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Physical Review A

Nonlinear atom optics: General formalism and atomic solitons

Georg Lenz, Pierre Meystre, Ewan M. Wright

Published Aug 1, 1994• 90 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.50.1681

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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 study an ensemble of N bosonic atoms coupled by dipole-dipole interaction and also interacting with an electromagnetic field. Effective single-atom nonlinear Schr\"odinger equations are derived. The implications of these equations are discussed in the context of some standard atom optics geometries, illustrating in particular how many-body effects modify the Pendell\"osung of Bragg scattering. In another regime, the problem is reduced to a classical massive Thirring model, with the possibility of generating atomic Thirring solitons.
Summary form only given. Nonlinear Schrodinger equations are ubiquitous in physics, and appear in problems ranging from Bose liquids and light propagation in nonlinear media to quantum chains, nonlinear atom optics, and atomic Bose condensation. A generic situation leading to a nonlinear Schrodinger equation description involves the effective single-particle dynamics of a many-particle system coupled by a two-body interaction, and with the manybody wave function truncated via a Hartree-Fock Ansatz. However, if this system is in addition coupled to one, or possibly several reservoirs, such a description is no longer appropriate, and must be superseded by a master equation.
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The availability of coherent matter waves from Bose-Einstein condensates had led to the extension of atom optics from the linear to the nonlinear regime, in analogy with the developments in optics following the invention of the laser. We provide a tutorial introduction to this emerging field, emphasizing the role of collisions as a “nonlinear medium” in the focusing and defocusing of matter waves, the launching of atomic solitons, the generation of vortices, and the demonstration of four-wave mixing both among matter waves and between optical and matter waves.
A system of N spin-1/2 particles interacting with a thermal reservoir is used as a pedagogical example for advanced undergraduate and graduate students. We introduce and illustrate some methods, approximations, and phenomena related to dissipation and nonlinearity in many-particle physics. We start our analysis from the dynamical Sudarshan-Lindblad quantum master equation for the density operator of a system S interacting with a thermal reservoir R. We derive the quantum version of the so-called Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations such that the master equation can be decomposed in a hierarchical set of N - 1 equations (N > 1). The hierarchy is broken by introducing the mean-field approximation and reducing the problem to a nonlinear single particle system. In this scenario, the Hamiltonian is nonlinear (i.e., it depends on the state of <S), although the superoperator responsible for the dissipation and decoherence of S remains unaffected. To provide a useful tool to students: (1) we discuss the physical approximations involved, (2) we derive the analytical solution to the mean values equations of motion resulting from the Hamiltonian, (3) we solve analytically the master equation in the stationary regime, (4) we obtain and discuss the solution of the nonlinear master equation, numerically, and finally, (5) we discuss the master equation beyond the mean-field approximation and show how to introduce higher order quantum correlations that have been previously neglected.
The dynamics of an atom-molecule-conversion system subject to dephasing noises is studied in this paper. Using a dephasing master equation and mean-field theory, we derive a Bloch equation for the system, and this equation is compared with the Bloch equation derived by the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy truncation approach. Fixed points of the system are calculated by solving both the Bloch equation and the master equation. Comparison between these two calculations suggests that while at short times the mean-field theory (MFT) is a good approximation for the atom-molecule-conversion system, a high-order hierarchy truncation approach is necessary for the system on a long time scale. Although the MFT cannot predict the fixed points correctly, its predictions for the stability of the fixed points are consistent with the BBGKY theory for a wide range of parameters.