Quasiparticle descriptions are a powerful tool in condensed matter physics as they provide an analytical treatment of interacting systems. In this thesis we will apply this tool to theoretically describe two systems: a superconductor interacting with cavity photons and a flowing Bose-Einstein condensate forming a sonic black hole. First we will consider a two-dimensional s-wave BCS superconductor coupled to microwave cavity photons. We show how a nonequilibrium occupation of the photons can induce a nonequilibrium distribution of superconductor Bogoliubov quasiparticles, yielding an enhancement of the superconducting gap. The analytic dependence of this enhancement is provided in terms of the photon spectral and occupation functions, offering a large parameter space over which enhancement exists. Next, we analyze the equilibrium properties of a similar superconductor-cavity structure which has strong sub-dominant d-wave pairing interaction. In this case there is a collective mode known as the Bardasis-Schrieffer mode, which is essentially an uncondensed d-wave Cooper pair. We show that by driving an external supercurrent through the sample the Bardasis-Schrieffer mode can be hybridized with cavity photons, forming exotic Bardasis-Schrieffer-polaritons. We then turn to consider a flowing Bose-Einstein condensate. In the presence of inhomogeneous flow, the long-wavelength motion of quasiparticles can be mapped onto the kinematics of matter fields in a curved spacetime. This mapping allows for the simulation of a black hole and its interactions with quantum fields via analogy. We show that in the case of a step-like jump in the condensate flow the emission of analogue Hawking radiation is accompanied by evanescent modes which are pinned to the event horizon. Finally, we generalize this setup to include pseudo-spin half spinor Bose condensates. In this case, we show that the analogue spacetime the quasiparticles experience can be of the exotic Newton-Cartan type. Newton-Cartan gravity -- the geometric formulation of Newtonian gravity -- is realized when the Goldstone mode disperses quadratically as opposed to linearly. The nature of the analogue spacetime is controlled by the presence or absence of an easy-axis anisotropy in the boson spin-exchange interaction. We conclude by arguing that this Newton-Cartan spacetime can be experimentally realized in current platforms.
Quasiparticle descriptions are a powerful tool in condensed matter physics as they provide an analytical treatment of interacting systems. In this thesis we will apply this tool to theoretically describe two systems: a superconductor interacting with cavity photons and a flowing Bose-Einstein condensate forming a sonic black hole. First we will consider a two-dimensional s-wave BCS superconductor coupled to microwave cavity photons. We show how a nonequilibrium occupation of the photons can induce a nonequilibrium distribution of superconductor Bogoliubov quasiparticles, yielding an enhancement of the superconducting gap. The analytic dependence of this enhancement is provided in terms of the photon spectral and occupation functions, offering a large parameter space over which enhancement exists. Next, we analyze the equilibrium properties of a similar superconductor-cavity structure which has strong sub-dominant d-wave pairing interaction. In this case there is a collective mode known as the Bardasis-Schrieffer mode, which is essentially an uncondensed d-wave Cooper pair. We show that by driving an external supercurrent through the sample the Bardasis-Schrieffer mode can be hybridized with cavity photons, forming exotic Bardasis-Schrieffer-polaritons. We then turn to consider a flowing Bose-Einstein condensate. In the presence of inhomogeneous flow, the long-wavelength motion of quasiparticles can be mapped onto the kinematics of matter fields in a curved spacetime. This mapping allows for the simulation of a black hole and its interactions with quantum fields via analogy. We show that in the case of a step-like jump in the condensate flow the emission of analogue Hawking radiation is accompanied by evanescent modes which are pinned to the event horizon. Finally, we generalize this setup to include pseudo-spin half spinor Bose condensates. In this case, we show that the analogue spacetime the quasiparticles experience can be of the exotic Newton-Cartan type. Newton-Cartan gravity -- the geometric formulation of Newtonian gravity -- is realized when the Goldstone mode disperses quadratically as opposed to linearly. The nature of the analogue spacetime is controlled by the presence or absence of an easy-axis anisotropy in the boson spin-exchange interaction. We conclude by arguing that this Newton-Cartan spacetime can be experimentally realized in current platforms.
