arXiv:0704.0146v1  [cond-mat.other]  2 Apr 2007
Vortices in Bose-Einstein Condensates: Theory
N. G. Parker1, B. Jackson2, A. M. Martin1, and C. S. Adams3
1 School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia.
ngparker@ph.unimelb.edu.au,amm@ph.unimelb.edu.au
2 School of Mathematics and Statistics, Newcastle University, Newcastle upon
Tyne, NE1 7RU, United Kingdom. brian.jackson@newcastle.ac.uk
3 Department of Physics, Durham University, South Road, Durham, DH1 3LE,
United Kingdom. c.s.adams@durham.ac.uk
1 Quantized vortices
Vortices are pervasive in nature, representing the breakdown of laminar fluid
flow and hence playing a key role in turbulence. The fluid rotation associated
with a vortex can be parameterized by the circulation Γ =
dr · v(r) about
the vortex, where v(r) is the fluid velocity field. While classical vortices can
take any value of circulation, superfluids are irrotational, and any rotation
or angular momentum is constrained to occur through vortices with quan-
tized circulation. Quantized vortices also play a key role in the dissipation of
transport in superfluids. In BECs quantized vortices have been observed in
several forms, including single vortices [1, 2], vortex lattices [3, 4, 5, 6] (see
also Chap. VII), and vortex pairs and rings [7, 8, 9]. The recent observation
of quantized vortices in a fermionic gas was taken as a clear signature of the
underlying condensation and superfluidity of fermion pairs [10]. In addition to
BECs, quantized vortices also occur in superfluid Helium [11, 12], nonlinear
optics, and type-II superconductors [13].
1.1 Theoretical Framework
Quantization of circulation
Quantized vortices represent phase defects in the superfluid topology of the
system. Under the Madelung transformation, the macroscopic condensate
‘wavefunction’ ψ(r, t) can be expressed in terms of a fluid density n(r, t) and a
macroscopic phase S(r, t) via ψ(r) =
n(r, t) exp[iS(r, t)]. In order that the
wavefunction remains single-valued, the change in phase around any closed
contour C must be an integer multiple of 2π,
∇S · dl = 2πq, (1)
http://arXiv.org/abs/0704.0146v1
2 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
where q is an integer. The gradient of the phase S defines the superfluid
velocity via v(r, t) = (h̄/m)∇S(r, t). This implies that the circulation about
the contour C is given by,
v · dl = q
. (2)
In other words, the circulation of fluid is quantized in units of (h/m). The
circulating fluid velocity about a vortex is given by v(r, θ) = qh̄/(mr)θ̂, where
r is the radius from the core and θ̂ is the azimuthal unit vector.
Theoretical model
The Gross-Pitaevskii equation (GPE) provides an excellent description of
BECs at the mean-field level in the limit of ultra-cold temperature [14]. It
supports quantized vortices, and has been shown to give a good description of
the static properties and dynamics of vortices [14, 15]. Dilute BECs require a
confining potential, formed by magnetic or optical fields, which typically varies
quadratically with position. We will assume an axially-symmetric harmonic
trap of the form V = 1
m(ω2rr
2 + ω2zz
2), where ωr and ωz are the radial and
axial trap frequencies respectively. Excitation spectra of BEC states can be
obtained using the Bogoliubov equations, and specify the stability of station-
ary solutions of the GPE. For example, the presence of the so-called anomalous
modes of a vortex in a trapped BEC are indicative of their thermodynamic
instability. The GPE can also give a qualitative, and sometimes quantitative,
understanding of vortices in superfluid Helium [11, 12].
Although this Chapter deals primarily with vortices in repulsively-inte-
racting BECs, vortices in attractively-interacting BECs have also received
theoretical interest. The presence of a vortex in a trapped BEC with attractive
interactions is less energetically favorable than for repulsive interactions [16].
Indeed, a harmonically-confined attractive BEC with angular momentum is
expected to exhibit a center-of-mass motion rather than a vortex [17]. The
use of anharmonic confinement can however support metastable vortices, as
well as regimes of center-of-mass motion and instability [18, 19, 20].
Various approximations have been made to incorporate thermal effects
into the GPE to describe vortices at finite temperature (see also Chap. XI).
The Popov approximation self-consistently couples the condensate to a normal
gas component using the Bogoliubov-de-Gennes formalism [21] (cf. Chap. I
Sec. 5.2). Other approaches involve the addition of thermal/quantum noise to
the system, such as the stochastic GPE method [22, 23, 24] and the classical
field/truncated Wigner methods [25, 26, 27, 28]. Thermal effects can also be
simulated by adding a phenomenological dissipation term to the GPE [29].
Basic properties of vortices
In a homogeneous system, a quantized vortex has the 2D form,
Vortices in Bose-Einstein Condensates: Theory 3
ψ(r, θ) =
nv(r) exp(iqθ). (3)
The vortex density profile nv(r) has no analytic solution, although approx-
imate solutions exist [30]. Vortex solutions can be obtained numerically by
propagating the GPE in imaginary time (t→ −it) [31], whereby the GPE con-
verges to the lowest energy state of the system (providing it is stable). By en-
forcing the phase distribution of Eq. (3), a vortex solution is generated. Figure
1 shows the solution for a q = 1 vortex at the center of a harmonically-confined
BEC. The vortex consists of a node of zero density with a width characterized
by the condensate healing length ξ = h̄/
mn0g, where g = 4πh̄
2a/m (with a
the s-wave scattering length) and n0 is the peak density in the absence of the
vortex. For typical BEC parameters [3], ξ ∼ 0.2 µm. For a q = 1 vortex at
the center of an axially-symmetric potential, each particle carries h̄ of angular
momentum. However, if the vortex is off-center, the angular momentum per
particle becomes a function of position [15].
1.2 Vortex structures
Increasing the vortex charge widens the core due to centrifugal effects. In
harmonically-confined condensates a multiply-quantized vortex with q > 1 is
energetically unfavorable compared to a configuration of singly-charged vor-
tices [32, 33]. Hence, a rotating BEC generally contains an array of singly-
charged vortices in the form of a triangular Abrikosov lattice [3, 4, 5, 6, 34]
(see also Chap. VII), similar to those found in rotating superfluid helium
[11]. A q > 1 vortex can decay by splitting into singly-quantized vortices via
a dynamical instability [35, 36], but is stable for some interaction strengths
[37]. Multiply-charged vortices are also predicted to be stabilized by a suitable
localized pinning potential [38] or the addition of quartic confinement [33].
Two-dimensional vortex-antivortex pairs (i.e. two vortices with equal but
opposite circulation) and 3D vortex rings arise in the dissipation of superflow,
and represent solutions to the homogeneous GPE in the moving frame [39, 40],
with their motion being self-induced by the velocity field of the vortex lines.
When the vortex lines are so close that they begin to overlap, these states are
no longer stable and evolves into a rarefaction pulse [39].
Having more than one spin component in the BECs (cf. Chap. IX) pro-
vides an additional topology to vortex structures. Coreless vortices and vortex
‘molecules’ in coupled two-component BECs have been probed experimentally
[41] and theoretically [42]. More exotic vortex structures such as skyrmion ex-
citations [43] and half-quantum vortex rings [44] have also been proposed.
2 Nucleation of vortices
Vortices can be generated by rotation, a moving obstacle, or phase imprinting
methods. Below we discuss each method in turn.
4 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
2.1 Rotation
As discussed in the previous section, a BEC can only rotate through the
existence of quantized vortex lines. Vortex nucleation occurs only when the
rotation frequency Ω of the container exceeds a critical value Ωc [15, 32, 46].
Consider a condensate in an axially-symmetric trap which is rotating about
the z-axis at frequency Ω. In the Thomas-Fermi limit, the presence of a vortex
becomes energetically favorable when Ω exceeds a critical value given by [47],
0.67R
. (4)
This is derived by integrating the kinetic energy density mn(r)v(r)2/2 of the
vortex velocity field in the radial plane. The lower and upper limits of the
integration are set by the healing length ξ and the BEC Thomas-Fermi radius
R, respectively. Note that Ωc < ωr for repulsive interactions, while Ωc > ωr
for attractive interactions [16]. In a non-rotating BEC the presence of a vortex
raises the energy of the system, indicating thermodynamic instability [48].
