Observation and control of quantized scattering halos
G. Chatelain, N. Dupont, M. Arnal, V. Brunaud, J. Billy, B. Peaudecerf, P. Schlagheck, D. Guéry-Odelin
OObservation and control of quantized scattering halos
G. Chatelain , N. Dupont , M. Arnal , V. Brunaud , J. Billy , B. Peaudecerf , P. Schlagheck and D. Guéry-Odelin Laboratoire Collisions, Agrégats, Réactivité, IRSAMC,Université de Toulouse, CNRS, UPS, F-31062 Toulouse, France CESAM research unit, University of Liege, 4000 Liege, Belgium
We investigate the production of s − wave scattering halos from collisions between the momentumcomponents of a Bose-Einstein condensate released from an optical lattice. The lattice periodicitytranslates in a momentum comb responsible for the quantization of the halos’ radii. We report onthe engineering of those halos through the precise control of the atom dynamics in the lattice: we areable to specifically enhance collision processes with given center-of-mass and relative momenta. Inparticular, we observe quantized collision halos between opposite momenta components of increasingmagnitude, up to 6 times the characteristic momentum scale of the lattice. Introduction - Scattering experiments act as a probe re-vealing at a macroscopic scale the properties of collisionalprocesses that occur at a microscopic scale. Since theseminal works of H. Geiger, E. Marsden and E. Ruther-ford [1–3], such experiments have remained a method ofchoice to probe atoms, molecules and their interactions.In short, the description of scattering in quantum me-chanics gives rise to two remarkable features : firstly thequantum description of the collision process leads to itsdecomposition in terms of partial scattering waves [4],secondly, each of the collisional partners can itself be de-scribed as a matter-wave, which can be in a superposi-tion of several components with well defined momenta,leading to multiple elementary collisional processes hap-pening all “at once". As a result, quantum scatteringexemplifies a key feature of quantum mechanics: wave-particle duality.With the advent of cold atom samples, this topic hasbeen revisited with only a few partial waves involved.The characterization of the collisional properties, andin particular of the s -wave scattering length, was per-formed either using photoassociation measurements [5–8]or studying the kinetics towards equilibrium of an atomicsample [9, 10]. In this latter type of measurements, theinterplay between partial waves turns out to be subtle:the thermalization rate involves partial waves interfer-ences while the collision rate does not [11]. Using a1D collider geometry, the experiments of Refs. [12, 13]have captured quantum scattering in its purest form:at low energy, the s -wave collisions create a sphericalshell of pair-correlated atoms, and at slightly higher en-ergy the volume occupied by the scattered atoms reflectthe interference between partial waves. Such collider-like experiments have also been carried out with differentspecies [14, 15]. More recently, second- and third-ordercorrelations between momentum-correlated atoms in acollisional halo have been investigated [16, 17], openingquantum-nonlocality tests to ensembles of massive parti-cles [18].In this article, we engineer the collisions betweenatomic ensembles in a multiple 1D collider, using an out-of-equilibrium Bose-Einstein condensate (BEC) of Rbreleased from an optical lattice. The collisions occur
Figure 1: Scheme of the experimental protocol. A Bose-Einstein condensate (red clouds) is initially loaded in an opti-cal lattice (dashed blue line). a The lattice is suddenly phase-shifted by an amount ϕ = ϕ (solid blue line); here ϕ = 180 ◦ .The subsequent dynamics in the lattice modifies the momen-tum distribution. b After a sufficiently long time-of-flight, themomentum distribution exhibits diffraction peaks, separatedby (cid:126) k L = h/d , characteristic of the wave nature of the BECin the lattice (red disks) along with isotropic scattering halosdue to s -wave scattering between atoms (light red disk). through s -wave scattering, between the discrete momen-tum components of the BEC, in the course of the time-of-flight, leading to the appearance of scattering halos[19]. Through accurate control of the lattice phase andamplitude, we show that it is possible to engineer thedynamics of the BEC in the lattice before the release,and tailor the wavefunction of the atoms after releaseand expansion, in order to selectively enhance the colli-sion processes between specific momentum componentsof the BEC. We can thus produce quantized scatteringhalos from collisions with a chosen relative and/or center-of-mass (c-o-m) momentum. a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov t hold ( s ) s bcdefg a = 140 o k Visibility Center00.25 -4.5 k k k k k b c d e -6 -4 -2 0 2 4 60.00.5 f -6 -4 -2 0 2 4 6 p / k L g -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 p / k L Figure 2:
Collision halos of × h/d diameter and varying center-of-mass momentum. a. Simulated visibility V of the orders of diffraction (see text) as a function of the lattice depth s , and holding time t hold , for a sudden phase shift of ϕ = 140 ◦ . The center-of-mass momentum of the expected dominant collision halo is color-coded. Black markers indicateexperimentally tested parameters, with the vertical error bar showing the standard deviation of the independent measurementof s . b-g Left: experimental diffraction orders probability distributions { π j } (blue, offset to the right), with black error barsshowing one standard deviation, and simulated ones for the same parameters ϕ , s, t hold (red, offset to the left). The red shadedhorizontal bands represent the visibilities for the theoretically simulated distributions, and stretch between the probabilitiesof the second and third most populated diffraction orders. Right: single-shot experimental absorption images from which themomentum distributions are extracted. The collision halo between the two most populated orders are clearly visible. The colorcode of the absorption images indicates the c-o-m momentum for the scattering halo, with the same color code as Fig. 2 a .The color scale is set to enhance the collision halos (clipping high values on some diffraction orders). The parameters used fordata b-g are { s, t hold [ µ s] } = { . ± . , } , { . ± . , } , { . ± . , } , { . ± . , . } , { . ± . , . } and { . ± . , . } , respectively. Background - When a BEC is loaded in an opticallattice, the periodic structure of the lattice imprints onthe wavefunction. In particular the resulting momentumdistribution is made up of equally spaced peaks, sepa-rated by an interval h/d , where d is the lattice spac-ing and h the Planck constant. The shape of each in-dividual peak is set by atomic interactions and the ex-tra confinement superimposed to the lattice. This mo-mentum distribution can be measured by releasing theatoms from the lattice and allowing the atomic cloudto ballistically expand for a sufficiently long time-of-flight (TOF), t T OF : the spatial density n ( r ) then repro-duces the initial momentum density ˜ n ( p ) up to a scaling: n ( r , t T OF ) = ˜ n ( p = m r /t T OF , t = 0) , with m the massof the atom (see Fig. 1).However, this comb pattern resulting from the wavenature of the BEC is only a partial description of thefinal atomic density distribution: for interacting atoms,collisions originating from the particle nature of mattermay occur during the ballistic expansion. In the low-energy regime characteristic of ultracold atoms, s -waveelastic scattering dominates which, due to energy andmomentum conservation in the collision process, resultsin spherical halos centered on the center of mass of thetwo colliding atomic wavepackets (see [20, 21] and Ap-pendix A).Thus, spherical collision halos are expected to ap-pear in the momentum distribution between the diffrac- tion orders after time-of-flight, as depicted in Figure 1.Each diffraction order, centered on a position x j = j × h/ ( md ) × t T OF ( j ∈ Z ) , contains a fraction π j ofthe number of atoms N initially in the BEC. These frac-tions determine the atomic density in the halos : in aperturbative approach, the number of collisions betweenorders j and k , and therefore the number of atoms scat-tered in the corresponding halo, is proportional to theproduct π j π k ([22], Appendix A).In this work, we engineer the (dominant) scatteringhalos, by precisely controlling the state of the BEC inthe lattice before release. To this end, we apply a suddendisplacement to the lattice on a scale smaller than the lat-tice spacing. This triggers out-of-equilibrium dynamicsof the BEC inside the lattice. Controlling the duration ofthis evolution in the lattice prior to release (see Fig. 1),we effectively tailor the momentum distribution { π j } . Methods - We perform our experiments in a hybrid trap[23] in which we obtain pure rubidium-87 Bose-Einsteincondensates of · atoms in the lowest hyperfine state | F = 1 , m F = − (cid:105) . These BECs are loaded in a one-dimensional optical lattice produced by two counterprop-agating laser beams of wavelength λ = 1064 nm super-imposed to the optical dipole beam of the hybrid trap.In the optical lattice, the atoms experience the followingpotential : t hold ( s)01020304050 s bcdefg a = 180 o k Visibility Diameter00.25 10 k k k k k -6 -4 -2 0 2 4 6 p / k L b -6 -4 -2 0 2 4 6 p / k L c -6 -4 -2 0 2 4 6 p / k L d -6 -4 -2 0 2 4 6 p / k L e -6 -4 -2 0 2 4 6 p / k L f -6 -4 -2 0 2 4 6 p / k L g Figure 3:
Collision halos of j × h/d diameter (1 ≤ j ≤ and center-of-mass momentum 0. a. Simulated visibility V of the orders of diffraction (see text) as a function of the lattice depth s , and holding time t hold , for a shift of ϕ = 180 ◦ . Thec-o-m momentum of the collision is zero, and the relative momentum is color-coded. Black markers indicate experimentallytested parameters, with the vertical error bar showing the standard deviation of the independent measurement of s . Fullsymbols relate to the data shown in b-g , and all symbols including empty ones relate to figure 4 d . b-g Left: experimentaldiffraction orders probability distributions { π j } (blue, offset to the right), with black error bars showing one standard deviation,and simulated ones for the same parameters ϕ , s, t hold (red, offset to the left). The red shaded horizontal bands representthe visibilities for the theoretically simulated distributions, and stretch between the probabilities of the second and third mostpopulated diffraction orders. Right: corresponding single-shot experimental absorption images. The collision halos betweenthe two most populated orders are clearly visible. The color code of the absorption images indicates the diameter for thescattering halo, with the same code as Fig. 3 a . The color scale is set to enhance the collision halos (clipping high values onsome diffraction