matter waves in time-modulated complex light potentialsmatter waves in time-modulated complex light...

PHYSICAL REVIEW A, VOLUME 62, 023606

Matter waves in time-modulated complex light potentials

S. Bernet,1 R. Abfalterer,2 C. Keller,3 M. K. Oberthaler,4 J. Schmiedmayer,2 and A. Zeilinger31Institut fur Medizinische Physik, Universita¨t Innsbruck, Mu¨llerstraße 44, A-6020 Innsbruck, Austria

2Institut fur Experimentalphysik, Universita¨t Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria3Institut fur Experimentalphysik, Universita¨t Wien, Boltzmanngasse 5, A-1090 Wien, Austria

4Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom~Received 19 January 2000; published 17 July 2000!

Temporal light modulation methods which are of great practical importance in optical technology, areemulated with matter waves. This includes generation and tailoring of matter-wave sidebands, using amplitudeand phase modulation of an atomic beam. In the experiments atoms are Bragg diffracted at standing light fields,which are periodically modulated in intensity or frequency. This gives rise to a generalized Bragg situationunder which the atomic matter waves are both diffracted and coherently shifted in their de Broglie frequency.In particular, we demonstrate creation of complex and non-Hermitian matter-wave modulations. One interest-ing case is a potential with a time-dependent complex helicity@V}exp(ivt)#, which produces a purely lopsidedenergy transfer between the atoms and the photons, and thus violates the usual symmetry between absorptionand stimulated emission of energy quanta. Possible applications range from atom cooling over advancedatomic interferometers to a new type of mass spectrometer.

PACS number~s!: 03.75.Be, 03.75.Dg, 32.80.Pj

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I. INTRODUCTION

The interaction of atomic matter waves with standilight fields has been established as a very useful tool in aoptics. In standing light fields atoms can be trapped, coodiffracted, manipulated, and arranged in so called opticaltices @1,2#. The simplicity of the systems recently alloweone to investigate many relevant quantum optical phenomwhich are closely related to solid-state physics, like quantchaotic behavior@3#, Bloch oscillations@4#, Wannier-Starkladders and Landau-Zener tunneling@5#, spectroscopy of theBloch bands@6,7#, and temporal interference effects@8,9#.An important subgroup of these interactions consistsBragg scattering of atomic beams at standing light fie@10–14#, which can then be viewed in analogy to condensmatter physics as ‘‘crystals of light’’@15#. Atomic Braggdiffraction has been demonstrated at both refractive andsorptive light crystals, revealing interesting features whare analogous to standard Bragg diffraction in crystallogphy, like Pendello¨sung-oscillations, anomalous transmissi~the Borrmann effect! through absorptive crystals, or thspectral dependence of the scattering phase@12,14,16#. Eventhe interaction withcomplexlight potentials has been investigated@17#, resulting in asymmetric Bragg diffraction, i.ein a lopsided momentum transfer between atoms and ptons, which violates an empirical rule called Friedel’s law

In practical applications the coherence of the Bragg stering process allowed for its use as an efficient cohebeam-splitting mechanism in atom interferometers@18,19#,and more recently, as an output coupler for coherent mawaves from a Bose-Einstein condensate of atoms@20,21#.Due to the extremely high momentum and energy selectiof Bragg scattering, it is also used as a very sensitive sptroscopic tool to investigate such degenerate quantum g@22#. These methods frequently exploit the possibilitymodulate the standing light field in order to get a travelinor a pulsed light crystal. Thus it becomes important to inv

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tigate in detail the phenomena which arise during the inaction of atomic matter waves with such modulated ligfields. One more motivation of our investigations is the sucess of frequency shifters and sideband modulators in trtional light optics in both scientific and practical applictions. Thus we may expect corresponding breakthroughmatter-wave optics, particularly in combination with the rcent advances in the generation of atomic Bose-Einstein cdensates which can act as ‘‘atom lasers.’’

In previous publications we have already demonstrathat the effect of atomic scattering at traveling Bragg grings ~which arise in an intensity modulated light crystal! re-sults in coherently frequency shifted atomic matter wavThis can be interpreted as an atom-optic analog toacousto-optic frequency shifter in photon optics@23,24#, oras a process of sideband generation similar to radioquency techniques. In the next sections we will first theorcally and then experimentally investigate this basic effecmore detail.

Furthermore, we extend our investigations to generaltential modulations, which can be even complex, and deonstrate how to realize them experimentally with twcomplementary methods, i.e., either by independently molating real and imaginary parts of the light potential@25#, orby controlling its amplitude and complex phase@26#. Usingthese methods we realize potential modulations of the foV}exp(6ivMt), which represent a complex helix in the timdomain. These modulations lead either to a subtraction oan addition of a quantized energy amount\vM to the kineticenergy of the atomic matter waves, depending on the sigthe exponent.

For an explanation of the observed diffraction effectspresent an intuitive Ewald-type model which extendsclassical Bragg situation to the case where the crystamodulated. Such a model allows for the derivation of tnew Bragg condition, under which a coupled processBragg diffraction and atomic matter-wave frequency shifti

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S. BERNETet al. PHYSICAL REVIEW A 62 023606

~‘‘atomic sideband generation’’! is allowed. For calculatingthe diffraction efficiencies at the new Bragg angles thesults of standard Bragg scattering theory are used, as deindependently in dynamical diffraction theory@27#, or in acoupled wave formalism@28#, though with additional consideration of the effects of complex, non-Hermitian potentia

Finally we briefly sketch another viewpoint of our expements, which points out its relation to recently publishinteresting experiments, where the Bloch band structuremagneto-optically trapped atoms in optical lattices has binvestigated@4–7#. We show that atomic Bragg scatteringmodulated potentials can be interpreted as a transition withe energy-band structure of the light crystal, which isduced by the temporal modulation process. Thus our expments represent a highly sensitive spectroscopic investion of the energy-band structure~‘‘Bloch bands’’! of thelight crystal. The main result within this viewpoint is thaspecial types of complex potential modulations, that aretually realized in our experiments, lead todirectedtransitionswithin the energy bands of the light crystal. This asymmeenergy transfer corresponds to a very unusual interactiontween atoms and light fields, which, in particular, doeslead to thermal equilibrium~like, e.g., in blackbody radiation!. Besides its fundamental interest, this might have prtical applications in atom cooling.

II. BRAGG DIFFRACTION IN SPACE AND TIME

The interaction of atoms and standing light fields candescribed by two complementary pictures. In atomic physoften an energy-momentum picture is used. There, atabsorb a photon from one mode of the light field, followby stimulated reemission to another mode. Thus the aacquires a momentum change which corresponds to twoton recoil momenta. The process is allowed if the energythe atom is conserved, i.e., if the absolute value of the atovelocity does not change in the laboratory frame~supposedthat the atom does not change its internal state duringinteraction!. The efficiency of such an allowed process dpends on the probability of the coupled absorption/stimulaemission process, and thus on the light frequency detunfrom an atomic transition line. The picture can be geneized for the description of processes, where the atom chaits internal state, or where the atom interacts with light fieof multiple frequencies~‘‘Doppler-sensitive Raman transitions’’!. There the atom can change both its momentumits kinetic energy, if a generalized condition of total enerand momentum conservation is fulfilled, which dependsthe geometry of the interaction process, on the participalight frequencies, and on the internal state energies. Atailed overview over atom-light interactions in this picturegiven in Ref.@29#.

In other fields of diffraction physics, e.g., in light anx-ray diffraction, electron, and neutron scattering, oftencomplementary picture is used which is based on the inaction of an incident wave with a periodic medium. Thethe diffraction is described according to the Huygens prciple as an interference process of all partial waves scattcoherently from the periodic structure. This can be also g

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eralized to a situation where an energy change of thefracted wave is allowed, if the scattering medium is temprally modulated or moving.

Both the energy-momentum picture and the Huygens pture lead to equivalent results, i.e. to a dynamic diffractitheory. Here we chose the description in the wave picturesulting in a so-called Ewald construction, which is wknown in standard diffraction physics. One advantage is tthe allowed diffraction geometry can be constructed direcin the laboratory frame. We generalize this picture to tsituation of scattering at a time modulated periodic mediuthough with reference to the energy-momentum picturethe end of the paper by pointing out the relation of our eperiments to the field of so-called Bloch band spectrosco

In general, a light field represents a frequency-dependcomplexrefractive index for the atom@30#. Consequently, anextended standing light field corresponds to a periodicfractive index modulation with the ability to diffract aatomic matter wave—a scattering ‘‘crystal’’ made of lighThe fact that the scattering of atomic matter waves at slight crystals can be described as a standard Bragg difftion process, like, e.g., x-ray diffraction at solid crystals, hbeen well investigated in the literature@10–13#. Analogieshave been found even in very fine details@14–16#.

The main characteristic of standard Bragg diffraction isquantizedmomentum exchange between the incident waand the scattering crystal, which can occur only in multipof the ‘‘grating momentum’’\GW , where GW is the gratingvector of the crystal with the absolute valueuGW u52p/d5..G, inversely proportional to the grating periodd. Verysimilar, a potential which is periodically modulated in thtime domain with frequencyvM can, under certain circumstances, exchange energy quanta in multiples of\vM withthe incident particle. Such an energy exchange, which althe optical frequency of a scattered photon, or the de Brofrequency of a scattered particle, is technically describedfrequency sideband generation. Due to the analogy betwdiffraction and sideband generation, both characterized bquantized exchange of momentum or energy, respectivthe process of sideband generation is sometimes denote‘‘diffraction in time’’ @31#.

The close analogy between spatial and ‘‘temporal’’ dfraction also results in a similar classification of the diffration regimes. In standard scattering there is a significantference between diffraction at thin gratings~the Raman-Nathregime! and thick crystals~the Bragg regime!. In the Braggregime the grating isspatially thick enough to define asharply fixed orientation of the grating vector. In a thin graing regime these orientations are undefined in a certain ctinuous range, allowing for diffraction at a correspondingcontinuous range of incidence angles. The borderline issically given by Heisenberg’s position-momentum unctainty limit.

Similarly, in temporal diffraction the interaction time between the crystal and particle generates two significantlyferent regimes: a ‘‘short’’~temporal Raman-Nath! and a‘‘long’’ ~temporal Bragg! diffraction regime. If the interac-tion time between the incident particle and a periodica

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MATTER WAVES IN TIME-MODULATED COMPLEX . . . PHYSICAL REVIEW A 62 023606

modulated potential is long enough, energy can only bechanged in sharply defined energy quanta, whereas in a s~e.g., pulsed light crystal! interaction regime, energy can bexchanged in a continuous range, which allows the gention of spectrallybroad sidebands. Again the transition between the two regimes is given basically by an uncertairelation, though now by the time-energy uncertainty.

