Table Of ContentLowest Landau Level vortex structure of a Bose-Einstein condensate rotating in a
harmonic plus quartic trap
Xavier Blanc, Nicolas Rougerie
Universit´e Paris 6, Laboratoire Jacques-Louis Lions,
175 rue du Chevaleret, 75013 Paris, France.
(Dated: February 2, 2008)
We investigate the vortex patterns appearing in a two-dimensional annular Bose-Einstein con-
8 densaterotatingin aquadraticplusquarticconfiningpotential. Weshowthat in thelimit ofsmall
0 anharmonicity the Gross-Pitaevskii energy can be minimized amongst the Lowest Landau Level
0
wave functions and use this particular form to get theoretical results in the spirit of [A. Aftalion
2
X. Blanc F. Nier, Phys. Rev. A 73, 011601(R) (2006)]. In particular, we show that the vortex
n pattern is infinitebut not uniform. We also compute numerically thecomplete vortex structure: it
a isan Abrikosov lattice strongly distorted neartheedgesof thecondensatewith multiplyquantized
J vortices appearing at thecenter of the trap.
3
2 PACSnumbers: 03.75.Lm,67.85.Bc
]
h I. INTRODUCTION force creates a hole in the condensate at the center of
c
the trap[12,13, 14]. Forrotationspeeds Ω&Ω thereis
e c
m When rotated, superfluids are expected to exhibit a a vortex lattice in an annular region and a central hole
rich structure of quantized vortex states. This has been where the density vanishes and around which there is a
-
t experimentally confirmed for Bose-Einsteincondensates: circulation. However, this hole is not usually referred to
a
t the techniques developed to create a single vortex state as a giant vortex as not all the circulation is contained
s [1, 2] rapidly led to arrays containing up to several hun- in a central vortex [12].
.
t dreds of vortices [3, 4, 5, 6]. Typically these condensates Finally, for even larger rotation speeds the condensate
a
m areconfinedbyharmonicpotentials,andthelargevortex continues to expand radially with constant area so that
arrays are obtained for angular velocities Ω approaching the width of the annulus decreases and all the vortices
-
d the radial trap oscillator frequency ω. are expected to retreat into the central hole, resulting
n With these type of potentials, the limit Ω ω is singu- in a pure irrotational state with macroscopic circulation
o lar: as the centrifugal force balances the t→rapping force [12, 13, 15], often called a giant vortex [16, 17].
c the Thomas-Fermiradiusofthe condensatedivergesand In this paper we address the regime Ω & Ωc. We would
[
the central density decreases towards zero. like to know the precise repartition of vortices, both
1 In the experiments [7, 8], a blue detuned laser directed in the annular Thomas-Fermi region where we expect
v alongtheaxialdirectionaddsaquarticcomponenttothe that they form a regular lattice, and in the region of
1 usual harmonic potential, resulting in a potential of the low density. Indeed, we believe that there should be
7
form : ”invisible” vortices beyond the external radius of the
5
condensate (as is the case for a harmonically trapped
3 1 k
. V(r)= mω2r2+ r4 (I.1) condensate in fast rotation [14]) and in the central hole
1 2 4 [12].
0
In the limit of weak anharmonicity, the single particle
8 where m is the mass of the atoms. The nice feature of
states are restricted to the Lowest Landau Level (LLL)
0 this potential is that the centrifugal force, varying as
: Ω2r can always be compensated by the trapping force wave functions [14]. Thus our analysis consists in mini-
v varying as (mω2r+kr3), so that one can explore the mizingtheGross-PitaevskiienergyintheLLL.Usingthe
i
X region Ω −ω. explicitexpressionofthe projectionontothe LLLweget
≥ theoreticalresultsinthe spiritof[18, 19]. We provethat
r Condensates rotating in this kind of trap present a
a a wave function minimizing the Gross-Pitaevskii energy
very rich variety of vortex phases. At sufficiently slow
in the LLL cannot have a finite number of zeroes, thus
rotation speeds, the system is not expected to have any
the vortices of the condensate cannot lie in a bounded
vortex. For increasing Ω there is a sequence of states
domain. This implies that there should be an infinite
with more andmore vorticity,where singly andmultiply
number of invisible vortices. We also construct critical
quantized vortices may be present at the same time
points for the Gross-Pitaevskii energy by distorting a
[9, 10, 11].
