Post-Markov master equation for the dynamics of open quantum systems

7 February 2000

Physics Letters A 265 Ž2000. 331–336 www.elsevier.nlrlocaterphysleta

Post-Markov master equation for the dynamics of open quantum systems Ting Yu a , Lajos Diosi ´

b,c

, Nicolas Gisin a , Walter T. Strunz

d

a

c

Group of Applied Physics, UniÕersity of GeneÕa, 1211 GeneÕa 4, Switzerland b Institute for AdÕanced Study, Wallotstrasse 19, D-14193 Berlin, Germany Research Institute for Particle and Nuclear Physics, POB 49, 1525 Budapest 114, Hungary d Fachbereich Physik, UniÕersitat ¨ GH Essen, 45117 Essen, Germany Received 29 November 1999; accepted 24 December 1999 Communicated by P.R. Holland

Abstract A systematic first-order correction to the standard Markov master equation for open quantum systems interacting with a bosonic bath is presented. It extends the Markov Lindblad master equation to the more general case of non-Markovian evolution. The meaning and applications of our ‘post’-Markov master equation are illustrated with several examples, including a damped two-level atom, the spin-boson model and the quantum Brownian motion model. Limitations of the Markov approximation, the problem of positivity violation and initial slips are also discussed. q 2000 Published by Elsevier Science B.V. All rights reserved.

The fundamental approach to open quantum systems relies on a closed system-plus-reservoir model whose time evolution is governed by the standard Schrodinger or von-Neumann equation. Due to the ¨ coupled dynamics of system and environment, effectively, the evolution of the system’s reduced density operator r t depends on its past, it is non-MarkoÕian. For a quantum particle coupled to an environment of harmonic oscillators, this is obvious from its path integral propagator as derived by Feynman and Vernon w1x. It is also apparent from Zwanzig’s projection approach w2x where the evolution equation for r t takes the form of an integro-differential equation involving an integral over the whole history of the density operator. Exactly how far back the integration over the past has to be extended defines the environmental ‘memory’ time scale t .

A finite memory time t leads to severe problems for the treatment of open system dynamics which is even more true for attempts to find numerical solutions. Growing experimental advances on mesoscopic scales w3x, however, demand an efficient theory for open system dynamics beyond the standard Markov approximation: these are experiments with high-Q microwave cavities, investigations of the spontaneous emission from atoms in a structured radiation continuum Že.g. ‘photonic band gap’ materials. or the output coupling from a Bose–Einstein condensate to create an atom laser, to name a few. Also, the important phenomenon of decoherence which takes place on time scales that can be of the same order as the correlation time of the environment requires theories beyond the standard Markov approximation.

0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 0 1 4 - 1

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T. Yu et al.r Physics Letters A 265 (2000) 331–336

There are a few derivations of exact nonMarkovian master equations for model systems Žsee e.g. w4–7x.. Remarkably, despite the underlying non-Markovian dynamics, these exact evolution equations may be cast into the form of a time-local master equation involving the reduced density operator r t at time t only. A finite memory time and thus non-Markovian effects are exactly taken into account by suitable time dependent coefficients entering the resulting master equation. The derivation of a useful master equation for general open systems beyond the Markov regime remains an outstanding open problem. In this Letter we present a first order ‘post’-Markov master equation for an arbitrary quantum system coupled to an environment of harmonic oscillators. As in the case of the above-mentioned soluble models, our result takes the form of a time-local master equation with time dependent coefficients involving the bath correlation function. To zeroth order in the memory time Žt s 0. we recover the standard Markov master equation. To begin with, let us recall the Markov case. From an axiomatic approach, Lindblad w8x derived the most general Markov evolution equation for a density operator Ž " s 1.:

r˙ t s yi w H , r t x q 12 Ž L, r t L† q L r t , L†

.,

Ž 1.

system interacting with a bosonic oscillator environment with total Hamiltonian, Htot s H q Ý g l Ž La†l q L†al . q Ý vl a†l al . l

Ž 2.

l

Here, H is the Hamiltonian of the system and L a system operator describing the coupling to the environment. The standard zero temperature bath correlation function of model Ž2. is

a Ž t , s . s Ý g l2 eyi vlŽ tys. .

