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Dynamical vortex phases in a Bose-Einstein condensate driven by a rotating optical

lattice

Kenichi Kasamatsu1 and Makoto Tsubota21Department of General Education, Ishikawa National College of Technology, Tsubata, Ishikawa 929-0392, Japan

2Department of Physics, Osaka City University, Sumiyoshi-Ku, Osaka 558-8585, Japan

(Dated: May 6, 2021)

We present simulation results of the vortex dynamics in a trapped Bose-Einstein condensate inthe presence of a rotating optical lattice. Changing the potential amplitude and the relative rotationfrequency between the condensate and the optical lattice, we find a rich variety of dynamical phasesof vortices. The onset of these different phases is described by the force balance of a driving force,a pinning force and vortex-vortex interactions. In particular, when the optical lattice rotates fasterthan the condensate, an incommensurate effect leads to a vortex liquid phase supported by thecompetition between the driving force and the dissipation.

PACS numbers: 03.75.Lm, 03.75.Kk, 67.40.Vs

Driven vortex lattices interacting with periodic pin-ning sites have attracted considerable interests, becausethey exhibit novel phase transitions between dynami-cal vortex states as a function of driving force. Thisproblem has been studied in a wide variety of con-densed matter systems including type-II superconductorsand Josephson-junction arrays, described by the Frenkel-Kontrova-type models of friction [1]. Recently, alkali-atomic Bose-Einstein condensates (BECs) in periodic po-tentials have been proposed to be another possible, su-perior system to study the vortex state caused by theinterplay between pinning effects and vortex-vortex in-teractions [2, 3, 4, 5, 6]. In the condensed matter systemssuch as superconductors, the pinning effects incidental tounavoidable impurities play an essential role in the vortexdynamics. Since the BEC system is free from impurities,the intrinsic phenomena associated with quantized vor-tices can be studied through direct visualization [7]. Inaddition, a laser-induced optical lattice (OL) constructsan ideal periodic potential, in which the microscopic pin-ning parameters, such as depth, periodicity, and symme-try of the lattice can be carefully controlled [8].

In this Letter, we study the dynamic properties of vor-tices in a trapped BEC driven by a rotating OL. A ro-tating OL has already been realized experimentally byusing a laser beam passing through a rotating mask [6].In the previous studies [2, 3, 4, 5], the authors consid-ered the equilibrium properties of a rotating BEC in thepresence of a corotating OL with the same rotation fre-quency as the BEC. They obtained the phase diagramof the vortex lattice structure as a function of the pin-ning parameters through variational [2] or numerical [3]analysis. For a strongly interacting limit, this systemcan be mapped into the Bose-Hubbard Hamiltonian inthe rotating frame [4, 5], exhibiting a series of quantumphase transition [5]. Tung et al. reported the experimen-tal demonstration of the vortex pinning and the evidenceof the structural phase transition [6]. Beside these inter-esting studies, we can also consider a problem in which

the rotation frequency of the OL is different from that ofthe condensate. Then, the moving optical lattice yieldsa rich variety of nonequilibrium vortex phases that arecharacterized by distinct vortex dynamics. The onsetof these phases can be understood by the simple argu-ment of a balance between the driving force, the pin-ning force, and the force coming from vortex-vortex in-teractions. In particular, when the optical lattice rotatesfaster than the condensate, the competition between thedriving force and the dissipation causes a vortex liquid-like phase. This is an example of dissipative structuressustained by energy input and output [9], which offers anavenue to study them in atomic-gas BECs.This system allows us to run ab-initio simulations

based on the Gross-Pitaevskii (GP) equation [10],whereas the other system has relied on phenomenolog-ical theory [1]. Suppose that the condensate is createdin a rotating harmonic trap with an angular velocity Ωz.Here, we confine ourselves to the pancake geometry byassuming a tight axial confinement of a harmonic trapand high rotation rates which reduce the radial trap fre-quency ω⊥. We then give the Gaussian for the frozenaxial component of the condensate wave function, obtain-ing a quasi-two-dimensional GP equation with a renor-malized coupling constant. In a frame rotating with Ω,the transverse wave function ψ(x, y, t) (normalized to beunity) obeys

(i − γ)∂ψ

∂t=

[

−∇2 +r2

4+ VOL + U |ψ|2 − µ− ΩLz

]

ψ;

(1)we choose our units for time, length, and energy as1/ω⊥, a⊥ =

√

h/2mω⊥, and hω⊥, respectively. Here,r2 = x2 + y2, m the atomic mass, µ the chemical po-tential, and Lz the z component of the angular mo-mentum operator. The coupling constant is given byU = 4

√πλNa/a⊥ with the particle number N , the s-

wave scattering length a, and the aspect ratio λ = ωz/ω⊥

of the trapping potential. The pinning effect is causedby the OL potential VOL = V0[sin

