Entropy Production of Open Quantum System in Multi-Bath Environment
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Entropy Production of Open Quantum System in Multi-Bath Environment
Cheng-Yun Cai,
1, 2
Sheng-Wen Li,
3, 2
Xu-Feng Liu, and C. P. Sun
3, 2, ∗ State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics andUniversity of the Chinese Academy of Sciences, Beijing 100190, People’s Republic of China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Beijing Computational Science Research Center, Beijing 100084, China Department of Mathematics, Peking University, Beijing 100871, China
We study the entropy production of an open quantum system surrounded by a complex environ-ment consisting of several heat baths at different temperatures. The detailed balance is elaboratedin view of the distinguishable channels provided by the couplings to different heat baths, and arefined entropy production rate is derived accordingly. It is demonstrated that the entropy produc-tion rates can characterize the quantum statistical property of the baths: the bosonic and fermionicbaths display different behaviors in the high-temperature limit while they have the same asymptoticbehavior at low temperature.
PACS numbers: 05.10.Gg, 05.40.Ca, 05.70.Ln,
Introduction. —The concept of entropy plays an impor-tant role in our understanding of complex physical sys-tems. For a closed system its entropy will never decreaseif the unitarity condition is satisfied whether the systemenjoys time reversal invariance or not [1, 2]. For an opensystem contacting with its environment entropy produc-tion is a pivotal concept. The so called entropy produc-tion rate (EPR) is usually regarded as a signature of theirreversibility associated with such a system [3]. In fact,EPR is a proper physical quantity tagging the steadystate of an open system. A lot of open systems canbe well studied in the framework of time-homogeneousMarkov process. The steady states of such systems fallinto two categories: equilibrium steady states and non-equilibrium ones. In a non-equilibrium steady state thedetailed balance is broken, or equivalently, the EPR doesnot vanish. Thus one can say that the system is in anequilibrium steady state if and only if the accompanyingEPR is zero.It should be noted that from purely mathematicalpoint of view the reversibility of a time-homogeneousMarkov process and the detailed balance condition hasbeen thorough studied by Kolmogorov [4].For a Markov process determined by the Pauli masterequation ˙ p i = P j p j L ji − p i L ij , the breaking of detailedbalance is quantitively characterized by a non-vanishingEPR [5–8] R = 12 X ij ( p i L ij − p j L ji ) ln p i L ij p j L ji . (1)Here, p j is the probability of the open system with whichthe open system appears in state- j and L ij is the transi-tion rate from state- i to state- j . If the detailed balance issatisfied, i.e., p i L ij = p j L ji , there is no entropy produc-tion when the system reaches the steady state. This ob-servation has been the starting point of a fruitful study ofsome non-equilibrium biochemical reactions (see Ref. [9] OS...
Figure 1: (color online). A complicated non-equilibrium pro-cess of an open system (OS) contacting with n heat baths,where the temperature of the i -th heat bath is T i . and references therein).The above formula of EPR is valid for a classical sys-tem contacting with a single canonical heat bath. Forthe system in a complex environment consisting of twoor more heat baths at different temperatures (see Fig. 1),which allows more complicated non-equilibrium processes[10], the formula should be generalized. In fact, if theconventional EPR formula (1) were applied in this case,we would obtain a vanishing EPR whenever the systemreaches the steady state. This contradicts the intuitionalphysical picture. We notice that the non-equilibrium pro-cesses induced by the multi-bath environment emerge inmany practical systems, such as an opto-mechanical sys-tem with quantum cavity field coupled to two movingcavity wells at different temperatures [11], and a quan-tum dot system gated by two electrodes [12, 13] at dif-ferent temperatures.In this letter, we first derive a quantum EPR formulafor the above mentioned multi-bath case from the masterequation of the open quantum system. When the quan-tum