Numerical Results for BEC shells of arbitrary thickness

We complete our discussion of vorticity in three-dimensional BEC shells by presenting results of two numerical studies. They access more realistic physical models of the shell’s equilibrium density and complement the approximate approach presented above. Calculations of the ground state of the hollow condensate shell obtained by the imaginary-time method, previously discussed for defect-less BECs in Sec. 4.3, have been conducted by collaborator Prof. C. Lannert while work centering numerical solutions to the two-dimensional GP equation on a curved surface and the subsequent local density approximation (LDA) was carried out by collaborator Dr. K. Sun. Both approaches are presented and detailed in Ref. [79]. With these approaches, we aim to refine and strengthen our conclusions following Eq. (8.56) as well as the critical rotation rate estimate. Through imaginary time numerics we corroborate the fact that lower rotation rates result in nucleation of a vortex line in the case of hollow condensate shells, as opposed to fully-filled spherical BECs. The LDA further identifies a relationship between the critical rotation rate and condensate shell thickness.

Critical rotation speed from imaginary time numerics

Numerical solutions of the Gross-Pitaevskii equation found by using an imaginary-time method enable us to extend our analysis beyond the Thomas-Fermi approximation, and to consider shells of arbitrary thickness. We specifically seek numerical solutions corresponding to condensate shells having a vortex along the z-axis with vorticity `. The resulting energy equation for the condensate wavefunction magnitude,

|ψ|, can be minimized using an imaginary-time algorithm [104]. By using the bubble-trap of Eq. (4.3), Vbubble(r) = mω20p(r2− ∆)2/4 + Ω2, we numerically find the ground state wavefunction amplitude and

energy for a harmonically-trapped filled spherical condensate, as well as a shell of arbitrary size and thickness. To make contact with previous discussion, we present results concerning a fairly thin shell having thickness- to-radius ratio δ/R ≈ 0.3 and trap confinement frequency ωsh= ω0. We compare these results to the case

of a condensate sphere confined by the same frequency, ω0. In these calculations the effective interaction

constant U0is set by 8πN as/Sl= 10, 000 where Slis the oscillator length of the harmonic confining trap. In

other words, the numerical solutions obtained here correspond to the strong interaction regime of the BEC as well.

By numerically obtaining the ground state energy for the no-vortex (` = 0) and single-vortex (` = 1) cases, we find the critical rotation rate for nucleation of a vortex. For the thin (t/R ≈ 0.3) condensate shell, we find Ωsh

c = 0.02ω0, whereas for the fully-filled spherical BEC we find Ωspc = 0.2ω0. The factor of ten

difference in critical rotation speeds bears out and illustrates the much lower energy cost for a vortex in a hollow shell, compared with a filled sphere BEC. This conclusion is in agreement with the more analytical, albeit approximate, approach of Sec. 8.3.1.

Layered two-dimensional shells and the local density approximation

As a complementary calculation, in this Section, we build on the numerical scheme used in Sec. 8.2.2 and approximate a thick condensate shell as a series of concentric two-dimensional shells. We assume that the superfluid flow along the radial direction and any possible interactions between the layered two-dimensional shells are negligible. These assumptions constitute a local density approximation (LDA), schematically represented in Fig. 8.5.

We denote the shell thickness δ so that the concentric two-dimensional shells have radii ranging between the three-dimensional BEC’s inner (Thomas-Fermi) radius Rin= R − δ and its outer (Thomas-Fermi) radius

R. The energy functional is evaluated as

ELDA3D =

X

i

E2D(ri), (8.60)

where E_2D(ri) is the energy of each 2D shell, taking the same form as Eq. (8.38) with the shell radius being

equal to ri. The effective rotation speed for each two-dimensional shell is here scaled by its radius so that

˜

ΩLDA(ri) = (r2i/R

Figure 8.5: Schematic reperesentation of the local density approximation (LDA) discussed in Sec. 8.3.2. Here, a condensate shell having non-negligible thickness is approximated as concentric layers of two-dimensional condensate shells. We assume that flows and interactions between the layers are negligible.

The rotational term in the energy functional of each concentric layer therefore has r-dependence. Therefore, there ought to exist a radius rc, such that the effective rotation speed is large enough to stabilize a vortex pair

on the poles for the layers with ri > rc (the ‘outer’ layers) but not for those with ri < rc (the ‘inner’ layers).

Since the vortex line cannot break into parts, its stability is determined by the layers that energetically dominate and thus depends on both the rotation speed ˜Ω (which determines rc) and the shell thickness δ

(which determines the relative energetic contributions between the outer and inner layers).

Given a fixed rotation speed, the thicker the shell, the more the inner layers contribute to the system’s energy. Thus, we expect to identify a critical thickness beyond which the local energy minimum for a vortex line piercing through the poles disappears. Figure 8.6(a) shows the energy functional of shell BECs having various thicknesses at ˜Ω = 0.6 (with the vortex location fixed at θ = α on each 2D layer). The critical thickness above which the vortex line is no longer stable at the poles in this case corresponds to δ/R = 0.222.

Conversely, for a given thickness, the LDA calculations show a critical rotation speed that determines the stability of the vortex line through the poles. In Fig. 8.6(b), we plot the critical rotation speed as function of BEC shell thickness, and find an approximately linear relationship. This result suggests a non-destructive

way to experimentally probe the thickness of a BEC shell by finding the lowest rotation speed that stabilizes a single vortex aligned with the rotation axis and passing through the poles.

Figure 8.6: (a) Energy ELDA

3D [in Eq. (8.60)] of 3D shell BECs as a function of vortex location for various

shell thickness δ and ˜Ω = 0.6. The data are from the LDA calculation and are levelled up at α = 0 (b) Critical rotation speed vs shell thickness from the LDA calculation. The leftmost data point is for a pure 2D shell.

In connecting these results with realistic experiments, we note that the LDA ignores coupling between the two-dimensional layer shells. In other words, we ignore any possible radial superfluid flow and associated radial kinetic energy. For a realistic shell BEC having thickness near the critical value, we posit that the competition between the vortex’s outer and inner parts (with respect to the radial coordinate r) could not only reduces the force on the whole vortex but also cause bending or tilting of the line-like vortex cores. This effect, somewhat analogus to assigning the vortex core a tension along its length, might increase the life time of the vortex pair staying at a finite angle α in sufficiently thick condensate shells. However, for very thin shells, we do not expect such an effect as radial flow would increase condensate shell energy.

We complete this Section by summarizing our findings. First, the approximate method of “slicing and stacking” of two-dimensional planes in order to model a thin condensate shell and numerical solutions of the Gross-Pitaevskii equation found by an imaginary time algorithm both show that the rotation rate at which a vortex enters a thin condensate shell i.e. a vortex line is nucleated is smaller than in the case of a fully-filled, spherical condensate geometry. Numerical results concretely show this difference to be non- negligible. Further, numerical calculations within the LDA show that the stability of a vortex-antivortex pair at opposite poles of the condensate shell depends on both the rotation rate Ω and its thickness δ. While a vortex line extending along the rotation axis of a hollow BEC shell may nucleate for relatively low system rotation rates, its dynamical behavior and stability depend on the details of the hollow condensate geometry. In this way, as in our discussion of collective modes of hollow spherically-symetric condensates, the existence of the inner surface and the vanishing denisty at the condensate’s center (its finite thickness) significantly change the physics of the system when compared to similar fully-filled BECs.