In experiments, vortices are formed only when the trap is rotated at a
much higher frequency than Ωc [3, 4, 5], demonstrating that the energetic
criterion is a necessary, but not sufficient, condition for vortex nucleation.
There must also be a dynamic route for vorticity to be introduced into the
condensate, and hence Eq. (4) provides only a lower bound for the critical
frequency.
The nucleation of vortices in rotating trapped BECs appears to be linked to
instabilities of collective excitations. Numerical simulations based on the GPE
have shown that once the amplitude of these excitations become sufficiently
large, vortices are nucleated that subsequently penetrate the high-density bulk
of the condensate [23, 27, 29, 49, 50].
One way to induce instability is to resonantly excite a surface mode by
adding a rotating deformation to the trap potential. In the limit of small
perturbations, this resonance occurs close to a rotation frequency Ωr = ωℓ/ℓ,
where ωℓ is the frequency of a surface mode with multipolarity ℓ. In the
Thomas-Fermi limit, the surface modes satisfy ωℓ =
ℓωr [51], so Ωr =
ℓ. For example, an elliptically-deformed trap, which excites the ℓ = 2
quadrupole mode, would nucleate vortices when rotated at Ωr ≈ ωr/
This value has been confirmed in both experiments [3, 4, 5] and numerical
simulations [23, 27, 29, 49, 50]. Higher multipolarities were resonantly excited
in the experiment of Ref. [6], finding vortex formation at frequencies close to
the expected values, Ω = ωr/
ℓ, and lending further support to this picture.
A similar route to vortex nucleation is revealed by considering stationary
states of the BEC in a rotating elliptical trap, which can be obtained in the
Thomas-Fermi limit by solving hydrodynamic equations [52]. At low rotation
rates only one solution is found; however at higher rotations (Ω > ωr/
bifurcation occurs and up to three solutions are present. Above the bifurcation
point one or more of the solutions become dynamically unstable [53], leading
Vortices in Bose-Einstein Condensates: Theory 5
to vortex formation [54]. Madison et al. [55] followed these stationary states
experimentally by adiabatically introducing trap ellipticity and rotation, and
observed vortex nucleation in the expected region.
Surface mode instabilities can also be induced at finite temperature by
the presence of a rotating noncondensed “thermal” cloud. Such instabilities
occur when the thermal cloud rotation rate satisfies Ω > ωℓ/ℓ [56]. Since all
modes can potentially be excited in this way, the criterion for instability and
hence vortex nucleation becomes Ωc > min(ωℓ/ℓ), analogous to the Landau
criterion. Note that such a minimum exists at Ωc > 0 since the Thomas-Fermi
result ωℓ =
ℓωr becomes less accurate for high ℓ [57]. This mechanism may
have been important in the experiment of Haljan et al. [34], where a vortex
lattice was formed by cooling a rotating thermal cloud to below Tc.
2.2 Nucleation by a moving object
Vortices can also be nucleated in BECs by a moving localized potential. This
problem was originally studied using the GPE for 2D uniform condensate flow
around a circular hard-walled potential [58, 59], with vortex-antivortex pairs
being nucleated when the flow velocity exceeded a critical value.
In trapped BECs a similar situation can be realized using the optical dipole
force from a laser, giving rise to a localized repulsive Gaussian potential. Under
linear motion of such a potential, numerical simulations revealed vortex pair
formation when the potential is moved at a velocity above a critical value [60].
The experiments of [61, 62] oscillated a repulsive laser beam in an elongated
condensate. Although vortices were not observed directly, the measurement
of condensate heating and drag above a critical velocity was consistent with
the nucleation of vortices [63].
An alternative approach is to move the laser beam potential in a circular
path around the trap center [64]. By “stirring” the condensate in this way one
or more vortices can be created. This technique was used in the experiment
of Ref. [6], where vortices were generated even at low stirring frequencies.
2.3 Other mechanisms and structures
A variety of other schemes for vortex creation have been suggested. One of
the most important is that by Williams and Holland [65], who proposed a
combination of rotation and coupling between two hyperfine levels to create
a two-component condensate, one of which is in a vortex state. The non-
vortex component can then either be retained or removed with a resonant
laser pulse. This scheme was used by the first experiment to obtain vortices
in BEC [1]. A related method, using topological phase imprinting, has been
used to experimentally generate multiply-quantized vortices [66].
Apart from the vortex lines considered so far, vortex rings have also been
the subject of interest. Rings are the decay product of dynamically unstable
dark solitary waves in 3D geometries [7, 8, 67, 68]. Vortex rings also form
6 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
in the quantum reflection of BECs from surface potentials [69], the unstable
motion of BECs through an optical lattice [70], the dragging of a 3D object
through a BEC [71], and the collapse of ultrasound bubbles in BECs [72].
The controlled generation of vortex rings [73] and multiple/bound vortex ring
structures [74] have been analyzed theoretically.
A finite temperature state of a quasi-2D BEC, characterized by the ther-
mal activation of vortex-antivortex pairs, has been simulated using classical
field simulations [75]. This effect is thought to be linked to the Berezinskii-
Kosterlitz-Thouless phase transition of 2D superfluids, recently observed ex-
perimentally in ultracold gases [76]. Similar simulations in a 3D system have
also demonstrated the thermal creation of vortices [77, 78].
3 Dynamics of vortices
The study of vortex dynamics has long been an important topic in both clas-
sical [79] and quantum [12] hydrodynamics. Helmholtz’s theorem for uniform,
inviscid fluids, which is also applicable to quantized vortices in superfluids
near zero temperature, states that the vortex will follow the motion of the
background fluid. So, for example, in a superfluid with uniform flow velocity
vs, a single straight vortex line will move with velocity vL, such that it is
stationary in the frame of the superfluid.
Vortices similarly follow the “background flow” originating from circulat-
ing fluid around a vortex core. Hence vortex motion can be induced by the
presence of other vortices, or by other parts of the same vortex line when it is
curved. Most generally, the superfluid velocity vi due to vortices at a partic-
ular point r is given by the Biot-Savart law [12], in analogy with the similar
equation in electromagnetism,
(s − r) × ds
|s− r|3
; (5)
where s(ζ, t) is a curve representing the vortex line with ζ the arc length.
Equation (5) suffers from a divergence at r = s, so in calculations of vortex
dynamics this must be treated carefully [80]. Equation (5) also assumes that
the vortex core size is small compared to the distance between vortices. In
particular, it breaks down when vortices cross during collisions, where recon-
nection events can occur. These reconnections can either be included manually
[81], or by solving the full GPE [82]. The latter method also has the advantage
of including sound emission due to vortex motion or reconnections [83, 84].
In a system with multiple vortices, motion of one vortex is induced by the
circulating fluid flow around other vortices, and vice-versa [11]. This means
that, for example, a pair of vortices of equal but opposite charge will move
linearly and parallel to each other with a velocity inversely proportional to
the distance between them. Two or more vortices of equal charge, meanwhile,
Vortices in Bose-Einstein Condensates: Theory 7
will rotate around each other, giving rise to a rotating vortex lattice as will be
discussed in Chap. VII. When a vortex line is curved, circulating fluid from
one part of the line can induce motion in another. This effect can give rise to
helical waves on the vortex, known as Kelvin modes [85]. It also has interesting
consequences for a vortex ring, which will travel in a direction perpendicular
to the plane of the ring, with a self-induced velocity that decreases with in-
creasing radius. Classically, this is most familiar in the motion of smoke rings,
though similar behavior has also been observed in superfluid helium [86].
This simple picture is complicated in the presence of density inhomo-
geneities or confining walls. In a harmonically-trapped BEC the density is
a function of position, and therefore the energy, E, of a vortex will also de-
pend on its position within the condensate. To simplify matters, let us con-
sider a quasi-2D situation, where the condensate is pancake-shaped and the
vortex line is straight. In this case, the energy of the vortex depends on its
displacement r from the condensate center [87], and a displaced vortex feels a
force proportional to ∇E. This is equivalent to a Magnus force on the vortex
[88, 89, 90] and to compensate the vortex moves in a direction perpendicular
to the force, leading it to precess around the center of the condensate along a
line of constant energy. This precession of a single vortex has been observed
experimentally [2], with a frequency in agreement with theoretical predictions.
