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Quantum vortices in optical lattices P. Vignolo,1,2 R. Fazio,3,1 and M.P. Tosi1 1NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy 2INFN, largo B. Pontecorvo 3, I-56127 Pisa, Italy 3International School for Advanced Studies (SISSA), via Beirut 2-4, I-34014 Trieste, Italy A vortex in a superfluid gas inside an optical lattice can behave as a massive particle moving in 7 a periodic potential and exhibiting quantum properties. In this Letter we discuss these properties 0 and show that the excitation of vortex motions in a two-dimensional lattice can lead to striking 0 measurable changes in its dynamic response. It would be possible by means of Bragg spectroscopy 2 tocarryoutthefirstdirectmeasurementoftheeffectivevortexmass,thepinningtotheunderlying n lattice, and thedissipative damping. a J PACSnumbers: 05.30.Jp;74.78.w;74.81.Fa 8 1 Theunderstandingofthe staticanddynamicalbehav- theon-siterepulsionbetweenthebosonsarecompetitive. ] ior of vorticeshas been crucialto describe numerous dif- Quantumfluctuationsduetotheinterplayofthelocalre- t f ferent situations in superfluids ranging from liquid He- pulsions and of the hopping have dramatic consequences o lium to high-temperature superconductors [1, 2]. These for vortex dynamics. In this case a vortex behaves as s . defects can be created by means of an applied magnetic a macroscopic quantum particle, moving in a periodic t a field in superconductors and by putting the sample into potential with a mass that we evaluate and show to be m rotation in superfluid Helium, or they can be thermally directly measurable by Bragg spectroscopy. At variance - excited in low-dimensional systems where the unbinding fromotherrecentstudiesofvorticesinfrustratedoptical d ofvortex-antivortexpairsisatthecoreoftheBerezinskii- lattices [18, 19, 20, 21, 22, 23], we discuss the dynamical n o Kosterlitz-Thouless transition. Low-dimensional super- properties of an individual vortex. In order to achieve c conductors and in particular Josephson Junction Arrays this regime one can either apply a very low frustration [ (JJAs)havebeenformanyyearsanaturalplaygroundfor by means ofa rotationof the lattice [24] generating only 1 studyingclassicalandquantumpropertiesofvortices[3]. a few and very weakly interacting vortices, or create a v A vortex in a JJA behaves as a massive particle moving vortex excitation by means of phase inprinting [25, 26]. 9 inaperiodicpotentialandsubjecttodissipation[4],and Of particular relevance is the very recent observation of 3 underappropriateconditionsvorticescanshowquantum vortex pinning in co-rotating optical lattices by Tung et 4 properties such as interference or tunneling. Among the al. [24], which indicates that what we propose here is 1 0 most interesting experiments performed with vortices in within reach of experimental verification. 7 JJAs we mention the observation of ballistic motion [5], The model - We considera Bosegasatzerotemper- 0 themeasurementoftheAharonov-Cashereffectforavor- ature inside a square lattice with lattice constant a and / t tex going around a charge [6], and the Mott-Anderson Ns lattice sites. We assume that the system can be de- a insulator of vortices [7]. scribedby asingle-bandBose-HubbardHamiltonian[27] m Optical lattices for atomic gases, which currently are d- under intense investigation[8, 9, 10], can behave as tun- H =−J2 ˆb†iˆbj+H.c.+U nî(nî−1)−µ nî, (1) n neling junction arrays. In this Letter we analyze vortex Xhiji Xi Xi o excitations in an optical lattice and show that a super- :c fluid gasin an opticallattice offers a unique opportunity whereˆb†i andˆbi are the creation and annihilation opera- v for a direct measurement of vortex properties (such as torsforabosononthei-thsiteandnˆ =ˆb†ˆb isthenum- i i i i X the mass, the coupling to its environmentor the pinning beroperator. ThecouplingconstantU describesthelocal potential) via a Bragg spectroscopy experiment. This is interaction between bosons, µ is the chemical potential, r a in contrast to the JJA case, where only indirect mea- and J the matrix element for hopping between nearest- surements based on transport properties are available. neighbors sites. The on-site interaction energy and the The Bragg spectroscopy technique [11, 12, 13] has been hopping energy are given by U = g dr w (r R )4, 0 i | − | appealed to for a variety of experiments on ultra-cold J = (h¯2/2m)−1 drw∗(r R ) 2wR(r R ), interms − 0 − i ∇ 0 − j atomic gases, and in optical lattices has been