Optical control of the density and spin spatial profiles of a planar Bose gas
Y.-Q. Zou, ?. Le Cerf, B. Bakkali-Hassani, C. Maury, G. Chauveau, P.C.M. Castilho, R. Saint-Jalm, S. Nascimbene, J. Dalibard, J. Beugnon
OOptical control of the density and spin spatialprofiles of a planar Bose gas
Y.-Q. Zou ∗ , ´E. Le Cerf ∗ , B. Bakkali-Hassani , C. Maury ,G. Chauveau , P.C.M. Castilho , R. Saint-Jalm , S.Nascimbene , J. Dalibard , J. Beugnon Laboratoire Kastler Brossel, Coll`ege de France, CNRS, ENS-PSL University,Sorbonne Universit´e, 11 Place Marcelin Berthelot, 75005 Paris, France Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, CP 369,13560-970 S˜ao Carlos, Brazil Department of Physics, Ludwig-Maximilians-Universit¨at M¨unchen,Schellingstr. 4, D-80799 M¨unchen, GermanyE-mail: [email protected]
Abstract.
We demonstrate the arbitrary control of the density profile of a two-dimensional Bose gas by shaping the optical potential applied to the atoms. Weuse a digital micromirror device (DMD) directly imaged onto the atomic cloudthrough a high resolution imaging system. Our approach relies on averaging theresponse of many pixels of the DMD over the diffraction spot of the imagingsystem, which allows us to create an optical potential with arbitrary grey levelsand with micron-scale resolution. The obtained density distribution is optimizedwith a feedback loop based on the measured absorption images of the cloud.Using the same device, we also engineer arbitrary spin distributions thanks to atwo-photon Raman transfer between internal ground states. a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b ptical control of the density and spin spatial profiles of a planar Bose gas
1. Introduction
Ultracold quantum gases are ideal platforms tostudy physical phenomena, thanks to their highflexibility and their isolation from the environment.They are widely used for quantum simulations [1]and metrological applications [2]. Various trapgeometries have been realized to confine atomic clouds.Historically, harmonic confinements have been thenorm in cold atom experiments due to their easeof implementation [3, 4]. The recent realization ofuniform systems opened new perspectives to explorethe thermodynamic properties and dynamical behaviorof quantum gases [5, 6, 7, 8]. Other trap potentials havebeen applied to explore physics in specific geometries,such as supercurrents in ring potentials [9, 10, 11, 12],analog sonic black holes in more complex potentials[13], and low-entropy phases in lattice systems [14].In the past years, several approaches havebeen developed to generate complex optical potentialprofiles [15, 16, 17, 18, 19, 20, 21, 22]. Most of them relyon the development of spatial light modulators (SLMs),which can modulate the phase or the intensity of a lightbeam. Digital micromirror devices (DMDs) are one ofthe most widely used in cold atom experiments thanksto their low cost, simple use and high refresh rates.They consist of millions of individual micromirrorswhich can be set in two different orientations, hencecorresponding to a “black” or “white” signal in achosen image plane of the DMD chip. They havebeen used to correct optical aberrations when workingas a programmable amplitude hologram in a Fourierplane [23], and to produce different potential profilesby direct imaging [21, 24, 25, 26].In this article, we demonstrate arbitrary controlof the density profile of two-dimensional (2D) Bosegases by tailoring the in-plane trapping potential usingDMDs. We program a pattern on the DMD chipand simply image it onto the atomic cloud. Thelimitation due to the binary status of the DMD pixels(black or white) is overcome by realizing a spatialaverage of the response of ∼
25 pixels over the pointspread function of the imaging system. This givesus access to several levels of grey for the opticalpotential at a given position in the atomic plane. TheDMD pattern is computed thanks to an error diffusionalgorithm combined with a feedback loop to directlyoptimize the measured atomic density distribution.The method is proved to be efficient and robust tooptical imperfections. In addition, we demonstrate therealization of arbitrary spin distributions with the sameprotocol by using spatially resolved two-photon Ramantransitions. DMD1 DMD2PBSObjective 1AtomsObjective 2 Camera
Figure 1.
