Microscopic reversibility of quantum open systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Microscopic reversibility of quantum open systems
Takaaki Monnai ∗ ∗ Department of Applied Physics, Osaka City University,3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
The transition probability for time-dependent unitary evolution is invariant under the reversalof protocols. In this article, we generalize the expression of microscopic reversibility to externallyperturbed large quantum open systems. The time-dependent external perturbation acts on thesubsystem during a transient duration, and subsequently the perturbation is switched off so thatthe total system would thermalize. We concern with the transition probability for the subsystembetween the initial and final eigenstates of the subsystem. In the course of time evolution, the energyis irreversibly exchanged between the subsystem and reservoir. The time reversed probability is givenby the reversal of the protocol and the initial ensemble. Microscopic reversibility equates the timeforward and reversed probabilities, and therefore appears as a thermodynamic symmetry for openquantum systems.
PACS numbers: 05.30.-d,05.70.Ln
I. INTRODUCTION
The time reversal invariance of the equations of mo-tion amounts to universal symmetry of fluctuation the-orems and related equalities[1–19] for the fluctuationof entropy production and particle current for meso-scopic systems, and thus plays fundamental role in thenonequilibrium statistical mechanics. The symmetriesconnect the probabilities of positive and negative entropyproduction[9, 20], which are calculated from the timeforward and reversed transition probabilities for quan-tum systems[4, 5, 13, 15–17]. In the classical Markovianstochastic dynamics, the conditional probability func-tional of the trajectories satisfies a symmetry expressedby the probability functionals and heat. This relation isalso called microscopic reversibility[9], and generalized tothe quantum open systems by concerning the heat calcu-lated from set of transitions of the reservoir states[10, 11].On the contrary, we are interested in the transition prob-abilities between the system states instead of the statis-tics for dissipative quantities, and give an expression ofmicroscopic reversibility. It is known that the inducedabsorption and emission by an external electric field areequally probable within the realm of the Fermi’s Goldenrule provided that the initial and final Fock states areexchanged. Absorption process can be seen as the timereversal of the corresponding emission process.In this article, we show that a symmetry similar tothat of the induced absorption holds for the macroscopictime-dependent open systems where the system couplesto the reservoir. Since the expression appears as a genericequation for the forward and reversed protocols, we callit as microscopic reversibility.This article is organized as follows. First, we describeour model and forcing protocol. And an expression ofmicroscopic reversibility for open systems is derived in ∗ Electronic address: [email protected]
Eq.(14). Then the microscopic reversibility is numeri-cally verified.
II. MODEL
Let us consider a finite system interacting with amacroscopically large reservoir at an inverse tempera-ture β . The system is externally controlled by a timedependent parameter λ ( t ), which is for example a springconstant for the case of a harmonic oscillator. Thereforethe total energy change of the system is caused by theexternal work done and the heat flow from the reservoir.The total Hamiltonian is H ( t ) = H s ( λ ( t )) + H r + H sr , (1)where H s ( λ ( t )), H r , and H sr are the Hamiltonians of thesystem, the reservoir, and the interaction between them,respectively.Let us prepare the initial state as ρ (0) = ρ s ( λ (0)) ⊗ ρ r ; ρ s ( λ (0)) = e − βH s ( λ (0)) Z ( λ (0)) (2)which is the product of the canonical