  Quantum fluctuations in time-dependent, harmonically-trapped Bose-Einstein
condensates are studied within Bogoliubov theory. An eigenmode expansion of the
linear field operators permits the diagonalization of the Bogoliubov-de Gennes
equation for a stationary condensate. When trap frequency or interaction
strength are varied, the inhomogeneity of the background gives rise to
off-diagonal coupling terms between different modes. This coupling is
negligible for low energies, i.e., in the hydrodynamic regime, and an effective
space-time metric can be introduced. The influence of the inter-mode coupling
will be demonstrated in an example, where I calculate the quasi-particle number
for a quasi-one-dimensional Bose-Einstein condensate subject to an exponential
sweep of interaction strength and trap frequency.