Sketch of the experimental setup for arbitrary densitycontrol. Two DMDs are used to project an optical potential ontothe atoms with a high NA microscope objective (Objective 1).Both of them are illuminated by a blue-detuned 532 nm laser.DMD1 provides the hard-wall potential, while DMD2 adds anadditional potential for density control. The light fields fromthe two DMDs are mixed on a polarizing beam splitter (PBS)with orthogonal polarizations so that they do not interfere witheach other. The atoms are imaged onto the camera with a secondidentical objective (Objective 2). We use absorption imaging tomeasure the 2D density profiles on a CCD camera.
2. Apparatus and main results
We work with a degenerate 2D Bose gas of Rb atoms.The main experimental setup has been describedpreviously in [27, 28]. Briefly, about 10 Rb atomsin the F = 1 , m = 0 hyperfine ground state are loadedinto a 2D box potential. The vertical confinement isprovided by a vertical lattice. All atoms are trappedaround a single node of the lattice in an approximatelyharmonic potential with a measured trap frequency ω z / π = 4 . ‡ . All laser beams used for creatingthe 2D box potential have a wavelength of 532 nmand thus repel Rb atoms from high intensity regions.The cloud temperature is controlled by lowering thein-plane potential height, thus enabling evaporativecooling. We reach temperatures below 30 nK and anaverage 2D atom density of ∼ µ m − , correspondingto a regime where the cloud is well described by theThomas-Fermi approximation. Both the interactionenergy and thermal energy are smaller than the verticaltrapping frequency and the atom cloud is thus in theso-called quasi-2D regime.We show in figure 1 a sketch of the experimental ‡ All DMDs used in this work are DLP7000 from TexasInstruments and interfaced by Vialux GmbH. ptical control of the density and spin spatial profiles of a planar Bose gas (a) (b) (c) (d) µ m 0 . . . . . . . . . . . (e) (f ) (g) (h) . x, y (pixel) O D xy . x, y (pixel) O D xy . r (pixel) O D . θ ( ◦ ) O D Figure 2.
Various density profiles realized in our experiment. From left to right, we show a uniform profile and linearly varyingdensity profiles along x , along the radial direction and along the azimuthal direction. (a)-(d) Averaged absorption images (50, 99,50, 20 shots respectively). (e)-(h) Corresponding OD profiles integrated over one direction ( x and y in (e)-(f), azimuthal in (g) andradial in (h)). The solid lines represent the OD profiles of the target density distributions. setup for arbitrary density control. We modify thedensity distribution by using another DMD (DMD2)to impose an additional repulsive optical potential tothe hard-wall potential made by DMD1. The patternon DMD2 is imaged onto the atomic plane thanks toan imaging system of magnification ≈ /
70. The pixelsize of DMD2 is 13 . µ m, leading to an effective sizeof 0.2 µ m in the atomic plane. The numerical aperture(NA ∼ µ m. Consequently,the area defined by the diffraction spot of the imagingsystem typically corresponds to a region where 5 × w ∼ µ m in the atomic plane. The intensityof the beam is set to provide a maximum repulsivepotential around 2 µ where µ is the chemical potentialof the gas for a density of 80 µ m − . The potential isadded before the final evaporation stage in the boxpotential.The 2D atomic density profile is obtained byabsorption imaging with a second identical microscopeobjective placed below the glass cell. This imagingsystem has a similar optical resolution and the effectivepixel size of the camera in the atomic plane is 1.15 µ m.We probe the atoms in the trap using a 10 µ s pulseof light on the D line resonant between the F = 2ground state and the F (cid:48) = 3 excited state. Before detection, a microwave pulse is applied to transfer acontrolled fraction of atoms into the ground level from F = 1 , m = 0 to F = 2 , m = 0, which thus absorbslight from the imaging beam. The transferred fractionis controlled so that the measured optical depth (OD)is always smaller than 1.5 to reduce nonlinear imagingeffects.Figure 2 presents a selection of 2D density profilesrealized in our experiment. For each example, weshow in figure 2(a-d) averaged absorption images andin figure 2(e-h) the corresponding mean OD integratedalong one or two spatial directions. Figure 2(a) showsa uniform profile in which we have corrected theinhomogeneities caused by residual defects of theoverall box potential created by the combination ofDMD1 and vertical lattice beams. Figures 2(b-d)correspond to linearly varying density distributionsrespectively along the x direction, along the radialdirection and along the azimuthal direction.
