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arXiv:cond-mat/0702045 · Physical Review A

Limits to the analog Hawking temperature in a Bose-Einstein condensate

Sebastian Wüster, Craig M. Savage

Published Jul 9, 2007• 28 citations
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Abstract

Quasi-one-dimensional outflow from a dilute gas Bose-Einstein condensate reservoir is a promising system for the creation of analog Hawking radiation. We use numerical modeling to show that stable sonic horizons exist in such a system under realistic conditions, taking into account the transverse dimensions and three-body loss. We find that loss limits the analog Hawking temperatures achievable in the hydrodynamic regime, with sodium condensates allowing the highest temperatures. A condensate of $30\phantom{\rule{0.2em}{0ex}}000$ atoms, with transverse confinement frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}=6800\ifmmode\times\else\texttimes\fi{}2\ensuremath{\pi}\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$, yields horizon temperatures of about $20\phantom{\rule{0.3em}{0ex}}\mathrm{nK}$ over a period of $50\phantom{\rule{0.3em}{0ex}}\mathrm{ms}$. This is at least four times higher than for other atoms commonly used for Bose-Einstein condensates.

DOI
10.1103/physreva.76.013608

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Quasi-one-dimensional outflow from a dilute gas Bose-Einstein condensate reservoir is a promising system for the creation of analog Hawking radiation. We use numerical modeling to show that stable sonic horizons exist in such a system under realistic conditions, taking into account the transverse dimensions and three-body loss. We find that loss limits the analog Hawking temperatures achievable in the hydrodynamic regime, with sodium condensates allowing the highest temperatures. A condensate of 3000030\phantom{\rule{0.2em}{0ex}}00030000 atoms, with transverse confinement frequency \ensuremathω\ensuremath⊥=6800\ifmmode×\else\texttimes\fi2\ensuremathπHz{\ensuremath{\omega}}_{\ensuremath{\perp}}=6800\ifmmode\times\else\texttimes\fi{}2\ensuremath{\pi}\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}\ensuremathω\ensuremath⊥​=6800\ifmmode×\else\texttimes\fi2\ensuremathπHz, yields horizon temperatures of about 20nK20\phantom{\rule{0.3em}{0ex}}\mathrm{nK}20nK over a period of 50ms50\phantom{\rule{0.3em}{0ex}}\mathrm{ms}50ms. This is at least four times higher than for other atoms commonly used for Bose-Einstein condensates.
We report numerical evidence of Hawking emission of Bogoliubov phonons from a sonic horizon in a flowing one-dimensional atomic Bose–Einstein condensate. The presence of Hawking radiation is revealed from peculiar long-range patterns in the density–density correlation function of the gas. Quantitative agreement between our fully microscopic calculations and the prediction of analog models is obtained in the hydrodynamic limit. New features are predicted and the robustness of the Hawking signal against a finite temperature discussed.
The sonic analog of a gravitational black hole in dilute-gas Bose-Einstein condensates is investigated. It is shown that there exist both dynamically stable and unstable configurations which, in the hydrodynamic limit, exhibit behaviors completely analogous to that of gravitational black holes. The dynamical instabilities involve the creation of quasiparticle pairs in positive and negative energy states. We illustrate these features in two qualitatively different one-dimensional models, namely, a long, thin condensate with an outcoupler laser beam providing an ``atom sink,'' and a tight ring-shaped condensate. We also simulate the creation of a stable sonic black hole by solving the Gross-Pitaevskii equation numerically for a condensate subject to a trapping potential which is adiabatically deformed. A sonic black hole could, in this way, be created experimentally with state-of-the-art or planned technology.
The sonic analog of Hawking radiation can now be experimentally recreated in Bose-Einstein condensates that contain an acoustic black hole. In these experiments the signal strength and approximate analog Hawking temperature increase for denser condensates, which however also suffer increased atom losses from inelastic collisions. To determine how these affect analog Hawking radiation, we numerically simulate creation of the latter in a Bose-Einstein condensate in the presence of atomic losses, including nonunitary quantum field dynamics using the truncated Wigner approximation. In particular we explore modifications of density-density correlations through which the radiation has been analyzed so far. We find no evidence that losses directly alter the basic picture of the analog Hawking effect; instead all consequences that we find are indirect: Losses increase the contrast of the correlation signal, which we attribute to condensate heating by the losses, in turn leading to a component of stimulated radiation in addition to the spontaneous one. Other indirect consequences are the modification of the white-hole instability pattern and a change of slope of the Hawking tongue.
One can use transsonic Bose-Einstein condensates of alkali atoms to establish the laboratory analog of the event horizon and to measure the acoustic version of Hawking radiation. We determine the conditions for supersonic flow and the Hawking temperature for realistic condensates on waveguides where an external potential plays the role of a supersonic nozzle. The transition to supersonic speed occurs at the potential maximum and the Hawking temperature is entirely determined by the curvature of the potential.
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.
Taking advantage of the known exact mapping of the one-dimensional hard core Bose (HCB) fluid onto a non-interacting spinless fermion gas, we examine in full detail a thought experiment on cold atoms confined in a quasi-one-dimensional trap, in order to investigate the emergence of the analogue Hawking radiation. The dynamics of a gas of interacting bosons impinging on an external potential is exactly tracked up to the reach of a stationary state. Under few strict conditions on the experimental parameters, the stationary state is shown to be described asymptotically by a thermal distribution, precisely at the expected (analogue) Hawking temperature. However, we find that in most experimental conditions the emerging ``Hawking-like radiation'' is not thermal. This analysis provides a novel many-body microscopic interpretation of the Hawking mechanism, together with useful limits and conditions for the design of future experiments in Bose-Einstein condensates.