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arXiv:0810.3088 · Physical Review A

Spacetime analog of Bose-Einstein condensates: Bogoliubov–de Gennes formulation

Yasunari Kurita, Michikazu Kobayashi, Takao Morinari, Makoto Tsubota, Hideki Ishihara

Published Apr 15, 2009• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.79.043616

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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.
In this paper, we formulate quantum field theory on an analogue curved spacetime in a Bose-Einstein condensate where quantum particles on curved spacetime are equated with Bogoliubov quasi-particle excitations on the Bose-Einstein condensate. By using our formulation, we numerically investigate a particle creation in analogue expanding Universe which can be expressed as Bogoliubov quasiparticles in an expanding Bose-Einstein condensate and obtain its spectrum which shows the thermal Maxwell-Boltzmann distribution, the temperature of which can be experimentally accessible. This is the first direct numerical simulation for the particle creation and its spectrum on analogue curved spacetime by using the system of Bose-Einstein condensates.
We formulate particle-creation phenomena in Bose-Einstein condensates in terms of conserving gapless mean-field theory for weakly interacting Bose gases. The particle-creation spectrum is calculated by rediagonalizing the Bogoliubov--de Gennes (BdG) Hamiltonian in mean-field theory. The conservation implies that quasiparticle creation is accompanied by quantum back reaction to the condensates. Particle creation in this mean-field theory is found to be equivalent to that in quantum field theory (QFT) in curved space-time. An expression is obtained for an effective metric affected by quantum back reaction. The formula for the particle-creation spectrum obtained in terms of QFT in curved space-time is shown to be the same as that given by rediagonalizing the BdG Hamiltonian.
A Bose gas in a double-well potential, exhibiting a true Bose-Einstein condensate (BEC) amplitude and initially performing Josephson oscillations, is a prototype of an isolated, nonequilibrium many-body system. We investigate the quasiparticle (QP) creation and thermalization dynamics of this system by solving the time-dependent Keldysh-Bogoliubov equations. We find avalanchelike QP creation due to a parametric resonance between BEC and QP oscillations, followed by slow, exponential relaxation to a thermal state at an elevated temperature, controlled by the initial excitation energy of the oscillating BEC above its ground state. The crossover between the two regimes occurs because of an effective decoupling of the QP and BEC oscillations. This dynamics is analogous to elementary particle creation in models of the early universe. The thermalization in our setup occurs because the BEC acts as a grand canonical reservoir for the quasiparticle system.
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.
Pair creation is a fundamental prediction of quantum field theory in curved spacetimes. While classical aspects of this phenomenon have been observed, the experimental confirmation of its quantum origin remains elusive. In this article, we quantify the entanglement produced by pair creation in two dimensional Bose-Einstein condensate (BEC) analogs of expanding universes and examine the impact of various experimental factors, including decoherence from thermal noise and losses. Our analysis evaluates the feasibility of detecting entanglement in these systems and identifies optimal experimental configurations for achieving this goal. Focusing on the experimental setup detailed in Viermann et al. [Quantum field simulator for dynamics in curved spacetime, Nature (London) 611, 260 (2022)], we demonstrate that entanglement can be observed in these BEC analogs at a significance level of $\ensuremath{\sim}2\ensuremath{\sigma}$ with current capabilities, and at $\ensuremath{\gtrsim}3.3\ensuremath{\sigma}$ with modest improvements. Achieving this would provide unequivocal evidence of the quantum nature of pair creation and validate one of the most iconic predictions of quantum field theory in curved spacetimes.
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.