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.
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.
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.
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.
Generalizing the one-component case, we demonstrate that the propagation of sound waves in two-component Bose-Einstein condensates can also be described in terms of effective sonic geometries under appropriate conditions. In comparison with the one-component case, the two-component setup offers more flexibility and several advantages. In view of these advantages, we propose an experiment in which the evolution of the inflaton field, and thereby the generation of density fluctuations in the very early stages of our universe during inflation, can be simulated, realizing a quantum simulation via analogue gravity models.
The vacuum of quantum field theory is not empty space but filled with quantum vacuum fluctuations, which give rise to many intriguing effects. The first part of this Thesis addresses cosmic inflation, where the quantum fluctuations of the inflaton field freeze and get amplified in the expanding universe. Afterwards, we turn our attention towards Bose-Einstein condensates, a laboratory system. Since most of our calculations are performed using a mean-field expansion, we will study the accuracy of a finite-range interaction potential onto such an expansion. Exploiting the universality of quantum fluctuations, several aspects of cosmic inflation will be identified in ballistically expanding Bose-Einstein condensates. The effective action technique for calculating the quantum backreaction will be scrutinized. Finally, we consider dynamic quantum phase transitions in the last part of this Thesis. To this end two specific scenarios will be investigated: firstly, the structure formation during the superfluid to Mott-insulator transition in the Bose-Hubbard model; and secondly, the formation of spin domains as a two-dimensional spin-one Bose gas is quenched from the (polar) paramagnetic to the (planar) ferromagnetic phase. During this quench, the symmetry of the ground state is spontaneously broken and vortices (topological defects) form.
We analyze the phenomena of condensate collapse, as described by Donley et al. [Nature 412, 295 (2001)] and N. Claussen [Ph. D thesis, University of Colorado, 2003 (unpublished)] by focusing on the behavior of excitations or fluctuations above the condensate, as driven by the dynamics of the condensate, rather than the dynamics of the condensate alone or the kinetics of the atoms. The dynamics of the condensate squeezes and amplifies the quantum excitations, mixing the positive and negative frequency components of their wave functions thereby creating particles that appear as bursts and jets. By analyzing the changing amplitude and particle content of these excitations, our simple physical picture explains well the overall features of the collapse phenomena and provides excellent quantitative fits with experimental data on several aspects, such as the scaling behavior of the collapse time and the number of particles in the jet. The prediction of the bursts at this level of approximation is less than satisfactory but may be improved by including the backreaction of the excitations on the condensate. The mechanism behind the dominant effect---parametric amplification of vacuum fluctuations and freezing of modes outside of horizon---is similar to that of cosmological particle creation and structure formation in a rapid quench (which is fundamentally different from Hawking radiation in black holes). This shows that Bose-Einstein condensate dynamics is a promising venue for doing ``laboratory cosmology.''
Abstract Early during the era of cosmic inflation, rotational invariance may have been broken, only later emerging as a feature of low-energy physics. This motivates ongoing searches for residual signatures of anisotropic space-time, for example in the power spectrum of the cosmic microwave background. We propose that dipolar Bose–Einstein condensates (BECs) furnish a laboratory quantum simulation platform for the anisotropy evolution of fluctuation spectra during inflation, exploiting the fact that the speed of dipolar condensate sound waves depends on direction. We construct the anisotropic analogue space-time metric governing sound, by linking the time-varying strength of dipolar and contact interactions in the BEC to the scale factors in different coordinate directions. Based on these, we calculate the dynamics of phonon power spectra during an inflation that renders the initially anisotropic Universe isotropic. We find that the expansion speed provides an experimental handle to control and study the degree of final residual anisotropy. Gravity analogues using dipolar condensates can thus provide tuneable experiments for a field of cosmology that was until now confined to a single experiment, our Universe.