In this thesis we consider the application of phase-space methods to Bose-Einstein condensates; the work comprises of three main parts: Part I: A phase-space method for the Bose-Hubbard model; Part II: An analogue model of an expanding universe in Bose-Einstein condensates. and Part III: An analogue model of an acoustic Black Hole in Bose-Einstein condensates. In part I we present a phase-space method for the Bose-Hubbard model based on the Qfunction representation. In particular, we consider two model Hamiltonians in the mean-field approximation; the first is the standard “one site” model where quantum tunneling is approximated entirely using mean-field terms; the second “two site” model explicitly includes tunneling between two adjacent sites while treating tunneling with other neighbouring sites using the meanfield approximation. The ground state is determined by minimising the classical energy functional subject to quantum mechanical constraints, which take the form of uncertainty relations. For each model Hamiltonian we compare the ground state results from the Q-function method with the exact numerical solution. The results from the Q-function method, which are easy to compute, give a good qualitative description of the main features of the Bose-Hubbard model including the superfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem and show there are some limitations of the method particularly in the weak lattice regime. Analogue models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In particular, the prediction of quasiparticle creation in ultra-cold Bose gases in specific configurations can be viewed as an analogue to either cosmological particle creation or the Hawking effect. In part II of this thesis we investigate the analogue of cosmological particle creation in an expanding universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we simulate a 2D homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for several different expansion scenarios the calculated particle production is consistent with the underlying theory. For inflationary models we find the particle production for long wavelength modes coincides with the analytic theory within the acoustic approximation, whereas the particle production is suppressed for short wavelength (ie. free-particle like) modes. Moreover, particle production is enhanced for faster expansions, approaching the analytic result for the sudden expansion in the limit of a very fast expansion. For the case of a cyclic expansion, particle production peaks for a mode frequency that is approximately half of the driving frequency as expected for parametric resonance. In part III of this thesis we investigate an acoustic black hole in a Bose-Einstein condensate, formed by two de Laval nozzles in a ring configuration — a system we refer to as the quantum de Laval nozzle. Our model is formulated in one dimension with a sinusoidal potential. For nonzero superfluid flow, this system can exhibit stable transonic flow with both black and white hole sonic horizons. Stationary states are found by solving the time-independent Gross-Pitaevskii equation subject to a phase quantisation constraint. By solving the projected Bogoliubov-de Gennes equations for the system, we also find the discrete spectrum and quasiparticle modes. There are dynamical instabilities for certain values of winding number and potential depth, for which it is possible to construct pairs of normalisable modes. We further investigate the dynamics of the system using a classical field method based on the truncated Wigner approximation. For a low winding number and unstable configuration, we find exponential growth for the pair of unstable modes, whereas there is no growth in these modes for a stable configuration. This can be interpreted as non-degenerate parametric amplification, valid for short times. In contrast, for a large winding number, there is significant growth in modes for both stable and unstable configurations. This is indicative of higher order processes neglected in the quasiparticle picture, which is further reinforced by that fact that large winding number solutions require large nonlinearities. Finally, we consider the connection of our results with the usual semi-classical prediction of the Hawking effect. For an unstable configuration, the normalised unstable modes couple equal and opposite real frequencies, so that the growth in these modes represents the closest analogy with the Hawking effect for our quantum system.
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 thesis covers three applications of Bose-Einstein condensates and related phenomena, in the theme of pair creation and pair annihilation. First, Bose-Einstein condensates (BEC) are viewed as one of the candidates to implement a sonic black hole. This can lead to the observation of analog Hawking radiation, resulting from a phonon pair creation at a black-hole horizon (BH). Such implementation has been achieved in a resent experiment by J. Steinhauer, in which a black-hole/white-hole pair has been produced. He also reported the observations of self-amplifying Hawking radiation, via a lasing mechanism operating between the black and white-hole horizons. Through our simulations, we find that the observations should be attributed not to the black hole laser effect, but rather to a growing zero-frequency bow wave, generated at the white-hole horizon. The relative motion of the two horizons produces a Doppler shift of the bow wave at the BH, where it stimulates a monochromatic Hawking radiation. We also find that shot-to-shot atom number variations and quantum fluctuations give density-density correlations consistent with those reported in the experiments. In particular, atom number variations can produce a spurious correlation signal. Secondly, a sonic black hole/white hole pair and phonon pair creation can also be realized using a ring-shaped condensate. Here we focus on the phonon spectroscopy of a ring condensate. We probe the phonon excitation spectrum by applying a harmonically driven barrier to a 23Na Bose-Einstein condensate in a ring-shaped trap. When excited resonantly, these wavepackets display a regular periodic structure. The resonant frequencies depend upon the particular configuration of the barrier, but are commensurate with the orbital frequency of a sound wave traveling around the ring. Energy transfer to the condensate over many cycles of the periodic wavepacket motion causes enhanced atom loss from the trap at resonant frequencies. Solutions of the Gross-Pitaevskii (GP) equation exhibit quantitative agreement with the experimental data. Thirdly, positronium (Ps) BECs have been of experimental and theoretical interest due to their potential application as the gain medium of a gamma-ray laser. Ps BECs are intrinsically spinor due to the presence of ortho- (o-Ps) and para-positronium (p-Ps), whose annihilation lifetimes differ by three orders of magnitude. We study the spinor dynamics and annihilation processes in the p-Ps/o-Ps system using both solutions of the GP equations and a rate-equation approach. For an initially unpolarized condensate, there is a threshold density at which spin mixing between o-Ps and p-Ps occurs. Beyond this threshold, there are unstable spatial modes accompanied by spin mixing. To ensure a high production yield above the critical density, a careful choice of external field must be made to avoid the spin mixing instability.
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.
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.
Controlling the states of quantum systems is a subject of major interest for both fundamental science and technological development. In this thesis, we carry out theoretical studies on how this can be achieved using light, both of classical and of quantum mechanical nature. We study a driven cavity system, where the driving field is classical and the cavity field is quantum mechanical. We demonstrate how virtual scattering of driving photons inside the cavity via two-photon processes can induce controllable long-range electron interactions in two-dimensional materials. We show that both the strength and the sign of the interactions can be tuned by the driving parameters and that the interactions are not screened effectively except at very low frequencies. For realistic cavity parameters, driving-induced heating of the electrons by inelastic photon scattering is suppressed and coherent electron interactions dominate. When the interactions are attractive, they cause an instability in the Cooper channel at a temperature proportional to the square root of the driving intensity. Our results open up a new avenue for engineering quantum materials on-demand and the possibility of studying the effect of long-range interactions in condensed matter systems. Following that, we take a more general approach to investigate the effect of long-range density-density electron attractive interactions on a superconductor. We show that the low-lying excitations of a 2D conventional superconductor can be significantly altered by these long-range interactions. Using BCS theory, we find that these interactions combine non-linearly with the intrinsic local attractions of the superconductor to increase the Bogoliubov quasiparticle excitation energies, thus enlarging the superconducting gap. Moreover, we show how the long-range nature of the driven-cavity-induced attractions qualitatively changes the collective excitations of the superconductor. Specifically, they lead to the appearance of additional collective excitations of the excitonic modes. They further push the Higgs mode "in gap": it lies below the Bogoliubov quasiparticle continuum such that it cannot decay through the quasiparticle excitations.