Atom interferometry with Bose-Einstein condensates in a double-well potential

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1 Atom interferometry with Bose-Einstein conensates in a ouble-well potential Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchar, an A. E. Leanhart Department of Physics, MIT-Harvar Center for Ultracol Atoms, an Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambrige, Massachusetts, 2139 (Date: October 8, 23) A trappe-atom interferometer was emonstrate using gaseous Bose-Einstein conensates coherently split by eforming an optical single-well potential into a ouble-well potential. The relative phase between the two conensates was etermine from the spatial phase of the matter wave interference pattern forme upon releasing the conensates from the separate potential wells. Coherent phase evolution was observe for conensates hel separate by 13 µm for up to 5 ms an was controlle by applying ac Stark shift potentials to either of the two separate conensates. PACS numbers: 3.75.Dg, q, b, 3.75.Lm Atom interferometers have been use to sense accelerations [1, 2] an rotations [3, 4], monitor quantum ecoherence [5], characterize atomic an molecular properties [6], an measure funamental constants [1, 7]. Demonstrating atom interferometry with particles confine by magnetic [8 11] an optical [12] microtraps an waveguies woul realize the matter wave analog of optical interferometry using fiber-optic evices. Current proposals for confine-atom interferometers rely on the merger an separation of two potential wells to split an recombine atomic wavepackets [13 15]. Atom-atom interactions ten to localize particles in either potential well an reuce the coherence of the splitting an recombination processes [16, 17], whereas tunnelling serves to elocalize the atomic wavepackets an maintain a wellefine relative phase between the potential wells [16]. Bose-Einstein conensates are to matter wave what lasers are to optical interferometry, i.e., a coherent, single moe an highly brilliant source. Conensates have been coherently elocalize over multiple sites in optical lattices where the tunnelling energy ominates the onsite atom-atom interaction energy ue to the sub-micron barrier between neighboring potential wells [2, 18 21]. Such a thin barrier helps to maintain phase coherence across the lattice, but also prevents aressing iniviual lattice sites. Large separation of the split conensates an iniviual aressability of each arm are requirements for a versatile an useful interferometer, able, e.g., to sense spatially-varying forces. These apparently simple requirements represent a consierable challenge when it comes to splitting a Bose-Einstein conensate with a thick barrier that prevents tunneling an moving it by large (compare to the conensate size) istances without affecting its quantum mechanical phase. It is not even theoretically clear if the two conensates generate by splitting will share the same phase (a phasecoherent state) or have well-efine number of particles each but without relative phase coherence (a numbersqueeze state) [16, 22 24]. In this Letter, we emonstrate a trappe-atom interferometer with gaseous Bose-Einstein conensates confine in an optical ouble-well potential. Conensates were coherently split by eforming an initially single-well potential into two wells separate by 13 µm. The relative phase between the two conensates was etermine from the spatial phase of the matter wave interference pattern forme upon releasing the atoms from the separate potential wells [25, 26]. An ac Stark phase shift was applie to either conensate by temporarily turning off the laser beam generating its potential well. The spatial phase of the matter wave interference pattern shifte linearly with the applie phase shift. In aition, the large well separation suppresse tunnelling such that the phase of each conensate evolve inepenently. Without the ai of tunnelling to preserve phase coherence, the measure coherence time of the separate conensates was 5 ms. Bose-Einstein conensates containing over Na atoms were create in the F =1,m F = 1 state in a magnetic trap, capture in the focus of a 164 nm optical tweezers laser beam, an transferre into an auiliary science chamber as escribe in Ref. [27]. In the science chamber, the conensate was loae from the optical tweezers into a seconary optical trap forme by a counter-propagating, orthogonally-polarize 164 nm laser beam shifte in frequency from the tweezers by 1 MHz to avoi interference effects. The seconary optical trap was forme by a collimate laser beam that passe through an acousto-optic moulator (AOM) an was focuse onto the conensate with a lens [Fig. 1(a)]. The AOM was riven simultaneously by two raio frequency (rf) signals to tailor the shape the potential from single-well [Fig. 1] to ouble-well [Fig. 1(c)]. The separation between the potential wells was controlle by the frequency ifference between the rf rives. The 1/e 2 raius of each focuse beam was 5 µm. For typical optical powers, this resulte in a single beam trap epth U = h 5 khz, where h is Planck s constant, an a raial (aial) trap frequency f r = 615 Hz (f z = 3 Hz). Conensates were initially loae from the tweezers into a single-well trap [Fig. 1]. After holing the clou in this trap for 15 s to amp ecitations, the conensate

