid:105) is the energy eigenstate of H S ( α S ),(Π b • ) nm = (cid:88) k,l (Π b ( α )) nm,kl ( • ) kl (4.5)and ( • ) kl def = (cid:104) k | • | n (cid:105) . This supposition implies (3.20). A sufficient condition by which p n ( t ) obeys (4.1)is below: (1) H S ( α S ) is non-degenerate and (2) { α n ∈ α S | ∂∂α n | n (cid:105) (cid:54) = 0 } are fixed. The eigenenergy candepend on { α n ∈ α S | ∂∂α n | n (cid:105) = 0 } . We show that our average entropy production (2.37) is given by asimilar expression of (4.3): σ = (cid:90) τ dt (cid:88) n,m σ nm ( α t ) p m ( t ) . (4.6)Here, σ nm ( α ) def = (cid:88) b K ( b ) nm ( α ) θ ( b ) nm ( α ) = − (cid:88) b K ( b ) nm ( α ) ln K ( b ) nm ( α ) K ( b ) mn ( α ) , (4.7)with θ ( b ) nm ( α ) def = − ln K ( b ) nm ( α ) K ( b ) mn ( α ) K ( b ) nm ( α ) (cid:54) = 00 K ( b ) nm ( α ) = 0 . (4.8)Because of (2.75), (2.77) and (2.78), the particle and energy currents are given by i X b = Tr S [ W X b ρ ( t )]with W X b = − (Π † b X S ) † ( X = H, N ). (2.76) leads( W X b ) nm = − (cid:88) k,l (Π b ) lk,mn ( X S ) kl . (4.9)We suppose ( X S ) nm = ( X S ) nn δ nm for X = N, H . Since ( X S ) kl is a diagonal matrix, ( W X b ) nm is also adiagonal matrix. Then, i X b = (cid:88) m ( W X b ) mm p m ( t ) , (4.10)holds. Substituting ( W X b ) mm = − (cid:80) n K ( b ) nm ( X S ) nn into (4.10), we obtain i X b = − (cid:88) n,m K ( b ) nm ( X S ) nn p m ( t )= (cid:88) n,m K ( b ) nm [( X S ) mm − ( X S ) nn ] p m ( t ) . (4.11)This equation leads˙ σ ( t ) = − (cid:88) n,m (cid:88) b K ( b ) nm β b ( t ) { [( H S ) mm − ( H S ) nn ] − µ b ( t )[( N S ) mm − ( N S ) nn ] } p m ( t ) . (4.12)17sing the local detailed balance conditionln K ( b ) nm ( α ) K ( b ) mn ( α ) = β b { [( H S ) mm − ( H S ) nn ] − µ b [( N S ) mm − ( N S ) nn ] } , (4.13)we obtain (4.6).Now we introduce a matrix K λ ( α ) by[ K λ ( α )] nm def = (cid:88) b K ( b ) nm ( α ) e iλθ ( b ) nm ( α ) . (4.14)Then, we obtain ∂∂ ( iλ ) (cid:12)(cid:12)(cid:12) λ =0 (cid:88) n,m (cid:104) T exp (cid:2) (cid:90) τ dt K λ ( α t ) (cid:3)(cid:105) nm p m (0) = (cid:90) τ dt (cid:88) n,m σ nm ( α t ) p m ( t ) = σ. (4.15) K λ was originally introduced by Sagawa and Hayakawa [19]. About averages, our entropy production isthe same with Sagawa and Hayakawa.We show that the difference between σ C nm ( α ) and σ nm ( α ) is O ( ε ): σ C nm ( α ) = σ nm ( α ) + O ( ε ) . (4.16)In fact, K ( b ) nm can be expanded as K ( b ) nm = γ b ¯ K nm + (cid:88) i =1 , ε i,b K i,bnm + O ( ε ) , (cid:88) b γ b = 1 , (4.17)then we obtain σ C nm ( α ) = σ (0 , nm + σ C(2) nm ( α ) + O ( ε ) , (4.18) σ nm ( α ) = σ (0 , nm + σ (2) nm ( α ) + O ( ε ) , (4.19)with σ (0 , nm def = − ¯ K nm ln ¯ K nm ¯ K mn + (cid:88) i,b ε i,b (cid:104) K i,bnm ln ¯ K nm ¯ K mn + K i,bnm − K i,bmn ¯ K nm ¯ K mn (cid:105) . (4.20) σ C(2) nm ( α ) and σ (2) nm ( α ) are quadratic orders of ε i,b . While the former includes ε i,b ε i (cid:48) ,b (cid:48) ( b (cid:54) = b (cid:48) ) terms, thelatter dose not. (4.16) leads σ Cex = σ ex + O ( ε δ ) . (4.21)Here, σ Cex is given by (C.13). Then, (C.12), the result of Ref. [21], coincides with (3.22) when p n ( t ) = (cid:104) n | ρ ( t ) | n (cid:105) is governed by the master equation (4.1).18 Time-reversal operations
In this section, we define the time-reversal operation and examine the dependence of the excess entropyproduction on the time-reversal symmetry. We denote the time-reversal operator of the system by θ . Wethen define ˜ Y def = θY θ − , (5.1)for all Y ∈ B and ˜ J ˜ Y def = θ ( J Y ) θ − , (5.2)for a superoperator J of the system. The time-reversal of ˆ K ( α ) ρ ( α ) = 0 is given by i [ ˜ H L ( α ) , ˜ ρ ( α )] + (cid:88) b ˜Π b ( α ) ˜ ρ ( α ) = 0 , (5.3)using (2.74). If ˜ H L ( α ) = H L ( α ) , ˜Π b ( α ) = Π b ( α ) , (5.4)hold, the above equation coincides with the equation of ρ ( − ( α ) since [ H S , ρ ( κ )0 ] = 0, then˜ ρ ( α ) = ρ ( − ( α ) def = ˘ ρ ( α ) , (5.5)holds. If the total Hamiltonian is time-reversal invariant, (5.4) holds [38]. If (5.4) holds and we neglectthe Lamb shift Hamiltonian, the instantaneous steady state is time-reversal invariant: ˜ ρ (0)0 = ρ (0)0 .For time-reversal symmetric system, ∂∂α n S sym ( ρ ( α )) = − Tr S (cid:104) ln ˜ ρ ( α ) ∂ρ ( α ) ∂α n (cid:105) + O ( ε ) , (5.6)holds. Here, S sym ( ρ ) def = − Tr S (cid:2) ρ
12 (ln ρ + ln ˜ ρ ) (cid:3) , (5.7)is the symmetrized von Neumann entropy. Combining (3.18) with (5.5), we obtain A σn ( α ) = ∂∂α n S sym ( ρ ( α )) + O ( ε ) , (5.8)then, the equation (3.22) with S vN → S sym holds. As analogy, we consider S (cid:48) ( α ) def = − Tr S (cid:2) ρ ( α ) 12 (ln ρ ( α ) + ln ˘ ρ ( α )) (cid:3) , (5.9)for generally non-time-reversal symmetric system. The difference between ∂S (cid:48) ( α ) /∂α n and the first termof the RHS of (3.18) is ∂S (cid:48) ( α ) ∂α n − (cid:16) − Tr S (cid:104) ln ˘ ρ ( α ) ∂ρ ( α ) ∂α n (cid:105)(cid:17) = −
12 Tr S (cid:2) ∂ρ ∂α n (ln ρ − ln ˘ ρ ) (cid:3) −
12 Tr S (cid:2) ρ ∂∂α n ln ˘ ρ (cid:3) . (5.10)19o calculate the RHS of this equation, we use formulasln( A + ηB ) = ln A + (cid:90) ∞ ds (cid:16) η A + s B A + s − η A + s B A + s B A + s + O ( η ) (cid:17) , (5.11) ∂∂α n ln A ( α ) = (cid:90) ∞ ds A ( α ) + s ∂A ( α ) ∂α n A ( α ) + s , (5.12)where A, B, A ( α ) ∈ B and η is small real number. ρ − ˘ ρ = εψ + O ( ε ) holds because ρ ( κ )0 = ρ gc ( α S ; β, βν ).Then, the first term of the RHS of (5.10) is given by −
12 Tr S (cid:2) ∂ρ ∂α n (ln ρ − ln ˘ ρ ) (cid:3) = − ε (cid:90) ∞ ds Tr S (cid:104) ∂ρ ∂α n ρ + s ψ ρ + s (cid:105) + O ( ε ) . (5.13)The second term of the RHS of (5.10) is given by −
