Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

The second class of stationary solutions comprises configurations containing vortex lines
around which the circulation takes non-zero quantized values
dl · v = κ 2π¯h
, κ = ±1, ±2,... . (1.14)
m
The corresponding velocity field is irrotational except on the vortex line, where it is singular.
The density on the vortex line is zero, and the radius of the vortex core is of the order of the
healing length ξ =(8πan) −1/2 [19] (chapter III), where a is the scattering length characterizing
2-body interaction, and n is the density of the condensate in absence of the vortex. The
quantization of the circulation is a general property of superfluids. It is the consequence of
the existence of a single-valued order parameter [20, 21]. States with κ>1 are unstable and
fragment into κ vortices each with a unit quantum circulation (see [19], chapter III). In a
stationary condensate, a single vortex line passes through the center of the trap while several
vortex lines form a regular vortex lattice free of any major distortions, even near the boundary.
Such “Abrikosov” lattices were first predicted for superconductors [22]. Tkachenko showed
that their lowest energy structure should be triangular for an infinite system [23]. Stationary
vortex lattice configurations in dilute-gas Bose-Einstein condensates have for example been
studied theoretically in [24, 25, 26] (see also [27] and references therein). The experimental
generation of vortex states has been described in [28, 6, 7, 8, 9, 10, 12, 11, 29, 30, 13].
Vortex lattices containing up to ∼ 130 vortices have been reported [12]. Their life time
can extend up to the one of the condensate itself [13]. In most experiments vortices have
been identified by detecting the vortex cores in the density distribution after expansion (see
[31, 25, 32] for related theoretical calculations). A visibility of the density reduction of up
to 95% [13] has been reported. An alternative technique consists in the measurement of the
angular momentum [7, 10, 9]. This method exploits the fact that the quadrupole surface
modes with angular momentum ±2¯h are not degenerate in the presence of vortices [64]. In
this way it is possible to observe the jump of the angular momentum from 0 to ¯h per particle
when a vortex line moves from the edge of the cloud to its stable position at the center of the
trap [7]. The angular momentum associated with a single vortex line has also been measured
by exciting the scissors mode [33]. Moreover, phase singularities due to vortex excitations have
been observed as dislocations in the interference fringes formed by the stirred condensate and
a second unterperturbed condensate [34].
In the past years, issues of primary experimental interest have been the nucleation and
stabilization of vortices and vortex lattices (see next chapter), the properties of vortex lattices
made up of a large number of vortex lines [8, 12, 9, 11, 37, 35], the decay of vortex configurations
[6, 8, 12, 37, 13], the bending of vortex lines [11, 36], excitations of vortex lines [38, 29]
and of vortex lattices [39] and the behavior of vortex lattices under rapid rotation [40, 41, 42].
The theory of vortices in trapped dilute Bose-Einstein condensates has been reviewed in [27]
(see also [1, 15]).
A condensate which is initially in the groundstate of the non-rotating trap will evolve
according to (1.9) with Vext given by (1.3). To understand for what value of Ω the system
will start responding to the rotation, it is useful to study small perturbations δΨ(r,t) of the
groundstate Ψ0 = |Ψ0| exp(−iµ0t/¯h) in a static axi-symmetric trap (Ω =0,ε=0)
Ψ(r,t)=Ψ0 + δΨ(r,t) . (1.15)
Linearizing the GPE in the small perturbation δΨ(r,t) we can study the conditions under which
the system becomes unstable when the rotating trap is switched on. This analysis is done by
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