Vortex-lattice melting in a one-dimensional optical lattice

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by  Michiel Snoek

Vortex-lattice melting in a one-dimensional optical lattice
Michiel Snoek and H. T. C. Stoof
Institute for Theoretical Physics, Utrecht University,
Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
6 We investigate quantumfluctuationsof a vortex lattice in a one-dimensional optical lattice. Our
0 methodgivesfullaccesstoallthemodesofthevortexlatticeandwediscussinparticulartheBloch
0 bands of the Tkachenko modes. Because of the small number of particles in the pancake Bose-
2 Einstein condensates at every site of the optical lattice, finite-size effects become very important.

n Therefore,thefluctuationsinthevortexpositionsareinhomogeneousandthemeltingofthelattice
a occurs from the outside inwards. Tunneling between neighbouring pancakes substantially reduces
J theinhomogeneity as well as the size of thefluctuations.

1
3 PACSnumbers: 03.75.Lm,32.80.Pj,67.40.-w,67.40.Vs

]
l Introduction. — Very rapidly rotating ultracold ations are much greater. Moreover, by varying the cou-
l
a
bosonic gases have been predicted to form highly- plingbetweenthepancakes,itispossibletostudythedi-
h
correlated quantum states [1, 2, 3]. In these states, the mensional crossover between decoupled two-dimensional
-
s Bose-Einstein condensate has been completely depleted melting, and the strong-coupling limit where the three-
e
by quantum fluctuations, and quantum liquids appear dimensional situation is recovered. Recently, the density
m
withexcitationsthatcancarryfractionalstatistics. Some profilesforquantumHallliquidsinsuchageometryhave
.
t of these states have been identified with (bosonic) frac- alsobeencalculated[11]. Becauseofthesmallnumberof
a
m tional quantum Hall states [2, 4, 5]. It is by now a long particles in each pancake shaped Bose-Einstein conden-
standing goal to observe the experimental signatures of sate, finite-size effects become very pronounced in this
-
d these very interesting states in the context of ultracold setup. In particular, the critical filling factor for the
n quantum gases. The conditions for these states to form melting of the lattice ν changes compared to the ho-
c
o
have been expressed in the requirement that the ratio mogeneous situation. Moreover, melting is not expected
c
ν =N/N of the number of atoms N and the number of to occur homogeneously but starts at the outside and
[ v
vortices N , should be smaller than some critical value then gradually moves inwards as the rotation speed in-
v
1
ν . The ratio ν plays the role of the filling factor and creases. Therefore, phase coexistence is expected, where
v c
estimates for the critical ν are typically around 8 [2, 6]. a vortex crystalis surroundedby a vortexliquid. In this
5 c
9 These estimates are made for infinitely large systems. Letter we study this interesting physics by investigating
6 However,observedfillingfactorsareuptillnowalways the quantum fluctuations of the vortex lattice for realis-
1 greaterthan100,wherealmostperfecthexagonallattices tic numbers of particles and vortices. To decide whether
0
form and no sign of melting can be seen [7]. These ex- ornotthe vortexlatticeis meltedweusethe Lindemann
6
0 perimentsarecarriedoutwithBose-Einsteincondensates criterion, which in this case has to be applied locally.
/ consisting of typically 105 particles, whereas the maxi- Besides the possibility of observing highly-correlated
t
a mum number of vortices observed is around 300. De- quantum states after the vortex lattice has melted, also
m creasing the number of particles results in loss of experi- the vortices themselves have very interesting properties.
- mental signal, whereas the number of vortices is limited For moderately rotating gases, the vortices are observed
d
by the rotation frequency that has to be smaller than toorderinaAbrikosovlattice [12,13]. Thelowestmode
n
o the transverse trapping frequency. Adding a quartic po-
c tential, which stabilizes the condensate also for rotation
v: frequencies higher than the transverse trap frequency,
i has until now not improved this situation [8], although
X
it has opened up the possibility of forming a giant vor-
r
tex in the center of the cloud [9]. In recent work, how-
a
ever, it was realized that quantum fluctuations of the
vortices can be greatly enhanced, without loosing exper-
imentalsignal,byusingaone-dimensionalopticallattice
y
[10]. The optical lattice divides the Bose-Einstein con-
x
densate into a stackoftwo-dimensionalpancakeconden-
z
sates that are weakly coupled by tunneling as schemat- FIG.1: (Coloronline)Setupinwhichthemeltingofthevor-
ically shown in Fig. 1. The number of particles in a texlattice isstudied. Anoptical lattice along thez-direction
single pancake is much smaller than in a Bose-Einstein (indicated in blue) divides the condensate into pancake con-
condensate in a harmonic trap, and therefore the fluctu- densate which are coupled by tunnelingprocesses.
2
ofthelatticeistheso-calledTkachenkomode[14],which
0.8 0.9
recentlyhasbeeninvestigatedexperimentally[7,15]and
theoretically [16, 17, 18]. Since the study of the fluctu-
ations of the vortex lattice requires full information on
all the modes of the lattice, we also can identify this
mode. Because of the optical lattice, this mode actually
becomesaBlochbandandwealsodetermine the disper-
sionoftheTkachenkomodesinthe axialdirectionofthe
optical lattice.
Latticevibrations. —Theactiondescribingthesystem
in the rotating frame is given by S = dt d3x (x,t), 0.95 0.97
L
with the Lagrange density given by R R
~2 2 g
=Ψ∗ i~∂ + ∇ V(x)+ΩL Ψ2 Ψ, (1)
t z
L (cid:18) 2m − − 2| | (cid:19)
where Ψ(x,t) is the Bose-Einstein condensate wavefunc-
tion, m is the mass of the atoms, Ω is the rotation fre-
quency, L = i~(y∂ x∂y) is the angular momentum
z x
operator, and g = 4π~−2a/m is the interaction strength,
withathethree-dimensionalscatteringlength. Thecom-
FIG. 2: (Color online) Classical vortex lattice and density
bination of an optical lattice in the axial direction and
profile for rotation frequencies Ω/ω⊥ = .8, .9, .95 and .97.
harmonicconfinementintheradialdirectiongivesanex-
Here U = 10, which corresponds to N = 250. White means
ternalpotential V(x)=Vzcos2(λz/2)+12mω⊥2(x2+y2), high density, black low density. The vortex positions are in-
where λ is the wavelengthof the laser producing the op- dicated by a red dot, such that also the vortices outside the
tical lattice. Assuming that the optical lattice is deep condensate are visible.
enough we can perform a tight-binding approximation
and write Ψ(x,t) = Ψ(z z )Ψ (x,y,t), where Ψ(z)
i − i i define a dimensionless interaction strength by means of
ischosentobethelowPestWannierfunctionoftheoptical
U = Ng′/4πℓ2~ω = Na/√2πℓ . In the calculations
lattice. In this way we obtain the Lagrange density ⊥ z
we take the scattering length of 87Rb, λ = 700 nm and
= Ψ∗(i~∂ + ~2∇2 mω⊥2 r2+ΩL g′ Ψ 2)Ψ Vz/Ez = 16, which gives U = 25N. The on-site en-
L i t 2m − 2 z− 2| i| i ergy part of the Lagrange density depends in this ap-
Xi proximation only on the particle density and becomes
+t Ψ∗iΨj, (2) (ω+1/ω 2Ω)r2n(r)/2+2πωUn2(r). The lowest Lan-
−
Xhiji dau level wavefunctions and, therefore, also the atomic
density, are fully determined by the location of the vor-
where ij denotes that the sum is taken over nearest
tices. To consider the quantum mechanics of the vortex
h i
neighbouring sites. We have defined the interaction co-
lattice we, therefore, replace the functional integral over
efficient g′ = 4π~2a/√2πℓ m and hopping amplitude
z thecondensatewavefunctionsbyapathintegraloverthe
t=4Vz3/4Ez1/4exp[ 2 Vz/Ez], where Ez =2π2~2/λ2m vortex positions.