Linking diffraction in space and time, all combinationsspatial and temporal diffraction regimes can be realized, le.g., sharply defined momentum transfer but continuousergy exchange~diffraction of atoms at a pulsed thick lighgrating in Ref.@13#!, corresponding in our terms to a ‘‘spatial Bragg and temporal Raman-Nath regime,’’ or continuomomentum transfer and quantized energy exchange~diffrac-tion of atoms and neutrons at vibrating mirrors in Refs.@32#and @33#, respectively!, corresponding to ‘‘spatial RamanNath and temporal Bragg scattering.’’ However, all of thecombinations exhibit different scattering effects.

In our experiments we investigate coupled spatial diffrtion and sideband generation in a spatial and temporal Brregime. In contrast to all other combinations this gives risenew discreteBragg angles, under which the atomic matwaves are simultaneously diffracted and frequency shifThe periodic light potential for the atoms can be tailoredsuperimposing different light fields, or by modulating thlight frequency across an atomic resonance line. In thelowing we present an intuitive model for deriving the neBragg condition for scattering at temporally modulated crtals.

A. Generalized Ewald construction for modulated crystals

In static Bragg diffraction, an incident wave is scatteronly under specificdiscreteincidence conditions—the Bragangles. This is a consequence of both energy conservafor the incident wave, and quantization of the momentexchange between crystal and wave in multiples of\GW ,whereGW is the grating vector (uGW u52p/d). In the case of alight crystal set up by two counterpropagating light wavwith wavelength l and corresponding wave vectokWL (ukWLu52p/l), the grating constantd corresponds tol/2,and the grating vectors areGW 562kWL.

Energy conservation and quantized momentum exchaare the basis of the well-known Ewald construction, whthe tips of incident and scattered wave vectors are locatea common circle around their origin~equal length corre-sponding to energy conservation!; additionally, they are con-nected by a grating vector~corresponding to a quantized momentum change of\GW ). A construction for first-order Braggdiffraction, corresponding to a momentum exchange ofactly \GW is sketched in Fig. 1~a!. Obviously, for a givenlength of the incident wave vectorkWA5mvW A /\ ~wherem and

vW A are the atomic mass and velocity, respectively!, for dif-fraction at thenGth order, the Ewald construction yields onone possible incidence angle; the Bragg angleuB,nG

, satisfy-ing

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However, since the grating vector is usually not orienti.e., it exists in both opposite directions, the orientationsincident and scattered wave vectors can be exchanged, wyields two symmetrically arranged Bragg angles for scating at the two conjugate6nGth Bragg orders.

The situation changes in the case of scattering at acused standing light wave~thin grating!, as sketched in Fig1~b!. Due to focusing in thez direction, thez component ofthe grating vector,Gz is no longer defined sharply, but has auncertainty in an angular range inversely proportional tosize of the focusDGz'(2Dz)21 @34#. Although the condi-tion of energy conservation~elastic scattering! still holds, thedirection of the transferred grating momentum can nowchosen in a certain continuous range, and the Ewald cstruction yields a corresponding range of incidence angwhere elastic diffraction is possible. As an example, twolowed pairs of incident and scattered wave vectorssketched in Fig. 1~b!. Thus, focusing of a transversely extended~plane! standing light wave results in a continuoutransition from a Bragg regime, where scattering appeonly at discrete incidence conditions, to a thin-grating regi~the Raman-Nath regime!, where a broad range of incidencangles~and momenta! is scattered@35#.

The Ewald construction can be generalized to the sittion of a temporally modulated crystal. Such a tempomodulation provides the possibility to transfer kinetic enerto the scattered wave. If the interaction timet is short~the‘‘temporal Raman-Nath regime’’!, e.g., by pulsing the lightcrystal, then there results a continuous energy uncertaDEkin'\/t of the scattered wave, which enables Bragg dfraction at much less stringent incidence conditions. Thisdemonstrated in Fig. 1~c! by a modified Ewald constructionwhere the circle of the original Ewald construction is substuted by a circular band with a thickness of 1/tvA , corre-

FIG. 1. Ewald constructions for different situations of first-ordscattering at a given direction~‘‘to the right side’’!. The discussedcases are~A! a thick static grating,~B! a thin static grating,~C! athick pulsed grating, and~D! a thick, harmonically modulated grating.

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sponding to an energy uncertainty ofDEkin'\/t. Similar tothe spatial Raman-Nath situation of Fig. 1~b!, diffraction isnow again allowed for a whole range of incidence anglHowever, in contrast there is now afixed momentum transfebetween incident and outgoing wave vectors, accompaby acontinuous energy transfer. Thus the length of the scattered wave vector can change, and scattering becomes intic. As mentioned above, such a situation might be chaterized as a spatial Bragg regime and a temporal RamNath regime.

However, in our experiment we stay in a spatial and teporal Bragg regime by employing a temporally harmonmodulation with frequencyvM , and allowing for a suffi-ciently long interaction timet ~several modulation periods!.Now the energy exchange between the crystal and wavquantized in multiples of\vM , with an uncertainty in-versely proportional to the interaction time (DEkin'\/t).Consequently, both momentum and energy transfer betwcrystal and wave are quantized. The Ewald constructmodified for this situation, is sketched in Fig. 1~d!. Now thepeak of the incident wave vector defines the radius ofmiddle circle. The lengths of the atomic wave vectors shifby the quantized energy offsets6\vM are given by newcircles with smaller and larger radii, offset byvM /vA . Asharply defined grating vectorGW connects incident and outgoing wave vectors, which, however, can now lie on tdifferent circles. Obviously, these conditions are satisfionly for discrete incidence angles, although there are nthree possibilities for first-order diffraction in one directioas indicated in the figure. The situation thus correspondsgeneralization of Bragg scattering to the time domain. Csequently, the standard static Bragg condition results aspecial case settingvM50. The waves scattered at the neincidence angles are expected to be coherently frequeshifted by the modulation frequency of the crystal. Note tin our situation of scattering at a Bragg crystal, the diffracbeam contains only one single-frequency component atime, because only a certain frequency shift, which depeon the incidence angle, can fulfill the generalized Bragg cdition derived above. This means that a modulated Brcrystal indicates the frequency shift of a diffracted atom bunique Bragg angle. Thus, in the spatial and temporal Brregime, a modulated crystal acts simultaneously as a sband generator and an analyzer.

B. Calculation of the generalized Bragg condition

In order to calculate the new Bragg angles for diffractiwith quantized energy exchange we analyze the situasketched in Fig. 1~d!. Both momentum and energy can onbe changed by quantized amounts, requiring

kWA85kWA1nGGW ,

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2m1nM\vM .

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TherenM andnG are the numbers of frequency modultion quanta\vM and grating momenta\GW , exchanged be-tween wave and crystal, and thus correspond to the tempand spatial diffraction orders, respectively.kWA andkW A8 denotethe incident and diffracted atomic wave vectors. An adtional condition is imposed by the fact that each exchangea modulation quantum~i.e., each ‘‘diffraction process intime’’ ! has to be accompanied by an exchange of a gramomentum~i.e., a spatial diffraction process!, in contrast tospatial diffraction, which can also happen without enerexchange~‘‘standard’’ Bragg diffraction!.1 Therefore, thepossible temporal diffraction ordersnM are restricted by thespatial diffraction ordernG to only three cases:

nM50 or nM56nG . ~3!

This means that there is always the possibility for elasscattering (nM50) or, that there is the possibility to exchange a number of energy quanta corresponding to thetial diffraction order (nM56nG).

For solving Eqs.~2! it can be exploited that due to momentum conservation only the component of the atomwave vectorkA,uu parallel to the grating vector can changbut not the perpendicular componentkA,' @this is alsostraightforwardly seen from the geometry sketched in F1~d!#. This means that

kA,'8 5kA,' ,

kA,uu8 5kA,uu1nGG, ~4!

\kA,uu82

2m5

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2m1nM\vM .

These equations can be solved easily. As an abbreviawe introduce the matter-wave frequency of an atom,vA

5\ukWAu2/2m, and the two-photon recoil frequencyvRec

5\uGW u2/2m an atom acquires with respect to its original reframe after addition of one grating vectorGW . ThusvRec is ageometric constant of the setup. As a result, for the paracomponents of an allowed pair of incident and diffractwave vectors we obtain

kA,uu51

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kA,uu8 51

2nGGS nMvM

nG2 vRec

11D ,

and

1This is a consequence of the Schro¨dinger equation, where spatiaand temporal coordinates are not symmetric.

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ukWA8 u5ukWAuAnMvM

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Thus an incidence angleu In , where an incoming atomcan be scattered, is

sin~u In!5kA,uu

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!1nMvMuGW u

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Obviously, the relation is a generalization of the static Bracondition, yielding the static Bragg angle in the case ofunmodulated crystal (nM50).

Similarly, the angle of the scattered atom,uOut , is givenby

sin~uOut!5kA,uu8

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!

A11nMvM /vA

. ~8!

In our experimental situation of very small incidence ascattering angles~below 200mrad!, all the sinus functionscan be replaced by their arguments. Therefore, from Eq.~7!,in a very good approximation, we obtain the differenceDuof the new Bragg anglesu In ~with frequency shift! from theoriginal static Bragg angles,uB,nG

, for the cases where Eq

~3! holds, i.e.,nM56nG :

Du5u In2uB,nG56uB

vM

vRec. ~9!

ThereuB5G/2kA is the static Bragg angle for first-ordediffraction. This shows that the angular offset of a neBragg angle from the static Bragg angle, obtained at a molated crystal, is a direct measure of the frequency shiftquired by the scattered atoms. Therefore, a modulated Bcrystal is able to produce sidebands of matter waves,simultaneously, to identify the frequency offset of the sidbands by their offset from the static Bragg angle. This retion will be used in several parts of our experiment.

Additionally, our experimentally applied modulation frequencies are far below the matter-wave frequencyvA of theincident atoms. Considering this, from Eqs.~7! and ~8! weobtain the total scattering angle of the atoms asuOut2u In52uB,nG

, which is identical to the scattering angle of staBragg diffraction. Additionally, according to Eq.~7!, the an-gular offset of the new allowed incidence angles fromstatic Bragg angle does not depend on the diffraction or~assumingnM56nG), i.e., the incidence angles of all atomfrequency shifted bynG\vM are offset by the same amounThis shows that the total far-field diffraction pattern

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frequency-shifted atoms looks exactly like the diffractiopattern in ‘‘static’’ Bragg diffraction, though with an angulaoffset given by Eq.~9!. The situation thus resembles scatteing at a moving crystal, which would result in a scatteringfrequency-shifted atoms with a diffraction pattern lookinlike a static diffraction picture, although centered aroundnew incidence angle. In fact, in a previous publicationinterpretation based on atomic scattering at traveling scrystals within the modulated light crystal was introduc@23#.