coherence is neglected, our general result reduces tothe multi-channel expression of EPR as given in Ref. [10].Then we show that the EPRs of non-equilibrium systemswith bosonic and fermionic baths have a similar behaviorat low temperature but behave quite differently in thehigh temperature region. This leads to the conclusion:non-equilibrium process is important as it can character-istically reflect the quantum statistical property of thebath. Entropy Production Rate for Open Quantum Sys-tems. —The Markovian evolution of an open quantumsystem in contact with its environment is described bya dynamical semigroup Λ t . To be precise, we have ρ ( t ) = Λ t ρ (0) and the density matrix ρ ( t ) satisfies themaster equation ˙ ρ = L [ ρ ] with Lindblad super-operator L (we always work in the interaction picture hereafter) [14].For a heat bath at temperature T , it has been shownthat the EPR can be formulated by the relative en-tropy as R (0) = − dd t S ( ρ || ρ th ) [8, 15]. Here, the referencestate ρ th = exp[ − βH ] /Z is the steady state given by L [ ρ th ] = 0 , and it is the thermal equilibrium state at thetemperature T = β − . The EPR formula is decomposedinto two terms, one of which is − dd t tr[ ρ ln ρ ] , representingthe entropy changing rate of the open system itselfandthe other of which takes the form: dd t tr( ρ ln ρ th ) = dd t tr[ ρ ( − βH )] = − T d h H i d t . (2)Since − dd t h H i := ˙ Q is the rate of the heat dissipatinginto the heat bath [16], this term describes the entropychanging rate of the heat bath. Thus this EPR formulaactually counts the total entropy changing rate of boththe open system and its environment.For an open quantum system interacting with N reservoirs (Fig. 1), the master equation assumes theform ˙ ρ = P Nl =1 L l [ ρ ] , where L l is the Lindblad super-operator corresponding to the l -th reservoir. In timeinterval d t , the energy dissipating into reservoir- l is ¯d Q l := − tr (cid:2) L l [ ρ ] H (cid:3) d t [16], thus the entropy changingrate through the l -th reservoir is ˙ Q l T l = − T l tr (cid:2) L l [ ρ ] H (cid:3) , (3)where ρ th l := exp[ − β l H ] /Z l is the thermal state ofreservoir- l with temperature β − l . Then the entropy pro-duction rate for a non-equilibrium quantum system isobtained as R = dd t tr[ ρ ln ρ ] + P l ˙ Q l /T l , i.e, R = − N X l =1 tr (cid:2) L l [ ρ ](ln ρ − ln ρ th l ) (cid:3) , (4)which represents the total entropy changing rate of theopen system and its multi-bath environment. It has beenproved that the quantity R l := − tr (cid:2) L l [ ρ ](ln ρ − ln ρ th l ) (cid:3) is non-negative [8], so we always have R = P l R l > .The dynamics of the quantum coherence is usually de-coupled from that of the populations (when we say quan-tum coherence, we mean the effect contributed from the Figure 2: (color online). The transition diagram of a four-level system contacting with four heat baths. We use differentcolors to distinguish the channels corresponding to differentheat baths. off-diagonal terms h i | ρ | j i of ρ in the eigen energy rep-resentation) [15]. When the quantum coherence is ne-glected (the effect of the quantum coherence will be stud-ied later), the Lindblad equation reduces to Pauli masterequation, and the above Eq. (4) reduces to R ≃ − N X l =1 X i h i |L l [ ρ ] | i ih i | (ln ρ − ln ρ th l ) | i i . (5)Replacing h i |L l [ ρ ] | i i by P j ( p j L ( l ) ji − p i L ( l ) ij ) , we obtain R = 12 N X l =1 X i,j ( p j L ( l ) ji − p i L ( l ) ij )(ln p i p j + E i − E j T l ) Here L ( l ) ij is the transition rate from state- i tostate- j resulted from the coupling to the l -th heatbath, and we have used the fact that h i | ρ th l | i i =exp( − E i /T l ) /Z l . Considering the microscopic reversibil-ity condition L ( l ) ij /L ( l ) ji = exp[( E i − E j ) /T l ] , we then have R = 12 X i = j N X l =1 ( p i L ( l ) ij − p j L ( l ) ji ) ln p i L ( l ) ij p j L ( l ) ji , (6)a refined entropy production rate. This is the same asEq. (10) in Ref. [10]. If the quantum coherence is takeninto account, there would be extra entropy production.Note that in the above refined entropy production rate(REPR) transitions caused by different heat baths (seeFig. 