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arXiv:2202.10441 · Physical Review A

Curved and expanding spacetime geometries in Bose-Einstein condensates

Mireia Tolosa-Simeón, Álvaro Parra-López, Natalia Sánchez-Kuntz, Tobias Haas, Celia Viermann, Marius Sparn, Nikolas Liebster, Maurus Hans, Elinor Kath, Helmut Strobel, Markus K. Oberthaler, Stefan Floerchinger

Published Sep 14, 2022• 30 citations
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Abstract

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.

DOI
10.1103/physreva.106.033313

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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.
We consider excitations of a spin-1 Bose-Einstein-condensate in the vicinity of different mean-field configurations and derive mappings to emergent relativistic quantum field theories minimally coupled to curved acoustic spacetimes. The quantum fields are typically identified with Nambu-Goldstone bosons, such that the structure of the analog quantum field theories on curved spacetimes depends on the (spontaneous) symmetry breaking pattern of the respective ground state. The emergent spacetime geometries are independent of each other and exhibit bimetricity in the polar and antiferromagnetic phase, whereas one has trimetricity in the ferromagnetic phase. Compared to scalar BECs, the spinor degrees of freedom allow us to investigate massive vector and scalar fields where the former is a spin-nematic rotation mode in the polar phase, which can be cast into a Proca field that is minimally coupled to a curved spacetime that emerges on length scales larger than the spin-healing length. Finally, we specify the Zeeman couplings and the condensate trap to be spacetime-dependent such that a cosmological Friedmann-Lemaître-Robertson-Walker metric can be achieved. This work enables a pathway towards quantum-simulating cosmological particle production of Proca quanta via quenching the quadratic Zeeman coefficient or via magnetic field ramps, which both result in the creation of spin-nematic squeezed states.
Abstract We study the low energy phonon dynamics of a Bose–Einstein condensate (BEC) with a density profile that is equivalent, via a coordinate transformation, to phonons traveling in a “spherical” curved spacetime that realizes the Friedman–Lemaître–Robertson–Walker (FLRW) metric. The metric of this BEC is characterized by its curvature κ\kappa κ κ and a time-dependent scale factor a ( t ), with an increase in the latter corresponding to an expansion of the analog FLRW universe. We study the propagation of classical phonons in such BECs, finding that a sudden change in the scale factor induces ripples in the wave motion. In addition, we study quantum phonon creation (or vacuum amplification) due to the scale-factor modification and quantify their entanglement.
We experimentally study cosmological particle production in a two-dimensional Bose-Einstein condensate, whose density excitations map to an analog cosmology. The expansion of spacetime is realized with tunable interactions. The particle spectrum can be understood through an analogy to quantum mechanical scattering, in which the dynamics of the spacetime metric determine the shape of the scattering potential. Hallmark scattering phenomena such as resonant forward scattering and Bragg reflection are connected to their cosmological counterparts, namely linearly expanding space and bouncing universes. We compare our findings to a theoretical description that extends beyond the acoustic approximation, which enables us to apply the model to high-momentum excitations.
Analog models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In this paper, we investigate the analog of cosmological particle creation in a Friedmann-Robertson-Walker universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we focus on a two-dimensional homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for various forms of the scaling function the particle production is consistent with the underlying theory in the long wavelength limit. In this context, we further discuss the implications of modified dispersion relations that arise from the microscopic theory of a weakly interacting Bose gas.
In the expanding universe, relativistic scalar fields are thought to be attenuated by friction, which results from the dilation of the underlying spacetime metric. By contrast, in a contracting universe this pseudo-friction would lead to amplification. Here, we experimentally measure both Hubble attenuation and amplification in expanding and contracting toroidally-shaped Bose-Einstein condensates, in which phonons are analogous to cosmological scalar fields. We find that the observed attenuation or amplification depends on the temporal phase of the phonon field, which is only possible for non-adiabatic dynamics, in contrast to the expanding universe in its current epoch, which is adiabatic. The measured strength of the Hubble friction disagrees with recent theory [J. M. Gomez Llorente and J. Plata, Phys. Rev. A 100 043613 (2019) and S. Eckel and T. Jacobson, SciPost Phys. 10 64 (2021)], suggesting that our model does not yet capture all relevant physics. While our current work focuses on coherent-state phonons, it can be extended to regimes where quantum fluctuations in causally disconnected regions of space become important, leading to spontaneous pair-production.
The production of quantum field excitations or particles in cosmological spacetimes is a hallmark prediction of curved quantum field theory. The generation of cosmological perturbations from quantum fluctuations in the early Universe constitutes an important application. The problem can be quantum-simulated in terms of structure formation in an interacting Bose-Einstein condensate (BEC) with time-dependent s-wave scattering length. Here, we explore a mapping between cosmological particle production in general (D+1D+1D+1)-dimensional spacetimes and scattering problems described by the nonrelativistic stationary Schr\"odinger equation in one dimension. Through this mapping, intuitive explanations for emergent spatial structures in both the BEC and the cosmological system can be obtained for analogue cosmological scenarios that range from power-law expansions to periodic modulations. The investigated cosmologies and their scattering analogs are tuned to be implemented in a (2+12+12+1)-dimensional quantum field simulator.