2 (a) F F (a) Integrate Optical Density AOM g f 1, f 2 (c) Position U FIG. 1: (a) Schematic iagram of the optical setup for the ouble-well potential. An acousto-optic moulator (AOM) was riven by two frequencies, f 1 an f 2, an iffracte a collimate beam into two beams. The AOM was place in the focal plane of a lens of focal length F so that the two beams propagate parallel to each other. The raial separation of the potential wells,, was controlle by the frequency ifference, f = f 1 f 2. g enotes the irection of gravitational acceleration. The absorption image shows two well-separate conensates confine in the ouble-well potential iagrame in (c). The fiel of view is 7 µm 3 µm. Energy iagrams for initial single-well trap with =6µm an(c) final ouble-well trap with =13µm. In both an (c), U = h 5 khz an the peak atomic mean fiel energy was h 3 khz. The potential imple in was <h 5 Hz which was much less than the peak atomic mean fiel energy allowing the trap to be characterize as a single-well. The potential barrier in (c) was h 4.7 khz which was larger than the peak atomic mean fiel energy allowing the resulting split conensates to be characterize as inepenent. containe 1 5 atoms with a peak atomic mean fiel energy µ h 3 khz. The single-well trap was eforme into a ouble-well potential [Fig. 1(c)] by linearly increasing the frequency ifference between the rf signals riving the AOM over 5 ms. The amplitue of the rf signals were tailore uring the splitting process to guarantee an even ivision of the conensate atoms an nearly equal trap epths after splitting. Releasing the conensates from the ouble-well potential allowe them to ballistically epan, overlap, an interfere (Fig. 2). Each realization of the eperiment prouce a matter wave interference pattern with the same spatial phase. This reproucibility emonstrate that eforming the optical potential from a single-well into a ouble-well coherently split the conensate into two clous with eterministic relative phase, i.e., therel- ative phase between the two conensates was the same from shot-to-shot. This eperiment erive its ouble-well potential from a single laser beam passing through an AOM. Thus, vibrations an fluctuations of the laser beam were common moe to both wells, an a clean an rapi trap turn off was achieve by switching off the rf power riv- g FIG. 2: Matter wave interference. (a) Absorption image of conensates release from the ouble-well potential in Fig. 1(c) immeiately after splitting an allowe to overlap uring 3 ms of ballistic epansion. The imaging ais was parallel to the irection of gravitational acceleration, g. The fiel of view is 6 µm 35 µm. Raial ensity profiles were obtaine by integrating the absorption signal between the ashe lines, an typical images gave > 6% contrast. The soli line is a fit to a sinusoially-moulate Gaussian curve from which the phase of the interference pattern was etracte (see tet). This figure presents ata acquire in a single realization of the eperiment. ing the AOM. In contrast, past eperiments create a ouble-well potential by splitting a magnetically trappe conensate with a blue-etune laser beam [25]. This previous work was unable to observe a reproucible relative phase between the split conensates, ue to pointing fluctuations in the blue-etune laser beam an irreproucible turn off of the high current magnetic trap that initiate ballistic epansion. The relative phase between the two separate conensates was etermine by the spatial phase of their matter wave interference pattern. For a ballistic epansion time t 1/f r, each conensate ha a quaratic phase profile [28], ψ ± ( r, t) = n ± ( r, t)ep(i m 2 ht r ± /2 2 + φ ± ), where ± enotes one well or the other, n ± is the conensate ensity, m is the atomic mass, is a vector connecting the two wells, φ ± is the conensate phase, an h = h/2π. Interactions between the two conensates uring ballistic epansion have been neglecte. The total ensity profile for the matter wave interference pattern takes the form n( r, t) =(n + + n +2 n + n cos( m ht + φ r)), (1) where φ r = φ + φ is the relative phase between the two conensates an = ˆ. To etract φ r, an integrate cross section of the matter wave interference pattern [Fig. 2] was fitte with a sinusoially-moulate Gaussian curve, G() = A ep( ( c ) 2 /σ 2 )(1 +

