Stochastic process behind nonlinear thermodynamic quantum master equation
aa r X i v : . [ qu a n t - ph ] M a y epl draft Stochastic process behind nonlinear thermodynamic quantummaster equation
Hans Christian ¨Ottinger
ETH Z¨urich, Department of Materials, Polymer Physics, HCI H 543, CH-8093 Z¨urich, Switzerland
PACS – Decoherence; open systems; quantum statistical methods
PACS – Nonequilibrium and irreversible thermodynamics
PACS – Markov processes
Abstract. - We propose a piecewise deterministic Markovian jump process in Hilbert space suchthat the covariance matrix of this stochastic process solves the thermodynamic quantum masterequation. The proposed stochastic process is particularly simple because the normalization ofthe vectors in Hilbert space is preserved only on average. As a consequence of the nonlinearityof the thermodynamic master equation, the construction of stochastic trajectories involves thedensity matrix as a running ensemble average. We identify a principle of detailed balance and afluctuation-dissipation relation for our Markovian jump process.
Introduction. –
Dissipative quantum systems areoften described in terms of quantum master equations,where linear master equations of the Lindblad form aremost popular in the literature [1–3]. However, it has beenknown for some 30 years that these linear master equa-tions have a fundamental problem because they invoke aninappropriate “quantum regression hypothesis” [4–6]. Forquantum systems in contact with a heat bath, this prob-lem has been overcome by the projection-operator deriva-tion of a nonlinear quantum master equation associatedwith a “modified quantum regression hypothesis” [4]. Formore general thermodynamic environments, a generalizednonlinear quantum master equation has been obtained bya most natural extension of the geometric formulation ofnonequilibrium thermodynamics from classical to quan-tum systems [7–9] inspired by Dirac’s method of classicalanalogy [10].The present letter addresses a possible route to thenumerical solution of nonlinear thermodynamic quantummaster equations. Stochastic simulation techniques havebeen established as a particulary versatile and convenienttool for solving quantum master equations [1,11]. The ba-sic idea is the unraveling of the master equation (see, forexample, chap. 6 of [1]). The time-dependent density ma-trix or statistical operator ρ ( t ) solving the master equationis obtained as a second moment or expectation, ρ ( t ) = E (cid:0) | ψ ( t ) i h ψ ( t ) | (cid:1) , (1)where ψ ( t ) is a suitably defined stochastic process in the underlying Hilbert space. Such a process consists of ran-dom quantum jumps and a deterministic Schr¨odinger-typeevolution modified by a friction term.We begin with a brief summary of the general formof nonlinear quantum master equations based on ther-modynamic principles. After introducing a general classof piecewise deterministic Markovian jump processes inthe underlying Hilbert space, we identify the parametersrequired to reproduce a thermodynamic master equationand we discuss the form and significance of the jump oper-ator in detail. In the conclusions, we discuss the usefulnessof the Markovian jump process both for stochastic simula-tions and for gaining fundamental insights into dissipativequantum systems. Thermodynamic quantum master equation. –
Based on purely thermodynamic considerations and ageneralization from classical to quantum systems inspiredby a geometric formulation of nonequilibrium thermody-namics, the following master equation for the evolution ofthe density matrix ρ on a suitable Hilbert space has beenproposed to characterize a quantum subsystem in contactwith an arbitrary classical nonequilibrium system actingas its environment [7–9]: dρdt = i ¯ h [ ρ, H ] − k B [[ H e , S e ]] Qx [ Q, [ Q, H ] ρ ] − [[ H e , H e ]] Qx [ Q, [ Q, ρ ]] . (2)p-1ans Christian ¨OttingerIn this equation, ¯ h and k B are Planck’s constant (dividedby 2 π ) and Boltzmann’s constant, respectively. The firstterm describes the reversible contribution to the evolutiongenerated by the Hamiltonian H via the commutator [ , ].The remaining terms are of irreversible nature and resultfrom a coupling of the quantum subsystem to its