We numerically solve an effective single-atom Hartree-Fock master equation in order to investigate the effects of spontaneous emission on atomic Thirring solitons and atomic gap solitons. We show that spontaneous emission detrimentally effects the existence of Thirring solitons, as expected from the fact that they only occur on resonance. In contrast, gap solitons, which can also exist off resonance, survive in the presence of spontaneous emission in weakly allowed transitions, albeit at high atomic densities. This result is illustrated in the case of the [2 ^{1}$${\mathit{S}}_{0}\ensuremath{\leftrightarrows}2 ^{3}$${\mathit{P}}_{1}]- intercombination line of magnesium.
We present a many-body theory of nonlinear atom optics, and discuss some of its physical implications in the coherent regime. Considering a system of N identical two-level atoms interacting with classical and quantum-mechanical electromagnetic fields, we derive a Fock-space many-particle master equation. Introducting a Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy and a Hartree-Fock factorization to truncate this hierarchy, we obtain an effective nonlinear single-particle master equation that forms the basis of nonlinear atom optics. In the second part of the paper, we concentrate on the coherent part of that master equation, and derive an effective single-atom nonlinear Schr\"odinger equation. This equation leads to the prediction of a number of effects, and, in particular, several kinds of atomic solitons. We discuss and numerically study two such kinds of solitons, Thirring solitons and gap solitons. Finally, the axial containment of an atomic gap soliton is illustrated.
We have applied the transformation of the slow light equations to Liouville theory that we developed in our previous work, to study the influence of relaxation on the soliton dynamics. We solved the problem of the soliton dynamics in the presence of relaxation and found that the spontaneous emission from the upper atomic level is strongly suppressed. Our solution proves that the spatial shape of the soliton is well preserved even if the relaxation time is much shorter than the soliton time length. This fact is of great importance for applications of the slow-light soliton concept in optical information processing. We also demonstrate that the relaxation plays a role of resistance to the soliton motion and slows the soliton down even if the controlling field is constant.
We have applied the transformation of the slow-light equations to the Liouville theory that we developed in our previous work, to study the influence of relaxation on the soliton dynamics. We solved the problem of the soliton dynamics in the presence of relaxation and found that the spontaneous emission from the upper atomic level is strongly suppressed. Our solution proves that the spatial shape of the soliton is well preserved even if the relaxation time is much shorter than the soliton time length. This fact is of great importance for applications of the slow-light soliton concept in optical information processing. We also demonstrate that relaxation plays a role of resistance to the soliton motion and slows the soliton down even if the controlling field is constant.
An extended Bose-Einstein condensate (BEC) in an optical lattice provides a kind of periodic dielectric and causes band gaps to occur in the spectrum of light propagating through it. We examine the question whether these band gaps can modify the spontaneous emission rate of atoms excited from the BEC, and whether they can lead to a self-stabilization of the BEC against spontaneous emission. We find that self-stabilization is not possible for BECs with a density in the order of 10 14 cm −3. However, the corresponding non-Markovian behavior produces significant effects in the decay of excited atoms even for a homogeneous BEC interacting with a weak laser beam. These effects are caused by the occurrence of an avoided crossing in the photon (or rather polariton) spectrum. We also predict a new channel for spontaneous decay which arises from an interference between periodically excited atoms and periodic photon modes. This new channel should also occur in ordinary periodic dielectrics. 1
A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from
N identical two-level atoms is investigated further. It is shown that the equation of motion for the
p(<N)-atom density matrix is coupled to the (
p+1)-atom density matrix. We first consider the initial excitation of the system to state $|{\ensuremath{\theta}}_{0},{\ensuremath{\phi}}_{0}〉$ (${\ensuremath{\theta}}_{0}<\ensuremath{\pi}$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time
t can be described by a single parameter $\ensuremath{\theta}(t)$ ($\ensuremath{\theta}<\ensuremath{\pi}$), with ${sin}^{2}\frac{1}{2}\ensuremath{\theta}$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈({S}_{i}^{Z}\ensuremath{-}〈{S}_{i}^{Z}〉)({S}_{j}^{Z}\ensuremath{-}〈{S}_{j}^{Z}〉)〉$ for any pair of atoms is of the order of
N1. The case when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of
2N coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.