Classical and quantum dynamics of Bose-Einstein condensates

Abstract

After the first experimental realization of a Bose-Einstein condensate (BEC) in 1995, BECs have become a subject of intense experimental and theoretical study. In this dissertation, I present our results on the classical and quantum dynamics of BECs at zero temperature under different scenarios. First, I consider the analog of slow light in the collision of two BECs near a Feshbach resonance. The scattering length then becomes a function of the collision energy. I derive a generalization of the Gross-Pitaevskii equation for incorporating this energy dependence. In certain parameter regimes, the group velocity of a BEC traveling through another BEC decreases. I also study the feasibility of an experimental realization of this phenomena. Second, I analyze an experiment in which a BEC in a ring-shaped trap is stirred by a rotating barrier. The phase drop across and current flow through the barrier is measured from spiral-shaped density profiles created by interfering the BEC in the ring-shaped trap and a concentric reference BEC after release from all trapping potentials. I show that a free-particle expansion is sufficient to explain the origin of the spiral pattern and relate the phase drop to the geometry of a spiral. I also bound the expansion times for which the phase drop can be accurately determined and study the effect of inter-atomic interactions on the expansion time scales. Third, I study the dynamics of few-mode BECs when they become dynamically unstable after preparing an initial state at a saddle point of the Hamiltonian. I study the dynamics within the truncated Wigner approximation (TWA) and find that, due to phase-space mixing, the expectation value of an observable relaxes to a steady-state value. Using the action-angle formalism, we derive analytical expressions for the steady-state value and the time evolution towards this value. I apply these general results to two systems: a condensate in a double-well potential and a spin-1 (spinor) condensate. Finally, I study quantum corrections beyond the TWA in the semiclassical limit. I derive general expressions for the dynamics of an observable by using the van Vleck-Gutzwiller propagator and find that the interference of classical paths leads to non-perturbative corrections. As a case study, I consider a single-mode nonlinear oscillator; this system displays collapse and revival of observables. I find that the interference of classical paths, which is absent in the TWA, leads to revivals.