classicalenvironment. They are expressed through double com-mutators involving the self-adjoint coupling operator Q sothat the normalization condition, tr ρ = 1, is automati-cally preserved in time. The subscript ρ on an arbitraryoperator A indicates the modified operator A ρ = Z ρ λ A ρ − λ dλ, (3)which is basically the product of A and ρ , but with acompromise between placing ρ to the left or the right of A . If A is self-adjoint, this property is inherited by A ρ .We further have the useful identity[ A ρ , ln ρ ] = [ A, ρ ] , (4)which can be established by evaluating the matrix ele-ments for the eigenstates of ρ [7].Whereas the type of the coupling to the environment isgiven by the observable Q , the strength of the coupling isexpressed in terms of a dissipative bracket [[ , ]] Q defined asa binary operation on the space of observables for the clas-sical environment (in this letter, boldface bracket symbolsare used to distinguish classical dissipative brackets fromquantum commutators). If the equilibrium or nonequilib-rium states of the environment are characterized by statevariables x , classical observables are functions or function-als of x , and their evaluation at a particular point of thestate space is indicated by a subscript x . The classicalobservables H e and S e in eq. (2) are the energy and theentropy of the environment, respectively. For arbitrary en-vironmental variables A e and B e , the dissipative bracketoccurring in the master equation (2) is of the general form[12–14] [[ A e , B e ]] Qx = ∂A e ( x ) ∂x M Q e ( x ) ∂B e ( x ) ∂x , (5)where M Q e is a positive-semidefinite symmetric matrix.The size of the square matrix M e is given by the num-ber of state variables x .As a straightforward generalization of eq. (2), severalcoupling operators Q j can be incorporated easily [4, 7, 8].The proposed construction of an underlying stochasticprocess can be generalized accordingly. The master equa-tion (2) describes the influence of a classical environmenton a quantum subsystem. The equation describing the re-verse influence of the quantum system on its environmenthas been given in [7–9]. Stochastic process. –
In order to achieve a represen-tation (1) for the solution of the quantum master equation(2), we consider a piecewise deterministic Markovian jump process in the underlying Hilbert space. The construc-tion described here is inspired by the work of Breuer andPetruccione [1, 15, 16], with the pioneering precursor pa-pers [17, 18], however, we do not insist on preserving thenormalization of the wave function for each realization ofa stochastic trajectory. For nonlinear master equations, itis perfectly natural to think in terms of mean field interac-tions and to preserve certain properties only on average.
Definition of Markovian jump process.
A key step inthe construction of the underlying stochastic process is theintroduction of the jump operator, which we assume to beof the general form˜ Q = α ( Q + β [ Q, H ] ρ ρ − ) , (6)where the real coefficients α and β remain to be deter-mined. Note that, contrary to the coupling operator Q ,the jump operator ˜ Q is not self-adjoint. Once we havedefined the jump operator ˜ Q , we only need to specify thejump rate and the deterministic evolution between jumpsto obtain a complete definition of our piecewise determin-istic Markovian jump process in Hilbert space. We per-form quantum jumps ψ → ˜ Qψ with rate γ, (7)where the rate parameter γ remains to be determined.Note that the wave function ˜ Qψ is not normalized. Fi-nally, we propose the following modified Schr¨odinger equa-tion for the deterministic evolution of the state ψ betweenthe random jumps, dψdt = − i ¯ h Hψ + Λ ψ, (8)where, contrary to the Hamiltonian H , the linear operatorΛ need not be self-adjoint. Again, the normalization of thewave function ψ is not preserved. Whereas the jumps ineq. (7) correspond to the noise represented by the diffusionterm in a classical Fokker-Planck equation, the modifica-tion of the Schr¨odinger equation by the linear operatorΛ in eq. (8) is the counterpart of deterministic friction.The striking simplicity of the proposed stochastic processdefined by eqs. (6)–(8) should be noted. Master equation for second moments.
