This thesis covers three applications of Bose-Einstein condensates and related phenomena, in the theme of pair creation and pair annihilation.

First, Bose-Einstein condensates (BEC) are viewed as one of the candidates to implement a sonic black hole. This can lead to the observation of analog Hawking radiation, resulting from a phonon pair creation at a black-hole horizon (BH). Such implementation has been achieved in a resent experiment by J. Steinhauer, in which a black-hole/white-hole pair has been produced. He also reported the observations of self-amplifying Hawking radiation, via a lasing mechanism operating between the black and white-hole horizons. Through our simulations, we find that the observations should be attributed not to the black hole laser effect, but rather to a growing zero-frequency bow wave, generated at the white-hole horizon. The relative motion of the two horizons produces a Doppler shift of the bow wave at the BH, where it stimulates a monochromatic Hawking radiation. We also find that shot-to-shot atom number variations and quantum fluctuations give density-density correlations consistent with those reported in the experiments. In particular, atom number variations can produce a spurious correlation signal.

Secondly, a sonic black hole/white hole pair and phonon pair creation can also be realized using a ring-shaped condensate. Here we focus on the phonon spectroscopy of a ring condensate. We probe the phonon excitation spectrum by applying a harmonically driven barrier to a 23Na Bose-Einstein condensate in a ring-shaped trap. When excited resonantly, these wavepackets display a regular periodic structure. The resonant frequencies depend upon the particular configuration of the barrier, but are commensurate with the orbital frequency of a sound wave traveling around the ring. Energy transfer to the condensate over many cycles of the periodic wavepacket motion causes enhanced atom loss from the trap at resonant frequencies. Solutions of the Gross-Pitaevskii (GP) equation exhibit quantitative agreement with the experimental data.

Thirdly, positronium (Ps) BECs have been of experimental and theoretical interest due to their potential application as the gain medium of a gamma-ray laser. Ps BECs are intrinsically spinor due to the presence of ortho- (o-Ps) and para-positronium (p-Ps), whose annihilation lifetimes differ by three orders of magnitude. We study the spinor dynamics and annihilation processes in the p-Ps/o-Ps system using both solutions of the GP equations and a rate-equation approach. For an initially unpolarized condensate, there is a threshold density at which spin mixing between o-Ps and p-Ps occurs. Beyond this threshold, there are unstable spatial modes accompanied by spin mixing. To ensure a high production yield above the critical density, a careful choice of external field must be made to avoid the spin mixing instability.