Vortex knots in a Bose-Einstein condensate.

Abstract

We present a method for numerically building a vortex knot state in the superfluid wave function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how vortex knots break up into vortex rings.

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PHYSICAL REVIEW E 85, 036306 (2012)
Vortex knots in a Bose-Einstein condensate
Davide Proment,1,2,*Miguel Onorato,1,2and Carlo F. Barenghi3
1Dipartimento di Fisica, Universit` a degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy, EU
2INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy, EU
3School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom, EU
(Received 26 October 2011; published 19 March 2012)
We present a method for numerically building a vortex knot state in the superfluid wave function of a
Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution
and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We
find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the
knot travels faster. Finally, we show how vortex knots break up into vortex rings.
DOI: 10.1103/PhysRevE.85.036306PACS number(s): 47.32.C−, 03.75.Lm
I. INTRODUCTION
In 1867, following the works of Helmholtz on vortices
and of Riemann on Abelian functions, Lord Kelvin modeled
atoms as knotted vortex tubes in ether [1], effectively giving
birth to knot theory [2]. This discipline has fascinated
mathematicians and physicists since. More recently, knots
have been the studied in different branches of physics, ranging
from classical fluid dynamics [3,4], magneto-hydrodynamics
[5], and classical field theory [6], to superconductors
[7,8], excitable media [9], optics [10,11], and liquid-crystal
colloids [12].
Knots in superfluids are identified with closed vortex
lines, regions of fluid around which the circulation assumes
nonzero (quantized) value. Vortex rings have been studied
experimentally in superfluid liquid helium [13,14] and in
Bose-Einstein condensates [15]. Numerical simulations have
revealed that superfluid turbulence contains linked vortex
lines [16], but, to the best of our knowledge, individual
vortices with nontrivial topology have never been observed
directly. To shed light on this problem, energy, motion, and
stability of vortex knots have been examined theoretically and
numerically using the classical theory of thin-cored vortex
filaments. In this approach, the governing incompressible
Euler dynamics is expressed by the Biot-Savart law or by
its local induction approximation (LIA). Considering the LIA
limit, it has been conjectured [17] and recently proved [18,19]
that any closed curve more (topologically) complex than a
ring is linearly unstable to perturbation or changes its knot
type during the evolution. However, when the full Biot-Savart
model is considered, a stabilization effect is observed and,
under certain conditions, it is found that some vortex knots
travel and preserve the knot type without breaking up for
distances larger than their own diameters [20,21].
In superfluid helium, the validity of the classical theory
of thin-core vortex filaments is based on the large separation
of scales between the vortex core radius a0(approximately
10−8cmin4Heand10−6cmin3He-B)andthetypicaldistance
? between vortices. In turbulence experiments, ? ≈ 10−3
to 10−4cm; the last value is also the typical diameter of
experimental vortex rings [14]. The situation is very different
*davideproment@gmail.com; www.to.infn.it/∼proment
inatomicBose-Einsteincondensates,where?isonlyfewtimes
larger than a0. In this context, the Gross-Pitaevskii equation
(GPE) is clearly a more realistic model [22], particularly at
very low temperatures, as thermal effects can be neglected.
TheadvantageoftheGPEisthatitdoesnotneedthecut-off
parameter required by the classical vortex filament theory
to de-singularize the Biot-Savart integral [23]. The second
advantage is that the GPE naturally describes vortex recon-
nections [24], which must be implemented algorithmically in
the Biot-Savart model. Any prediction about the evolution,
the shape preservation, and the breakup of a vortex structure
which is not orders of magnitude bigger than a0is therefore
more reliable if obtained using the GPE. The third advantage
of searching for vortex knots in a Bose-Einstein condensate is
that direct images of individual vortex structures are possible
without the use of tracer particles which will certainly disturb
these structures. The disadvantage is that atomic condensates
are small, and thus the motion of these structures will be
affected by the boundaries and by the nonuniformity density
of the background condensate. Before investigating these
effects, however, it is essential to establish whether vortex
knot solutions of the governing GPE exist, and, if they do, if
they are sufficiently long-lived structures. This is the limited
aim which we set in this work.