ensembles of thesystem ρ s ( λ (0)) and of the reservoir ρ r at the same in-verse temperature β . The partition function of the sys-tem is Z ( λ (0)) ≡ Tr s e − βH s ( λ (0)) . (3)Through out this paper, we assume that the interactionenergy H sr is small compared to the energy of the sub-system H s and the reservoir H r . This assumption is rea-sonable for macroscopic systems, since H s and H r arethe bulk energy, while H sr would be proportional to thesurface area. Note that the assumption is different fromthe weak coupling limit. Namely, the weak coupling as-sumes that interaction is negligible, and the perturbativeanalysis is available. On the other hand, in our case theinteraction is not necessarily vanishing, while we requirethat the ratio between the interaction energy and bulkenergy is negligible. III. TWICE MEASUREMENTS SCHEME
The twice measurements scheme consists of the initialand final observations, (i) and (ii).(i) At t = 0, we measure the energy operator H s ( λ (0)),and gain an eigenenergy E n (0). The system statebecomes the corresponding eigenstate | n (0) i , H s ( λ (0)) | n (0) i = E n (0) | n (0) i . (4)Subsequently the total system unitarily evolves un-til t = T as ρ ( T ) = U ( | n (0) i e − βE n (0) Z ( λ (0)) h n (0) | ⊗ ρ r ) U + . (5)Here U = T { e − i ~ R T dsH ( s ) } is the unitary time evo-lution operator. For the time evolution, we requirethat the external forcing is switched off well in ad-vance t = T and the total density matrix wouldrelax to an equilibrium state at t = T , U ρ (0) U + ∼ = ρ s ( λ ( T )) ⊗ ρ r ; ρ s ( λ ( T )) = e − βH s ( λ ( T )) Z ( λ ( T )) . (6)Here H s ( λ ( T )) is the corresponding energy opera-tor at t = T and Z ( λ ( T )) is the partition function.Eq.(6) comes from the equation at the level of ma-trix elements[25]Tr r U ρ (0) U + ∼ = ρ s ( λ ( T )) (7)and the assumption of smallness of the interactionenergy, and holds when acting on the local state ofthe subsystem as in Eq.(14).(ii) At t = T , we measure the energy H s ( λ ( T )) and ob-tain some eigenenergy E m ( T ) with the correspond-ing eigenvector | m ( T ) i . Regarding the reservoir, wedon’t perform any measurements . The transitionprobability that the initial and final system statesare | n (0) i and | m ( T ) i is then P F ( | n (0) i → | m ( T ) i )= Tr r {h m ( T ) | U | n (0) i e − βE n (0) Z ( λ (0)) ρ r h n (0) | U + | m ( T ) i} . (8)Here the Kraus operator A nm ≡ p Z ( λ (0)) e − βEn (0)2 h m ( T ) | U | n (0) i (9) describes the transition of the system state, andsatisfies X n,m A + nm A nm = X n h n (0) | Z ( λ (0)) e − βH s ( λ (0)) U + X m | m ( T ) ih m ( T ) | U | n (0) i = 1 . (10)Note that the matrix elements such as h m ( T ) | U | n (0) i contain the reservoir variables.The second equality follows from the complete-ness P m | m ( T ) ih m ( T ) | = 1, and the unitarity U + U = 1.Similarly the probability of the time-reversed dy-namics is calculated as well. Firstly, let us definethe reversed dynamics by reversing the time de-pendence of the system Hamiltonian in Eq.(2) as H s ( λ ( T − t )), i.e. the reversal at time T . Also, westart with the initial state Θ e − βHs ( λ ( T )) Z ( λ ( T )) ⊗ ρ r Θ − ,where Θ is the anti unitary time reversal opera-tor of the total system. The partition function isdefined as Z ( λ ( T )) ≡ Tr s e − βH s ( λ ( T )) . (11)(i-2) We measure the energy Θ H s ( λ ( T ))Θ − at t = 0and consider the case that the eigenenergy E m ( T )corresponding to (i), and the system state becomesΘ | m ( T ) i . Θ | m ( T ) i is proportional to | m ( T ) i whenthe energy E m ( T ) does not degenerate.(ii-2) At the final time t = T , we again measure theenergy Θ H s ( λ (0))Θ − and obtain the eigenenergy E n (0). The system state is Θ | n (0) i .Then the probability that the initial and final statesare Θ | m ( T ) i and Θ | n (0) i for the reversed dynamicsis P R (Θ | m ( T ) i → Θ | n (0) i )= Tr r {h n (0) |←− Θ U + Θ | m ( T ) i e − βE m ( T ) Z ( λ ( T )) ρ r h m ( T ) |←− Θ U Θ | n (0) i} . (12)Here h n (0) |←− Θ U + Θ | m ( T ) i = h m ( T ) | U | n (0) i is theinner product of Θ | n (0) i and U + Θ | m ( T ) i . IV. MICROSCOPIC REVERSIBILITY