3. Detailed implementation
One could naively think that for a given target densityprofile, the suitable pattern on DMD2 could be directlycomputed and imaged onto the atoms. However,several features prevent such a simple protocol. First,the DMD is a binary modulator. Then, for a finitenumber of pixels, it is not possible to create anarbitrary grey-level pattern with perfect accuracy.Here, we use the well-known error diffusion technique ptical control of the density and spin spatial profiles of a planar Bose gas (a) Start Grey-levelprofileG n Binary DMDpatternImage theatomsA n Target adaptionT n Has A n converged?Stop Yes No (b) (c) . . . . . . Figure 3. (a) Diagram of the iterative algorithm. (b) Exampleof grey-level profile G n obtained during the optimization loopused to create the linearly varying profile shown in figure 2(b).(c) Corresponding dithered image computed with the errordiffusion algorithm and programmed on the DMD. The grey levelranges from 0 to 1, with an effective pixel size of 1.15 µ m equalto the one of the absorption image. The DMD pattern is binarywith an effective pixel size of 0.2 µ m. to generate the binary pattern for a given grey-levelprofile [29, 30]. Second, the imaging system fromDMD2 to the atoms has an optical response that leadsto a modification of the ideal image, mainly becauseof the finite aperture of the optical elements. Third,any imperfection on the optical setup (inhomogeneityof the laser beam, optical aberrations...) also degradesthe imaging of the DMD pattern onto the atomic cloud.Finally, the atomic density distribution is obtainedthrough absorption imaging, which adds noise mostlycoming from the photonic shot noise induced by theimaging beam. Hence, an iterative method is needed toobtain the optimal DMD pattern that gives a densitydistribution as close as possible to the target. Theworking principle of the optimization loop is simplyto add (remove) light at the positions where there aremore (fewer) atoms than the target until the densityprofile converges to the target one.Figure 3(a) shows the steps of the iterative loop.The basic idea of each step n consists in computing thedifference between the measured density distributionA n and the target image T n , and adding it with asuitable gain K to the previous grey-level intensityprofile G n . This gives the grey-level profile of iteration n + 1 (see figure 3(b)),G n +1 = G n + K (A n − T n ) , (1)which is then discretized thanks to the error diffusionalgorithm (see figure 3(c)) and imaged onto the atoms. Besides this general idea, we detail below some specificfeatures of our loop:- We initialize the optimization with a grey-levelprofile G which can either be uniformly 0 or 1.- To avoid border effects, we select on theabsorption images a region slightly inside the boxpotential (two pixels smaller in each direction) fordensity control and we extrapolate the grey-levelprofile G n outside the box. The extrapolationis done by simply duplicating the value of theoutermost pixels of G n by three more pixels alongeach side for a square box or along the radialdirection for a disk.- The image A n of the density distribution isobtained from the average of several repetitionsof the experiment with the same parameters tolimit the contribution of detection noise.- The measured image of the atomic distribution isconvoluted with a Gaussian function of rms width1 pixel of the camera of the imaging system. Thisconvolution removes some high frequency noise inthe absorption image, such as detection noise, thatour protocol cannot compensate.- Considering the Gaussian shape of the beamilluminated on DMD, we choose K to be positiondependent K ( x, y ) = K × e x − x y − y w ,where w is the waist of the beam in the atomicplane and x and y are the coordinates of thecenter of the beam. It makes the effective gainapproximately the same for all the pixels.- At each iteration, we rescale the amplitude of thetarget profile to obtain the same mean opticaldepth as the one of A n . This avoids takinginto account errors coming from the shot-to-shotvariation of the atom number which would lead toa global error that we are not interested in. Notethat this variation is smaller than 10 % during theoptimization loop.
4. Characterization of the loop
We stop the optimization loop when the measureddensity distribution has converged to the target one,up to a predefined precision. To estimate thedeviation from the target, we define a figure of merit F m corresponding to the measured root-mean-squaredeviation: F m = (cid:115) (cid:80) ( i,j ) ∈ A (OD( i, j ) − OD T ( i, j )) N pix (cid:80) ( i,j ) ∈ A OD( i, j ) , (2)where A is the region of interest containing N pix pixelsand OD( i, j ) (resp. OD T ( i, j )) is the measured averageOD (resp. target OD). The value of the figure of merit F m results from two kinds of contributions. Obviously, ptical control of the density and spin spatial profiles of a planar Bose gas (a) . . F i g u r e o f m e r i t F m N d F (b)
50 10000 . . N a F i g u r e o f m e r i t F m N d F (c)
10 2000 . . F K = 0 . K = 0 . K = 0 . Figure 4.