We stress that we do not intend to propose a mechanism
to experimentally create vortex knots in condensates, but only
to study the possible existence and preservation of knot type
of these solutions of the GPE. We shall see that even setting
up a topologically nontrivial structure in the wave function
numerically is not a minor task; indeed, to the best of our
knowledge, this is the first time it has been done for a single
scalar field describing the condensate order parameter. For
completeness, we emphasize that the existence and stability of
vortex knots in more complicated Bose systems have already
been discussed using the Faddeev-Skyrme model [25–27], as
in the case of a charged two-condensate Bose system [28], an
interacting mixture of charged and neutral superfluid [29], and
spinor condensates [30,31]. Related work on vortex unknots,
notably, vortex rings perturbed by Kelvin waves, was carried
out recently by Helm et al. [32] and Sonin [33].
The manuscript is organized as follows. Section II explains
howtocreateanelementaryvortexknotintheinitialconditions
of the condensate wave function. Section III deals with
the analysis of the dynamical properties of vortex knots.
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DAVIDE PROMENT, MIGUEL ONORATO, AND CARLO F. BARENGHIPHYSICAL REVIEW E 85, 036306 (2012)
Section IV describes the breakup of vortex knots. Finally, the
conclusions are in Sec. V.
II. VORTEX KNOT INITIAL CONDITIONS
We consider the defocusing GPE written in the following
dimensionless form:
2i ∂tψ + ∇2ψ − |ψ|2ψ = 0,
wherenoexternalconfiningpotentialispresent.Thecharacter-
istic length scale of perturbations of the uniform condensate,
called healing length, is defined as
1
√?ρ?, where ?ρ? =1
is the mean density of the condensate. Besides the energy, the
GPE conserves the total number of particles, and therefore
ξ is a conserved quantity too. Without loss of generality, we
choose to deal with a system that has an unperturbed density
(the density field at infinity) equal to unity and assume that
perturbations are localized in a small region of the sample. In
this hypothesis, ξ ? 1 in our units.
We now explain how to numerically build a vortex knot.
First we construct a vortex. Consider the two-dimensional
plane sOz, that is to say, defined by the axes s and z lying
on it. A stable vortex is a hole (zero value) in the density field
aroundwhichthephaseofthewavefunctionchanges by±2π.
Asufficientlyaccuratedescriptionofatwo-dimensionalvortex
centered in the origin of the sOz plane is given by the wave
function ?2D(s,z) =√ρ(R)e−i θ(s,z), where R =√s2+ z2:
R2(a1+ a2R2)
1 + b1R2+ b2R4,
θ(s,z) = atan2(z,s),
atan2(...)beingtheextensionofthearctangentfunctionwhose
principal value is in the range (−π;π], and the coefficients
a1= 11/32,a2= 11/384,b1= 1/3,andb2= 11/384arising
from a second-order Pad´ e approximation [34]. Figure 1 shows
how the density field behaves around the axisymmetric vortex
center.
(1)
ξ =
V
?
V
|ψ|2dV
(2)
ρ(R) =
(3)
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10
ρ
R
FIG. 1. The density field ρ around an axisymmetric two-
dimensional vortex. Radial distances R are in units of the healing
length ξ.
It is clear from the plot that the vortex core is of the order
of the healing length, and the bulk value of the density ρ = 1
is recovered at larger distances.
We now come back to vortex knots in a three-dimensional
system. We define a knot as a closed curve over a torus,
characterized by the toroidal radius R0 and the poloidal
radius R1. More precisely, a closed curve Tp,q on the torus
is determined by counting the number of toroidal wraps p and
the number of poloidal wraps q. For example, the curves T1,1
and T2,2describe, respectively, the unknot (the simple vortex
ring) and two unlinked rings. The first topologically nontrivial
curve is the trefoil T2,3. In this work we shall focus on the two
simplest knots, the trefoil T2,3and its dual T3,2.