However, for many situations another picture is advangeous. An alternative Bragg condition can be derivedconsidering the ondulation or ‘‘wobbling’’ frequency,vWob,with which an atom incident at one of the new Bragg angu In traverses the grating planes of a crystal:

vWob5vA sin~u In!uGW u52nM

nGvM1nGvRec. ~10!

If we take into account the additional conditionnM56nG @Eq. ~3!# for diffraction with energy exchange, this cabe rewritten as

nG2 \vRec5nG\~vWob6vM !. ~11!

This central equation can be used as an alternative forlation of the Bragg condition@Eq. ~7!# for inelastic scatteringat modulated crystals. An interpretation is straightforwardthe atom is regarded in itsincident rest frame. There theenergy of the atom is zero. However, after a static Brascattering process, the atom moves. The correspondingnetic energy of the atomwith respect to the original resframecan only be supplied from a time-dependent modution ‘‘seen’’ by the atom in its frame. In fact, even in statscattering the atom experiences a light intensity modulawith frequencyvWob due to the apparently moving lighcrystal~with respect to the atomic rest frame!, and thereforecan change its kinetic energy in multiples of\vWob. Analternative interpretation of the static Bragg condition noresults from the requirement that this energy change mmatch the atom’s recoil energynG

2 \vRec, linked to the ad-dition of nG grating momenta~i.e., nGth-order diffraction! tothe resting atom~note that the quadratic dependence of trecoil energy on the diffraction order is due to the mattwave dispersion relation in vacuum!. This means that thestatic Bragg condition for matter-wave scattering can beformulated asnG

2 \vRec5nG\vWob.However, if the crystal is temporally modulated with

frequency ofvM , the atom experiences this frequency cotribution in addition to the ondulation frequencyvWob.These two~‘‘spatial’’ and temporal! frequency contributionsresult in beating frequencies ofvWob6vM , correspondingto new modulation quanta which can be exchanged betwthe atom and crystal. Therefore, the meaning of Eq.~11! isthat the recoil frequency fornGth-order diffraction(nG

2 \vRec) has to match a multiple of the new energy quatum \(vWob6vM) which can be exchanged between tatom and crystal. This alternative formulation of the Bra

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condition is particularly useful for discussing the angular avelocity selectivity of the inelastic Bragg diffraction, as wbe shown below.

C. Diffraction efficiency at Bragg incidence

In the next step we investigate the efficiency of sidebaproduction. In the Ewald picture of Fig. 1, this might be seas a determination of the weights of the different circlwhich then represent the wave-vector magnitudes of elacally and inelastically scattered atoms. In this short discsion we will limit ourselves to the situation of scatteringfirst order. For static first-order Bragg scattering, it canshown that the amplitude of the diffracted wave function,A,is given by@10,12,16,28#

A5sinS pDz

L D . ~12!

ThereL is the so-called Pendello¨sungs length dependinon the grating constantd, on the atomic recoil energyERec5\2G2/2m, and on the coefficientVG of the spatial Fouriertransform of the grating potentialV, which corresponds tothe amplitude of the grating component with periodd:

L5dERec

VG sin~uB!. ~13!

Obviously, the diffracted amplitude oscillates betwe21 and 1 as a function of the crystal length,Dz, or of thegrating amplitude. This is denoted as the so-called ‘‘Pendlosung.’’ However, if scattering is only limited to lowdiffraction efficiencies well below the first Pendello¨sungslength ~this condition is fulfilled in many of our experiments!, it is possible to expand Eq.~12!:

A'pDz

L. ~14!

The corresponding diffraction efficiencyP, i.e., the ratioof diffracted atoms to incident atoms, is then given by tsquared modulus of Eq.~14!:

P'S pDz

L D 2

. ~15!

The situation changes if the potential is additionamodulated in time. Now a two-dimensional Fourier tranform has to be performed, dividing the total potential in cotributions with grating constantsnGG and modulation periodnMvM , i.e.,

V~x,t !5( ( VnG ,nMexp~ inGGx2 inMvMt !. ~16!

The amplitude of first-order Bragg scattering at a subging with grating vectornGG, modulated with one of the frequenciesnMvM within the temporal Fourier spectrum of thmodulation, is now given by inserting the respective Four

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the atomic incidence occurs under the corresponding Brangles@given by Eq.~7!#.

Specifically, the diffraction efficiencies for first-order sptial diffraction at the basic grating with the exchange of ze~elastic scattering! or one modulation quanta are

PnG51,nM505S pDz sin~uB!

ERecdD 2

uV1,0u2,

~17!

PnG51,nM5615S pDz sin~uB!

ERecdD 2

uV1,61u2,

respectively.There it is assumed that the condition of small diffracti

efficiencies holds, such that the approximation of Eq.~15!can be applied. The first term of each equation only contageometric constants of the setup. Thus it turns out thatdiffraction efficiencies depend only on the squared moduof the coefficients of a two-dimensional~spatial and tempo-ral! Fourier transform. This dependence will be confirmedSec. III.

III. EXPERIMENTS

We start by describing the scheme of our experiment@36#.Then we will demonstrate the basic features of scatterintime-modulated crystals, before investigating the effectscomplex potential modulations.

A. Setup

Our setup is sketched in Fig. 2. Argon atoms in a mestable state are diffracted at a standing light wave, followby spatially resolved detection.

The argon atoms are first excited to a metastable s~lifetime .30 s! in a gas discharge. Only these metastab

FIG. 2. Experimental setup~not to scale!. A collimated thermalbeam of metastable argon atoms crosses a standing light wavediffraction pattern can be registered with spatial resolution by scning a narrow slit in front of the extended channeltron detector. Tlaser light intensity is modulated in some parts of the experimusing an acousto-optic modulator~AOM!. The two graphs show theresults for off-Bragg incidence~only one peak of transmitted atoms!and Bragg incidence~a second peak of Bragg-diffracted atomarises! in the case of an unmodulated laser beam. Additionally,scattering efficiency can be registered as a function of the atoincidence angle at the light crystal by fixing the detection slit atposition of the diffraction peak and tilting the retroreflection mirro

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can be detected by our ‘‘channeltron’’ detector, by releastheir high excitation energy~12 eV! in a collision with thedetector surface. The beam of metastable argon atemerges into the high vacuum chamber, with an averagelocity of 700 ms21, and a thermal velocity distribution o60% ~full width at half maximum!. The beam is collimatedby a set of two slits~first slit 10 mm, second slit 5mm,distance 1.4 m! to a divergence of less than 5mrad, i.e.,considerably better than the diffraction angle of 2uB

'36 mrad.After collimation the atoms enter a 5-cm-long interacti

region with a standing light field, set up by retroreflecting tcentral part of a collimated, expanded laser beam at a m~surface flatness,50 nm!, located in the vacuum beamlineThe frequency of the diode laser is actively locked at an oatomic transition of metastable argon~at 801.7 nm! usingsaturation spectroscopy. The frequency can be shifted bwell-defined offset using an acousto-optic frequency shifBoth the intensity and frequency of the laser can be molated with rates up to 200 kHz, using an acousto-opswitch, or by directly modulating the diode laser currerespectively. Details of the light crystal generation varythe different parts of the experiment, and are explainedlow.

The standing light field represents a light crystal withgenerally complex index of refraction for the atoms. Ththe atomic beam is diffracted if its incidence angle is a Braangle, but not affected at arbitrary incidence. The crysangle can be controlled by tilting the retroreflection mirrwith a piezo actuator.

In the far field behind the crystal, at a distance of 1.4the diffraction pattern can be spatially resolved by scanninthird 10-mm slit in front of the large area ‘‘channeltron’detector. However, in most of the experiments the detecslit is just located at the position of a Bragg diffraction peaand the diffraction efficiency is then measured as a funcof the mirror angle~‘‘rocking curves’’!. Note that even in thecase of modulated light crystals, the detection slit positremains on the same position, since both the direction ofincident beam and the total diffraction angle are always cstant.

B. Interaction of metastable argon atoms with light fields

Obviously, the whole experiment depends on the intertion between the argon atoms and the light field. The cosponding theory is well established, and thus we pointonly features important for our actual experiments.

A light field represents a positive, a negative, or evenimaginary potential for an atom, depending on the detunof the light frequency from an atomic transition line. Inteaction of atoms with detuned light gives rise to the weknown dipole potential. The spectral shape of the potentiaa typical dispersion curve, as displayed in Fig. 3. Its specdependence is given by@30#

Vreal~v!5\VRabi

2

4

v2v0

~v2v0!21~g/2!2. ~18!

02360

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VRabi5DE/\ is the Rabi frequency of the populatiooscillations between the two levels with a transition dipomomentD, driven by the electrical strength of the light fielE at resonance;v0 is the center frequency of the transitioline, andg is the effective line width of the transition.

From quantum mechanics it is known that an exterpotential changes the phase velocity of a wave function. Dto the fact that such a spectrally dependent phase changeas a spectral filter on matter waves, it has to satisfy cercausality requirements defined by the Kramers-Kronig retions. They demand that a dispersion shaped spectral dedence of a real potential has to be accompanied by a Loreshaped imaginary contribution:

Vimag~v!52 i\VRabi

2

4

g/2

~v2v0!21~g/2!2. ~19!

Such an imaginary potential corresponds to an absorpcoefficient with a Lorentzian spectral profile. In fact, suchimaginary contribution is contained in the interaction btween atoms and light fields, since resonant excitation ofatom consequently leads to spontaneous emission of aton, which kicks the atom out of its path—i.e., the atomeffectively removed from its original state~similarly, absorp-tion of a light beam in a material can be due to diffuscattering!. However, since in a two-level atom at opticfrequencies the deflection due to diffuse scattering of a pton is smaller than the detector angular resolution, thisflection is typically not resolved in Bragg diffraction experments. The situation changes in our three-level argon atFor realizing detectable ‘‘absorption,’’ the most importafeature of the metastable 1s5 atoms is theiropen transition1s5→2p8 (4s@3/2#2→4p@5/2#2) at 801.7 nm~the effectiveline width is '9 MHz!. If the atoms are excited at thi

FIG. 3. Complex potential of a light field for an atom nearopen atomic transition line. The real part corresponds to the dippotential, with the spectral shape of a dispersion line centered atransition frequency. The~negative! imaginary part corresponds t‘‘absorption’’ of the atoms, realized by pumping to the~undetected!ground state. Its spectral shape corresponds to a Lorentzian. Bethe complex phase angle is drawn as a function of the lightquency. It has the shape of an arctangent function with an offse2p/2.

