D&#8217

Fattori, Marco

Inguscio, Massimo

Jona Lasinio, M.

Modugno, Giovanni

Modugno, Michele

Roati, G.

Zaccanti, Matteo

English

Florence Research

Atom Interferometry with a Weakly Interacting Bose-Einstein CondensateM. Fattori,1,2 C. D’Errico,1,3 G. Roati,1,3 M. Zaccanti,1 M. Jona-Lasinio,1M. Modugno,1,3,4,5 M. Inguscio,1,3 and G. Modugno1,31LENS and Dipartimento di Fisica, Universita` di Firenze, and INFM-CNR Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy2Museo Storico della Fisica e Centro Studi e Ricerche ‘‘Enrico Fermi,’’ Compendio del Viminale, 00184 Roma, Italy3INFN, Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino, Italy4Dipartimento di Matematica Applicata, Universita` di Firenze, Italy5BEC-INFM Center, Universita` di Trento, I-38050 Povo, Italy(Received 26 October 2007; published 28 February 2008)We demonstrate the operation of an atom interferometer based on a weakly interacting Bose-Einsteincondensate. We strongly reduce the interaction induced decoherence that usually limits interferometersbased on trapped condensates by tuning the s-wave scattering length almost to zero via a magneticFeshbach resonance. We employ a 39K condensate trapped in an optical lattice, where Bloch oscillationsare forced by gravity. The fine-tuning of the scattering length down to 0:1 a0 and the micrometric sizes ofthe atomic sample make our system a very promising candidate for measuring forces with high spatialresolution. Our technique can be in principle extended to other measurement schemes opening newpossibilities in the field of trapped atom interferometry.DOI: 10.1103/PhysRevLett.100.080405 PACS numbers: 03.75.Dg, 03.75.Gg, 03.75.LmA Bose-Einstein condensate (BEC) is the ideal candi-date for atom interferometry because it offers the ultimatecontrol over the phase and amplitude of a matter wave. Itsmacroscopic coherence length [1] allows high phase con-trast in interferometric experiments. Its confinement inatomic traps or waveguides can lead to the realization ofatom interferometers with high spatial resolution [2,3].Unfortunately in high density trapped condensed clouds,interaction induces phase diffusion [4] and can cause sys-tematic frequency shifts due to uncontrolled atomic densitygradient, thus seriously limiting the performances of aBEC atom interferometer. In order to avoid the deleteriouseffect of interaction, atom interferometers for high preci-sion measurement use free falling dilute samples of non-degenerate atoms [5]. The main drawbacks are the limitedinterrogation time (0.5 s) due to the finite size of theapparatus and the poor spatial resolution of this type ofsensors. Using fermionic atoms instead of bosons repre-sents one possibility to have access to trapped interferom-etry with a degenerate gas [6]. In fact, because of the Pauliexclusion principle, the atomic scattering cross section atsufficiently low temperatures is fully suppressed. However,the quantum pressure limits the spatial resolution, and themomentum spread reduces the interference contrast.Another way to reduce the effect of the interaction is torealize a number squeezed splitting [7,8]. In this way,coherence times are increased at the expense of the inter-ference signal visibility. Despite the fundamental limitrepresented by interaction induced decoherence, severalgroups are performing experiments with trapped BECs[7–10], in the challenging search for the ‘‘ideal’’interferometer.In this Letter, we demonstrate the conceptually simplestsolution to the long-standing problem of interaction in-duced decoherence in BEC interferometers. We showhow, by properly tuning the interaction strength in a quan-tum degenerate gas of 39K [11] by means of a broadmagnetic Feshbach resonance [12], we can greatly increasethe coherence time of an atom interferometer. By achievingalmost vanishing values of the s-wave scattering length, wedemonstrate trapped atom interferometry with a weaklyinteracting BEC.The interferometer we adopted, commonly known asBloch oscillations