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arXiv:1707.09664 · Physical review. D/Physical review. D.

Low-frequency analogue Hawking radiation: The Bogoliubov-de Gennes model

Antonin Coutant, Silke Weinfurtner

Published Jan 8, 2018• 34 citations
arXivHTMLPDF

Abstract

We analytically study the low-frequency properties of the analogue Hawking effect in Bose-Einstein condensates. We show that in one-dimensional flows displaying an analogue horizon, the Hawking effect is dominant in the low-frequency regime. This happens despite nonvanishing grey-body factors, that is, the coupling of the Hawking mode and its partner to the mode propagating with the flow. To show this, we obtained analytical expressions for the scattering coefficients, in general flows and taking into account the full Bogoliubov dispersion relation. We discuss the obtained expressions for the grey-body factors. In particular, we show that they can be significantly decreased if the flow obeys a conformal coupling condition. We argue that in the presence of a small but non-zero temperature, reducing grey-body factors greatly facilitates the observation of entanglement, that is, establishing that the state of the Hawking mode and its partner is non-separable.

DOI
10.1103/physrevd.97.025006

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•Jan 1, 2005•0 citations
We analytically study the low-frequency properties of the analogue Hawking effect in Bose-Einstein condensates. We show that in one-dimensional flows displaying an analogue horizon, the Hawking effect is dominant in the low-frequency regime. This happens despite nonvanishing grey-body factors, that is, the coupling of the Hawking mode and its partner to the mode propagating with the flow. To show this, we obtained analytical expressions for the scattering coefficients, in general flows and taking into account the full Bogoliubov dispersion relation. We discuss the obtained expressions for the grey-body factors. In particular, we show that they can be significantly decreased if the flow obeys a conformal coupling condition. We argue that in the presence of a small but non-zero temperature, reducing grey-body factors greatly facilitates the observation of entanglement, that is, establishing that the state of the Hawking mode and its partner is non-separable.
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We apply the microscopic Bogoliubov theory of dilute Bose-Einstein condensates to analyze quantum and thermal fluctuations in a flowing atomic condensate in the presence of a sonic horizon. For the simplest case of a step-like horizon, closed-form analytical expressions are found for the spectral distribution of the analog Hawking radiation and for the density correlation function. The peculiar long-distance density correlations that appear as a consequence of the Hawking emission features turns out to be reinforced by a finite initial temperature of the condensate. The analytical results are in good quantitative agreement with first principle numerical calculations.
We study the phonon fluxes emitted when the condensate velocity crosses the speed of sound, i.e., in backgrounds which are analogous to that of a black hole. We focus on elongated one-dimensional condensates and on stationary flows. Our theoretical analysis and numerical results are based on the Bogoliubov--de Gennes equation without further approximation. The spectral properties of the fluxes and of the long distance density-density correlations are obtained, both with and without an initial temperature. In realistic conditions, we show that the condensate temperature dominates the fluxes and thus hides the presence of the spontaneous emission (the Hawking effect). We also explain why the temperature amplifies the long distance correlations which are intrinsic to this effect. This confirms that the correlation pattern offers a neat signature of the Hawking effect. Optimal conditions for observing the pattern are discussed, as well as correlation patterns associated with scattering of classical waves. Flows associated with white holes are also considered.
Hawking radiation, the spontaneous emission of thermal photons from an event horizon, is one of the most intriguing and elusive predictions of field theory in curved spacetimes. A formally analogue phenomenon occurs at the supersonic transition of a fluid: in this respect, ultracold gases stand out among the most promising systems but the theoretical modelling of this effect has always been carried out in semiclassical approximation, borrowing part of the analysis from the gravitational analogy. Here we discuss the exact solution of a one-dimensional Bose gas flowing against an obstacle, showing that spontaneous phonon emission (the analogue of Hawking radiation) is predicted without reference to the gravitational analogy. Long after the creation of the obstacle, the fluid settles into a stationary state displaying the emission of sound waves (phonons) in the upstream direction. A careful analysis shows that a precise correspondence between this phenomenon and the spontaneous emission of radiation from an event horizon requires additional conditions to be met in future experiments aimed at identifying the occurrence of the Hawking-like mechanism in Bose-Einstein condensates.
We study double-barrier interfaces separating regions of asymptotically subsonic and supersonic flow of Bose-condensed atoms. These setups contain at least one black hole sonic horizon from which the analogue of Hawking radiation should be generated and emitted against the flow in the subsonic region. Multiple coherent scattering by the double-barrier structure strongly modulates the transmission probability of phonons, rendering it very sensitive to their frequency. As a result, resonant tunneling occurs with high probability within a few narrow frequency intervals. This gives rise to highly non-thermal spectra with sharp peaks. We find that these peaks are mostly associated with decaying resonances and only occasionally with dynamical instabilities. Even at achievable non-zero temperatures, the radiation peaks can be dominated by spontaneous emission, i.e. enhanced zero-point fluctuations, and not, as is often the case in analogue models, by stimulated emission.
The propagation of fluctuations in the velocity potential of an inviscid, irrotational fluid is governed by a wave equation very similar to that governing the propagation of massless scalar fields in curved spacetime. This similarity allows certain fluid systems to be used as experimentally realisable models for objects that cannot be studied under controlled conditions, such as black holes and ultrarelativitsic rotating stars. The analogy applies only to the propagation of fields in a fixed background, not to the behaviour of the background itself. However, even this limited similarity has excited a great deal of interest from researchers, since it suggests that phenomena such as superradiance and Hawking radiation may have analogues in superfluids. In recent years, experimental advances in the creation and manipulation of Bose-Einstein condensates have provided researchers with very low temperature systems possessing a uniquely simple microscopic theory. Furthermore, under certain long-length-scale approximations, Bose-Einstein condensates can be described as classical irrotational inviscid fluids, and therefore can be treated as analogues of appropriate gravitational systems. We present a comprehensive and detailed critical review of the mean-field theory of Bose-Einstein condensates, the conditions under which Bose-Einstein condensates can be treated as an analogue gravity system, the construction of analogue black holes, and several derivations of Hawking radiation and analogue Hawking radiation. The interdisciplinary nature of analogue gravity means that the relevant results are widely scattered in the literature, and we consider this critical review and discussion of the relevant literature to be a central and noteworthy element of the project. We then analyse in detail a recent derivation of analogue Hawking radiation in Bose-Einstein condensates as a consequence of quantum depletion. We elucidate the methodology and the physical meaning of the results by comparison with the derivations of Hawking radiation considered previously, and discuss possible extensions and improvements. We also present an original investigation of analogue superradiance in simple vortices, with the vortex density profile chosen to mimic a realistic vortex in a Bose-Einstein condensate. We find that superradiant amplification occurs and is likely to be significant, but our long-length-scale approximation is of dubious validity near the scattering point.