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The Journal of Chemical Physics

Non-Markovian stochastic Schrödinger equations in different temperature regimes: A study of the spin-boson model

Inés de Vega, Daniel Alonso, Pierre Gaspard, Walter T. Strunz

Published Mar 22, 2005• 40 citations

Abstract

Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.

DOI: 10.1063/1.1867377

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Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.

In recent years, there has been tremendous focus on identifying whether effective descriptions of open quantum systems such as master equations can accurately describe the dynamics of open quantum systems. One particular question is whether they provide the correct steady state in the long time limit. The transient regime is also of interest. Description of evolution by various master equations---some of them being not completely positive---is benchmarked against exact solutions [see, e.g., R. Hartmann and W. T. Strunz, Phys. Rev. A 101, 012103 (2020)]. An important property of true evolution is its non-Markovian features, which are not captured by the simplest completely positive master equations. In this paper we consider a non-Markovian yet completely positive evolution (known as refined weak coupling or cumulant equation) for the spin-boson model with an overdamped Drude-Lorentz spectral density and arbitrary coupling. We benchmark it against a numerically exact solution, as well as against other master equations, for different coupling strengths and temperatures. We find the cumulant to be a better description in the weak-coupling regime where it is supposed to be valid. For the examples considered it shows superiority at moderate and strong couplings in the low-temperature regime for all examples considered. In the high-temperature regime, however, its advantage vanishes. This indicates that the cumulant equation is a good candidate for simulations at weak to moderate coupling and low temperature. Our calculations are greatly facilitated due to our concise formulation of the cumulant equation by means of representation of the density matrix in the SU(N) basis.

Non-Markovian stochastic Schrodinger equations (NMSSE) are important tools in quantum mechanics, from the theory of open systems to foundations. Yet, in general, they are but formal objects: their solution can be computed numerically only in some specific cases or perturbatively. This article is focused on the NMSSE themselves rather than on the open-system evolution they unravel and aims at making them less abstract. Namely, we propose to write the stochastic realizations of linear NMSSE as averages over the solutions of an auxiliary equation with an additional random field. Our method yields a non-perturbative numerical simulation algorithm for generic linear NMSSE that can be made arbitrarily accurate for reasonably short times. For isotropic complex noises, the method extends from linear to non-linear NMSSE and allows to sample the solutions of norm-preserving NMSSE directly.

The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.

Non-Markovian stochastic Schrödinger equations (NMSSE) are important tools in quantum mechanics, from the theory of open systems to foundations. Yet, in general, they are but formal objects: their solution can be computed numerically only in some specific cases or perturbatively. This article is focused on the NMSSE themselves rather than on the open-system evolution they unravel and aims at making them less abstract. Namely, we propose to write the stochastic realizations of linear NMSSE as averages over the solutions of an auxiliary equation with an additional random field. Our method yields a non-perturbative numerical simulation algorithm for generic linear NMSSE that can be made arbitrarily accurate for reasonably short times. For isotropic complex noises, the method extends from linear to non-linear NMSSE and allows to sample the solutions of norm-preserving NMSSE directly.

One longstanding difficult problem in quantum dissipative dynamics is to solve the spin-boson model in a non-Markovian regime where a tractable systematic master equation does not exist. The spin-boson model is particularly important due to its crucial applications in quantum noise control and manipulation as well as its central role in developing quantum theories of open systems. Here we solve this important model by developing a non-Markovian quantum Langevin approach. By projecting the quantum Langevin equation onto the coherent states of the bath, we can derive a set of non-Markovian quantum Bloch equations containing no explicit noise variables. This special feature offers a tremendous advantage over the existing stochastic Schr\"odinger equations in numerical simulations. The physical significance and generality of our approach are briefly discussed.

The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.