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arXiv:0809.3524 · Journal of Physics Condensed Matter

A unitary perturbation theory approach to real-time evolution problems

Andreas Hackl, Stefan K. Kehrein

Published Dec 1, 2008• 36 citations
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Abstract

We discuss a new analytical approach to real-time evolution in quantum many-body systems. Our approach extends the framework of continuous unitary transformations such that it amounts to a novel solution method for the Heisenberg equations of motion for an operator. It is our purpose to illustrate the accuracy of this approach by studying dissipative quantum systems on all timescales. In particular, we obtain results for non-equilibrium correlation functions for general initial conditions. We illustrate our ideas for the exactly solvable dissipative oscillator and, as a non-trivial model, for the dissipative two-state system.

DOI
10.1088/0953-8984/21/1/015601

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We discuss a new analytical approach to real-time evolution in quantum many-body systems. Our approach extends the framework of continuous unitary transformations such that it amounts to a novel solution method for the Heisenberg equations of motion for an operator. It is our purpose to illustrate the accuracy of this approach by studying dissipative quantum systems on all timescales. In particular, we obtain results for non-equilibrium correlation functions for general initial conditions. We illustrate our ideas for the exactly solvable dissipative oscillator and, as a non-trivial model, for the dissipative two-state system.
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.
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A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.
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(†).