We present a study of analog cosmological models in Bose-Einstein condensates (BEC) and in graphene, and superfluidity in a box-shaped traps. We start by examining the dynamics of a Bose-Einstein condensate (BEC) trapped inside an expanding toroid that can realize an analog inflationary universe. The expanding condensate forces phonons to undergo redshift and damping due to quantum pressure, owing to the thinness of the ring. We predict that such expanding BECs can exhibit spontaneous phonon creation from the vacuum state and show how it would manifest in the atom density and density correlations and discuss connections with the inflationary theory. We then extend this work to study entanglement of the spontaneously generated phonon pairs, for which we use the techniques of quantum continuous variables. We then develop a protocol to experimentally measure the correlations entering the covariance matrix, allowing an experimental quantification of the entanglement properties of the inflationary BEC. We then present a of how a spatially-varying quasiparticle velocity in honeycomb lattices, achievable using strained graphene or in engineered cold-atom optical lattices that have a spatial dependent tunneling amplitude, can yield the Rindler Hamiltonian embodying an observer accelerating in Minkowski spacetime. Within this setup, a sudden switch-on of the spatially-varying strain yields a spontaneous production of electron-hole pairs, an analog version of the Unruh effect characterized by the Unruh temperature. We discuss how this thermal behavior, along with Takagi's statistics inversion, can manifest themselves in photo-emission and scanning tunneling microscopy experiments. We show that the electronic conductivity grows linearly with frequency $\omega$, vanishing in the DC limit. Finally, we find that the total electronic energy at zero environment temperature looks like Planck's blackbody result for photons due to the aforementioned statistics inversion, whereas for an initial thermally excited state of fermions, the total internal energy undergoes stimulated particle reduction. Finally, we study ultracold Fermi gases trapped inside a three-dimensional box with vanishing boundary conditions, with interactions modeled using the Fermi-Huang pseudopotential. Taylor expanding the pairing to linear order in position, solves the gap equation yielding the transition temperature and local pairing amplitude that vanishes at the edges.
We present a study of analog cosmological models in Bose-Einstein condensates (BEC) and in graphene, and superfluidity in a box-shaped traps. We start by examining the dynamics of a Bose-Einstein condensate (BEC) trapped inside an expanding toroid that can realize an analog inflationary universe. The expanding condensate forces phonons to undergo redshift and damping due to quantum pressure, owing to the thinness of the ring. We predict that such expanding BECs can exhibit spontaneous phonon creation from the vacuum state and show how it would manifest in the atom density and density correlations and discuss connections with the inflationary theory. We then extend this work to study entanglement of the spontaneously generated phonon pairs, for which we use the techniques of quantum continuous variables. We then develop a protocol to experimentally measure the correlations entering the covariance matrix, allowing an experimental quantification of the entanglement properties of the inflationary BEC. We then present a of how a spatially-varying quasiparticle velocity in honeycomb lattices, achievable using strained graphene or in engineered cold-atom optical lattices that have a spatial dependent tunneling amplitude, can yield the Rindler Hamiltonian embodying an observer accelerating in Minkowski spacetime. Within this setup, a sudden switch-on of the spatially-varying strain yields a spontaneous production of electron-hole pairs, an analog version of the Unruh effect characterized by the Unruh temperature. We discuss how this thermal behavior, along with Takagi's statistics inversion, can manifest themselves in photo-emission and scanning tunneling microscopy experiments. We show that the electronic conductivity grows linearly with frequency ω, vanishing in the DC limit. Finally, we find that the total electronic energy at zero environment temperature looks like Planck's blackbody result for photons due to the aforementioned statistics inversion, whereas for an initial thermally excited state of fermions, the total internal energy undergoes stimulated particle reduction. Finally, we study ultracold Fermi gases trapped inside a three-dimensional box with vanishing boundary conditions, with interactions modeled using the Fermi-Huang pseudopotential. Taylor expanding the pairing to linear order in position, solves the gap equation yielding the transition temperature and local pairing amplitude that vanishes at the edges.
We present a theoretical study of the dynamics of a Bose-Einstein condensate (BEC) trapped inside an expanding toroid that can realize an analog inflationary universe. As the system expands, we find that phonons in the BEC undergo redshift and damping due to quantum pressure effects, owing to the thinness of the ring. We predict that rapidly expanding toroidal BEC's can exhibit spontaneous particle creation, and study this phenomenon in the context of an initial coherent state wave function. We show how particle creation would be revealed in the atom density and density correlations, and discuss connections to the cosmological theory of inflation.
Recent interest and realization of ultracold bosons confined in shell-shaped, or bubble, traps have opened up several directions in the study of Bose-Einstein condensates (BECs) in intrinsically curved geometries. Here, we explore the prospect of using thin spherical shell-shaped BECs as analog gravity simulators of an expanding or contracting curved universe. We study an interacting BEC bubble subject to a temporal quench of the interaction strength. Our theoretical approach, which uses a Bogoliubov treatment following a dimensional reduction of a spherical BEC to the low-energy excitations of a shell-shaped structure, demonstrates how this system serves as a laboratory analog for a positively curved universe having a time-varying scale factor. We investigate spontaneous pair production triggered by quenches associated with exponentially decaying or growing interaction strengths, focusing on the emergence of oscillations in the structure factor---analogous to Sakharov oscillations in the Cosmic Microwave Background angular power spectrum. We quantify the entanglement of the produced pairs via logarithmic negativity, and show how this measure may be extracted from the power spectrum. These findings establish a bridge between quantum information science and early-universe cosmology in a laboratory setting.
