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Physical Review A

Hierarchy of equations of multiple-time correlation functions

Daniel Alonso, Inés de Vega

Published May 17, 2007• 34 citations
Publisher

Abstract

In this paper we derive the evolution equations for non-Markovian multiple-time correlation functions of an open quantum system without using any approximation. We find that these equations conform an open hierarchy in which $N$-time correlation functions are dependent on $(N+1)$-time correlations. This hierarchy of equations is consistently obtained with two different methods: A first one based on Heisenberg equations of system operators, and a second one based on system propagators. The dependency on higher order correlations, and therefore the open hierarchy structure, only disappears in certain particular cases and when some hypothesis or approximations are considered in the equations. In this paper we consider a perturbative approximation and derive the general evolution equation for $N$-time correlations. This equation turns to depend only on $N$-time and lower order correlation functions, conforming a closed hierarchy structure that is useful for computational purposes.

DOI
10.1103/physreva.75.052108

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In this paper we derive the evolution equations for non-Markovian multiple-time correlation functions of an open quantum system without using any approximation. We find that these equations conform an open hierarchy in which NNN-time correlation functions are dependent on (N+1)(N+1)(N+1)-time correlations. This hierarchy of equations is consistently obtained with two different methods: A first one based on Heisenberg equations of system operators, and a second one based on system propagators. The dependency on higher order correlations, and therefore the open hierarchy structure, only disappears in certain particular cases and when some hypothesis or approximations are considered in the equations. In this paper we consider a perturbative approximation and derive the general evolution equation for NNN-time correlations. This equation turns to depend only on NNN-time and lower order correlation functions, conforming a closed hierarchy structure that is useful for computational purposes.
A method is presented for the systematic derivation of a hierarchy of coupled equations for the computation of two-time correlation functions of operators for open many-body quantum systems. We show how these systems of equations can be closed in mean-field and beyond approximations. Results for the specific example of the spectral weight functions are discussed. Our method allows one to access the full temporal evolution, not just the stationary solution, of non-equilibrium open quantum problems described by a Markovian master equation.
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(†).
We study the Lindblad master equation in the space of operators and provide simple criteria for closeness of the hierarchy of equations for correlations. We separately consider the time evolution of closed and open systems and show that open systems satisfying the closeness conditions are not necessarily of Gaussian type. In addition, we show that dissipation can induce the closeness of the hierarchy of correlations in interacting quantum systems. As an example we study an interacting optomechanical model, the Fermi–Hubbard model, and the Rabi model, all coupled to a fine-tuned Markovian environment and obtain exact analytic expressions for the time evolution of two-point correlations.
We derive the evolution equations for two-time correlation functions of a generalized non-Markovian open quantum system based on a modified stochastic Schrödinger equation approach. We find that the two-time reduced propagator, an object that used to be characterized by two independent stochastic processes in the Hilbert space of the system, can be simplified and obtained by taking ensemble average over one single noise. This discovery can save the cost of computation, and accelerate the converging process when taking the average over noisy trajectories. As a result, our method can be widely applied to many open quantum models, especially large-scale systems and extend the quantum regression theory to the non-Markovian case. In the short-time simulations, it is observed a significant difference between Markovian and non-Markovian cases, which can be applied to realize the environmental spectrum detection and enhance the measurement sensitivity in varying open quantum systems.
We introduce a formalism for time-dependent correlation functions for systems whose evolutions are governed by non-Hermitian Hamiltonians of general type. It turns out that one can define two different types of time correlation functions. Both these definitions seem to be physically consistent while becoming equivalent only in certain cases. Moreover, when autocorrelation functions are considered, one can introduce another function defined as the relative difference between the two definitions. We conjecture that such a function can be used to assess the positive semidefiniteness of the density operator without computing its eigenvalues. We illustrate these points by studying analytically a number of models with two energy levels.
We derive the dynamical equation of the reduced propagator, an object that evolves state vectors of the system conditioned to the dynamics of its environment, which is not necessarily in the vacuum state at the initial time. Such a reduced propagator is essential to obtain multiple-time correlation functions (MTCFs). We also study the evolution of MTCFs within the weak-coupling limit and show that the quantum regression theorem is, in general, not satisfied. We illustrate the theory in two different cases: first, solving an exact model, and, second, presenting the results of the numerical integration for a system coupled with a thermal environment through a nondiagonal interaction.