12 Tr S (cid:2) ρ ∂∂α n ln ˘ ρ (cid:3) = − (cid:90) ∞ ds Tr S (cid:104) ∂ ˘ ρ ∂α n ρ + s ( ˘ ρ + εψ ) 1˘ ρ + s (cid:105) + O ( ε )= −
12 Tr S (cid:2) ∂ ˘ ρ ∂α n (cid:3) − ε (cid:90) ∞ ds Tr S (cid:104) ∂ ˘ ρ ∂α n ρ + s ψ ρ + s (cid:105) + O ( ε )= − ε (cid:90) ∞ ds Tr S (cid:104) ∂ ˘ ρ ∂α n ρ + s ψ ρ + s (cid:105) + O ( ε )= − ε (cid:90) ∞ ds Tr S (cid:104) ∂ ( θ ˘ ρ θ − ) ∂α n ρ + s ˜ ψ ρ + s (cid:105) + O ( ε ) . (5.14)Here, we used ε ( ˘ ρ + s ) − = ε ( ρ + s ) − + O ( ε ) and Tr S • = Tr S ˜ • if Tr S • is real. In general, the RHSof (5.10) is not O ( ε ). However, if ˜ ρ = ˘ ρ holds, the RHS of (5.10) becomes O ( ε ) since ˜ ψ = − ψ ,then (5.6) holds. In the proof of (5.6), Yuge et al. [20] used incorrect equations ∂∂α n ln ˜ ρ = ˜ ρ − ∂ ˜ ρ ∂α n andln ρ − ln ˜ ρ = εψ ˜ ρ − + O ( ε ). In this paper, for open systems described by the quantum master equation (QME), we investigated theexcess entropy production under quasistatic operations between nonequilibrium steady states (NESSs).We propose a new definition of the average entropy production rate ˙ σ ( t ) using the average energy andparticle currents, which are calculated by using the full counting statistics (FCS) with QME (FCS-QME).Then, we introduced the generalized QMEs (GQMEs) providing ˙ σ ( t ). The GQMEs do not relate thehigher moments (thus and the FCS) of the entropy production, but we can calculate only the average of theentropy production. Using the GQME, in weakly nonequilibrium regime, we analyzed the Berry-Sinitsyn-Nemenman (BSN) vector for the entropy production, A σn ( α ), which provides the excess entropy production σ ex under quasistatic operations between NESSs as the line integral of A σn ( α ) in the parameter space. Wehave shown that the BSN vector A σn ( α ) for the entropy production is given by ρ ( α ), the instantaneoussteady state of the QME and ˘ ρ ( α ), that of the QME which is given by reversing the sign of the Lambshift term. If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, we obtainthat the excess entropy production is given by the difference of the von Neumann entropies at the initialand final times of the operation. In general, the potential S ( α ) such that A σn ( α ) = ∂ S ( α ) ∂α n + O ( ε ) dosenot exist, but for time-reversal symmetric system, we showed that S ( α ) is the symmetrized von Neumann20ntropy. Additionally, we pointed out that preceding expression of the entropy production in the classicalMarkov jump process [21, 37] is different from ours and showed that these approximately equivalent inthe weakly nonequilibrium regime. We also checked that the definition of the average entropy productionin the classical Markov jump process by Ref. [19] is equivalent to ours. Acknowledgements
We acknowledge helpful discussions with S. Okada. Part of this work is supported by JSPS KAK-ENHI (26247051).
