Vortex Lattice Structures of a Bose-Einstein Condensate in a Rotating Triangular Lattice Potential
aa r X i v : . [ c ond - m a t . o t h e r] J un Vortex Lattice Structures of a Bose-Einstein Condensate in a Rotating TriangularLattice Potential
T. Sato , T. Ishiyama , and T. Nikuni Institute for Solid State Physics, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-9601, Japan (Dated: October 22, 2018)We study the vortex pinning effect in a Bose-Einstein Condensate in the presence of a rotatinglattice potential by numerically solving the time-dependent Gross-Pitaevskii equation. We consider atriangular lattice potential created by blue-detuned laser beams. By rotating the lattice potential, weobserve a transition from the Abrikosov vortex lattice to the pinned vortex lattice. We investigate thetransition of the vortex lattice structure by changing conditions such as angular velocity, strength,and lattice constant of a rotating lattice potential. Our simulation results clearly show that thelattice potential has a strong vortex pinning effect when the vortex density coincides with thedensity of the pinning points.
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I. INTRODUCTION
Quantized vortices are one of the most character-istic manifestations of superfluidity associated with aBose-Einstein condensate (BEC) in atomic gases [1].Formations of triangular Abrikosov vortex lattice inBose condensates have been clearly observed by rotatinganisotropic trap potentials [2, 3, 4]. Microscopic mech-anism of the vortex lattice formation has been exten-sively studied both analytically and numerically usingthe Gross-Pitaevskii equation for the condensate wave-function [5, 6, 7, 8, 9, 10, 11, 12].Another interesting development in ultracold atomicgases is creating periodic potential using optical lat-tices [13]. The studies of a BEC in optical lattices havefound a lot of interesting phenomenon, such as the tran-sition from the superfluid to Mott insulator phase [14].More recently, combining the above two systems, i.e.,rotating BEC and optical lattices has attracted growingattention. In particular, the vortex phase diagrams ofa BEC in a rotating optical lattice potential have beentheoretically studied, since one expects structural phasetransition of vortex lattice structures due to vortex pin-ning [15, 16, 17, 18]. In the condensed matter systemssuch as superconductors, the vortex pinning due to im-purities and defects in solids have been extensively inves-tigated [19, 20]. In the atomic condensates, rotating op-tical lattices have been experimentally realized at JILAgroup, making use of a laser beam passing through arotating mask [21]. This experiment observed the struc-tural phase transition of vortex lattice structures in ro-tating Bose condensated gases due to vortex pinning bythe laser beam. In this system, one can control the pin-ning parameters by changing conditions such as angularvelocity, strength, and lattice constant of rotating opti-cal lattices. This allows one to investigate vortex pinningeffect in a quantitative manner..In this paper, stimulated by the JILA experiment, westudy the vortex pinning effect by numerically solving the time-dependent Gross-Pitaevskii equation. We observe astructural phase transition of vortex lattice structures ofa BEC in a rotating triangular lattice potential createdby blue-detuned laser beams.
II. FORMULATION
We consider a Bose-condensated gas trapped in a har-monic potential and a rotating periodic potential with anangular velocity Ω. The dynamics of the condensate isdescribed by the time-dependent Gross-Pitaevskii equa-tion. Assuming a pancake-shaped trap, we use a two-dimensional Gross-Pitaevskii equation. In the rotatingframe, the Gross-Pitaevskii equation is given by (for adetailed derivation, see for example Ref. [7])( i − γ ) ~ ∂ψ ( r , t ) ∂t = (cid:20) − ~ m ∇ + V ext ( r ) + g | ψ ( r , t ) | − Ω L z (cid:21) ψ ( r , t ) . (1)Here, V ext ( r ) = mω ( x + y ) / V lattice ( r ) describesthe total external potential, L z = − i ~ ( x∂/∂y − y∂/∂x )denotes the z component of the angular momentum op-erator, g = 4 π ~ a s /m is the strength of interaction with a s being the s -wave scattering length, and γ is the phe-nomenological dissipative parameter [22, 23, 24, 25, 26].The lattice potential created by blue-detuned laser beamsarranged in the lattice geometry is expressed as V lattice ( r ) = X n ,n V exp (cid:20) − | r − r n ,n | ( σ/ (cid:21) , (2)where r n ,n = n a + n a describes lattice pin-ning points and V describes the strength of the laserbeam. We consider the triangular lattice geometrywith two lattice unit vectors given by a = a (1 , a = a ( − / , √ / a is the lattice constant. FIG. 1: Density profile (left) and Structure factor profile(right) of a BEC in a triangular lattice potential without ro-tation by setting Ω = 0. The lattice potential geometry istriangular lattice at a/a ho = 2 . V / ~ ω = 5 . Throughout this paper, we scale the length and energyby a ho = p ~ /mω and ~ ω . We set the dimensionlessinteraction strength as C = 4 πa s N/R = 1000, where N is the total particle number and R is size of the conden-sate along the z -axis direction [7], and the width of thelaser beam as σ/a ho = 0 .
