The analogy between the propagation of a massless scalar field in curved spacetime and the propagation of sound waves in a Bose-Einstein Condensate (BEC) gives rise to the possibility of observing the analogues of black hole radiance effects, such as Hawking Radiation and superradiance, in the laboratory. In order to provide a solid foundation for realising such analogue gravity models in an experimental setting, we have conducted a numerical investigation of phonon scattering from a BEC vortex by solving the time-dependent Gross-Pitaevskii (GP) equation, describing the dynamics of dilute gas BECs. The consideration of phonon superradiance in BECs in advance of Hawking Radiation is motivated by the fact superradiance may be easier to detect. ¶ We first present the results of one-dimensional simulations of sound waves propagating in a BEC and reflecting from a vortex-like density profile. Using these results as a guide, two-dimensional simulations of sound propagation in a BEC, and reflection from a BEC vortex, are performed. The technique of propagating the wavefunction in imaginary time was employed to find the vortex ground state. The accuracy of the simulations and the convergence of the solution to the vortex ground state was verified using a range of numerical diagnostics. Finally, preliminary simulations were performed within the superradiant frequency regime for the purposes of determining whether the amplification of incoming waves occurs given an appropriate choice of parameters. Overall, the two-dimensional modelling of reflection of cylindrical waves from a BEC vortex provides a basis for a systematic study of phonon superradiance in BECs.
The analogy between the propagation of a massless scalar field in curved spacetime and the propagation of sound waves in a Bose-Einstein Condensate (BEC) gives rise to the possibility of observing the analogues of black hole radiance effects, such as Hawking Radiation and superradiance, in the laboratory. In order to provide a solid foundation for realising such analogue gravity models in an experimental setting, we have conducted a numerical investigation of phonon scattering from a BEC vortex by solving the time-dependent Gross-Pitaevskii (GP) equation, describing the dynamics of dilute gas BECs. The consideration of phonon superradiance in BECs in advance of Hawking Radiation is motivated by the fact superradiance may be easier to detect. ¶ We first present the results of one-dimensional simulations of sound waves propagating in a BEC and reflecting from a vortex-like density profile. Using these results as a guide, two-dimensional simulations of sound propagation in a BEC, and reflection from a BEC vortex, are performed. The technique of propagating the wavefunction in imaginary time was employed to find the vortex ground state. The accuracy of the simulations and the convergence of the solution to the vortex ground state was verified using a range of numerical diagnostics. Finally, preliminary simulations were performed within the superradiant frequency regime for the purposes of determining whether the amplification of incoming waves occurs given an appropriate choice of parameters. Overall, the two-dimensional modelling of reflection of cylindrical waves from a BEC vortex provides a basis for a systematic study of phonon superradiance in BECs.
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.
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.
Abstract The propagation of acoustic perturbations in the velocity potential of a Bose–Einstein condensate (BEC) behave like minimally coupled massless scalar fields in a curved (1 + 1) dimensional Lorentzian space-time, where their propagation is governed by the Klein–Gordon wave equation. For linearized perturbations, this geometric picture can still apply in the presence of a single vortex state of the BEC. Thus far, the amplified scattering of axisymmetric perturbations from a vortex, as a manifestation of the acoustic superradiance, has been investigated by assuming a constant background density of the condensate. This paper addresses the validity of this approximation within the same theoretical workframe, by employing a self-consistent density profile that is obtained by solving the Gross–Pitaevskii equation for an unbound BEC and an approximation to the numerical solution is used throughout the paper. The resulting radial density profile around the vortex implies a radially varying speed of sound, which modifies the entire propagation dynamics as well as the loci of the event horizon and the ergosphere. We investigate the superradiance in temporal domain and, through an independent asymptotic formulation, in the spectral domain. The main conclusions are that the self-consistent density profile remedies the overshoot of the superradiance dynamics temporally and that the spectral profile of the superradiance differs significantly in the vortex-scaled low frequency regime between the constant density and self-consistent density formulations.
Analogue gravity explores how collective excitations in condensed matter systems can reproduce the behavior of fields in curved spacetimes. An important example is the acoustic black holes that can occur for sound in a moving fluid. In these lecture notes, we focus on atomic Bose-Einstein condensates (BECs), quantum fluids that provide an interesting platform for analogue gravity studies thanks to their accurate theoretical description, remarkable experimental control, and ultralow temperatures that allow the quantum nature of sound to emerge. We give a pedagogical introduction to analogue black holes and the theoretical description of BECs and their elementary excitations, which behave as quantum fields in curved spacetimes. We then apply these tools to survey the current understanding of black-hole superradiance and analogue Hawking radiation, including explicit examples and numerical methods.
This paper investigates the acoustic superradiance of the density and phase fluctuations from the single vortex state of a Bose-Einstein condensate, by employing full time-domain and asymptotic frequency domain numerical calculations. The draining bathtub model of an incompressible barotropic fluid is adopted to describe the vortex. The propagation of the axisymmetric density and phase fluctuations in the condensate are governed by the massless scalar Klein-Gordon wave equation, which establishes the rotating black-hole analogy. Hence, the amplified scattering of these fluctuations from the vortex comprise the superradiance effect. A particular coordinate transformation is applied to reveal the event horizon and the ergosphere termwise in the metric and the respective asymptotic spectral solutions. A comparative analysis of the time domain and asymptotic frequency domain results are given for a range of rotational speed of the vortex and the frequency of the impinging fluctuations. The agreement at low rotational speeds of the vortex is shown to be very good, which starts to deteriorate at higher rotational speeds due to increasing constraint violations of the time-domain calculations. We further demonstrate an asymptotic upper bound for the superradiance as a function of vortex rotational speed, provided that the vortex remains stable.
Abstract A vortex formed in the superfluid state of a Bose-Einstein condensate may exhibit superradiance a la blackhole for radially propagating acoustic fluctuations. The analogy is usually based on the so-called draining bathtub model of the vortex, in which an event horizon and ergosphere emerges when the radial velocity of the superfluid exceeds the propagation speed of sound in the condensate. The acoustic fluctuations mimic a massless scalar field in the curved Lorentzian space-time of the vortex and are governed by the Klein-Gordon wave equation. One common main approximation is the constant background density of the superfluid even in the presence of the vortex. This sets a constant relativistic sound speed. However, the vortex state solution of the Gross-Pitaevskii equation clearly shows that both the density and the speed of sound vary radially near the vortex core, where the event horizon and thus the superradiance will take place. What changes would this complex interdependence bring to the formulation and to the outcomes of the superradiance based on constant density approximation? Here, we recount this question posed under the guidance of Prof. Tekin Dereli and present recent results. We show that the self-consistent density modifies the amplification dynamics near the event horizon significantly, thereby altering the temporal and spectral fingerprint of the superradiance of the vortex.