.. A Momentum Filter for Atomic Gas
Wei Xiong, Xiaoji Zhou, ∗ Xuguang Yue, Yueyang Zhai, and Xuzong Chen
School of Electronics Engineering & Computer Science, Peking University, Beijing 100871, China (Dated: October 31, 2018)We propose and demonstrate a momentum filter for atomic gas based on a designed Talbot-Lau interferometer. It consists of two identical optical standing-wave pulses separated by a delayequal to odd multiples of the half Talbot time. The one-dimensional momentum width along thelong direction of a cigar-shaped condensate is rapidly and greatly purified to a minimum, whichcorresponds to the ground state energy of the confining trap in our experiment. We find goodagreement between theoretical analysis and experimental results. The filter is also effective fornon-condensed cold atoms and could be applied widely.
PACS numbers: 67.85.Hj, 67.85.Jk, 03.75.Kk
I. INTRODUCTION
Atomic sources with long coherence lengths, corre-sponding to a narrow momentum width, can be used toobserve new physical phenomena and contribute to theimproved precision measurement spectroscopy. Atomicclocks [1, 2] are for example greatly improved with nar-rower momentum width and so are the atomic inter-ferometers [3, 4]. Large correlation lengths have beenachieved with Bose-Einstein condensate [5, 6]. Manytechniques have been used to actually reduce the mo-mentum width of atomic gases. Some of them, suchas the velocity-selective coherent population trapping(VSCPT) [7] and Raman filter [8], behave as momen-tum filters to select atoms with specific momenta and todiscard the others. Other techniques benefit from thethought of filter to achieve lower temperature, such asevaporative cooling [9, 10] and Raman cooling [11].In this paper, we report a momentum filter schemeby precisely discriminating the different momenta-relatedphase evolutions during a matter wave Talbot-Lau inter-ference sequence. The Talbot effect was observed as anear-field diffraction with optical waves [12] and later ob-served with atoms [13–18]. In those works on the matterwave Talbot effect, the initial matter waves were approx-imated as a mono-energetic plane wave. In our work,the momentum distribution of a practical matter wave isconsidered. The effect of the momentum filter appearswhen an atomic gas is diffracted by a designed tempo-ral Talbot-Lau interferometer with specific standing wavepulses and time intervals between the pulses. As a re-sult, the interferometer generates different interferencepatterns for different initial momenta and behaves as a fil-ter. When the filter is applied on condensates, we choosethe intervals to be short enough (less than 11 T T / ∗ Electronic address: [email protected] interaction between atoms. The redistributing and puri-fying effects of the filter are demonstrated in the exper-iments and also analyzed theoretically. With the filter,we are able to rapidly purify one dimensional momen-tum width of the atomic gas to a minimum, which is themomentum width of atoms in the ground state of theconfining trap. Additionally, we discuss the influence ofatomic interaction in the experiments above. The mo-mentum filter is effective for both non-condensed coldatoms and condensate as shown in the experiment. Wealso discuss some possible applications of the filter.This paper is organized as follows. In Sec. II, a physi-cal model to interpret the momentum filter is presented.We derive a concise expression with Raman-Nath approx-imation [19] to reveal the physics of the filter, and thentreat the standing wave as a one dimensional optical lat-tice for more precise description. We also discuss theeffect of interaction between atoms for filters on conden-sates. Section III and IV present the effect of momentumredistribution and purification by the filter respectively.In Sec. V, the flexibility of the filter is demonstrated.Section VI contains discussion and conclusions.
