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arXiv:0903.2261 · Physical Review A

Stability of a Bose-Einstein condensate with higher-order interactions near a Feshbach resonance

Nikolaj Thomas Zinner, Mathias Kirk Thøgersen

Published Aug 14, 2009• 46 citations
arXivHTMLPDF

Abstract

We consider the stability of an ultracold trapped Bose-Einstein condensate near a Feshbach resonance. Using a modified Gross-Pitaevskii equation that includes higher-order terms and a multichannel model of Feshbach resonances, we find regions where stability can be enhanced or suppressed around experimentally measured resonances. We suggest a number of ways to probe the stability diagram. Using scattering length zero crossings huge deviations are found for the critical particle number. Effects are enhanced for narrow resonances or tighter traps. Macroscopic tunneling of the condensate is another possible probe for higher-order interactions; however, to see this requires very narrow resonances or very small particle numbers.

DOI
10.1103/physreva.80.023607

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We consider the stability of an ultracold trapped Bose-Einstein condensate near a Feshbach resonance. Using a modified Gross-Pitaevskii equation that includes higher-order terms and a multichannel model of Feshbach resonances, we find regions where stability can be enhanced or suppressed around experimentally measured resonances. We suggest a number of ways to probe the stability diagram. Using scattering length zero crossings huge deviations are found for the critical particle number. Effects are enhanced for narrow resonances or tighter traps. Macroscopic tunneling of the condensate is another possible probe for higher-order interactions; however, to see this requires very narrow resonances or very small particle numbers.
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 ≈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 take into account the higher-order corrections in two-body scattering interactions within a mean-field description, and investigate the stability conditions and collective excitations of a harmonically trapped Bose–Einstein condensate (BEC). Our results show that the presence of higher-order corrections causes drastic changes to the stability condition of a BEC. In particular, we predict that with the help of the higher-order interaction, a BEC can now collapse even for positive scattering lengths; whereas, a usually unstable BEC with a negative scattering length can be stabilized by positive higher-order effects. The low-lying collective excitations are significantly modified as well, compared to those without the higher-order corrections. The conditions for a possible experimental scenario are also proposed.
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.
We present semianalytical many-body results for energies and excitation frequencies for an inhomogeneous Bose-Einstein condensate over a wide range of atom numbers NNN for both small sss-wave scattering lengths, typical of most laboratory experiments, and large scattering lengths, achieved by tuning through a Feshbach resonance. Our dimensional perturbation treatment includes two-body correlations at all orders and yields analytical results through first order by taking advantage of the high degree of symmetry of the condensate at the zeroth-order limit. Because NNN remains a parameter in our analytical results, the challenge of calculating energies and excitation frequencies does not rise with the number of condensate atoms. In this proof-of-concept paper the atoms are confined in a spherical trap and are treated as hard spheres. Our many-body calculations compare well to Gross-Pitaevskii results in the weakly interacting regime and depart from the mean-field approximation as the density approaches the strongly interacting regime. The excitation frequencies provide a particularly sensitive test of beyond-mean-field corrections. For example, for N=2000N=2000N=2000 atoms and an experimentally realized large scattering length of $a=0.433{a}_{ho}\phantom{\rule{0.3em}{0ex}}({a}_{ho}=\sqrt{\ensuremath{\hbar}∕m{\ensuremath{\omega}}_{ho}})$ we predict a 7575%75 shift from the mean-field breathing mode frequency.