2(kX)+ sin2(kY )] with

2

δω = 0 δω = 0.1δω =−0.1(a)

(c)

(b)

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

2

4

6

8

V0

δω

U=1000Ω = 0.85 γ = 0.01

fully pinned vortex lattice

inner-pinned outer-depinnedvortex lattice

vortex latticesliding

pre-meltingvortex lattice

vortexliquid

vortex lattice sliding

FIG. 1: (a) The profile of the OL for k = π/3 at t = 0. Bright(Dark) regions represent the potential maxima (minima). (b)Condensate density for each δω at a certain time in the quasi-stationary stage. The parameter values are U = 1000 andΩ = 0.85. The middle panel (δω = 0) is the initial state ofthe simulations. The presented box dimensions along x and yare −11.8 to +11.8 (The simulations were done in the region−14.0 to +14.0). (c) δω (relative rotation frequency) - V0

(potential amplitude) phase diagram of the dynamical vortexstates. The phase boundary is determined by calculating thestructure factor and the angular momentum per atom, as de-scribed in the text. The dashed boundary for the depinningof all vortices is obtained by solving Eq. (4).

square symmetry. We consider the situation in which theOL rotates with a rotation frequency ω different from Ωof the condensate. Then, VOL has an explicit time de-pendence through

(

XY

)

=

(

cos δωt sin δωt− sin δωt cos δωt

)(

xy

)

. (2)

Here, δω = ω − Ω is the relative rotation frequency be-tween the condensate and the OL; the OL rotates faster(slower) than the condensate for δω > 0 (δω < 0). Fi-nally, we introduce the phenomenological parameter γ inthe left side of Eq. (1), which represents the dampingof the condensate motion caused by the noncondensedatoms rotating with Ω [11]. The damping is necessary toobtain the phase diagram of the quasi-stationary states.We use the small value γ = 0.01 which is small enoughto ensure the underdamped motion of the vortices; theoverdamped motion is unlikely to occur in an ultracolddilute atomic BEC.

Our simulations start from the initial state in which theBEC is equilibrated under rotation Ω and a corotatingOL with the same rotation frequency ω = Ω. Although a

0 100 200 300 400 500t

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500t

S (

ksq

)

0 100 200 300 400 500t

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500

S (

ksq

)

t

(a) δω = - 0.075 (b) δω = - 0.25

(c) δω = 0.075 (d) δω = 0.125

FIG. 2: (Color online) Time development of the structurefactor S(kSQ, t) for V0=2.0 and for different values of δω. Theaverage of the obtained data is given by the solid curve. Theinset show the corresponding vortex trajectories within theThomas-Fermi radius of the initial state in a frame rotatingwith the OL.

rich variety of equilibrium vortex structures emerge de-pending on the system parameters [3], we focus on theinitial state in which vortices are commensurate with thepinning sites. For example, given Ω = 0.85, the wavenumber k = π/3 for the OL [see Fig. 1(a)] and the cou-pling constant U = 1000, we can obtain such an initialstate through the imaginary time propagation of Eq. (1),as shown in the middle of Fig. 1(b). The nucleated vor-tices are pinned at the potential maxima of the OL andform a square lattice. We do not consider the very smallvalues of V0 either, because the competition between tri-angular vortex lattice and the square OL potential occurs[6]. Next, we tune the rotation frequency of the OL byδω adiabatically so as not to excite the collective motion[12]. We then calculate the time evolution from this ini-tial state for a given set of parameters (δω, V0), using theCrank-Nicolson implicit scheme.

After a sufficiently long time, the condensate motionbecomes quasi-stationary because of the damping pa-rameter γ, with the remaining durable vortex dynamicsdriven by the OL. These dynamics exhibit nontrivial be-havior depending on the values of (δω, V0), summarizedin the δω - V0 phase diagram in Fig. 1(c). To characterizeclearly the dynamics and the underlying structures, wecalculate the vortex trajectory in the frame rotating with

the OL and the corresponding structure factor defined asS(k, t) = 1

Nc

∑

j exp[ik · rj(t)] [3], where j labels individ-ual vortices, rj(t) is the position of the jth vortex andNc is the total number of vortices. If vortices are pinnedby the OL and form a perfect square lattice, S(k, t) be-comes unity if we choose k = kSQ = 2ky, which is one

3

0

0.2

0.4

0.6

0.8

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Vo=0.5Vo=1.0Vo=2.0Vo=4.0Vo=8.0Vo=1.0 (0.05)

(a)

δω

<S

(k

)>S

Q

5

10

15

20

25

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Vo=0.5Vo=1.0Vo=2.0Vo=4.0Vo=8.0

rigidlz

(b)