In more 3D situations, such as spherical or cigar-shaped condensates, the vor-
tex can bend [91, 92, 93, 94] leading to more complicated motion [15]. Kelvin
modes [95, 96] and vortex ring dynamics [88] are also modified by the density
inhomogeneity in the trap.
In the presence of a hard-wall potential, a new constraint is imposed such
that the fluid velocity normal to the wall must be zero, vs ·n̂ = 0. The resulting
problem of vortex motion is usually solved mathematically [79] by invoking
an “image vortex” on the other side of the wall (i.e. in the region where there
is no fluid present), at a position such that its normal flow cancels that of the
real vortex at the barrier. The motion of the real vortex is then simply equal
to the induced velocity from the image vortex circulation.
4 Stability of vortices
4.1 Thermal instabilities
At finite temperatures the above discussion is modified by the thermal oc-
cupation of excited modes of the system, which gives rise to a noncondensed
normal fluid in addition to the superfluid. A vortex core moving relative to the
normal fluid scatters thermal excitations, and will therefore feel a frictional
force leading to dissipation. This mutual friction force can be written as [11],
fD = −nsΓ{αs′ × [ s′ × (vn − vL)] + α′s′ × (vn − vL)}, (6)
where ns is the background superfluid density, s
′ is the derivative of s with
respect to arc length ζ, α and α′ are temperature dependent parameters,
8 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
while vL and vn are the velocities of the vortex line and normal fluid respec-
tively. The mutual friction therefore has two components perpendicular to the
relative velocity vn − vL.
To consider an example discussed in the last section, an off-center vortex in
a trapped BEC at zero temperature will precess such that its energy remains
constant. In the presence of a non-condensed component, however, dissipation
will lead to a loss of energy. Since the vortex is topological it cannot simply
vanish, so this lost energy is manifested as a radial drift of the vortex towards
lower densities. In Eq. (6) the α term is responsible for this radial motion,
while α′ changes the precession frequency. The vortex disappears at the edge
of the condensate, where it is thought to decay into elementary excitations
[97]. Calculations based upon the stochastic GPE have shown that thermal
fluctuations lead to an uncertainty in the position of the vortex, such that
even a central vortex will experience thermal dissipation and have a finite
lifetime [24]. This thermodynamic lifetime is predicted to be of the order of
seconds [97], which is consistent with experiments [1, 3, 94].
4.2 Hydrodynamic instabilities
Experiments indicate that the crystallization of vortex lattices is temperature-
independent [5, 98]. Similarly, vortex tangles in turbulent states of superfluid
Helium have been observed to decay at ultracold temperature, where thermal
dissipation is virtually nonexistent [99]. These results highlight the occurrence
of zero temperature dissipation mechanisms, as listed below.
Instability to acceleration
The topology of a 2D homogeneous superfluid can be mapped on to a (2+1)D
electrodynamic system, with vortices and phonons playing the role of charges
and photons respectively [100]. Just as an accelerating electron radiates ac-
cording to the Larmor acceleration squared law, a superfluid vortex is inher-
ently unstable to acceleration and radiates sound waves.
Vortex acceleration can be induced by the presence of an inhomogeneous
background density, such as in a trapped BEC. Sound emission from a vortex
in a BEC can be probed by considering a trap of the form [45],
Vext = V0
1 − exp
mω2rr
2. (7)
This consists of a gaussian dimple trap with depth V0 and harmonic frequency
component ωd, embedded in an ambient harmonic trap of frequency ωr. A 2D
description is sufficient to describe this effect. This set-up can be realized with
a quasi-2D BEC by focussing a far-off-resonant red-detuned laser beam in the
center of a magnetic trap. The vortex is initially confined in the inner region,
where it precesses due to the inhomogeneous density. Since sound excitations
Vortices in Bose-Einstein Condensates: Theory 9
Fig. 1. Profile of a singly-quantized (q = 1) vortex at the center of a harmonically-
confined BEC: (a) condensate density along the y = 0 axis (solid line) and the
corresponding density profile in the absence of the vortex (dashed line). (b) 2D
density and (c) phase profile of the vortex state. These profiles are calculated nu-
merically by propagating the 2D GPE in imaginary time subject to an azimuthal
2π phase variation around the trap center.
−6 −4 −2 0 2 4 6
x (ξ)
(i) (ii)
Fig. 2. Vortex path in the dimple trap geometry of Eq. (7) with ωd = 0.28(c/ξ).
Deep V0 = 10µ dimple (dotted line): mean radius is constant, but modulated by the
sound field. Shallow V0 = 0.6µ dimple and homogeneous outer region ωr = 0 (dot-
ted line): vortex spirals outwards. Outer plots: Sound excitations (with amplitude
∼ 0.01n0) radiated in the V0 = 0.6µ system at times indicated. Top: Far-field distri-
bution [−90, 90]ξ×[−90, 90]ξ. Bottom: Near-field distribution [−25, 25]ξ×[−25, 25]ξ,
with an illustration of the dipolar radiation pattern. Copyright (2004) by the Amer-
ican Physical Society [45].
have an energy of the order of the chemical potential µ, the depth of the dimple
relative to µ leads to two distinct regimes of vortex-sound interactions.
V0 ≫ µ: The vortex effectively sees an infinite harmonic trap - it precesses
and radiates sound but there is no net decay due to complete sound reabsorp-
tion. However, a collective mode of the background fluid is excited, inducing
slight modulations in the vortex path (dotted line in Fig 2).
V0 < µ: Sound waves are radiated by the precessing vortex. Assuming ωr =
0, the sound waves propagate to infinity without reinteracting with the vortex.
10 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
The ensuing decay causes the vortex to drift to lower densities, resulting in a
spiral motion (solid line in Fig. 2), similar to the effect of thermal dissipation.
The sound waves are emitted in a dipolar radiation pattern, perpendicularly
to the instantaneous direction of motion (subplots in Fig. 2), with a typical
amplitude of order 0.01n0 and wavelength λ ∼ 2πc/ωV [15], where c is the
speed of sound and ωV is the vortex precession frequency. The power radiated
from a vortex can be expressed in the form [45, 101, 102],
P = βmN
, (8)
where a is the vortex acceleration, N is the total number of atoms, and β is
a dimensionless coefficient. Using classical hydrodynamics [101] and by map-
ping the superfluid hydrodynamic equations onto Maxwell’s electrodynamic
equations [102], it has been predicted that β = π2/2 under the assumptions
of a homogeneous 2D fluid, a point vortex, and perfect circular motion. Full
numerical simulations of the GPE based on a realistic experimental scenario
have derived a coefficient of β ∼ 6.3 ± 0.9 (one standard deviation), with the
variation due to a weak dependence on the geometry of the system [45].
When ωr 6= 0, the sound eventually reinteracts with the vortex, slowing but
not preventing the vortex decay. By varying V0 it is possible to control vortex
decay, and in suitably engineered traps this decay mechanism is expected to
dominate over thermal dissipation [45].
Vortex acceleration (and sound emission) can also be induced by the pres-
ence of other vortices. A co-rotating pair of two vortices of equal charge has
been shown to decay continuously via quadrupolar sound emission, both an-
alytically [103] and numerically [104]. Three-body vortex interactions in the
form of a vortex-antivortex pair incident on a single vortex have also been sim-
ulated numerically, with the interaction inducing acceleration in the vortices
with an associated emission of sound waves [104].
Simulations of vortex lattice formation in a rotating elliptical trap show
that vortices are initially nucleated in a turbulent disordered state, before
relaxing into an ordered lattice [50]. This relaxation process is associated
with an exchange of energy from the sound field to the vortices due to these
vortex-sound interactions. This agrees with the experimental observation that
vortex lattice formation is insensitive to temperature [5, 98].
Kelvin wave radiation and vortex reconnections
In 3D a Kelvin wave excitation will induce acceleration in the elements of
the vortex line, and therefore local sound emission. Indeed, simulations of
the GPE in 3D have shown that Kelvin waves excitations on a vortex ring
lead to a decrease in the ring size, indicating the underlying radiation process
[84]. Kelvin wave excitations can be generated from a vortex line reconnection
[83, 84] and the interaction of a vortex with a rarefaction pulse [105].