interferometer, is based on a multiplewell scheme [2,6,13,14]. The condensate is adiabaticallyloaded in a sinusoidal potential with period =2, realizedwith an optical standing wave of wavelength . In thepresence of an external force F, the macroscopic wavefunction  of the condensate can be described as a coherentsuperposition of Wannier Stark states i [15], parame-trized with the lattice site index i, characterized by com-plex amplitudes of module ipand phase i,  Piipexpjii [16]. In the absence of interaction, thephase of each state evolves according to the energy shiftinduced by the external potential, i.e., i  Fit=2@. Byreleasing the cloud from the lattice, the macroscopic inter-ference between different Wannier Stark states gives rise tothe well known Bloch oscillations of the density pattern,with period tbloch  2h=F. A measurement of the fre-quency of such oscillations allows a direct measurementof the external force. Generally, interactions give rise to acomplex system of nonlinear equations for i and i. In theweakly interacting limit, the i do not change and, inaddition to i, extra phase terms 0i, proportional to thelocal interaction energy, are accumulated, i.e., 0i / asit,where as is the atomic scattering length. This causes phasediffusion and destruction of the interference pattern. In thisLetter, we demonstrate that by tuning as almost to zero, itPRL 100, 080405 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending29 FEBRUARY 20080031-9007=08=100(8)=080405(4) 080405-1 © 2008 The American Physical Societyis possible to reduce the values of 0i and extend thecoherence time of the interferometer.Our experimental apparatus has been described indetail elsewhere [11]. Using sympathetic cooling with87Rb and other techniques similar to the ones developedfor other potassium isotopes [11,17–19], we producea 39K BEC with 4 104 atoms in the absolute ground statejF  1; mF  1i. Because of the negative backgroundscattering length of 39K (abg  33 a0), the final stageof evaporation is performed in a crossed optical dipoletrap at a magnetic field of 395 G. At this field, a broadFeshbach resonance (width of the resonance   52 G,center of the resonance Bc  402:4 G [12]) allows thehomo-nuclear scattering length of K to be large and posi-tive (200 a0) and a stable condensate can form. Takinginto account the dependence of as on the magnetic fieldasB  abgf1 =B Bcg, the possibility of tuningwith high accuracy the interaction from repulsive to attrac-tive is evident. In particular, around Bzc  Bc 	 , i.e., thepoint where as vanishes, asB  abg=B Bzc and afine control of the scattering length (0:6 a0=G) is possible.Our magnetic field stability, better than 100 mG, allows areduction of as by nearly a factor of 103, down to the0:06a0 level. This high degree of tunability is possibleonly in few other atomic species where the ratio abg= isfavorably small [20,21].The condensate is produced in a horizontal crossedoptical dipole trap with frequencies x; y; z 85; 105; 100 Hz. After production of a degenerate sam-ple, we tune as to the desired value in 50 ms and we loadthe atoms in a vertical optical lattice with depth sER, wheres 6 and ER  @2k2=2m is the recoil energy from absorp-tion of a lattice photon (k  2=1032 nm), with m themass of a potassium atom. In order to avoid an abruptchange of the horizontal confinement from the combinedpotential, i.e., the trap plus the lattice, to the lattice only, weapply a levitating magnetic field gradient and decrease thetrapping confinement to (44, 76, 43) Hz in 40 ms. Note thatthe time scales for the variations of the scattering lengthand the external confinement have been chosen to alwayskeep the condensate in the ground state of the trappingpotential [22]. When the crossed dipole trap and the levi-tating field are switched off, Bloch oscillations start in thevertical lattice with radial confinement x  y  44 Hz.We have intentionally chosen this rather strong confine-ment to work in a regime where interaction is the mainsource of