orders). The parameters used for data b-g are { s, t hold [ µ s] } = { . ± . , } , { . ± . , } , { . ± . , } , { . ± . , . } , { . ± . , . } and { . ± . , . } , respectively. U ( x, t ) = − s E L cos ( k L x + ϕ ) , (1)where E L = (cid:126) k L / (2 m ) ( E L = 4 E R , with E R the re-coil energy) and k L = 2 π/d are respectively the energyand the wavevector associated to the lattice. The di-mensionless depth of the lattice s is independently andprecisely calibrated [24] for each experiment presentedhere. The phase ϕ is set by the relative phase betweenthe two phase-locked acousto-optic modulators control-ling the lattice beams. In Rb, higher-order d − wavecollisions occur for energies of ∼ µ K [12, 13], while inthe experiments shown here we impart at most ∼ µ K of collisional energy. We are therefore in the purely s -wave scattering regime. Within this regime, in orderfor the changes in momentum distribution to directlyrelate to the number of atoms in the halos, the scat-tering cross section needs to be independent from therelative speed across all our experiments (see AppendixA), and not be affected by scattering suppression dueto the superfluid onset [25]. We have checked that theminimum relative velocity we can impart in a collisionis of h/ ( md ) = 8 . · s − , which is much larger thanthe peak sound velocity for the BEC in one of the collid-ing diffraction orders ( c ∼ . · s − ). Therefore thescattering cross-section is the same for all collision halospresented here.In order to tailor the momentum distribution after expansion, we use the following general procedure: wefirst load adiabatically the BEC for ϕ = 0 ; then asudden phase shift ϕ is applied [31], triggering out-of-equilibrium dynamics ( t = 0 + ); in the following holdingtime, the momentum distribution { π j ( t ) } evolves in thelattice; finally, the lattice is released at t = t hold . Theatom cloud then expands for a duration t T OF , and anabsorption image of the resulting density n ( r ) is taken tomeasure the momentum distribution and the scatteringhalos.As a guide to our engineering, we numerically computethe evolution of a one-body wavefunction, initially in theground state of an infinite optical lattice of depth s . Afterthe phase shift ϕ , the wavefunction is decomposed onthe lattice bands, and evolves during the holding time t hold . We then extract the final momentum distribution(see for example Fig 2 b-g , red bars and Appendix B). Results - In a first series of experiments, we demon-strate control of the c-o-m momentum of the scatteringhalo. For that purpose, we use the phase shift ϕ = 140 ◦ ,which puts the atom clouds in the lattice on the side slopeof each well (see Fig. 1). As the atoms reach the bottomof the well in the following dynamics, we expect a high c-o-m momentum, which is higher the deeper the lattice is,with a small dispersion of the distribution over diffractionorders.As the dominant collision halo will occur between thetwo mostly populated diffraction orders, we define a visi- bility parameter V as the difference between the fractions π j of the second and third most populated diffraction or-ders in the momentum distribution . It is a measure ofhow well the two diffraction orders contributing to themain halo "stand out", and varies between 0 (the thirdhighest order is as populated as the second) and 0.5 (thetwo orders contributing to the main halo are the onlyones populated). In Fig. 2 a , the values of the visibility V obtained from the wavefunction simulation are plottedover a large range of lattice depths s and holding times t hold . As we are here interested in controlling mainly thec-o-m momentum, the visibility V is only plotted if thetwo dominant orders are next to each other (separated by × h/d in momentum), and is otherwise set to 0. Finallywe also indicate the value of the c-o-m momentum bya color code. As intuitively expected, higher c-o-m mo-menta can be reached for deeper lattices, for a holdingtime that gets shorter the deeper the lattice gets.At a given depth, our analysis in terms of visibilityshows that "patches" of lower c-o-m momentum valuescan be obtained for longer holding times. However thevisibility in these patches is less pronounced, meaningthere are likely more than two significantly populatedorders of diffraction for these parameters, and we canexpect that several collisional processes will occur simul-taneously in the experiment. Therefore we have not rep-resented experimental data for these values.In Fig. 2 b-g , we represent a few snapshots of the den-sity distributions obtained after time-of-flight for param-eters that are indicated in Fig. 2 a . For each of thesemeasurements, the lattice depth was calibrated inde-pendently [24]. The disk-shaped halo due to collisionsbetween the two most populated orders, separated by × h/d , is clearly visible, and is the main feature besidesthe regular structure of the diffraction orders. Along-side the absorption pictures we represent the measuredhistograms { π j } of the populations in the diffraction or-der p j = j × h/d . These are compared to the calculatedhistograms from the wavefunction simulation, and thevisibility parameter from the simulation is indicated bya red-shaded area. Note that the sign of the c-o-m mo-mentum can be easily changed by changing the sign ofthe shift ϕ .Even though the fraction of atoms in the halo can getrather large (see below), the agreement between the sim-ulated