consid- of the Wannier fuRnction w0(r) (Ri is the coordinate of ered for a measurement of the excitation spectrum of the i-th site). Here g =4π¯h2a /(√2πml ) is the repul- sc ⊥ a Bose gas in the Mott-insulator phase [14], and of its siveinteractionstrengthinthe casewherethetransverse coexistence with a superfluid phase ina dishomogeneous size l of the lattice is larger than the scattering length ⊥ cloud [15]. a . sc We consider a Bose gas in the superfluid phase inside Iftheaveragenumberofbosonspersiteismuchlarger a lattice [16, 17], in a regime where the hopping and thanone,thefieldoperatorscanbeapproximatedasˆb i ≃ 2 √n¯exp(iφˆ), with φˆ being the phase operator on the i- in 2D we have a = 426 nm, w 96 nm, l = 5µm, i i ⊥ ≃ th site. The Bose-Hubbard model can be recast into the a =5.5 nm and L=75 µm, and the vortex mass is sc quantum phase Hamiltonian M 29mln(L/a) 150m 2.2 10−20gr. (5) v Hˆ = Jn¯ cos(φˆ φˆ )+U δnˆ2 µ˜ δnˆ (2) ≃ ≃ ≃ × − i− j i − i hXi,ji Xi Xi The behavior of the vortex mass as a function of the potential well depth (which affects w) and of the size of where µ˜ = 2U µ 1. The number operator has been the lattice is depicted in Fig. 1. − − expressedin terms ofthe fluctuations aroundits average value n¯, nˆ = n¯ +δnˆ . The number fluctuation opera- i i MMvv//mm tor and the phase are canonically conjugate variables, [δnî,e±iφj] = δije±iφj. The regime that we consider 600 335000 throughout this work is Jn¯ U: the system is deep 250 in the superfluid region, but≫quantum fluctuations are 400 200 150 present and play a crucial role in the vortex dynamics. 100 Vortex properties - The presence of a static vortex 200 50 inside the lattice can be described to a good approxi- 250 mbyation by a phase distribution of the boson field given 01 1.5 2 2.5 3 3.5 4 4.5 5 50 150 L/a V0/Er y y i φ =arctan − , (3) i (cid:18)xi x(cid:19) FIG. 1: The vortex mass Mv (in units of the boson mass m) − as a function of V0/Er and L/a, for the case a = 426 nm, wherex,y arethecoordinatesofthecenterofthevortex. l⊥ =5µm, and asc =5.5 nm. Deepinthe superfluidregimeandattemperaturesmuch lower than Jn¯/K , phase rigidity ensures that again to B a good approximation, a moving vortex can still be de- The effective potential seen by the vortex has been scribed by Eq. (3) but with a time-dependent position numerically evaluated in the context of JJAs and, in of the vortex center. The existence of a vortex mass can the case of a vortex moving in the x direction inside be understood qualitatively by noting that if a vortex a large two-dimensional array the effective potential is moves of a distance of the order of a in a time δt=a/v, periodic [28], vbeingitsvelocity,thephasedifferenceδφ ateachbond ij changesintimeasδφ =φ (t+a/v) φ (t). Duetothe ij ij ij U (x)=0.1Jn¯[cos(2πx/a) 1]. (6) − v commutationrelationbetweenthenumberandphaseop- − erators, a time-dependent phase leads to a contribution In the presence of a vortexthe whole arraythus behaves to the energy which is quadratic in the vortex velocity asamacroscopicparticleofmassM movinginaperiodic v (seethesecondtermofther.h.sofEq.(2)). Theproblem potential. For such a macroscopic object one has to also of calculating the vortex mass can then be reduced to take into account the interaction with the environment, find the phasedifferences acrossjunctions at times t and andthemainsourceofdampingisdue totheinteraction t+a/v. with the long-wavelength phase modes that are excited An effective action for a vortex in a lattice can then during vortex motion. This damping has been analyzed be obtained by inserting Eq. (3) in the Bose-Hubbard indetailinthecontextofJJAs(seeforexample[29,30]). modelinEq. (2)andthenexpressingtheresultingaction InthisLetterforsimplicityweshalljustcommentonhow in terms of the vortex coordinates r(t)= x(t),y(t) [4]. our results are modified in accounting for dissipation. { } Theon-siterepulsiontermprovidesakineticenergyterm Dynamic structure factor - We turnto a calculation T =(Mv/2)r˙2, where the vortex mass in a lattice of size of the dynamical response of the Bose gas inside a lat- L is tice, bothinthe absenceandinthe presenceofa vortex. The centralquantity ofinterestis the dynamic structure √2πl w2 ⊥ Mv = 4a a2 m ln(L/a). (4) factor, which in a tight-binding scheme takes the form sc Thevortexmassthusscaleslinearlywiththebosonmass, S(q,ω)= dteiωt e−iq·(Ri−Rj) δnˆ (t)δnˆ (0) . (7) i j increases with the width w of the Wannier function, and Z Xi,j h i decreases with the scattering length. Letusconsiderforanillustrationthe87Rblatticereal- This spectrum canbe measuredin experiments ofBragg ized by Greiner and coworkers[17], in the caseV =4E spectroscopy [11, 12]: two probe laser beams, with fre- 0 r where V and E are the well depth of the optical lat- quencies ω and ω = ω + ω and wave-vectors k 0 r 1 2 1 1 ticeandtherecoilenergyrespectively. Forsuchasystem and k = k + q, scatter on the boson gas and the 2 1 3 spectrum measures the probability of momentum trans- Instead of a q-dependent resonance as in Eq.