Vortices in Bose-Einstein Condensates: Theory 11
Vortex lines which cross each other can undergo dislocations and reconnec-
tions [106], which induce a considerable burst of sound emission [83]. Although
they have yet to be probed experimentally in BECs, vortex reconnections are
hence thought to play a key role in the dissipation of vortex tangles in Helium
II at ultra-low temperatures [11].
5 Dipolar BECs
A BEC has recently been formed of chromium atoms [107], which feature a
large dipole moment. This opens the door to studying of the effect of long-
range dipolar interactions in BECs.
5.1 The Modified Gross-Pitaevskii Equation
The interaction potential Udd(r) between two dipoles separated by r, and
aligned by an external field along the unit vector ê is given by,
Udd(r) =
êiêj
(δij − 3r̂ir̂j)
. (9)
For low energy scattering of two atoms with dipoles induced by a static electric
field E = Eê, the coupling constant Cdd = E
2α2/ǫ0 [108, 109], where α is the
static dipole polarizability of the atoms and ǫ0 is the permittivity of free space.
Alternatively, if the atoms have permanent magnetic dipoles, dm, aligned in
an external magnetic field B = Bê, one has Cdd = µ0d
m [110], where µ0 is the
permeability of free space. Such dipolar interactions give rise to a mean-field
potential
Φdd(r) =
d3rUdd (r − r′) |ψ (r′) |2, (10)
which can be incorporated into the GPE to give,
ih̄ψt =
∇2 + g|ψ|2 + Φdd + V
ψ. (11)
For an axially-symmetric quasi-2D geometry (ωz ≫ ωr) rotating about the
z -axis, the ground state wavefunction of a single vortex has been solved numer-
ically [111]. Considering 105 chromium atoms and ωr = 2π × 100Hz, several
solutions were obtained depending on the strength of the s-wave interactions
and the alignment of the dipoles relative to the trap.
For the case of axially-polarized dipoles the most striking results arise
for attractive s-wave interactions g < 0. Here the BEC density is axially
symmetric and oscillates in the vicinity of the vortex core. Similar density
oscillations have been observed in numerical studies of other non-local inter-
action potentials, employed to investigate the interparticle interactions in 4He
12 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
[112, 113, 114, 115], with an interpretation that relates to the roton structure
in a superfluid [115]. For the case of transversely-polarized dipoles, where the
polarizing field is co-rotating with the BEC, and repulsive s-wave interactions
(g > 0), the BEC becomes elongated along the axis of polarization [116] and
as a consequence the vortex core is anisotropic.
5.2 Vortex Energy
Assuming a dipolar BEC in the TF limit (cf. Sec. 5.1 in Chap. I), the en-
ergetic cost of a vortex, aligned along the axis of polarization (z-axis), has
been derived using a variational ansatz for the vortex core [117], and thereby
the critical rotation frequency Ωc at which the presence of a vortex becomes
energetically favorable has been calculated. For an oblate trap (ωr < ωz),
dipolar interactions decrease Ωc, while for prolate traps (ωr > ωz) the pres-
ence of dipolar interactions increases Ωc. A formula resembling Eq. (4) for
the critical frequency of a conventional BEC can be used to explain these
results, with R being the modified TF radius of the dipolar BEC. Indeed,
using the TF radius of a vortex-free dipolar BEC [118, 119] and the conven-
tional s-wave healing length ξ, it was found that Eq. (4) closely matches the
results from the energy cost calculation. Deviations become significant when
the dipolar interactions dominate over s-wave interactions. In this regime the
s-wave healing length ξ is no longer the relevant length scale of the system,
and the equivalent dipolar length scale ξd = Cddm/(12πh̄
) will characterize
the vortex core size.
For g > 0 and in the absence of dipolar interactions, the rotation frequency
at which the vortex-free BEC becomes dynamically unstable, Ωdyn, is always
greater than the critical frequency for vortex stabilization Ωc. However in
the presence of dipolar interactions, Ωdyn can become less than Ωc, leading
to an intriguing regime in which the dipolar BEC is dynamically unstable
but vortices will not enter [117, 120]. As with attractive condensates [17], the
angular momentum may then be manifested as center of mass oscillations.
6 Analogs of Gravitational Physics in BECs
There is growing interest in pursuing analogs of gravitational physics in con-
densed matter systems [121], such as BECs. The rationale behind such models
can be traced back to the work of Unruh [122, 123], who noted the analogy
between sound propagation in an inhomogeneous background flow and field
propagation in curved space-time. This link applies in the TF limit of BECs
where the speed of sound is directly analogous to the speed of light in the
corresponding gravitational system [124]. This has led to proposals for exper-
iments to probe effects such as Hawking radiation [125, 126] and superradiance
[127]. For Hawking radiation it is preferable to avoid the generation of vortices
[121, 128], and as such will not be discussed here. However, the phenomena
Vortices in Bose-Einstein Condensates: Theory 13
of superradiance in BECs, which can be considered as stimulated Hawking
radiation, relies on the presence of a vortex [129, 130, 131, 132], which is
analogous to a rotating black hole.
Below we outline the derivation of how the propagation of sound in a BEC
can be considered to be analogous to field propagation [121]. From the GPE
it is possible to derive the continuity equation for an irrotational fluid flow
with phase S(r, t) and density n(r, t), and a Hamilton-Jacobi equation whose
gradient leads to the Euler equation. Linearizing these equations with respect
to the background it is found that
′ = − 1
∇S · ∇S′ − gn′ + h̄
, (12)
′ = − 1
∇ · (n∇S′) − 1
∇ · (n′∇S) , (13)
where n′ and S′ are the perturbed values of the density n and phase S respec-
tively. Neglecting the quantum pressure ∇2-terms, the above equations can
be rewritten as a covariant differential equation describing the propagation of
phase oscillations in a BEC. This is directly analogous to the propagation of
a minimally coupled massless scalar field in an effective Lorentzian geometry
which is determined by the background velocity, density and speed of sound in
the BEC. Hence, the propagation of sound in a BEC can be used as an analogy
for the propagation of electromagnetic fields in the corresponding space-time.
Of course one has to be aware that this direct analogy is only valid in the TF
regime, which breaks down on scales of the order of a healing length, i.e. the
theory is only valid on large length scales, as is general relativity.
6.1 Superradiance
Superradiance in BECs relies on sound waves incident on a vortex structure
and is characterized by the reflected sound energy exceeding the incident
energy. This has been studied using Eqs. (12) and (13) for monochromatic
sound waves of frequency ωs and angular wave number qs incident upon a
vortex [129] and a ‘draining vortex” (a vortex with outcoupling at its center)
[130, 131, 132].
For the vortex case, a vortex velocity field v(r, θ) = (β/r)θ̂ and a density
profile ansatz was assumed. Superradiance then occurs when βqs > Ac∞,
where A is related to the vortex density ansatz and c∞ is the speed of sound
at infinity [129]. Interestingly, this condition is frequency independent.
For the case of a draining vortex, an event horizon occurs at a distance
a from the vortex core, where the fluid circulates at frequency Ω. Assuming
a homogeneous density n and a velocity profile v(r, θ) =
−car̂ +Ωa2θ̂
where c is the homogeneous speed of sound, superradiance occurs when 0 <
ωs < qsΩ [130, 131, 132].
The increase in energy of the outgoing sound is due to an extraction of
energy from the vortex and as such it is expected to lead to slowing of the
14 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
vortex rotation. However, such models do not include quantized vortex angular
momentum, and as such it is expected that superradiance will be suppressed
[132]. This raises tantalizing questions, such as whether superradiance can
occur if vorticity is quantized, if such effects can be modeled with the GPE,
and whether the study of quantum effects in condensate superradiance will
shed light on quantum effects in general relativity.
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  Vortices are pervasive in nature, representing the breakdown of laminar fluid
flow and hence playing a key role in turbulence. The fluid rotation associated
with a vortex can be parameterized by the circulation $\Gamma=\oint {\rm d}{\bf
r}\cdot{\bf v}({\bf r})$ about the vortex, where ${\bf v}({\bf r})$ is the
fluid velocity field. While classical vortices can take any value of
circulation, superfluids are irrotational, and any rotation or angular momentum
is constrained to occur through vortices with quantized circulation. Quantized
vortices also play a key role in the dissipation of transport in superfluids.