regular hexagonal vortex lattice in the following way:
At rotation speeds Ω ω, a large triangular array
∼ first, using a theta function as is done for the Abrikosov
of singly quantized vortices appears, which is not
problem [20], we construct a function u which modulus
qualitatively different from what happens in a purely
is periodic over an infinite regular hexagonal lattice and
harmonic trap. The difference is when Ω is taken larger
which vanishes only at the points of the lattice with
than ω: the condensate continues to expand radially
simple zeroes. Next we multiply u by a slow varying
and there is a critical speed Ω at which the centrifugal
c
2
profile α (which corresponds to the square root of the ent studies on harmonically trapped condensates rotat-
coarse-grain average of the atom density over several ingatspeedsclosetothetrapfrequency(seeforexample
cells [14, 21]). We then project αu onto the LLL. It can [14, 22, 23]). In [9, 10] the LLL approximation is used
be shown[19]thatprojectingthisfunction ontothe LLL to determine the phase diagramofa condensate trapped
implies a distortion of the vortex lattice. byaharmonicplus quarticpotentialinthe limit ofweak
The state of a condensate in the LLL is entirely known interactions and small anharmonicity.
from the positions of its vortices. This allows us to The LLL consists of functions of the form ψ(z) =
adapt the numerical method of [14] and minimize the f(z)e Ωz2/2 where z is the complex variable x +ix
− | | 1 2
Gross-Pitaevskii energy functional of the condensate by andf isaholomorphicfunction. ForanLLLfunctionψ,
varying the locations of its vortices. As is expected in the Gross-Pitaevskiienergy reduces to
the range of parameters we chose, the condensate lies
in an annular domain in which vortices are regularly (ψ)=Ω+ (ψ)
GP LLL
E E
distributed [12, 13]. We also find that there are invisible 1 Ω2 k G
vortices both beyond the outer radius of the condensate LLL(ψ)= − z 2+ z 4 ψ 2+ ψ 4 dz,
and in the central hole. The vortex lattice is strongly E ZC(cid:18)(cid:18) 2 | | 4| | (cid:19)| | 2| | (cid:19)
(II.2)
distorted at the inner and outer edges of the condensate
and we find that multiply quantized vortices appear at so that we will minimize the energy in the LLL
LLL
the center of the trap. under the mass constraint ψ 2 =1. E
| |
The paper is organized as follows: in Section II we Minimizing (II.2) without the assumption that ψ is in
describe our model and general approach. We show the LLL gives the main scaRles of the problem. One gets
how reducing the minimization of the Gross-Pitaevskii the Thomas-Fermi distribution
efonrerwghyicthowtehedeLrLivLealenaEdsulteor-Laangerwangveareiqautiaotnioanl psarotibsfileemd ψ (r)2 =max µ+ (Ω22−1)r2− k4r4,0 (II.3)
TF
by the minimizer. We then use this equation in Section | | G !
III toshowthattheminimizer necessarilyhasaninfinite
where r is the radial coordinate. We refer to [12, 14]
number of zeroes and to construct critical points for
for the detailed study of such a distribution. In particu-
the energy. In Section IV we describe our numerical
lar one gets the expression of Ω (see equation (III.19)),
method and comment our results, relating them to our c
which allows to deduce that for 1 Ω at most of the
theoretical approach. Finally we give our conclusions in
order of k2/3G1/3 we have Ω Ω| .−Th|en (ψ) has
Section V. c LLL
∼ E
theorderofk1/3G2/3 andthe spatialextensionofψ is
TF
of the order of (G/k)1/6.
