Analog models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In this paper, we investigate the analog of cosmological particle creation in a Friedmann-Robertson-Walker universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we focus on a two-dimensional homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for various forms of the scaling function the particle production is consistent with the underlying theory in the long wavelength limit. In this context, we further discuss the implications of modified dispersion relations that arise from the microscopic theory of a weakly interacting Bose gas.
Analog models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In this paper, we investigate the analog of cosmological particle creation in a Friedmann-Robertson-Walker universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we focus on a two-dimensional homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for various forms of the scaling function the particle production is consistent with the underlying theory in the long wavelength limit. In this context, we further discuss the implications of modified dispersion relations that arise from the microscopic theory of a weakly interacting Bose gas.
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.
We experimentally study cosmological particle production in a two-dimensional Bose-Einstein condensate, whose density excitations map to an analog cosmology. The expansion of spacetime is realized with tunable interactions. The particle spectrum can be understood through an analogy to quantum mechanical scattering, in which the dynamics of the spacetime metric determine the shape of the scattering potential. Hallmark scattering phenomena such as resonant forward scattering and Bragg reflection are connected to their cosmological counterparts, namely linearly expanding space and bouncing universes. We compare our findings to a theoretical description that extends beyond the acoustic approximation, which enables us to apply the model to high-momentum excitations.
Phonons have the characteristic linear dispersion relation of massless relativistic particles. They arise as low-energy excitations of Bose-Einstein condensates and, in nonhomogeneous situations, are governed by a space- and time-dependent acoustic metric. We discuss how this metric can be experimentally designed to realize curved spacetime geometries, in particular, expanding Friedmann-Lema\^{\i}tre-Robertson-Walker cosmologies, with negative, vanishing, or positive spatial curvature. A nonvanishing Hubble rate can be obtained through a time-dependent scattering length of the background condensate. For relativistic quantum fields, this leads to the phenomenon of particle production, which we describe in detail. We explain how particle production and other interesting features of quantum field theory in curved spacetime can be tested in terms of experimentally accessible correlation functions.
We construct quantum field theory in an analog curved spacetime in Bose-Einstein condensates based on the Bogoliubov--de Gennes equations by exactly relating quantum particles in curved spacetime with Bogoliubov quasiparticle excitations in Bose-Einstein condensates. Here, we derive a simple formula relating the two, which can be used to calculate the particle creation spectrum by solving the time-dependent Bogoliubov--de Gennes equations. Using our formulation, we numerically investigate particle creation in an analog expanding universe which can be expressed as Bogoliubov quasiparticles in an expanding Bose-Einstein condensate. We obtain its spectrum, which follows the thermal Maxwell-Boltzmann distribution, the temperature of which is experimentally attainable. Our derivation of the analogy is useful for general Bose-Einstein condensates and not limited to homogeneous ones, and our simulation is an example of particle creations by solving the Bogoliubov--de Gennes equation in an inhomogeneous condensate.
We study the truncated Wigner method applied to a weakly interacting spinless Bose-condensed gas which is perturbed away from thermal equilibrium by a time-dependent external potential. The principle of the method is to generate an ensemble of classical fields ψ( r ) which samples the Wigner quasi-distribution function of the initial thermal equilibrium density operator of the gas, and then to evolve each classical field with the Gross–Pitaevskii equation. In the first part of the paper we improve the sampling technique over our previous work (Sinatra et al 2000 J. Mod. Opt. 47 2629–44) and we test its accuracy against the exactly solvable model of the ideal Bose gas. In the second part of the paper we investigate the conditions of validity of the truncated Wigner method. For short evolution times it is known that the time-dependent Bogoliubov approximation is valid for almost pure condensates. The requirement that the truncated Wigner method reproduces the Bogoliubov prediction leads to the constraint that the number of field modes in the Wigner simulation must be smaller than the number of particles in the gas. For longer evolution times the nonlinear dynamics of the noncondensed modes of the field plays an important role. To demonstrate this we analyse the case of a three-dimensional spatially homogeneous Bose-condensed gas and we test the ability of the truncated Wigner method to correctly reproduce the Beliaev–Landau damping of an excitation of the condensate. We have identified the mechanism which limits the validity of the truncated Wigner method: the initial ensemble of classical fields, driven by the time-dependent Gross–Pitaevskii equation, thermalizes to a classical field distribution at a temperature T class which is larger than the initial temperature T of the quantum gas. When T class significantly exceeds T a spurious damping is observed in the Wigner simulation. This leads to the second validity condition for the truncated Wigner method, T class − T ≪ T , which requires that the maximum energy max of the Bogoliubov modes in the simulation does not exceed a few k B T .
The production of quantum field excitations or particles in cosmological spacetimes is a hallmark prediction of curved quantum field theory. The generation of cosmological perturbations from quantum fluctuations in the early Universe constitutes an important application. The problem can be quantum-simulated in terms of structure formation in an interacting Bose-Einstein condensate (BEC) with time-dependent s-wave scattering length. Here, we explore a mapping between cosmological particle production in general (D+1)-dimensional spacetimes and scattering problems described by the nonrelativistic stationary Schr\"odinger equation in one dimension. Through this mapping, intuitive explanations for emergent spatial structures in both the BEC and the cosmological system can be obtained for analogue cosmological scenarios that range from power-law expansions to periodic modulations. The investigated cosmologies and their scattering analogs are tuned to be implemented in a (2+1)-dimensional quantum field simulator.