Now let us show the relation between probabilities ofthe forward and reversed transitions P F ( | m (0) i → | n ( T ) i ) ∼ = P R (Θ | n ( T ) i → Θ | m (0) i ) . (13)Here the difference between the forward and reversedprobabilities goes to zero in the macroscopic limit. Forthis purpose, we apply the methodology developed inRefs.[10, 11] and thermalization property Eq.(6). Werewrite the forward probability as P F ( | n (0) i → | m ( T ) i )= Tr r ρ r h n (0) | p Z ( λ (0)) e − βH s ( λ (0)) U + | m ( T ) i ρ − r ρ r ρ − r h m ( T ) | U p Z ( λ (0)) e − βH s ( λ (0)) | n (0) i ρ r = Tr r h n (0) | U + ( { U ρ s ( λ (0)) ρ r U + } ρ s ( λ ( T )) − ρ − r ) ρ s ( λ ( T )) | m ( T ) i ρ r h m ( T ) | ρ s ( λ ( T )) ( ρ s ( λ ( T )) − ρ − r { U ρ s ( λ (0)) ρ r U + } ) U | n (0) i∼ = Tr r h n (0) | U + p Z ( λ ( T )) e − βH s ( λ ( T )) | m ( T ) i ρ r h m ( T ) | p Z ( λ ( T )) e − βH s ( λ ( T )) U | n (0) i = 1 Z ( λ ( T )) e − βE m ( T ) Tr r h n (0) | U + | m ( T ) i ρ r h m ( T ) | U | n (0) i = P R (Θ | m ( T ) i → Θ | n (0) i ) . (14)The first equality derives from the cyclic property of thetrace. In the second equality, U + U = 1 is inserted. Also,with the use of the relaxation property of the densitymatrix Eq.(13) which comes from Eq.(7) and the small-ness of interaction energy H sr with respective to thebulk energy, the contents of the curly brackets are justthe inverse of the remaining quantities in the brackets ρ s ( λ ( T )) − ρ − r . V. NUMERICAL DEMONSTRATION OFMICROSCOPIC REVERSIBILITY
In this section, we numerically show the microscopicreversibility Eq.(14). See also the detailed numerical ver-ification in Ref.[22] for various parameters and measure-ment basis. We consider N = 8 site spin chain in thespatially inhomogeneous time dependent magnetic field.Note that thermalization property has been observed inrelatively small system sizes[23, 24]. In the context ofquantum derivation of thermal state, the dimension ofthe Hilbert space, which exponentially depends on thesystem size plays essential role[25]. The Hamiltonian isgiven as H ( t ) = H s ( t ) + H r ( t ) + H sr H s ( t ) = − J s N s − X j =1 σ zj σ zj +1 + α s N s − X j =1 σ xj + h ( t ) N s − X j =1 σ zj H r = − J r N X j = N s +1 σ zj σ zj +1 + α r N X j = N s +1 σ xj + γ r N X j = N s +1 σ zj H sr = − J sr σ zN s σ zN s +1 + α sr σ xN s + γ sr σ zN s , (15)where σ ij is the i component of the Pauli matrix. Thesites j = 1 , N s = 2) and 3 < j ≤ N are regarded P F HÈ m H L >® È n H t L > L FIG. 1: The time dependence of the forward and re-versed probabilities P F ( | m (0) i → | n ( t ) i )(blue line) and P R (Θ | n ( t ) i → Θ | m (0) i )(black line) is shown where the ini-tial and states are the eigenstates with the first to secondlargest eigenvalues. as the subsystem and reservoir. The external magneticfield h ( t ) = B − B tanh( µ ( t − τ )) satisfies h (0) ∼ = B andswitched off after t = τ . The initial state is prepared as ρ (0) = ρ s (0) ρ r ; ρ s (0) = 1 Z s (0) e − βH s (0) ρ r = 1 Z r e − βH r . (16)We calculated the forward and reversed transition prob-abilities in Fig.1. The parameters are chosen as follows.The exchange interactions at each site are J s = J r = J sr = 1, the x component of the magnetic field is ex-pressed by the parameters α s = α r = 1, α sr = 0 .
2, andsimilarly the z component of the magnetic field is givenby h ( t ) with B = 3, µ = 5, γ r = 1, and γ sr = 0 .
2. Theinverse temperature is β = 0 .
01 and the switching timeis τ = 5.The unitary time evolution U = T { e − i R T H ( t ) dt } is discretized as e − i ∆ tH ( N ∆ t ) e − i ∆ tH (( N − t ) · · · e − i ∆ tH (∆ t ) e − i ∆ tH (0) with the time step ∆ t = 0 .
