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Spacetime analogue of Bose-Einstein condensates: Bogoliubov-de Gennes formulation

We construct quantum field theory in an analogue curved spacetime in Bose-Einstein condensates based on the Bogoliubov-de Gennes equations, by exactly relating quantum particles in curved spacetime with Bogoliubov quasiparticle excitations in Bose-Einstein condensates. Here, we derive a simple formula relating the two, which can be used to calculate the particle creation spectrum by solving the time-dependent Bogoliubov-de Gennes equations. Using our formulation, we numerically investigate particle creation in an analogue expanding Universe which can be expressed as Bogoliubov quasiparticles in an expanding Bose-Einstein condensate. We obtain its spectrum, which follows the thermal Maxwell-Boltzmann distribution, the temperature of which is experimentally attainable. Our derivation of the analogy is useful for general Bose-Einstein condensates and not limited to homogeneous ones, and our simulation is the first example of particle creations by solving the Bogoliubov-de Gennes equation in an inhomogeneous condensate.