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wavelength, they decay spontaneously with a branchingtion of 72% to the ground state which is not detected bychanneltron, i.e., the atoms are absorbed with respect tomeasurement.

Both real and imaginary parts of the potential can be cobined in a resulting complex potential:

V~v!5\VRabi

2 /4

v2v01 ig/2. ~20!

Figure 3~A! shows the completecomplexpotential of a lightfield for an atom near an absorption line, split into tLorentzian imaginary part and the dispersion profile ofreal part.

The arctangent-shaped complex phase angle of the potial is plotted below as a function of the light frequency. Thspectral shape of the phase angle is important, since it dmines the phase of the diffracted atoms. This is due tofact that, in the case of a complex potential, the Pendello¨sunglength becomes complex@Eq. ~13!#, and the complex potential phase then directly determines the phase of the scattwave according to Eq.~14!. This behavior was experimentally verified earlier@15#, and investigated in detail in Re@16#. Interesting theoretical investigations of matter-waBragg scattering at complex potentials were presentedRefs. @37,38#. The most important aspect for our actual eperiments is the fact that the phase of the scattered wavebe adjusted in a range from2p to 0 by changing the lightfrequency from the far red detuned side to the far bluetuned side of an atomic transition.

C. Diffraction at modulated real light potentials

The basis of our investigations is the fact that a modulalight crystal produces new Bragg peaks, and that the atomthese new peaks are coherently shifted by the modulafrequencyvM to higher or lower matter-wave frequencieThe coherence of the frequency shifting process was alredemonstrated in a previous publication@24#. Here we dem-onstrate quantitative investigations about the angular dedence, peak width, and velocity selectivity of the diffractiprocess, in order to verify our Ewald-type diffraction modWe also investigate a special case where Bragg diffractioa modulated crystal occurs at a perpendicular incidencethe atoms, a situation which can be never achieved in sdard static Bragg scattering. This special situation mihave practical applications as a sensitive method for mspectroscopy.

1. Intensity-modulated light crystal

In the first experiment combining spatial and tempoBragg diffraction, an atomic beam was scattered at a licrystal which was periodically switched on and off. Athough this situation was already investigated in Re@23,24#, here we briefly present the basic effects. Typiresults of such an experiment are shown in Fig. 4.

In that figure we compare the efficiencies of first-ordBragg diffraction as a function of the atomic incidence anfor two cases of an unmodulated light crystal~upper graph!,

02360

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and a periodically intensity modulated light crystal~lowergraph!. The light frequency was detuned far enough ('10linewidths! from the open transition at 801.7 nm such ththe potential could be assumed to be purely real. The expment was performed by locating the detection slit at thesition of the first diffraction order, and then detecting tnumber of diffracted atoms as a function of the retrorefltion mirror angle. In the case of an unmodulated crystalobtained only one peak of diffracted atoms at the staBragg angle. This is a typical result demonstrating thatmeasurements were performed in the Bragg diffractiongime ~note that the result of such a measurement in the cof diffraction at a thin grating would consist in an almosconstant efficiency, i.e., without any angular dependence!.

In the next experiment~lower graph!, the light crystal wasswitched on and off periodically with a modulation frequency ofvM52p360 kHz. As a result we found two newpeaks of Bragg scattered atoms, arranged symmetricaround the static Bragg peak. In a previous publication it wshown that these peaks consist of atoms whose matter-wfrequency iscoherentlyshifted by the modulation frequenc@24#. This was demonstrated by interferometric superpositof the diffracted frequency shifted matter wave with ttransmitted unshifted wave, resulting in traveling interfeence fringes, which were detected. The sign of the frequeshift depends on the side on which the new peaks ariserespect to the central Bragg peak. Frequency-upshifted atappear at positions closer to perpendicular incidence,vice versa. According to Eq.~9! the angular separationDufrom the static Bragg angle is proportional to the modulat

FIG. 4. Generation of new Bragg peaks at a modulated crysIn the upper graph, the efficiency of first-order Bragg diffractionan unmodulatedlight crystal is plotted as a function of the atomincidence angle~‘‘rocking curve,’’ see inset!, i.e., the angle of theretroreflection mirror~the solid line is a Gaussian fit to the data!.Only one peak of diffracted atoms appears at the Bragg anglethe lower graph the measurement is repeated, using an intenmodulated light crystal, which is switched on and off periodicawith a frequency ofvM52p360 kHz. Two new peaks arise at neincidence angles. They are arranged symmetrically around the sBragg peak at the center. Note the slight difference in the widththe new peaks.

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frequency (Du/uB56vM /vRec, where vRec5\uGW u2/2m52p330 kHz!. This relation is verified in a sequenceexperiments where the modulation frequency was changea range from 25 to 200 kHz, and ‘‘rocking curves’’ werecorded in the same way as before. In Fig. 5 the positionthe new Bragg peaks is plotted as a function of the modtion frequency.

The graph clearly demonstrates the linear behavior ofangular offset of the new Bragg peaks from the static Braangle. The absolute values of the corresponding slopestained from the data (s50.6160.01 mrad kHz21) agreewith the expected value @Eq. ~9!: s5uB /vRec50.60 mrad kHz21] within the experimental resolution~in-serting the experimentally measured Bragg angle of 18mrad,which corresponds to an average atomic velocity of 6ms21).

As an additional test of our model, we examined twidths of the new Bragg peaks at different modulation fquencies. From our previous theoretical considerations,expected that the peak width increases with increasmodulation frequency, due to the broad longitudinal velocdistribution of the atoms in the beam. The reason for thismost easily seen from the alternative Bragg conditionmodulated crystals, formulated in Eq.~11!. This condition isimposed on the two new Bragg peaks, frequency shifted6vM for first-order spatial diffraction (nG51), and requresvWob5vRec6vM , respectively. Since both the intensimodulation frequencyvM and the recoil frequencyvRec areconstants, the only dispersive parameter depending onatomic velocity and on the incidence angle is the ‘‘wobling’’ frequency vWob with which the atoms rattle acrosthe grating planes, given byvWob52puvA

W usin(u)/d, whereu

is the incidence angle of the atoms, anduvAW u is the modulus

of the atomic velocity. A first-order Taylor expansion theshows, that the wobbling frequency has a certain wiDvWob, depending on the width of the velocity distributioDuvAu:

FIG. 5. Offset of the two new Bragg peaks~see Fig. 4, lowergraph! from the static Bragg angle as a function of the intensmodulation frequency. The linear behavior expected from Eq.~9! isclearly demonstrated.

02360

in

of-

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0

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yisr

y

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h

DvWob52pDuvAu

dsin~u!. ~21!

This relation shows that the width of the wobbling frquency, and correspondingly the size of the angular indence range, around which the alternative Bragg condi@Eq. ~11!# can be fulfilled, depends on both the corresponing incidence angleu, where the new Bragg peaks are cetered, and on the broad velocity distributionDuvAu('350-ms21 full width at half maximum! of the atomicbeam. With increasing incidence angle, and thus withcreasing modulation frequency, the dependence from thelocity distribution becomes larger, increasing the widththe new Bragg peaks in the ‘‘rocking curves.’’ The samrelation also reveals that the two peaks of frequency-shiatoms appearing in each rocking curve~like in Fig. 4! shouldhave different widths. Note that this asymmetry betweenwidths of the left and right peaks is already visible in tdata of Fig. 4. The reason for this is that the two new Brapeaks arise at different absolute incidence angles@see Eq.~9!# u615uB(16vM /vRec). Therefore, the width of thewobbling frequency distribution, and accordingly the angurangeDu in which Eq.~11! is fulfilled, becomes different forthe left and right peaks, respectively:

Du61'd

2puvAuDvWob5

DuvAuuvAu

uB~16vM /vRec!.

~22!

In fact, the widths of the peaks are not symmetric.special case with the most pronounced asymmetry istained if the modulation frequency corresponds to the refrequency. In this case the angular width of one of the tpeaks should not be influenced by the velocity distributionall (12vM /vRec50), whereas the other peak should shoa doubled velocity dependence (11vM /vRec52) with re-spect to the static peak width obtained by settingvM50 inEq. ~22!.

In Fig. 6 the width of the peaks in the rocking curvesplotted as a function of the incidence angle at which the npeaks are centered. The graph agrees with the behaviopected from our model@Eq. ~22!#. The individual data pointsfor the left~negative angles! and right peaks~positive angles!are taken at different modulation frequencies~the same dataset as used for Fig. 5!; however, the width is not plotted asfunction of the modulation frequency but as a function of tcorresponding incidence angles. First, the data show thapeak width increases with increasing incidence angle~modu-lation frequency!, in accordance with Eq.~21!. Second, thegraph shows that a minimum of the width is obtainedperpendicular incidence of the atoms, and not at Bragg indence~the two linear lines are drawn to indicate the centersymmetry!. Thus the width of the peaks does not depedirectly on their angular distance to the static Bragg pebut rather on the absolute incidence angle. In the next stion, we pay some special attention to the extreme cwhere Bragg diffraction occurs at perpendicular incidenc

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2. Bragg diffraction at perpendicular incidence

In ‘‘normal’’ static experiments the Bragg condition canever be fulfilled at perpendicular incidence of a wave acrystal. The situation changes, however, in the case otime-modulated potential. In order to be diffracted the atohave to ‘‘feel’’ a potential modulation in their rest framewhich is equal to the two-photon recoil frequency@Eq. ~11!#.This modulation can be produced either by the atomictling across the grating planes (vWob), or by an externallyapplied intensity modulation (vM), or by a beating of both(vWob6vM). This viewpoint suggests that at perpendicuincidence, where the incident atoms move parallel tograting planes and therefore the ‘‘rattling’’ frequencyvWobvanishes, the necessary modulation must be delivered byexternally applied intensity modulation, i.e.,vM5vRec.Due to symmetry considerations, no diffraction direction cthen be preferred and scattering is expected to occurequal intensities at the two conjugate first diffraction ordeThus the diffraction pattern resembles one obtained at agrating, since the typical asymmetry of the Bragg diffractipatterns is repealed~typically only one peak of diffractedatoms is expected, as, e.g., shown in Fig. 2!.