distributions, which do not include interactions,and the measured distributions, for which collisions haveoccurred, is very good. This can seem surprising, asatoms are removed from the diffraction peaks during thecollisions. However in any collision between two ordersof diffraction, an equal number of atoms is removed fromthe involved peaks, after which the measured distribu-tion (as shown in blue in Fig. 2) is the new normalizeddistribution in the peaks. We have simulated that evenwith a fraction of colliding atoms as high as 30%, theexpected change between the non-interacting theoreticaldistribution and the distribution obtained after account-ing for the removal of the colliding atoms and normaliz- ing remains small (<5%). This means that the one-bodywavefunction simulation is a surprisingly robust guide inour investigation of collision halos, as is further evidencedbelow (see Fig. 4).In a second series of experiments, our aim was to con-trol specifically the relative momentum of the collisions.To that end, we apply a shift ϕ = 180 ◦ : the atom cloudsin the lattice are placed at the top of the lattice poten-tial, and split in two identical clouds that fall on eitherside (see Fig. 1). In a classical picture, after a holdingtime that allows the split clouds to reach the bottom ofthe wells, momentum orders of opposite sign should beequally populated, with zero c-o-m momentum. Theirtypical magnitude should also increase with lattice depth s . In Fig. 3 a , we plot the values of the visibility V , asdefined before, over a range of values of lattice depths s and holding times t hold , for ϕ = 180 ◦ , as calculatedfrom the wavefunction simulation. We do not impose acondition anymore on the separation between the mostpopulated orders (note that in this case, the theoreticaldistributions are symmetrical and the two highest peaksare equally populated). The diameter of the expected col-lision halo (with zero c-o-m momentum) is indicated by acolor code. As expected, the relative collisional momen-tum increases with lattice depth. In the main feature ofthis figure, the relative momentum of the most populatedorders increases in steps of × h/d as the lattice depth isincreased, for decreasing holding times (a behaviour thatis expected intuitively).In Fig. 3 b-g we represent snapshots of the densitydistributions obtained after time-of-flight for parametersthat are indicated in Fig. 3 a . Disk-shaped collision halosof increasing diameters, quantized in units of h/d , areclearly visible. To the left of the absorption images, werepresent the momentum distributions π j as extractedfrom absorption images. We compare the experimentallymeasured histograms with those from the wavefunctionsimulation for the same parameters s and t hold , and findonce again very good agreement. Since large collision ha-los become very dilute at the long time-of-flight neededfor the reconstruction of the histograms (a few tens of ms ), we used a different time-of-flight for the visualiza-tion of the halos (typically a few ms ).The observed fraction of colliding atoms we measure isin general larger than predicted by a perturbative theory(see Appendix A). To measure it, we count the atoms inthe collision halos when the diffraction orders are masked,as in Fig. 4 a . We account for the portion of disk hid-den, and calculate the ratio of the number of atoms inthe halo to the sum of the numbers in the halo and inthe diffraction orders. We measure typically a fractionof ± in the data presented Fig. 2 b and Fig. 3 b (from a statistical average of the fractions in 15 absorp-tion images), where the theory predicts 20%. This isin contrast to the results in [22] which were well de-scribed with a similar prediction; but while these resultsinvolved typically a few percent of the atoms of the BEC a R a d i a l d i s t r i b u t i o n ( a r b . un i t s ) b
10 20TOF (ms)100200 R a d i u s ( m ) b c E L )0123456 E x p a n s i o n v e l o c i t y ( v L ) c d Figure 4:
Quantization of the scattering halo radius. a
Typical absorption image from which the radius of the maincollision halo is measured. The red rectangle indicates the mask used to hide the diffraction orders in order to extract thehalo’s characteristics. b From the masked image a , an angular average is performed (blue markers) and a sigmoid fit (red line)to the resulting radial distribution allows us to extract the radius of the collisional halo (dashed red line). c The procedureshown in b is repeated for multiple TOF values (blue markers). A linear fit (solid red line) yields the velocity expansion of thescattering halo. d Measured expansion velocities as a function of the lattice depth s (blue markers). The parameters { s, t hold } corresponding to these data points are indicated by the disks in Fig. 3 a . The quantization of the expansion velocity of themain scattering halo in terms of v L = h/ ( md ) is apparent. The blue line indicates the expected expansion velocity of the halooriginating from the diffraction orders with the highest visibility at depth s (see text). The model and the observed steps arein very good agreement. in the collisions between two orders, we observe collisionsbetween highly populated orders, and get typically fivetimes larger atom fractions involved in the collisions, atwhich point a non-perturbative approach may be neededto get quantitative agreement. We also independentlychecked that this higher number is not due to collisionshappening inside