Numerical energy estimates

In this Section, we numerically calculate the wavefunction ψ2D(θ, φ) of a rotating two-dimensional spherical

BEC. This work was primarily carried out by collaborator Dr. K. Sun and it consists of minimizing the energy functional that describes a two-dimensional condensate shell in its rotating frame. For shells rotating

at the angular rate Ω, we have E2D[ψ] = Z sin θdθdφ  |∇ψ2D| 2 +U2D 2 |ψ2D| 4 + ˜Ωψ_2D∗ (i∂φ)ψ2D  , (8.38)

where U2D is the effective interaction strength in two dimensions. Details of the minimization calculation,

addressing mathematical issues with the use of polar coordinates, and the determination of the U2D value

are presented in Appendix C. In our calculations, U2D is chosen so that the typical ratio of kinetic energy

to interaction energy is ∼ 5% i.e. we work in the (Thomas-Fermi) limit of strong inter-atomic interactions. This method – minimizing Eq. (8.38) – is equivalent to solving the GP equation, and we keep all the physical quantities dimensionless by using ER = ~2/(2M R2) as the unit of energy and BEC shell radius R as unit

length. As in previous Sections, we focus on the vortex-antivortex pair configuration. This structure is imposed by fixing the wavefunction zeros (vortex cores) at θ = α and θ = π − α respectively, as in Fig. 8.2. While Eq. (8.35) does account for the energy cost of maintaining vortex cores empty of condensate atoms, this minimization approach captures the details of vortex shape and its significance more precisely.

This feature of the numerical calculation enhances the understanding we have gained from the analytical approach leading to Eq. (8.35). Concretely, in Fig. 8.4, we compare energy values for a rotating two- dimensional condensate shell obtained through analytical estimates neglecting vortex cores in panel (a) and values of the energy functional obtained through GP equation calculations that include them in panel (b). In the former case, for clarity of comparison, Fig. 8.4 (a) shows the plot of the energy functional per particle in the rotating frame in a dimensionless form as

ε(α) = Evor−vor− Erot N ER

=1

2ln(cos α) − ˜Ω cos α, (8.39)

where N = 4πR2_ρ

2D is the total number of particles and ˜Ω = 2m_~ ΩR2 is a dimensionless angular velocity.

This curve and the numerically calculated one in Fig. 8.4 (b) exhibit same behaviors above and below a critical angular velocity ˜Ωc

α=0 ∼ 1

2. Hence, the GP approach confirms the stability of a vortex-antivortex

pair at poles of the shell (α = 0) when system rotation is fast enough.

Numerical results also show that the sharp decrease of the energy functional to the global minimum at the shell’s equator occurs at a larger α than implied by the analytic results in Eq. (8.39). In other words, a vortex-antivortex configuration could be stable against pair annihilation at the equator of the BEC shell for a larger range of α than analytical estimates of the previous Section may imply. We consequently conjecture that the effect of density depletion at the vortex cores moderates the intervortex interaction until the vortices are very close to each other. Notably, for angles near α = π, we cannot disregard vortex core effects, and

the energy functional of Eq. 8.39 is no longer appropriate. 0 10 20 30 40 50 60 0.00 0.05 0.10 0.15

α (Degree)

Ω=0.4-

0.5-

0.6-

α (Degree)

Ω=0.45-

0.49-

0.54_-

(a)

(b)

ε

(α

)

E

Figure 8.4: (a) Energy ε(α) in Eq. (8.39) of a rotating 2D spherical BEC for angular velocity ˜Ω above, equal to, and below the critical value ˜Ωc = 1₂. (b) Energy E2D in Eq. (8.38) obtained from the numerical

GP equation calculations. For a convenient comparison, the energy curves in both panels are levelled up at α = 0.

Focusing on the superfluid flow obtained from the GP wavefunction, for any finite α we find that the flow pattern resembles one that rotates about a string through both vortex cores, as illustrated in Fig. 8.2. We employ a variational approach to approximate the wavefunction by considering the shape of such a string, parameterized in a convenient way. This allows us to determine the condensate density and phase, and the associated superfluid flow. We set an arbitrary conic-section curve on the x-z plane (φ = 0 or π) through the vortex cores on the sphere which takes the general form z2_{= (λ − 1)x}2_{+ 2bx + (1 − λsin}2_{α − 2b sin α) with}

two variational parameters b and λ. We numerically minimize the energy functional of Eq. (8.38) with a variational wavefunction ψvar(θ, φ) = fvareiSvar, where the phase Svaris now the azimuthal angle with respect

to the variational curve, and the amplitude fvar has a variational depletion at the vortex cores. Specifically,

fvar(θ, φ) = A√ σα ζ2_+σα2×

σπ−α √

ζ2+σπ−α2

, where A is the normalization constant, ζ a variational parameter, and σα= cos−1(cos α cos θ +sin α sin θ cos φ) is the distance from the vortex core at α. We find that the minimum

energy state corresponds to a circle string (λ = 1) passing through the vortex cores and perpendicular to the sphere’s surface. Further, these variational results qualitatively agree with the GP results with respect to energy functional curves as well as critical rotation speed.

The variational assumption of a curved string connecting the vortex-antivortex pair relates our analysis to studies of vortex dynamics in filled 3D spherical BECs where a physical vortex string (line) may move off-axis [9, 39] or bend [64, 126, 127] due to interaction or density inhomogeneity effects. Namely, a vortex- antivortex pair on opposite poles of a 2D condensate shell corresponds to a straight on-axis vortex in a 3D BEC, while the pair at angle α > 0 corresponds to a bent vortex. This correspondence additionally informs the understanding of vortices on a spherically symmetric BEC undergoing dimensional crossover from a hollow 2D geometry to a shell-shaped condensate having finite thickness and, finally, to a fully filled

spherical BEC. Such crossover could be experimentally achieved by the bubble trap discussed in Sec. 4.2 and Ref. [98].