II. PHYSICAL MODEL
Similarly to the optical Talbot-Lau effect, the temporalTalbot-Lau effect on matter wave behaves as shown inFig. 1(a). Two standing wave (optical lattice) pulses withthe same duration τ and induced optical potential U ,behave as two gratings. The matter wave accumulates aphase shift during the time interval τ f as what happens toa light wave along its optical path. The two effects bothlead to the matter (optical) wave interference patterns.Fig. 1(b) shows the mechanism of a band-pass momen-tum filter centered at zero momentum. The time interval τ f is designed as odd multiples of the half Talbot time T T /
2, where the Talbot time T T [18] is the minimumtime for the matter wave front’s reconstruction after thefirst pulse. An atomic gas with its momenta distributedaround zero is diffracted into the components with mo- a r X i v : . [ phy s i c s . a t m - c l u s ] J un FIG. 1: (color online) The temporal Talbot-Lau effect on mat-ter wave as a momentum filter. (a) The similarity betweenthe optical and the matter wave Talbot-Lau effect. The inputwave source is an optical (matter) wave. The two standingwave pulses behave as the two gratings. The phase of thelight wave evolutes along the space distance similarly as thematter wave during the time interval. (b) The scheme of themomentum filter. An atomic cloud with a momentum distri-bution around zero is scattered to be around momenta ± n (cid:126) k ( n = 0 , , , ... ) by the first pulse. During a time delay T T / ± n (cid:126) k acquire aphase shift around n π . After the second pulse, the atomswith the initial momenta closer to zero will be more probablyscattered back into the initial state. (c) The relation betweenthe probability P q and its initial momentum q . The solid,dashed and dotted line correspond to the interval of T T / T T / T T / menta around ± n (cid:126) k ( n = 0 , , , ... , the Plank constant (cid:126) , the wave vector of the light for the standing wave k )by the first pulse. With the interval T T /
2, the atomswith momenta ± n (cid:126) k , which originate from the atomswith initial zero momentum, obtain the accurate phaseevolution n π (which can be simplified as nπ because n and n have the same parity), and the second pulse en-tirely diffracts them back to their initial states. For theatoms with non-zero initial momenta, the larger the ini-tial momentum is, the larger the phase deviation from n π is and the more imperfect the recurrence will be,which results in a momentum filtering.For a brief description of the filter, we consider thestanding wave pulses to be short enough for the Raman-Nath approximation and obtain the probability of anatom with initial momentum q returning to the initial state as: P q = + ∞ (cid:88) n = −∞ J − n (Ω τ ) J n (Ω τ ) e − iE n,q τ f / (cid:126) , (1)with the Bessel function of the first kind J n , the two-photon Rabi frequency Ω [20], the kinetic energy E n,q =(2 n (cid:126) k + q ) / M and the atomic mass M . While the in-terval is designed as τ f = (2 N + 1) T T / , ( N = 0 , , ... ),the atoms with initial zero momentum accumulate thephase as E n, τ f / (cid:126) = n (2 N + 1) π , and the probabilityreaches the maximum as P q =0 = 1.Afterwards, we apply the Bloch theorem for a moreprecise description of the scattering process as describedin [21], where the standing wave is considered as an op-tical lattice and re-calculate the probability P q as shownin Fig. 1(c). It can be seen from the figure that themomentum filter actually consists of a series of filterswith different centers. The filter can be described aslong as the width and position of each filter are defi-nite. The 1 /e width 2 q of a single filter can be evalu-ated based on P q = 1 /e . Since the probability is princi-pally influenced by the phase evolution during the inter-val, we introduce a phase shift index α , which satisfies απ = ( E ,q − E , ) τ f / (cid:126) ≈ kq τ f /M , while q (cid:28) (cid:126) k is easily satisfied. The index α indicates the phase de-viation between the atoms with the momentum 2 (cid:126) k and2 (cid:126) k + q during the interval. It reveals the relation be-tween the interval and the width of a single filter as:2 q ≈ M απ/kτ f (2)Since τ f is equal to (2 N + 1) T T /
