We report on the study of the momentum distribution of a one-dimensional Bose gas in an optical lattice. From the momentum distribution we extract the condensed fraction of the gas and thereby measure the depletion of the condensate and compare it with a theoretical estimate. We have measured the coherence length of the gas for systems with average occupation n̄>1 and n̄<1 per lattice sit

Esslinger, T.

Köhl, M.

Moritz, H.

Schori, C.

Stöferle, T.

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DOI: 10.1007/s00340-004-1662-8Appl. Phys. B 79, 1009–1012 (2004)Lasers and OpticsApplied Physics Bm. ko¨hlt. sto¨ferleh. moritzc. schorit. esslinger1D Bose gases in an optical latticeInstitute for Quantum Electronics, ETH Zürich Hönggerberg, 8093 Zürich, SwitzerlandReceived: 17 June 2004/Revised version: 7 September 2004Published online: 20 October 2004 • © Springer-Verlag 2004ABSTRACT We report on the study of the momentum distribu-tion of a one-dimensional Bose gas in an optical lattice. Fromthe momentum distribution we extract the condensed fraction ofthe gas and thereby measure the depletion of the condensate andcompare it with a theoretical estimate. We have measured thecoherence length of the gas for systems with average occupationn > 1 and n < 1 per lattice site.PACS 05.30.Jp; 03.75.Kk; 03.75.Lm; 73.43.Nq1 The one-dimensional Bose gas in an optical latticeA one-dimensional gas can be created in a trapwhen the confining potential restricts the motion of the par-ticles to one dimension with the transverse motional degreesof freedom being frozen out. A cigar shaped harmonic trap-ping geometry is characterized by the frequencies ω⊥ in thetwo strongly confining axes and ωax in the weakly confiningaxis. A kinematically one-dimensional situation is achievedwhen all particles occupy only the ground state in the radialdirections, which implies that both the thermal energy kBTand the interaction energy µ have to be much smaller than thetransverse energy level spacing. In general, one-dimensionalsystems exhibit increased quantum fluctuations of the phase,such that for a homogeneous 1D gas Bose–Einstein con-densation is prevented. For trapped low-dimensional gasesthe cross-over to a Bose–Einstein condensate takes place ata finite temperature kBT1D = Nhωax/ ln(2N), where N is thenumber of particles in the 1D system [1, 2]. The fluctuatingphase alters the properties of the gas [3–7]. One-dimensionalquantum systems exhibit a wealth of fascinating phenomenawhose explanations go beyond the mean-field description [8].One-dimensional trapped Bose–Einstein condensateswere recently created and studied [9] using an optical latticeconsisting of two mutually perpendicular standing wave laserfields. In this geometry the optical lattice forms an array ofone-dimensional tubes, each filled with a Bose–Einstein con-densate. This experiment revealed the distinctively differentexcitation spectrum of a one-dimensional quantum system ascompared to its three-dimensional counterpart [10]. In a pre- Fax: +41-1-6331254, E-mail: esslinger@phys.ethz.chvious experiment a Bose condensates was loaded into a two-dimensional optical lattices to study the coherence betweenthe tubes. In that experiment the tunnel-coupling between ad-jacent tubes was larger than the axial oscillation frequency,thereby an array of strongly coupled tubes was created [11].Very recently the lifetime of one-dimensional gases createdin an optical lattice were studied [12]. In elongated magneticand optical traps a regime was accessed where a Bose con-densate with µ ≤ hω⊥ coexisted with a three-dimensionalthermal cloud [13, 14]. In similar elongated traps studies ofsolitons [15] and of enhanced phase fluctuations have beenperformed [16, 17].When the one-dimensional Bose gas is exposed to an ad-ditional optical lattice potential in axial direction the bosonsmay become localized in the minima of a periodic potentialand the system can then be described by the Bose–HubbardHamiltonian [18, 19]:H = − J˜∑〈i, j〉aˆ†j aˆi +∑iεini + U2∑ini(ni −1) . (1)J˜ denotes the hopping matrix element between neighboringlattice sites and determines the rate of which a particle dis-appears from lattice site i and tunnels to the adjacent latticesite j (aˆi and aˆ†i are the annihilation and creation operators foran atom at lattice site i, respectively). The total tunnel coup-ling is J = z J˜ with z being the coordination number of thelattice. The inhomogenity of the atom trap is characterized byεi = m2 (iωaxd)2 where m is the atomic mass and d the latticespacing. The occupation number of lattice site i is denotedni and U is the onsite-interaction energy between two bosonson the same lattice site. The one-dimensional Bose–HubbardHamiltonian exhibits a transition from a superfluid phase toa Mott insulating phase for a ratio (U/J)c  2 [3–6], whichwe experimentally demonstrated recently [7]. Due to strongerquantum fluctuations of the phase in a one-dimensional quan-tum system this value is lower than the corresponding criticalvalue in three dimensions (U/J)c = 5.8 [18–20]. Tuning