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Physical Review A

Decay of an atom coupled strongly to a reservoir

B. M. Garraway

Published Jun 1, 1997• 188 citations
Publisher

Abstract

An approach to the decay of a two-state quantum system coupled to a resonant environment is developed in terms of exact (nonperturbative) master equations. Starting from a quantum optical model of a two-level system coupled to a heatbath, it is shown that the effect of the bath can be replaced by one or more pseudomodes. This description is valid for a wide class of analytic density-of-state functions and it leads to exact master equations which fully describe the non-Markovian decay of the quantum system without the use of perturbation theory, the Born approximation, or the Markov approximation. Two examples are given: a simple model of a density-of-states gap which has two poles, and a non-Lorentzian resonance.

DOI
10.1103/physreva.55.4636

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An approach to the decay of a two-state quantum system coupled to a resonant environment is developed in terms of exact (nonperturbative) master equations. Starting from a quantum optical model of a two-level system coupled to a heatbath, it is shown that the effect of the bath can be replaced by one or more pseudomodes. This description is valid for a wide class of analytic density-of-state functions and it leads to exact master equations which fully describe the non-Markovian decay of the quantum system without the use of perturbation theory, the Born approximation, or the Markov approximation. Two examples are given: a simple model of a density-of-states gap which has two poles, and a non-Lorentzian resonance.
Atoms can nowadays be placed in increasingly exotic environments such as microscopic cavities and materials with photonic band gaps. High-Q cavities can now easily result in a strong coupling between an atom and its environment where perturbation theory should no longer be appropriate. The purpose of this paper is to describe the dynamics of a multilevel V-type atomic system (including the case of a two-level system) which interacts with a reservoir modeled by a generalized density of states. A theoretical construct, the pseudomode, is utilized to develop general methods for solution. Without using perturbation theory the equivalent master equation is developed and the relationship between the master equation, the pseudomodes, and the generalized density of states function is explored with examples. Utilizing a straightforward definition of the pseudomode, it is found that many functions for the density of states lead to problematic non-Lindblad master equations. Several examples are given, and it is shown how to convert the non-Lindblad master equations into a Lindblad form in these cases. The examples include a non-Lorentzian resonance and a simple model of a photonic band gap.
The dynamics of a three-level atom in a cascade configuration with both transitions coupled to a single structured reservoir of quantized field modes is treated using Laplace transform methods applied to the coupled amplitude equations. Results are also obtained from master equations by two different approaches, that is, involving either pseudomodes or quasimodes. Two different types of reservoir are considered, namely a high-Q cavity and a photonic band-gap system, in which the respective reservoir structure functions involve Lorentzians. Non-resonant transitions are included in the model. In all cases non-Markovian behaviour for the atomic system can be found, such as oscillatory decay for the high-Q cavity case and population trapping for the photonic band-gap case. In the master equation approaches, the atomic system is augmented by a small number of pseudomodes or quasimodes, which in the quasimode approach themselves undergo Markovian relaxation into a flat reservoir of continuum quasimodes. Results from these methods are found to be identical to those from the Laplace transform method including two-photon excitation of the reservoir with both emitting sequences. This shows that complicated non-Markovian decays of an atomic system into structured EM field reservoirs can be described by Markovian models for the atomic system coupled to a small number of pseudomodes or quasimodes.
The dynamics of a three-level atom in a cascade configuration with both transitions coupled to a single structured reservoir of quantized field modes is treated using Laplace transform methods applied to the coupled amplitude equations. Results are also obtained from master equations by two different approaches, that is, involving either pseudomodes or quasimodes. Two different types of reservoir are considered, namely a high-Q cavity and a photonic band-gap system, in which the respective reservoir structure functions involve Lorentzians. Non-resonant transitions are included in the model. In all cases non-Markovian behaviour for the atomic system can be found, such as oscillatory decay for the high-Q cavity case and population trapping for the photonic band-gap case. In the master equation approaches, the atomic system is augmented by a small number of pseudomodes or quasimodes, which in the quasimode approach themselves undergo Markovian relaxation into a flat reservoir of continuum quasimodes. Results from these methods are found to be identical to those from the Laplace transform method including two-photon excitation of the reservoir with both emitting sequences. This shows that complicated non-Markovian decays of an atomic system into structured EM field reservoirs can be described by Markovian models for the atomic system coupled to a small number of pseudomodes or quasimodes.
We develop an exact framework for describing the non-Markovian dynamics of an open quantum system interacting with an environment modeled by a generalized spectral density function.The approach relies on mapping the initial system onto an auxiliary configuration, comprising the original open system coupled to a small number of discrete modes, which in turn are each coupled to an independent Markovian reservoir.Based on the connection between the discrete modes and the poles of the spectral density function, we show how expanding the system using the discrete modes allows for the full inclusion non-Markovian effects within an enlarged open system whose dynamics is governed by an exact Lindblad master equation.Initially we apply this result to obtain a generalization of the pseudomode method [B.M. Garraway, Phys. Rev. A. 55, 2290 (1997)] in cases where the spectral density function has a Lorentzian structure.For many other types of spectral density function, we extend our proof to show that an open system dynamics may be modeled physically using discrete modes which admit a non-Hermitian coupling to the system and for such cases determine the equivalent master equation to no longer be of Lindblad form.For applications involving two discrete modes, we demonstrate how to convert between pathological and Lindblad forms of the master equation using the techniques of the pseudomode method.
The master equation for a two-level atom driven by a strong classical field and damped into a tailored reservoir with nonflat density of modes is derived under the Born-Markov approximation. To derive the master equation, the dressing transformation on the atomic operators is performed first, and, next, the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The effects of a strong field and/or structured reservoir are seen as nonstandard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs. The master equation leads to the generalized optical Bloch equations that can easily be solved for the steady state and, together with the quantum regression theorem, allow for analytical expressions for the fluorescence, as well as absorption spectra.
Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. By employing the exact solution for the model, analytical expressions are determined for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows an explicit comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, the structure of widely used phenomenological master equations with a memory kernel may be incompatible with a nonperturbative treatment of the underlying microscopic model. Several physical implications of the results on the microscopic analysis and the phenomenological modeling of non-Markovian quantum dynamics of open systems are discussed.