A. The T2,3knot (trefoil)
The vortex line of a T2,3knot lays on the torus as shown in
Fig. 2. Any plane sOz passing through the z axis intercepts
the curve T2,3at four different points, which correspond to
four two-dimensional point vortices on the plane sOz. The
positions of these two-dimensional vortices vary with respect
to the choice of the plane sOz; in other words, these positions
are functions of the angle variable φ introduced in Fig. 2. For
example,thevortexpositionsofthewavefunctionfortheangle
φ = 0areshowninFig.3.Byconstruction,thesepointvortices
are located on the circumference defined by the intersection of
the plane with the torus, and rotate on it following a particular
function f(φ). To assure continuity of the vortex line and to
describe the trefoil knot, the function f(φ) must have the form
f(φ) = 3φ/2, with φ ∈ [0,π).
Wearenowreadytowritethethree-dimensionalwavefunc-
tionwhichdescribesthetrefoilknotT2,3.Intheapproximation
that the healing length ξ is much smaller than the inter-vortex
distance, the two-dimensional wave function in the plane sOz
is given by the superposition (multiplication) of the wave
function ?2Dof each two-dimensional vortex centered in the
correct position, where the opposite circulation is obtained by
FIG. 2. (Color online) Construction of the trefoil knot T2,3.
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VORTEX KNOTS IN A BOSE-EINSTEIN CONDENSATE PHYSICAL REVIEW E 85, 036306 (2012)
FIG. 3. (Color online) Positions of the four vortices on φ = 0
usedtoconstruct the wave function of the trefoilknotT2,3. Clockwise
andanticlockwisearrowsdescribethemotionofvortexpositionswith
respect to the function f(φ) = 3φ/2.
applying the complex conjugation operator (...)∗. Thus, the
three-dimensional wave function results in
ψ2,3(x,y,z)
= ?2D{s(x,y) − R0− R1cos[α(x,y)],z − R1sin[α(x,y)]}
×?2D{s(x,y) − R0− R1cos[α(x,y) + π],
z − R1sin[α(x,y) + π]}
×?∗
z − R1sin[α(x,y)]}
×?∗
z − R1sin[α(x,y) + π]},
with s(x,y) = sgn(x)?x2+ y2, where sgn(...) is the sign
2D{s(x,y) + R0+ R1cos[α(x,y)],
2D{s(x,y) + R0+ R1cos[α(x,y) + π],
(4)
function, and α(x,y) = 3/2atan2(y,x).
B. The T3,2knot
Thetechniqueusedtodefinethewavefunctionofthetrefoil
knot can be extended to any other knot built on a torus. The
T3,2knot can be represented on the torus as shown in Fig. 4.
In this case the generic plane sOz intersects the knot in six
FIG. 4. (Color online) Construction of the trefoil knot T3,2.
FIG. 5. (Color online) Positions of the six vortices on φ = 0
used to construct the wave function of the knot T3,2. Clockwise and
anticlockwise arrows describe the motion of vortex positions with
respect to the function g(φ) = 2φ/3.
points,wherethecentersareafunctiong(φ)oftheangleφ and
rotate around the circumference defined by the plane and the
torus intersection. An example of the configuration for φ = 0
is shown in Fig. 5. The function g(φ) is g(φ) = 2φ/3, with
φ ∈ [0,π).
Again, using the two-dimensional vortex description ?2D,
in the limit of intervortex distance much greater than the
healing length ξ, the three-dimensional wave function of a
T3,2knot is
ψ3,2(x,y,z)
=?2D{s(x,y) − R0− R1cos[α(x,y)],z − R1sin[α(x,y)]}
×?2D{s(x,y) − R0− R1cos[α(x,y) + 2π/3],
z − R1sin[α(x,y) + 2π/3]}
×?2D{s(x,y) − R0− R1cos[α(x,y) + 4π/3],
z − R1sin[α(x,y) + 4π/3]}
×?∗
z − R1sin[α(x,y)]}
×?∗
2D{s(x,y) + R0+ R1cos[α(x,y)],
2D{s(x,y) + R0+ R1cos[α(x,y) + 2π/3],
FIG. 6. (Color online) Isosurfaces of the density field at the
threshold level ρth= 0.2 for T2,3knots of various knot ratios R1/R0
(seeTableI).Snapshotsattimest = 0,400,800,1200.Unstableknots
are not shown. For complete evolution movies refer to [37].
