Superfluidity without Symmetry-Breaking: The Time-Dependent Hartree-Fock Approximation for Bose-Condensed Systems

We develop a time-dependent Hartree-Fock approximation that is appropriate for Bose-condensed systems. Defining a {\it depletion Green's function} allows the construction of condensate and depletion particle densities from eigenstates of a single time-dependent Hamiltonian, guaranteeing that our approach is a conserving approximation. The poles of this Green's function yield the energies of number-changing excitations for which the condensate particle number is held fixed, which we show has a gapped spectrum in the superfluid state. The linearized time-dependent version of this has poles at the collective frequencies of the system, yielding the expected zero sound mode for a uniform infinite system. We show how the approximations may be expressed in a general linear response formalism.