2, the width is also ofdiscrete values. As shown in Fig. 1(c), when the intervalis increased, the width of the filter decreases and so doesthe distance δq between adjacent centers. This distancemeans that the atoms with an initial momentum differ-ence δq will obtain the same interference pattern afterthe filter, since the phase deviation ( E n,q + δq − E n,q ) τ f / (cid:126) is integral multiple of 2 π . According to our analysis, δq satisfies δq = 2 (cid:126) k/ (2 N + 1), which can also be drawnfrom the figure.We analyze the momentum filter with single-atommodel above, which works for non-condensed cold atoms.When the atomic source is a condensate, interaction be-tween atoms needs to be considered. When condensatesare scattered by one dimensional filter sequence, the in-teraction between atoms takes effect. The widths of lightpulses (several µs generally) are much shorter than thecharacteristic time of interaction energy (magnitude as ms ), so interaction during light pulses can be neglected.Atomic interaction can not be ignored so easily duringintervals between light pulses, because intervals are pos-sibly comparable with the characteristic time of the inter-action shift. For a condensate in a harmonic trap exposedto a one-dimensional momentum filter, its interaction en-ergy can be averaged in the plane perpendicular to thelattice direction and get a distribution along the latticedirection as E I = E (1 − z /R Z ) , with the interactionenergy maximum E in lattice direction Z , the coordi-nate of lattice direction z and the Thomas-Fermi radius R Z of the condensate along this direction.According to the Gross-Pitaevskii equation, the phaseof 2 (cid:126) k component after the first lattice pulse evolves as e − i ( E k + V + E I ) τ f / (cid:126) , with the kinetic energy E k along Zdirection and magnetic trap V . The frequency of themagnetic trap along Z direction is only 20 Hz , so itcan be neglected, compared with the other two com-ponents. The kinetic energy is related to the Talbottime as E k / (2 π (cid:126) ) = 1 /T T . For the initial momentum q corresponding to a momentum width introduced bymomentum filter, the rate of its induced phase shift is(2 (cid:126) k + q ) − (2 (cid:126) k ) / M h ≈ (cid:126) k × q/M h .On the other hand, for interaction between atomsalong the Z direction, the average of interaction energyis 80% of the maximum. The rate of the phase shift in-duced by interaction can be approximated as E / h . Ifthe rate of the phase shift induced by the initial momen-tum is larger, i. e. 2 (cid:126) k × q/M > E /
5, momentum filterplays a main role in the scattering process. Otherwise,interaction energy plays a major role.In our experiment, the Talbot time is T T = 76 µs , andthe initial momentum q leads to a rate of phase shiftas q/ (cid:126) k × . kHz . The interaction energy is about1 . kHz at the center of the atomic cloud in our experi-ment and its distribution along Z direction after averageis 1 kHz × (1 − z /R Z ) . The rate of phase shift inducedby atomic interaction is about 200 Hz in our experiment.While interval increases, the corresponding momentumwidth decreases, and the phase evolution rate relevant tothe momentum width also decreases. On the other hand,the phase evolution rate induced by interaction is inde-pendent of the interval, so these two rates can be equalwith a certain interval. When the rates introduced byinitial momentum and interaction are equal, the intervalis 6 T T and it means that the effect of momentum filterplays bigger role than the interaction when the intervalis shorter than that, and things are opposite when theinterval is longer. The interaction induced phase shift isposition dependent and the interaction energy will accel-erate the phase evolution of atoms around the center ofatomic cloud. The filtered fraction results of both mo-mentum filter and interaction energy effects. As a result,we demonstrate the momentum filter mainly within theregime, where intervals are less than 6 T T . III. MOMENTUM REDISTRIBUTION
We performed the experiments on a Rb Bose-Einstein condensate (BEC) system [22, 23]. After pre-cooling, a cigar shaped Rb condensate of 1 × atomsin 5 S / | F = 2 , M F = 2 (cid:105) state was achieved by radiofrequency (RF) cooling in the a harmonic magnetic trap(MT), with 20 Hz axial frequency and 220 Hz radial fre-quency. The initial atomic gas is prepared as a conden-sate without observed non-condensed cold atoms. A one FIG. 2: (color online) Redistributing effect of the momentumfilter. (a) Sequence of the experiment. (b) The ratio of thefiltered atoms ( N ) over the total number of the atoms ( N A )versus the interval τ f . The round solid dots are the experi-mental results and the hollow square dots are the theoreticalanalysis, considering that the effect of atomic gases’ practi-cal temperature and the momentum expansion induced by s wave scattering are equivalent to an initial momentum width0 . (cid:126) k . dimensional designed far-red-detuned (the wavelength is852 nm ) standing wave pulse sequence was applied ontothe condensate along the axial direction. The latticedepth is 80 E R , which is calibrated by Kapitza-Dirac scat-tering experimentally. In the experiments, we choose thewidth of a single pulse to be 3 µs , so that the first pulseis able to couple most of the atoms (about 60%) to themomenta ± (cid:126) k , some atoms are populated around ± (cid:126) k (about 36%), others are around 0 (cid:126) k and other momenta.The second pulse is the same as the first one. In ourexperiment, the Talbot time is 76 µs and the index α isabout 0 .