Recent experiments have employed rapidly expanding toroidal Bose-Einstein condensates (BECs) to mimic the inflationary expansion in the early universe. One expected signature of the expansion in such experiments is spontaneous particle creation (of phonons) which is observable in density-density correlations. We study entanglement of these particles, which are known to result in a two-mode squeezed state. Using techniques for Gaussian states of continuous variable systems, we quantify the entanglement generated in this system, including effects such as decoherence and the use of an initially squeezed state, which can suppress and enhance entanglement, respectively. We also describe a protocol to experimentally measure the correlations entering the covariance matrix, allowing an experimental quantification of the entanglement properties of the inflationary BEC.
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.
Cosmic inflation is the theory that the universe underwent a violent, rapid expansion at the beginning of time, giving rise to much of the observed structure in the universe 1. This theory is supported by indirect observational evidence from many different sources (e.g. see 2). However, direct experimental investigation of inflation is of course impossible. For this reason there has been considerable interest in analog models for astrophysics (e.g. see 3), in which analogous behavior is studied in systems that are experimentally convenient. Due to their cleanness and great flexibility, dilute ultra-cold atomic gases (particularly Bose-Einstein condensates) form an attractive system for realizing such analog models. Indeed, a number of such table-top-astrophysics experiments have been conducted with Bose-Einstein condensates. For example, supernova-like collapse and explosion dynamics 4, controlled by modifying inter-atomic interactions using Feshbach resonances, and later interpreted as cosmological particle creation 5; Faster-than-sound flow of a condensate has been used to produce a sonic black hole analog 6; The in situ measurements of density fluctuations in a quenched quasi-two-dimensional condensate have demonstrated Sakharov like oscillations, analogous to the process thought to give rise to the anisotropy in the cosmic microwave background radiation 7. In Quantum simulations of the early universe Opanchuk and collaborators 8 propose using a two component condensate to simulate the inflationary phase of the early universe (see Fig. 1). In appropriate limits the relative phase between the two fields describing the condensate components is known to obey the sine-Gordon equation 9. The sine-Gordon model (SGM) is one of many scalar models for inflation that have been investigated in cosmology. It is thought to provide an acceptable model of the beginning of inflation where, starting from an unstable vacuum state, quantum fluctuations seed the cosmological inhomogeneities that lead to structure formation. However, the SGM likely fails to describe the end of inflation as its scalar potential does not generate efficient parametric resonances needed to ensure reheating of the universe 10, 11. The SGM also has applications in cosmology beyond inflation, and is a model for the “axion” matter field considered as a promising candidate for dark matter12. An important innovation in the scheme proposed by Opanchuk et al. is the introduction of a microwave (linear) coupling between the components that provides a practical way to initialize the system into an unstable vacuum state and controllably explore the subsequent evolution. Approximate numerical simulations, performed using the truncated Wigner approach, verify dynamical clustering can occur within experimentally realistic parameter regimes. The main challenge to realizing the proposed scenario will be finding an atom, and a pair of internal states, with suitable interaction parameters. It is likely that such an atom will be available given the increasing diversity of atomic species that have been Bose-condensed, and the great control over interactions afforded by Feshbach resonances. Aided by the rich toolbox of techniques available to ultra-cold atomic physics an array of opportunities present themselves for developing this line of research, such as: Engineering gravitational-like effects through including long range interactions, e.g. through electromagnetically induced gravity 13; Enhancing quantum fluctuations through stronger interactions or by confining the system in an optical lattice; Engineering richer models by including more spin states and other species of atoms.
Superfluidity is a remarkable phenomenon. Superfluidity was initially characterized by flow without friction, first seen in liquid helium in 1938, and has been studied extensively since. Superfluidity is believed to be related to, but not identical to Bose-Einstein condensation, a statistical mechanical phenomena predicted by Albert Einstein in 1924 based on the statistics of Satyendra Nath Bose, where bosonic atoms make a phase transition to form a Bose-Einstein condensate (BEC), a gas which has macroscopic occupation of a single quantum state. Developments in laser cooling of neutral atoms and the subsequent realization of Bose-Einstein condensates in ultracold gases have opened a new window into the study of superfluidity and its relation to Bose-Einstein condensation. In our atomic sodium BEC experiment, we studied superfluidity and dissipationless flow in an all-optical toroidal trap, constructed using the combination of a horizontal ``sheet''-like beam and vertical ``ring''-like beam, which, like a circuit loop, allows flow around the ring. On inducing a single quantum of circulation in the condensate, the smoothness and uniformity of the toroidal BEC enabled the sustaining of a persistent current lasting 40 seconds, limited by the lifetime of the BEC due to background gas pressure. This success set the stage for further experiments studying superfluidity. In a first set of experiments, we studied the stability of the persistent current by inserting a barrier in the flow path of the ring. The superflow stopped abruptly at a barrier strength such that the local flow velocity at the barrier exceeded a critical velocity, which supported decay via the creation of a vortex-antivortex pair. Our precise control in inducing and arresting superflow in the BEC is a first step toward studying other aspects of superfluidity, such as the effect of temperature and dimensionality. This thesis discusses these experiments and also details partial-transfer absorption imaging, an imaging technique developed in the course of this work.