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 present a topical review of the development of finite-temperature field theories of Bose-Einstein condensation in weakly interacting atomic gases. We highlight the difficulties in obtaining a consistent finite-temperature theory that has a gapless excitation spectrum in accordance with Goldstone's theorem and which is free from both ultraviolet and infrared divergences. We present results from the two consistent theories developed so far. These are the Hartree-Fock-Bogoliubov theory within the Popov approximation and a many-body T -matrix approach which we have termed gapless-Hartree-Fock-Bogoliubov (GHFB). Comparison with the available experimental results is made and the remaining difficulties are highlighted.
We formulate a conserving gapless mean-field theory for Bose-Einstein condensates on the basis of a Luttinger-Ward thermodynamic functional. It is applied to a weakly interacting uniform gas with density
n and s-wave scattering length
a to clarify its fundamental thermodynamic properties. It is found that the condensation here occurs as a first-order transition. The shift of the transition temperature
ΔTc from the ideal-gas result
T0 is positive and given to the leading order by
ΔTc=2.33an1/3T0, in agreement with a couple of previous estimates. The theory is expected to form a new theoretical basis for trapped Bose-Einstein condensates at finite temperatures.
This paper presents a conserving gapless mean-field theory for weakly interacting Bose gases. We first construct a mean-field Luttinger-Ward thermodynamic functional in terms of the condensate wave function
Ψ and the Nambu Green's function
G^ for the quasiparticle field. Imposing its stationarity respect to
Ψ and
G^ yields a set of equations to determine the equilibrium for general non-uniform systems. They have a plausible property of satisfying the Hugenholtz-Pines theorem to provide a gapless excitation spectrum. Also, the corresponding dynamical equations of motion obey various conservation laws. Thus, the present mean-field theory shares two important properties with the exact theory: ``conserving'' and ``gapless.'' The theory is then applied to a homogeneous weakly interacting Bose gas with s-wave scattering length
a and particle mass
m to clarify its basic thermodynamic properties under two complementary conditions of constant density
n and constant pressure
p. The superfluid transition is predicted to be first-order because of the non-analytic nature of the order-parameter expansion near
Tc inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature
Tc shows quite a different interaction dependence between the
n-fixed and
p-fixed cases. In the former case
Tc increases from the ideal gas value
T0 as
Tc/T0=1+2.33an1/3, whereas it decreases in the latter as
Tc/T0=1−3.84a(mp/2πℏ2)1/5. Temperature dependences of basic thermodynamic quantities are clarified explicitly.
In this tutorial we present an introduction to some theoretical methods of quantum field theory applied to the description of a trapped Bose–Einstein condensate. First of all, we give a brief account of the main characteristics of the phenomenon of condensation and present the many-body Hamiltonian of the system. We outline some of the most important approaches used in the characterization of a condensed Bose gas, including the mean-field theory and the Hartree–Fock–Bogoliubov method. Finally we illustrate the use of these techniques addressing some important issues in quantum atom optics. We characterize the quantum state of a Bose–Einstein condensate (BEC) at zero temperature. We also describe a process of Beliaev coupling between quasiparticles using a method that includes terms beyond the usual Bogoliubov approach.