Since the recoil frequency is equal for all atoms, indepdent of their velocity, and since additionally the wobblinfrequency vanishes for all atoms, there should be no veloselectivity of the Bragg diffraction process@this is also sug-

FIG. 6. Angular width of the new Bragg peaks as a functionthe incidence angles belonging to different modulation frequen~same data set as in Fig. 5!. Negative and positive angles correspond to the left and right peaks of frequency-downshifted-upshifted atoms, respectively. With increasing modulation fquency, and correspondingly growing incidence angle, the pwidth enlarges, due to the increased dependence on the broadgitudinal velocity distribution of the atomic beam. The plot alshows that the broadening of upshifted and downshifted diffracpeaks is almost symmetric with respect to theperpendicular inci-dence direction~indicated in the plot!, and not with respect toBragg incidence~zero angular offset!. In particular, the minimalpeak width is not obtained for static diffraction~i.e., for atomswhich are incident at the static Bragg angle!, but for atoms whichare frequency upshifted by 30 kHz, corresponding to exactly ppendicular incidence.

02360

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gested by Eq.~22! setting 12vM /vRec50#. Thus, differentdiffraction efficiencies can only be due to different interation times with the light crystal.

Experimental investigations of Bragg diffraction at pependicular incidence are plotted in Fig. 7. In the experimwe recorded far-field diffraction patterns for various atomvelocities by scanning the detection slit. The velocities wselected using a time-of-flight method, where the gas dcharge of the atomic source was pulsed~pulse width 0.5 ms!and the atomic intensities were recorded as a functiontheir arrival time at the detector, at a distance of'2.9 mfrom the source. In the first experiment@Fig. 7, graph~a!# thecrystal was adjusted for perpendicular incidence ofatomic beam, and the light crystal was unmodulated. Aspected for static Bragg diffraction,~almost! no scattered at-oms were registered. Some remaining scattering is due tolimited angular collimation of our beam, and to the limiteBragg selectivity of our crystal. Both restrictions becommore important for faster atoms, and therefore some hig‘‘background’’ is measured there. In Fig. 7~b! the same ex-periment was performed, though now we modulated thetensity of our light crystal with the recoil frequencyvRec

52p330 kHz. As expected from our model, the data sha symmetric diffraction of atoms at the two conjugated fidiffraction orders. The matter-wave frequency of the atoin both new Bragg peaks is shifted by the modulation fquency in the same direction, that is to higher absolute vues. The diffraction efficiency shows only a small depedence on the atomic velocity. This becomes especievident when compared to a more general situation of Brscattering at modulated crystals, as presented in Fig. 7~c!.There we chose an ‘‘arbitrary’’ modulation frequency of 7kHz, and adjusted the corresponding new incidence anglefrequency-downshifted atoms, that is'60 mrad. The corre-sponding diffraction patterns show the typical featureBragg diffraction, i.e., there appears only one peak of dfracted atoms. Additionally, the diffraction efficiency depends strongly on the atomic velocity. A comparison of tdiffraction efficiencies as a function of the atomic velocitifor perpendicular incidence, for Bragg incidence, and forcidence at'60 mrad is presented in Fig. 8.

In this figure the difference in the velocity dispersion~elastic and inelastic! diffraction processes at increasing asolute incidence angles becomes obvious. In particular,minimal velocity selectivity of Bragg diffraction at a modulation frequency of 30 kHz, and the increased velocity dpendence of elastic scattering and of scattering at mhigher modulation rates~0 kHz and 75 kHz in the figure! isin agreement with Eq.~22!.

The independence of the diffraction condition at perpedicular incidence,vM5vRec52h/ml2, from parameterslike atomic velocity, light intensity, and interaction potentiaalso suggests an interesting application. In fact, the modtion frequencyvM at which the new Bragg peaks appedepends only on the atomic mass, and on the light walength which can be measured very accurately, and whcan be chosen arbitrarily in a broad range in the vicinity otransition line~real potential!, as long as a small interactio

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FIG. 7. Bragg diffraction at perpendicular incidence: The graphs show far-field diffraction patterns~recorded by scanning the detectioslit! for atoms with different velocities~time of flight through our beam line!. ~A! is taken for an unmodulated light crystal at perpendicuincidence of the atoms. Only the peak of transmitted atoms in the center of each scan is obtained, with almost no diffracted atomsBragg condition can never be fulfilled at perpendicular incidence. In~B!, the light crystal was modulated with the atomic recoil frequenof 30 kHz, again at perpendicular incidence. Now atoms are diffracted symmetrically to both sides of the central peak of transmitteAlthough the experiment was performed in the Bragg regime, the results are similar to diffraction patterns of a thin grating,diffraction efficiency shows only a small dependence on the atomic velocity. In~C! a more ‘‘general’’ situation of scattering at modulatecrystals is investigated for comparison. There, a modulation frequency of 75 kHz was applied, and the atomic incidence angle wato one of the two new corresponding Bragg angles at'60 mrad. Now, as expected for the Bragg case, the diffraction patterns showone peak of scattered atoms, and the diffraction efficiency depends strongly on the atomic velocity.

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potential between the atoms and the light exists. This mimply a practical application as a highly sensitive methodmass spectroscopy, e.g., in the case of a molecular beammeasuring the intensity modulation frequency at which dfraction of a collimated, perpendicularly incoming beamobserved. The accuracy of the method can be significaimproved by increasing the interaction time between the pticles and the light crystal, using a slower beam or a thiccrystal. For larger molecules, where the modulation fquency would be very low, the resolution of the method cbe improved by observing diffraction into higher spatBragg ordersnG.1 ~i.e., larger deflection angles!, whichrequires a correspondingly higher modulation frequennGvRec.

The basic experiments demonstrated so far are the fodation for the advanced investigations, which will be psented in the next section. Mainly, they confirm our modand the usefulness of the alternative formulation of the Brcondition in Eq.~11! for the diffraction of frequency-shiftedatoms. This frequency shifting can be interpreted as a sband generation of the matter waves. However, in contrastypical situations of sideband generation~e.g., in the radio-frequency technique! in our case the diffracted beams geneally contain only one frequency component at a time. Tmeans that a modulated Bragg crystal cannot only be useshift the frequency of diffracted atoms, but also to indicatheir frequency shift by the corresponding new Bra

02360

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angles.2 This feature will now be applied for investigatinthe sideband structure generated by unconventional~evennon-Hermitian! potential modulations.

D. Diffraction at complex modulated light potentials

In the preceding section we demonstrated the basic efof atomic matter-wave frequency shifting by diffraction atintensity-modulated light crystal. We now extend our invetigations to complex temporal modulation schemes whboth real and imaginary parts of the potential, or its amptude and complex phase, are manipulated independeThere we use the fact that by adjusting the laser frequewe can change our potential within a range of positive anegative real values, to imaginary values. This will be usto Fourier synthesize specific temporal modulation functioof the potential. According to our model@Eq. ~17!#, the Fou-

2In contrast, in a thin grating regime the effect of frequency shing ~or sideband generation! would also be possible; however, theall sidebands would appear simultaneously in the diffracted beaSuch an experiment in the Raman-Nath regime would lose theture to indicate the sideband generation, e.g., a spatially resofar-field diffraction picture would not reveal the sideband structuHowever, in our case of a combined spatial and temporal Brregime our modulated crystal acts simultaneously as both a sband generator and an analyzer.

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rier coefficients should directly determine the ability of omodulated crystal to create sidebands of the corresponfrequencies. For the sideband detection we use the factaccording to Eq.~9!, the appearance of a new Bragg angautomatically indicates the corresponding frequency offsethe diffracted atoms.

1. Complex light potential with temporal helicity

In this experiment we exploit the fact that it is possibletailor potentials by superimposing light from different laseWe start from the same setup as in the previous experim~Fig. 2!; though now we use two diode lasers to createlight crystal @25#. The light of the two lasers is first collinearly superposed at a beam-splitter cube. Then thecombined beams enter the same collimation optics as beThis results in two exactly coinciding light crystals generain front of the retroreflection mirror.3 However, the intensi-ties of the two lasers can now be modulated independe

3The light crystals can be regarded as exactly coinciding atposition where the atomic beam passes. The reason is that the mmal experimentally used frequency difference is on the order ofMHz, resulting in a spatial beating period of 1.5 m, whereasdistance of the atomic beam to the mirror surface is only 2 mm

FIG. 8. Velocity selectivity of Bragg diffraction at intensitmodulated light crystals. The efficiencies of first-order Bragg dfraction are plotted as a function of the atomic velocity for thrdifferent modulation frequencies~0, 30 kHz, and 75 kHz!. In onecase the modulation frequency was equal to the recoil frequency~30kHz!, and the corresponding incidence angle of the atoms waspendicular to the light crystal. In the other case an ‘‘arbitrarmodulation frequency of 75 kHz was chosen, and the corresponnew Bragg angle of scattering of frequency-downshifted atoms60 mrad was adjusted. The middle curve shows the diffractionficiency of elastically scattered atoms, incident at the static Brangle. Obviously, the data taken at a modulation frequency okHz shows the smallest dependence on the atomic velocity.residual velocity dependence is due to the increasing interactime, but not to a velocity selectivity of the diffraction process.contrast, the data taken at 0- and 75-kHz modulation frequenshow a larger dependence on the velocity, due to the growingfluence of the longitudinal velocity distribution at higher absoluincidence angles.

02360

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using for each laser a separate acousto-optic modul~AOM! for periodically switching their intensity. In the experiments we used the same modulation frequencies forlasers, though with an adjustable temporal phase relatThis could be achieved by using two function generatwith one common time basis to drive the two AOM’s.

Using different frequency detunings of the two light crytals from resonance, and additionally, using different relatmodulation phases, we can generate a potential of the fo

V~x,t !5„11cos~Gx!…@VL1„11cos~vMt !…

1VL2„11cos~vMt1f t!…#. ~23!

The first term of the equation describes a fully modulaspatial grating, which is formed in front of the mirror suface, whereas the second term describes the applied temmodulation. There it is assumed that the temporal intenmodulation of the two lasers is harmonical and fully modlated. The independent variables which can be controlledour experiment are the individual crystal potentialsVL1 andVL2, and the relative temporal phasef t between the twoharmonic potential modulations, which can be adjustedrectly at the frequency generators driving the AOM’s. Tcrystal potentials can be chosen real~positive and negative!or imaginary~only negative! by adjusting the light frequencies. The absolute values of the potentials can be controby changing the corresponding light intensities.

In the experiment the absolute values of the two potentwhere always equalized by adjusting the light intensity oftwo individual crystals independently for equal relative dfraction efficiencies, i.e.,uVL1u5uVL2u. Therefore, we can se

VL1ªV0 ,

~24!

VL2ªV0 exp~ ifc!.

