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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Stochastic wavefunction methods beyond the Born - Markov and rotating-wave approximations

P Stenius, Ataç İmamoğlu

Published Feb 1, 1996• 30 citations
Publisher

Abstract

We develop a stochastic wavefunction approach to analyse non-Markovian system - reservoir interactions. Provided that the reservoir density operator is Gaussian, this approach can be used to treat strong reservoir coupling beyond the Born and rotating-wave approximations. The memory effects associated with the finite reservoir correlation time are incorporated into the dynamics by expanding the initial system using fictitious harmonic oscillator modes, so that the new enlarged system has Markovian dissipative interactions. We apply the new approach to the analysis of exciton - phonon interaction in a quasi-zero-dimensional GaAs structure and predict a strong deviation from the Born - Markov result.

DOI
10.1088/1355-5111/8/1/021

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We develop a stochastic wavefunction approach to analyse non-Markovian system - reservoir interactions. Provided that the reservoir density operator is Gaussian, this approach can be used to treat strong reservoir coupling beyond the Born and rotating-wave approximations. The memory effects associated with the finite reservoir correlation time are incorporated into the dynamics by expanding the initial system using fictitious harmonic oscillator modes, so that the new enlarged system has Markovian dissipative interactions. We apply the new approach to the analysis of exciton - phonon interaction in a quasi-zero-dimensional GaAs structure and predict a strong deviation from the Born - Markov result.
We extend the quantum Monte Carlo wave-function method of quantum optics to non-Markovian system-reservoir interactions. The finite correlation time of the reservoir and the associated memory effects are incorporated into the dynamics by expanding the initial system using fictitious harmonic-oscillator modes, which have Markovian dissipative interactions. The underlying principle is the fact that a class of reservoir spectral functions can be approximated by a finite superposition of Lorentzian functions with positive coefficients.
The exact non-Markovian dynamics of a microcavity strongly coupled to a general reservoir at arbitrary temperature are studied. With the exact master equation for the reduced density operator of the cavity system, we analytically solve the time evolution of the cavity state and the associated physical observables. We show that the non-Markovian dynamics are completely determined by the propagating (retarded) and correlation Green functions. Comparing the non-Markovian behavior at finite temperature with those at the zero-temperature limit or Born-Markov limit, we find that the non-Markovian memory effect can dramatically change the coherent and thermal dynamics of the cavity. We also numerically study the dissipation dynamics of the cavity through the mean mode amplitude decay and the average photon number decay in the microwave regime. It is shown that the strong coupling between the cavity and the reservoir results in a long-time dissipationless evolution to the cavity field amplitude, and its noise dynamics undergo a critical transition from the weak to the strong coupling due to the non-Markovian memory effect.
This thesis investigates two thematic lines of research, both underpinned by non-Markovian system-reservoir interactions in quantum optics. The overarching focus is on modelling the open system dynamics in a non-perturbative fashion, broadly on - though not restricted to - instances when the environment is structured. A theory is developed by means of enlarging the open system over environmental degrees of freedom to include memory effects in its dynamics. This is achieved using an established technique that involves mapping a bosonic environment onto a 1D chain of harmonic oscillators. Within this setting, we apply a Heisenberg equation-of-motion approach to derive an exact set coupled differential equations for the open system and a single auxiliary oscillator of the chain. The combined equations are shown to have their interpretation rooted in a quantum Markov stochastic process. Including the auxiliary chain oscillator as part of the original system then enables us to obtain an exact master equation for the enlarged system, avoiding any need for the Born-Markov approximations. Our method is valid for a dissipative two-state system, with cases of multiple excitations and added driving discussed. Separately, we apply the framework of quantum Darwinism to an atom-cavity system, and, subsequently, to a more general multiple-environment model. In both cases, the time-dependent spread of correlations between the open system and fractions of the environment is analysed during the course of the decoherence process. The degree to which information is redundant across different fractions is checked to infer the emergence of classicality. In the second case, we go further and present a decomposition of information in terms of its quantum and classical correlations. A quantitative measure of redundancy is also studied with regard to its ability to witness non-Markovian behaviour. Besides fundamental interest, our results have application to quantum information processing and quantum technologies, keeping in mind the potential beneficial use of non-Markovian effects in reservoir engineering.
We propose a formulation to obtain the exact susceptibility for system arbitrary operators to the external fields by means of the whole-system Hamiltonian (system plus reservoir) diagonalization methods, where the dissipative effects directly reflect the nature of the structured non-Markovian reservoir. This treatment does not make the Born-Markovian approximation in structured non-Markovian reservoir. The relations between linear response function and bound-states for the system as well as structured reservoir are found, which shows the photon bound-states and continuous energy spectrum can be readout from the susceptibility, respectively. These results are then used to examine the validity of second-order Born-Markovian approximation, where we find interesting features (e.g., bound-states) are lost in the approximate treatments for open systems. We study the dependence of the response function on the type (spectrum density) of interaction between the system and structured reservoir. We also give the physical reasons behind the disappearance of the bound-states in the approximation method. Finally, these results are also extended to a more general quantum network involving an arbitrary number of coupled-bosonic system without rotating-wave approximation. The presented results might open a new door to understand the linear response and the energy spectrum for non-Markovian open systems with structured reservoirs.
Open quantum systems are often encountered in many different physical situations. From quantum optics to statistical mechanics, they are fundamental in the understanding of a great variety of different phenomena. Some of the most common examples are the relaxation to equilibrium, the existence of nonequilibrium stationary states, and the dynamics of atoms in interaction with electromagnetic fields. A crucial step in the analysis is to consider the quantum open system and its environment as the two mutually interacting components of a larger isolated system. Thereafter, the so‐called Markov approximation is often considered, which consists on assuming that the time scales associated to the dynamics of the quantum open system are larger than those of the environment. It is the interplay of the different time scales associated with the system and the environment what determines the validity of the different approximations made. In this paper we will discuss the dynamics of a open quantum system in contact with a reservoir when the Markov approximation is not valid, and we have to include some non‐Markovian or memory effects.
Interaction of a small system S with a large reservoir R amounts to thermal relaxation of the reduced system density operator ρS(t). The presence of the reservoir enters the equation of motion for ρS(t) through the reservoir correlation functions fkl(τ) (defined in the text), which decay to zero for τ→∞ on a time scale τc. Commonly, this τc is much smaller than the inverse relaxation constants for the time evolution of ρS(t). Then a series of approximations can be made, which lead to a Markovian equation of motion for ρS(t). In this paper the assumption of a small reservoir correlation time is removed. The equation of motion for ρS(t) is solved, and it appears that the memory effect, due to τc≂/0, can be incorporated in a frequency dependence of the relaxation operator Γ̃(ω). Subsequently, (unequal-time) quantum correlation functions of two system operators are considered, where explicit expressions for (the Laplace transform of) the correlation functions are obtained. They involve again the relaxation operator Γ̃(ω), which accounts for the time regression. Additionally it is found that an initial-correlation operator Υ̃(ω) arises, as a consequence of the fact that the equal-time correlation functions do not factorize as ρS(t) times the reservoir density operator. It is pointed out that the frequency dependence of Γ̃(ω) and the occurrence of a nonzero T̃(ω) both arise as a result of τc≂/0, and should therefore be treated on an equal footing. Explicit evaluation of Γ̃(ω) and Υ̃(ω) shows that their matrix elements can be expressed entirely in f̃kl(ω), just as in the Markov approximation. Hence no essential complications appear if one should go beyond the limits of a small reservoir correlation time τc.