Fringe spacing and phase of interfering matter waves O. Vainio,1,2 C. J. Vale,1,∗ M. J. Davis,1 N. R. Heckenberg,1 and H. Rubinsztein-Dunlop1 1School of Physical Sciences, University of Queensland, St Lucia, Qld 4072, Australia 2Department of Physics, University of Turku, FIN-20014, Turku, Finland (Dated: February 6, 2008) We experimentally investigate the outcoupling of atoms from Bose-Einstein condensates using 6 0 two radio-frequency (rf) fields in the presence of gravity. We show that the fringe separation in 0 the resulting interference pattern derives entirely from the energy difference between the two rf 2 fields and not the gravitational potential difference. We subsequently demonstrate how the phase and polarisation of the rf radiation directly control the phase of the matter wave interference and n providea semi-classical interpretation of theresults. a J PACSnumbers: 03.75.Hh,03.75.Be,39.20.+q,03.75.Pp 1 3 Ever since the first realisations of Bose-Einstein con- and B(r) is the magnetic field. ] r densates (BECs) in dilute atomic gases, their coherence Consideracondensatetrappedinacigar-shapedmag- e properties have been the subject of much investigation. neticpotentialoftheformU(r)=mω2(κ2x2+y2+z2)/2 h z The first clear demonstration that BECs possess long where m is the mass of the atom, ω = ω is the trap- t z y o range phase coherence was through the interference of ping frequency in the tight directions of the trap and t. twospatiallyseparatedcondensates[1]. Sincethen,other κ = ωx/ωz. Atoms can be outcoupled from the surface a experiments have studied the coherence properties us- of an ellipsoid of the magnetic equipotential which sat- m ing atom laser output from an array of tunnel coupled isfies the resonance condition (1). However, gravity will - condensates in an optical standing wave [2], Bragg spec- cause a displacement of the minimum of the total po- d troscopy [3], density fluctuations [4] and intereferometry tential from the magnetic field minimum. This gravita- n o [5,6]. Anelegantschemetoprobecondensatecoherence, tionalsagmeansa harmonicallytrappedcondensatewill c based on interfering atom laser beams, was reported by be displaced from the magnetic field minimum by a dis- [ Bloch et al. [7, 8]. This used two radio frequency (rf) tance, z = −g/ω2, where ω is the trapping frequency 0 z z fields to outcouple atoms from different locations within inthe directionofgravity. Thisdisplacementistypically 1 v a condensate. A high contrast matter wave interference greater than the size of the condensate, so that the el- 0 pattern was observed at temperatures well below the lipsoidal equipotential surfaces can be approximated by 9 BEC transition temperature, confirming the phase co- planeswhichintersectthecondensateatdifferentheights, 6 herence of the condensate. A numerical model of two z. Inthissituation,the dependence onthexandy coor- 1 outcoupled modes agreed with the experimental obser- dinates can be neglected for many quantitative purposes 0 vations [9]. More recently, the atom-by-atom build up and only the z dimension need be considered. 6 0 of a matter wave interference pattern has been observed In previous work [7, 8], two rf fields of frequencies, ω1 / using single atom detection [10]. To date, experiments andω2,wereusedtooutcoupleatomsfromacondensate. t a have primarily focussed on the visibility of the interfer- The two spatially separated resonances were interpreted m ence patterns. In this paper, we describe experiments as creating two slits from which atoms were extracted - which address the fringe spacing, phase and nature of from the condensate. The outcoupled atoms formed two d the interference. matter waves which interfered, in close analogy with a n Outcoupling atoms from Bose-Einstein condensates Young’s double slit experiment. The visibility of the in- o with rf fields has been used extensively to produce terference pattern provided a measure of the first order c : beams of atoms, generally referred to as “atom lasers” phase coherence of the condensate. v [11, 12, 13]. The rf radiation drives resonant (stimu- The outcoupling points, z and z , used in [7, 8], i 1 2 X lated) transitions from a trapped Zeeman sublevel to an were chosento be centred around the middle of the con- untrapped state in which the atom falls under gravity. densate, located at z , the minimum of the combined r 0 a Outcoupling occurs at locations where the total energy magnetic(harmonic)andgravitational(linear)potential. difference