Ž 3.

l

Its decay, as a function of the time delay t y s, defines the ‘memory’ or correlation time t of the environment. We choose to normalize the coupling constants g l in Ž2. such that a Ž t, s . is normalized Ž Ha Ž t, s . ds s 1. which means that the overall coupling strength is determined by a parameter hidden in the coupling operator L in Ž2.. A lengthy derivation w10x based on a nonMarkovian quantum trajectory approach developed recently w11x, shows that to first order in the memory time t the reduced density operator of model Ž2. evolves according to the first-order ‘post’-Markov master equation,

r˙ t s yi w H , r t x q g 0 Ž t . L, r t L† q g 0) Ž t . L r t , L† q ig 1 Ž t . L† , w H , L x r t y ig 1) Ž t . r t w L† , H x , L

where, for simplicity, we only consider the case of a single Lindblad operator L representing the influence of the environment. Such Markov Lindblad master equations are widely used in quantum optics w9x. We identify two ‘system’ time scales in Ž1.: the ordinary dynamic time scale determined by the Hamiltonian H Žwhich we call vy1 ., and a damping or relaxation time scale Žwhich we call gy1 . determined by the operator L†L. The true open system dynamics will be well described by Eq. Ž1. as long as terms of the order vt and gt can be neglected, where t is again the environmental memory time. Our aim is a ‘post’-Markov master equation valid to first order in vt , gt . In what follows, we explain the main result and illustrate it with several prominent physical problems, including a damped two-level system, the spin-boson model and the quantum Brownian motion model. We start with a quantum

q g 2 Ž t . L† , w L† , L x L r t q g 2) Ž t . r t L† w L† , L x , L ,

Ž 4.

where g0 Ž t . s g 1Ž t . s

t

H0 a Ž t , s . ds, t

H0 a Ž t , s . Ž t y s . ds,

g2 Ž t . s

t

Ž 5. Ž 6.

s

H0 H0 a Ž t , s . a Ž s,u . Ž t y s . duds.

Ž 7.

The master Eq. Ž4. is the central theme of this paper. Note first that according to our convention, the function g 0 Ž t . is of the order one. Therefore, as in Ž1., the first two lines in Ž4. define the time scales vy1 Ž H . and gy1 Ž L†L.. The two functions g 1Ž t .

T. Yu et al.r Physics Letters A 265 (2000) 331–336

and g 2 Ž t ., however, are of the order of the memory time t . Therefore, the third and fourth line in Ž4. are smaller than the first and second line, respectively, by a factor gt . As t 0, the third and forth line in Ž4. become negligible, the real part of g 0 Ž t . tends to 1 2 , and the contribution of the imaginary part of g 0 Ž t . can be absorbed by a renormalized Hamiltonian: our ‘post’-Markov master Eq. Ž4. reduces to the Lindblad master Eq. Ž1. for vanishing memory time t . The third and fourth lines, which arise due to short but finite correlation time, represent the new first-order non-Markovian corrections. The time-dependent coefficients g i Ž t . describe an important initial slip on the time scale of the memory time t w4x. Note that Ž4., assuming that the initial state is factorable, is valid for zero temperature and also for finite temperature if L s L†. General finite temperature post-Markov master equation can be obtained directly from the first order perturbation of the finite temperature quantum state diffusion ŽQSD. equation w11x.