decoherence even when as is reduced down to the1a0 level.A first demonstration of our capability to extend thecoherence times of the interferometer by tuning as ispresented in Fig. 1. Here, we report the absorption imagesof BECs released from the lattice, after different times ofoscillations performed at 100a0 and 1a0 scattering length.The as  100 a0 measurement shows the performances ofthe interferometer with a typical interacting condensate.After two Bloch periods, the interference pattern is drasti-cally broadened. Instead, the measurement with as  1 a0shows no discernible broadening on this time scale, dis-playing how the interaction induced decoherence can bestrongly suppressed with our method.To analyze quantitatively the effect of the tuning of theinteraction, we repeat the same experimental sequence fordifferent as, measuring the vertical width of the centralpeak at integer times of the Bloch period (Fig. 2). Thisstudy is limited to a range of values of as (29 a0  1:3 a0)where interaction is the main source of decoherence (seebelow). Notice that initially the widths increase linearlywith time. This is a direct consequence of the phase terms0i / asit, with i  const, that evolve linearly in time[16]. Later on, when the momentum distribution of the2hkb)a)FIG. 1 (color online). Bloch oscillations from 0 to 4 ms, in steps of 0.4 ms, for a condensate with (a) 100 a0 and (b) 1 a0 scatteringlength. The picture shows absorption images of the cloud after release from the lattice. The expansion lasts 12.5 ms, and the scatteringlength is changed to 33 a0 only 3 ms before image acquisition.PRL 100, 080405 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending29 FEBRUARY 2008080405-2condensate occupies the whole first Brillouin zone, thewidths saturate. From a numerical calculation of i andthe theoretical analysis described in [16], we derive theinterference pattern between the Wannier Stark states inthe momentum space and determine the width of thecentral peak as a function of time. In Fig. 3, we comparethe theoretical decoherence rates with the measured onesas a function of as. Rates are defined as the slope of a linearfit, at short times, of the curves shown in Fig. 2 in unit of2@k. For this range of values of as, an almost linearbehavior is found. The possibility to increase the coherencetime of the interferometer by almost a factor of 100, whenas is tuned down to 1 a0, is remarkable. Notice that ourtheoretical analysis does not take into account the expan-sion of the cloud in the presence of interaction when theatoms are released from the lattice. In addition, it com-pletely neglects energetic and dynamical instability effects[23]. These are good approximations for small values of as.Below 1a0, a laser induced decoherence of 1 s1 startsto significantly contribute to the evolution of the system,preventing a quantitative comparison of the observed de-coherence rates with theory. We have however furtherinvestigated the effect of interaction on the decoherenceof the interferometer around the zero crossing. In thisregion, we have used a cloud, dense enough to make theeffect of the interaction visible, but sufficiently diluted inorder to completely exclude the effect of three body losses(see below) and to prevent the condensate from collapse forsmall negative as [11]. A condensate with as  ain  0 isinitially prepared. Right after the beginning of Bloch os-cillations, the external magnetic field is tuned to a finalvalue in 2 ms. This value is kept constant for 180 ms ofBloch oscillations and for 12 ms of expansion from thelattice. We have probed the decoherence for two differentdensities that were obtained by using two different valuesof ain, 3a0 and 1a0. In both cases, the width of the inter-ference peak reveals a minimum at 350G (see Fig. 4).Notice that this point is in agreement with the expectedposition of the zero crossing (350:4 0:4 G) [12]. Thesymmetric trend of the data confirms that the decoherencedepends on the magnitude and not on the sign