Convergence of the iterative algorithm. (a) Plot of F m , N d and F with iteration number. Target profile is a lineardensity distribution along x in a square box (of figure 2(b)). F converges very fast and stays around 0.06 after iteration 6. N d decreases suddenly at iteration 8 and 15 because N a (numberof absorption images for averaging) changes from 5 to 10 atiteration 8 and to 99 at iteration 15. (b) For the last iteration(iteration 15), we plot F m , N d and F versus the number ofimages N a used for averaging. Both F m and N d decrease with N a while N d does not depend on N a . (c) Evolution of F fordifferent K (cid:48) s. there is the actual deviation of the density distributionfrom the target. In addition, several features of themeasurement method give an undesired contributionto F m . Indeed, thermal fluctuations of the atomiccloud, projection noise due the partial transfer imagingdiscussed above and photonic shot noise in absorptionimaging lead to unavoidable residual noise. Forour parameters, the two dominant mechanisms arephotonic and projection noise with a similar weight,whose exact values depend on the studied densitydistribution.The contributions coming from photonic shot noise and projection noise can be reduced by averagingmore images. However, for the typical repetition rateof our experiment ( ∼
30 s), the number of averagedimages has to be limited to a few tens for realisticapplications. To characterize the optimization loop, wecompute this noise contribution N d so as to remove itfrom the measured F m . We directly estimate N d fromthe set of images taken with the same parameters bycomputing the dispersion of the measured absorptionimages from the averaged image, N d = (cid:118)(cid:117)(cid:117)(cid:116) (cid:80) k (cid:80) ( i,j ) ∈ A (OD k ( i, j ) − OD( i, j )) N pix N a (cid:80) ( i,j ) ∈ A OD( i, j ) , (3)where the index k refers to the k -th absorption imageamong the N a pictures taken for the average. We thusdefine the corrected figure of merit: F = (cid:113) F m − N d , (4)which quantifies the distance of the density profile fromthe target while removing measurement noise.In figure 4(a), we show the evolution of F m , N d and F as a function of the number of iterations inthe example case of a linear profile in a square box(as shown in figure 2(b)). We initialize the loop with agrey-level profile equal to zero and we choose K = 0 . N d with the iteration number. Interestingly, we see that F converges almost monotonously to about 0.06 after thefirst 6 iterations and then stays approximately constantwhatever the value of N a is. This indicates that thecontribution of measurement noise is well subtracted.This is confirmed in figure 4(b), where we plot F m , N d and F as a function of N a using the data of the finaliteration of figure 4(a). As expected, both F m and N d decrease with N a while F does not change.We also studied the behavior of the iterative loopwith different K (cid:48) s varying from 0.1 to 0.6. Theconvergence of F is plotted in figure 4(c). The iterativealgorithm works well for a large range of values of K . We observe that increasing K speeds up theconvergence, but too large values of K lead to stronglocal variations in the measured images. In practice,for most target distributions, we use K = 0 . F . Themain limitation comes from the number of iterationsused in the experiment ( ∼ F decreases slowly down to ∼ . ptical control of the density and spin spatial profiles of a planar Bose gas (a) (b) . . r ( µ m)OD 0 10 200 . . r ( µ m) . . . Figure 5.
Imprinting a spatial spin texture. We show thedensity distribution of atoms in | (cid:105) immersed in a bath ofatoms in | (cid:105) . The total density of the gas is uniform in a20 µ m disk ( ∼ µ m − , corresponding to OD ∼ | (cid:105) in semilog scalefor (a) a Gaussian profile and (b) a solitary Townes profile.The solid lines are the target radial profiles. Insets show thecorresponding averaged absorption images (20 shots). Thedashed lines represent the edges of the bath of atoms in | (cid:105) .