thedepth of the periodic potential changes the parameter U/J ,which leads to increased effects of interparticle interactionsand small filling of the lattice sites may turn the system intoa gas of hardcore bosons [21, 22]. In the inhomogeneous sys-tem Mott insulating regions with commensurate filling coexistwith superfluid regions with incommensurate filling and theinsulator is formed in a crossover type transition [4].1010 Applied Physics B – Lasers and Optics2 Experimental setup2.1 Bose condensates in an optical latticeIn the experiment, we collect up to 2×109 87Rbatoms in a vapor cell magneto-optical trap. After polarizationgradientcoolingandopticalpumping into the |F = 2, mF = 2〉hyperfine ground state the atoms are captured in a mag-netic quadrupole trap. After magnetic transport of the trappedatoms over a distance of 40 cm the magnetic trapping poten-tial is converted into the harmonic and elongated potential ofa QUIC trap [23]. Subsequently, we perform radio frequencyinduced evaporation of the cloud over a period of 25 s. Afterevaporation we observe almost pure three-dimensional Bose–Einstein condensates of up to 1.5(0.2)×105 atoms. Follow-ing the condensation we adiabatically change the trappinggeometry to an approximately spherical symmetry with trap-ping frequencies of ωx = 2π ×18 Hz, ωy = 2π ×20 Hz, andωz = 2π ×22 Hz. This reduces the peak density by a factor offour and allows us to load the optical lattice more uniformly.The optical lattice is formed by three retro-reflected laserbeams. Each beam is derived from a laser diode at a wave-length of λ = 826 nm. At the position of the condensate thebeams are circularly focused to 1/e2-radii of 120µm (x andy axes) and 105µm (z). The three beams possess mutuallyorthogonal polarizations and their frequencies are offset withrespect to each other by several ten MHz. We stabilize thelasers to a high-finesse Fabry–Pe´rot cavity, thereby reducingtheir line width to ∼ 10 kHz. In order to load the condensateinto the ground state of the optical lattice, the intensities ofthe lasers are slowly increased to their final values using anexponential ramp with a time constant of 25 ms and a dura-tion of 100 ms. The resulting optical potential depths Vx,y,zare proportional to the laser intensities and are convenientlyexpressed in terms of the recoil energy ER = h2k22m with k = 2πλ .2.2 Preparation of one-dimensional quantum gasesUsing the optical lattice we realize one-dimensionalquantum gases ordered in an array of one-dimensional tubes.Two lattice axes are ramped to a fixed value V⊥ ≡ Vx = Vz =30 ER. In this configuration the transverse tunnelling rates Jxand Jz are small compared to the duration of the experiment.We therefore neglect tunnelling between neighboring tubesand assume an array of isolated 1D gases. An additional opti-cal lattice is applied along the symmetry axis with a potentialdepth Vax ≡ Vy  V⊥. The potential depth Vax controls thevalue of U/J .3 Results3.1 Momentum distributionWe have studied the momentum distribution of thegas by imaging the expanding atom cloud after being releasedfrom the optical lattice. Prior to switching off the lattice thelaser intensity of the axial lattice is increased to  25 ER for40µs. Then all laser beams are extinguished simultaneouslywithin 5µs and the magnetic trapping potential is switched offwithin 300µs. This procedure may be regarded as a projectionof the wave function of the atoms onto the Bloch momen-tum states for a lattice depth of 25 ER. This method offersthe advantage of controlling adiabatic processes during theswitch-off of the lattice. These might affect the momentumdistribution of the released atoms [22].For our one-dimensional gases the collisional interactionduring expansion – which would give rise to a Thomas–Fermitype envelope of the density distribution in the y-direction –is small. This is related to the extremely fast radial expan-sion of the clouds, which are initially confined to the harmonicoscillator ground state a⊥ = √h/(mω⊥) with a trapping fre-quency of ω⊥ = 2π × 33 kHz. The peak density decreasesproportional to 1/√1+ (ω⊥t)2 which is much faster than thetimescale for axial expansion that is given by the axial trap-ping frequency of ωax = 2π ×80 Hz. This axial confinementis introduced by the gaussian waist of the laser beams formingthe array of one-dimensional tubes.We have studied the resulting momentum distributionafter 15 ms of free expansion of the cloud. The atoms wereimaged by resonant absorption imaging and we determine theoptical density in the center of the cloud by averaging overa 62µm wide section in the image. Figure 1 shows the meas-ured momentum distribution for a sample with an averagefilling of n¯ = 1.2 atoms per lattice site. We calculate the aver-age filling from the mean density of atoms which is derivedknown atom number and the measured trapping frequenciesand includes a modified coupling strength due to the localiza-tion of the atoms in the optical lattice potential [26]. The atomnumber distribution at a given lattice site in the