of thescattering length. Notice that for ain  1 a0, the atomsare loaded over fewer lattice sites, resulting in larger i.In this case, a slightly larger width of the peak on theminimum is measured. This confirms that interaction in-duced decoherence is small but still present. The increase0 5 10 15 20 25 3001020304050607080Decoherence rate (s-1 )Scattering length (units of a0)FIG. 3 (color online). Black circles indicate decoherence rates(see text) as a function of the scattering length. Rates areobtained from the slope of a linear fit, at short times, of thecurves shown in Fig. 2 in unit of 2@k. The region put in evidencecomes from theoretical predictions of the rates, for 2 104 to5 104 atoms, in order to account for number fluctuations.0.5 a0 Bzc348.5 349.0 349.5 350.0 350.5 351.0 351.5 352.00.240.260.280.300.320.340.360.380.400.42Peak Width (units of 2hk)Magnetic Field (G)FIG. 4 (color online). Interference peak width as a function ofthe scattering length hold for 180 ms of Bloch oscillations. Thecondensate is prepared for two different values of as, 3a0(squares) and 1a0 (circles), and tuned to the final value, in2 ms, right after Bloch oscillations have started. The linesrepresent a Gaussian fit to the data.0 20 40 60 80 100 120 1400.00.20.40.60.81.029 a 013 a06 a02.5 a01.3 a0Peak Width (units of 2hk)Time (ms)FIG. 2 (color online). Vertical 1= ep width of the central peakof the density profile as a function of the Bloch oscillation time,at integer multiples of the Bloch period after 12.5 ms of expan-sion from the trap. The measurement is performed for differentvalues of the scattering length, listed on the plot, with 4 104atoms on average. The lines are a fit to the data excluding thepoint at t  0 ms, where imaging saturation effects can occur, upto times when the peak width is approximately 0:5 2@k.PRL 100, 080405 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending29 FEBRUARY 2008080405-3in the width is compatible with our expected resolution of0:06 a0. Further experiments will be necessary to clarifywhether dipole-dipole interaction between the magneticdipoles of the atoms starts to play a significant role whenas is tuned to such small values.The possibility of tuning the scattering length allowsalso to control the dimensions of a trapped condensate.This can have important consequences for what concernsthe spatial resolution of atom interferometers where atunable BEC is implemented. In the following, we describethis aspect in the case of a Bloch oscillations interferome-ter. During Bloch oscillations, the in-trap extension of thesample results from the spatial interference of differentWannier Stark states i. Therefore, the size of the cloudcan have at most a variation of the order of the extension lof the single i, which then set the ultimate space resolu-tion of the interferometer. In our case, for s  6 and F mg, where g  9:8 m=s2, l 2 m [15]. This limit can beachieved preparing condensates with a size of the order of lwhich is possible, in our case, using typical trapping po-tential. To verify the in-trap size of our nearly ideal BEC,we have performed an experiment in which the condensateis prepared at as  0 and then released from the combinedpotential, i.e., the trap plus the vertical lattice. In fact, thecoherent splitting of the condensate over several latticesites results in an interference pattern in the momentumspace with a central peak width in units of 2@k approxi-mately equal to the inverse of the number of occupied sites[24]. Notice that, when as  0, the in-trap momentumdistribution is accessible experimentally because it is ex-actly mapped in the atomic density after a free expansion.We measure a peak width of  0:1 2@k, and this corre-sponds to 10 sites occupied. This is in agreement with the1=e2 spatial width of 4:5 m for a condensate confined inour combined potential which has, for this measurement, avertical trapping frequency of 100 Hz. It is important tostress that for small values of as, the peak density of thecloud