65. In our parameter set, thehealing length is ξ/a ho = 0 .
12. We numerically solve theGross-Pitaevskii equation in Eq. (1) using Fast-Fourier-Transform (FFT) technique.We first determine the ground-state condensate wave-function without rotation by setting Ω = 0 in Eq. (1).Starting with this wavefunction as a initial state, we nu-merically evolve the Gross-Pitaevskii equation with a ro-tating lattice potential with a fixed angular velocity Ω,until the system relaxes into equilibrium. We investigatethe vortex lattice structure from the equilibrium conden-sate density profile n ( r ) = | ψ ( r ) | . In order to quan-tify the vortex lattice structure, we calculate the densitystructure factor defined by S ( k ) = Z d r n ( r ) e − i k · r . (3)The structure factor given in Eq. (3) contains informa-tion about the periodicity of the condensate density. Fortriangular lattice geometry, S ( K ) exhibits periodic peaksof the regular hexagonal geometry. In order to distin-guish between the Abrikosov lattice and the pinned vor-tex lattice, we calculate the peak intensity of the struc-ture factor | S ( K ) | at the lattice pinning point, namely, K = 2 π/a (1 , / √ K = 2 π/a (0 , / √ E lattice = Z d r ψ ∗ ( r ) V lattice ( r ) ψ ( r ) , (4)which can also be used to quantify the transition of thevortex structure. As we see below, the vortex pinning issignified by a marked decrease of E lattice . III. SIMULATION RESULTS
In this section, we show our numerical simulation re-sults for vortex lattice structures of a BEC in a rotating
FIG. 2: Density profiles (left) and Structure factor pro-files (right) in equilibrium state after rotating condensateswith a fixed angular velocity Ω /ω = 0 .
70 at a/a ho = 2 .
2. Thefigures show vortex lattice structures for different values of thestrength of a triangular lattice potential; (a) V / ~ ω = 0 . V / ~ ω = 0 .
3, (c) V / ~ ω = 0 . triangular lattice potential. Fig. 1 shows the equilibriumcondensate density profile and the density structure fac-tor profile in the presence of a triangular lattice potentialwithout rotation.Fixing the lattice constant with a/a ho = 2 .
2, we in-vestigate the transition of the vortex lattice structure bychanging the strength V for various angular velocitiesΩ. Fig. 2 shows the density profiles and the structurefactor profiles in equilibrium state after rotating conden-sates with a fixed angular velocity Ω /ω = 0 .
70. Onecan see that for weak lattice potential V / ~ ω = 0 . V / ~ ω = 0 . V / ~ ω = 0 . | S ( K ) | and E lattice against the lat-tice strength V . As shown in Fig. 3 (a), for Ω /ω =0 .
70, the lattice potential energy E lattice decreases grad-ually, which indicates that vortices are partially pinned, FIG. 3: Lattice potential energy ( N ) and peak intensity of thestructure factor at the lattice pinning point ( (cid:7) ) with a/a ho =2 . /ω = 0 . /ω = 0 .
55. Here E lattice0 and | S ( K ) | is the latticeenergy and the peak intensity of the structure factor at thelattice pinning point of the ground state (Ω = 0) for eachlattice strength. and reaches constant when all vortices are pinned for V / ~ ω ≥ .
4. Correspondingly the structure factor | S ( K ) | increases gradually and finally becomes constantwhen all vortices are pinned. From these results, togetherwith directly looking at the condensate density profile,we conclude that the structural phase transition occursat V / ~ ω = 0 .
4, which we define as the strength for thestructural phase transition V c / ~ ω . Fig. 3 (b) shows theanalogous result for Ω /ω = 0 .
55. In this case, there isan intermediate domain where the Abrikosov lattice andthe pinned lattice coexist. In this domain, the vortex lat-tice structure cannot be categorically determined becauseof the competition between the vortex-vortex interactionand the lattice pinning effect.From these results, we map out the phase diagrams ofvortex lattice structures against Ω and V , as we plot inFig. 4 (a). The lower domain is the Abrikosov latticedomain, while the upper domain is the pinned latticedomain. The intermediate domain represents coexist-ing state of the Abrikosov lattice and the pinned lattice.Looking at the phase boundary of the pinned lattice do-main in Fig. 4 (a) as V c = V c (Ω), we find that V c takes aminimum value as a function of Ω, which we define as theminimum pinning strength V c , min = V c (Ω = Ω min ). From FIG. 4: (a) Phase diagrams for vortex lattice structures for a/a ho = 2 .
2. (b) Lattice constant a against Ω min giving theminimum pinning strength. The solid line represents Eq. (5). Fig. 4 (a) for a/a ho = 2 .
