This Thesis develops in the field of analogue gravity, a branch of physics which relies on a specific correspondence that exists between fluid dynamics and curved spacetimes; thanks to that, effects which are hard to study in the gravitational context are investigated in the realm of hydrodynamics with the hope that the consequent results could be transferred back to the cosmological domain and give us better insights in the gravitational context. In this Thesis, I try to introduce the reader to the realm of analogue gravity and to the concepts of Bose-Einstein condensation, Hawking radiation and its analogue counterpart (Chapter 1 and 2); then in Chapter 3 and 4 I describe the details of our work. In the past years, we have developed two directions of research, both aimed at the characterization of the analogue Hawking radiation in Bose-Einstein condensates. One side of our work had the purpose of investigating the results obtained in recent experiments apt to the detection of the analogue Hawking radiation in a quantum gas. Our results are very important in the description of the phenomena and give insights on how to develop future experiments (Chapter 3). The other side of our work aimed at the characterization of this effect from a theoretical point of view by means of an innovative approach, which allows for a much deeper and more detailed description of the radiation and helps shine new light on this effect (Chapter 4). In Chapter 5 we briefly discuss the possible developments of our studying, highlighting the missing pieces of this field and suggesting possible ways to approach the existing problems.
This Thesis develops in the field of analogue gravity, a branch of physics which relies on a specific correspondence that exists between fluid dynamics and curved spacetimes; thanks to that, effects which are hard to study in the gravitational context are investigated in the realm of hydrodynamics with the hope that the consequent results could be transferred back to the cosmological domain and give us better insights in the gravitational context. In this Thesis, I try to introduce the reader to the realm of analogue gravity and to the concepts of Bose-Einstein condensation, Hawking radiation and its analogue counterpart (Chapter 1 and 2); then in Chapter 3 and 4 I describe the details of our work. In the past years, we have developed two directions of research, both aimed at the characterization of the analogue Hawking radiation in Bose-Einstein condensates. One side of our work had the purpose of investigating the results obtained in recent experiments apt to the detection of the analogue Hawking radiation in a quantum gas. Our results are very important in the description of the phenomena and give insights on how to develop future experiments (Chapter 3). The other side of our work aimed at the characterization of this effect from a theoretical point of view by means of an innovative approach, which allows for a much deeper and more detailed description of the radiation and helps shine new light on this effect (Chapter 4). In Chapter 5 we briefly discuss the possible developments of our studying, highlighting the missing pieces of this field and suggesting possible ways to approach the existing problems.
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.
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.
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 focus on investigating the many-body physics of Bose-Einstein condensates of atoms and light. The many-body phenomena that we discuss in this thesis contains two different cases. The first case is the space-time crystal in a Bose-Einstein condensate of atoms. We show the observation of a space-time crystal using ultra-cold atoms, where the periodic structure in both space and time are directly visible in the experimental images. The underlying physics in our superfluid can be described ab initio and allows for a clear identification of the mechanism that causes the spontaneous symmetry breaking. Our results pave the way for the usage of space-time crystals for the discovery of novel nonequilibrium phases of matter. Moreover, we investigate the formation of the space-time. We theoretically model this space-time crystal that contains the high-order axial mode driving by the radial breathing mode. Furthermore, we provide the comparisons of the theoretical prediction, the experiment and the numerical simulation for the mode function, the flux and then the mode frequency. We obtain accurate agreements between the experiment, the numerical simulation and our theoretical prediction. The other case is the analogue Schwarzschild black hole in condensates of light. By etching a hole in the mirrors or by placing a scatterer in the center of a cavity, we can create a sink for light. In a Bose-Einstein condensate of photons this sink results in the creation of a so-called radial vortex, which is a two-dimensional analogue of a Schwarzschild black hole. We theoretically investigate the Hawking radiation and the associated greybody factor of this Schwarzschild black hole. In particular, we determine the density-density and velocity-velocity correlation functions of the Hawking radiation, which can be measured by observing the spatial correlations in the fluctuations in the light emitted by the cavity.
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 Hawking effect can be understood as a broad kinematic phenomenon associated with mode behavior near a horizon. While astrophysical black holes produce one specific realization of this radiation, this perspective inspires extensive theoretical and experimental efforts to create event horizons in diverse physical systems to observe the resulting analogue Hawking emission. One of the most successful realizations is the fiber-optical analogue, based on nonlinear quantum optics. In these notes, we introduce and motivate this system while outlining the theoretical concepts underlying the gravitational analogy. Finally, we review key experiments and discuss their impact on the field.