Explicit Finite-Difference and Particle Method for the Dynamics of Mixed Bose-Condensate and Cold-Atom Clouds

We present a new numerical method for studying the dynamics of quantum fluids composed of a Bose–Einstein condensate and a cloud of bosonic or fermionic atoms in a mean-field approximation. The method combines an explicit time-marching algorithm, previously developed for Bose–Einstein condensates in a harmonic or optical-lattice potential, with a particle-in-cell approach to the equation of motion for the one-body Wigner distribution function in the cold-atom cloud. The method is tested against known analytical results on the free expansion of a fermion cloud from a cylindrical harmonic trap and is validated by examining how the expansion of the fermionic cloud is affected by the simultaneous expansion of a condensate. We then present original calculations on a condensate and a thermal cloud inside a harmonic well and a superposed optical lattice, by addressing the free expansion of the two components and their oscillations under an applied harmonic force. These results are discussed in the light of relevant theories and experiments.