05 whichis much shorter than the time scale of external perturba-tion. Initially, we measure the eigenstate of H s (0) cor-responding to the largest eigenvalue. The measurementbasis at t = T is chosen as the eigenstate of H s ( T ) whichcorresponds to the second largest eigenvalue. Note thatthere are no crossing of time dependent energy levels ofthe subsystem, and Eq.(13) holds also for the transitionsto other three eigenstates. It is remarkable that the timedependence of the forward and reversed transition prob-abilities are similar as a function of time even after thequench at t ∼ = τ . VI. SUMMARY
We have derived an expression of microscopic re-versibility for macroscopic quantum open systems. Themicroscopic reversibility is trivial for the transition prob-ability of unitary time evolution for the total system.Similar reversibility is well-known for the transition prob-abilities of induced absorption and emission under theinfluence of an electric field in equilibrium. The micro-scopic reversibility is regarded as a generalization to thecase of generic time-dependent perturbation. Therefore,it is a symmetry holds in generic macroscopic quantumopen systems. The microscopic reversibility is numeri-cally verified for a spin chain with time-dependent per-turbation. In the context of the quantum generalizationof the reversibility for the classical conditional probabil-ity functional, Ref.[10] derives another symmetry for the quantum trajectories. The main difference from Ref.[10]is the quantity which we measure, number of measure-ments, and the definition of the reversed process. Here wepursue and measure the system states. This is a usefulproperty of the present scheme, since the system vari-ables are expected to be much easier to measure withsufficient accuracy compared to those of the spatially ex-tended large reservoir.
VII. ACKNOWLEDGMENT
T.M. is grateful to Professor S.Tasaki for his encour-agement and Professor A.Sugita for fruitful discussions.This work is supported by the JSPS research programunder the Grant 22 · [1] C.Jarzynski, J.Stat.Phys. (2000) 77-102[2] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys.Rev. Lett. , (1993) 2401[3] G. Gallavotti, E.G.D. Cohen, J. Stat. Phys. 80 (1995)931[4] J.Kurchan, e-print:cond-mat:0007360 (2000)[5] S.Tasaki, and T.Matsui, Quantum Prob. White NoiseAnal.,17 (2003) 100-119[6] I.Callens, W.De Roeck, T. Jacobs, C.Maes, K. Netoˇcn´y,Physica D (2004) 383-391[7] M.F.Gelin, and D.S.Kosov, Phys.Rev.E (2008)011116[8] C.Jarzynski, Phys.Rev.Sett. (1997) 2690 ; Phys.Rev.E (1997) 5018[9] G.E.Crooks, Phys.Rev.E, (1999) 2721; (2000) 2361[10] G.E.Crooks, Phys.Rev.A, (2008) 034101[11] G.E.Crooks, J. Stat. Mech.: Theor. Exp. (2008) P10023[12] M.Campisi, P.Talkner, and P.Hanggi, Phys.Rev.Lett. (2009) 210401[13] D.Andrieux, and P.Gaspard, Phys. Rev. Lett. (2008)230404[14] D.Andrieux, P.Gaspard, T.Monnai, and S.Tasaki, New Journal of Physics, (2009) 043014[15] T.Monnai, Phys. Rev. E , (2005) 027102[16] Takaaki Monnai, Phys. Rev. E , (2010) 011129[17] K.Saito, and Y.Utsumi, Phys.Rev.B , (2008) 115429[18] M.Esposito, U.Harbola, and S.Mukamel, Phys. Rev. E (2007) 031132[19] M. Esposito, U. Harbola and S. Mukamel, Rev. Mod.Phys. , (2009) 1665[20] M.Esposito, K.Lindenberg, and C.Van den Broeck, NewJournal of Physics (2010) 013013[21] M.Esposito, and T.Monnai, J.Phys.Chem.B, (2011)5144[22] T.Kawamoto, arXiv:1011.3788[23] K.Saito, S.Takesue, and S.Miyashita, J. Phys. Soc. Jpn. , 1243-1249 (1996)[24] R.V.Jensen and R.Shankar, Phys. Rev. Lett. , (1985)1879[25] S.Goldstein, J.L.Lebowitz, R.Tumulka, and N.Zanghi,Phys.Rev.Lett.96