(9), the fer h¯q at energy h¯ω [13]. The f-sum rule gives the Bragg scattering acquires a resonance at a well defined first spectral moment M (q) as M S(q,ω)ωdω = frequency Ω indicating that the whole lattice responds 1 1 v ≡ 1 0[δnˆ ,[Hˆ,δnˆ†]0 , δnˆ being the FoRurier transform collectivelyhavingthe propertiesofa singlemacroscopic 2h¯h | q q | i q of δnˆ . particle, the vortex. This is the main result of this pa- i Wefirstconsiderthecaseinwhichnovortexispresent. per. Under the conditions specified above, the presence Inside the superfluid regime (Jn¯ U), it is enough to of a vortex induces a resonance at a frequency that al- ≫ considerlong-wavelengthphasefluctuations,asdescribed lows access to the vortex mass. Let us remark that this by expansion of the cosine in the phase Hamiltonian up resonantbehaviorisrelatedtothepresenceofthelattice to second order. In this limit the Hamiltonian is eas- and to the existence of quantum fluctuations originating ily diagonalized in Fourier space by means of the trans- fromthelocalrepulsion. TheBraggspectrumofavortex formations φˆ = [UN /(h¯Ω )]1/2 aˆ +aˆ† /√2 and in a Bose-Einsteincondensate is otherwise determinated k s k (cid:16) k −k(cid:17) by a dispersion relation [13]. The peculiar dependence nˆk =(Ns¯hΩk/U)1/2(cid:16)aˆk−aˆ†−k(cid:17)/i√2, with the result of the spectral strength in Eq. (12) on the transferred momentumq isdue to the coupling betweenthe exciting 1 Hˆ = ¯hΩ aˆ†aˆ + (8) radiation and the lattice: at low momentum all phases kX∈BZ k(cid:18) k k 2(cid:19) in the lattice are excited and the dynamic response is enhanced. We finally should comment on the fact that whereΩ2 =(2Jn¯U/¯h2)[2 cos(k a) cos(k a)]. Herethe k − x − y Eq.(12)doesnotfulfillthef-sumrule: thisshouldcome quasi-momentumk=(k ,k )is inside the firstBrillouin x y as no surprise, as this expression is valid only at low en- zone. By taking into account the time dependence of ergy. the particle numberfluctuationoperatordictatedby Eq. The coupling to long-wavelength phase fluctuations (8),itisstraightforwardtoobtainthedynamicstructure provides the main dissipation mechanism for the vortex factor as motions[29,30,31]. Toafirstapproximationthisresults ¯hΩ S(q,ω)= qδ(ω Ω ). (9) in Ohmic damping on the vortex. In the presence of dis- q 2U − sipation the delta function in the dynamical response is The physical interpretation of Eq. (9) is clear: small-q smeared and acquires a finite width proportional to the absorption occurs at a frequency corresponding to the dissipation strength. dispersion relation of the Goldstone sound mode in the In summary, in this Letter we have discussed some lattice. In this case M (q) = (Jn¯/¯h)[2 cos(q a) 1 x main aspects of quantum dynamics of a vortex in an − − cos(q a)]. y atomic superfluid gas inside an optical lattice. We have The presence of a vortex can induce, besides changes specifically considered the situation in which the vor- in the sound wave spectrum, specific contributions asso- tex is pinned by the lattice potential and only executes ciated with excitations of vortex motions. Severaldiffer- small oscillations around its energy minimum. In this ent situations can be envisaged for the latter, but here case Bragg spectroscopy should allow a measurement of we discuss the interesting case in which the hopping pa- the vortex mass. One can envisage other situations in rameter J is sufficiently large that the vortex is pinned which to study vortex dynamics. Experiments on quan- to aminimumofthe periodicpotentialgivenbyEq. (6). tum tunneling/coherence of vortices seem to be within Inthis casethe vortexdynamics isassociatedwith small experimental reach. oscillations around its equilibrium position and can be WeacknowledgefruitfuldiscussionwithC.Bruderand described by the harmonic oscillator Hamiltonian M.Polini. ThisworkhasbeensupportedbyMIUR-PRIN 1 1 H = M r˙2+ M Ω2r2, (10) andbyECthroughgrantsEUROSQIPandRTNNANO. v 2 v 2 v v where we have defined Ω = (0.1Jn¯/M )1/22π/a. 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