TheLLLapproximationisvalidwhentwoconditionsare
fulfilled[14,21]: (i)theexcessenergy (ψ )issmall
II. MODEL-APPROACH ELLL TF
compared to the splitting between the LLL and the first
excited Landau level ( 2 in our units), and (ii) the
A. Gross-Pitaevskii energy and LLL reduction ∼
spatial extension of the condensate is large compared to
the vortices interdistance, so that coarse-grain averages
We assume a strong confinement along the rotation of atomic and vortex densities over several vortex cells
axis, so that we are in an almost 2-D situation. We take make sense. Using the scalings derived from (II.3), this
ω, ~ω and ~/(mω) as units of frequency, energy and conditions reduce to
lengthrespectivelyandconsiderthe Gross-Pitaevskiien-
1
ergy of thepcondensate in the rotating frame, k G and 1 Ω .k2/3G1/3. (II.4)
≪ ≪ √k | − |
1 2 In Section IIIC we will give more details on how the
(ψ)= ψ iΩx ψ
GP ⊥ scalings (II.4) are derived and see that taking into ac-
E R2 2 ∇ −
Z (cid:18) count the LLL constraint does not modify the scales of
+ 1−Ω2 x2(cid:12)(cid:12)+ k x4 ψ 2(cid:12)(cid:12)+ G ψ 4 dx, (II.1) the problem. Indeed, we show in Section IIIB that al-
2 | | 4| | | | 2| | though a function suchas (II.3) cannot be in the LLL, a
(cid:18) (cid:19) (cid:19)
properdistributionofvorticesallowsonetoapproximate
where x = (x1,x2), x⊥ = ( x2,x1) and G is a dimen- very closely such a profile in the LLL, modifying only
−
sionless coefficient characterizing the strength of atomic slightly the averagedatomic density of the condensate.
interactions(see[12,14]forexample). Weminimize
GP
under the mass constraint ψ 2 = 1. The main ideEa of
| |
ouranalysisis to restrictthe minimizationof to the B. Mathematical framework for the small
GP
first eigenspace of the opeRrator iΩxE 2, corre- anharmonicity regime
⊥
− ∇−
spondingtotheeigenvalueΩ. ThisistheLowestLandau
(cid:0) (cid:1)
Level, introduced in the context of Bose-Einstein con- We now describe the mathematical framework we are
densationbyHo[21]andthensuccessfullyusedindiffer- going to use in Section III. It was introduced in [18,
3
19] for the theoretical study of an harmonically trapped Indeed,theweakderivativeoftheenergyF atf along
LLL
condensate. g is given by
We define a small parameter
ε=k1/3 (II.5) DFLLL(f) g = β z 2+ 1 z 4 f¯ge−|z|2/ε
· C − | | 4| |
Z (cid:18)(cid:18) (cid:19)
corresponding to our small anharmonicity regime and +G f 2f¯ge 2z2/ε dz.
study the asymptotics of the problem as ε 0. We 2| | − | |
→ (cid:19)
take
Usingintegrationbypartsand∂ f =∂ g =0onthefirst
z¯ z¯
Ω=1+βk2/3 (II.6) termandthe factthatΠ (g)=g onthe secondterm,we
ε
obtain (II.11).
with β G1/3 so that we are in the range of pa-
∼ Wegivetwoequivalentformsof(II.11)thatweshallneed
rameters (II.4) where the LLL approximation is justi-
in the sequel:
fied. We rescale distances by making the change of vari-
ables φ(z)=√εψ(√εz), the energy becomes ELLL(ψ)= βM f + 1 M2f +εM f + Gf¯(ε∂ )[f2(z/2)]=µf,
εELLL(φ) with − ε 4 ε ε 2 z
(II.12)
(cid:0) (cid:1)
1 G
E (φ)= β z 2+ z 4 φ2+ φ4 dz.