in presence of the lattice exceeds 1014 atoms=cm3.Three body losses over few hundreds of ms are neg-ligible thanks to a low value of K3  1:3 0:5 1029 cm6 s1 for 39K, that we have measured in theexperiment in the zero crossing region.The proved capability of drastically reduce the interac-tion induced decoherence by a fine-tuning of the scatteringlength, and the micrometric spatial resolution achievable(almost 1 order of magnitude better than in previous inter-ferometers of this kind [6,25]), make our system a verypromising candidate for precise measurements of CasimirPolder forces and gravitational forces between atoms andsurfaces at small distances in the 1–10 m region [25,26].Further studies on the effect of the residual interaction onthe coherence and experimental tests in a new systemwhere technical noise is suppressed will be necessary todetermine the ultimate performances of this interferometer.In conclusion, we have performed atom interferometrywith a nearly noninteracting BEC of 39K. We proved thatinteraction induced decoherence can be suppressed by al-most two orders of magnitude by tuning the scatteringlength almost to zero, with a broad Feshbach resonance.We expect that this technique can be extended to otherinterferometric schemes, opening new exciting perspec-tives in the field of trapped atom interferometry. We finallyshowed that a Bloch oscillations interferometer where atunable BEC is implemented is a promising candidate formeasurements of forces at small distances from surfaces.While writing this manuscript, we became aware of asimilar work performed with 133Cs in Innsbruck [27].We acknowledge contributions by F. Marin, discussionswith A. Simoni, A. Smerzi, S. Stringari, and the rest of thequantum gases group at LENS. This work was supportedby MIUR (PRIN 2006), EU (MEIF-CT-2004-009939),INFN, Ente CRF, and Int. Proj. SCALA.[1] M. R. Andrews et al., Science 275, 637 (1997).[2] B. P. Anderson and M. A. Kasevich, Science 282, 1686(1998).[3] Y. Shin et al., Phys. Rev. Lett. 92, 050405 (2004).[4] Y. Castin and J. Dalibard, Phys. Rev. A 55, 4330 (1997).[5] A. Peters, K. Y. Chung, and S. Chu, Nature (London) 400,849 (1999).[6] G. Roati et al., Phys. Rev. Lett. 92, 230402 (2004).[7] G. B. Jo et al., Phys. Rev. Lett. 98, 030407 (2007).[8] W. Li et al., Phys. Rev. Lett. 98, 040402 (2007).[9] A. Gu¨nther et al., Phys. Rev. Lett. 98, 140403 (2007).[10] T. Schumm et al., Nature Phys. 1, 57 (2005).[11] G. Roati et al., Phys. Rev. Lett. 99, 010403 (2007).[12] C. D’Errico et al., New J. Phys. 9, 223 (2007).[13] M. BenDahan et al., Phys. Rev. Lett. 76, 4508 (1996).[14] R. Battesti et al., Phys. Rev. Lett. 92, 253001 (2004).[15] J. Zapata et al., Phys. Rev. A 63, 023607 (2001).[16] D. Witthaut et al., Phys. Rev. E 71, 036625 (2005).[17] L. De Sarlo et al., Phys. Rev. A 75, 022715 (2007).[18] G. Modugno et al., Science 294, 1320 (2001).[19] G. Roati et al., Phys. Rev. Lett. 89, 150403 (2002).[20] L. Khaykovich et al., Science 296, 1290 (2002);K. Strecker et al., Nature (London) 417, 150 (2002).[21] T. Weber et al., Science 299, 232 (2003).[22] Yu. Kagan, E. Surkov, and G. Shlyapnikov, Phys. Rev.Lett. 79, 2604 (1997); T. Gericke et al., J. Mod. Opt. 54,735 (2007).[23] L. Fallani et al., Phys. Rev. Lett. 93, 140406 (2004).[24] P. Pedri et al., Phys. Rev. Lett. 87, 220401 (2001).[25] G. Ferrari et al., Phys. Rev. Lett. 97, 060402 (2006).[26] S. Dimopoulos and A. A. Geraci, Phys. Rev. D 68, 124021(2003); I. Carusotto et al., Phys. Rev. Lett. 95, 093202(2005); J. Obrecht et al., Phys. Rev. Lett. 98, 063201(2007).[27] M. Gustavsson et al., Phys. Rev. Lett. 100, 080404 (2008).PRL 100, 080405 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending29 FEBRUARY 2008080405-4

Atom interferometry with a weakly interacting Bose-Einstein condensate

https://flore.unifi.it/bitstream/2158/317267/1/IdealInterferometry.pdf

Atom interferometry with a weakly interacting Bose-Einstein condensate

Abstract

Similar works

Full text

Available Versions

Florence Research