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Fourth-Order Quantum Master Equation and Its Markovian Bath Limit Seogjoo Jang, Jianshu Cao, and Robert J

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Fourth-Order Quantum Master Equation and Its Markovian Bath Limit Seogjoo Jang, Jianshu Cao, and Robert J JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 7 15 FEBRUARY 2002 Fourth-order quantum master equation and its Markovian bath limit Seogjoo Jang, Jianshu Cao, and Robert J. Silbey Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ͑Received 14 September 2001; accepted 28 November 2001͒ Fourth-order quantum master equations ͑FQMEs͒ are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1445105͔ I. INTRODUCTION In the present paper, we derive a general and a detailed form of the fourth-order QME ͑FQME͒. The expression for 1–17 Quantum master equations ͑QMEs͒ offer direct pre- the FQME where the superoperators are given by Liouville scriptions on how to evolve an open quantum system, and operators, without explicit averaging over the bath degrees of have provided canonical frameworks for the description of freedom, are well known.5–7,23–28 Our new contribution is to various quantum relaxation processes. Although formally ex- perform the explicit averaging over the bath, based on few act equations are available,3–7 in practical applications18,19 necessary assumptions, and to derive general expressions for the usual choices have been second-order equations such as the kernels of the FQME, where the bath correlation func- Bloch–Redfield equation1–4 and its generalizations.8–14 tions are decoupled from the system superoperators. This These retain the necessary ingredients of the dissipative quantum dynamics while being simple enough, and can be result leads to the ability to perform routine FQME calcula- related to the microscopic properties of the system, an im- tions for various systems and general formal analyses in portant asset hardly attainable by axiomatic approaches15–17 some limiting situations. Our main focus here is on the latter only. aspect, in one particular limit. We derive more detailed ex- However, the second-order QME ͑SQME͒1–4,8–14 has a pressions of the kernels for the model of linearly coupled limitation, the assumption of weak system-bath interaction. harmonic oscillator bath and then make a careful examina- With the advance of ultrafast spectroscopy and submicros- tion of the Markovian bath limit. cale experiments, increasing number of studies are being per- For the model of linearly coupled harmonic oscillator formed for condensed phase systems where the weak system- bath, the path integral formalism29–32 provides a powerful bath interaction does not necessarily hold. If suitable alternative to the perturbation approaches. Caldeira and 20–22 transformations can be found such that the weak inter- Leggett ͑CL͒30,31 applied the Feynman–Vernon influence action regime is recovered, applications of the SQME to functional formalism33 to a system of a quantum particle in a such systems after the transformation can be justified. Oth- continuous potential field. In the limit where the bath be- erwise, explicit considerations of the higher order terms may comes Markovian, they derived a quantum Fokker–Planck be necessary. Recent theoretical studies on the higher order equation30 which has the correct classical limit. Hereafter, QME23–26 or its stochastic Schro¨dinger equation version27,28 this will be called CL master equation ͑CLME͒. Quite inter- have important implications in this respect. estingly, the CLME can also be derived from the SQME for Due to the perturbative nature, the interaction strength the same model.10,11,14 Does this mean that consideration up regime where a higher order QME is applicable may not be dramatically larger than that of the SQME. However, model to the second order is exact for the CL model in the Markov- calculations26–28 demonstrate that higher order corrections ian bath limit? Although various works on the model of har- 34–36 can bring substantial improvements in intermediate coupling monic oscillator system seem to advocate this view, no regimes of physical relevance. In other cases, even new rigorous clarification is available for a general system Hamil- 37 qualitative features can arise.24 These examples indicate that tonian, as has been recently pointed out. Although a satis- a systematic extension of the SQME to a higher order is factory answer for this remains a challenge, the more modest worthwhile and that it may provide a more solid theoretical goal of examining the CLME at the fourth-order level can be understanding of open system quantum dynamics. achieved through our Markovian FQME. 0021-9606/2002/116(7)/2705/13/$19.002705 © 2002 American Institute of Physics Downloaded 03 Jun 2002 to 18.60.2.110. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp 2706 J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Jang, Cao, and Silbey A two-level system coupled to the harmonic oscillator The total density operator at tϭ0 is assumed to be the bath has been studied numerous times, since it is the simplest following product form: nontrivial model mimicking an open discrete quantum 32 ␳ ͒ϭ␴ ͒␳ ͑ ͒ system. However, straightforward perturbative studies be- ͑0 ͑0 b , 4 yond the second order are rare.23–26 Important contributions were made by Skinner and co-workers23,24 who showed that ␴͑ ͒ ␳ where 0 is an arbitrary system density operator and b is the dephasing time (T ) can be different from twice the 2 an equilibrium bath density operator commuting with Hb . population relaxation time (T1) even when the system-bath The bath part of the interaction Hamiltonian satisfies the fol- coupling is off-diagonal. More surprisingly, T2 can be greater lowing condition: than 2T1 for a certain complex coupling. These results are obtained for finite bath correlation time. According to their Tr ͕␳ B͖ϭ0, ͑5͒ formal expression for the pure dephasing time,24 however, b b the second-order result of T ϭ2T is recovered in the limit 2 1 which is always possible for a properly defined system where the bath timescale becomes zero. Thus for the model Hamiltonian. of two-level system off-diagonally coupled to the Markovian For the conditions as specified above, formally exact bath, the fourth-order corrections should appear only in the quantum master equations with simple structure can be de- coherence terms. Confirmation of this and calculation of the rived. From these formally exact equations, the FQMEs can coherence terms can be done quite easily using our general be obtained by making approximations valid up to the fourth expression for the Markovian FQME. order of the system-bath interaction. There are two well- The paper is organized as follows. In Sec. II, we derive known distinctive procedures. One uses the projection opera- FQMEs for a general system plus bath Hamiltonian. Section tor technique,3,5–10,38,39 which leads to time-nonlocal ͑TN͒ III considers the special case of linearly coupled harmonic equation, and the other employs the generalized cumulant oscillator bath. In Sec. IV, a Markovian FQME is derived in theorem,5–7,23,40 which leads to time-local ͑TL͒ equation. the high temperature Ohmic bath limit, which is then applied When considered up to the infinite order, these two equations to the above two examples of the system Hamiltonian, the are equivalent.5–7 Within the perturbative approximations, CL model and a two-level system off-diagonally coupled to however, they become different. the bath. Section V summarizes conclusions and the impli- cations of the present work. A. Time-nonlocal fourth-order quantum master equation „TN-FQME… II. GENERAL EXPRESSIONS FOR THE FOURTH- The TN equation is often called chronological ordering ORDER EQUATION prescription6 or time convolution7 equation. The derivation Consider a closed composite of system plus bath with can be made using the projection operator technique as men- the Hamiltonian given by tioned above. Although this procedure has been demon- strated many times,3,5–10,38,39 we provide our own derivation HϭH ϩH ϩH ϭH ϩH , ͑1͒ s b 1 0 1 in Appendix A to make the present work self-contained. ͑ ͒ where Hs is the system Hamiltonian, Hb the bath Hamil- Equation A8 is the formally exact TN-QME for the re- ␴ ϵ ␳ tonian, and H1 represents the interaction between the system duced system density operator (t) Trb͕ (t)͖. The TN- and the bath. For the clarity of presentation, we assume that FQME can be found by making a second-order approxima- Ϫ ͐t ␶ ϪP L ␶ the interaction is a single product of a system operator S and tion for exp(ϩ)͕ i ␶ d Ј(1 ) 1( Ј)͖. The result is Eq. a bath operator B as follows: ͑A9͒, which can be equivalently written as ϭ ͑ ͒ H1 SB. 2 t ͑ ͒ ͑ ͒ All the following results are, however, general enough to be ␴˙ ͑t͒ϷϪ ͵ d␶͑K 2 ͑t,␶͒ϪK 4 ͑t,␶͒͒␴͑␶͒, ͑6͒ extended to the cases where the interaction is given by a sum 0 of terms like Eq.
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