V0 is the potential of the first laser, andfc is the complexpotential phase difference depending on the difference oftwo laser detunings from resonance~see Fig. 3!. In order tocalculate the Bragg diffraction efficiencies for first-ordelastic Bragg scattering, and sideband production of posior negative order~corresponding to positive or negative frequency shift!, a two-dimensional~temporal and spatial! Fou-rier analysis of the total potential has to be performed,cording to Eq.~16!. The Fourier coefficientsVnG51,nM50 and

VnG51,nM561, then determine the first-order diffracted wavamplitudesfor elastic and inelastic scattering, respectiveThe corresponding diffraction efficiencies are then proptional to the squares of these Fourier coefficients, accordto Eq. ~17! ~valid for small efficiencies!. Fortunately, such atwo-dimensional Fourier transform is easily performed in tcase of harmonic potential modulations. Since spatially atemporally dependent functions are just multiplied in E~17!, the time-dependent and spatial Fourier transformsbe separated. The two-dimensional transform is then justproduct of the two individual ones. The spatial part of E~17! describes a normal potential grating with well-knowBragg diffraction properties, i.e., it enables diffraction at t

exi-0e

-

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than

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positive or negative first diffraction orders. In our caseare only interested in the sideband generation efficienciethe first spatial diffraction order~as detected in our ‘‘rockingcurves’’!, such that it is sufficient to analyze only the temporal part of the modulation. This can be done easilysubstituting all cosine functions according to cosa)5„exp(ia)1exp(2ia)…/2. Evaluation of the products ansorting the terms then yields

V~x,t !5V0„11cos~G•x!…$„11exp~ ifc!…

1 12 @11exp„i ~fc1f t!…exp~ ivMt !#

1 12 @11exp„i ~fc2f t!…exp~2 ivMt !#%. ~25!

This indicates that we have a ‘‘normal’’ static Bragg graing ~first term! of modulusuV0„11exp(ifc)…u, and two time-modulated components with absolute values ofuV0@11exp„i (fc1f t)…#u/2, anduV0@11exp„i (fc2f t)…#u/2, pro-ducing frequency-downshifted and -upshifted sidebands,spectively. In a typical ‘‘rocking curve’’ as in Fig. 4, thsquared modulus of these three components determineintensities of the center peak, and the negative and possideband peaks, respectively:

P1,0}uV0@11exp~ ifc!#u2,~26!

P1,61}uV0@11exp„i ~fc6f t!…#/2u2.

Here it should be mentioned that for technical reasonsmodulations actually applied were not harmonic, but rectgular functions~the lasers were switched on and off!. Thisresults in additional Fourier components of higher ordHowever, the first of these components is already the thharmonic of the modulation frequency, leading to new pein the rocking curves which are far separated from the invtigated peaks of first-order sidebands, and thus do not disthe measurements.

In order to verify the predictions of Eq.~26! for the dif-fraction efficiencies, we chose different frequency detunin(fc) and modulation phases (f t) of the two superimposedlight crystals, both intensity modulated with a frequencyvM52p3100 kHz. Results of these experiments are plotin Fig. 9.

This figure shows ‘‘rocking curves,’’ i.e., the efficiencieof first-order Bragg diffraction, as a function of the incidenangle~angle of the retroreflection mirror!, the same kind ofmeasurement as performed in the previous experiment~seeFig. 4!. The center of each plot corresponds to the staBragg angle, i.e., the center peak always consists of elcally scattered atoms. As discussed above, side peaks aleft or right side of the center are due to atoms whichcreased or decreased their kinetic energy by one modulaquantum\vM in the course of the diffraction process, rspectively. The inset of each plot indicates the appliedtential modulations of the two superimposed light crystaAs mentioned above, the absolute values of the two cosponding light potentials were equalized by adjusting thesuperimposed crystals individually for equal relative diffration efficiencies of about 20%.

02360

of

y

e-

theve

e-

r.ds

s-rb

s

fd

icti-the-on

-.e-o-

In Fig. 9~a!, the two lasers were both far detuned~tenlinewidths! on the red side of the resonance line (V0 is nega-tive real, andfc50), and they were switched on and osimultaneously (f t50). The situation thus correspondseffect to one single laser which is modulated. Thereforeobtain a result which is identical to the result of our ‘‘basicsideband modulation experiment, sketched in the lowgraph of Fig. 4, i.e., one central peak of elastically scatteatoms, and two side peaks of frequency-upshifted a-downshifted sidebands, respectively. The relative intensiof the four peaks should be 1:4:1, according to Eq.~26!.However, it has to be mentioned that these quantitative emates are only valid for small~relative! diffraction efficien-cies. In the case of higher efficiencies, the sideband peaksexpected to gain efficiency with respect to the peak of etically scattered atoms.

In Fig. 9~b!, we performed a second control experime

FIG. 9. Generation of atomic matter-wave sidebands with dferent potential modulation schemes. The figure shows the fiorder diffraction efficiencies as a function of the crystal ang~‘‘rocking curves’’!, for different temporal modulation functionsThe time dependence of the two overlapping light crystal potentare indicated in the insets. In~G! and~H!, temporal potential modu-lations with a ‘‘helicity in complex space’’ result in completelasymmetric sideband generation.

6-13

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hth

est

e

thothts

th

acti

stallys

ystivuaa

erm

e

allysasbo

.

thinaca

n

o

twoak.

theion

lexera-

st

nsity

nessedperi-

de-s theeec-the

bled

iesonrs isandion

areac-

therim-


by keeping the two lasers far red detuned (V0 is negativereal, andfc50, as before!, but by switching them on and ofalternately (f t5p), i.e., one laser was switched off whethe other was switched on. In effect, this yields a constared detuned light field. Therefore, such an experiment cosponds to Bragg diffraction at an unmodulated static cryswhich yields only one peak of elastically scattered atomsthe Bragg angle, similar to the upper graph in Fig. 4. Tsuppression off the two side peaks is in agreement, withpredictions of Eq.~26!.

In Fig. 9~c!, the two lasers were detuned on different sidof the resonance line, i.e., the corresponding two light crypotentials were positive real and negative real (V0 is nega-tive real, andfc5p), respectively. The two lasers wermodulated with equal phase (f t50). As a result, scatteringis completely suppressed. The reason for this is thatnegative potential due to the red detuned laser and the ptive potential due to the blue detuned laser cancel each oat any time. Therefore, no resulting crystal potential exisand no scattering can be observed.

An interesting situation is sketched in Fig. 9~d!, where thedifferent detunings of the two lasers were the same as inlast experiment (V0 is negative real, andfc5p), but nowthe lasers were switched again alternately (f t5p). Obvi-ously, this results in a suppression of the static Bragg pewhereas the sideband efficiencies are equal to the efficienin Fig. 9~a!. The reason for the suppression of the staBragg peak is that, during its passage through the cryeach atom ‘‘sees’’ the crystal potential switch periodicabetween positive and negative values~each atom experienceabout six switching periods inside the crystal!. Therefore, onthe time average, no crystal results, and no atoms canelastically scattered. On the other hand, there exists a crpotential which is modulated between positive and negavalues, and which is thus able to generate sidebands. Qtitatively, Eq.~26! predicts the same sideband efficienciesin Fig. 9~a!, in agreement with our experimental result.

In the next parts of the experiment, we created a gencomplex potential modulation. This was achieved by cobining a negativeimaginary crystal potential~generated bytuning one of the lasers exactly on resonance! with a nega-tive real potential~obtained by tuning the other laser on thfar red side of the resonance frequency, i.e.,V0 is negativereal, andfc5p/2). The data of these experiments appemore noisy, since the total diffraction efficiency is drasticareduced with respect to the previous situations. The reafor this is that only about 10% of the incident atoms can pthe absorptive crystal. From these transmitted atoms, a25% are diffracted at the first order~static diffraction!, re-sulting in an absolute diffraction efficiency of only 2.5%Furthermore, some of the metastable atoms ('28%), whichshould be absorbed after excitation by decaying toground state, rather decay spontaneously to their origmetastable state, which contributes to some additional bground. Nevertheless, the expected diffraction effectsclearly visible.

In Figs. 9~e! and in 9~f!, the two lasers were switched oand off simultaneously@in Fig. 9~e!, f t50#, and alternately@in Fig. 9~f!, f t5p#, respectively. The results of the tw

02360

t,e-l,t

ee

sal

esi-er,

e

k,iescl,

betalen-

s

al-

r

onsut

ealk-re

experiments are approximately the same: They showsymmetric side peaks in addition to the central Bragg peThe symmetric sideband generation in Figs. 9~e! and 9~f! ispredicted by our model@Eq. ~26!#, as well as the symmetrybetween the two situations. The reason for this is thatsum and the difference of the relative temporal modulatphasesf t , and the complex phasesfc , in both situationsyields only the values2p/2 or 1p/2, which leads to nodifference in the absolute values of theP1,61 terms in Eq.~26!. However, these are the only situations where a comppotential modulation leads to a symmetric sideband gention.

The sideband symmetry is maximally broken in the latwo experiments@Figs. 9~g! and 9~h!#, where the laser fre-quencies were the same as before, but the temporal intemodulation phases were chosen to be2p/2 and1p/2, re-spectively. Obviously, this results in a suppression of osideband and an enhancement of the other. The suppreand enhanced sidebands are exchanged in the two exments. This is due to the fact that both the temporal phasef tand the complex phasefc take the magnitudep/2. However,whereas the temporal phase enters into Eq.~26! with differ-ent signs for frequency-upshifted and -downshifted sibands, the sign of the complex phase stays constant. Thutwo phases add top or cancel each other to 0 for thfrequency-upshifted and -downshifted sidebands, resptively, resulting in a suppression or an enhancement ofrespective peaks. According to Eq.~26! the efficiencies ofthe enhanced sideband should be approximately douwith respect to the results of Figs. 9~e! and 9~f!.

A more detailed investigation of the sideband efficiencas a function of the relative temporal intensity modulatiphasef t between the red detuned and the resonant lasepresented in Fig. 10. The data indicate that the sidebefficiency depends sinusoidally on the relative modulatphase. Frequency-upshifted and -downshifted sidebandscreated alternately, i.e., a suppression of one sideband is

FIG. 10. Sideband generation efficiency as a function ofrelative phase between the intensity modulations of two supeposed real and imaginary light crystal potentials.

6-14

sean

ion

ttio-inin

aseesnstioth

petie

enoncofi

ssun.vele.y

t

c

r tin

roanutin

of

enth atemant ofwein

nlynt,ofa

ses.ure

sthe

totheen-lat-im-welex-

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encenc-t-

mine.avebeer.tinghem-f the

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companied by a maximal efficiency of the other. All theobservations confirm our model of sideband generation,are in agreement with Eq.~26!.