20. According to our calculation, the influencesfrom momentum filter and atomic interaction are equalwhen the interval is 6 T T , so we choose the interval from T T / T T / ms free fall (see Fig. 2(a)). As the inter-val increases, the bandwidth of the filter decreases andthe relative population of the atoms returning to theirinitial states also decreases as shown in Fig. 2(b). Forthe non-condensed-cold-atom background of the cloud,its momentum distribution is classically Gaussian. How-ever, for the condensate, when its momentum can notbe neglected, Thomas-Fermi approximation is no longervalid and it is reasonable that its momentum distributionis analogous to Gaussian profile. Therefore, we describecondensate’s momentum distribution approximatively as ce − q / ∆ (the normalization coefficient c , the momentum q and the 1 /e width 2∆). In this way, we calculate the rel-ative population (square hollow dots in the figure) of thefiltered atoms, compare the calculation with the experi-ment results (round solid dots) and find that the calcula-tion generally pictures the trend of experimental results.The decrease of filter bandwidth, derived from Eq. (2) as δ = 4 α (cid:126) k/ (2 N + 1)(2 N + 3) ≈ α (cid:126) k/N , shrinks rapidlyas a function of N . As a result, the number of the filteredatoms changes more and more slowly while the interval τ f increases by units of T T in Fig. 2(b).For the theoretical curve (see square hollow dots inFig. 2(b)), atomic gases to be filtered are not pure con-densates with zero temperature and s wave scattering[24] also introduces momentum dispersion during the re-coil after the first pulse. We approximate these two ef-fects as an average initial momentum width 0 . (cid:126) k [21].When the intervals are short, the simulated filtered frac-tions are slightly higher than experimental ones, becausethe effective initial momentum widths are lower than0 . (cid:126) k . While the intervals get longer, the effective initialmomentum widths are higher, so the filtered fractions arelower.There are still some deviations between the calculationand the experiment, although the experimental resultsare normalized to minimize the uncertainty of the ini-tial atom number. One of the possible reasons may bethe assumed purity of the condensate in the theory. Apractical condensate is always surrounded by a cloud ofnon-condensed cold atoms with much wider momentumwidth. The non-condensed cold atoms contribute littleto the filtered component, but much more to the totalatom number. As a result, the fluctuation of the non-condensed cold atom number leads to the deviation inour experiment with respect to theory.In the experiment about momentum redistribution bya pulse sequence, when the interval τ f is 17 T T /
2, we ob-served two holes at the position of the momenta ± (cid:126) k inTOF signal as shown in Fig. 3(c). With this interval, therate of phase shift induced by momentum correspondingto the width of the filter is 141 Hz and that induced byinteraction is still 200 Hz , so that the two holes originatefrom both position selection by atomic interaction andmomentum filtering. We numerically simulated the pro-cess with the only difference that the initial condensateis with (see Fig. 3(b)) or without (see Fig. 3(a)) atomicinteraction. The two simulations clearly show the effectintroduced by atomic interaction.As shown in Fig. 3(a), for condensate without atomicinteraction, many peaks appear in the component around0 (cid:126) k , and each peak corresponds to a certain center of thefilter, because the absence of atomic interaction leads tolittle expansion in TOF signal and none of other redis-tributing effects. Consequently, only the momenta partsproduced by the filter can be seen and the valleys inthe ± (cid:126) k components also illustrate that. In Fig. 3(b),for condensate with atomic interaction, only one peak FIG. 3: (color online) Redistribution introduced by both mo-mentum filter and atomic interaction. (a) A numerical sim-ulation of TOF signal considering momentum filter withoutinteraction. (b) A numerical simulation of TOF signal in-cluding momentum filter and atomic interaction. (c) A TOFsignal obtained in our experiment. Two holes around ± (cid:126) k components can be observed. emerges around 0 (cid:126) k , and it is because that interactionleads to expansion and blur the peaks in TOF signal.However, two valleys arise around ± (cid:126) k separately. Theyare wider than the ones in condensate without interac-tion and obviously come from phase gradient introducedby atomic interaction, except for the filter. We numer-ically analyzed the process and found that the result isgenerally consistent with the experimental result. Theholes in Fig. 3(c) are even larger, and the reasons may bedispersion induced by s-wave scattering, deviation fromimaging system and so forth. IV. MOMENTUM PURIFICATION
