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arXiv:2509.09403 · Physical review. D/Physical review. D.

Backreaction equations for 1 + 1 dimensional BEC sonic black holes

Roberto Balbinot, Alessandro Fabbri, Giorgio Ciliberto, Nicolas Pavloff

Published Nov 19, 2025• 0 citations
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Abstract

As in the gravitational context, one of the most challenging open questions in analogue black holes formed in Bose-Einstein condensates concerns the backreaction of Hawking-like radiation on the condensate and its subsequent evolution. In this work we derive the basic equations describing this backreaction within the density-phase formalism, which avoids infrared divergences and is particularly well suited to one-dimensional configurations.

DOI
10.1103/9m8k-ml6s

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There are a number of interesting phenomena that are predicted to be a result of the behaviour of general relativity close to a black hole. The most well known of these is Hawking radiation. Although well established, the theory of Hawking radiation is unsupported by experiment, since black holes cannot be investigated in the lab. However, there is a strong similarity between the equations governing the behaviour of a sound wave in inviscid irrotational fluid systems and a massless scalar field in a gravitational system. This similarity provides hope that effects like Hawking radiation can be investigated in analogous fluid systems. The requirement that potential fluid analogues be barotropic, inviscid and irrotational indicates that superfluids would be the logical object of experiments designed to analyse such systems. The most easily controlled superfluid to which experimentalists have access is a Bose Einstein Condensate in an appropriately designed trap, and there is interest in designing an experiment to look for Hawking radiation in a BEC. A BEC is also ideal for that purpose because of the extremely low temperatures at which the condensation occurs, given the very low temperature of Hawking radiation. However, Hawking radiation itself is a very subtle effect, and it would be very hard to detect. This means that a related but more conspicuous effect is interesting as a stepping stone to a test for the more well known Hawking radiation. This related effect is Superradiance, a result of the reflection of an incoming scalar field with appropriate angular characteristics from the ergoregion of a massive rotating body such as a black hole. An ergoregion is a region around a rotating black hole or supermassive rotating star within which it is impossible to remain stationary with respect to an observer at infinity without exceeding the speed of light. Before an experiment can be designed to test for superradiance in a BEC it is necessary to establish whether the theory describing Bose-Einstein Condensates predicts that it will actually occur. Work on this problem thus far has either failed to consider whether the effect occurs, and instead focused on the consequences it would have, or has involved approximations that are not strictly valid. Here therefore, after an overview of both the analogy between fluid and gravitational systems and the specific theory related to Bose Einstein Condensates is an analysis using the full quantum theory of BECs. The initial consideration is analytic in character, but the equations do not allow for an exact solution within the approximations attempted that predict superradiance, so a change is made to a numerical simulation using the XMDS numerical simulation package. Unfortunately, this attempt also fails, since ongoing problems encountered in running the simulation consumed all the time available. A discussion of these problems is included.

•Jan 1, 2005•0 citations
As in the gravitational context, one of the most challenging open questions in analogue black holes formed in Bose-Einstein condensates concerns the backreaction of Hawking-like radiation on the condensate and its subsequent evolution. In this work we derive the basic equations describing this backreaction within the density-phase formalism, which avoids infrared divergences and is particularly well suited to one-dimensional configurations.
We consider simple one dimensional models of acoustic black holes formed by Bose-Einstein condensates where the flow is stepwise homogeneous. We shall concentrate on the case in which an extended sonic region is present, i.e. a region where the flow velocity equals exactly the speed of sound. These models can be thought of as acoustic black holes with a thick horizon. Particular attention will be devoted to the number of created particles and to the density-density correlation functions, these latter being at present time the experimental tool to identify Hawking like radiation in acoustic black holes.
We extend the Gross-Pitaevskii equation to incorporate the effect of quantum fluctuations onto the flow of a weakly interacting Bose-Einstein condensate. Applying this framework to an analog black hole in a quasi-one-dimensional, transonic flow, we investigate how acoustic Hawking radiation back-reacts on the background condensate. Our results point to the emergence of stationary density and velocity undulations in the supersonic region (analogous to the black hole interior) and enable to evaluate the change in upstream and downstream Mach numbers caused by Hawking radiation. These findings provide new insight into the interplay between quantum fluctuations and analog gravity in Bose-Einstein condensates.
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.
There are a number of interesting phenomena that are predicted to be a result of the behaviour of general relativity close to a black hole. The most well known of these is Hawking radiation. Although well established, the theory of Hawking radiation is unsupported by experiment, since black holes cannot be investigated in the lab. However, there is a strong similarity between the equations governing the behaviour of a sound wave in inviscid irrotational fluid systems and a massless scalar field in a gravitational system. This similarity provides hope that effects like Hawking radiation can be investigated in analogous fluid systems. The requirement that potential fluid analogues be barotropic, inviscid and irrotational indicates that superfluids would be the logical object of experiments designed to analyse such systems. The most easily controlled superfluid to which experimentalists have access is a Bose Einstein Condensate in an appropriately designed trap, and there is interest in designing an experiment to look for Hawking radiation in a BEC. A BEC is also ideal for that purpose because of the extremely low temperatures at which the condensation occurs, given the very low temperature of Hawking radiation. However, Hawking radiation itself is a very subtle effect, and it would be very hard to detect. This means that a related but more conspicuous effect is interesting as a stepping stone to a test for the more well known Hawking radiation. This related effect is Superradiance, a result of the reflection of an incoming scalar field with appropriate angular characteristics from the ergoregion of a massive rotating body such as a black hole. An ergoregion is a region around a rotating black hole or supermassive rotating star within which it is impossible to remain stationary with respect to an observer at infinity without exceeding the speed of light. Before an experiment can be designed to test for superradiance in a BEC it is necessary to establish whether the theory describing Bose-Einstein Condensates predicts that it will actually occur. Work on this problem thus far has either failed to consider whether the effect occurs, and instead focused on the consequences it would have, or has involved approximations that are not strictly valid. Here therefore, after an overview of both the analogy between fluid and gravitational systems and the specific theory related to Bose Einstein Condensates is an analysis using the full quantum theory of BECs. The initial consideration is analytic in character, but the equations do not allow for an exact solution within the approximations attempted that predict superradiance, so a change is made to a numerical simulation using the XMDS numerical simulation package. Unfortunately, this attempt also fails, since ongoing problems encountered in running the simulation consumed all the time available. A discussion of these problems is included.