™

333

As a first example, we consider a damped twolevel system: H s 12 vsz , L s g sy. For the sake of simplicity, we phenomenologically choose an exponentially decaying correlation function a Ž t, s . s 21t ey< tys < r t with memory time t . Since this model can be solved exactly w7,11x, we are able to compare results from our post-Markov master Eq. Ž4., v r˙ t s yi w sz , r t x y i v Ž g g 1 Ž t . . w sq sy , r t x 2

'

q g Ž g0 Ž t . q g g2 Ž t . . = Ž 2 sy r t sqy  sq sy , r t 4 . ,

Ž 8.

with the exact result. It here happens that the postMarkov master Eq. Ž8. is of Lindblad form Ž1. with time-dependent coefficients Žthe real g 1Ž t . gives rise to a time-dependent frequency shift.. Therefore we know that r t remains a proper density operator for all times and all parameters. As seen in the next two examples this property is not a generic feature of the post-Markov master Eq. Ž4.. To illustrate the limits

Fig. 1. Illustration of the limit of the Markov approximation Ždotted curve. for a damped two-level system. The dashed curve is our post-Markov result, the solid curve is the exact result. The initial state is < c 0 : s < y :. The parameters are v s g ,gt s 0.2.

T. Yu et al.r Physics Letters A 265 (2000) 331–336

334

of the Markov approximation we compare in Fig. 1 the average ² sy : obtained from the post-Markov master equation Ždashed curve., from the Markov master equation Ždotted curve. and from the exact master equation Žsolid curve. for the parameters v s g and gt s 0.2. We see that the post-Markov master equation gives a much better result than the Markov master equation as expected for a significant memory time of gt s 0.2. So far, our results assume a zero temperature environment. As stated before, it turns out w11x, that for a selfadjoint coupling operator L s L† the same analysis holds, with a Ž t, s . in Ž3. replaced by the finite temperature expression

a Ž t , s . s Ý g l2 coth l

vl

ž / 2 kT

yisin vl Ž t y s .

Ž " s 1. .

A finite temperature introduces an additional time scale 1rkT but we here focus on the high-temperature limit kT 4 ty1 , thus disregarding the new time scale. For an Ohmic bath we obtain the standard correlation function

Ž 9.

where DŽ t . is a smeared out delta function, decaying on the memory time scale t Žthe inverse of the cutoff frequency., and D˙ is its time derivative. The precise shape of DŽ t . depends on the type of high-frequency cutoff chosen in the sum Ž3.. Due to the somewhat singular D˙Ž t . term in Ž9., the coefficient g 1Ž t . in Ž6. is no longer of the order of the memory time t but turns out to be of the same order as the zero order term g 0 , and both terms in Ž4. are relevant in the t 0 limit. The forth line in Ž4. involving the function g 2 Ž t . vanishes for L s L†. Note that we have slightly changed our convention in order to be in line with standard notation: the correlation function Ž9. is here normalized such that the coefficient of the imaginary part, relevant for the damping, is unity. Let us consider the high-temperature spin-boson model w12x. The system Hamiltonian is H s y 12 vsx q 12 Vsz , where v is the tunneling matrix element and V depicts the bias of the system. The coupling

™

r˙ t s yi w H , r t x q g g 0 Ž t . sz r t sz y g g 0 Ž t . r t q H.c.y i vg g 1 Ž t . sx r t y vg g 1 Ž t . sy r t sz q H.c.

Ž 10 .

After an initial slip Ž t 4 t ., the g i Ž t . approach their asymptotic values and the density operator evolves according to

r˙ t s yi w H , r t x q 2 kTgsz r t sz y 2 kTgr t q vg  sx , r t 4 y i vgsy r t sz q i vgsz r t sy .

Ž 11 .

cos vl Ž t y s .

a Ž t , s . s 2 kTD Ž t y s . q i D˙ Ž t y s . ,

'

operator is L s g sz . The ‘post’-Markov master equation can be obtained directly from Ž4.:

Obviously, both Eq. Ž10. and its asymptotic form Ž11. are not of Lindblad form Ž1. due to the presence of the last three terms. Therefore, as a Markov equation with constant coefficients, we cannot expect Ž11. to preserve the positivity of r t if applied to an arbitrary initial density operator. Our derivation shows that for the asymptotic Eq. Ž11. to be meaningful, one has to use an effectiÕe initial condition w4x obtained by fully determining the initial slip arising from the time dependent coefficients g i Ž t . in the true master Eq. Ž10.. We show numerically that Ž10. preserves the positivity of the density matrix while the asymptotic Eq. Ž11. applied to the same initial condition fails to do so. The positivity of r t is equivalent to the condition <