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Journal of Modern Optics

Coherent Atomic Excitation in a Cavity. I. Low Density

M.J. Shaw, Bruce W. Shore

Published May 1, 1990• 5 citations
Publisher

Abstract

We present a procedure for treating coherent selective excitation (and subsequent ionization) of atomic vapour in a radiation-filled cavity. Specifically, we describe a technique for solving the coupled Maxwell-Bloch equations for atoms in a one-dimensional cavity by means of a spatial quasi-Fourier expansion that introduces a sequence of increasing spatial harmonics for atomic populations and coherences. We discuss the resulting coupled equations and present examples of solutions for a simple model of stationary and moving two-state atoms in the limit of very low density. The atomic excitation exhibits a travelling wave behaviour, but with significant modulation.

DOI
10.1080/09500349014550961

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We present a procedure for treating coherent selective excitation (and subsequent ionization) of atomic vapour in a radiation-filled cavity. Specifically, we describe a technique for solving the coupled Maxwell-Bloch equations for atoms in a one-dimensional cavity by means of a spatial quasi-Fourier expansion that introduces a sequence of increasing spatial harmonics for atomic populations and coherences. We discuss the resulting coupled equations and present examples of solutions for a simple model of stationary and moving two-state atoms in the limit of very low density. The atomic excitation exhibits a travelling wave behaviour, but with significant modulation.
This memo documents a portion of our work developing methods to treat selective excitation (and subsequent ionization) of atomic vapor in a cavity. Specifically, it describes a technique for solving the coupled Maxwell-Bloch equations for atoms in a one-dimensional cavity by means of a spatial quasi-Fourier expansion that introduces a sequence of increasing spatial harmonics for atomic populations and coherences. We present this method and the algorithm that forms the basis of a computer code for solving these equations. We present examples of solutions for a simple model of stationary two-state atoms in the limit of very low density. These examples illustrate both the usefulness and the limitations of spatial quasi-Fourier expansion techniques. In particular, the examples presented here show the importance of retaining a succession of spatial harmonics when the excitation is coherent (as evidenced by Rabi oscillations). 12 refs., 5 figs.
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Coherent manipulations of atoms using laser light The internal structure of a particle - an atom or other quantum system in which the excitation energies are discrete - undergoes change when exposed to pulses of near-resonant laser light. This tutorial review presents basic concepts of quantum states, of laser radiation and of the Hilbert-space statevector that provides the theoretical portrait of probability amplitudes - the tools for quantifying quantum properties not only of individual atoms and molecules but also of artificial atoms and other quantum systems. It discusses the equations of motion that describe the laser-induced changes (coherent excitation), and gives examples of laser-pulse effects, with particular emphasis on two-state and three-state adiabatic time evolution within the rotating-wave approximation. It provides pictorial descriptions of excitation based on the Bloch equations that allow visualization of two-state excitation as motion of a three-dimensional vector (the Bloch vector). Other visualization techniques allow portrayal of more elaborate systems, particularly the Hilbert-space motion of adiabatic states subject to various pulse sequences. Various more general multilevel systems receive treatment that includes degeneracies, chains and loop linkages. The concluding sections discuss techniques for creating arbitrary pre-assigned quantum states, for manipulating them into alternative coherent superpositions and for analyzing an unknown superposition. Appendices review some basic mathematical concepts and provide further details of the theoretical formalism, including photons, pulse propagation, statistical averages, analytic solutions to the equations of motion, exact solutions of periodic Hamiltonians, and population-trapping "dark" states.
We derive a set of cavity-modified Maxwell-Bloch equations for a two-level atom in a cavity driven by a coherent field from the single-atom Fokker-Planck equation in the bad-cavity limit. These equations have the same form as the Maxwell-Bloch equations for a two-level atom in free space interacting with a coherent field. The presence of a cavity is reflected in three cavity-modified parameters: decay rate ${\ensuremath{\gamma}}_{\mathrm{eff}}$, Rabi frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{eff}}$, and detuning ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{eff}}$. We show that the cavity-modified Maxwell-Bloch equations provide an easy way to study cavity-modified spontaneous emission, cavity-induced radiative energy level shifts, and vacuum Rabi splitting.
We use a basis of Bloch wave functions to solve the problem of spontaneous emission from an atom inside a standing-wave cavity. In this way we incorporate the effects of the atom’s quantized center-mass-motion in the periodic light-force potential of the one-quantum dressed states during the spontaneous emission process. We calculate emission spectra and the momentum distribution of the atom. Under strong-coupling conditions, “vacuum” Rabi spectra become asymmetric, with an increased relative weight given to the red side of the spectrum because of the transfer of energy to the center-of-mass motion of the atom. Because of the nonperturbative interaction between the atom and the cavity mode, many momentum quanta are transferred to the atom from the cavity mirrors during the irreversible emission of just one photon. Thus, the atom is diffracted by its own radiation reaction field. We analyze these effects using the Bloch-state basis. Use of the secular approximation provides a simple rate equation description of transitions from the Bloch states excited in the initial state to the atom-cavity ground state. We compare this approach with other ways of doing the calculations and discuss the accuracy of the secular approximation. We consider the extension of these calculations to treat the effects of photon statistics on the momentum distribution of atoms traversing a standing-wave cavity.
The interaction of a two-level atom with a single mode of the radiation field is examined. The cavity mode is damped by coupling to a reservoir in a squeezed state, and the atom is also driven by an external laser field. A master equation for the atomic density operator in the bad-cavity limit is derived. The effect of the squeezing in the stationary population inversion and in the resonance spectrum of the light emitted by the atom into the background modes is studied. The results are compared with the numerical solution of the full master equation for the atom-plus-cavity mode-density operator.