2, we find that V c , min / ~ ω ≈ . min /ω ≈ . V c , min on the lattice constant a can be understood asfollows. When vortices form the triangular lattice andundergo rigid rotation with angular velocity Ω, the latticeconstant is expressed as a function of angular velocity by a v (Ω) a ho = s √ π /ω . (5)In Fig. 4 (b), we plot lattice constant a against Ω min .We find that the formula in Eq. (5) well fits the numeri-cal data. This means that the pinning strength V c takesminimum value when a v (Ω) matches to the lattice con-stant a . When this “matching relation” is satisfied, weaklattice potential has a stronger pinning effect than vortex-vortex interaction, which leads to a sharp transition ofthe vortex lattice structure.In the case when this “matching relation” is satisfied,one may presume that the vortex lattice array matchthe lattice potential array without vortex pinning effect.However, from Fig. 4 (a), we find that minimum pinninglattice strength V c , min does not fall to zero, but takes afinite value. Actually in obtaining the phase diagram inFiq. 4 (a), we solved the Gross-Pitaevski equation start-ing with the initial ground state wavefunction without FIG. 5: (a) Phase diagrams for vortex lattice structures for a/a ho = 2 . V or decreasing V from pinnedlattice domain ( (cid:7) ). At Ω ≥ Ω min , phase boundary of actuallythe pinned lattice phase is solid line. (b) Total free energy F with a/a ho = 2 . /ω = 0 .
70 ( ♦ ), Ω /ω = 0 .
71 ( (cid:13) ) andΩ /ω = 0 .
725 ( N ). Here F increase is in the case of increase of V , while F decrease is in the case of decrease of V from pinnedlattice domain. (c) Total free energy F with a/a ho = 2 . /ω = 0 .
65 ( ♦ ), Ω /ω = 0 .
675 ( (cid:13) ) and Ω /ω = 0 .
69 ( N ). vortices. As evolving the Gross-Pitaevski equation, vor-tices are nucleated, forming vortex lattices. In this dy-namical process, it may be possible that vortices relaxinto metastable configurations, and this may be the rea-son why V c , min takes a finite value. In order to investigatethis in more detail, we solve the Gross-Pitaevski equationstarting with a perfectly pinned vortex lattice at largeenough V . We then slowly decrease V . In Fig. 5 (a), we plot the phase boundary of the pinned lattice phaseobtained in this way, together with phase boundaries pre-viously shown in Fiq. 4 (a). We find that at Ω = Ω min ,the perfectly pinned vortex lattices is stable down to in-finitesimally small lattice potential V →
0. In order tosee whether the pinned state obtained here has the lowerenergy than the tilted Abrikosov lattice obtained in theprevious calculation, we calculate the total free energy F = Dh − ~ m ∇ + V ext ( r ) + g | ψ ( r , t ) | − Ω L z i ψ ( r , t ) E . (6)In Figs. 5 (b) and (c), we plot the free energy F for a/a ho = 2 . V . Here, F increase is the free energyobtained by solving the Gross-Pitaevski equation start-ing with non-vortex initial state and increasing V , while F decrease is the free energy obtained by decreasing V frompinned lattice domain. We make the separate plots forΩ ≥ Ω min and Ω < Ω min in Figs. 5 (b) and (c), respec-tively. One can see from Fig. 5 (b) that, at Ω = Ω min , F decrease < F increase holds down to V →
0. This meansthat the Abrikosov lattice found for 0 < V < V c , min inthe phase diagram in Fig. 4 (a) is actually metastable.Similarly for Ω > Ω min , we find a finite region where theAbrikosov lattice is metastable. However, for Ω < Ω min ,we find F decrease > F increase as shown in Fiq. 5 (c). Thismeans that the pinned vortex lattice does not have thelowest energy in the weak lattice domain.From the above results, we found that property of vor-tex pinning is quite different for Ω ≥ Ω min and Ω < Ω min .For Ω ≥ Ω min , the vortex lattice structure exhibitssharp transition. In contrast, for Ω < Ω min , there is nosharp transition, but the vortex lattice structure exhibitscrossover through the intermediate coexisting region. IV. SUMMARY AND DISCUSSION
In summary, we have studied the vortex pinning effectby observing the structural phase transition of vortex lat-tice structures of a Bose-Einstein condensate in a rotatingtriangular lattice potential. We observed the transitionof the vortex lattice structure from the Abrikosov vortexlattice to the pinned lattice where all vortices are pinnedto lattice points. The transition is determined dependingon the competition between strength and density of thelattice potential, and vortex density and vortex-vortexinteraction. From our simulation, we found that the lat-tice potential has a strong vortex pinning effect when a v (Ω) matches to the lattice constant a of the triangu-lar lattice potential, which means that the vortex den-sity coincides the density of pinning points. In the casewhen this “matching relation” is satisfied, we observedthat minimum pinning lattice strength V c , min takes a fi-nite value. By investigating the structural transition inmore detail, we found that for Ω ≥ Ω min , there are re-gions where the Abrikosov lattice is metastable, while forΩ < Ω min , the pinned vortex lattice phase is metastablestate. From the above results, we found that propertyof vortex pinning is quite different for Ω ≥ Ω min andΩ < Ω min . For Ω ≥ Ω min , the vortex lattice structure ex-hibits sharp transition. In contrast, for Ω < Ω min , thereis no sharp transition, but the vortex lattice structure ex-hibits crossover through the intermediate coexisting re-gion.In a separate paper, we will discuss the pinning effectof a rotating square lattice potential. We will show thatthe property of vortex pinning is quite different from thetriangular lattice and more complicated for a rotating square lattice potential. V. ACKNOWLEDGMENTS
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