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DAVIDE PROMENT, MIGUEL ONORATO, AND CARLO F. BARENGHI PHYSICAL REVIEW E 85, 036306 (2012)
TABLE I. Vortex knot parameters of T2,3and T3,2used in the
simulations.
Case
T2,3–T3,2
(a)–(a)
(b)–(b)
(c)–(c)
(d)–(d)
Knot ratio
R1/R0
Max size
2(R0+ R1)
44ξ
48ξ
56ξ
64ξ
Min size
2R1
Break up
Yes/No
1/10
1/5
2/5
3/5
4ξ
8ξ
16ξ
24ξ
N–N
N–N
Y–Y
Y–Y
z − R1sin[α(x,y) + 2π/3]}
×?∗
z − R1sin[α(x,y) + 4π/3]},
with s(x,y) = sgn(x)?x2+ y2and α(x,y) = 2/3atan2(y,x).
III. VORTEX KNOT DYNAMICS
2D{s(x,y) + R0+ R1cos[α(x,y) + 4π/3],
(5)
To study the dynamics and shape preservation of the knots
T2,3 and T3,2 with different geometries, we have to find a
compromise between the accessible numerical resolution and
the box size: we need to resolve small scales near the vortex
coresand,atthesametime,minimizethefinitesize(boundary)
effects.Werecallthattheparameterswhichidentifyourvortex
knots, the toroidal and poloidal radii R0and R1, are expressed
in units of the healing length ξ.
We chose to uniformly discretize physical space using a
Cartesian grid with steps ?x = ?y = ?z = 0.5ξ spanning
over the knot ratios R1/R0= 1/10,1/5,2/5,3/5. We expect
vortex knots to behave similarly to vortex rings, that is to say,
we expect that they travel along the direction of the torus axis
ofsymmetry(thezaxis).Takingourcomputationalconstraints
intoaccount,weuse192 × 192 × 512gridpoints(Lx= Ly=
96ξ and Lz= 256ξ) and the toroidal radius R0= 20ξ. This
choice allows us to have a minimum value of R1= 2ξ (when
R1/R0= 1/10), acceptable to observe the small intervortex
interactions, and a maximum knot size of 2(R0+ R1) = 64ξ
(when R1/R0= 3/5), which gives tolerable boundary effects.
Table I summarizes the simulation parameters.
In order to let the knot travel for the maximum distance
in the z direction, at the start of the calculation (t = 0) the
FIG. 7. (Color online) Isosurfaces of the density field at the
threshold level ρth= 0.2 for T3,2knots of various knot ratios R1/R0
(see Table I). Snapshots at times to t = 0,400,800. Unstable knots
are not shown. For complete evolution movies refer to [37].
0
50
100
150
200
250
0 400 800
t
1200 1600
zCM
Finite system size
Estimated boundary effects
T2, 3 cases
R1/R0=3/5
R1/R0=2/5
R1/R0=1/5
R1/R0=1/10
FIG. 8. (Color online) Position along the z axis (in units of
the healing length) of the center of mass of T2,3knots of various
knot ratios R1/R0 as a function of time. The filled points denote
the position where vortex knots break up. The horizontal lines
denote, respectively, the distance where boundary effects become
non-negligible and the finite system size along z.
vortex knot is centered at the point [Lx/2,Ly/2,2(R0+ R1)].
With this choice, the knot can propagate for a distance of
3 to 53/11 ? 4.8 times its maximum size before hitting the
opposite side of the computational domain corresponding to
z = Lz. The GPE is integrated in time using a split-step
methodwithantiperiodic(reflective)boundaryconditions.The
integrationtimestepis?t = 0.02smallerthatthefastestlinear
period Tc? 0.032. This value allows us to conserve the initial
energy and mass up to 3% and 1%, respectively, in all the
simulations. Details on the numerical algorithm can be found
in Refs. [35,36].
