The work in this article extends the optimal control framework of variational calculus to optimize the conversion of a Bose-Einstein condensate of atoms to one of molecules. It represents the derivation of the closed form optimal control equations for a system governed by a nonlinear Schr\"odinger equation and its successful application. It was necessary to derive a density matrix formulation of the coupled Gross-Pitaevskii equations to optimize STIRAP-like Raman light fields, to overcome dissipation.
This work is an up-to-date and partially improved analysis of the basics of the mathe- matical description of Bose-Einstein condensation in terms of first principles of Quantum Mechanics. The main aims are: 1) to provide a compact, yet coherent overview: - of the basic mathematical tools used to formalise Bose-Einstein condensation, - of the mathematical techniques for studying several features of this physical phenomenon, - and of how such means emerge as the natural ones in connection with their physical interpretation; 2) to discuss and to place in the above perspective some new contributions and improvements: - on equivalent characterizations of Bose-Einstein condensation, - on the strength of the convergence for some currently available asymptotic results, - and on the effects of interparticle correlations on the energy and the dynamics of the many-body condensate.
We propose a stochastic description of the dynamics of a Bose-Einstein condensate within the context of Nelson stochastic mechanics. We start from the
N interacting conservative diffusions, associated with the
N Bose particles, and take an infinite particle limit. We address several aspects of this formulation. First, we consider the problem of extending to a system with self-interaction the variational formulation of Nelson stochastic mechanics due to Guerra and Morato. In this regard we discuss two possible extensions, one based on a doubling procedure and another based on a constraint Eulerian type variational principle. Then we consider the infinite particle limit from the point of view of the
N-particles Madelung equations. Since conservative diffusions can be identified with proper infinitesimal characteristics pairs
(ρN(t),vN(t)), a time marginal probability density and a current velocity field, respectively, we consider a finite Madelung hierarchy for the marginals pairs
(ρN,n(t),vN,n(t)), obtained by properly conditioning the processes. The infinite Madelung hierarchy arises from the finite one by performing, for each fixed
n, a mean-field scaling limit in
N. Finally, we introduce a
n-particle conditioned diffusions which naturally parallels the quantum mechanical approach and is a new approach within the context of Nelson stochastic mechanics. We then prove the convergence, in the infinite particle limit, of the law of such a conditioned process to the law of a self-interacting diffusion which describes the condensate.
We develop theory for a flattened dipolar Bose–Einstein condensate produced by harmonic confinement along one direction. The role of both short-ranged contact interactions and long-ranged dipole–dipole interactions is considered, and the dipoles are allowed to be polarized along an arbitrary direction. We discuss the symmetry properties of the condensate and the part of the excitation spectrum determining stability, and introduce two effective interaction parameters that allow us to provide a general description of the condensate properties, rotons, and stability. We diagonalize the full theory to obtain benchmark results for the condensate and quasiparticle excitations, and characterize the exact mean field stability of the system. We provide a unified formulation for a number of approximate schemes to describe the condensate and quasiparticles, including the standard quasi-two-dimensional approximation, two kinds of variational ansatz, and a Thomas–Fermi approximation. Some of these schemes have been widely used in the literature despite not being substantiated against the exact theory. We provide this validation and establish the regimes where the various theories perform well.