superfluidphase can be approximated by a Poisson distribution. There-fore, even with an average occupation number of n¯ = 0.6 theprobability of finding doubly or multiply occupied lattice sitesis on the order of 10%.3.2 Quantum depletion of the condensateWhen the potential depth of the lattice is raised thestrength of the atom-atom interaction increases. This is ac-companied by quantum depletion of the condensate [24]. TheFIGURE 1 Measured momentum distribution for N = 1.5×105 atoms fordifferent depths of the axial lattice potential. Top to bottom on the left handsite: Vax = 5, Vax = 7, Vax = 9, Vax = 12, Vax = 18 (in units of ER). The in-set shows an absorption image after 10 ms time of flight where the area ofaveraging is indicatedKÖHL et al. 1D Bose gases in an optical lattice 1011FIGURE 2 Coherent fraction of the 1D Bose gas as measured from thetime-of-flight image. One-dimensional system with n¯ = 1.2 (solid squares),one-dimensional system with n¯ = 0.6 (open squares), three-dimensional gaswith n¯ = 1 (triangles). Part of the experimental data are taken from [7]. Thesolid line shows the calculated quantum depletion according to (2) usingNw = 40reduced condensate fraction of the system diminishes the con-trast in the matter wave interference pattern after the atomshave been released from the optical lattice. To extract the num-ber of coherent atoms Ncoh from the interference pattern, theinterference peaks at 0hk, ±2hk and ±4hk are fitted by gaus-sians. Incoherent atoms, both due to localization of the atomsin the lattice or due to depletion of the condensate, give rise toa broad gaussian background centered at zero quasi momen-tum which dominates for higher Vax [25]. Taking this fit asa measure of the number of incoherent atoms Nincoh, we cal-culate the coherent fraction fc = NcohNcoh+Nincoh [7]. The measureddata are shown in Fig. 2. For the one-dimensional Bose gas inan optical lattice the quantum depletion is calculated by [26]nn0=√12π2mm∗1a1Dn1Dln(4Nwπ). (2)Here m is the atomic mass, m∗ is the effective mass in thelattice, n1D is the 1D density, a1D is the one-dimensional scat-tering length [27] and Nw is the number of potential wellsoccupied by atoms. This expression holds in the tight bindingregime but for small depletion. Figure 2 displays the measuredcoherent fraction together with the calculated values for thequantum depletion, which show good agreement in the rangeof validity of (2).3.3 Coherence lengthFrom the width of the central coherent momen-tum component the coherence length of the gas in the opticallattice can be inferred. A Bose–Einstein condensate exhibitsa long coherence length which leads to a small width of the in-terference peak. Due to the inhomogeneity of the trap the tran-sition to the Mott insulator sets in at those positions, where thelocal density becomes commensurable with the spacing of thepotential wells in the optical lattice. This process breaks up thecondensed cloud into smaller units and therefore reduces thecoherence length of the system. In Fig. 3 we show the changeof the coherence length as a function of the parameter U/J .For the one-dimensional gas with an average occupancy ofFIGURE 3 Width of the central momentum peak as a measure of thephase coherence length of the system. One-dimensional system with n¯ = 1.2(solid squares), one-dimensional system with n¯ = 0.6 (open squares), three-dimensional gas with n¯ = 1 (triangles). Part of the data is taken from [7]n¯ = 1.2 we observe a kink in the coherence length at U/J  2which indicates the onset of the Mott insulating phase. For thethree-dimensional gas with n¯ = 1 we find the Mott insulatortransition at U/J  6, in agreement with the mean field the-ory. For very small one-dimensional systems with n¯ = 0.6 weobserve that the coherence length of the sample starts to in-crease around U/J = 4. This is in qualitative agreement withthe theoretical prediction that the Mott-insulator transition oc-curs for U/J  2 only for a chemical potential correspondinga density n¯ = 1 but at larger values for n¯ < 1 [3].4 ConclusionWe have studied the momentum distribution ofone-dimensional Bose gases in an optical lattice for differentaverage occupation of the lattice sites. We find that the co-herent fraction of atoms is independent of the atomic densitywhen we tune the interaction strength U/J from the superfluidto the Mott-insulating regime. The change of the coherencelength appear to indicate that the system enters the Mott-insulator for small atomic densities only for larger values ofU/J .ACKNOWLEDGEMENTS We acknowledge stimulating dis-cussions with Thierry Giamarchi, Corinna Kollath and Stefan Wessel andsupport from SNF and QSIT.REFERENCES1 W. Ketterle, N.J. van Druten: Phys. Rev.A 54, 61 (1996)2 D.S. Petrov, G.V. Shlyapnikov, J.T.M. Walraven: Phys. Rev. Lett. 85,3745 (2000)3 T.D. 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1D Bose gases in an optical lattice

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1D Bose gases in an optical lattice

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