LLL
C − | | 4| | | | 2| | 1 1
Z (cid:18)(cid:18) (cid:19) (cid:19)(II.7) ( β+ ε)Πε(z 2f)+ Πε z 2Πε(z 2f)
− 4 | | 4 | | | |
We minimize this new energy amongst all functions φ in +GΠ ((cid:0)z2/ε f 2f)=(cid:1)µf. (II.13)
the LLL satisfying the mass constraint φ2 =1. ε −| | | |
| |
For every φ in the LLL we define the function f(z) =
φ(z)ez2/2ε. By definition of the LLL, Rf belongs to the The operator f¯(ε∂z) is defined by
| |
Fock-Bargmannspace
+
f¯(ε∂ )[g]= ∞a (ε∂ )kg
z k z
Fε = f holomorphic , C|f|2e−|z|2/εdz <∞ . (II.8) Xk=0
(cid:26) Z (cid:27)
if f(z) = a zk. Equation (II.12) is obtained
The space is a Hilbert space for the scalar product k
Fε from (II.11) as in [18] with some algebra on the
P
non-linear term. To get (II.13) we use an integra-
f,g = f(z)g(z)e−|z|2/εdz. tion by parts to show that M f = Π (z 2f), then
h i ZC M2f =Π (z 2Π (z 2f)) and weεget the rεes|ul|t.
ε ε | | ε | |
The pointofintroducingsuchaspaceisthatthe orthog- In the following section we prove our theoretical results.
onal projection of any function g onto is explicitly Using (II.12) we show below that any minimizer f
ε
F
known [24, 25]: of (II.10) (and thus any minimizer φ of (II.7)) has
an infinite number of zeroes for ε small enough. The
Π (g)(z)= 1 ezz¯′/εe z′2/εg(z )dz , (II.9) construction of critical points is done using (II.13).
ε −| | ′ ′
πε C
Z
so that if we write the variational problem (II.7) for f
rather than for φ, namely if we look for f minimiz-
∈Fε III. ANALYTICAL STUDY
ing
In this section we present the results we are able to
FLLL(f)= β z 2+ 1 z 4 f 2e−|z|2/ε derive from equations (II.12) and (II.13).
C − | | 4| | | |
Z (cid:18)(cid:18) (cid:19)
+G f 4e 2z2/ε dz (II.10)
− | |
2| | A. Infinite number of zeroes
(cid:19)
undertheconstraint f,f =1,weareabletoderivethat
h i We show that any minimizer f of (II.10) (and there-
f satisfies the equation
fore any minimizer ψ of (II.2)) has an infinite number of
βM f + 1 M2f +εM f +GΠ (e z2/ε f 2f)=µf zeroes if ε is small enough. The argument is by contra-
− ε 4 ε ε ε −| | | | diction and in two steps.
(II.11)
(cid:0) (cid:1)
1. Suppose f has a finite number of zeroes. Then one
where µ is the chemical potential coming from the mass
may write f(z) = P(z)eϕ(z) where P is a polyno-
constraint and M is the operator defined by
ε
mial andϕ is a holomorphic function. Nowf
ε
M =ε∂ z. and the condition f 2e z2/εdz < im∈plFies
ε z C| | −| | ∞
R
4
that Re(ϕ(z)) z 2/(2ε). It is well-known (see and
≤ | |
[26] for example) that a holomorphic function can
satisfy this condition only if it is a polynomial of 6 3bG 2/3 βε 7ε2
β2
degree less than 2. Therefore we know that 5 8π − − 2 − 16
(cid:18) (cid:19)
f(z)=P(z)eα1z+α2z2 (III.1) 5 Ge 1/12 4/5
− ε 2/5+O(ε 1/5) (III.6)
− −
≥ 4 2π
and the integrability condition on f implies α (cid:18) (cid:19)
2
≤
1/(2ε). Injecting (III.1) in (II.11) and comparing which is a contradiction if ε is small enough.
the exponential growth of the different terms of
(II.11) as in [19] yields α = α = 0. So, if f
1 2
Intherestofthepaperwetakeanεsmallenoughforthe
has a finite number of zeroes, it is a polynomial.
inequation (III.6) not to be verified. We then conclude
2. Now, suppose f is a polynomial of degree thatf hasaninfinitenumberofzeroes. Asthereisalimit
n and inject this in (II.12). The term onhowclosevorticescanbeinthisregime(see[21]),they
−βMεf + 41 Mε2f +εMεf is a polynomial of de- cannot lie in a bounded domain and the vortex pattern
gree n, therefore (II.12) implies that the term extends to infinity.
