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arXiv:1710.05800 · Physical Review X

A Rapidly Expanding Bose-Einstein Condensate: An Expanding Universe in the Lab

Stephen Eckel, A. Kumar, Ted Jacobson A, I. B. Spielman, Gretchen K. Campbell

Published Apr 19, 2018• 218 citations
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Abstract

We study the dynamics of a supersonically expanding ring-shaped Bose-Einstein condensate both experimentally and theoretically. The expansion redshifts long-wavelength excitations, as in an expanding universe. After expansion, energy in the radial mode leads to the production of bulk topological excitations - solitons and vortices - driving the production of a large number of azimuthal phonons and, at late times, causing stochastic persistent currents. These complex nonlinear dynamics, fueled by the energy stored coherently in one mode, are reminiscent of a type of "preheating" that may have taken place at the end of inflation.

DOI
10.1103/physrevx.8.021021

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•Jan 1, 2018•3 citations
We study the dynamics of a supersonically expanding ring-shaped Bose-Einstein condensate both experimentally and theoretically. The expansion redshifts long-wavelength excitations, as in an expanding universe. After expansion, energy in the radial mode leads to the production of bulk topological excitations - solitons and vortices - driving the production of a large number of azimuthal phonons and, at late times, causing stochastic persistent currents. These complex nonlinear dynamics, fueled by the energy stored coherently in one mode, are reminiscent of a type of "preheating" that may have taken place at the end of inflation.
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.
We systematically construct stationary soliton states in a one-component, two-dimensional, repulsive, Gross-Pitaevskii equation with a ring-shaped targetlike trap similar to the potential used to confine a Bose-Einstein condensate in a recent experiment [R. Mathew, A. Kumar, S. Eckel, F. Jendrzejewski, G. K. Campbell, M. Edwards, and E. Tiesinga, Phys. Rev. A 92, 033602 (2015)]. In addition to the ground-state configuration, we identify a wide variety of excited states involving phase jumps (and associated dark solitons) inside the ring. These configurations are obtained from a systematic bifurcation analysis starting from the linear, small atom density, limit. We study the stability and, when unstable, the dynamics of the most basic configurations. Often these lead to vortical dynamics inside the ring persisting over long time scales in our numerical experiments. To illustrate the relevance of the identified states, we showcase how such dark-soliton configurations (even the unstable ones) can be created in laboratory condensates by using phase-imprinting techniques.
The form and stability properties of axisymmetric and spherically symmetric stationary states in two and three dimensions, respectively, are elucidated for Bose-Einstein condensates. These states include the ground state, central vortices, and radial excitations of both. The latter are called ring solitons in two dimensions and spherical shells in three. The nonlinear Schr\"odinger equation is taken as the fundamental model; both extended and harmonically trapped condensates are considered. It is found that the instability times of ring solitons can be long compared to experimental time scales, making them effectively stable over the lifetime of an experiment.
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.
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.
This dissertation presents multiple results of our experiments with ring-shaped 23Na Bose-Einstein condensates. First, we measure the effect of temperature on the lifetime of quantized, persistent currents. We find that the persistent current lifetime decreases when the temperature is increased. We also extract the critical velocity by measuring the size of hysteresis loops. The critical velocity is found to be a strong function of temperature. Second, we implement a new technique of measuring the circulation state of a persistent current in-situ, which is minimally-destructive. This technique uses the Doppler effect. We then measure the current-phase relation of a weak link inserted in the ring-shaped superfluid and measure the excitation spectrum of the ring-shaped superfluid. Finally, we study the dynamics of rapidly expanding rings, and explore the analogy between our experimental system and the expansion of the universe.