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From rotating atomic rings to quantum Hall states SUBJECT AREAS: M. Roncaglia1,2, M. Rizzi2 & J. Dalibard3 QUANTUM PHYSICS THEORETICAL PHYSICS 1Dipartimento di Fisica del Politecnico, corso Duca degli Abruzzi 24, I-10129, Torino, Italy, 2Max-Planck-Institut fu¨r Quantenoptik, ATOMIC AND MOLECULAR Hans-Kopfermann-Str. 1, D-85748, Garching, Germany, 3Laboratoire Kastler Brossel, CNRS, UPMC, E´cole normale supe´rieure, PHYSICS 24 rue Lhomond, 75005 Paris, France. APPLIED PHYSICS Considerable efforts are currently devoted to the preparation of ultracold neutral atoms in the strongly Received correlated quantum Hall regime. However, the necessary angular momentum is very large and in experiments 15 March 2011 with rotating traps this means spinning frequencies extremely near to the deconfinement limit; consequently, the required control on parameters turns out to be too stringent. Here we propose instead to follow a dynamic Accepted path starting from the gas initially confined in a rotating ring. The large moment of inertia of the ring-shaped 4 July 2011 fluid facilitates the access to large angular momenta, corresponding to giant vortex states. The trapping potential is then adiabatically transformed into a harmonic confinement, which brings the interacting atomic Published gas in the desired quantum-Hall regime. We provide numerical evidence that for a broad range of initial 22 July 2011 angular frequencies, the giant-vortex state is adiabatically connected to the bosonic n 5 1/2 Laughlin state. hile coherence between atoms finds its realization in Bose–Einstein condensates1–3, quantum Hall Correspondence and states4 are emblematic representatives of the strongly correlated regime. The fractional quantum Hall requests for materials effect (FQHE) has been discovered in the early 1980s by applying a transverse magnetic field to a two- W 5 should be addressed to dimensional (2D) electron gas confined in semiconductor heterojunctions . Since then, FQHE has never stopped M.R. (roncaglia@bo. to intrigue the scientific community due to non-trivial transport properties and exotic topological quantum infn.it) phases6. Such interest has also influenced the research in ultracold atomic gases, which in the last decade have been successfully exploited as a highly controllable playground for quantum simulations of many-body physics3. The large versatility of these setups allows one to confine atoms in 2D harmonic traps and to impose an effective magnetic field either by rapid rotation7, 8 or by laser-induced geometric gauge potentials10. In principle, such opportunity should allow one to experimentally explore the bosonic version of QHE, even if unfortunately it has been hitherto elusive. From a theoretical point of view, a variety of interesting ground states (GS’s) have been identified for Bose gases as a function of the effective magnetic field7–9. At zero field, i.e., without rotation, the particles undergo Bose– Einstein condensation1, 2 and the atomic ensemble is superfluid. Differently from a rigid body, a superfluid of N particles reacts to rotation with the formation of quantized vortices, whose number Nw increases with the rotation frequency. At large filling factor v 5 N/Nw > 10 an ordered vortex lattice is formed. For v , 10, the lattice melts because of quantum fluctuations, which signals the breakdown of the mean-field description and the access into the FQHE regime. The filling factor is now better defined as v 5 N/mmax with mmax the maximum angular momentum occupied by single particles. FQHE states are obtained for values of v of order unity, which corre- spond to very large total angular momenta L / N2. Like in solid-state physics, most of the preparation procedures employed so far in rotating atomic ensembles approached the GS by cooling down the system with a fixed Hamiltonian. By contrast we explore in this paper an alternative method that consists in starting from an easily preparable state (typically uncorrelated), following a dynamic route by changing an external parameter, and eventually obtaining the desired state. This strategy has been successful for the experimental investigation of the superfluid to Mott insulator transition in optical lattices11. We propose to implement it to reach quantum Hall states with the following steps: (i) We engineer a Mexican-hat trapping potential by superposing a standard harmonic trap with the repulsive potential created by a ‘‘plug’’ laser beam, which is focused at the center of the trap. (ii) By stirring the gas, we prepare the N bosonic atoms in a giant vortex state, corresponding to the lowest energy state of the Mexican-hat potential for a given angular momentum