−
istherecoilenergyasspociatedwiththeopticallatticeand Next we want to determine the quadratic fluctuations
lz = λ/2π(Vz/Ez)1/4 is the harmonic length associated around the Abricosov lattice. To do so, we first have to
with the optical lattice. findtheclassicalgroundstate. Wecalculatedthisground-
Melting is only expected for a Bose-Einstein conden- stateforupto37vortices. Forsmallnumbersofvortices,
sate that is weakly interacting. The wavefunction can the groundstate is distorted from the hexagonal lattice
then be taken to be part of the lowest Landau level. [21]. In general, there are also vortices far outside the
That is to say we consider wavefunctions of the form condensate. The coarse-grained atomic density is well
i(z − zi)exp[−|z|2/2], where z = (x + iy)/ℓ, zi = approximated by a Thomas-Fermi profile [22]. For fixed
Q(xi + iyi)/ℓ and (xi,yi) is the position of the ith vor- interaction U and different rotation frequencies pictures
tex. Here ℓ is the ’magnetic length’. To increase the of the classical groundstate are given in Fig 2. Then we
validity of this study, we use ℓ as a variational param- studythequadraticfluctuationsbyexpandingtheaction
eter instead of fixed it to the radial harmonic length, up to second order in the fluctuations [20]. This yields
such that our results are also valid for stronger inter- an action of the form
actions [19]. The associated frequency is ω = ~/mℓ2.
From now on distances are rescaled by ℓ, frequencies S = u (Ti∂ E) u t u J u , (3)
i t i i j
· − · − · ·
are scaled by the radial trappig frequency ω⊥, and we Xi Xhiji
3
where ui = (δx1,i,δy2,i,δx2,i,δy2,i,...) (...,uni,...) 2.5
≡
is the total displacement vector of all the point vortices
on site i, and T, E and J are matrices depending on Ω,
2
U,andthe classicallattice positions. To diagonalizethis
action along the z-axis, we perform a Fourier transfor-
⊥
mation to obtain ω 1.5
/
)
k
S = u∗ (Ti∂ E t(1 cos[kλ/2]J) u (4) ω(
Xk k· t− − − · k 1
Finally, we completely diagonalize this action by the
transformationv =P u suchthat the actionbecomes 0.5
k k k
S = v∗ (i∂ ω (k))v whereω (k)arethemode
k,α kα t− α kα α
frequPencies of the vortex lattice. This means that the 0
0 0.5 1 1.5 2 2.5 3
v , where k labels the momentum in the z-direction kλ/2
kα
and α labels the mode, correspond to bosonic operators
FIG. 3: Dispersion of the Tkachenko modes along the direc-
with commutation relation [vkα,vk′α′„] = δkk′δαα′. This tionoftheopticallattice. Forthisplottheparameterswhere
allowsustocalculatetheexpectationvalueforthefluctu-
chosen as: U = 10, t = 1/10, and Ω = .97. One gapless
ationsinthevortexpositions,butalsoforthecorrelations
linear mode and fivegapped tight-binding like modes can be
between the various point vortices. identified. The gapless mode is the acoustic Goldstone mode
Tkachenko modes. — The Tkachenko modes are associated with the broken O(2)-symmetry due to the pres-
purely transverse modes of the vortex lattice. In a har- enceof thevortex lattice.
monic trapwithsphericalsymmetry they become modes
which are purely angular. In the radial direction their
spectrum is discretized, because of the finite lattice size. more.
The number ofradialTkachenkomodes equals the num- Vortex-latticemelting. —Quantumfluctuationsofthe
ber of vortex rings. For 37 vortices 6 Tkachenko modes vorticesultimatelyresultinmeltingofthevortexlattice.