the lattice during the non-equilibriumevolution time t hold (see Appendix C).We have also verified that the diameter of the dominantcollision halo in momentum space is indeed quantized interms of h/d (the momentum scale set by the lattice),and that larger diameters are reached for larger latticedepths. This analysis was performed independently fromthe known position of the diffraction orders, or any apriori visibility calculation. This is also performed for aphase shift of ϕ = 180 ◦ , and for the experimental param-eter values shown in Fig. 3 a . For each of the parametervalues { s , t hold } , we record a sequence of images with in-creasing TOF. On each of these images, as in Fig. 4 a , wemask the diffraction orders, to focus on the most visibledisk-shaped halo. By performing an angular average, weobtain a radial distribution, clearly showing the edge ofthe collision halo. We fit the position of this edge witha sigmoid function (Fig. 4 b ), which gives us the radiusof the scattering halo for a given TOF. We then extractthe speed of expansion of the scattering halo by fitting the linear growth of the radius with TOF (Fig. 4 c ).This procedure was repeated for multiple values of s ,choosing for each depth a holding time t hold yielding ascattering halo with the strongest signal. The results areshown in Fig. 4 d . We find indeed that the expansion ve-locity of the dominant halo, as measured from the haloonly, shows sharp jumps between discrete values that arean integer multiple of v L = h/ ( md ) , the velocity scaleset by the lattice. As the lattice depth increases, so doesthe collisional energy, which is converted from the latticepotential energy during t hold , and so does the expansionvelocity of the dominating halo. Alongside the experi-mental data in Fig. 4 d , we represent the expected colli-sion velocity for the dominant scattering halo, as givenby the point with highest visibility for a given depth s onthe map Fig. 3 a , obtained from the wavefunction simula-tion. We find that the experimentally observed steps forthe relative collision velocity in the dominant halos arein remarkably good agreement with the characterizationin terms of visibility of the diffraction orders. Discussion and conclusion - In this work, we havedemonstrated that we can engineer at will the collisionhalos from ultracold atoms released from an optical lat-tice. This is achieved through the control of the dynamicsof a BEC in the optical lattice, which allows us to tailorthe momentum distribution giving rise to the collisions t hold ( s)01020304050 s a = 100 o p / k L b -4 -3 -2 -1 0 1 2 3 4 p / k L Figure 5:
Collision halo of × h/d diameter and × h/d c-o-m momentum. a. Simulated visibility V of the orders ofdiffraction (see text), with the added constraint that the two most populated orders have momenta × h/d and × h/d , as afunction of the lattice depth s , and holding time t hold , for a shift of ϕ = 100 o . The black marker indicates the experimentallytested parameters, with the vertical error bar showing the standard deviation of the independent measurement of s . b Left:experimental diffraction orders probability distribution { π j } (blue, offset to the right), with black error bars showing onestandard deviation, and simulated one for the same parameters ϕ , s , t hold (red, offset to the left). The red shaded horizontalband represents the visibility for the theoretically simulated distribution, and stretches between the probabilities of the secondand third most populated diffraction orders. Right: corresponding single-shot experimental absorption image. The collisionhalo between the two most populated orders is clearly visible. The color scale is set to enhance the collision halo (clipping highvalues on some diffraction orders). The parameters used for data b are { s, t hold [ µs ] } = { . ± . , . } . after release from the trap. We have shown that we canselectively populate diffraction orders to impart a largemomentum either to the center-of-mass motion of collid-ing atom clouds, or to their relative motion. The searchfor appropriate experimental parameters has been guidedby the simulation of the dynamics of a non-interactingBEC in the lattice, and the identification of two highlypopulated diffraction orders with the desired characteris-tics. The energies involved in the collisions are quantizedin terms of the lattice characteristic momentum but canbe adjusted through a wide range of values.The protocol used here to demonstrate separate con-trol on center-of-mass and relative momentum can beextended to realize arbitrary combinations of the two. InFig. 5, we provide an example where we control simul-taneously the collision halo diameter ( × h/d ) and thecenter-of-mass momentum, ( × h/d ). This still relieson the simple control of the static lattice depth and asudden phase shift, and a visbility search for the desireddiffraction orders. This work could be extended furtherusing more elaborate dedicated schemes for the controlof the momentum distribution, by an appropriate time-dependent shaping of both the depth and phase of the lat-tice before time-of-flight expansion. This could provide afairly simple and general technique alongside more con-ventional atom-optics schemes [26]. Such an optimizationcould also be of interest for matter-wave interferometrywhere the engineering of momentum superpositions ofatomic ensembles is a key technique [27, 28]. Acknowledgments
