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.