To observe the momentum purifying effect of the fil-ter, we manage to keep the filtered component in themagnetic trap and discard the others. As described inFig. 4(a), the magnetic trap and the RF field are main-tained for 12 . FIG. 4: (color online) Purification effect of the momentumfilter. (a) Time sequence for momentum narrowing. (b) TOFsignals demonstrating one dimensional momentum narrowing.The figures from top to bottom show a condensate withoutfiltering, a condensate modified by a filter with interval T T / T T / ing this time. The 1 /e width Z (0) of the atomic gas atthe moment of being released from the magnetic trap canbe calculated based on the wave function of the conden-sate after the filter. The evolution of the atomic gas canbe figured out according to the time-dependent Gross-Pitaevskii equation. We are able to evaluate the momen-tum width after the free falling time t from the TOFsignal by working out the ballistic expansion equation Z ( t ) = Z (0) + (2 q /M ) t , with the size Z ( t ) of thecloud along the lattice direction.The TOF signal in Fig. 4(b) demonstrates the one di-mensional momentum purification with increased inter-val. The reduction of the atomic cloud size along thelattice direction from the TOF signals pictures clearlythe effect of the momentum purification. The figuresfrom top to bottom show a condensate without filter-ing, a condensate modified by a filter with interval T T / T T / × atoms. If itreached thermal equilibrium and were in the Thomas-Fermi regime, its initial size in the trap along Z direc-tion would be 2 R Z = (2 µ N /mω Z ) / ≈ µm ( µ N is thechemical potential related to s-wave scattering length a ,atomic number N and geometrical frequency of the mag- netic trap ω as µ N = (15 N (cid:126) a/ (4 √ M )) / ( M ω ) / ). After 30 ms expansion, the size of atomic cloud wouldreach 60 µm , which is much larger than 28 µm as observedin experiment. As a result, the size of atomic cloud’sshrinking mainly depends on momentum purifying, butnot the decrease of atomic number.The achieved one dimensional momentum width by thefilters with increased interval is shown in Fig. 4(c). Themomentum width predicted by the calculation (squarehollow dots) goes along with the experimental results(round solid dots) well until the interval of the filter ex-ceed 7 T T /
2, which corresponds to the expected width0 . (cid:126) k . When the atomic gas is abruptly released fromthe magnetic trap, its potential energy vanishes but thekinetic half remains, so the average kinetic energy of theatomic gas in free space can not get lower than that of theground state of the confining trap. The frequency 20 Hz of the axial magnetic trap corresponds to the momentumwidth 0 . (cid:126) k as (0 . (cid:126) k ) /M = h × Hz/ T T / T T / V. FLEXIBILITY OF THE FILTER
We show in this section that the filter can be appliedflexibly, as it can be used for both non-condensed coldatoms and condensate. Several filters can be combined toa new one and the center of a filter may also be adjusted.The effectiveness of the filter for different kinds ofatomic gases is demonstrated in Fig. 5. The effect of afilter for non-condensed cold atoms is shown in Fig. 5(a)with the interval T T / T T / FIG. 5: (color online) TOF signals showing the filtering ontwo kinds of atomic gases. Curve (1) is the initial atomicgas and curve (2) is the atomic gas after the filter. (a) Acloud of non-condensed cold atoms with temperature 513 nK modified by a filter with the interval T T /
2. (b) A mixture ofcondensate and non-condensed cold atoms with temperature346 nK modified by a filter with the interval 5 T T / T T /
