Density correlations and analog dynamical Casimir emission of Bogoliubov phonons in modulated atomic Bose-Einstein condensates

Abstract

We present a theory of the density correlations that appear in an atomic Bose-Einstein condensate as a consequence of the dynamical Casimir emission of pairs of Bogoliubov phonons when the atomatom scattering length is modulated in time. Different regimes as a function of the temporal shape of the modulation are identified and a simple physical picture of the phenomenon is discussed. Analytical expressions for the density correlation function are provided for the most significant limiting cases. This theory is able to explain some unexpected features recently observed in numerical calculations of Hawking radiation from analog black holes.The dynamical Casimir effect [1] is a very general prediction of quantum field theory: whenever the boundary conditions and/or the dispersion law and/or the background of a quantum field are quickly varied in time, pairs of quanta are generated off the vacuum state by parametric amplification of zero-point noise. The simplest and most celebrated example of dynamical Casimir effect was predicted for an optical cavity whose planeparallel mirrors are made to rapidly oscillate in time along the cavity axis [2,3,4]. Despite the significant effort devoted to these fascinating effects, no experimental observation of the dynamical Casimir effect has yet appeared, the main reason being the difficulty in moving the mirror at a fast enough speed [3]. Alternative schemes to modulate the effective optical length of a cavity on a very fast time-scale by acting on the refractive index of the cavity material have been proposed and shown to give a sizeable intensity of dynamical Casimir emission [5,6,7,8,9,10,11]. Experiments in this direction are in progress in several groups [12,13].Since the original proposal by Unruh [14], the advances in the field of the so-called analog models [15] have pointed out the possibility of simulating the physics of a quantum field on a generic curved space-time in tabletop condensed-matter experiments: the propagation of elementary excitations in spatially and temporally inhomogeneous systems can in fact be recast in terms of a relativistic wave equation on an effective curved spacetime. In the simplest case of acoustic waves in a fluid, the space-time metric is fixed by the spatial and temporal profiles of the sound speed and of the flow velocity. Upon quantization of the resulting field theory, an analog dynamical Casimir effect is then expected to appear whenever the sound speed in a spatially homogeneous system is made to quickly vary in time, which in the language of the analogy corresponds to the expansion or contraction of the underlying universe [16,17]. On the other hand, in the presence of an acoustic horizon separating an upstream region of sub-sonic flow from a downstream one of super-sonic flow, the emission of analog Hawking radiation has been predicted [14,18,19].As proposed in [20,21], a most promising way of experimentally investigating this physics involves the measurement of the correlation function of density fluctuations. A recent numerical e...

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