2) are treated separately while in the spirit of theconventional EPR Eq. (1) they should be merged. Thisessential difference naturally leads to different under-standings of equilibrium state. Elaborate Detailed Balance and Time Reversibility. —As we have argued above, if the environment is composedof two or more heat baths, the REPR (6), instead of theconventional EPR Eq. (1), should be used to investigatethe entroy production problem. Then, the condition forzero EPR is refined as p i L ( l ) ij = p j L ( l ) ji . (7) Figure 3: (color online). The transition diagram of a Λ -typesystem contacting with two heat baths. (a) A rough descrip-tion of the system’s possible transitions, ↔ and ↔ .(b) Each transition in (a) may be further divided into twochannels, each of which corresponds to a transition caused bya certain heat bath. This condition is subtler than the detailed balance con-dition p i L ij = p j L ji and justifies the name of elaboratedetailed balance (EDB). The quantity L ( l ) ij can be viewedas the transition rate from state- i to state- j through the l -th channel. From this point of view, the EDB requiresnot only the balance of transitions between any two statesof the system, but also the balance of each transitionchannel. This property has been suggested by Lewis asa criterion for equilibrium [17].Let us probe further the concept of transition channelwith the case of a Λ -type system contacting with twoheat baths. We consider the transition from state-1 tostate-2, which releases energy of amount ω to its envi-ronment. If there appears an energy increase of ω inthe first heat bath, then it can be judged that this tran-sition results from the first heat bath. Thus it is phys-ically justified to understand the transition as realizedthrough two distinguishable channels, each of which cor-responds to a transition caused by a certain heat bath.As a consequence of the validity of the concept of dis-tinguishable channels it can be argued that a transitionchain like · · · → → → · · · is not a complete de-scription. A complete description should be somethinglike · · · (1) −→ (2) −→ (2) −→ · · · , where (1) −→ denotes atransition from state-1 to state-2 via the first channel and (2) −→ denotes a transition from state-1 to state-3 viathe second channel. All the possible channels togethermake up a transition diagram as shown in Fig. 3b.Under certain conditions the equivalence between theconventional detailed balance and the time reversibilityof a Markov process has been proved by Kolmogorov[4]. As there is no generic way to model the evolutionof the open system with distinguishable transition chan-nels as a mathematically well defined Markov process,Kolmogorov’s result is not immediately applicable to theopen system with two or more baths. Nevertheless, fromphysical intuition, one may well expect that the time re-versibility of such systems requires that the likelihood oftransitions (2) −→ be the same in the forward processand backward process: p L (2)13 = p L (2)31 . Furthermore, it is also intuitively correct that the EDB will guaranteethe time reversibility of the dynamics of such systems.Since the open systems considered here are mesoscopicor microscopic, the evolution should be subjected toquantum dynamics. There exist mainly two kinds ofquantum effects to be considered in the entropy produc-tion, related to quantum statistics and quantum coher-ence respectively. Quantum Statistical Effect on Entropy ProductionRate. — Let us first consider the factor of quantum statis-tics. Bosonic and fermionic environments are fundamen-tal in the study of quantum open systems. Bosonic heatbaths usually appear in opto-mechanical systems whilefermionic ones are common in the study of quantum dots.What is the essential difference between these two basickinds of environments? In this section we try to answerthis question from entropy production point of view. Ourstarting point is the REPR formula (6), applied to a two-level system.For a two-level system contacting with two heat baths,the REPR [Eq. (6)] for the steady state is R = Ω L + L · T − T T T ( L (2)21 L (1)12 − L (2)12 L (1)21 ) . (8)Here, we have adopted the labeling: state- and state- denote the excited state and the ground state respec-tively. Different kinds of environments will lead to dif-ferent forms of transition rates. Specifically, the transi-tion rates caused by the l -th heat bath have the followingforms [15], L ( l )12 = γ l (1 ± N ( β l )) , (9) L ( l )21 = γ l N ( β l ) , (10)where ‘ + ’ and ‘ − ’ correspond to the bosonic andfermionic cases respectively. N ( β ) = 1 / (exp( β Ω) ∓ isthe distribution function, and γ l is the coupling strengthbetween the system and the l -th heat bath. One canverify that these transition rates satisfy the microscopicreversibility L ( l )12 = exp( − β l Ω) L ( l )21 .The REPRs for bosonic and fermionic environmentscan be calculated directly. The results are R boson = Ω γ γ γ (2 N + 1) + γ (2 N + 1) · T − T T T ( N − N ) ,R fermion = Ω γ γ γ + γ · T − T T T ( N − N ) , (11)where N l = N ( β l ) . If the two temperatures are nearlyequal, T ≃ T ≃ T , then we have the estimation R boson ≃ γ γ γ + γ · Ω T · N (1 + N )2 N + 1 · (cid:18) ∆ TT (cid:19) ,R fermion ≃ γ γ γ + γ · Ω T · N (1 − N ) · (cid:18) ∆ TT (cid:19) . (12)In this case both of the entropy production rates are pro-portional to the square of ∆ T /T . Thus, the entropy pro-duction rate can be regarded as a response to the “drivingforce” (∆ T /T ) , and the ratio between the response andthe “driving force” C = R/ (∆ T /T ) as a “conductance”in some sense [18]. In the low temperature region, the“conductances” corresponding to bosonic and fermionicenvironments have the same asymptotic behavior: C boson / fermion ≃ γ γ γ + γ · Ω T e − Ω T . (13)They both exponentially decays to zero as T → . In thehigh temperature region, a remarkable difference arises.In this region we have C boson ≃ γ γ γ + γ · Ω2 T ,C fermion ≃ γ γ γ + γ · (cid:18) Ω2 T (cid:19) . (14)Thus, the bosonic “conductance” C boson ∝ /T as T →∞ , while the fermionic “conductance” C fermion ∝ /T as T → ∞ . Since the “conductance” C tends to zeroin both of the limits T → and T → ∞ , there existsa maximum conductance at a certain finite temperature T m (see the peak of blue line in Fig. 4). The numericalsimulation results for the bosonic and fermionic environ-ments are presented in Fig. 4. It is clearly illustrated thatthe conductances, namely, the ratio between REPR and (∆ T /T ) , are indeed different for bosonic and fermionicenvironments, especially in the high temperature region. Quantum Coherence Effect on Entropy ProductionRate. — Now we consider the factor of quantum coher-ence. Hereafter, for simplicity we assume the Marko-vian property of the dynamics of the quantum open sys-tem and the validity of the rotating wave approximationto the quantum master equation. Under this assump-tion, the evolutions of the diagonal and the off-diagonalparts of the open system’s density matrix are decoupled[15]. In other words, L l [ ρ coh ] should have vanishing diag-onal elements. Here, ρ coh is the off-diagonal part of thedensity matrix, which represents the quantum coherenceof the open system. Thus, tr (cid:2) L l [ ρ coh ] ln ρ th l (cid:3) vanishes.This means that the quantum coherence exerts no influ-ence on the heat flow between the open system and itsenvironment. The entropy change of the open system − dtr[ ρ ln ρ ] / d t can be divided into two parts: − dd t tr[ ρ ln ρ ] = − tr[ d ρ dia d t ln ρ ] − tr[ d ρ coh d t ln ρ ] , (15)where ρ dia is the diagonal part of the density matrix.Correspondingly, the REPR in Eq.(4) can be divided into C b / f / γ γ γ + γ T/ Ω Figure 4: (color online). The ratio C between the refinedentropy production rate R and (∆ T /T ) vs the temperature T . The red solid line corresponds to the boson case whilethe blue dashed line corresponds to the fermion case. Bothcases have the same behavior in the low temperature region.In the high temperature region the ratio C is proportionalto /T in the boson case, while it is proportional to /T inthe fermion case. The peaks of the two curves correspond tothe maximum conductances in boson case and fermion caserespectively. two parts: R = − dd t tr[ ρ dia ln ρ ] + N X l =1 tr (cid:2) L l [ ρ dia ] ln ρ th l (cid:3)! − tr[ d ρ coh d t ln ρ ] . (16)When t is larger than the time scale of the decoherence,we have ρ ≃ ρ dia and the term in the first line of the aboveequation is none other than the entropy production ratedue to the diagonal part of the quantum open system,which is equal to the classical REPR inEq.