Figures 6 and 7 show the isosurfaces of the density field
corresponding to the threshold value ρth= 0.2 at the initial
conditions and at successive times for the T2,3 and T3,2
knots, respectively (broken knots will be discussed in the next
section). As expected, vortex knots move along the z direction
(the axis of symmetry of the torus), but also twist around it.
Qualitatively, vortex knots with small knot ratio R1/R0are
fast and long-lived, as they propagate along the z direction
0
50
100
150
200
250
0 400 800 1200
zCM
t
Finite system size
Estimated boundary effects
T3, 2 cases
R1/R0=3/5
R1/R0=2/5
R1/R0=1/5
R1/R0=1/10
FIG. 9. (Color online) As in Fig. 8 but for T3,2.
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0
0.5
1
1.5
2
2.5
0 400 800
t
1200 1600
vz/vring(R0)
T2, 3 cases
vring of radius R0
R1/R0=3/5
R1/R0=2/5
R1/R0=1/5
R1/R0=1/10
FIG. 10. (Coloronline)VelocitycomponentvzofT2,3versustime
of knots with various knot ratios R1/R0 before destroying (filled
points). Velocities are expressed in units of vortex ring velocity (7)
with quantum number n = 1 and radius R = R0.
without breaking. During the evolution, Kelvin waves [13]
appear; such waves are visible at the last stages of cases (a)
and (b).
In order to quantify the evolution of vortex knots and
compare one knot with others, we define the knot center of
mass rCM= (xCM,yCM,zCM) as
?
rCM=
VrH(ρth− |ψ|2)dV
?
VH(ρth− |ψ|2)dV
,
(6)
where H(...) is the Heaviside step function. Figures 8 and 9
show the z component zCMof the knot center of mass (shifted
with respect to the initial position) for the T2,3and T3,2cases,
respectively.Inbothcases,knotswithsmallerknotratioR1/R0
movefasterandpropagateforlongerdistancesbeforebreaking
up (a filled point at the end of each curve marks the breakup
point).
The z component of the velocity of a vortex knot is esti-
mated by evaluating vz(t) = [zCM(t + τ) − zCM(t)]/τ (where
0
0.5
1
1.5
2
2.5
3
3.5
0 400 800 1200
vz/vring(R0)
t
T3, 2 cases
vring of radius R0
R1/R0=3/5
R1/R0=2/5
R1/R0=1/5
R1/R0=1/10
FIG. 11. (Color online) As in Fig. 10 but for T3,2.
0
0.5
1
1.5
2
0 0.1 0.2 0.4 0.6 1
vz/vring(R0)
R1/R0
A2, 3=-1.38±0.08, B2, 3=1.55±0.02
A3, 2=-1.41±0.25, B3, 2=1.95±0.08
T2, 3
T3, 2
FIG. 12. (Color online) Averaged velocity components vzof T2,3
and T3,2vortex knots with various knot ratios R1/R0. Velocities are
expressed in units of vortex ring velocity (7) with quantum number
n = 1 and radius R = R0. Error bars correspond to one standard
deviation.
τ = 4 for numerical convenience). Figures 10 and 11 show
vz(t) measured in units of the vortex ring velocity [38]:
vring(R) =
nκ
4πR
?
ln
?8R
ξ
?
− 0.615
?
,
(7)
havingquantumnumbern = 1andradiusR = R0(notethatin
our nondimensional system the quantum of circulation is κ =
2π). It is apparent that vortex knots move with approximately
constant z velocity before either breaking up or reaching the
boundary of the computational domain, where the interaction
with the image slows them down.
It is instructive to analyze the mean and the standard
deviation of the vortex knots’ velocities measured in the
constant-velocity regimes. The results, expressed in units of
vring(R0), are shown in Fig. 12. Three conclusions can be
drawn from this figure:
FIG. 13. (Color online) Three successive snapshots showing how
the T2,3 vortex knot with knot ratio R1/R0= 2/5 breaks up into
two vortex rings. Here we plot two perspectives (up and to the
side of the vortex propagation) of the isosurfaces of the density
field corresponding to the threshold level ρth= 0.2. For a complete
evolution movie refer to [37].
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DAVIDE PROMENT, MIGUEL ONORATO, AND CARLO F. BARENGHIPHYSICAL REVIEW E 85, 036306 (2012)
FIG. 14. (Color online) Two sequences of three successive snapshots showing how the T3,2vortex knot with knot ratio R1/R0= 3/5 breaks
up into three vortex rings. Here we plot two perspectives (up and to the side of the vortex propagation) of the isosurfaces of the density field
corresponding to the threshold level ρth= 0.2. For a complete evolution movie refer to [37].
(1) T2,3 knots are slower than T3,2 knots with the same
knot ratio. This is physically expected as the velocity field
of torus knots at large distance is similar to the velocity field
of vortex rings with multiple circulation: T2,3corresponds to
circulation of 2κ and T3,2to 3κ. According to Eq. (7), the
velocity is directly proportional to the circulation, and so T2,3
knots should be slower than T3,2ones. However, this simple
consideration does not apply well to knots, because we would
haveexpected,forthesmallknotratiotested(R1/R0= 1/10),
a scaled velocity of vz/vring(R0) ? 2 and vz/vring(R0) ? 3
for T2,3and T3,2respectively, and this is not the case.
(2) Thez-velocitycomponentscaleswiththeknotratioand
can be parametrized as
?R1
whereAp,qandBp,qarecoefficientswhichrefertothegeneric
torus knot Tp,q. Values of Ap,q and Bp,q for the knots T2,3
and T3,2are reported in Fig. 12. It is interesting to observe
that A2,3? A3,2, indicating an evidence of universality for a
generic knot which will be studied in future works.
(3) The z-velocity component of short-lived knots (i.e.,
knots that decay before reaching the computational bound-
aries) is less or similar to vring(R0). On the contrary, knots
that preserve their shapes within the computational domain
are characterized by vz> vring(R0).
vz
R0
?
= Ap,qR1
R0
+ Bp,q,
(8)
IV. THE BREAKING OF A KNOT
In our simulations we have observed that some vortex
knotsbreakupintotopologicallysimplerobjects[37].Wefirst
analyze the unstable T2,3knots; these are knots corresponding
to knot ratios R1/R0= 2/5,3/5. In Fig. 13 we show three
snapshotsofthedecayoftheT2,3knotwithratioR1/R0= 2/5.
It is apparent that the knot breaks into two vortex rings via
threesimultaneousself-reconnectionevents (see,inparticular,
the snapshot corresponding to t = 224). The decay of the
vortex knot T2,3with ratio R1/R0= 3/5, not shown here, is
similar.
On the contrary, T3,2 vortex knots break in a different
manner. As shown in Fig. 14, the vortex knot T3,2with knot
ratio R1/R0= 3/5 first decays in one vortex ring and two
linked vortex rings via two simultaneous self-reconnection
events (snapshot at time t = 192). Subsequently, the small
free ring escapes from the other rings, which undergo two
simultaneous reconnection events that create two unlinked
vortex rings (snapshot at time t = 428). The last step is
remarkable: there is no apparent reason why two linked
vortex knots should in principle unlink into two vortex rings
(by making two simultaneous reconnection events) without
forming a single ring (by one reconnection event).
The T3,2vortex knot with ratio R1/R0= 2/5 qualitatively
decays in the same way, producing a set of three unlinked
vortex rings, but the steps are quite different. In the first step, a
freevortexringandtwolinkedvortexringsareagainproduced.
However,thefreering,whichisinitiallylocatedbehindthetwo
linked vortex rings, is smaller and faster than the other rings.
