Expansion of a mesoscopic Fermi system from a harmonic trap

We study the dynamics of time-of-flight expansion of an atomic Fermi system with a finite number of particles N after it is released from a harmonic trapping potential. We consider two different initial states: the Fermi sea state and the paired state. In the former case, we derive exact and simple analytic expressions for the dynamics of the particle density and density-density correlation functions, taking into account the level quantization and possible anisotropy of the trap. To describe the paired state, we use the projection of the grand-canonical BCS wave function onto the subspace with a fixed number of particles, and obtain analytic expressions for the density and its correlators in the leading order with respect to the ratio of the trap frequency and the superconducting gap (the ratio is assumed small). We discuss several dynamic features, such as the time evolution of the peak due to pair correlations, which may be used to distinguish between the Fermi sea and the paired state.