This work was supported by Programme Investisse-ments d’Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT, and researchfunding Grant No. ANR-17-CE30-0024. M.A. acknowl-edges support from the DGA (Direction Générale del’Armement), and N.D. support from Région Occitanieand Université Paul Sabatier.
Appendix A: Perturbative theory of scattering halos
In this section, we provide a quantitative theoreticaldescription of the atom-atom scattering processes givingrise to the halo structures that are observed in the ex-periments. The perturbative framework that we developis inspired from previous works that were investigatingcolliding Bose-Einstein condensates [20, 29]. The resultsthat it yields are consistent with the calculations under-taken in Ref. [22].Our starting point is a perfect Bose-Einstein conden-sate that is prepared within a one-dimensional opticallattice. The atoms of the condensate are sharing thesingle-particle wavefunction Φ ( r ) = φ ( r ) ∞ (cid:88) l = −∞ ψ l e ilk L z (A1)with r ≡ ( x, y, z ) and the lattice being oriented alongthe z axis of the coordinate system. ψ l are the Fouriercomponents of the periodic condensate wavefunction inthe lattice. The function φ ( r ) = 1 √ R ϕ ( r /R ) (A2)with ϕ ( ρ ) = (cid:118)(cid:117)(cid:117)(cid:116) π (cid:32) − ω ⊥ ¯ ω ( ρ x + ρ y ) − ω || ¯ ω ρ z (cid:33) (A3)is a Thomas-Fermi envelope that accounts for the pres-ence of a weak overall elliptic confinement with the lon-gitudinal and transverse frequencies ω || and ω ⊥ , respec-tively, and with ¯ ω = ( ω || ω ⊥ ) / . The associated Thomas-Fermi radius R is straightforwardly evaluated as R = (15 N a s ¯ a ) / (A4)with ¯ a = [ (cid:126) / ( m ¯ ω )] / , where N is the number of atomsin the condensate and a s denotes the s -wave scatteringlength.At time t = 0 the lattice and confinement potentialsare switched off and the atomic cloud is thereby allowedto freely expand. If atom-atom interactions could becompletely neglected from that instant on, the momen-tum distribution of the atoms would be simply given bythe Fourier transform of the condensate wavefunction,i.e., we would have n ( p , t ) = N | ˜Φ ( p ) | with ˜Φ ( p ) = ∞ (cid:88) l = −∞ ψ l ˜ φ ( p − l (cid:126) k L e z ) (A5)where ˜ φ ( p ) = 1 √ π (cid:126) (cid:90) d rφ ( r ) e − i p · r / (cid:126) (A6)is the Fourier transform of the Thomas Fermi profile(A2). As we generally have k L R (cid:29) in the experiment,Eq. (A5) suggests the appearance of tighly localized mo-mentum density peaks centered about integer multiplesof the lattice momentum l (cid:126) k L , each one of those popu-lated with N | ψ l | atoms.The key approximation that we undertake here is toassume that we can account for the presence of atom-atom interaction during the free expansion process of thecondensate in a perturbative manner, using first orderquantum perturbation theory. Employing the interac-tion representation, the time evolution of the many-bodystate | Ψ t (cid:105) describing the atomic cloud is approximatelywritten as | Ψ t (cid:105) (cid:39) | Ψ (cid:105) − i (cid:126) (cid:90) t dt (cid:48) ˆ U ( t (cid:48) ) | Ψ (cid:105) + O ( a s ) . (A7)in linear order in the s -wave scattering length a s . Here, ˆ U ( t ) = g π (cid:126) ) (cid:90) d p (cid:90) d p (cid:90) d p (cid:48) (cid:90) d p (cid:48) × δ ( p + p (cid:48) − p − p (cid:48) ) × ˆ ψ † t ( p ) ˆ ψ † t ( p (cid:48) ) ˆ ψ t ( p (cid:48) ) ˆ ψ t ( p ) (A8) with g = 4 π (cid:126) a s /m is the two-body interaction Hamil-tonian expressed in terms of the atomic creation and an-nihilation operators in momentum space which evolveaccording to ˆ ψ † t ( p ) = ˆ ψ † ( p ) exp[ itp / (2 m (cid:126) )] as well as ˆ ψ t ( p ) = ˆ ψ ( p ) exp[ − itp / (2 m (cid:126) )] , respectively, and whichfulfill the bosonic commutation relation (cid:104) ˆ ψ t ( p ) , ˆ ψ † t ( p (cid:48) ) (cid:105) = (cid:104) ˆ ψ ( p ) , ˆ ψ † ( p (cid:48) ) (cid:105) = δ ( p − p (cid:48) ) . (A9)We can therefore rewrite Eq. (A8) as ˆ U ( t ) = a s π m (cid:126) (cid:90) d p (cid:90) d p (cid:90) d p (cid:48) (cid:90) d p (cid:48) × δ ( p + p (cid:48) − p − p (cid:48) ) × exp (cid:20) itm (cid:126) ( p − p ) · ( p (cid:48) − p ) (cid:21) × ˆ ψ † ( p ) ˆ ψ † ( p (cid:48) ) ˆ ψ ( p (cid:48) ) ˆ ψ ( p ) (A10)and use the standard definition of the momentum-spacefield operators giving rise to ˆ ψ ( p ) | Ψ (cid:105) = ˜Φ ( p )ˆ b | Ψ (cid:105) (A11)where ˆ b is the annihilation operator associated with thecondensate orbital.In the following, we focus on regions in momentumspace located far away from the lattice density peaks | ˜ φ ( p − l (cid:126) k L e z ) | which we obtained in zeroth order in theinteraction strength. For a momentum p being in sucha scarcely populated region we can safely set ˜Φ ( p ) = 0 and hence ˆ ψ ( p ) | Ψ (cid:105) = 0 . The atom density detected atsuch a momentum is therefore determined as n ( p , t ) = (cid:104) Ψ t | ˆ ψ † ( p ) ˆ ψ ( p ) | Ψ t (cid:105) (cid:39) (cid:104) Π t ( p ) | Π t ( p ) (cid:105) (A12)with | Π t ( p ) (cid:105) = − i (cid:126) (cid:90) t dt (cid:48) (cid:104) ˆ ψ ( p ) , ˆ U ( t (cid:48) ) (cid:105) | Ψ (cid:105) (A13)in lowest nonvanishing order in the s -wave scatteringlength. Evaluating (cid:104) ˆ ψ ( p ) , ˆ U ( t (cid:48) ) (cid:105) = a s π m (cid:126) (cid:90) d p (cid:90) d p (cid:48) × exp (cid:20) it (cid:48) m (cid:126) ( p − p ) · ( p (cid:48) − p ) (cid:21) × ˆ ψ † ( p + p (cid:48) − p ) ˆ ψ ( p (cid:48) ) ˆ ψ ( p ) (A14)and using Eq. (A11) in combination with