As a consequence, it reconnects with the two linked vortex
rings, as shown in Fig. 15 (snapshot t = 752). At this point
the reformed knot breaks up, undergoing the same sequence
previously described forthe T3,2withratioR0/R1= 3/5 case,
FIG. 15. (Color online) Three successive snapshots showing how
the T3,2knot with knot ratio R1/R0= 2/5 breaks into three vortex
rings. Here we plot two perspectives (up and to the side of the vortex
propagation) of the isosurfaces of the density field corresponding to
the threshold ρth= 0.2. For a complete evolution movie refer to [37].
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VORTEX KNOTS IN A BOSE-EINSTEIN CONDENSATEPHYSICAL REVIEW E 85, 036306 (2012)
and the outcome is a set of three vortex rings (snapshot at time
t = 952). Note that, in the last snapshot, the first knot (in the
sense of the position) has split into two smaller vortex rings
via a self-reconnection event which is probably a consequence
of its Kelvin wave oscillations.
V. CONCLUSIONS
We have numerically analyzed the existence and evolution
ofvortexknotsintheGPEmodelofasinglescalarcondensate.
We have proposed a novel numerical technique for creating ab
initio vortex knots in the wave function of the condensate. In
particular,wehavefocusedournumericalcomputationsonthe
two simplest (in the topological sense) vortex knots, T2,3and
the T3,2.
We have analyzed the shape preservation of such knots
with respect to the knot ratio R1/R0. We have found that a
knot can break up into simple rings during the propagation,
or preserve its knot type within our computational domain.
Our numerical experiments clearly show that a small knot
ratio (R1/R0= 1/10,1/5) increases the lifetime, whereas a
large knot ratio (R1/R0= 2/5,3/5) decreases it, in agreement
with[20].IntheLIAlimititisprovedthatanytorusknotsTp,q
with p > q > 1 are unstable, while in the case q > p a knot
could be neutrally stable but quickly changes its knot type
during the evolution [18]. However, in our results using the
Gross-Pitaevskiimodel,thesedifferencesdonotseemtooccur,
as for small knot ratios both T2,3and T3,2behave similarly,
are long-lived structures, and preserve their shape within our
computational box.
We have found that vortex knots propagate essentially as
vortex rings. We have measured the vortex knot velocities
along the torus symmetry axis and shown that the velocity
depends linearly on the knot ratio for both T2,3and T3,2.
Finally, we have studied the details of the breakup of vortex
knots. Although we do not have a theoretical explanation for
the breakup, we have observed evidences of generic breaking
behavior: T2,3knots always break into two vortex rings via a
threesimultaneousself-reconnectionevent,whereasT2,3knots
first decay into three vortex rings via two simultaneous self-
reconnections which create a free ring and two linked rings,
then undergo two simultaneous reconnections which split the
resulting link.
Webelievethatourworkopensupnewinterestingproblems
in the field of fluid topology applied to superfluids and Bose-
Einstein condensates. The natural developments of our study
will be a theoretical investigation of the stability of vortex
knots and an experimental study of the creation of a knotted
initial condition in an atomic condensate.
ACKNOWLEDGMENTS
The authors acknowledge G. Boffetta, F. De Lillo,
A. L. Fetter, and A. J. Youd for comments and suggestions,
and a CINECA Award (No. HP10BQW4X9) 2011 for the
availability of high-performance computing resources and
support. C.F.B. is grateful to the Leverhulme Trust and to
the EPSRC for financial support. D.P. expresses his gratitude
to LLNS VISIT, POV-RAY, and GNUPLOT developers for the free
software used for the visualization of numerical results.
[1] W. H. Thomson, Trans.-R. Soc. Edinburgh 25, 217 (1869).
[2] C. Adams, The Knot Book (W.H. Freeman Publishers,
New York, 1994).
[3] H. K. Moffatt, J. Fluid Mech. 35, 117 (1969).
[4] H. K. Moffatt, Nature (London) 347, 367 (1990).
[5] A. Y. K. Chui and H. K. Moffatt, Proc. Math. Phys. Sci. 451,
609 (1995).
[6] L. Faddeev and A. J. Niemi, Nature (London) 387, 58
(1997).