3 Fringe Phase (eg) Stanar eviation (eg) (a) (c) Hol Time (ms) FIG. 3: Phase coherence of the separate conensates. (a) The spatial phase of the matter wave interference pattern is plotte versus hol time after splitting the conensate. Each point represents the average of eight measurements. The phase evolution was ue to unequal trap epths for the two wells, which was etermine from the linear fit to be h 7 Hz or 1% of the trap epth. Stanar eviation of eight measurements of the relative phase between separate conensates versus hol time after splitting. A stanar eviation 14 egrees (ashe line) is epecte for ranom relative phases. (c) an () Typical matter wave interference patterns after ms an 5 ms holing. The curvature of the interference fringes increase with hol time limiting the coherence time of the separate conensates to 5 ms. B cos( 2π λ ( )+φ f )), where φ f is the phase of the interference pattern with respect to a chosen fie. Ieally, if was set at the center of the two wells, then φ r = φ f. However, misalignment of the imaging ais with the irection of gravitational acceleration create a constant offset, φ f = φ r + δφ. With t = 3 ms the measure fringe perio, λ =41.5 µm, was within 4% of the point source formula preiction [Eq. (1)], ht/m = 39.8 µm. The relative phase between the separate conensates was observe to evolve linearly in time [Fig. 3(a)]. This evolution was primary ue to applie ac Stark shift potentials an coul be tailore by ajuste the ifference in trap epths between the two wells. The stanar eviation of eight measurements of φ r was < 9 egrees for conensates split then hel separate for up to 5 ms [Fig. 3]. Furthermore, for hol times up to 1 ms, the stanar eviation was substantially smaller, < 4 egrees. Since a ranom istribution of phases between 18 an +18 egrees woul have a stanar eviation of 14 egrees, the measure results quantitatively confirm the reproucible nature of the splitting process an the coherent evolution of the separate conensates. 4 () 5 Funamental limits on the phase coherence between isolate conensates arise ue to the number fluctuations associate with the escription of the conensate [16, 22 24, 29]. For Poissonian number fluctuations, we woul epect a phase iffusion time 25 ms. Atom-atom interactions may localize particles in either potential well an reuce the number fluctuations. This woul reuce the measure coherence of the split conensates, but eten the phase iffusion time [16, 22 24, 3]. The uncertainty in etermining φ r at hol times > 5 ms is attribute to aial an breathing-moe ecitations create uring the splitting process. These ecitations lea to interference fringes that were angle an ha substantial curvature, renering a etermination of φ r impossible. Splitting the conensate more slowly in an effort to minimize ecitations, but still fast compare to the phase iffusion time, i not improve the measure stability of φ r. Since controlling aial ecitations appears critical for maintaining phase coherence, splitting conensates that are freely propagating in a waveguie potential may be more promising [1]. The phase sensitivity of the trappe-atom interferometer was emonstrate by applying ac Stark phase shifts to either (or both) of the two separate conensates. Phase shifts were applie to iniviual conensates by pulsing off the optical power generating the corresponing potential well for a uration τ p 1/f r. Figure 4(a) shows that the spatial phase of the matter wave interference pattern shifte linearly with the pulse uration, as epecte. Due to the inhomogeneous optical potential, U(r), the applie ac Stark phase shifts varie across the conensate as φ(r) = U(r)τ p / h. Averaging this phase shift over the inhomogeneous conensate ensity, n( r), approimates the epecte spatial phase shift of the matter wave interference pattern as φ = 1 N 3 rn( r) φ( r) =(U 2 7 µ) t/ h, where N is the number of atoms, an U is the potential well epth, an µ is the atomic mean fiel energy. The measure phase shifts yiele U = h 5 khz [Fig. 4], which was consistent with calculations base on the measure trap frequencies. The measure phase shifts of the matter wave interference epene only on the time-integral of the applie ac Stark phase shifts [Fig. 4], as epecte for uncouple conensates. The final relative phase, φ r, shoul be the same on ifferent phase trajectories because the history of phase accumulation oes not affect the total amount of accumulate phase. For couple conensates, Josephson oscillations between the wells woul cause the relative phase to vary nonlinearly with time [28, 31] an prouce a time epenent signal in Fig. 4. Due to the large well separation an mean fiel energy h 1.7 khz below the barrier height, the single-particle tunnelling rate in our system was etremely low ( < 1 3 Hz) [28], an the conensates were effectively uncouple. This claim is supporte qualitatively by the absorption image