can be identified. A close comparison with continuum To decide wether or not the lattice is melted, we use
theoryforafinite-sizesystem,wherealsoadiscretespec- the Lindemann criterion, which in this inhomogeneous
trum was found [23], is possible but beyond the scope situation has to be applied locally. The Lindemann cri-
of this Letter. Moreover, the Tkachenko modes also terion means that the lattice is melted, when the ratio
have a dispersion in the z-direction. Without the op- u2 /∆2 ofthedisplacementfluctuationsofvortexnat
h nii ni
tical lattice some aspects of these modes were recently site i and the square of the averagedistance to the neig-
investigated [24]. For typical parameters this disper- bouringvortices∆ exceeds acriticalvalue c2,whichis
ni L
sion is plotted in Fig. 3. As is clearly visible, there known as the Lindemann parameter. For easy compar-
is one gapless mode, which is linear at long wavelengths, ison with previous work on this topic, we use the value
whiletheothermodesareroughlyjusttight-bindinglike. c2 = .02. Because the coarse-grained particle density
L
Moreover,variousavoidedcrossingsbetweenthesemodes decreases with the distance to the origin, vortices on
are clearly visible. The gapless mode is the Goldstone the outside are already melted, while the inner part of
mode associated with the spontaneously broken rota- the crystal remains solid. Therefore a crystal phase in
tional O(2)-symmetry due to the presence of the vor- the inside coexists with a liquid phase on the outside.
tex lattice. When the tunneling rate is very small, the In Fig. 4 we compute the radius of the crystal phase
gapped modes have exactly a tight-binding dispersion R normalizedtothecondensateradiusR,asafunction
cr
and the gapless mode gets a dispersion proportional to of the rotation frequency for fixed numbers of particles
sin[kλ/2]. This can be understood by observing that in and an interaction strength U, and for various hopping
this case the modes are decoupled and the Hamiltonian strengths t. Here we define the condensate radius R as
for the gapless mode reduces to the Josephson Hamilto- the radius for with the angularly averageddensity drops
nian = E ∂2/∂φ2 +E cos(φ φ )2, which below .003. The crystal radius R is defined as the ra-
H − c i i J i − j cr
indeed has thisPdispersion upon qPuadratic expansion. dius of the innermost vortex ring that is melted accord-
It is interesting to note that for a small rotation ing to the Lindemann criterion. When according to this
frequency, which implies a small vortex lattice, the definition R > R, we set the crystal radius equal to
cr
Tkachenko modes are not the lowest-lying modes. For the condensate radius, i.e., R = R. The ratio R /R
cr cr
U = 10, a Tkachenko mode becomes the lowest-lying shows discrete steps because of the ring-like structure in
gapped mode when Ω>.978,but there are many modes which the vortices order themselves. We compare this
in between the second and the third Tkachenko mode. with a simple local density calculation, where the crite-
This confirms the expectation that increasingthe vortex rionN/N =n(r)/n (r)=8derivedinRef. [6]isapplied
v v
latticewillbringdowntheTkachenkospectrummoreand locally,using a Thomas-Fermidensity profile to describe
4
1 occurrenceofmeltingbasedonenergyconsiderationsand
thus shed more light on the accuracy of the application
of the Lindemann criterion in this inhomogeneous situa-
0.8
tion. In the future we want to make a closer connection
with continuum theory. Finally we want to extend the
R
/ 0.6 analysistoincludeaquarticpotential,inwhichcasegiant
Rcr vortexformationisprediction. Inthiscasetheliquidcan
37 also form in the center, instead of at the outer edge. Fi-
0.4
Nv 31 nallywealsowanttoinvestigatedthemeltingatnonzero
temperatures, which is experimentally relevant, because
25
0.2 the zero-temperature limit is difficult to reach [25].
19
0.92 0.94 0.96
Ω/ω⊥
0 We thank Masud Haque, Jani Martikainen, and Nigel
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98
Ω/ω⊥ Cooperforhelpfuldiscussions. Thisworkissupportedby
the Stichting voorFundamenteel Onderzoekder Materie
FIG. 4: Crystal radius Rcr normalized to the condensate ra-
(FOM) and the Nederlandse Organisatie voor Weten-
dius R as a function of the rotation frequency for N = 300,
schappelijk Onderzoek (NWO).
U = 10. The full and dashed line are for t = 0 and 1/100,
respectively. The dotted line is the result of a local-density
approximation. IntheinsetthenumberofvorticesNv thatis
within thecondensateisplottedasafunction oftherotation
frequency.
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