Gf¯(ε∂ )[f2(z(cid:0)/2)] is also(cid:1)of degree n. But
2 z
(ε∂ )k[f2(z/2)] is of degree 2n k, so that f must
z
beoftheformf(z)=czn. Injec−tingthisalasttime B. Construction of critical points
in (II.12), using the improved Stirling [27] formula
and the condition f,f = 1 yields a condition on As is the case for harmonically trapped condensates
h i
n: in fast rotation, we expect that in the range of parame-
ters we explore the scales of the problem will decouple.
Ge 1/12
µ+βε βnε+ − Namely we expect any minimizer of (II.7) to be of the
≥− 2πε√n formαuwhereuvariesonthescaleofthevortexpattern
ε2 n2ε2 3nε2 (which is small compared to the size of the condensate)
+ + + . (III.2)
and α is a slow varying profile giving the general shape
2 4 4
ofthe condensate. In this sectionweshow thatalthough
We now bound the chemical potential µ: taking
suchafunctionisnotintheLLL,onecanapproachitby
the -scalarproductofeachside of (II.11) with f
Fε anLLL function whichis analmostcriticalpoint for the
yields
energy (II.7). We also relate the effect of the projection
βε ε2 onto the LLL to the distortion of a regular lattice in the
µ 2FLLL(f)+β2 + . (III.3) region of low atomic density.
≤ − 2 16
More precisely we introduce
Hereweusedthefactthatthespectrumof βM +
ε
14 Mε2+εMε is bounded below uniform−ly with u (z)=e z2/2εf (z), f (z)=ez2/2εΘ τI z,τ
respect to ε. We shall see in Section IIIC (see τ −| | τ τ πε
(cid:0) (cid:1) (cid:18)r (cid:19)
(III.22)) that (III.7)
where τ =τ +iτ is any complex number and
R I
FLLL(f)=ELLL(fe−|z|2/2ε)
+
3 3bG 2/3 Θ(v,τ)= 1 ∞ ( 1)neiπτ(n+1/2)2e(2n+1)πiv. (III.8)
≤ε−1ELLL = 5(cid:18) 8π (cid:19) −β2!. i n=X−∞ −
The Θ function has the property (see [28] for more de-
Thus we have
tails) Θ(v+k+lτ,τ) = ( 1)k+le 2iπlve iπlτΘ(v,τ) so
− −
−
βε 7ε2 that u (z) is periodic overthe lattice πεZ πεZτ,
2ε−1 LLL+β2 | τ | τI ⊕ τI
E − 2 − 16
and vanishes at each point of the latticqe. q
Ge−1/12 n2ε2 3nε2 The interest of introducing such functions is twofold.
βnε+ + + (III.4)
≥− 2πε√n 4 4 Firstly itis known[28]that anyfunction v whosemodu-
lus is periodic overthe lattice πεZ πεZτ, vanishes
and the left-hand side of (III.4) is bounded uni- τI ⊕ τI
formly with respect to ε. Minimizing the right- exactly on the points of the lqattice wiqth simple zeroes
hand side of (III.4) with respect to n (taken as a and such that g = vez2/2ε is holomorphic must be pro-
| |
continuousvariableasitshouldbe verylargewhen portional to u . Secondly, the function f is a solution
τ τ
ε is small) for fixed ε yields to the Abrikosov problem (see [18, 19, 20, 29])
n∼ Ge2−π1/12 2/5ε−6/5 (III.5) Π(|fτ|2e−|z|2/εfτ)=λτfτ, with λτ = |uτ|2 b((τI)I,I.9)
(cid:18) (cid:19) (cid:10) (cid:11)
5
and contribution. With such a profile and taking µ = ν in
(III.14) we get
b(τ)= |uτ|4 = e πkτ l2/τI. (III.10)
h(cid:10)|uτ|2i(cid:11)2 kX,l∈Z − | −| ( β+ 1ε)Πε(z 2fα,τ)+ 1Πε z 2Πε(z 2fα,τ)
− 4 | | 4 | | | |
Equation(III.9)issimilarto(II.13)withoutthepotential +GΠ (e z2/ε f 2f )=ν(cid:0)f +O(ε1/4) (cid:1)(III.17)
ε −| | α,τ α,τ α,τ
term and with µ = λ so that one can expect to obtain | |
τ
a solution of (II.13) by a slight modification of f . We
τ so that f is almost a critical point for the energy
α,τ
refer to [18] and the references therein for details on the
(II.10).