In order toidentify the proper choice of the parameters α , β , γ and ofthe friction operator Λ we write the evolution equation forthe second moments which, in the spirit of the unravelingidea expressed in eq. (1), can be regarded as an evolutionequation for the density matrix, dρdt = i ¯ h [ ρ, H ] + Λ ρ + ρ Λ † + γ ( ˜ Qρ ˜ Q † − ρ ) , (9)where the jumps with rate γ lead to a loss of the currentconfiguration and the creation of new states distorted bymeans of the jump operator ˜ Q . After inserting the defini-tion (6) of ˜ Q , eq. (9) can be rewritten as dρdt = i ¯ h [ ρ, H ] − γα β [ Q, [ Q, H ] ρ ] − γα [ Q, [ Q, ρ ]] , (10)p-2tochastic process behind thermodynamic quantum master equationprovided that the linear operator Λ is chosen asΛ = γ h − α Q + α β (cid:0) [ Q, H ] ρ ρ − (cid:1) i . (11)The occurrence of the desired dissipative terms in eq. (10)justifies the selection of the jump operator ˜ Q accordingto eq. (6). It is this jump operator that expresses thethermodynamic nature of the nonlinear quantum masterequation on the level of the stochastic process.Note that both the jump operator ˜ Q in eq. (6) and thefriction operator Λ in eq. (11) are expressed in terms of Q and [ Q, H ] ρ ρ − . The implied relationship between ˜ Q andΛ may be regarded as a fluctuation-dissipation relation .We can now compare eqs. (2) and (10) to determine theremaining coefficients. We find γα = 2 [[ H e , H e ]] Qx , (12)and β = [[ H e , S e ]] Qx k B [[ H e , H e ]] Qx . (13)Whereas β can be identified uniquely, only the combina-tion γα is fixed by the comparison, not the parameters α and γ separately. This is a direct consequence of a scalingproperty of the thermodynamic master equation (2): anyrescaling of the coupling operator Q can be compensatedby a corresponding rescaling of the dissipative bracket sothat Q does not have an absolute meaning. From eqs. (6)and (7) one realizes that the normalization of ˜ Qψ is pro-portional to α . We can hence choose α such that thenormalization of the state vector ψ does not change onaverage during a jump process, that is,tr( ˜ Qρ ˜ Q † ) = 1 , (14)thus obtaining an absolute meaning of the jump operator˜ Q (and of the coupling operator Q ). Then, the normal-ization of the state vector ψ does not change on averageduring the deterministic evolution either, that is,tr(Λ ρ ) = 0 , (15)because a master equation of the form (10) preserves thetrace of ρ and hence, according to eq. (1), the averagenormalization of ψ . Indeed, both conditions (14) and (15)are equivalent to α h tr( QρQ ) − β tr (cid:0) [ Q, H ] ρ ρ − [ Q, H ] ρ (cid:1)i = 1 . (16)Note that the sign of α is still not fixed by eq. (16). Ac-tually, one could even introduce a random phase in anyjump without any effect on the resulting quantum masterequation.Whereas, from a theoretical point of view, eq. (16) pro-vides a most satisfactory condition for determining α , forpractical purposes like computer simulations it may bemore convenient to work with an approximate value of α and to fix γ accordingly to fulfill eq. (12). For example,one could choose the equilibrium value of α or the solutionof the approximate equation α tr( QρQ ) = 1.
Jump operator at equilibrium.
As the jump operator˜ Q is the key ingredient to our construction of a piecewisedeterministic Markovian jump process, we next considerit in more detail for an equilibrated quantum system, thatis, for [8, 9] ρ ∝ exp (cid:26) − Hk B T e (cid:27) and T e [[ H e , S e ]] Qx = [[ H e , H e ]] Qx . (17)By means of eq. (4), we obtain the following simplifiedexpression for ˜ Q ,˜ Q = α (cid:18) Q −
12 [ Q ρ , ln ρ ] ρ − (cid:19) = α Q + ρQρ − ) , (18)together withΛ = γ (cid:20) − α (cid:0) Q − ρQ ρ − + QρQρ − + ρQρ − Q (cid:1)(cid:21) . (19)By evaluating the matrix elements of ˜ Q for the eigen-states | n i of the Hamiltonian with eigenvalues E n , h m | ˜ Q | n i = α (cid:18) (cid:26) E n − E m k B T e (cid:27)(cid:19) h m | Q | n i , (20)we obtain an illuminating interpretation of the jump op-erator ˜ Q : The coupling operator Q describes the possiblejumps between quantum states and the modified operator˜ Q introduces the proper transition probabilities by a gen-eralized form of detailed balance for matrix elements ratherthan transition probabilities. This situation is very simi-lar to the well-known procedure in classical Monte Carlosimulations where one first defines the allowed moves andthen introduces transition probabilities according to theMetropolis method in order to satisfy detailed balance(see, for example, p. 12 of [19] or p. 337 of [14]). Harmonic oscillator at equilibrium.
