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 multi-channel 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 founds 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.
Deep Dive into Stability of a BEC with Higher-order Interactions near a Feshbach Resonance.
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 multi-channel 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 founds 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.
Introduction The stability of Bose-Einstein condensates (BECs) in ultracold alkali gases is determined by the sign of the scattering length a [1]. For a < 0, one has effectively attractive interactions and the condensate will collapse to a dense state when the number of condensed particles N is larger than a critical number N c [1,2,3,4]. This has been beautifully demonstrated in experiments with 7 Li [5], 87 Rb [6,7], and recently with a dipolar 52 Cr BEC [8]. The findings indicate that the theory based on the surprisingly simple Gross-Pitaevskii (GP) equation can reproduce and describe most features of the experiments.
The GP equation includes two-body terms through a contact interaction which is parametrized by a. This is equivalent to a Born approximation but with an effective coupling that is obtained by replacing a born by the physical scattering length a. However, higher-order terms in the expansion of the phase shifts at low momenta, determined by the effective range r e , the shape parameter etc., give corrections to the simple GP equation. In this paper we explore the influence of the effective range term on the properties of a BEC. In particular, we show that the critical number of condensed atoms depends strongly on the higher-order scattering term when the scattering length approaches zero (zero-crossing). We also show how the macroscopic quantum tunneling (MQT) rate, in which the entire BEC tunnels as a coherent entity, can be modified for small condensate samples.
The considered effects depend on a combination of a and r e which yield different behavior for wide and narrow Feshbach resonances. Recent measurements on 39 K found many both wide and narrow resonances which allow for tuning of a over many orders of magnitude [9,10]. We therefore consider a selected example from 39 K in order to elucidate the general behavior for realistic experimental parameters.
The paper is organized as follows: We first introduce the modified GP equation with effective range dependence. Using both a variational and numerical approach we find a phase diagram describing the stability of the BEC. We then consider a Feshbach resonance model including effective range variations. The behavior of the critical particle number near a scattering length zerocrossing is derived. We discuss MQT and show numerically how the rate is modified. We finally discuss other possibilities for probing the higher-order interactions.
Modified GP Equation We assume that the condensate can be described by the GP equation and we focus on the a < 0 attractively interacting case. Since we are interested in the ultracold regime, where the temperature is much smaller than the critical temperature for condensation, we adopt the T = 0 formalism. In order to include higher-order effects in the two-body scattering dynamics, we use the modified GP equation derived in [11] for which the equivalent energy functional is
where m is the atomic mass, V is the external trap, U 0 = 4π 2 a/m, and g 2 = a 2 /3 -ar e /2 with a and r e being the s-wave scattering length and effective range, respectively. We are interested in the stability properties of the ground-state and we therefore perform a variational calculation using the mean-field trial wavefunction
where q is the dimensionless variational parameter and b = /mω is the trap length. The normalization is N = dr|Ψ(r)| 2 . For simplicity we only consider isotropic traps with V (r) = 1 2 mω 2 r 2 . However, the effects found should hold for deformed traps as well (along the lines of the analysis in [12]). The variational energy is E(q) N ω = 3 4 q 2 + 3 4
In Fig. 1 we plot E(q) for different parameters. As shown, there are many possibilities for g 2 = 0, including stable, unstable, and metastable systems. We see that the g 2 term modifies the barrier for N |a|/b = 0.5, implying that tunneling rates will be altered. For N |a|/b = 0.7, the g 2 = 0 case has no barrier at all, and here the g 2 term can in fact produce a small barrier on its own. The q -5 dependence of the term means that the effect is small. However, the plot clearly shows that a new stability analysis is needed. In addition to the variational approach we have numerically solved the full time-independent GP equation corresponding to eq. ( 1).
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FIG. 1: Energy of a BEC with fixed N |a|/b as function of the variational parameter q, i.e. the size of the condensate. The higher-order interaction term g2 modifies the height and shape of the barrier.
To determine the groundstate stability one looks for the vanishing of the barrier towards q = 0. For g 2 = 0 eq. ( 3) leads to N c |a|/b ≈ 0.671 [1]. The full integration of the GP equation gives N c |a|/b ≈ k 0 , k 0 = 0.5746 [2,3]. These values are indicated by filled points in Fig. 2.
In the general g 2 = 0 case we first take the variational energy eq. ( 3) and solve for multiple roots of dE(q)/dq. The resulting “phase diagram” is plotted in Fig. 2 (solid line). In the upper le
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