quantity b(τ). Let us just mention that it is minimum
Wenowexplaintheeffectofourprocedureonthe vortex
(b(τ) 1.16) for τ = e2iπ/3, which corresponds to a pattern. The property u =αu +O(ε1/4) shows that
∼ α,τ τ
hexagonal lattice.
on the support of α the zeroes of u are distributed
α,τ
We define
close to those of u , on a regular hexagonal lattice. To
τ
Π (αf ) get information on the ”invisible vortices” lying in the
ε τ
fα,τ = Π (αf ),Π (αf ) 1/2 (III.11) region of low density, we evaluate the number N(R) of
h ε τ ε τ i zeroes of fα,τ in a ball of radius R. The Cauchy formula
yields
and
uα,τ =fα,τe−|z|2/2ε (III.12) N(R)= R 2πdθ eRz′/εe−|z′|2/εz′α(z′)fτ(z′e−iθ)dz′.
2πε eRz′/εe z′2/εα(z )f (z e iθ)dz
where α C0,1/2(C,C) is a slow varying profile with Z0 R −| | ′ τ ′ − ′
(III.18)
compact s∈upport and α2 = 1. We use Lemma 5.4 in R
Using a Laplace method to evaluate the integrals, we
| |
[19], which states that
see that their ratio is bounded for large R, so that
R
N(R) R/ε. AregularlatticewouldgiveN(R) R2/ε,
u =αu +O(ε1/4). (III.13) ∝ ∝
α,τ τ so we deduce that the lattice is strongly distorted out-
This, together with the fact that u 2 is periodic over a sidetheexternalradiusofthecondensate. Notethatthis
τ
lattice of period bounded by O(ε1|/2)|yields method does not give information on the distribution of
vortices in the central hole when the condensate has an
annular form, for this would correspond to R small, and
1 1
( β+ ε)Πε(z 2fα,τ)+ Πε z 2Πε(z 2fα,τ) the Laplacemethod is not efficientin this case. We refer
− 4 | | 4 | | | |
toournumericalsimulationsinSectionIVforthe vortex
+GΠ (e z2/ε f 2f(cid:0) ) µf = (cid:1)
ε −| | α,τ α,τ α,τ structure in the central hole.
| | −
1
Π (( µ β z 2+ z 4+Gb(τ)α2)αΠ f )+O(ε1/4)
ε ε τ
− − | | 4| | | |
(III.14) C. Evaluation of physical quantities
and
Here we analyze the TF profile (III.16), or rather
1 1
F (f )=E (u )= α˜(z)=√ε− α(z√ε− ), to evaluate some relevant physi-
LLL α,τ LLL α,τ
cal quantities in the original scaling.