In BECs quantized vortices have been observed in several forms, including
single vortices, vortex lattices, and vortex pairs and rings. The recent
observation of quantized vortices in a fermionic gas was taken as a clear
signature of the underlying condensation and superfluidity of fermion pairs. In
addition to BECs, quantized vortices also occur in superfluid Helium, nonlinear
optics, and type-II superconductors.

arXiv:0704.0146v1  [cond-mat.other]  2 Apr 2007
Vortices in Bose-Einstein Condensates: Theory
N. G. Parker1, B. Jackson2, A. M. Martin1, and C. S. Adams3
1 School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia.
ngparker@ph.unimelb.edu.au,amm@ph.unimelb.edu.au
2 School of Mathematics and Statistics, Newcastle University, Newcastle upon
Tyne, NE1 7RU, United Kingdom. brian.jackson@newcastle.ac.uk
3 Department of Physics, Durham University, South Road, Durham, DH1 3LE,
United Kingdom. c.s.adams@durham.ac.uk
1 Quantized vortices
Vortices are pervasive in nature, representing the breakdown of laminar fluid
flow and hence playing a key role in turbulence. The fluid rotation associated
with a vortex can be parameterized by the circulation Γ =
dr · v(r) about
the vortex, where v(r) is the fluid velocity field. While classical vortices can
take any value of circulation, superfluids are irrotational, and any rotation
or angular momentum is constrained to occur through vortices with quan-
tized circulation. Quantized vortices also play a key role in the dissipation of
transport in superfluids. In BECs quantized vortices have been observed in
several forms, including single vortices [1, 2], vortex lattices [3, 4, 5, 6] (see
also Chap. VII), and vortex pairs and rings [7, 8, 9]. The recent observation
of quantized vortices in a fermionic gas was taken as a clear signature of the
underlying condensation and superfluidity of fermion pairs [10]. In addition to
BECs, quantized vortices also occur in superfluid Helium [11, 12], nonlinear
optics, and type-II superconductors [13].
1.1 Theoretical Framework
Quantization of circulation
Quantized vortices represent phase defects in the superfluid topology of the
system. Under the Madelung transformation, the macroscopic condensate
‘wavefunction’ ψ(r, t) can be expressed in terms of a fluid density n(r, t) and a
macroscopic phase S(r, t) via ψ(r) =
n(r, t) exp[iS(r, t)]. In order that the
wavefunction remains single-valued, the change in phase around any closed
contour C must be an integer multiple of 2π,
∇S · dl = 2πq, (1)
http://arXiv.org/abs/0704.0146v1
2 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
where q is an integer. The gradient of the phase S defines the superfluid
velocity via v(r, t) = (h̄/m)∇S(r, t). This implies that the circulation about
the contour C is given by,
v · dl = q
. (2)
In other words, the circulation of fluid is quantized in units of (h/m). The
circulating fluid velocity about a vortex is given by v(r, θ) = qh̄/(mr)θ̂, where
r is the radius from the core and θ̂ is the azimuthal unit vector.
Theoretical model
The Gross-Pitaevskii equation (GPE) provides an excellent description of
BECs at the mean-field level in the limit of ultra-cold temperature [14]. It
supports quantized vortices, and has been shown to give a good description of
the static properties and dynamics of vortices [14, 15]. Dilute BECs require a
confining potential, formed by magnetic or optical fields, which typically varies
quadratically with position. We will assume an axially-symmetric harmonic
trap of the form V = 1
m(ω2rr
2 + ω2zz
2), where ωr and ωz are the radial and
axial trap frequencies respectively. Excitation spectra of BEC states can be
obtained using the Bogoliubov equations, and specify the stability of station-
ary solutions of the GPE. For example, the presence of the so-called anomalous
modes of a vortex in a trapped BEC are indicative of their thermodynamic
instability. The GPE can also give a qualitative, and sometimes quantitative,
understanding of vortices in superfluid Helium [11, 12].
Although this Chapter deals primarily with vortices in repulsively-inte-
racting BECs, vortices in attractively-interacting BECs have also received
theoretical interest. The presence of a vortex in a trapped BEC with attractive
interactions is less energetically favorable than for repulsive interactions [16].
Indeed, a harmonically-confined attractive BEC with angular momentum is
expected to exhibit a center-of-mass motion rather than a vortex [17]. The
use of anharmonic confinement can however support metastable vortices, as
well as regimes of center-of-mass motion and instability [18, 19, 20].
Various approximations have been made to incorporate thermal effects
into the GPE to describe vortices at finite temperature (see also Chap. XI).
The Popov approximation self-consistently couples the condensate to a normal
gas component using the Bogoliubov-de-Gennes formalism [21] (cf. Chap. I
Sec. 5.2). Other approaches involve the addition of thermal/quantum noise to
the system, such as the stochastic GPE method [22, 23, 24] and the classical
field/truncated Wigner methods [25, 26, 27, 28]. Thermal effects can also be
simulated by adding a phenomenological dissipation term to the GPE [29].
Basic properties of vortices
In a homogeneous system, a quantized vortex has the 2D form,
Vortices in Bose-Einstein Condensates: Theory 3
ψ(r, θ) =
nv(r) exp(iqθ). (3)
The vortex density profile nv(r) has no analytic solution, although approx-
imate solutions exist [30]. Vortex solutions can be obtained numerically by
propagating the GPE in imaginary time (t→ −it) [31], whereby the GPE con-
verges to the lowest energy state of the system (providing it is stable). By en-
forcing the phase distribution of Eq. (3), a vortex solution is generated. Figure
1 shows the solution for a q = 1 vortex at the center of a harmonically-confined
BEC. The vortex consists of a node of zero density with a width characterized
by the condensate healing length ξ = h̄/
mn0g, where g = 4πh̄
2a/m (with a
the s-wave scattering length) and n0 is the peak density in the absence of the
vortex. For typical BEC parameters [3], ξ ∼ 0.2 µm. For a q = 1 vortex at
the center of an axially-symmetric potential, each particle carries h̄ of angular
momentum. However, if the vortex is off-center, the angular momentum per
particle becomes a function of position [15].
1.2 Vortex structures
Increasing the vortex charge widens the core due to centrifugal effects. In
harmonically-confined condensates a multiply-quantized vortex with q > 1 is
energetically unfavorable compared to a configuration of singly-charged vor-
tices [32, 33]. Hence, a rotating BEC generally contains an array of singly-
charged vortices in the form of a triangular Abrikosov lattice [3, 4, 5, 6, 34]
(see also Chap. VII), similar to those found in rotating superfluid helium
[11]. A q > 1 vortex can decay by splitting into singly-quantized vortices via
a dynamical instability [35, 36], but is stable for some interaction strengths
[37]. Multiply-charged vortices are also predicted to be stabilized by a suitable
localized pinning potential [38] or the addition of quartic confinement [33].
Two-dimensional vortex-antivortex pairs (i.e. two vortices with equal but
opposite circulation) and 3D vortex rings arise in the dissipation of superflow,
and represent solutions to the homogeneous GPE in the moving frame [39, 40],
with their motion being self-induced by the velocity field of the vortex lines.
When the vortex lines are so close that they begin to overlap, these states are
no longer stable and evolves into a rarefaction pulse [39].
Having more than one spin component in the BECs (cf. Chap. IX) pro-
vides an additional topology to vortex structures. Coreless vortices and vortex
‘molecules’ in coupled two-component BECs have been probed experimentally
[41] and theoretically [42]. More exotic vortex structures such as skyrmion ex-
citations [43] and half-quantum vortex rings [44] have also been proposed.
2 Nucleation of vortices
Vortices can be generated by rotation, a moving obstacle, or phase imprinting
methods. Below we discuss each method in turn.
4 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
2.1 Rotation
As discussed in the previous section, a BEC can only rotate through the
existence of quantized vortex lines. Vortex nucleation occurs only when the
rotation frequency Ω of the container exceeds a critical value Ωc [15, 32, 46].