(6) as we havepointed out before. The term in the second line can beinterpreted as the entropy production rate due to theevolution of the off-diagonal part of the quantum opensystem. Thus we reach the qualitative conclusion thatquantum coherence effect contributes an additional partto the entropy production.To quantitatively study the effects of quantum coher-ence, let us concretely analyze a two-level system coupledwith a single heat bath. The Markovian quantum masterequation of this open system reads [15] d ρ d t = Γ − ( σ − ρσ + − σ + σ − ρ − ρσ + σ − )+ Γ + ( σ + ρσ − − σ − σ + ρ − ρσ − σ + ) , (17)where σ + = | e ih g | , σ − = | g ih e | , are the raising andlowering operators of the two-level system respectively.The evolution of the off-diagonal elements ρ eg ( t ) = exp( − Γt / ρ eg (0) , as expected, is decoupled from the di-agonal part of the system where Γ = Γ + + Γ − is the in-verse evolution timescale of the off-diagonal part. Thusthe entropy production due to the quantum coherenceeffect is R con = − α ( t ) ln 1 + α ( t )1 − α ( t ) d | ρ eg | d t , (18)where α ( t ) = s(cid:18) Γ − − Γ + Γ − + Γ + (cid:19) + 4 | ρ eg | . (19)For a long-time evolution such that | ρ eg | ≪ (Γ − − Γ + ) / Γ , R con can be estimated as R con ⋍ Γ β Ω coth β Ω2 e − Γ t | ρ eg (0) | . (20)It decays exponentially as t → ∞ .If initially the diagonal part has already reached itsstable value, the entropy production rate due to the di-agonal part of the quantum open system would remainvaninishing in the evolution. Thus, if we can “kick” anopen system, which has been already stabilized to a ther-mal state, so that its off-diagonal part becomes non-zerowhile its diagonal part remains “untouched” , we may beable to observe the entropy production due to the quan-tum coherence effect. Conclusions and Discussions. — In this letter, we try toprobe open quantum systems in contact with two or morebaths from entropy production point of view. We de-rive a refined formula for the entropy production rate forsuch systems. This foumula can well reflect the effects ofstatistics and quantum coherence on the entropy produc-tion. In the two-bath case, it turns out that the REPRs inbosonic and fermionic environments are proportional tothe square of temperature difference (∆ T /T ) betweenthe two heat baths, but the behaviors of the so calledconductances are quite different in the high temperatureregion. This reveals a connection between the entropyproduction of the open quantum system and the quan-tum statistical property of the baths. The results in this letter are applicable to a non-equilibrium system weaklycoupled to its environment. If the system-bath couplingis too strong or the coupling spectrum has some exoticstructure, the non-Markovian effects may dominate thelong-time behavior of the systems [19], and the entropyproduction behavior for such non-Markovian processesdeserves further investigations.This work was supported by the National Natural Sci-ence Foundation of China (Grant No. 11121403) and theNational 973 program (Grant No. 2012CB922104 andNo. 2014CB921403). ∗ Electronic address: [email protected]; URL: [1] C. N. Yang and C. P. Yang , Trans. NY Acad. Sci. ,267 (1980).[2] J. S. Thomsen, Phys. Rev. ,1263 (1953).[3] G. Nicolis, I. Prigogine, Self-organization in Non-equilibrium Systems: From Dissipative Structures to Or-der Through Fluctuation , Wiley, New York, 1977.[4] A. N. Kolmogorov, Math. Ann. , 155 (1936).[5] J. Schnakenberg, Rev. Mod. Phys. , 571–585 (1976).[6] X. J. Zhang, H. Qian, and M. Qian, Physics Reports ,1 (2012).[7] D. Q. Jiang, M. Qian, and M. P. Qian, Mathematical The-ory of Non-equilibrium Steady States, in: Lecture Notesin Mathematics , vol. 1883, Springer-Verlag, Berlin, 2004.[8] H. Spohn, J. Math. Phys. (N.Y.) , 1227 (1978).[9] H. Qian and M. Qian, Phys. Rev. Lett. 84, 2271 (2000).[10] M. Esposito, U. Harbola, and S. Mukamel, Phy. Rev. E, , 031132 (2007).[11] H. Ian et al. , Phys. Rev. A , 013824 (2008).[12] S. A. Gurvitz, Phys. Rev. B , 6602–6611 (1998).[13] M. Esposito, K. Lindenberg, and C. Van den Broeck,Europhys. Lett. , 60010 (2009).[14] G. Lindblad, Commun. math. Phys. , 119 (1976).[15] H. P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems , Oxford University Press, New York,2002.[16] H. T. Quan, P. Zhang, and C. P. Sun, Phys. Rev. E ,056110 (2005).[17] G. N. Lewis, Proc. Natl. Acad. Sci. USA
179 (1925).[18] L. Onsager Phys. Rev. , 405 (1930).[19] C. Y. Cai, L. P. Yang, and C. P. Sun, Phys. Rev. A89