1 b(τ)G
β z 2+ z 4 α2+ α4 dz+O(ε1/4). Theanalysisofaprofilesuchas(III.16)hasalreadybeen
C − | | 4| | | | 2 | | done in[12, 14], so weonly adaptandsummarizethe re-
Z (cid:18)(cid:18) (cid:19) (cid:19)
(III.15) sults. The critical rotation speed for the condensate to
develop a central hole is
Equations (III.14) and (III.15) are the scale decoupling
we expected: the only contribution of the lattice to the 3k2bG 1/3
energy is throughthe coefficientb(τ), which is minimum Ω =1+ . (III.19)
c
8π
for the hexagonal lattice, τ = e2iπ/3. From now on we (cid:18) (cid:19)
shallnoteb=b(e2iπ/3) 1.16. Wenowhavetominimize
∼ For subcritical velocities, the behavior of the condensate
(III.15),whichisanenergysimilarto(II.2),withrespect
is not qualitatively different from that of a harmonically
to the profile α which is not in the LLL, so that the
trapped condensate, so we focus on velocities Ω Ω
c
minimization is performed in the usual way. We obtain ≥
which is equivalent to β 3bG 1/3. Then the inner
the Thomas-Fermi profile ≥ 8π
and outer radius of the condensate R are given by the
ν+β z 2 1 z 4 relations (cid:0) (cid:1) ±
α2(z)=max | | − 4| | ,0 (III.16)
| | bG
(cid:18) (cid:19) 24bG 1/3
whereνisthechemicalpotentialassociatedwiththecon- R+2 +R−2 =4βk−1/3, R+2 −R−2 = k ,
(cid:18) (cid:19)
straint α2 = 1 and b takes into account the vortices (III.20)
| |
R
6
the chemical potential is trapped condensate) would be to take
3bG 2/3 n
µ= (cid:18) 8π (cid:19) −β2!k1/3 (III.21) φ(z)=A bjzje−Ω|z|2/2
j=0
X
and the energy is
andvarythecoefficientsb . Theinterestofourapproach
j
2/3 is to give a direct access to the exact repartition of vor-
3 3bG
LLL = β2 k1/3. (III.22) tices, whereas the alternative method would require to
E 5 8π − !
(cid:18) (cid:19) compute the roots of a polynomial of degree n, which is
a delicate task for large n. In particular, varying the co-
This is slightly different from the result obtained in [12]
efficients couldprobably notgive the precise locationsof
with the solid-body approximation. Indeed, our analysis
invisible vortices.
allowsone,throughthecoefficientb,totakeintoaccount
We used a conjugate gradient method with a Goldstein
more preciselythe contributionofthe vortexpatterns to
and Price line-search. The integrals are computed using
the energy and atomic density, whereas the solid-body
the Gauss-Hermite method, and we take enough Gauss
approximationimpliesaninfiniteregularlatticewithpre-
points for the computations to be exact. This results
scribedvolumeofthecell. Theseresultsandthefactthat
in quite expensive calculations, but we have been able
b 1.16implythatouranalysisisvalidwithintherange
∼ to numerically construct condensates with up to 120
of parameters we announced in Section 2.1 : equations ∼
vortices.
(III.20)yieldthatthespatialextensionofthecondensate
is large compared to 1 (which is the order of the vortex
pattern spacing) if
B. Results
k G.
≪
We now see that the condition β G1/3 (or 1 Ω .
k2/3G1/3) is equivalent to Ω Ω ∼and ensures| th−at |the
c
∼
energy isoforderk1/3G2/3,sothattheenergycon-
LLL
E
dition for the LLL approximation to be valid reduces to
1
G .
≪ √k
Now, as we have both k G and G 1 , we see that
≪ ≪ √k
necessarily k 1, which justifies our study of a small
≪
anharmonicity regime.
FIG. 1: Vortex structure and atomic density for G = 3,k =
10−4,β =1(Ω−1=2.210−3). Thereare67vorticesintotal,
thecentral vortex is constituted of 11 single vortices.
IV. NUMERICAL SIMULATIONS
We show in Fig. 1 and Fig. 2 typical examples of
A. Approach
configurations we numerically computed. The qualita-
tive features of the vortex patterns and atomic densities
We want to numerically approach the minimizer φ of
confirmourtheoreticalresultsandareingoodagreement
in the LLL. We write
LLL with existing theoretical and numerical studies [12, 13].