Consider a condensate in an axially-symmetric trap which is rotating about
the z-axis at frequency Ω. In the Thomas-Fermi limit, the presence of a vortex
becomes energetically favorable when Ω exceeds a critical value given by [47],
0.67R
. (4)
This is derived by integrating the kinetic energy density mn(r)v(r)2/2 of the
vortex velocity field in the radial plane. The lower and upper limits of the
integration are set by the healing length ξ and the BEC Thomas-Fermi radius
R, respectively. Note that Ωc < ωr for repulsive interactions, while Ωc > ωr
for attractive interactions [16]. In a non-rotating BEC the presence of a vortex
raises the energy of the system, indicating thermodynamic instability [48].
In experiments, vortices are formed only when the trap is rotated at a
much higher frequency than Ωc [3, 4, 5], demonstrating that the energetic
criterion is a necessary, but not sufficient, condition for vortex nucleation.
There must also be a dynamic route for vorticity to be introduced into the
condensate, and hence Eq. (4) provides only a lower bound for the critical
frequency.
The nucleation of vortices in rotating trapped BECs appears to be linked to
instabilities of collective excitations. Numerical simulations based on the GPE
have shown that once the amplitude of these excitations become sufficiently
large, vortices are nucleated that subsequently penetrate the high-density bulk
of the condensate [23, 27, 29, 49, 50].
One way to induce instability is to resonantly excite a surface mode by
adding a rotating deformation to the trap potential. In the limit of small
perturbations, this resonance occurs close to a rotation frequency Ωr = ωℓ/ℓ,
where ωℓ is the frequency of a surface mode with multipolarity ℓ. In the
Thomas-Fermi limit, the surface modes satisfy ωℓ =
ℓωr [51], so Ωr =
ℓ. For example, an elliptically-deformed trap, which excites the ℓ = 2
quadrupole mode, would nucleate vortices when rotated at Ωr ≈ ωr/
This value has been confirmed in both experiments [3, 4, 5] and numerical
simulations [23, 27, 29, 49, 50]. Higher multipolarities were resonantly excited
in the experiment of Ref. [6], finding vortex formation at frequencies close to
the expected values, Ω = ωr/
ℓ, and lending further support to this picture.
A similar route to vortex nucleation is revealed by considering stationary
states of the BEC in a rotating elliptical trap, which can be obtained in the
Thomas-Fermi limit by solving hydrodynamic equations [52]. At low rotation
rates only one solution is found; however at higher rotations (Ω > ωr/
bifurcation occurs and up to three solutions are present. Above the bifurcation
point one or more of the solutions become dynamically unstable [53], leading
Vortices in Bose-Einstein Condensates: Theory 5
to vortex formation [54]. Madison et al. [55] followed these stationary states
experimentally by adiabatically introducing trap ellipticity and rotation, and
observed vortex nucleation in the expected region.
Surface mode instabilities can also be induced at finite temperature by
the presence of a rotating noncondensed “thermal” cloud. Such instabilities
occur when the thermal cloud rotation rate satisfies Ω > ωℓ/ℓ [56]. Since all
modes can potentially be excited in this way, the criterion for instability and
hence vortex nucleation becomes Ωc > min(ωℓ/ℓ), analogous to the Landau
criterion. Note that such a minimum exists at Ωc > 0 since the Thomas-Fermi
result ωℓ =
ℓωr becomes less accurate for high ℓ [57]. This mechanism may
have been important in the experiment of Haljan et al. [34], where a vortex
lattice was formed by cooling a rotating thermal cloud to below Tc.
2.2 Nucleation by a moving object
Vortices can also be nucleated in BECs by a moving localized potential. This
problem was originally studied using the GPE for 2D uniform condensate flow
around a circular hard-walled potential [58, 59], with vortex-antivortex pairs
being nucleated when the flow velocity exceeded a critical value.
In trapped BECs a similar situation can be realized using the optical dipole
force from a laser, giving rise to a localized repulsive Gaussian potential. Under
linear motion of such a potential, numerical simulations revealed vortex pair
formation when the potential is moved at a velocity above a critical value [60].
The experiments of [61, 62] oscillated a repulsive laser beam in an elongated
condensate. Although vortices were not observed directly, the measurement
of condensate heating and drag above a critical velocity was consistent with
the nucleation of vortices [63].
An alternative approach is to move the laser beam potential in a circular
path around the trap center [64]. By “stirring” the condensate in this way one
or more vortices can be created. This technique was used in the experiment
of Ref. [6], where vortices were generated even at low stirring frequencies.
2.3 Other mechanisms and structures
A variety of other schemes for vortex creation have been suggested. One of
the most important is that by Williams and Holland [65], who proposed a
combination of rotation and coupling between two hyperfine levels to create
a two-component condensate, one of which is in a vortex state. The non-
vortex component can then either be retained or removed with a resonant
laser pulse. This scheme was used by the first experiment to obtain vortices
in BEC [1]. A related method, using topological phase imprinting, has been
used to experimentally generate multiply-quantized vortices [66].
Apart from the vortex lines considered so far, vortex rings have also been
the subject of interest. Rings are the decay product of dynamically unstable
dark solitary waves in 3D geometries [7, 8, 67, 68]. Vortex rings also form
6 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
in the quantum reflection of BECs from surface potentials [69], the unstable
motion of BECs through an optical lattice [70], the dragging of a 3D object
through a BEC [71], and the collapse of ultrasound bubbles in BECs [72].
The controlled generation of vortex rings [73] and multiple/bound vortex ring
structures [74] have been analyzed theoretically.
A finite temperature state of a quasi-2D BEC, characterized by the ther-
mal activation of vortex-antivortex pairs, has been simulated using classical
field simulations [75]. This effect is thought to be linked to the Berezinskii-
Kosterlitz-Thouless phase transition of 2D superfluids, recently observed ex-
perimentally in ultracold gases [76]. Similar simulations in a 3D system have
also demonstrated the thermal creation of vortices [77, 78].
3 Dynamics of vortices
The study of vortex dynamics has long been an important topic in both clas-
sical [79] and quantum [12] hydrodynamics. Helmholtz’s theorem for uniform,
inviscid fluids, which is also applicable to quantized vortices in superfluids
near zero temperature, states that the vortex will follow the motion of the
background fluid. So, for example, in a superfluid with uniform flow velocity
vs, a single straight vortex line will move with velocity vL, such that it is
stationary in the frame of the superfluid.
Vortices similarly follow the “background flow” originating from circulat-
ing fluid around a vortex core. Hence vortex motion can be induced by the
presence of other vortices, or by other parts of the same vortex line when it is
curved. Most generally, the superfluid velocity vi due to vortices at a partic-
ular point r is given by the Biot-Savart law [12], in analogy with the similar
equation in electromagnetism,
(s − r) × ds
|s− r|3
; (5)
where s(ζ, t) is a curve representing the vortex line with ζ the arc length.
Equation (5) suffers from a divergence at r = s, so in calculations of vortex
dynamics this must be treated carefully [80]. Equation (5) also assumes that
the vortex core size is small compared to the distance between vortices. In
particular, it breaks down when vortices cross during collisions, where recon-
nection events can occur. These reconnections can either be included manually
[81], or by solving the full GPE [82]. The latter method also has the advantage
of including sound emission due to vortex motion or reconnections [83, 84].
In a system with multiple vortices, motion of one vortex is induced by the
circulating fluid flow around other vortices, and vice-versa [11]. This means
that, for example, a pair of vortices of equal but opposite charge will move
linearly and parallel to each other with a velocity inversely proportional to
the distance between them. Two or more vortices of equal charge, meanwhile,
Vortices in Bose-Einstein Condensates: Theory 7
will rotate around each other, giving rise to a rotating vortex lattice as will be
discussed in Chap. VII. When a vortex line is curved, circulating fluid from
one part of the line can induce motion in another. This effect can give rise to
helical waves on the vortex, known as Kelvin modes [85]. It also has interesting
consequences for a vortex ring, which will travel in a direction perpendicular
to the plane of the ring, with a self-induced velocity that decreases with in-
creasing radius. Classically, this is most familiar in the motion of smoke rings,
though similar behavior has also been observed in superfluid helium [86].