Eq. (A5) , weobtain the expression | Π t ( p ) (cid:105) = − ia s π m (cid:126) ∞ (cid:88) l,l (cid:48) = −∞ ψ l ψ l (cid:48) (cid:90) d p (cid:90) d p (cid:48) × ˜ φ ( p ) ˜ φ ( p (cid:48) ) (cid:90) t dt (cid:48) e iω l,l (cid:48) ( p − p , p − p (cid:48) ) t (cid:48) (A15) × ˆ ψ † [( l + l (cid:48) ) (cid:126) k L e z + p + p (cid:48) − p ]ˆ b | Ψ (cid:105) where we define ω l,l (cid:48) ( q , q (cid:48) ) = 1 m (cid:126) ( l (cid:126) k L e z − q ) · ( l (cid:48) (cid:126) k L e z − q (cid:48) ) . (A16)The time integral appearing in Eq. (A15) is straightfor-wardly evaluated yielding (cid:90) t dt (cid:48) e iω l,l (cid:48) ( q , q (cid:48) ) t (cid:48) = t sinc[ ω l,l (cid:48) ( q , q (cid:48) ) t/ e iω l,l (cid:48) ( q , q (cid:48) ) t/ (A17)with sinc( ξ ) ≡ sin( ξ ) /ξ . It has a similar effect as Dirac’sdelta distribution for large t → ∞ insofar as it wouldyield vanishing contributions within Eq. (A15) for valuesof ω l,l (cid:48) ( q , q (cid:48) ) that are different from zero. Hence, giventhe fact that the prefactor ˜ φ ( p ) ˜ φ ( p (cid:48) ) restrains the ef-fective integration domain of p and p (cid:48) to a very narrowregion about the origin, we can infer that for a sufficientlylong evolution time t a nonvanishing momentum densitycan be encountered only in the immediate vicinity of thesurface of spheres in momentum space that have theirnorth and south poles at the points l (cid:126) k L e z and l (cid:48) (cid:126) k L e z and are therefore characterized by the equation ( p − l (cid:126) k L e z ) · ( p − l (cid:48) (cid:126) k L e z ) = 0 (A18)for any pair of integers l, l (cid:48) ∈ Z .To quantitatively calculate the momentum density ac-cording to Eq. (A12), we can set ˆ ψ [( l + l (cid:48) ) (cid:126) k L e z + p + p (cid:48) − p ] | Ψ (cid:105) = 0 (A19)by the choice that we made for p , since we effectivelyhave | p + p (cid:48) | (cid:28) (cid:126) k L according to the above reasoning.Similarly, we can evaluate (cid:104) ˆ ψ [( l + l (cid:48) ) (cid:126) k L e z + p + p (cid:48) − p ] , ˆ ψ † [( l + l (cid:48) ) (cid:126) k L e z + p + p (cid:48) − p ] (cid:105) = δ l + l (cid:48) ,l + l (cid:48) δ ( p + p (cid:48) − p − p (cid:48) ) (A20)when this commutator is multiplied with the prefactor ˜ φ ∗ ( p ) ˜ φ ∗ ( p (cid:48) ) ˜ φ ( p ) ˜ φ ( p (cid:48) ) . The two spheres on which p has to be simultaneously located in order to yield a signif-icant momentum density would therefore have the samecenter at ( l + l (cid:48) ) (cid:126) k e z = ( l + l (cid:48) ) (cid:126) k e z , which impliesthat they are actually identical and we have either l = l and l (cid:48) = l (cid:48) or l = l (cid:48) and l (cid:48) = l .Using (cid:104) Ψ | ˆ b † ˆ b † ˆ b ˆ b | Ψ (cid:105) = N ( N − (cid:39) N for N (cid:29) and δ ( p + p (cid:48) − p − p (cid:48) ) = (cid:90) d r (2 π (cid:126) ) e i ( p + p (cid:48) − p − p (cid:48) ) · r / (cid:126) (A21)we can, consequently, write the momentum density (A12)as n ( p , t ) (cid:39) ∞ (cid:88) l,l (cid:48) = −∞ | ψ l | | ψ l (cid:48) | × n (cid:18) p − l + l (cid:48) (cid:126) k e z , l − l (cid:48) (cid:126) k e z , t (cid:19) (A22) with n ( p , p , t ) = (cid:90) d r | χ ( r , p , p , t ) | (A23)where χ ( r , p , p , t ) = N a s √ π (cid:126) m (cid:90) d p (cid:90) d p (cid:48) × (cid:90) t dt (cid:48) e it (cid:48) ( p + p − p ) · ( p (cid:48) − p − p ) / ( m (cid:126) ) × ˜ φ ( p ) ˜ φ ( p (cid:48) ) e i ( p + p (cid:48) ) · r / (cid:126) (A24)represents some sort of phase-space wavefunction thatdescribes the collision process between two momentumcomponents l and l (cid:48) of the condensate wavefunction, with p = ( l − l (cid:48) ) (cid:126) k L e z and with the origin in momen-tum space being set at ( l + l (cid:48) ) (cid:126) k L e z . In the contextof scattering halos, we are specifically interested in thecase l (cid:54) = l (cid:48) , which implies a finite relative momentum p ∼ (cid:126) k L of the two wavefunction components. We cantherefore neglect the p · p (cid:48) term arising in the exponentwithin Eq. (A24) as it will become negligibly small withrespect to the other terms in this exponent, owing to thepresence of the prefactor ˜ φ ( p ) ˜ φ ( p (cid:48) ) . This consequentlyyields χ ( r , p , p , t ) (cid:39) N a s √ π (cid:126) m (cid:90) t dt (cid:48) e it (cid:48) ( p − p ) · ( p + p ) / ( m (cid:126) ) × φ [ r − t (cid:48) ( p + p ) /m ] × φ [ r − t (cid:48) ( p − p ) /m ] . (A25)Making use of the fact that the Thomas-Fermi profile(A3) characterizing the condensate wavefunction is of fi-nite extent, we can safely take the limit t → ∞ in theabove expression and obtain n ( p , p ) (cid:39) N a s π (cid:126) p R (cid:90) ∞ dτ (cid:90) ∞ dτ (cid:48) (A26) × exp (cid:20) iR (cid:126) p ( p − p )( τ − τ (cid:48) ) (cid:21) (cid:90) d ρ × ϕ [ ρ − τ ( p /p + e z )] ϕ [ ρ − τ ( p /p − e z )] × ϕ [ ρ − τ (cid:48) ( p /p + e z )] ϕ [ ρ − τ (cid:48) ( p /p − e z )] for the momentum density of scattered atoms in the long-time limit, where we use p = ± p e z with p = | p | . Thisexpression can be further simplified by using the fact thatfor p (cid:54) = p the term exp[ iR ( p − p )( τ − τ (cid:48) ) / ( (cid:126) p )] has, inthe limit Rp / (cid:126) → ∞ , a similar effect as a delta functionin τ − τ (cid:48) within Eq. (A26). We are therefore entitled torewrite this expression as n ( p , p ) (cid:39) N a s πp R (cid:90) ∞ dτ sin[ τ R ( p − p ) / ( (cid:126) p )] p − p × (cid:90) d ρϕ ( ρ − τ e z ) ϕ ( ρ + τ e z ) , (A27)which for Rp / (cid:126) → ∞ yields a tight and isotropic con-centration of the momentum density around the sphere Figure 6: Momentum density profiles (plotted in arbitrarybut for all panels fixed units) as a function of p = | p | for p = (cid:126) k L / (corresponding to | l − l (cid:48) | = 1 ) with k L = 2 π/d and d = 532 nm . The profiles are obtained from a numericalevaluation of Eq. (A27) for the case of Rb, where we assumethe presence of a spherical trap ( ω ⊥ = ω || = ¯ ω ) with theconfinement frequencies (a,c) ¯ ω = 2 π ×