The extreme situations occur at relative modulatphases of6p/2, as already shown in Figs. 9~G! and 9~H!.Investigating these situations in more detail, one finds thathese situations the temporal part of the potential modulais of the form V(t)}exp(6ivMt). Such an extremely nonHermitian potential modulation represents a kind of helixcomplex space, i.e., the modulus of the potential remaconstant, whereas the complex phase rotates linearlyfunction of time. The orientation of the helix is given by thsign of the exponent. Thus our experimental result suggthat such a temporally helical potential is only able to trafer energy in one direction, which depends on the orientaof the helix. Detailed analysis shows that a modulation ofform exp(2ivMt) is able to emit energy in quanta of\vM ,whereas an opposite orientation of the helix enables thetential to absorb energy quanta. Thus the normal symmbetween absorption and stimulated emission probabilitiesduced by time-varying fields is broken. The symmetry btween absorption and stimulated emission is normally duthe fact that any real potential is Hermitian, which meathat conjugate Fourier coefficients are pairs of complex cjugates, i.e., any contribution with negative helicity is acompanied by another contribution with positive helicitythe same intensity. This symmetry can only be brokenopen systems, interacting with an environment which acta reservoir, since only in these cases the consideredsystem can be described by a non-Hermitian Hamiltonia

Interestingly, an analogous effect was recently obserin the spatial domain@17#. In a similar setup a light crystapotential with a complex helicity in space was tailored, i.V(x)}exp(iGx). Although periodic, such a potential is verdifferent from a real crystal [email protected].,V(x)}cos(Gx)#,since the amplitude of the potential is constant at any poinspace, though the complex phase changes. It has beenserved that such a potential with complex helicity in spatransfers a grating momentum\G only in one direction, i.e.,the spatial diffraction pattern becomes asymmetric, similaour actual experiment where a complex potential helicitytime transfers a modulation quantum\vM only in one direc-tion. Note that this analogy is not trivial, since the Sch¨-dinger equation is not symmetric with respect to spacetime coordinates.4 Together, the two results clearly point othe role of non-Hermitian potential modulations, both

4One consequence of this is that it is always possible to chathe direction of the atomic momentum without changing the ene~elastic scattering!, but it is impossible to change the energy withoadjusting the momentum. Therefore, in order to produce aquency sideband by a temporal potential modulation, there alwhas to be an additional mechanism that provides the related motum transfer. In our experiment we obtained this by combintemporal and spatial potential modulations, where the temporalcreated the energy shift, and the spatial part supplied the remomentum change through its grating vector.

02360

d

inn

sa

ts-ne

o-ryn--tos-

-

nasb-

d

,

inob-e

o

d

space and time, which give rise to asymmetric transfersmomentum or energy, respectively.

The asymmetric energy transfer in our actual experimsuggests an interesting practical application: Using sucmodulation scheme it might be possible to cool a subsysat the cost of its environment, which in our case would mea cooling of the atoms in the metastable state at the costhe atomic population in the ground state. However, herehave to mention that up to now this cannot be achievedour system, since our potential modulation does not oconsist of the desired complex helix, but also of a constanegative imaginary term causing continuous absorptionthe atoms. A working scheme for cooling thus will needrepumping transition, i.e., a resonance line which cau‘‘negative absorption’’~that is gain! at resonant excitationIn such a system, it is formally possible to generate a pcomplex helical potential acting on one of the levels. Atomin this level can then be cooled at the cost of the atoms inother level, which acts as a reservoir.

2. Frequency-modulated complex light potential

In the following, we demonstrate a different approachcontrol the energy transfer to the diffracted atoms. Inpreceding section, we showed how to control the time depdence of the complex potential by independently manipuing real and imaginary parts of the potential using superposed on-resonant and off-resonant light fields. Nowshow that a complementary method to manipulate a compfunction, i.e., by directly controlling its magnitude and complex phase, can also be realized, and yields results that aagreement with our model. In the following, we directmodulate the phase of the diffracted wave using thequency dependence of the scattering phase, rather thanrier synthesizing the frequency spectrum with differentsers, as before.

As already mentioned, the phase of the complex potendetermines the phase of the scattered matter waves. Thisbe seen directly using the potential dependence of thefracted waveamplitudegiven by Eq.~14! ~valid for smalldiffraction efficiencies!, allowing for a complex Pendello¨-sung length given by Eq.~13!. The spectral shape of thcomplex phase of a light potential near an atomic resonaline is plotted in Fig. 3. It corresponds to an arctangent fution, with an offset of2p/2. Thus a phase shift of the scatered matter wave in a range of2p to 0 can be achieved byjust changing the light frequency in an interval ranging frothe far red to the far blue detuned side of the absorption lThus we can achieve a phase modulation of the matter wby modulating the light frequency, which can technicallyachieved easily by modulating the current of a diode lasNevertheless, it has to be considered that, just by modulathe laser frequency without adjusting the light intensity, tamplitude of the light potential is also modulated. The coplete spectral dependence of the amplitude and phase olight potential is best expressed by reformulating Eq.~20!:

V~v!52 i\VRabi

2 /4

A~v2v0!21~g/2!2expF iarctanS v2v0

g/2 D G .~27!

gey

-ysn-

rted

6-15

thuc

lenn

laoic

ctas

et

v

uancaptiohe

xa

net

du-eom

rier

frea-a

freas

-

erch

bynaltheana-

ped

ftedncee ofan-

ab-thelgnq.entme

ithictualwasanso-of athency

dif-the

cynt

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rier


Here the first factor denotes the real amplitude, andsecond factor the complex phase of the potential. The eqtion shows that a general modulation of the laser frequenv(t), yields a rather complicated modulation of the comppotential. Nevertheless, a controlled phase modulation caachieved by simultaneously controlling the frequency aintensity of the laser light. One interesting kind of modution is suggested by the arctangent dependence of the cplex potential phase from the light frequency. If a periodfrequency modulation of the form

v~ t !5v01g

2tan~vMt ! ~28!

is applied, then this compensates for the arctangent speshape of the potential phase, and results in a linear phshift as a function of time. Such a linearly changing phashift just corresponds to a frequency offset ofvM , which isadded to the matter-wave frequency. Such frequency gention, obtained by adding a linear phase shift, acts similarlya Doppler frequency shift, obtained by reflecting a wafrom a continuously moving object which also produceslinearly changing phase offset. The difference in our sittion is that the phase shift cannot remain linear forever, sithe tangent function is periodic. Therefore, the actual shof the potential phase obtained by the frequency modulaof Eq. ~28! becomes a sawtooth function of time, i.e., tphase changes linearly in an interval between2p and 0, andthen jumpsback to its starting point. This corresponds eactly to the jump which the laser frequency has to performthe poles of the tangent function in order to jump from oend of the resonance line to the other. The complete effecthis kind of frequency modulation can be seen if the molation function@Eq. ~28!# is inserted into the complex potential @Eq. ~27!#. After some trigonometric manipulations wthen obtain, for the time dependence of the modulated cplex potential,

V~ t !52 i\VRabi

2

4gcos~vMt !exp~ ivMt !

52 i\VRabi

2

4g„11exp~ i2vMt !…. ~29!

This representation corresponds directly to a Foutransform of the temporal potential modulation, i.e., theexist a static Fourier component and a component withquency 2vM , with equal intensities. Therefore, this modultion is supposed to create both elastically scattered atomsa single frequency sideband offset by22vM . Note that thesideband frequency offset deviates from the modulationquencyvM . This is due to the phase jump after each phchange ofp in the tangent function~which performs twooscillations in an interval of 2p). Therefore, the actual frequency of the modulation is in fact 2vM , although the argu-ment of the tangent function is onlyvMt.

In order to experimentally investigate the effects of lasfrequency modulations, we used the same setup as sketin Fig. 2, with one single diode laser~without any acousto-

02360

ea-y,xbed-m-

ralsee

ra-oea-een

-t

of-

-

re-

nd

-e

-ed

optic modulator!. The laser frequency could be modulateddriving the diode laser current with a programmable siggenerator. The frequency offset obtained by changinglaser current was measured with a Fabry-Perot spectrumlyzer.

In the experiment we demonstrate that a tangent-shafrequency modulation function according to Eq.~28! pro-duces only one single sideband of either frequency-upshior frequency-downshifted atoms. This is done in a sequeof experiments where we successively increased the slopthe tangent function by increasing the amplitude of the tgent shaped frequency modulation~Fig. 11!. The results arecompared with numerical calculations, where the squaredsolute values of the respective Fourier coefficients ofpotential modulation~convoluted with a phenomenologicaline-shape function! were computed. Note that the precedinanalytic calculation holds only for a modulation functiowhich is exactly the inverse function of the argument in E~27!. Since we were now using different slopes of the tangfunction, the calculation of the Fourier composition becamore complicated and is done numerically.

In the experiment the laser frequency was modulated wa frequencyvM550 kHz across the 801.7-nm open atomtransition line. Nevertheless, as mentioned above, the acfrequency of the laser sweep across the resonance line100 kHz, since the tangent function is already periodic ininterval of p. The laser frequency was centered at the renance line and then periodically modulated in the shapesmooth tangent function from the red to the blue side ofresonance, followed by a sudden jump back. The frequeinterval in which the laser was swept was about65 line-widths across the resonance frequency. The first-orderfraction efficiency was measured again as a function ofcrystal angle~‘‘rocking curves’’!. The upper graph@Fig.

FIG. 11. Experimental realization of a lopsided frequenshifter using a frequency-modulated light crystal. In the differeplots, a tangent-shaped frequency modulation function (vM

52p350 kHz! according to Eq.~28! was applied, with differentslopes of 0, 3.5 MHz, 7 MHz, and 10 MHz, respectively. The slopcorrespond to the prefactorg/2 in Eq. ~28!. The experimental re-sults are reproduced by the numerical calculations of the Foucomposition~squared absolute values! of the potential modulations~right side of the graph!.

6-16

tet

lly

siddu

ift

oexEic

r-

roa

aan

ti

cathtia

lt

heaather

ecbe

ndwssetab

id

ch

roa

na

r,al

aser-e-ltinge-othu-

tterthe

ofelyofele-in

if-nd-der-toter-ich

dicslyef-ortin

lcet is

at-a

by

o

ul-

nect

pa-as-tinger,factfre-

gfne


11~a!# shows a rocking curve obtained at an unmodulalight crystal. The peak of elastically scattered atoms denothe position of the Bragg angle. In Figs. 11~b!, 11~c!, and11~d!, the light frequency was modulated symmetricaacross the atomic resonance frequency according to Eq.~28!with increasing amplitude of the tangent function.