between the trapped and untrapped states is Under this condition, the gravitationalenergy difference equalto~ωrf. ThisisusuallydeterminedbytheZeeman between the two outcoupling points, determined by the potential so that the resonantcondition may be written slit separation, ∆z =z −z , is exactly equal to the dif- 1 2 ference in energy between the two applied rf fields. This ~ω =µ g |B(r)| (1) rf B F caneasilybeseenfromthederivativeofthemagneticpo- whereµB istheBohrmagneton,gF istheLand´eg-factor tential, where ∆E ≈ mωz2z∆z. At the central position, z , we find ∆E =~(ω −ω )=mg∆z. 0 1 2 However, this result is only true when z¯= z , (where 0 z¯ = (z + z )/2, is the distance from the magnetic 1 2 ∗Electronicaddress: [email protected] field minimum to the centre of the two slits). If the 2 pled) state |1,0i experiences negligible magnetic poten- tial, but, while still within the condensate, experiences both the mean field and gravitationalpotentials. In 1D, the total energy of a particular substate can be written as 1 E (z)=−m ( mω2z2+µ g B )−mgz+g |ψ(z)|2 mF F 2 z B F 0 1d (3) where B is the magnetic field at the minimum of the 0 trap, g is the 1D effective interaction strength (as- 1d sumedtobethesameforallm ,whichisapproximately F true but not an essential point in this discussion) and |ψ(z)|2 = PmF |ψmF(z)|2 is the total atomic density. The energies of the trapped |1,−1i state and the un- trapped |1,0i state are plotted (solid lines) in Fig. 1(a) for the parametersused in our experiments. The shaded regions indicate the mean field contribution to the total FIG.1: (a)Totalenergy(solidlines)ofatomsinthetrapped, energy. Also shownare two pairs of rf fields, dashedand mF = −1 and untrapped mF = 0 states, for the parame- dash-dottedlines,withthesame∆ω,chosentoliewithin ters used in our experiments. The dashed and dash-dotted the width of the condensate. As the two pairs are cen- lines represent two pairs of rf fields, with equal ∆ω, used to tred around different z¯, the resulting ∆z for each pair is outcouple atoms from the BEC at different locations. The different, but the total energy difference is ~∆ω. gravitational energy difference mg∆z may vary for a fixed IntheTFlimittheinteractionenergybetweenthecon- ∆ω,however,when theinteraction energy (shaded)of atoms densate and the outcoupled state exactly compensates intheuntrappedstateisincluded,thetotalenergydifference for the difference in gravitational potential at different between the two outcoupled matter waves is always equal to ~∆ω. (b) The measured outcoupled matter wave beams dis- slit locations. The density profile, |ψ(z)|2, mirrors the play an interference pattern with a constant fringe spacing shape of the magnetic trapping potential so that the en- λ(z)givenbyequation4forafixed∆ω (inthiscaseequalto ergy splitting between the two states is always given by 2π×1000s−1), independentof ∆z. the difference in their magnetic potentials. Additionally, the energy of trapped atoms within the condensate is independent of z so that only the final energies on the tworesonantpoints arenotsymmetricallylocatedabout m = 0 curve determine the energy difference between F the centre of the trap, the z2 dependence of the mag- the two outcoupled beams. netic potential means that the slit separation for a fixed Havingestablishedthatthe fringe spacing,λ, depends ∆ω =ω1−ω2 varies inversely with z¯. Thus the gravita- only on ∆ω, it can easily be shown that tional energy difference between the two resonance posi- tions,isnotnecessarilyequalto~∆ωand∆zmaychange p2g(z−z0) λ(z)= , (4) significantlyacrossthewidthofacondensate. Forahar- ∆f monic potential, the slit separation is approximately where∆f =∆ω/2π. λvarieswithz becausetheoutcou- ~∆ω pledatomsaccelerateinthez-directionundergravity,as ∆z ≈ mω2z¯. (2) can be seen in Fig. 1(b). z We have performed a range of experiments to verify InaYoung’sdoubleslitexperiment,thefringespacing, this for several values of ∆ω with the resonant points λ, of the interference pattern is proportional to ∆z−1. centred around various z positions. Our experimental However,indualrfoutcouplingexperimentswithafixed procedure for producing condensates has been described ∆ω, λ is independent of ∆z. While the gravitational elsewhere [14] and was used here with only slight mod- energydifference,mg∆z,betweenthe tworesonantloca- ifications. An atom chip is used to produce near pure tions can change,this is not the only energy to consider. condensatescontaining2×105 87Rbatomsinthe|1,−1i