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arXiv:1505.00541

Gravity, Bose-Einstein Condensates and Gross-Pitaevskii Equation

Patrick Das Gupta

Published May 4, 2015• 0 citations
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Abstract

We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as $\approx 10^{20}$, there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a $1/r^3 $ gravity at short scales.

DOI
10.48550/arxiv.1505.00541

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We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as ≈1020\approx 10^{20}≈1020, there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a 1/r31/r^3 1/r3 gravity at short scales.
We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as ≈1020\approx 10^{20}≈1020, there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a $1/r^3 $ gravity at short scales.
We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small-amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as ≈10 20 , there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a 1/r 3 gravity at short scales.
We investigate the properties of modulational instability in the Salerno equation in quasi-one dimension in Bose-Einstein condensate (BEC). We analyzed the regions of modulational instability of nonlinear plane waves and determine the conditions of its existence in BEC. The existence and stability of strongly localized modes in discrete media is investigated with the framework of the Salerno model by using a linear analysis method. The regions of stability and instability are determined. Also the existence of localized modes for different values of parameters is shown numerically by homoclinic orbits intersection method. The response of Bose–Einstein condensates is studied with strong dipole–dipole atomic interactions to periodically varying perturbation. The dynamics is governed by the Gross–Pitaevskii equation with an additional nonlinear term, corresponding to nonlocal dipolar interactions. A mathematical model, based on the variational approximation, has been developed and applied to study parametric excitation of the condensates due to coefficient of nonlocal nonlinearity varying periodically. The model predicts the waveform of solitons in dipolar condensates and describes their small amplitude dynamics quite accurately. Theoretical predictions are verified by numerical simulations of the nonlocal Gross–Pitaevskii equation and a good agreement between them is found. The results can lead to better understanding of the properties of ultra-cold quantum gases, such as 52Cr, 164Dy and 168Er, where the long-range dipolar atomic interactions dominate the usual contact interactions. Dynamics of a matter wave soliton bouncing on the reflecting surface (atomic mirror) under the effect of gravity has been studied by analytical and numerical means. The iv analytical description is based on the variational approach. Resonant oscillations of the soliton's center of mass and width, induced by appropriate modulation of the atomic scattering length and the slope of the linear potential are analyzed. In numerical experiments, we observe the Fermi type acceleration of the soliton when the vertical position of the reflecting surface is periodically varied in time. Analytical predictions are compared with the results of numerical simulations of the Gross-Pitaevskii equation and a qualitative agreement between them is found.
Abstract. The motto of this paper is: Let's face Bose-Einstein condensation through nonUnear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose-Einstein condensates via the time-dependent Gross-Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonUnear dynamics. We consider in particular cold quantum gases where long-range interactions between the neutral atoms are present, in addition to the conventional short-range contact interaction, viz. gravity-Uke interactions, and dipole-dipole interactions. The residts obtained serve as a useful giude in the search for nonlinear dynamics effects in numerically exact quantum calcinations for Bose-Einstein condensates. A main residt is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which coidd be checked experimentally. Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, nonlinear dynamics effects PACS: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky 1.
Bose–Einstein condensates (BECs), a state of matter formed when a low-density gas of bosons is cooled to near absolute zero, continue to motivate novel work in theoretical and experimental physics. Although BECs are most commonly studied in stationary ground states, time-varying BECs arise when some aspect of the physics governing the condensate varies as a function of time. We study the evolution of time-varying BECs under non-autonomous Gross–Pitaevskii equations (GPEs) through a mix of theory and numerical experiments. We separately derive a perturbation theory (in the small-parameter limit) and a variational approximation for non-autonomous GPEs on generic bounded space domains. We then explore various routes to obtain time-varying BECs, starting with the more standard techniques of varying the potential, scattering length, or dispersion, and then moving on to more advanced control mechanisms such as moving the external potential well over time to move or even split the BEC cloud. We also describe how to modify a BEC cloud through evolution of the size or curvature of the space domain. Our results highlight a variety of interesting theoretical routes for studying and controlling time-varying BECs, lending a stronger theoretical formulation for existing experiments and suggesting new directions for future investigation.
We review our recent theoretical advances in the dynamics of Bose-Einstein condensates with tunable interactions using Feshbach resonance and external potential. A set of analytic and numerical methods for Gross-Pitaevskii equations are developed to study the nonlinear dynamics of Bose-Einstein condensates. Analytically, we present the integrable conditions for the Gross-Pitaevskii equations with tunable interactions and external potential, and obtain a family of exact analytical solutions for one- and two-component Bose-Einstein condensates in one and two-dimensional cases. Then we apply these models to investigate the dynamics of solitons and collisions between two solitons. Numerically, the stability of the analytic exact solutions are checked and the phenomena, such as the dynamics and modulation of the ring dark soliton and vector-soliton, soliton conversion via Feshbach resonance, quantized soliton and vortex in quasi-two-dimensional are also investigated. Both the exact and numerical solutions show that the dynamics of Bose-Einstein condensates can be effectively controlled by the Feshbach resonance and external potential, which offer a good opportunity for manipulation of atomic matter waves and nonlinear excitations in Bose-Einstein condensates.