30 Hz and (b,d) ¯ ω =2 π ×
100 Hz , and where we consider a condensate populationcontaining (a,b) N = 10 and (c,d) N = 2 × atoms. Theresulting Thomas-Fermi radii are evaluated as (a) R (cid:39) µ m ,(b) R (cid:39) . µ m , (c) R (cid:39) µ m , (d) R (cid:39) . µ m . The fullwidth at half maximum (FWHM) of the momentum densitypeaks equals roughly π (cid:126) /R . with the radius p = p . As is clearly seen from Eq. (A27),the width of these scattering halos in momentum spaceis of the order of ∆ p ∼ (cid:126) /R , which is in agreement withthe experimental findings.This is confirmed in Figure 6 which shows a numericalevaluation of Eq. (A27) for the experimental parametersunder consideration and for various choices of the conden-sate population and the overall confinement frequency(assuming a spherical trap). The atomic density in mo-mentum space is concentrated about a sphere of radius p = | l − l (cid:48) | (cid:126) k L / (cid:126) k L / in this example ( | l − l (cid:48) | = 1 ).The width (FWHM) of this scattering halo is found toroughly equal π (cid:126) /R in all of the four studies cases. Notethat these density profiles cannot be directly comparedwith the profile shown in Fig. 4 b since the latter is theresult of the integration of the scattering sphere alongthe imaging axis, followed by angular averaging.The total number of atoms that are scattered as a con-sequence of the collision between the l and l (cid:48) componentsof the condensate wavefunction (with l (cid:54) = l (cid:48) ) is thenstraightforwardly calculated as N coll . at . = 2 | ψ l | | ψ l (cid:48) | N with N = (cid:90) d pn ( p , p ) (cid:39) N a s R (cid:90) ∞ dτ (cid:90) d ρϕ ( ρ − τ e z ) ϕ ( ρ + τ e z )= (cid:34) N a s R (cid:18) ω ⊥ ω || (cid:19) / (cid:35) (A28) = σ πR (cid:34) N (cid:18) ω ⊥ ω || (cid:19) / (cid:35) . (A29) In this last expression we have isolated the scatteringcross section σ , which is here independent of the collid-ing orders l and l (cid:48) . Eq. A28 is quantitatively rather sim-ilar to the analogous Eq. (2) for the number of collisionswithin Ref. [22], which was obtained there in the frame-work of a three-dimensional optical lattice. The scal-ing with ( ω ⊥ /ω || ) / translates the fact that two cigar-shaped condensates moving across each other along theirsymmetry axis give rise to more atom-atom collisionsthan two pancake-shaped condensates. For a mean an-gular frequency ¯ ω = 2 π × Hz, a number of atomsequal to × , and an ideal, equal-weights collision, | ψ l | = | ψ l (cid:48) | = 0 . , we find a fraction N coll . at . /N (cid:39) %of atoms that collide. This is somewhat smaller thanwhat is observed experimentally, which may indicate thata non-perturbative approach is required for a quantita-tive agreement. Appendix B: Wavefunction calculation of thedistribution of orders of diffraction
The eigenstates of the lattice hamiltonian with the po-tential Eq. 1 are Bloch functions of the form : | Φ n,q (cid:105) = (cid:88) l ∈ Z C ( n,q ) l | χ q + lk L (cid:105) , where the vectors | χ k (cid:105) are eigenstates of the momentumoperator with eigenvalue (cid:126) k (and χ k ( x ) = e ikx √ π ), thequasi-momentum q ∈ [ − π/d, π/d [ , and the coefficients C ( n,q ) l are the solution of the central equation (for a givenquasi-momentum q , and ϕ = 0 ) [30]: (cid:18) l + qk L (cid:19) C ( n,q ) l − s (cid:16) C ( n,q ) l +1 + C ( n,q ) l − (cid:17) = E n,q E L C ( n,q ) l . (B1)This equation can easily be solved in matrix form to ob-tain the eigenenergies of the system and the decomposi-tion of the eigenstates on the plane wave basis.The BEC is initially loaded in the lowest energy eigen-state of the lattice, corresponding to a quasi-momentum q = 0 : | Ψ( t = 0 (cid:105) ) = | Φ , (cid:105) . After the sudden phaseshift, the wavefunction in the reference frame of the lat-tice takes the form : | Ψ( t = 0 + ) (cid:105) = (cid:88) l ∈ Z C (0 , l e − ilϕ | χ lk L (cid:105) . Since the lattice preserves its periodicity in the shift,the quasi-momentum is conserved. The wavefunction af-ter the shift can therefore be decomposed onto the eigen-states of the shifted lattice with quasi-momentum q = 0 to obtain the evolution during the holding time t hold : | Ψ( t hold ) (cid:105) = (cid:88) n ∈ N α n e − i En, t hold (cid:126) | Φ n, (cid:105) , α n = (cid:104) Φ n, | Ψ( t = 0 + ) (cid:105) = (cid:80) l ∈ Z C ∗ ( n, l C (0 , l e − ilϕ .Finally the population in the diffraction order of mo-mentum j × h/d is obtained by calculating the projectionof the state | Ψ( t hold ) (cid:105) on the corresponding eigenstate ofthe momentum operator | χ jk L (cid:105) : π j = |(cid:104) χ jk L | Ψ( t hold ) (cid:105)| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ N α n C ( n, j e − i En, t hold (cid:126) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Appendix C: Impact of in-lattice dynamics on thecollisional halos
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