2, the dashedone is that for a combination of two same filters with interval T T / T T / the TOF signal. Actually, there is no restriction withrespect to the choice of the initial atomic cloud for thefilter, but the parameters of the filter still need to beoptimized for the observation of the filter effect.Combination of single filters is a possible choice forachieving narrower width. The probabilities of recur-rence after a single filter with the interval T T / ± n (cid:126) k still remain inthe atomic gas and play a role in the interference of thesecond filter sequence. However, the combination stillbehaves as a filter with a narrower momentum width.The center of the momentum filter is not confined tobe only zero, such as some value q c , we can design thefilter with time constant τ f as E n,q c τ f / (cid:126) = (2 N + 1) π ( N = 0 , , , ... ) with similar bandwidth. For instance, ifthe intervals are 2 T t / , T t / , .. , the minimum centers ofthe corresponding filters will be (cid:126) k/ , (cid:126) k/ , .. . VI. DISCUSSION AND CONCLUSIONS
This momentum filter actually utilizes the interferenceamong all the momentum states generated by the firstlattice pulse to form a momentum filter. Although the |± (cid:126) k (cid:105) and |± (cid:126) k (cid:105) states overwhelm the others after thefirst pulse, the filter will be less functional if the other mo-mentum states are neglected, let alone the |± (cid:126) k (cid:105) statesalso being canceled. The consequence of ignoring higherorders’ momentum states is that the filter will miss somezero momentum atoms and collect more non-zero compo-nents and the momentum distribution will be broadened.The momentum width achieved by the filter may belimited by the following factors. The first one is the ini-tial momentum width of the atomic gas. The initial mo-mentum distribution can cover several centers of the fil-ter. Although the increased interval can reduce the band-width, the non-zero centers will weaken the purifying ofthe momentum. The second one is the relative line width γ of the laser of filter, since the interval can not increasefor failure of discriminating the phase shift due to the linewidth, which means ((2 (cid:126) k (1 + γ )) − (2 (cid:126) k ) ) τ f / M (cid:126) ≤ απ . The third one is the de-coherence time. The fil-ter is a process of interference, so there will be no filterif the quantum states de-coherent during the interval ofthe filter.In our experiment, the narrowest momentum widthachieved with this filter is limited by the confining trap.If the confinement of the trap is weaker, the momen-tum filter will be able to generate narrower momentumwidths. The experiments can be extended to three di-mensions, since three dimensional optical lattice is widelyused at present. The three dimensional filter could be atool for rapidly cooling atomic gases.This filter is not a repetitive demonstration of Talbot-Lau interferometer, since the matter wave we dealt withis distributed in momentum space, not a simple plainwave. As a result, the momentum selecting effect comesout from the interferometer. Briefly compared with othermomentum filters, our filter is simple and robust.Firstly, requirements on lasers are different in our filterand other velocity selection methods. Our filter consistsof optical lattice pulse sequences, which are easily de-signed and controlled. The lattice laser is so far detunedfrom atomic resonance that the filter can work well with-out laser frequency lock. This means a good many kindsof atoms and molecules can be momentum purified bylasers with a very wide range of frequencies, through ourmethod. By comparison, optical setups a for Raman filterand VSCPT are more complicated. For Raman filter, inorder to preserve the atoms with specific momenta andto blow off the others, at least two different frequencylaser beams are needed. Otherwise, the selected atomshave to be charged a momentum 2 (cid:126) k . Furthermore, laserfrequencies have to be locked to make sure that Rabi fre-quency matches the pulse duration.Secondly, our filter preserves both the internal and ex-ternal states of selected atoms. It hardly changes themomenta or the internal states of the selected atoms,and simultaneously charges other atoms with