E
φ(z)=P(z)e Ωz2/2 (IV.1) Note however that the numerics become quite intricate
− | | for large number of vortices, which accounts for the rel-
ative lack of symmetry of Fig. 2.
withP aholomorphicfunction. Aspolynomialsaredense
As was expected, the condensate develops a central
in , it is reasonable to fix an integer n and to restrict
Fε hole and visible vortices are regularly distributed in the
the analysis to functions φ where P is a polynomial of
annularregionofsignificantatomicdensity. Ourcompu-
degree less than n. We write our trial functions as in
tations also show a distortion of the vortex pattern near
[14], where A=kφkL−21/2 is the normalization factor: the external radius of the condensate, resulting in invis-
ible vortices in the exterior region of low atomic density
n
φ(z)=A (z z )e Ωz2/2 (IV.2) as is the case for harmonically trapped condensates [14].
j − | |
− Some vortices also lie in the central hole as theoretically
j=1
Y predicted (see for example [12]) and we can get informa-
and vary the locations z of vortices. An alternative tionontheirpreciselocations: weobserveadistortionof
j
method (used for example in [22] for a harmonically the regular lattice near the inner radius of the conden-
7
cantly. We find good agreement of our numerical results
and analytical study, with energies typically differing by
10 3 to 10 2 in relative value.
− −
∼ ∼
FIG. 2: Vortex structure and atomic density for G = 3,k =
10−5,β = 1(Ω−1 = 4.610−4). There are 119 vortices in
total, thecentral vortex is constituted of 20 single vortices.
FIG.4: Minimum energy asafunction ofthenumberof vor-
tices in the trial wave function (G = 3,k = 10−4,β = 1).
V. CONCLUSION
FIG.3: Two exampleofvortex configurationsfor G=3,k=
10−4,β =1(Ω−1=2.210−3), respectively with n=60 and We have investigatedthe vortex structure of a two di-
n=67 vortices. mensional annular Bose-Einstein condensate rotating in
a quadratic plus quartic confining potential. We focused
on the regime where the state of reference is a vortex
sate,resultinginisolatedsingly-quantizedvorticesencir- lattice encircling a central hole carrying a macroscopic
cling a central multiply quantized vortex. We computed circulation. We developed a theoretical and numerical
configurations for which this central vortex has up to 20 approach based on a small anharmonicity regime and a
units of circulation while there is a total of 32 units of reduction to the Lowest Landau Level states. Using a
circulationinthe entirehole and83visiblevortices. The theoretical method developed in [18, 19] we showed that
number of vortices (both visible and invisible) increases there is an infinite number of vortices in the condensate
with increasing Ω or β, but since our scaling does not so that they cannot lie in a bounded domain. We then
allow to explore a large domain of Ω when G is fixed, analytically and numerically constructed critical points
we mainly varied k. The total vorticity of the system for the reduced Gross-Pitaevskii energy and get further
increases with decreasing k. informationon their vortices. We find that they are reg-
All vortices do not have the same contribution to the ularly distributed on a hexagonal lattice in the annular
energy: as n increases, the vortex pattern in the annu- Thomas-Fermiregionwheretheatomicdensitytakessig-
lar regionof significantatomic density remains the same nificantvalues,andthatthispatternisstronglydistorted
uptopossiblerotations,withtheadditionalvorticesfirst at both edges of the annulus. In particular we find nu-
gatheringinthecentralvortex,thenconstitutingthedis- merical evidence of multiply quantized vortices appear-
torted lattice near the inner boundary of the condensate ing at the center of the trap. Our results agree with
and finally occupying the distorted sites beyond the ex- and complete existing studies [12, 13, 14], showing how
ternal radius. With increasing n (see Fig. 4), the energy the distortion of the vortex pattern modifies the results
reaches a first plateau when the central figure is formed, obtained with the solid-body approximation.
constituted of the visible vortices and the vortices in the
central hole. A second plateau is reached when enough
distorted sites beyond the external radius are occupied.
Acknowledgments
For example, for the parameterscorresponding to Fig. 1
andFig. 4(k =10 4,G=3,β =1),theenergyvariesby
−
10 6 inrelativevaluewhennisincreasedfrom60to This work was supported by a grant from R´egion Ile-
−
∼±
67 (Fig. 3) and the atomic density does not vary signifi- de-France.
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