δω

FIG. 3: Time-averaged values of the structure factor 〈S(kSQ)〉(a) and the angular momentum per atom 〈lz〉 (b) as a functionof δω for several values of V0. In (a), the result for γ = 0.05and V0 = 1.0 is also shown. In (b), the plot of the rigid-body

estimation [13] lz = C(Ω + δω)/p

1− (Ω + δω) is shown bythe dotted curve; C = 6.7 is the fitting parameter.

of the two fundamental reciprocal lattices for the squarelattice. Figure 2 shows the time evolution of S(kSQ, t) aswell as the vortex trajectories. Here, we pick out the vor-tices inside the Thomas-Fermi radius of the initial stateto obtain Fig. 2. The time-averaged value of S(kSQ, t)and the angular momentum per atom lz are the usefulquantities to characterize each distinct phase, which areshown in Fig. 3.There is a remarkable difference in the dynamics be-

tween the positive side of δω and the negative one. Forδω < 0, the rotation of the OL is slower than Ω. If|δω| is small, the initially pinned vortices stay in theiroriginal pinning sites, being dragged by the rotating OL.This behavior is shown in the inset of Fig. 2(a); S(kSQ)stays about 0.8 in the stationary stage. Then, the con-densate radius and the angular momentum of the con-densate decreases [see Fig. 1(b) and Fig. 3(b)] becauseits rotation must be reduced by the dragged vortices. Anincrease in |δω| causes depinning of vortices from the con-densate periphery as shown in Fig. 2(b), which leads tothe decrease of the time-averaged value 〈S(kSQ)〉. Withfurther increase in |δω|, the vortices cannot catch up tothe motion of the OL and start to make a sliding mo-tion. The onset of the sliding is seen from the devia-tion of 〈lz〉 from the rigid-body estimation. When thesliding motion occurs, all vortices rest in the rotatingframe of Ω. We identify the vortex phase as follows:(i) “fully pinned vortex lattice” which is determined by〈S(kSQ)〉 > 0.7, (ii) “inner-pinned outer-depinned vortexlattice” by 0.25 < 〈S(kSQ)〉 < 0.7 and by 〈lz〉 followingthe rigid-body estimation, and (iii) “vortex lattice slid-ing” by 〈S(kSQ)〉 < 0.25 and by 〈lz〉 deviating from therigid-body estimation.For δω > 0, if vortices are pinned, the angular mo-

mentum of the condensate must increase because the OLrotates faster than Ω. This demands additional vortexnucleation to ensure the fast rotation. The nucleatedvortices give the incommensurate effect for the vortexlattice. These vortices wander interstitially, exchanging

their locations with the pinned vortices, where the vor-tex trajectories show that the vortices move to nearest-neighbor sites. For small δω, the vortex configurationkeeps the original square symmetry, 〈S(kSQ)〉 stayingabout 0.85. However, a slight increase in δω rapidly de-creases 〈S(kSQ)〉, which indicates depinning of some orall commensurate vortices. We find that the interstitialvortices induce melting of the vortex lattice from the in-side as shown in Fig. 2(c). If V0 is relatively small, 〈lz〉in the quasi-stationary stage is near the initial value andthere are few interstitial vortices. The vortices then makea sliding motion. For large V0 on the other hand, the sta-tionary value of 〈lz〉 is larger than the initial value andthe incommensurate effect becomes important. Then,as shown in Fig. 2(d), the vortex motion is fully disor-dered by the nucleated additional vortices which preventthe other vortices from pinning, exhibiting a liquid-likebehavior. We characterize the phase (i) “fully pinnedvortex lattice” by 〈S(kSQ)〉 > 0.85, (iv) “pre-meltingvortex lattice” by 0.25 < 〈S(kSQ)〉 < 0.85, (v) “vor-tex liquid” by 〈S(kSQ)〉 < 0.25 and 〈lz〉 − lz(t = 0) > 2,and (vi) “vortex lattice sliding” by 〈S(kSQ)〉 < 0.25 and〈lz〉 − lz(t = 0) < 2.The onset of these different phases can be described

using force balance arguments. The force acting on avortex is given by F = Fd + Fp + Fvv, where Fd is thedriving force, Fp the pinning force, and Fvv the forcefrom the other vortices. The commensurate vortex willremain pinned as long as the inequality |Fp| > |Fd|+|Fvv|is satisfied. The pinning potential Upin for the vortexdisplacement r = (x, y) was calculated by Reijnders andDuine [2] for the case without a harmonic trap. Usingtheir results, we obtain the magnitude of the pinningforce Fp = −∇Upin as

|Fp| =az4akV0Q(kξ)