For example, forthe one-dimensional damped harmonic oscillator with fre-quency ω , the coupling operator Q coincides with the po-sition operator [9] and hence describes jumps into one ofthe two neighboring energy states. The energy differences E n − E m in eq. (20) are given by ± ¯ hω and the equilib-rium detailed balance factors favor jumps into the lowerenergy states over jumps into higher energy states. How-ever, these jump rates are completely different from thoseof the Pauli master equation for the populations of theenergy levels (see, for example, sec. 3.4.6.1 of [1]) becausealso the deterministic influence of Λ in eq. (8) causes tran-sitions between the energy eigenstates. Summary and conclusions. –
We have introduceda piecewise deterministic Markovian jump process inHilbert space with the jumps (7) and the deterministicevolution (8). In these equations, the jump operator ˜ Q isdefined in eq. (6) and the modification of the Schr¨odingerequation with Hamiltonian H is given by the operator Λin eq. (11). The remaining free parameters α , β , and γ are defined in eqs. (12), (13), and (16), where actuallyp-3ans Christian ¨Ottingeronly β and the combination γα are physically relevant.For a proper choice of α , the normalization of the statevector is assumed to be preserved on average, both for thedeterministic evolution and for the jumps.The thermodynamic origin of our Markovian jump pro-cess is reflected in the specific form (6) of the jump oper-ator ˜ Q . We have realized that, by construction, the jumpoperator ˜ Q incorporates a generalized principle of detailedbalance. The jump operator ˜ Q and the friction operatorΛ are connected by a fluctuation-dissipation relation.The evolution of the second moments of the Markovianjump process is governed by the nonlinear thermodynamicquantum master equation (2). The proposed stochasticprocess hence provides an unraveling of the thermody-namic master equation. Such a stochastic description ofdissipative quantum systems in contact with a classicalenvironment is very useful, both for developing stochas-tic simulation techniques and for gaining fundamental in-sights.As a consequence of the nonlinear nature of the ther-modynamic quantum master equation, the simulationof stochastic trajectories in Hilbert space according toeqs. (7) and (8) requires knowledge of the evolving densitymatrix, that is, the second moment (1) of the stochasticprocess. This moment can be obtained as a running en-semble average. This fundamental difference compared tolinear quantum master equations does not cause any severeproblems, as is well-known from the theory of nonlinearFokker-Planck equations, which are also known as nonlin-ear diffusions, processes with mean-field interactions, orweakly interacting stochastic processes (see [20], sec. 3.3.4of [21], or secs. 3.7.2–3.7.4 of [1]). In every time step, asingle ensemble average needs to be evaluated in order topropagate all stochastic trajectories of the ensemble. Oth-erwise, the background and benefits of stochastic simula-tions are identical to the ones employed with impressivesuccess for linear quantum master equations (see, for ex-ample, secs. 6.3 and 6.4 on photodetection and sec. 7.3on the damped harmonic oscillator and driven two-levelsystems in the textbook [1]).From a fundamental point of view, the stochastic re-formulation might lead to a deeper understanding of theevolution of dissipative quantum systems by offering a dif-ferent perspective. In particular, the reformulation mightoffer a possibility to build a theory of two- and multi-timecorrelations. This would, of course, require that amongthe many possible unravelings of the nonlinear thermody-namic master equation there exists a physically preferableone. It has been shown in [15,16] that “the synthesis of thecontinuous Schr¨odinger-type evolution and the discontin-uous quantum jumps of the Bohr picture” arises naturallyand directly in the description of open quantum systems,without any reference to the quantum master equation. Adirect thermodynamic formulation of the piecewise deter-ministic Markovian jump process in Hilbert space mighthelp in identifying a unique physically convincing stochas-tic process associated with a thermodynamic quantum master equation. ∗ ∗ ∗ I wish to thank Francesco Petruccione for encouragingme to look deeper into the properties of thermodynamicquantum master equations.
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