This simple picture is complicated in the presence of density inhomo-
geneities or confining walls. In a harmonically-trapped BEC the density is
a function of position, and therefore the energy, E, of a vortex will also de-
pend on its position within the condensate. To simplify matters, let us con-
sider a quasi-2D situation, where the condensate is pancake-shaped and the
vortex line is straight. In this case, the energy of the vortex depends on its
displacement r from the condensate center [87], and a displaced vortex feels a
force proportional to ∇E. This is equivalent to a Magnus force on the vortex
[88, 89, 90] and to compensate the vortex moves in a direction perpendicular
to the force, leading it to precess around the center of the condensate along a
line of constant energy. This precession of a single vortex has been observed
experimentally [2], with a frequency in agreement with theoretical predictions.
In more 3D situations, such as spherical or cigar-shaped condensates, the vor-
tex can bend [91, 92, 93, 94] leading to more complicated motion [15]. Kelvin
modes [95, 96] and vortex ring dynamics [88] are also modified by the density
inhomogeneity in the trap.
In the presence of a hard-wall potential, a new constraint is imposed such
that the fluid velocity normal to the wall must be zero, vs ·n̂ = 0. The resulting
problem of vortex motion is usually solved mathematically [79] by invoking
an “image vortex” on the other side of the wall (i.e. in the region where there
is no fluid present), at a position such that its normal flow cancels that of the
real vortex at the barrier. The motion of the real vortex is then simply equal
to the induced velocity from the image vortex circulation.
4 Stability of vortices
4.1 Thermal instabilities
At finite temperatures the above discussion is modified by the thermal oc-
cupation of excited modes of the system, which gives rise to a noncondensed
normal fluid in addition to the superfluid. A vortex core moving relative to the
normal fluid scatters thermal excitations, and will therefore feel a frictional
force leading to dissipation. This mutual friction force can be written as [11],
fD = −nsΓ{αs′ × [ s′ × (vn − vL)] + α′s′ × (vn − vL)}, (6)
where ns is the background superfluid density, s
′ is the derivative of s with
respect to arc length ζ, α and α′ are temperature dependent parameters,
8 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
while vL and vn are the velocities of the vortex line and normal fluid respec-
tively. The mutual friction therefore has two components perpendicular to the
relative velocity vn − vL.
To consider an example discussed in the last section, an off-center vortex in
a trapped BEC at zero temperature will precess such that its energy remains
constant. In the presence of a non-condensed component, however, dissipation
will lead to a loss of energy. Since the vortex is topological it cannot simply
vanish, so this lost energy is manifested as a radial drift of the vortex towards
lower densities. In Eq. (6) the α term is responsible for this radial motion,
while α′ changes the precession frequency. The vortex disappears at the edge
of the condensate, where it is thought to decay into elementary excitations
[97]. Calculations based upon the stochastic GPE have shown that thermal
fluctuations lead to an uncertainty in the position of the vortex, such that
even a central vortex will experience thermal dissipation and have a finite
lifetime [24]. This thermodynamic lifetime is predicted to be of the order of
seconds [97], which is consistent with experiments [1, 3, 94].
4.2 Hydrodynamic instabilities
Experiments indicate that the crystallization of vortex lattices is temperature-
independent [5, 98]. Similarly, vortex tangles in turbulent states of superfluid
Helium have been observed to decay at ultracold temperature, where thermal
dissipation is virtually nonexistent [99]. These results highlight the occurrence
of zero temperature dissipation mechanisms, as listed below.
Instability to acceleration
The topology of a 2D homogeneous superfluid can be mapped on to a (2+1)D
electrodynamic system, with vortices and phonons playing the role of charges
and photons respectively [100]. Just as an accelerating electron radiates ac-
cording to the Larmor acceleration squared law, a superfluid vortex is inher-
ently unstable to acceleration and radiates sound waves.
Vortex acceleration can be induced by the presence of an inhomogeneous
background density, such as in a trapped BEC. Sound emission from a vortex
in a BEC can be probed by considering a trap of the form [45],
Vext = V0
1 − exp
mω2rr
2. (7)
This consists of a gaussian dimple trap with depth V0 and harmonic frequency
component ωd, embedded in an ambient harmonic trap of frequency ωr. A 2D
description is sufficient to describe this effect. This set-up can be realized with
a quasi-2D BEC by focussing a far-off-resonant red-detuned laser beam in the
center of a magnetic trap. The vortex is initially confined in the inner region,
where it precesses due to the inhomogeneous density. Since sound excitations
Vortices in Bose-Einstein Condensates: Theory 9
Fig. 1. Profile of a singly-quantized (q = 1) vortex at the center of a harmonically-
confined BEC: (a) condensate density along the y = 0 axis (solid line) and the
corresponding density profile in the absence of the vortex (dashed line). (b) 2D
density and (c) phase profile of the vortex state. These profiles are calculated nu-
merically by propagating the 2D GPE in imaginary time subject to an azimuthal
2π phase variation around the trap center.
−6 −4 −2 0 2 4 6
x (ξ)
(i) (ii)
Fig. 2. Vortex path in the dimple trap geometry of Eq. (7) with ωd = 0.28(c/ξ).
Deep V0 = 10µ dimple (dotted line): mean radius is constant, but modulated by the
sound field. Shallow V0 = 0.6µ dimple and homogeneous outer region ωr = 0 (dot-
ted line): vortex spirals outwards. Outer plots: Sound excitations (with amplitude
∼ 0.01n0) radiated in the V0 = 0.6µ system at times indicated. Top: Far-field distri-
bution [−90, 90]ξ×[−90, 90]ξ. Bottom: Near-field distribution [−25, 25]ξ×[−25, 25]ξ,
with an illustration of the dipolar radiation pattern. Copyright (2004) by the Amer-
ican Physical Society [45].
have an energy of the order of the chemical potential µ, the depth of the dimple
relative to µ leads to two distinct regimes of vortex-sound interactions.
V0 ≫ µ: The vortex effectively sees an infinite harmonic trap - it precesses
and radiates sound but there is no net decay due to complete sound reabsorp-
tion. However, a collective mode of the background fluid is excited, inducing
slight modulations in the vortex path (dotted line in Fig 2).
V0 < µ: Sound waves are radiated by the precessing vortex. Assuming ωr =
0, the sound waves propagate to infinity without reinteracting with the vortex.
10 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
The ensuing decay causes the vortex to drift to lower densities, resulting in a
spiral motion (solid line in Fig. 2), similar to the effect of thermal dissipation.
The sound waves are emitted in a dipolar radiation pattern, perpendicularly
to the instantaneous direction of motion (subplots in Fig. 2), with a typical
amplitude of order 0.01n0 and wavelength λ ∼ 2πc/ωV [15], where c is the
speed of sound and ωV is the vortex precession frequency. The power radiated
from a vortex can be expressed in the form [45, 101, 102],
P = βmN
, (8)
where a is the vortex acceleration, N is the total number of atoms, and β is
a dimensionless coefficient. Using classical hydrodynamics [101] and by map-
ping the superfluid hydrodynamic equations onto Maxwell’s electrodynamic
equations [102], it has been predicted that β = π2/2 under the assumptions
of a homogeneous 2D fluid, a point vortex, and perfect circular motion. Full
numerical simulations of the GPE based on a realistic experimental scenario
have derived a coefficient of β ∼ 6.3 ± 0.9 (one standard deviation), with the
variation due to a weak dependence on the geometry of the system [45].
When ωr 6= 0, the sound eventually reinteracts with the vortex, slowing but
not preventing the vortex decay. By varying V0 it is possible to control vortex
decay, and in suitably engineered traps this decay mechanism is expected to
dominate over thermal dissipation [45].
Vortex acceleration (and sound emission) can also be induced by the pres-
ence of other vortices. A co-rotating pair of two vortices of equal charge has
been shown to decay continuously via quadrupolar sound emission, both an-
alytically [103] and numerically [104]. Three-body vortex interactions in the
form of a vortex-antivortex pair incident on a single vortex have also been sim-
ulated numerically, with the interaction inducing acceleration in the vortices
with an associated emission of sound waves [104].
Simulations of vortex lattice formation in a rotating elliptical trap show
that vortices are initially nucleated in a turbulent disordered state, before
relaxing into an ordered lattice [50]. This relaxation process is associated
with an exchange of energy from the sound field to the vortices due to these
vortex-sound interactions. This agrees with the experimental observation that
vortex lattice formation is insensitive to temperature [5, 98].
