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Cosmic Dynamics of Bose-Einstein Condensates

James E. Lidsey

Published Jul 9, 2003• 45 citations
OpenAlex

Abstract

A dynamical correspondence is established between positively curved, isotropic, perfect fluid cosmologies and quasi-two-dimensional, harmonically trapped Bose-Einstein condensates by mapping the equations of motion for both systems onto the one-dimensional Ermakov system. Parameters that characterize the physical properties of the condensate wavepacket, such as its width, momentum and energy, may be identified with the scale factor, Hubble expansion parameter and energy density of the universe, respectively. Different forms of cosmic matter correspond to different choices for the time-dependent trapping frequency of the condensate. The trapping frequency that mimics a radiation-dominated universe is determined.

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Journal of Physics B Atomic Molecular and Optical Physics•Nov 23, 2001•27 citations
A dynamical correspondence is established between positively curved, isotropic, perfect fluid cosmologies and quasi-two-dimensional, harmonically trapped Bose-Einstein condensates by mapping the equations of motion for both systems onto the one-dimensional Ermakov system. Parameters that characterize the physical properties of the condensate wavepacket, such as its width, momentum and energy, may be identified with the scale factor, Hubble expansion parameter and energy density of the universe, respectively. Different forms of cosmic matter correspond to different choices for the time-dependent trapping frequency of the condensate. The trapping frequency that mimics a radiation-dominated universe is determined.
A dynamical correspondence is established between positively curved, isotropic, perfect fluid cosmologies and quasi-two-dimensional, harmonically trapped Bose–Einstein condensates by mapping the equations of motion for both systems onto the one-dimensional Ermakov system. Parameters that characterize the physical properties of the condensate wavepacket, such as its width, momentum and energy, may be identified with the scale factor, Hubble expansion parameter and energy density of the universe, respectively. Different forms of cosmic matter correspond to different choices for the time-dependent trapping frequency of the condensate. The trapping frequency that mimics a radiation-dominated universe is determined.
We analytically study the expansion of a Bose-Einstein condensate in a ring-shaped trap with an increasing central radius. The evolution of the ground state is described using a scaling transform. Additionally, the dynamics of excited azimuthal modes over the varying ground state is analyzed through a generalization of the Bogoliubov--de Gennes approach. Our results explain some of the features observed in recent experiments focused on testing the applicability of the system as a parallel of cosmological inflationary models. The radial dynamics, which corresponds to the inflaton field of the cosmological counterpart, is analytically characterized: The expansion is found to induce the oscillatory displacement of the condensate as well as the coupled variation of the radial and vertical widths. Our findings account also for the observed redshift and emergence of the Hubble friction in the evolution of initially prepared azimuthal modes. Our description, which traces the role of the different components of the setup in the expansion, enhances the controllability and therefore the potential of the system as a ground for emulating the inflationary dynamics of cosmological models.
A Gaussian ansatz for the wave function of two-dimensional harmonically trapped anisotropic Bose-Einstein condensates is shown to lead, via a variational procedure, to a coupled system of two second-order, nonlinear ordinary differential equations. This dynamical system is shown to be in the general class of Ermakov systems. Complete integrability of the resulting Ermakov system is proven. Using the exact solution, collapse of the condensate is analyzed in detail. Time dependence of the trapping potential is allowed.
Phonons in expanding Bose–Einstein condensates with wavelengths much larger than the healing length behave in the same way as quantum fields within a universe undergoing an accelerated expansion. This analogy facilitates the application of many tools and concepts known from general relativity (such as horizons) and the prediction of the corresponding effects such as the freezing of modes after horizon crossing and the associated amplification of quantum fluctuations. Basically the same amplification mechanism is (according to our standard model of cosmology) supposed to be responsible for the generation of the initial inhomogeneities—and hence the seeds for the formation of structures such as our galaxy—during cosmic inflation (i.e., a very early epoch in the evolution of our universe). After a general discussion of the analogy ( analogue cosmology ), we calculate the frozen and amplified density–density fluctuations for quasi-two-dimensional (Q2D) and three-dimensional (3D) condensates which undergo a free expansion after switching off the (longitudinal) trap.
We study the dynamics of a Bose-Einstein condensate wavepacket trapped in an optical lattice. In the tight-binding limit, the system maps onto a discrete nonlinear Schrödinger equation. Using a collective coordinate approximation, we investigate different dynamical regimes: diffusive (in which the boson wavepacket spreads out), self-trapped (with a self-induced pinning), breathing (where the centre of mass of the wavepacket moves with an oscillating width), and solitonic (in which the wavepacket shape is exactly preserved during the dynamics).