4 (a) 54 Spatial Phase (eg) splitting τ p 15 µs free epansion ing ballistically. Implementing a similar reaout scheme with magnetic potentials generate by microfabricate current carrying wires shoul be possible an woul eliminate eleterious mean fiel effects inherent in proposals using in-trap wavepacket recombination. Propagating the separate conensates along a waveguie prior to phase reaout woul create an atom interferometer with an enclose area, an hence with rotation sensitivity. Spatial Phase (eg) τ p (µs) splitting 1 ms free epansion τ 4 τ (µs) 5 µs FIG. 4: Trappe-atom interferometry. (a) ac Stark phase shifts were applie to either well eclusively (soli circles an open circles) or both wells simultaneously (crosses) by turning off the corresponing rf signal(s) riving the AOM for a uration τ p. The resulting spatial phase of the matter wave interference pattern scale linearly with τ p an hence the applie phase shift. Applying the ac Stark shift to the opposite well (soli versus open circles) resulte in an interference pattern phase shift with opposite sign. Applying ac Stark shifts to both wells (crosses) resulte in no phase shift for the interference pattern. This ata was taken with a slightly moifie eperimental setup such that the trap epth of the iniviual potential wells was U = h 17 khz. This is why a 5 µs pulse inuce a 27 egree phase shift. A 5 µs pulse inuce a 7 egree shift inepenent of the pulse elay, τ.theeperimental setup was as escribe in Fig. 1 (U = h 5kHz). Soli an open circles have the same meaning as in (a). The insets show the time sequence of the optical intensity for the well(s) temporarily turne off. in Fig. 1(a) an the observation of high-contrast matter wave interference patterns that penetrate the full atomic ensity profile with uniform spatial perio an no thick central fringe [32]. In conclusion, we have performe atom interferometry with Bose-Einstein conensates confine in an optical ouble-well potential. A coherent conensate beam splitter was emonstrate by eforming a single-well potential into a ouble-well potential. The large spatial separation between the potential wells guarantee that each conensate evolve inepenently an allowe for aressing each conensate iniviually. Recombination was performe by releasing the atoms from the oublewell potential an allowing them to overlap while epan- We thank W. Jhe, C. V. Nielsen, an A. Schirotzek for eperimental assistance an S. Gupta, Z. Hazibabic, an M. W. Zwierlein for critical comments on the manuscript. This work was fune by ARO, NSF, ONR, an NASA. M.S. acknowleges aitional support from the Swiss National Science Founation. URL: [1] A. Peters et al., Phil. Trans. R. Soc. Lon. A 355, 2223 (1997). [2] B. P. Anerson an M. A. Kasevich, Science 282, 1686 [3] A. Lenef et al., Phys. Rev. Lett. 78, 76 (1997). [4] T. L. Gustavson, P. Bouyer, an M. A. Kasevich, Phys. Rev. Lett. 78, 246 (1997). [5] M. S. Chapman et al., Phys. Rev. Lett. 75, 3783 (1995). [6] C. R. Ekstrom et al., Phys.Rev.A51, 3883 (1995). [7] S. Gupta, K. Dieckmann, Z. Hazibabic, an D. E. Pritchar, Phys. Rev. Lett. 89, 1441 (22). [8] H. Ott et al., Phys. Rev. Lett. 87, 2341 (21). [9] W. Hänsel, P. Hommelhoff, T. W. Hänsch, an J. Reichel, Nature 413, 498 (21). [1] A. E. Leanhart et al., Phys. Rev. Lett. 89, 441 (22). [11] S. Schneier et al., Phys. Rev. A 67, (23). [12] R. Dumke et al., Phys. Rev. Lett. 89, 2242 (22). [13] E. A. Hins, C. J. Vale, an M. G. Boshier, Phys. Rev. Lett. 86, 1462 (21). [14] W. Hänsel, J. Reichel, P. Hommelhoff, an T. W. Hänsch, Phys. Rev. A 64, 6367 (21). [15] E. Anersson et al., Phys. Rev. Lett. 88, 141 (22). [16] C. Menotti, J. R. Anglin, J. I. Cirac, an P. Zoller, Phys. Rev. A 63, 2361 (21). [17] J. A. Stickney an A. A. Zozulya, Phys. Rev. A 66, 5361 (22). [18] C. Orzel et al., Science 291, 2386 (21). [19] F. S. Cataliotti et al., Science 293, 843 (21). [2] G. et al., Phys. Rev. Lett. 87, 1645 (21). [21] M. Greiner et al., Nature 415, 39 (22). [22] J. Javanainen an M. Wilkens, Phys. Rev. Lett. 78, 4675 (1997). [23] A. J. Leggett an F. Sols, Phys. Rev. Lett. 81, 1344 [24] J. Javanainen an M. Wilkens, Phys. Rev. Lett. 81, 1345 [25] M. R. Anrews et al., Science 275, 637 (1997). [26] O. Manel et al., Phys. Rev. Lett. 91, 147 (23). [27] T. L. Gustavson et al., Phys. Rev. Lett. 88, 241 (22). [28] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, an S. Stringari,

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