momenta ± (cid:126) k , ± (cid:126) k and so on. Consequently, our filter can worksolo and cooperate with other techniques smoothly, suchas evaporative cooling. For Raman filter, if the internalstates of filtered atoms need to be preserved, they haveto be charged with a momentum 2 (cid:126) k ; if the momenta ofthe selected atoms are preserved, they have to be pumpedinto another internal state. Otherwise, the atoms bothpreserving momenta and internal states are difficult todistinguish from others.Thirdly, our filter is robust for atoms being in the darkduring most of the filter process. When lattice is on,lasers in our filter can work well with very large detun-ing (taking Rb for example, its D nm , our lattice light works at 852 nm ) and very shortpulses (in our experiment, pulse duration can be as shortas 1 µs while lattice depth reaches 120 E R ); during therest of the filter process (in our experiment, the intervalbetween two lattice pulses is about 300 µs to achieve themomentum width 0 . (cid:126) k ), atoms stay in the dark. Asa result, the filter can hardly be disturbed by most ofthe perturbations and drifts on lattices frequency, phaseand intensity, such as vibration noise on optical latticemirror, because these noise frequencies are mostly below1 M Hz . In contrast, in order to achieve similar momen-tum width, the pulse duration in Raman filter is usuallyabout 2 ms , and noises with frequency higher than 500 Hz will be received by the system. Dark states in VSCPTare much more fragile. Hence, our filter is of great poten-tial to achieve ultra-narrow momentum width because a lot of noises are shielded.In conclusion, we apply a designed temporal matterwave Talbot-Lau interferometer for realizing a momen-tum filter, which can keep the atoms with lower momentain their initial states and drive the others to states withhigher momenta. This kind of momentum filter makesuse of the momentum-state interference to discriminatebetween momenta and hardly depends on the transitionbetween the atomic internal states. With such filter, wemanage to prepare an atomic cloud with 0 . (cid:126) k momen-tum width within 300 µs . We consider the interactionbetween atoms for condensate during the filter process,and find that it affects the redistribution of atomic gas,but hardly momentum purifying. We show that it is aneffective method for rapidly compressing the momentumwidth of an atomic gas, and systematically discuss thespecifications of the filter. It may be of great potentialfor achieving ultra low temperatures. Acknowledgement
We appreciate H. W. Xiong, G. V. Shlyapnikov, G. J.Dong, L. B. Fu, B. Wu and H. zhai for stimulating dis-cussions. We thank Thibault Vogt for critical reading ofour paper. This work is supported by the National Fun-damental Research Program of China under Grant No.2011CB921501, the National Natural Science Foundationof China under Grant No. 61027016, No.61078026 andNo.10934010. [1] Mark A. Kasevich et al. , Phys. Rev. Lett. , 612(1989).[2] R. Wynands et al. , Metrologia , 64(2005).[3] Alexander D. Cronin et al. , Rev. Mod. Phys. , 1051(2009).[4] J. B. Fixler et al. , Science , 74 (2007).[5] M. H. Anderson et al. , Science , 198 (1995).[6] K. B. Davis et al. , Phys. Rev. Lett. , 3969(1995).[7] A. Aspect et al. , Phys. Rev. Lett. , 826(1988).[8] M. Kasevich et al. , Phys. Rev. Lett. , 2297(1991).[9] Wolfgang Ketterle et al. , Advances in atomic molecularand optical physics , 181-236(1996).[10] K. B. Davis et al. , Phys. Rev. Lett. , 5202(1995).[11] M. Kasevich and S. Chu Phys. Rev. Lett. , 1741(1992).[12] William B. Case et al. , Optics Express , 23 (2009) [13] Michael S. Chapman et al. , Phys. Rev. A , 1 (1995).[14] J. F. Clauser and M. W. Reinsch Appl. Phys. B , 380(1992).[15] S. Nowak et al. , Opt. Lett. , 1430 (1997).[16] J. F. Clauser et al. , Phys. Rev. A , R2213 (1994).[17] O. Carnal et al. , Phys. Rev. A , 3079 (1995).[18] L. Deng et al. , Phys. Rev. Lett. , 5407(1999).[19] M. Edwards et al. , Phys. Rev. A , 063613 (2010).[20] A. F. Linskens et al. , Phys. Rev. A , 6 (1996).[21] W. Xiong et al. , Phys. Rev. A , 043616 (2011).[22] X. Zhou et al. , Phys. Rev. A , 013615 (2010).[23] B. Lu et al. , Phys. Rev. A , 051608(R) (2011).[24] Y. B. Band et al. , Phys. Rev. Lett.84