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Journal of Mathematical Physics

Perturbative and nonperturbative master equations for open quantum systems

Wusheng Zhu, HERSCHEL A. RABITZ

Published Feb 1, 2005• 4 citations
Publisher

Abstract

This paper develops perturbative and nonperturbative master equations for open quantum systems based on time-dependent variational functionals. The perturbative equations are more concise and suitable for dealing with cases of weak system-environment coupling for short evolution time scales. The nonperturbative equations are valid for all time and appropriate to treat cases of strong system-environment coupling. When a system contains an external control field, both the perturbative and nonperturbative master equations reveal the embedded control field dependence upon the system decoherence, which provides a basis for decoherence management.

DOI
10.1063/1.1839275

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This paper develops perturbative and nonperturbative master equations for open quantum systems based on time-dependent variational functionals. The perturbative equations are more concise and suitable for dealing with cases of weak system-environment coupling for short evolution time scales. The nonperturbative equations are valid for all time and appropriate to treat cases of strong system-environment coupling. When a system contains an external control field, both the perturbative and nonperturbative master equations reveal the embedded control field dependence upon the system decoherence, which provides a basis for decoherence management.
An extremely useful evolution equation that allows systematically calculating the two-time correlation functions (CF's) of system operators for non-Markovian open (dissipative) quantum systems is derived. The derivation is based on perturbative quantum master equation approach, so non-Markovian open quantum system models that are not exactly solvable can use our derived evolution equation to easily obtain their two-time CF's of system operators, valid to second order in the system-environment interaction. Since the form and nature of the Hamiltonian are not specified in our derived evolution equation, our evolution equation is applicable for bosonic and/or fermionic environments and can be applied to a wide range of system-environment models with any factorized (separable) system-environment initial states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equation is valid for both L = L(†) and L ≠ L(†) cases, in contrast to those evolution equations valid only for L = L(†) case in the literature. The derived equation that generalizes the quantum regression theorem (QRT) to the non-Markovian case will have broad applications in many different branches of physics. We then give conditions on which the QRT holds in the weak system-environment coupling case and apply the derived evolution equation to a problem of a two-level system (atom) coupled to the finite-temperature bosonic environment (electromagnetic fields) with L ≠ L(†).
The time-convolutionless master equation provides a general framework to model non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to the perturbative time-convolutionless quantum master equations. We further introduce a nonperturbative approach that approximates the time-convolutionless generator as a nested time-ordered exponential function.
The time-convolutionless master equation provides a general framework to model the non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to perturbative time-convolutionless quantum master equations. We further introduce a general iterative approach to construct nonperturbative approximations for the time-local generator as nested time-ordered exponential operators.
The time-convolutionless quantum master equation is widely used to simulate reduced dynamics of a quantum system coupled to a bath. However, except for several special cases, applications of this equation are based on perturbative calculation of the dissipative tensor, and are limited to the weak system-bath coupling regime. In this paper, we derive an exact time-convolutionless quantum master equation from the path integral approach, which provides a new way to calculate the dissipative tensor nonperturbatively. Application of the new method is demonstrated in the case of an asymmetrical two-level system linearly coupled to a harmonic bath.
Manipulations with open quantum systems (such as qubits) are fundamental for any quantum technology. They are the focus of studies involving optimal control theory. Usually, control is achieved through the use of time-dependent external fields when driven system evolution is simulated employing the Davies construction (second-order Markov quantum master equation formulation). As a weak (second order) coupling scheme, the Davies construction is limited in its ability to account for bath-induced coherences. To overcome the limitation, we utilize the nonequilibrium Green’s function method and demonstrate that accounting for the coherences makes a qualitative impact on quantum control studies. We find that accounting for the coherences is especially important when dealing with system evolution involving mixed states.
The reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time-local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Choosing the most suitable one for a specific system is not straightforward. By focusing on the accuracy of the reduced dynamics we provide a thorough assessment for a selection of methods (Redfield equation, quantum optical master equation, coarse-grained master equation, a related dynamical map approach, and a partial-secular approximation). We consider secondary, here, whether or not an approach guarantees positivity. We use two qubits coupled to a Lorentzian environment in a spin-boson-like fashion modeling a generic situation with various system and bath time scales. We see that, independent of the initial state, the simple Redfield equation with time-dependent coefficients is significantly more accurate than all other methods under consideration. We emphasize that positivity violation in the Redfield equation formalism becomes relevant only in a regime where any of the perturbative master equations considered here are rendered invalid anyway. This implies that the loss of positivity should in fact be welcomed as an important feature: it indicates the breakdown of the weak-coupling assumption. In addition we present the various approaches in a self-contained way and use the behavior of their errors to provide further insight into the range of validity of each method.