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arXiv:2304.01310

Constraints on short-range gravity with self-gravitating Bose-Einstein condensates

S. Gödtel, Cláus Lämmerzahl

Published Apr 3, 2023• 1 citation
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Abstract

In this work, we study low-lying collective excitations of a Bose-Einstein condensate with Newtonian and Yukawa-like two-particle interaction and derive boundaries for both Yukawa parameters. Using a variational approach, we explicitly show for spherical condensate that the corresponding frequencies depend on the gravitational interaction strength. The acquired results are presented in contour plots and compared to experimentally verified data from other tests. Furthermore, we discuss experimental requirements to test our theoretical model as well as possibilities to improve the boundaries. In addition, we consider axisymmetric condensates, where it turns out that disk-shaped BECs lead to better constraints. We also show that in theory we can determine the values for both Yukawa parameters independently by a measurement of at least two collective frequencies.

DOI
10.48550/arxiv.2304.01310

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In this work, we study low-lying collective excitations of a Bose-Einstein condensate with Newtonian and Yukawa-like two-particle interaction and derive boundaries for both Yukawa parameters. Using a variational approach, we explicitly show for spherical condensate that the corresponding frequencies depend on the gravitational interaction strength. The acquired results are presented in contour plots and compared to experimentally verified data from other tests. Furthermore, we discuss experimental requirements to test our theoretical model as well as possibilities to improve the boundaries. In addition, we consider axisymmetric condensates, where it turns out that disk-shaped BECs lead to better constraints. We also show that in theory we can determine the values for both Yukawa parameters independently by a measurement of at least two collective frequencies.
We study the ground-state properties of a Bose-Einstein condensate with short-range repulsion and gravitylike 1/r1/r1/r interatomic attraction in two-dimensions (2D). Using the variational approach we obtain the ground-state energy and analyze the stability of the condensate for a range of interaction strengths in 2D. We also determine the collective excitations at zero temperature using the time-dependent variational method. We analyze the properties of the Thomas-Fermi-gravity and gravity regimes, and we examine the vortex states, finding the coherence length and monopole mode frequency for these regimes. Our results are compared and contrasted with those in 3D condensates.
An approximate many-body theory incorporating two-body correlations has been employed to calculate low-lying collective multipole frequencies in a Bose–Einstein condensate containing A bosons, for different values of the interaction parameter . Significant difference from the variational estimate of the Gross–Pitaevskii equation has been found near the collapse region. This is attributed to two-body correlations and finite range attraction of the realistic interatomic interaction. A large deviation from the hydrodynamic model is also seen for the second monopole breathing mode and the quadrupole mode for large positive λ.
The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.
We analytically and numerically study the ground state and collective dynamics of Bose-Einstein condensates in two traps: a Newtonian potential and a logarithmic potential inspired by Modified Newtonian Dynamics (MOND). In the ground state, the MOND potential supports bound states only in the deep-MOND regime, where the condensate becomes significantly larger than its Newtonian counterpart. The size increases with repulsive coupling parameter βββ in both potentials. A clear scaling law of the size with β1/3β^{1/3}β1/3 emerges in the MOND case and is confirmed numerically over a wide parameter range, while for the Newtonian potential no simple scaling law exists as the Thomas-Fermi approximation ceases to be valid. For the dynamics, we derive and solve equations for the monopole collective mode. The larger MOND-bound condensate oscillates at a lower frequency, which scales as β−1/3β^{-1/3}β−1/3 in the strong-interaction limit. These scaling laws provide insights for quantum-simulation experiments aiming to probe modified-gravity scenarios with cold atoms.
Abstract We investigate the collective mode of a self-gravitating Bose–Einstein condensate (BEC) described by the Gross–Pitaevskii–Poisson (GPP) equations. The self-gravitating BEC has garnered considerable attention in cosmology and astrophysics, being proposed as a plausible candidate for dark matter. Our inquiry delves into the breathing and anisotropic collective modes by numerically solving the GPP equations and using the variational method. The breathing mode demonstrates a reduction in period with increasing total mass due to the density dependence of the self-gravitating BEC, attributed to the density-dependent nature of self-gravitating BECs, aligning quantitatively with our analytical findings. Additionally, we investigate an anisotropic collective mode in which the quadrupole mode intertwines with the breathing mode. The period of the quadrupole mode exhibits similar total mass dependence to that of the breathing mode. The characteristics of these periods differ from those of a conventional BEC confined by an external potential. Despite the differences in density dependence, the ratio of their periods equals that of the BEC confined by an isotropic harmonic potential. Furthermore, an extension of the variational method to a spheroidal configuration enables the isolation of solely the quadrupole mode from the anisotropic collective mode.
Abstract Ultra-cold atom experiment in space with microgravity allows for realization of dilute atomic-gas Bose-Einstein condensate (BEC) with macroscopically large occupation number and significantly long condensate lifetime, which allows for a precise measurement on the shape oscillation frequency by calibrating itself over numerous oscillation periods. In this paper, we propose to measure the Newtonian gravitational constant via ultra-cold atom BEC with shape oscillation, although it is experimentally challenging. We also make a preliminary perspective on constraining the modified Newtonian potential such as the power-law potential, Yukawa interaction, and fat graviton. A resolution of frequency measurement of (1−100) nHz(1-100)\,\mathrm {nHz}(1−100)nHz ( 1 - 100 ) nHz at most for the occupation number 10910^9109 10 9 , just one order above experimentally achievable number N∼106−108N\sim 10^6{-}10^8N∼106−108 N ∼ 10 6 - 10 8 , is feasible to constrain the modified Newtonian potential with Yukawa interaction greatly beyond the current exclusion limits.