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Physical Review A

Fokker-Planck equation for a single two-level atom: Applications in the bad-cavity limit

Changxin Wang, Reeta Vyas

Published Mar 1, 1995• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.51.2516

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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.
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.
In the general framework of the system size expansion of Van Kampen and Kubo, we consider the Fokker-Planck equation for a model of absorptive bistability in the bad-cavity limit. The physical system is described by the reduced atomic density operators after adiabatic elimination of the cavity field variables. Mapping of the master equation into ccc-number form according to the normal-ordering mapping scheme yields known results for the atomic fluctuations and correlation functions; however, it also leads to a Fokker-Planck equation with a non-positive-definite diffusion matrix. The symmetrical-order-mapping scheme eliminates this difficulty. The leading contribution to the system size expansion yields a Fokker-Planck equation for the symmetrical-ordered density function having a positive-definite diffusion matrix. The atomic expectation values and fluctuations previously derived from the quantum Langevin equations emerge naturally from this Fokker-Planck equation.
We consider a single two-level atom in an optical cavity, in which the system is driven by a coherent driving field. In the bad-cavity limit, when the cavity loss rate is taken to be large compared to the atomic decay rate and atom-field coupling, the cavity field can be adiabatically eliminated from the equations of motion for the atomic operators. The main effect of the cavity in this limit is the enhancement of spontaneous emission into the cavity mode. For this model, the second-order intensity correlation function g(2)() has been shown to exhibit photon antibunching, sub-Poissonian photon number distributions, and other nonclassical effects.1 In this paper, analytic results are presented for g(2)() for both the transmitted and fluorescent light fields when broadband squeezed light is injected into the cavity. Effects of the squeezing on nonclassical behavior is examined.
The photon statistics of the transmitted light from a driven cavity containing a single resonant two-level atom are studied in the bad-cavity limit. For weak driving fields, the second-order intensity correlation function shows novel nonclassical behavior due to the interference of the driving field and forward reradiation from the atom. This behavior is related to squeezing in the cavity transmission. A physical interpretation is given in terms of the reduced quantum state of the coupled atom-field system following photodetection.>
A quantum treatment of a coherently driven cavity with a medium of two-level atoms is given. The system displays both absorptive and dispersive bistability. Quantum fluctuations are included via a Fokker-Planck equation in a generalized PPP representation, after adiabatic elimination of the atomic variables. The transmitted spectrum and correlation function is calculated in a linearized (Gaussian) approximation. In the limit of large detuning and radiative damping, the quantum fluctuations are identical to those of a simple nonlinear polarizability theory. An analysis is given of the nonclassical statistical properties of the intensity-correlation spectrum.
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.