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arXiv:0803.2887 · Physical Review A

Derivation of Maxwell-Bloch-type equations by projection of quantum models

Hideo Mabuchi

Published Jul 15, 2008• 26 citations
arXivHTMLPDF

Abstract

A simple algebraic procedure is described for deriving Maxwell-Bloch-type equations from single-atom cavity quantum electrodynamics (cavity QED) master equations via orthogonal projection onto a manifold of semiclassical states. In particular, the usual Maxwell-Bloch equations are obtained---up to a state-dependent correction factor of order unity---straightforwardly from the unconditional Jaynes-Cummings master equation. The technique of projecting onto a semiclassical manifold can also be applied with conditional master equations (quantum filters), leading to stochastic simulation models that include multiplicative noise terms associated with fluctuations of the atomic dipole. The utility of such models is briefly explored in the context of single-atom absorptive bistability.

DOI
10.1103/physreva.78.015801

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A simple algebraic procedure is described for deriving Maxwell-Bloch-type equations from single-atom cavity quantum electrodynamics (cavity QED) master equations via orthogonal projection onto a manifold of semiclassical states. In particular, the usual Maxwell-Bloch equations are obtained---up to a state-dependent correction factor of order unity---straightforwardly from the unconditional Jaynes-Cummings master equation. The technique of projecting onto a semiclassical manifold can also be applied with conditional master equations (quantum filters), leading to stochastic simulation models that include multiplicative noise terms associated with fluctuations of the atomic dipole. The utility of such models is briefly explored in the context of single-atom absorptive bistability.
Many quantum mechanical systems require large (potentially infinite) numbers of variables to exactly describe their state. In this thesis, I examine two approaches to develop simple, approximate models for such systems, which capture their essential dynamics. I use two bistable regimes of the Jaynes-Cummings model of cavity quantum electrodynamics as example systems to evaluate the effectiveness of each approach. In the phase bistable regime (which occurs with large driving field, and which I study in an on-resonance "bad cavity" regime to make numerical simulations tractable), the cavity field switches between two states with identical amplitude but opposite phase. In the absorptive bistable regime (which I study with small driving field in an on-resonance "good cavity"'), two stable regions of state space differ in cavity field amplitude as well as their shape and qualitative behavior. After introducing these two regimes and their dynamics, I give a short introduction to projecting dynamical equations onto linear subspaces. Proper Orthogonal Decomposition (POD) allows the algorithmic construction of subspaces onto which the dynamics may be projected. I demonstrate that the application of POD to phase bistability results in effective approximate filters, while the asymmetry of the absorptive bistable case requires extensions to POD, developed in this thesis, to create a functional filter. Local Tangent Space Alignment is one of a class of unsupervised manifold learning algorithms which use the local geometry of high-dimensional data, such as quantum trajectories, to calculate the coordinates of that data on a low-dimensional manifold. I show how this algorithm functions, and characterize the manifolds that result from phase and absorptive bistability. I fit the 3-dimensional phase bistable manifold with a small set of system observables, and create a three-dimensional set of equations (similar to the semi-classical Maxwell-Bloch equations) which perform very well as a filter. Absorptive bistability again proves to be more complicated, but I am able to show that the underlying manifold is small, and make some progress on characterizing its relations with system observables.
Interaction of atoms with quantum states of light is a long-standing problem that apart from fundamental physics has potential applications for optical quantum-state storage, quantum communication, and quantum information. A fully quantum mechanical treatment of this problem is usually very complicated mathematically. Here we show, however, that quantum mechanical evolution equations describing single-photon emission (absorption) by atomic ensembles can be written in a form equivalent to the semiclassical Maxwell-Bloch equations. This connection allows us to find exact analytical solutions of the fully quantum mechanical problem. We also found that semiclassical Maxwell-Bloch equations should be written in a form different from those commonly used. Namely, the classical limit of the quantum problem gives a propagation equation with the Laplacian operator on the right-hand side rather than with the second-order time derivative.
This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.
Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite-dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case.) One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a stongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.
Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite-dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case.) One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a stongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.