L. (iii) The stirring is removed and the plug is adiabatically switched off. (iv) In the final harmonic trap, we obtain the GS with the initially imparted angular momentum L, thanks to rotational symmetry. We show that if L 5 N(N 2 1) then the 2-body contact interactions drive the gas into the celebrated bosonic v 5 1/2 Laughlin state4. SCIENTIFIC REPORTS | 1 : 43 | DOI: 10.1038/srep00043 1 www.nature.com/scientificreports Results done by shining the center of the harmonic trap with a laser beam 12 Single particle physics. In experiments with rotating atomic gases, prepared in a circular, Gaussian TEM00 mode . When the laser particles are usually trapped by a harmonic potential and stirred by frequency is chosen larger than the atomic resonance frequency time-varying magnetic field or auxiliary laser beams7. In the frame (‘blue detuning’), the laser beam creates a repulsive dipole potential rotating at angular speed V 5 Vz, the Hamiltonian of a single particle proportional to the light intensity. The beam is chosen to be perpen- in the harmonic trap of frequencies (v, v, vz) can be written as dicular to the xy plane and the dipole potential is of the form 1 M ÀÁÀÁM ÂÃÀÁ H ~ ðÞp{A 2z v2{V2 x2zy2 z v2z2, ð1Þ U ðÞx,y ~a exp {2 x2zy2 w2 , ð2Þ trap 2M 2 2 z w with A 5 MV 3 r 5 MV(2y,x,0). In the following we suppose that where w is the laser waist and a is proportional to the laser intensity. The sum of the harmonic potential (x2 1 y2)/2 and of U (x,y) has a all relevant energies are much smaller than Bvz, so that the motion w 2 along the z direction is frozen and the problem is effectively two- bump in x 5 y 5 0 in the laboratory frame when a . w /4. dimensional. In the limit of centrifugal deconfinement V R v,the At moderate intensities of the plug, as the ones employed in our system is formally equivalent to bosons of charge q 5 1 in uniform preparation scheme, the classification of single-particle energy eigen- magnetic field B 5=3A 5 2MVz.p Fromffiffiffiffiffiffiffiffiffiffiffiffiffi now on, we express states in terms of LL remains valid (see Fig. 1 and Methods). In the energies and lengths in units of Bv and B=Mv, respectively. It is LLL the single-body energies are in good approximation: 7 { z well known that the problemÀÁ can be rewrittenÀÁ as two decoupled 2 ðÞm 1 ~ { z zd { { { em~mdza 1z : ð3Þ harmonic oscillators Htrap 2a a 1 b b a a ,intermsof w2 ladder operators a, b,andd 5 1 2 V/v is the frequency offset. Every state is labeled by the occupationE number na, nb of the two modes, At fixed laser parameters a and w, the angular momentum m that and it is denoted as y . Note that a gauge field similar to the one minimizes em is a decreasing function of the rotation frequency offset nb ,na d. We denote by dm the value for which the level crossing em11 5 em entering into (1) can also be induced by geometric phases instead of occurs. The LLL state with angular momentum m is thus the lowest rotation10. The scheme outlined in the present paper should work energy state when d is chosen in the interval dm , d , dm21, whose equally well in this case, the only significant difference being that (1) width is is now the single-particle Hamiltonian in the laboratory frame, 2 {ðÞmz2 instead of the rotating frame. 2 2 I ~d { {d ~a 1z : = m m 1 m 2 2 In the limit d 1, the quantum number na identifies different w w manifolds called Landau Levels (LL). Within each LL, the states Later on, we will be interested in choosing a specific value m~‘ (labeled by nb) are quasi-degenerate due to the small separation energy d. The quantity m 5 n 2 n is the angular momentum and in maximizing the width I‘ of the stability window. This can be b a done, at fixed intensity a, by choosing w2~‘. The central rotation of the particle. In the lowest Landau level (LLL), na 5 0 and the one-body eigenfunctions assume the simple expression frequency in the stability window for ‘ then corresponds to 2 ~ ffiffiffiffiffiffi1 m {jjz =2 {ðÞ‘z2 y , ðÞz p z e , where z now denotes the position in the ‘z m 0 pm! c~ 1 z ~ 1 z 2 d‘ ðÞd‘{1 d‘ 2a 1 : ð4Þ complex plane (z 5 x 1 iy), with energies Em 5 md and angular 2 ‘2 ‘ momentum m. c {1 The first key feature of our proposition is to replace the ordinary Notice that for large values of ‘,wegetd‘!a‘ , thus if we want to harmonic potential with a Mexican-hat one, like in Fig. 1. This can be keep it sizable, we have to choose a!‘. Figure 1 | Mexican hat potential. (a) Bosonic atoms are confined in a combined trap with (i) an isotropic harmonic confinement and (ii) the dipole potential created by a blue-detuned, gaussian laser beam that plugs the trap center and pushes the particles away from this point.

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