5. Arbitrary spin distribution
Using a similar protocol, we also demonstrate arbitraryspin distributions by shaping a pair of copropagatingRaman beams which couple the | F = 1 , m = 0 (cid:105) ( | (cid:105) ) and | F = 2 , m = 0 (cid:105) ( | (cid:105) ) states by a two-photon Raman transition. The two Raman beamsoriginate from the same laser and have a wavelengthof ∼
790 nm, in between the D and D line of Rbatoms. One beam is frequency shifted with respectto the other by ∼ ≈ /
40 and a waist of 40 µ m.Starting from a cloud of atoms in state | (cid:105) ofuniform density, we pulse the Raman beams with aduration of a few tens of µ s to coherently transfera controlled fraction of atoms to state | (cid:105) . Inthis protocol, the total density of the cloud remainsuniform. We then image the density distribution ofatoms in state | (cid:105) prior to any spin dynamics and applyan optimization protocol identical to the one developedfor creating arbitrary density distributions. We showin figure 5 two examples of spin profiles realized inour system at the end of the optimization loop: aGaussian profile (figure 5(a)) and the so-called Townesprofile (figure 5(b)), which is a solitonic solution ofthe 2D attractive non-linear Schr¨odinger equation thatdecreases almost exponentially with r at large r [31].The measured profiles are very close to the target overtypically two orders of magnitude in density.
6. Discussion and outlook
In conclusion, we have demonstrated the arbitrarycontrol of the density profile of an ultracold 2Dquantum gas by tailoring a repulsive optical potential.We have also demonstrated the arbitrary creationof spin textures using spatially resolved Ramantransitions. An iterative method was applied, makingthe method robust to technical imperfections. Theapproach described here can be straightforwardlyapplied to other atomic species (bosonic or fermionic).It opens new possibilities for studying the dynamicsof single or multi-component low-dimensional gaseswhere, for instance, the presence of scale-invariance orintegrability leads to a rich variety of non-trivial timeevolutions [32, 33, 34, 35].
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Acknowledgments ∗ These two authors contributed equally to this work.This work is supported by ERC (Synergy UQUAM andTORYD), European Union’s Horizon 2020 Programme(QuantERA NAQUAS project) and the ANR-18-CE30-0010 grant.
Appendix
In this section, we simulate the experiment tounderstand the various contributions to the obtainedvalue of the figure of merit F for the density correction.In the simulation, we start with a “test” density profileA , which is obtained from an experiment with DMD2being off. It is an averaged image of 100 experimentalshots so that the detection noise is mostly averaged out.We follow the same procedure which was describedin figure 3(a) but in a “numerical experiment”. Wesimulate the action of the potential shaped by theDMD by using the local density approximation in theThomas-Fermi regime. Thus, for each iteration n ofthe loop we compute the density profile asA n = A − α C n , (5)where C n is the light intensity profile given by theDMD pattern after a convolution step that simulatesthe finite numerical aperture of the optical system. Weuse here a Gaussian profile with an rms width σ =0 . µ m. The parameter α is introduced to representthe effect of the light potential on the atomic density.We use as an input to the simulation experimentalimages of the optical depth distribution (OD ∼ α = 2 to be as close as possible to thecalibrated experimental parameters. We add an offsetto A n to keep the mean OD constant. We also havethe possibility to add some noise to A n to simulate theexperimental fluctuations.We show in figure 6 the simulated evolution of F as a function of the iteration number. The targetis a linear profile along the x direction, same as theone studied in figure 2(b) and figure 4. The blueand red curves show the simulated results with theparameters used in the experiment: K = 0 . A n a Gaussian ptical control of the density and spin spatial profiles of a planar Bose gas
830 6000 . . . . F
500 60000 . Figure 6.
Numerical simulation of the experiment. Evolutionof F as a function of iteration number with (blue) or without(red) noise. The target distribution is a linear density profilealong x . The diamond corresponds to the number of iterationsused in figure 4(a). The inset shows the same curves at largeiteration number. noise corresponding to N d = 0 .
09, which is the typicalnoise obtained in the experiment for the average of10 repetitions of the sequence. For the red curve, nodetection noise is added, i.e. N d = 0. The markeron the red curve corresponds to the point when theiterative loop is terminated for the experimental datashown in figure 4(a). Here, F = 0 . ∼ .
02) are obtained forlarger number of iterations ( ∼∼