As in Eq. (3.19) the lowest energy state available to the rotating shell, assuming the rotation is about the z-axis, depends on the expectation value of the angular momentum operator Lz. To calculate this quantity
we follow Ref. [115] and consider a single vortex close to the top of the condensate shell so that
hLzi =
Z
f∗e−iS(−i~∂φ)f eiS= ~ρ2D
Z Z
Since ~ v = ~ m∇S = ~ m 1 R∂θS ˆθ + ~ m 1 R sin θ∂φS ˆφ (8.29)
we can rewrite the previous equation as
hLzi = ~2ρ2D Z π 0 R2sin θdθ Z R sin θvφdφ = ~2ρ2D Z π 0 R2sin θdθ Z (d~l · ~v)φ (8.30)
where the last term is a loop around some fixed angle θ. If the vortex is situated at θ = θv then condensate
density should vanish for θ > θcwith θcreflecting the size of the vortex θc≈ ξ/2R (see Sec. 3.2). Otherwise,
the integral should simply produce 2π`. In other words,
hLzi = ~
m2π`ρ2DR
2cos θ
c (8.31)
For the equatorially symmetric vortex antivortex pair we scale Eq. (8.31) by a factor of two and conclude that
Erot= ΩhLzi = ~
m4π`Ωρ2DR
2cos θ
c (8.32)
This term has a geometrical interpretation
Erot= ~
mΩ`ρ2D(Atot− Avort) (8.33)
where Atot is the total surface area of the shell and Avort is the are taken up by the two vortices. More
precisely, the difference in Eq. (8.33) mutliplied by the (two-dimensional) condensate density gives the number of atoms that are rotating, taking into account density depletions at the positions of the vortices. Approximating vortices as point-like, we take
Erot≈ 4~
mπ`Ωρ2DR
2cos α. (8.34)
This expression also lends itself to a simple interpretation as cos α gives the fraction of angular momentum about the z-axis an atom at θ = α would locally experience as the whole condensate shell is rotating.
between the pair and the rotation term calculated in Eq. (8.34)
Etot= 2Ecore+ Evortex−vortex− Erot
Etot= 2π~ 2 m2 ρ2D` 2log(R/ξ) + π~ 2 2m2ρ2Dlog(1 + cos(2α)) − ~ mΩρ2D4πR 2cos α. (8.35)
Here, Ecore denotes the energy cost of maintaining the vortex core, as discussed in Sec. 3.2. This quantity
is set by the healing length of the condensate and depends on the type of atoms used and the inter-atomic interaction so we take it to be the same as in a flat BEC. It is further appropriate to make this approximation since we are assuming that all vortices being discussed are point-like or at least significantly smaller than the size of the condensate.
A plot of this energy, Eq. (8.35), as a function of α and three rotation rates Ω, is shown in Fig. 8.3. Depending on the rotation rate, it can have at most two distinct minima: one for the configuration where the vortex anti-vortex pair is aligned with the poles of the shell and one when α = π/2 i.e. the two defects overlap at the shell equator and the flows associated with them cancel out. There are no minima corresponding to intermediate α for which the vortex antivortex configuration could be stable.
While numerical studies [79] show that a vortex antivortex pair characterized by α 6= 0 is typically not the lowest energy state of the rotating condensate shell, if such a pair were engineered through some experimental technique any increase in rotation speed of the whole system would push them closer and closer to the rotation axis. This follows from considering the angular momentum term in Eq. (8.35). This term – ΩhLzi – decreases the total energy of the condensate the most when the vortex antivortex pair is situated
on respective poles of the condensate shell. When Ω is large, this term is large as well and the vortices align with the rotation axis. More intuitively, given the geometrical interpretation of Eq. (8.31), increasing ΩhLzi
increases the number of atoms rotating around the z-axis which accounts for the vortices (sourcing a flow that is “tilted” compared to a swirl about the z-axis) being pushed towards it.
More rigorously, we identify a critical rotation frequency Ωcα=0 at which condensate shell energy first
features a local minimum for α = 0, in addition to α = π/2. To that end we expand the total energy of the condensate hosting a vortex antivortex pair for small α. As the energy cost of maintaining the two vortex cores does not depend on α, this calculation captures exactly the competition between the rotation (pushing the vortices toward the rotation axis) and the vortex-vortex interaction (pulling the pair closer to each other). From the expansion
E = const +π~ρ2D m −~ 2m+ 2ΩR 2 α2+ − ~ 12m− 1 6ΩR 2 α4 (8.36)
Figure 8.3: Plots of total energy of a rotating two-dimensional condensate shell hosting a vortex anti-vortex pair at θ = α and θ = π − α for increasing values of rotation frequency. Here, the BEC shell has radius on the order of a micron and consists of N = 105/87 atoms. We assume that interactions between atoms are
strong and calculate ξ to be on the order of 10−3microns. The curve depicted in blue corresponds to half of the critical rotation frequency identified in Eq. (8.37), the one depicted in orange represents a BEC rotating exactly at that frequency while the curve in green shows the total energy for a rotation rate of Ω = 1.5Ωc
α=0.
We note that as rotation frequency Ω is increased through Ωc
α=0, the more favorable it becomes for the
vortex anti-vortex pair to be pinned at the poles i.e. for the flow that they produce to exactly match that due to the rotation.
it is clear that the coefficient of α2 changes sign and there is a local energy minimum for Ωcα=0R = 1
4 ~
mR. (8.37)
Hence, the vortex and the antivortex align with the rotation axis instead of annihilating at the shell’s equator once the linear speed associated with the rotation of the whole system (vlin = ΩR) is of the same
order of magnitude as the speed of condensate flow for a vortex antivortex pair positioned exactly along the z-axis (vvort
α=0= ~/mR). This is consistent with studies of two-dimensional rotating BECs where vortex
configurations mimicking rigid body rotation have been identified as most energetically favorable [68]. A numerical study complementary to the conclusions of this Section is summarized below.