√

sin2(2kx) + sin2(2ky), (3)

where az is the size along the z-axis, ξ = (8πaρ)−1/2

the healing length with the condensate density ρ, andQ(kξ) =

∫

∞

ξdrJ0(2kr)/r + J1(2kξ)/2kξ with the lth-

order Bessel function Jl of the first kind. This analyticform is valid when the vortex core is much smaller thanan optical lattice period and the strength of the potentialis sufficiently weak.For δω < 0, because the incommensurate effect is

not important, the square symmetry of the pinning sitemakes the Fvv irrelevant for the vortex dynamics. Thedriving force is given by the sum of the Magnus forceFM and the mutual friction force FD; the mutual fric-tion originates from the interaction between the vortexcore and normal-fluid flow. If a vortex is pinned by therotating OL, the vortex experiences the background flowwith −δωθ in the rotating frame of ω. Then, the Mag-nus force is written as FM = azmρκδωrz × θ with thequantum circulation κ = h/m. Since the Magnus forcebecomes larger as r increases, vortices begin to depin

4

from the condensate periphery. At near zero tempera-ture, the thermal atoms and the resulting mutual fric-tion force can be negligible. Hence, the sliding motioncan occur when |FP | for vortices near the trap centeris less than |FD| on the same position. We focus on avortex located at the pinning site (x, y) = (π/2k, π/2k).By comparing the FM at r = π/

√2k and the maximum

of |Fp| at (x, y) = (π/4k, π/4k), we can obtain the criti-cal δω for the depinning of all vortices. To estimate thechemical potential µ and the healing length ξ, we use thecentrifugal-force modified Thomas-Fermi approximationthat neglect the density variation caused by the vortexcores. Using the central density ρ = mµ/4πh2a, the crit-ical δω can be obtained by solving the equation

2

πk2V0Q(kξ(δω)) =

√

U(1− δω − Ω)

2πδω. (4)

The result is shown by the dashed line in Fig. 1(c), whichis in reasonable agreement with the numerically obtainedboundary for the sliding motion.For δω > 0, the argument becomes more complex be-

cause the interactions caused by the interstitial vorticescannot be negligible; the solution of Eq. (4) overestimatethe boundary of the vortex-liquid phase because of theneglect of Fvv. Also, the Thomas-Fermi approximationbreaks down for Ω + δω = ω ≃ 1 (δω ≃ 0.15). We needfurther detailed consideration to obtain the onset of thevortex lattice melting caused by the incommensurate ef-fect.The melting of vortex lattices is one of the most im-

portant topics in atomic-gas BECs. The melting was pre-dicted to occur in the limit of fast rotation and low con-densate particle number through thermal [14] or quan-tum fluctuations [15]. The melting mechanism shownin this paper is different from those studies; the vortexliquid phase is supported by the driving force and thedissipation. This is one of the examples of dissipativestructures sustained by the balance between energy in-put and output [9].Using different values of the dissipation parameter γ,

we find no qualitative change in the results if the valueof γ is sufficiently small. However, with increasing γ, wefind that the region of the fully-pinned lattice is reducedfrom the γ = 0.01 results; an example of 〈S(kSQ)〉 withγ = 0.05 and V0 = 1.0 is shown in Fig. 3(a). This is dueto the increase of the driving force associated with themutual friction force FD; the damping parameter γ isrelated to the interaction between the condensate atomsand the thermal atoms [11].In conclusion, we study the dynamical vortex phases

of a trapped BEC in the presence of a rotating OL witha different rotation frequency from the condensate. Wefind several nonequilibrium dissipative structures, wherethe vortices exhibit complex dynamics due to the com-petition of the driving force, the pinning force and the

vortex-vortex interaction. When the rotation rate of theOL becomes higher than that of the condensate, the addi-tional nucleated vortices destroy the commensurate vor-tex lattice, leading to the vortex liquid-like phase.

The authors are grateful to P. Louis for a critical read-ing of this manuscript. K.K. acknowledges the supportsof Grant-in-Aid for Scientific Research from JSPS (GrantNo. 18740213). M.T. acknowledges the supports ofGrant-in-Aid for Scientific Research from JSPS (GrantNo. 18340109) and Grant-in-Aid for Scientific Researchon Priority Areas (Grant No. 17071008) from MEXT.

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[12] The relative rotation frequency is ramped up as δω =0.005tΘ(δωf − δω), where Θ(x) is a step function.Throughout the paper, the value of δω represents thefinal value δω.

[13] For rigid-body rotation, the angular momentum per atomis given by lz = Lz/hN = (1/2)

R

dr(Ω − δω)r2|ψ|2

in our dimensionless scale. Using the centrifugal-forcemodified Thomas-Fermi radius for the integration re-gion and neglecting the spatial variation of the den-sity due to the vortex cores and the OL, we obtainlz = (

p

2U/π/3)(Ω − δω)/p

1− (Ω− δω). This resultdiffers from the numerical results by a few factor.

[14] S.A. Gifford and G. Baym, Phys. Rev. A 70, 033602(2004).

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