[7] E. Babaev, Phys. Rev. Lett. 88, 177002 (2002).
[8] E. Babaev, Phys. Rev. B 79, 104506 (2009).
[9] P. M. Sutcliffe and A. T. Winfree, Phys. Rev. E 68, 016218
(2003).
[10] J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, Nature
(London) 432, 165 (2004).
[11] M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J.
Padgett, Nat. Phys. 6, 118 (2010).
[12] U. Tkalec, M. Ravnik, S. Copar, S. Zumer, and I. Musevic,
Science 333, 62 (2011), [http://www.sciencemag.org/content/
333/6038/62.full.pdf].
[13] R.Donnelly,
Quantized
3 (CambridgeUniversity
1991).
[14] P. M. Walmsley and A. I. Golov, Phys. Rev. Lett. 100, 245301
(2008).
Vorticesin Helium
Cambridge,
II,Vol.
UK, Press,
[15] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A.
Collins, C. W. Clark, and E. A. Cornell, Phys. Rev. Lett. 86,
2926 (2001).
[16] D. R. Poole, H. Scoffield, C. F. Barenghi, and D. C. Samuels,
J. Low Temp. Phys. 132, 97 (2003).
[17] S. Kida, J. Phys. Soc. Jpn. 51, 1655 (1982).
[18] A. Calini and T. Ivey, J. Phys. A: Math. Theor. 44, 335204
(2011).
[19] A. Calini, S. F. Keith, and S. Lafortune, Nonlinearity 24,
3555 (2011).
[20] R.Ricca,D.C.Samuels,andC.F.Barenghi,J.FluidMech.391,
29 (1999).
[21] F. Maggioni, S. Alamri, C. F. Barenghi, and R. L. Ricca, Phys.
Rev. E 82, 026309 (2010).
[22] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Vol.
116 (Oxford University Press, New York, 2003).
[23] P. G. Saffman, Vortex Dynamics (Cambridge University Press,
Cambridge, UK, 1991).
[24] J. Koplik and H. Levine, Phys. Rev. Lett. 71, 1375 (1993).
[25] J. Hietarinta, J. J¨ aykk¨ a, and P. Salo, in proceedings of
the Workshop on Integrable Theories, Solitons and Duality,
PoS(unesp2002) (SISSA, Trieste, 2002), article ID 017.
[26] M. Kobayashi, J. Phys.: Conf. Ser. 297, 012013 (2011).
[27] J. Hietarinta, J. Palmu, J. J¨ aykk¨ a, and P. Pakkanen, New J. Phys.
14, 013013 (2012).
036306-7

Page 8

DAVIDE PROMENT, MIGUEL ONORATO, AND CARLO F. BARENGHI PHYSICAL REVIEW E 85, 036306 (2012)
[28] E. Babaev, L. D. Faddeev, and A. J. Niemi, Phys. Rev. B 65,
100512 (2002).
[29] E. Babaev, Phys. Rev. D 70, 043001 (2004).
[30] Y. Kawaguchi, M. Nitta, and M. Ueda, Phys. Rev. Lett. 100,
180403 (2008).
[31] J. Liang, X. Liu, and Y. Duan, Europhys. Lett. 86, 10008
(2009).
[32] J. L. Helm, C. F. Barenghi, and A. J. Youd, Phys. Rev. A 83,
045601 (2011).
[33] E. B. Sonin, Europhys. Lett. 97, 46002 (2012).
[34] N. Berloff, J. Phys. A: Math. Gen. 37, 1617 (2004).
[35] D. Proment, S. Nazarenko, and M. Onorato, Special Issue on
Small Scale Turbulence, Physica D: Nonlinear Phenom. 241,
304 (2012).
[36] W. Bao and H. Wang, J. Comput. Phys. 217, 612 (2006).
[37] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevE.85.036306formoviesshowingknotdynam-
icsoffivedifferentsimulations,includingevolutionoflong-lived
knots and breakup of short-lived knots.
[38] P. Roberts and J. Grant, J. Phys. A: Gen. Phys. 4, 55 (1971).
036306-8