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Singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode

John Edward Gough, Ramon, Van Handel

Published Sep 13, 2013• 25 citations
OpenAlex

Abstract

Abstract. We consider a physical system which is coupled indirectly to a Markovian resevoir through an oscillator mode. This is the case, for example, in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics. In the strong coupling limit the oscillator can be eliminated entirely from the model, leaving an effective direct coupling between the system and the resevoir. Here we provide a mathematically rigorous treatment of this limit as a weak limit of the time evolution and observables on a suitably chosen exponential domain in Fock space. The resulting effective model may contain emission and absorption as well as scattering interactions. 1.

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Abstract. We consider a physical system which is coupled indirectly to a Markovian resevoir through an oscillator mode. This is the case, for example, in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics. In the strong coupling limit the oscillator can be eliminated entirely from the model, leaving an effective direct coupling between the system and the resevoir. Here we provide a mathematically rigorous treatment of this limit as a weak limit of the time evolution and observables on a suitably chosen exponential domain in Fock space. The resulting effective model may contain emission and absorption as well as scattering interactions. 1.
Various approaches are investigated in order to analyze numerous stochastic differential equations that can be encountered in quantum optics problems, especially, in the case of the driven single cavity mode. The passage to the ordinary coupled differential equations is presented and the treatment of the obtained coupled system is explored. Generalization of the problem to stimulate variable coefficients is discussed and the exact solutions are achieved in explicit forms under suitable conditions on the coefficients.
We examine the dynamics of bipartite quantum entanglement between a two-level atom and the electromagnetic field. We begin with the Jaynes-Cummings model with a single field mode and examine in detail the exact time evolution of entanglement, including cases where the atomic state is initially mixed and the atomic transition is detuned from resonance. We then explore the effects of other nearby modes by calculating the exact time evolution of entanglement in more complex systems with two, three, and five field modes. For these cases we can obtain exact solutions which include the strong-coupling regime. Finally, we consider the entanglement of a two-level atom with the infinite collection of modes present in the intracavity field of a Fabre-Perot cavity. In contrast to the usual treatment of atom-field interactions with a continuum of modes using the Born-Markov approximation, our treatment in all cases describes the full non-Markovian dynamics of the atomic subsystem under general couplings. Only when an analytic expression for the infinite mode case is desired do we need to make a weak coupling assumption which at late times approximates Markovian dynamics.
The Rabi model considers a two-level system (or spin 1/2) coupled to a quantized harmonic oscillator and describes the simplest interaction between matter and light. The recent experimental progress in solid-state circuit quantum electrodynamics has engendered theoretical efforts to quantitatively describe the mathematical and physical aspects of the light-matter interaction beyond the rotating-wave approximation. We develop a stochastic Schr\"odinger equation approach which enables us to access the strong-coupling limit of the Rabi model and study the effects of dissipation and ac drive in an exact manner. We include the effect of Ohmic noise on the non-Markovian spin dynamics, resulting in Kondo-type correlations, as well as cavity losses. We compute the time evolution of spin variables in various conditions. As a consideration for future work, we discuss the possibility of reaching a steady state with one polariton in realistic experimental conditions.
The derivation of a quantum Markovian model for an optomechanical system consisting of a quantum mechanical mirror interacting with quantum optical input fields via radiation pressure is a difficult problem which ultimately involves the scattering process of quantum stochastic calculus. We show that, while the scattering process may be approximated in a singular limit by regular processes using different schemes, the limit model is highly sensitive to the mathematical interpretation of the approximation scheme. We find two main types of stochastic limits of regular models, and illustrate the origin of this difference at the level of one-particle scattering. As an alternative modeling scheme, we consider models of mirrors as nontrivial dielectric media with boundaries that are themselves quantized. Rather than treating the plane waves for the electromagnetic field, we take the actual physical modes and quantize these. The input-output formalism is then obtained in the far zone where the plane-wave approximation is valid. Several examples are considered, and the quantum stochastic model is derived. We also consider the quantum trajectories problem for continual measurement of the reflected output fields, and derive the stochastic master equations for homodyning and photon-counting detection to estimate the mirror observables.
Abstract The Lindblad equation, which describes Markovian quantum dynamics under dissipation, is usually derived under the weak system-bath coupling assumption. Strong system-bath coupling often leads to non-Markov evolution. The singular-coupling limit is known as an exception: it yields a Lindblad equation with an arbitrary strength of dissipation. However, the singular-coupling limit requires high-temperature limit of the bath, and hence the system ends up in a trivial infinite-temperature state, which is not desirable in the context of quantum control. In this work, it is shown that we can derive a Markovian Lindblad equation for an arbitrary strength of the system-bath coupling by considering a new scaling limit that is called the singular-driving limit, which combines the singular-coupling limit and fast periodic driving. In contrast to the standard singular-coupling limit, an interplay between dissipation and periodic driving results in a nontrivial steady state.