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Physical Review A

Cavity-modified Maxwell-Bloch equations for the vacuum Rabi splitting

Changxin Wang, Reeta Vyas

Published Jan 1, 1997• 16 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.55.823

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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 \ensuremathγeff{\ensuremath{\gamma}}_{\mathrm{eff}}\ensuremathγeff​, Rabi frequency \ensuremathΩeff{\mathrm{\ensuremath{\Omega}}}_{\mathrm{eff}}\ensuremathΩeff​, and detuning \ensuremathΔeff{\mathrm{\ensuremath{\Delta}}}_{\mathrm{eff}}\ensuremathΔeff​. 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 revisit the problem of spontaneous emission by a two-level system in a cavity, concentrating in particular on those aspects associated with the finite propagation time of light between the atom and the cavity boundaries. We find that spontaneous emission always occurs first at the free space rate and there is then an abrupt change in these dynamics when light reflected by the cavity mirrors returns to the atom, leading to interference effects that result either in an enhancement or a reduction of this rate or in a periodic exchange of excitation between the atom and the cavity field. From this point of view, reversible spontaneous emission in terms of single-mode ``vacuum Rabi oscillations'' can be interpreted as resulting from the interference between the multimode partial waves reflected at the cavity mirrors and the instantaneously radiated light. \textcopyright{} 1996 The American Physical Society.
A generalized second-order Fokker-Planck equation is derived for a single two-level atom in a cavity driven by an external classical field without using system size expansion or any truncation. Effects of detuning, as well as atomic and cavity decays, are incorporated in this Fokker-Planck equation. This equation is used to study atomic inversion and the second-order intensity correlation function of light in the bad-cavity limit by adiabatically eliminating the field variables. Several novel features in atomic inversion and intensity correlations that arise due to atomic and cavity detuning are discussed. Curves are presented to illustrate the behavior.
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.
A single two-level atom in a coherently driven cavity and damped by a broadband squeezed vacuum centered about the atomic transition frequency is studied. A second-order Fokker-Planck equation for this system is obtained without using system size expansion. Effects of detunings and cavity decays are also incorporated in the Fokker-Planck equation. This equation is used to study atomic inversion, fluorescent spectrum, and the intensity correlations of the transmitted and fluorescent photons in the bad-cavity limit. Several interesting effects in the atomic inversion, spectrum, and intensity correlations due to the squeezed vacuum are presented. These results are also compared with an atom that is damped by a thermal reservoir.
We have shown that an ultranarrow dark region can be associated with the feature of significant amount of vaccuum-Rabi splitting in the absorption spectrum of a weak coherent field probing a two-level atom strongly coupled to two degenerate cavity-modes with preselected polarizations. The cavity-induced transparency is produced by the decay-interference of the two independent cavity modes interacting with the same reservoir. It has been shown that the resonant probe transparency is independent of the individual decay rates of the cavity modes. The present model exhibits the possibility of obtaining the narrow dark transparency associated with the significant amount of Rabi splitting in the absorption spectrum with the appreciable value of the cavity-losses.