A Derivative of the von Neumann entropy
We show that ∂S vN ( ρ ( α )) ∂α n = − Tr S (cid:104) ln ρ ( α ) ∂ρ ( α ) ∂α n (cid:105) . (A.1)From the definition of the von Neumann entropy, the LHS of the above equation is given by ∂S vN ( ρ ( α )) ∂α n = − Tr S (cid:104) ln ρ ( α ) ∂ρ ( α ) ∂α n (cid:105) − Tr S (cid:104) ∂ ln ρ ( α ) ∂α n ρ ( α ) (cid:105) . (A.2)Using (5.12), the second term of the RHS of the above equation becomes − Tr S (cid:104) ∂ ln ρ ( α ) ∂α n ρ ( α ) (cid:105) = − Tr S (cid:104) (cid:90) ∞ ds ρ ( α ) + s ∂ρ ( α ) ∂α n ρ ( α ) + s ρ ( α ) (cid:105) = − Tr S (cid:104) (cid:90) ∞ ds ρ ( α )( ρ ( α ) + s ) ∂ρ ( α ) ∂α n (cid:105) = − Tr S (cid:104) ∂ρ ( α ) ∂α n (cid:105) = 0 . (A.3)Then, we obtain (A.1). B Derivation of the relation between k i,b and ρ κi,b In this section, we examin the relation of the coefficients of the expansion of ρ κ ( α ) and l (cid:48) ( α ) in (3.14) of3.2.First, we investigate k i,b in (3.14). (3.1) can be rewritten asˆ K † ( α ) l (cid:48) ( α ) + [ K (cid:48) ( α )] † J σ ss ( α ) . (B.1)Here, J σ ss ( α ) = O ( ε ) holds because i H b ss ( α ) , i N b ss ( α ) = O ( ε ) and J σ ss ( α ) = (cid:88) b (cid:0) − i H b ss ( α ) ε ,b + i N b ss ( α ) ε ,b (cid:1) (B.2)since (cid:88) b i X b ss ( α ) = − Tr S [ X S (cid:88) b L b ( α ) ρ ( α )] = 0 , ( X = N, H ) . (B.3)21hen we obtain ∂ i,b K (cid:48)† K † k i,b + ∂ i,b L b † l (cid:48) = 0 , (B.4)in O ( ε i,b ). Here, ∂ i,b X def = ∂X/∂α i,b and K def = ˆ K . The first term of the LHS is ∂ i,b K (cid:48)† ∂ [ K (cid:48) ] † ∂α i,b (cid:12)(cid:12)(cid:12) α i,b = α i = ∂ L † b [ α ,b H S − α ,b N S ] ∂α i,b (cid:12)(cid:12)(cid:12) α i,b = α i = ∂ i,b L b † [ βH S − βµN S ] + Π b † ∂ [ α ,b H S − α ,b N S ] ∂α i,b . (B.5)The third term of the LHS becomes ∂ i,b L b † l (cid:48) = ∂ i,b L b † ( − βH S + βµN S + c (cid:48)∗ − ∂ i,b L b † ( βH S − βµN S ) . (B.6)Here, we used ∂ i,b L b † K † K † k ,b + Π b † H S = 0 , (B.7) K † k ,b − Π b † N S = 0 . (B.8)Next, we show the relation between k i,b and ρ ( − i,b . (3.11) leads K κ ρ ( κ ) i,b + ∂ i,b L b ρ gc = 0 , (B.9)in O ( ε i,b ). Here, K κ def = ˆ K κ . By the way, L b ρ gc ( α S ; β b , β b µ b ) = 0 , (B.10)holds. Differentiating this equation by α i,b , we obtain ∂ i,b L b ρ gc = −L b ρ gc ( α S ; β b , β b µ b ) ∂α i,b = L b ∂ [ α ,b H S − α ,b N S ] ∂α i,b ρ gc ( α S ; β, βµ ) . (B.11)Substituting these equations into (B.9), we obtain K κ ρ ( κ )1 ,b + Π b ( H S ρ gc ) = 0 , (B.12) K κ ρ ( κ )2 ,b − Π b ( N S ρ gc ) = 0 . (B.13)Now, we use Π b ( • ρ gc ) = (Π b † • ) ρ gc , (B.14)which is derived from KMS condition (2.71). Using this relation, we rewire (B.12) and (B.13) as K κ ρ ( κ )1 ,b + (Π b † H S ) ρ gc = 0 , (B.15) K κ ρ ( κ )2 ,b − (Π b † N S ) ρ gc = 0 . (B.16)22ultiplying ρ − from the right, we obtain( K κ ρ ( κ )1 ,b ) ρ − + Π b † H S = 0 , (B.17)( K κ ρ ( κ )2 ,b ) ρ − − Π b † N S = 0 . (B.18)(B.14) can be rewritten as (Π b Y ) ρ − = Π b † ( Y ρ − ) , (B.19)for any Y = • ρ gc ∈ B by multiplying ρ − from the right. (B.19) leads(Π ρ ( κ ) i,b ) ρ − = Π † ( ρ ( κ ) i,b ρ − ) , (B.20)where Π def = (cid:80) b Π b . By the way, [ H S ( α S ) , ρ ( κ )0 ( α )] = 0 holds similarly to (2.74). Differentiating thisequation by α i,b , we obtain [ H S ( α S ) , ρ ( κ ) i,b ] = 0 . (B.21)This relation leads ( H × κ ρ ( κ ) i,b ) ρ − = H × κ ( ρ ( κ ) i,b ρ − ) = H ×− κ † ( ρ ( κ ) i,b ρ − ) , (B.22)where H × κ • def = − i [ H S ( α S ) + κH L ( α ) , • ]. We used ( H × κ ) † = − H × κ . In the first equality, we used that ρ gc commutes with H S and H L . (B.20) and (B.22) lead( K κ ρ ( κ ) i,b ) ρ − = K †− κ ( ρ ( κ ) i,b ρ − ) . (B.23)Substituting this into (B.17) and (B.18), we obtain K †− κ ( ρ ( κ )1 ,b ρ − ) + Π b † H S = 0 , (B.24) K †− κ ( ρ ( κ )2 ,b ρ − ) − Π b † N S = 0 . (B.25)Subtracting (B.24) ((B.25)) for κ = − K † ( k i,b − ρ ( − i,b ρ − ) = 0 . (B.26)This means k i,b = ρ ( − i,b ρ − + c i,b , (B.27)where c i,b is an arbitrary complex number. C Definition of entropy production of the Markov jump process
Except (C.9), this section is based on Ref. [21]. We consider the Markov jump process on the states n = 1 , , · · · , N : n ( t ) = n k ( t k ≤ t < t k +1 ) , t = 0 < t < t · · · < t n < t N +1 = τ. (C.1)23here N = 0 , , , · · · is the total number of jumps. We denote the above path byˆ n = ( N, ( n , n , · · · , n N ) , ( t , t , · · · , t N )) . (C.2)The probability to find the system in a state n is p n ( t ) and it obeys the master equation (4.1). We supposethe trajectory of the control ˆ α = (cid:0) α ( t ) (cid:1) τt =0 is smooth. Now we introduce θ nm ( α ) def = (cid:40) − ln K nm ( α ) K mn ( α ) K nm ( α ) (cid:54) = 00 K nm ( α ) = 0 . (C.3)If n (cid:54) = m , this is entropy production of process m → n . The entropy production of process (C.2) is definedby Θ ˆ α [ˆ n ] = N (cid:88) k =1 θ n k n k − ( α t k ) . (C.4)Then the weight (the transition probability density) associated with a path ˆ n is T ˆ α [ˆ n ] = N (cid:89) k =1 K n k n k − ( α t k ) exp (cid:104) N (cid:88) k =0 (cid:90) t k +1 t k dt K n k n k ( α t ) (cid:105) . (C.5)The integral over all the paths is defined by (cid:90) D ˆ n Y [ˆ n ] def = ∞ (cid:88) N =0 n k − (cid:54) = n k (cid:88) n ,n , ··· ,n N (cid:90) τ dt (cid:90) τt dt (cid:90) τt dt · · · (cid:90) τt N − dt N Y [ˆ n ] , (C.6)and the expectation value of X [ˆ n ] is defined by (cid:104) X (cid:105) ˆ α def = (cid:90) D ˆ n X [ˆ n ] p ss n ( α ) T ˆ α [ˆ n ] . (C.7)Here, p ss n ( α ) is the instantaneous stationary probability distribution characterized by (cid:80) m K nm ( α ) p ss m ( α ) =0. We introduce a matrix K λ ( α ) by[ K λ ( α )] nm def = K nm ( α ) e iλθ nm ( α ) . (C.8)Then, the k -th order moment of the entropy production is given by (cid:104) (Θ ˆ α [ˆ n ]) k (cid:105) ˆ α = ∂ k ∂ ( iλ ) k (cid:12)(cid:12)(cid:12) λ =0 (cid:88) n,m (cid:104) T exp (cid:2) (cid:90) τ dt K λ ( α t ) (cid:3)(cid:105) nm p ss m ( α ) . (C.9)In particular, the average is given by σ C def = (cid:104) Θ ˆ α [ˆ n ] (cid:105) ˆ α = (cid:90) τ dt (cid:88) n,m σ C nm ( α t ) p m ( t ) , (C.10)where σ C nm ( α ) def = K nm ( α ) θ nm ( α ) = − K nm ( α ) ln K nm ( α ) K mn ( α ) . (C.11)24ccording to Ref. [21], for a quasi-static operation, σ Cex = S Sh [ p ss ( α τ )] − S Sh [ p ss ( α )] + O ( ε δ ) , (C.12)holds where σ Cex def = σ C − (cid:90) τ dt (cid:88) n,m σ C nm ( α t ) p ss m ( α t ) , (C.13)and S Sh [ p ] def = − (cid:80) n p n ln p n . References [1] Landauer, R.: dQ = T dS far from equilibrium. Phys. Rev. A , 255 (1978).[2] Oono, Y., Paniconi, M.: Steady State Thermodynamics. Prog. Theor. Phys. 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