Kelvin wave radiation and vortex reconnections
In 3D a Kelvin wave excitation will induce acceleration in the elements of
the vortex line, and therefore local sound emission. Indeed, simulations of
the GPE in 3D have shown that Kelvin waves excitations on a vortex ring
lead to a decrease in the ring size, indicating the underlying radiation process
[84]. Kelvin wave excitations can be generated from a vortex line reconnection
[83, 84] and the interaction of a vortex with a rarefaction pulse [105].
Vortices in Bose-Einstein Condensates: Theory 11
Vortex lines which cross each other can undergo dislocations and reconnec-
tions [106], which induce a considerable burst of sound emission [83]. Although
they have yet to be probed experimentally in BECs, vortex reconnections are
hence thought to play a key role in the dissipation of vortex tangles in Helium
II at ultra-low temperatures [11].
5 Dipolar BECs
A BEC has recently been formed of chromium atoms [107], which feature a
large dipole moment. This opens the door to studying of the effect of long-
range dipolar interactions in BECs.
5.1 The Modified Gross-Pitaevskii Equation
The interaction potential Udd(r) between two dipoles separated by r, and
aligned by an external field along the unit vector ê is given by,
Udd(r) =
êiêj
(δij − 3r̂ir̂j)
. (9)
For low energy scattering of two atoms with dipoles induced by a static electric
field E = Eê, the coupling constant Cdd = E
2α2/ǫ0 [108, 109], where α is the
static dipole polarizability of the atoms and ǫ0 is the permittivity of free space.
Alternatively, if the atoms have permanent magnetic dipoles, dm, aligned in
an external magnetic field B = Bê, one has Cdd = µ0d
m [110], where µ0 is the
permeability of free space. Such dipolar interactions give rise to a mean-field
potential
Φdd(r) =
d3rUdd (r − r′) |ψ (r′) |2, (10)
which can be incorporated into the GPE to give,
ih̄ψt =
∇2 + g|ψ|2 + Φdd + V
ψ. (11)
For an axially-symmetric quasi-2D geometry (ωz ≫ ωr) rotating about the
z -axis, the ground state wavefunction of a single vortex has been solved numer-
ically [111]. Considering 105 chromium atoms and ωr = 2π × 100Hz, several
solutions were obtained depending on the strength of the s-wave interactions
and the alignment of the dipoles relative to the trap.
For the case of axially-polarized dipoles the most striking results arise
for attractive s-wave interactions g < 0. Here the BEC density is axially
symmetric and oscillates in the vicinity of the vortex core. Similar density
oscillations have been observed in numerical studies of other non-local inter-
action potentials, employed to investigate the interparticle interactions in 4He
12 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
[112, 113, 114, 115], with an interpretation that relates to the roton structure
in a superfluid [115]. For the case of transversely-polarized dipoles, where the
polarizing field is co-rotating with the BEC, and repulsive s-wave interactions
(g > 0), the BEC becomes elongated along the axis of polarization [116] and
as a consequence the vortex core is anisotropic.
5.2 Vortex Energy
Assuming a dipolar BEC in the TF limit (cf. Sec. 5.1 in Chap. I), the en-
ergetic cost of a vortex, aligned along the axis of polarization (z-axis), has
been derived using a variational ansatz for the vortex core [117], and thereby
the critical rotation frequency Ωc at which the presence of a vortex becomes
energetically favorable has been calculated. For an oblate trap (ωr < ωz),
dipolar interactions decrease Ωc, while for prolate traps (ωr > ωz) the pres-
ence of dipolar interactions increases Ωc. A formula resembling Eq. (4) for
the critical frequency of a conventional BEC can be used to explain these
results, with R being the modified TF radius of the dipolar BEC. Indeed,
using the TF radius of a vortex-free dipolar BEC [118, 119] and the conven-
tional s-wave healing length ξ, it was found that Eq. (4) closely matches the
results from the energy cost calculation. Deviations become significant when
the dipolar interactions dominate over s-wave interactions. In this regime the
s-wave healing length ξ is no longer the relevant length scale of the system,
and the equivalent dipolar length scale ξd = Cddm/(12πh̄
) will characterize
the vortex core size.
For g > 0 and in the absence of dipolar interactions, the rotation frequency
at which the vortex-free BEC becomes dynamically unstable, Ωdyn, is always
greater than the critical frequency for vortex stabilization Ωc. However in
the presence of dipolar interactions, Ωdyn can become less than Ωc, leading
to an intriguing regime in which the dipolar BEC is dynamically unstable
but vortices will not enter [117, 120]. As with attractive condensates [17], the
angular momentum may then be manifested as center of mass oscillations.
6 Analogs of Gravitational Physics in BECs
There is growing interest in pursuing analogs of gravitational physics in con-
densed matter systems [121], such as BECs. The rationale behind such models
can be traced back to the work of Unruh [122, 123], who noted the analogy
between sound propagation in an inhomogeneous background flow and field
propagation in curved space-time. This link applies in the TF limit of BECs
where the speed of sound is directly analogous to the speed of light in the
corresponding gravitational system [124]. This has led to proposals for exper-
iments to probe effects such as Hawking radiation [125, 126] and superradiance
[127]. For Hawking radiation it is preferable to avoid the generation of vortices
[121, 128], and as such will not be discussed here. However, the phenomena
Vortices in Bose-Einstein Condensates: Theory 13
of superradiance in BECs, which can be considered as stimulated Hawking
radiation, relies on the presence of a vortex [129, 130, 131, 132], which is
analogous to a rotating black hole.
Below we outline the derivation of how the propagation of sound in a BEC
can be considered to be analogous to field propagation [121]. From the GPE
it is possible to derive the continuity equation for an irrotational fluid flow
with phase S(r, t) and density n(r, t), and a Hamilton-Jacobi equation whose
gradient leads to the Euler equation. Linearizing these equations with respect
to the background it is found that
′ = − 1
∇S · ∇S′ − gn′ + h̄
, (12)
′ = − 1
∇ · (n∇S′) − 1
∇ · (n′∇S) , (13)
where n′ and S′ are the perturbed values of the density n and phase S respec-
tively. Neglecting the quantum pressure ∇2-terms, the above equations can
be rewritten as a covariant differential equation describing the propagation of
phase oscillations in a BEC. This is directly analogous to the propagation of
a minimally coupled massless scalar field in an effective Lorentzian geometry
which is determined by the background velocity, density and speed of sound in
the BEC. Hence, the propagation of sound in a BEC can be used as an analogy
for the propagation of electromagnetic fields in the corresponding space-time.
Of course one has to be aware that this direct analogy is only valid in the TF
regime, which breaks down on scales of the order of a healing length, i.e. the
theory is only valid on large length scales, as is general relativity.
6.1 Superradiance
Superradiance in BECs relies on sound waves incident on a vortex structure
and is characterized by the reflected sound energy exceeding the incident
energy. This has been studied using Eqs. (12) and (13) for monochromatic
sound waves of frequency ωs and angular wave number qs incident upon a
vortex [129] and a ‘draining vortex” (a vortex with outcoupling at its center)
[130, 131, 132].
For the vortex case, a vortex velocity field v(r, θ) = (β/r)θ̂ and a density
profile ansatz was assumed. Superradiance then occurs when βqs > Ac∞,
where A is related to the vortex density ansatz and c∞ is the speed of sound
at infinity [129]. Interestingly, this condition is frequency independent.
For the case of a draining vortex, an event horizon occurs at a distance
a from the vortex core, where the fluid circulates at frequency Ω. Assuming
a homogeneous density n and a velocity profile v(r, θ) =
−car̂ +Ωa2θ̂
where c is the homogeneous speed of sound, superradiance occurs when 0 <
ωs < qsΩ [130, 131, 132].
The increase in energy of the outgoing sound is due to an extraction of
energy from the vortex and as such it is expected to lead to slowing of the
14 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams
vortex rotation. However, such models do not include quantized vortex angular
momentum, and as such it is expected that superradiance will be suppressed
[132]. This raises tantalizing questions, such as whether superradiance can
occur if vorticity is quantized, if such effects can be modeled with the GPE,
and whether the study of quantum effects in condensate superradiance will
shed light on quantum effects in general relativity.
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Lett. 85, 2857 (2000).
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