The fringe spacing of the interference pattern depends ground state. Our chip design facilitates the production on the total energy difference between the two indistin- of relatively large condensates in highly stable trapping guishableoutcouplingpathsandmustalwaysequal~∆ω fields. The final trapping frequencies are 160Hz in the to satisfy energy conservation. This is a general result tight direction and 6.7Hz in the weak direction. The whichwe discuss below,for a BEC inthe Thomas-Fermi elongated geometry of the trap means we must cool well (TF) regime. below the 3D critical temperature T to produce fully c Consider the specific case of an F = 1 87Rb TF con- phase coherent condensates (typically T for our param- φ densate. The total energy of the trapped state |F = eters is less than T /2 [15]). Outcoupling is induced by c 1,m =−1iconsistsofthesumofitsmagnetic,gravita- turning on two rf fields of the same amplitude, with fre- F tional and mean field energies. The untrapped (outcou- quencies ω and ω tuned to be resonant with atoms in 1 2 3 the condensate, and Rabi frequencies, Ω=µ g |B|/2~, B F of 50Hz for each rf source. After outcoupling for 10ms, the trap is left on for a further 3ms before being turned off abruptly. An absorption image is taken after 5.3ms of free expansion. We firstcheckedthe reproducibility ofthe fringespac- ing for a fixed ∆ω of 2π × 1000s−1 as ∆z was varied overtherange390nmto560nmbyvaryingz¯. Whilethe visibility of the interference pattern decreased near the edges of the condensate, the wavelength was consistent with the value predicted by equation (4) to within 1%. Experimental values of λ were determined by taking an image of the outcoupled atoms, similar to that shown in Fig. 1(b), converting the z spatial axis into a time axis through the relation t = p2(z−z0)/g, integrating the output over x and fitting a cos2 function to the data. FIG. 2: The phase of the interfering matter wave beams is determined by the phase of the beating rf fields used in to The uncertaintyinλis determinedby theuncertaintyin drivetheoutcoupling. (a) and (b) represent different runsof the fitted frequency of the cos2 function. Variations at the experiment under identical conditions apart from differ- the levelof1%arewithinourexperimentaluncertainties ent phases of the applied rf field. On the left are absorption andnotsignificantwhencomparedtowhatwouldbe ex- images of the condensate (top) and outcoupled atoms and pected ifλwasdeterminedby ∆z−1. This wouldleadto on the right is the beat note of the corresponding rf used to a variation of more than 30% across the range of values drivetheoutcoupling, measured on an oscilloscope. Thever- we measured. tical axis indicates the time before the image was taken and A semi-classical interpretation of these dual rf out- was obtained for the atom images through the relationship, coupling experiments, based on the interference of the t=p2(z−z0)/g applied rf fields, may also be used to understand these experiments. Defining the mean frequency, ω¯ = (ω + 1 ω2)/2, and the beat frequency, δ = (ω1 − ω2)/2, and It is clear from these images that the outcoupled atoms recalling the standard trigonometric identity correspond to the largest amplitude of the rf beat note. In order to quantify this, we have analysed a series of sinω t+sinω t=2sinω¯tcosδt (5) 1 2 similar data in which the relative phase of the rf fields we see that the sumof two oscillatingfields is equivalent was allowed to vary randomly over the range 0 to 2π. toanamplitudemodulatedcarrierwave. Inourcase,the The fitting procedure described earlier was applied to carrier frequency, ω¯, is typically three orders of magni- all of the absorption images to determine the phase of tude higher than the beat frequency, δ. Adding a phase, the outcoupled beam. A similar fit was applied to the φ, to one of the rf fields shifts the phase of both the square of the measured rf beat note and the two phases carrier and beating terms by half this amount. are plotted against each other (filled sircles) in Fig. 3. We may now consider the condensate interacting with Thedashedlinethroughthisdataisaplotofy =x. The a single rf field at the carrier frequency, which is ampli- phaseofthemodulatedatombeammatchesverywellthe tude modulated in time. The number of atoms outcou- phase of the applied rf field. pled is proportional to the Rabi frequency squared (ie. In these experiments both rf fields were provided by proportional to the amplitude squared of the rf field at passing the two rf currents through the same coil. This time t) and is modulated at 2δ =∆ω. Once outcoupled, means the rf field in the vicinity of the BEC was lin- the atoms fall under gravity and, provided they have a earlypolarised,perpendiculartothe quantisationaxisx. low spread of initial momenta, the outcoupled density We have also performed experiments using separate, or- will be modulated in time. thogonally mounted coils where each rf current was sent Todemonstratethis, asequenceofdualrfoutcoupling through a different coil. The rf interference is no longer experiments was performed using fixed values ω and ω linearly polarised but rather a field whose polarisation 1 2 (∆ω = 2π ×500s−1) but varying the relative phase of varies from vertical linear, to left hand circular, to hori- the two rffields. This hasthe effectofshifting the phase zontal linear, to right hand circular and back to vertical of the rf beat note. All other experimental parameters linear in a single beat period (T =1/∆ω). This is anal- were kept fixed. Two examples of the data obtained are ogous to the optical field used in lin ⊥ lin sub-Doppler shown in Fig. 2. Absorption images of the outcoupled (Sisyphus) laser cooling [16] but the field is periodic in atoms appear on the left and the beat note of the corre- time rather than space. sponding rf fields used for outcoupling (measured on an The outcoupling transition from |1,−1i to |1,0i re- oscilloscope) are shown on the right. The z axis of the quires+~ofangularmomentum, andcanonlybe driven absorptionimages has been rescaledby p2(z−z0)/g to by σ+ radiation. When the rf is derived from a single linearisethe time axisforeaseofcomparisonwiththerf. coil, it is easy to see that the maximum amplitude of 4 coils. The atomic output is phase shifted by approx- imately π/2 from the rf as expected. The slight mis- match between the measured shift of 0.55π and the ex- pected shift of 0.5π was due to imperfect alignment of the rf coils (precise perpendicular alignment would have impeded optical access in our setup). For coils mounted antiparallelmaximumoutcouplingwouldoccurwhenthe two rf fields are π out of phase. Inconclusion,wehavestudiedtheoriginsofthe fringe spacing and phase of matter wave interference patterns, produced by outcoupling atoms from a BEC with two rf fields. We have shown that the energy difference be- tween the two rf fields determines the spacing of the in- terference pattern, not the gravitationalpotential differ- ence determined fromthe classicalslit separation. Semi- classicalargumentsbasedoninterferingrffieldscorrectly FIG.3: Plotofthefittedphaseofthemodulatedatombeam predict the experimental observations. These also show against the phase of the beating rf field. Filled circles rep- how the phase and polarisation of the rf field determine resent experimentally measured data using a single coil and the phase of the observed matter wave interference pat- the dashed line is a plot of y = x. Open squares are exper- tern. This extends previous work which looked at the imentally measured phases when the rf is produced by two fringe visibility for the specific case where ~∆ω =mg∆z separate (near) orthogonally mounted coils and the dotted [7]. Our findings do not contradict the phase coherence line is a plot of y=x+0.55π. studies reported in ref. [7]. Indeed, any random phase gradients within the BEC would lead to random initial σ+ radiation occurs when the two rf fields are in phase. velocities that would degrade the observed interference patterns. Finally, we note that we have also performed The linearly polarisedfield may be decomposed into two counter rotating circular fields and it is the σ+ compo- experiments with cold thermal atoms and see (as in [7]) thatthe visibility ofinterferencepatterndiminishes, due nent of this whichcouples to the atom. With perpendic- to the thermal spread of velocities in the trapped gas. ularlyorientedcoilshowever,themaximumamplitudeof the σ+ fieldoccursduringthe circularpolarisedphaseof We acknowledge valuable discussions with Craig Sav- the beat note. This happens when the rf fields are π/2 age,AshtonBradleyandMurrayOlsenandtechnicalas- out of phase. sistance from Evan Jones. O. V. acknowledges financial Also shown in Fig. 3 (open squares) is a plot of the support from the Jenny and Antti Wihuri Foundation phase of the atom laser versus the phase of the beating andtheAcademyofFinland(project206108). 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