In any case, new Bragg peaks are found only on oneof the static Bragg peak, indicating that the potential molation can produce only lop-sided energy shifts. In Fig. 11~c!the peaks of elastically scattered atoms and frequency shatoms have approximately equal intensities, as predictedEq. ~29! for a slope of the tangent function ofg/2'4.5 MHz~the theoretical linewidth of the 801.7-nm transition isg'9 MHz!, which should result in a sawtooth phase shiftthe complex potential phase as a function of time. Aspected from the Fourier representation of the potential in~29!, we obtain only one side peak of atoms shifted by twthe modulation frequency (2350 kHz!. However, our actu-ally applied slope in this experiment was 7 MHz. The diffeence might be due to a change in theeffectivelinewidth bysaturation effects when scanning the laser frequency acresonance, or to an increased linewidth of the laser generby its fast modulation.

In the other parts of the experiment it is shown thsmaller slopes of the modulation result in smaller sidebefficiencies@Fig. 11~b!#, whereas larger slopes@Fig. 11~d!#create increased sidebands, while simultaneously populahigher sideband orders~a small additional side peak!. Theexperimental results are in agreement with the numericalculations plotted on the right side, which representsquared value of the Fourier composition of the potenmodulation functions@there an effective linewidthg514MHz is assumed, as suggested by the symmetric resuFig. 11~c!#.

We also checked the effect of flipping the sign of ttangent function, which resulted, as expected, in a sidebpeak with a different sign of the frequency offset, i.e., a peat the other side of the central Bragg peak. This meansscanning across an atomic transition line from the red sidthe blue side yields different results as compared to theverse direction. This symmetry violation is due to the sptral shape of the complex interaction potential determinedthe Kramers-Kronig dispersion relations, and thus fundamtally based on causality.

Note that although the resulting asymmetric sidebalook similar to those in the previous experiment whereused independent modulations of real and imaginary partthe potential, the principle is different. Actually, the pharelation between the generated single sideband and the sBragg peak also differs in the two situations, which mightdemonstrated in an interferometric experiment.

The demonstrated method to produce an asymmetric sband has an analogy in the spatial domain@26#. The methodresembles the principle of a blazed grating in space. Sublazed grating has a sawtooth spatial profile. This profileembossed on an incident plane wave front, i.e. the wave facquires a sawtooth phase shift. In the ideal case, the spphase changes in a range of 2p, and then jumps back to 0. Ithis case the blazed grating has the property to diffract

02360

des

e-

edby

f-

q.e

ssted

td

ng

l-el

in

ndkattoe--yn-

seof

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aisnttial

n

incoming wave exclusively at one spatial diffraction ordebut not at the conjugate order. This is in contrast to a normphase grating, consisting of a harmonic periodic phmodulation, which diffracts symmetrically at conjugate oders. Very similarly, in our experiment we imprinted a timdependent phase shift at an incoming matter wave, resuin frequency sideband production. By applying a timdependent periodic phase shift with the shape of a sawtofunction, which we obtain as a result of our tangent modlation, we observe only one sideband of the atomic mawave. Thus our experiment transfers in a certain senseprinciple of a spatially blazed grating to the time domainatomic matter waves. Although the analogy is not completperfect,5 this viewpoint helps to understand the principleoperation, and suggests building time analogs to otherments of standard diffractive optics, like a ‘‘Fresnel lenstime.’’

3. Bloch band spectroscopy

Here we briefly sketch an alternative model for the dfraction behavior of the argon atoms in the modulated staing light field, which opens up a different viewpoint anpoints out the relation of our experiments to recently pformed investigations of atoms in optical traps. In additionthe extended Ewald model, our experiments can be inpreted as a kind of spectroscopy of the Bloch bands whdescribe the dynamic behavior of a particle in a periopotential. Recently, this viewpoint was used advantageouto model test systems for demonstrating some quantumfects predicted in solid-state physics for electronic transpin periodic crystals, which were hard to demonstratecondensed-matter systems@5,4,7#. In our case, the modeshows that our complex potential modulations can indudirected transitions between energy levels—an effect tha‘‘normally’’ forbidden with ‘‘ordinary’’ ~Hermitian! poten-tials, due to basic concepts of quantum mechanics.

A one-dimensional band-structure description of theoms in a crystal is sketched in Fig. 12. Bloch bands inweakly modulated spatially periodic system are formedrepeating the free-particle dispersion parabola~indicated by adashed line in the figure!, which is normally centered at zermomentum, at distances given by the grating vectorGW , sincein periodic systems all momenta are only defined up to mtiples of the fundamental grating momentum\GW . The influ-ence of quantum mechanics is, then, to split and recon

5Our lopsided frequency shifter is not a perfect analog to a stially blazed grating, since it still produces a residual peak of eltically scattered atoms. This would correspond to a blazed grain space, which diffracts only 50% of the incoming wave. Howevthe deviation from the ideal behavior can be attributed to thethat we can change the temporal matter-wave phase by ourquency modulation only in an interval ofp instead of the ideallyrequired interval of 2p. Similarly, a spatial sawtooth phase gratinwhich changes the phase of a wave front only in an interval opwould have the property to diffract only 50% of the wave at oorder.

6-17

he

boathti

e

-thrithla

theeeeh

oenelta

twingrn-ctwiobo

-hurear

ic-asd

al-agga-

use-

two-e-it is

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the individual parabolas at their crossing points, such tbands are formed. However, for weak modulation the undlying structure of periodically repeated free-particle paralas is still clearly expressed, and, actually only broken inarea closely surrounding the crossing points. In Fig. 12corresponding atomic eigenenergies are plotted as a funcof the atomic wave-vector componentkAi parallel to the grat-ing vector GW . Therefore, in our case of small incidencangles~paraxial geometry!, thex axis is directly proportionalto the atomic incidence angle (kAi'ukWAuu). The dashed parabola indicates the dispersion of a free atom outside ofcrystal. At the crystal boundary the originally free atoms acoupled into the particular crystal bands which coincide wtheir free~kinetic! energy parabola. Since this free paraboat most positions directly overlaps with the crystal bands,resulting population of the bands is obvious: e.g., the lowBloch band is populated for atomic incidence angles betw0 ~perpendicular incidence! anduB , the next band in a rangbetweenuB and 2uB , and so on. At incidence angles whicare far enough separated from the~prevented! crossingpoints, the corresponding eigenfunctions simply consist~almost! free plane waves that correspond to the incidwave. Only in cases where the free atomic parabola crossband gap is it possible to populate two Bloch bands simuneously, for example at positiona in Fig. 12. The two cor-responding eigenfunctions at these positions consist ofsinusoidally modulated wave functions. Their correspondintensity distributions form two copies of the crystal gratinwith a relative phase shift ofp with respect to each othe@16#, i.e., one of the atomic intensity gratings directly coicides with the crystal potential, whereas the other is exaout of phase. In these special cases, the energy gap betthe two populated bands leads to a different time evolutof the two corresponding atomic wave functions. Recomnation of these two wave functions at the rear boundarythe crystal then results intwo outgoing propagation directions, i.e. a transmitted and a Bragg-diffracted beam. Tthe ~prevented! crossing points of adjacent parabola corspond to the Bragg angles. These crossing points appethe positions~see Fig. 12!: kAi5nG/2, resulting in the stan-dard Bragg conditionuB'nG/2kA .

FIG. 12. Band structure of atomic matter waves in a odimensional light crystal. Transitions within the band structure~ar-rows b andc) are induced by a resonant perturbation, and resunew Bragg peaks of frequency-shifted atoms.

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So far the preceding illustrations were a qualitative pture of the predictions of dynamical diffraction theory,originally developed for diffraction of x rays, electrons, anneutrons at solid crystals@27#. Applied to our situation ofscattering at a modulated crystal, this model suggests anternative explanation for the appearance of the new Brpeaks in the ‘‘rocking curves.’’ There the potential modultion is assumed to be a resonant perturbation.6 Even atoms,which enter the crystal far from a Bragg angle, and thoccupy only one crystal band, can be excited by this timdependent resonant perturbation into a superposition ofbands~e.g., arrowsb or c in Fig. 12!, and successively scattered. This results in a Bragg diffraction of coherently frquency shifted matter waves. From the band symmetriesobvious that resonant transitions to higher~e.g., arrow b! andlower bands ~e.g., arrow c! are arranged symmetricallaround the static Bragg angles~e.g., position a!, which ex-plains the symmetric distribution of the new Bragg peaksthe rocking curves of Fig. 4. Calculating the vertical dtances between two bands in a region far from a bandcan be done by just determining the energy differencetween the corresponding two horizontally displaced freparticle parabolas~since these parabolas are only disturbedthe region very close to a static Bragg angle, i.e., at a bgap!. This straightforward calculation delivers the samesult for the modulation frequencies belonging to the nBragg angles, as our generalized Ewald model.

The above viewpoint also indicates that our experimeare a direct spectroscopic method to measure the Bloch bstructure of the light crystal. In contrast to previously dscribed methods of Bloch band spectroscopy in atomic tr@6#, the well-collimated beam of incoming atoms is generacoupled only into one single band~with the exception of thestatic Bragg angles!, and even this band is populated onlya specific momentum position, which can be controlledrectly by adjusting the incidence angle. Starting from thwell-defined position the temporal modulation induces arect resonant transition to an upper or lower band, whichindicated by the appearance of a new Bragg peak at thespective side of the static peak. Thus scanning the mod

6It might be striking that a fully modulated crystal potentialtreated here as a perturbation. However, as explained abovemain structure of the crystal bands is just given by repeating frparticle parabola at distances of a grating vectorG. Thus the mostimportant property is only determined by the periodicity of the crytal, but not by the absolute potential strength~as long as an adiabatic evolution of the atomic wave function in a band is possib!.The only property influenced by the potential strength is the spting ~and the eigenfunctions! in the region very close to the bandgap positions~in the case of small enough potentials, which is atomatically fulfilled if the crystal shows Bragg diffraction peakwith a high angular selectivity!. However, in our modulation ex-periments we are mainly interested in atoms far from the band gi.e., at positions where the originally free atomic wave functionalmost uninfluenced by the crystal in any case. There, in fact,crystal intensity modulation acts only as a small perturbation, eif the crystal potential is modulated with an amplitude of 100%.

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tion frequency or the incidence angle measures the distabetween Bloch bands with high accuracy~note that the typi-cal transition energy\vM is in our case about ten ordersmagnitude lower than the kinetic energy of the atoms!.

This viewpoint also implies that the asymmetric sidebageneration demonstrated in Sec. II is in fact a lopsided trsition between two crystal bands. The creation of only osideband in such an energy-level scheme means thatone of the indicated transitions~e.g., arrow b! can be inducedby the potential modulation, whereas the same modulacannot drive the corresponding symmetric transition~arrowc!, although the modulation frequency is resonant with btransitions. The reason for this behavior is the temporallicity of the complex potential modulation. As mentioneabove, such an absorption without stimulated emission~orvice versa! within the band structure of an optical latticsuggests applications like a cooling of the atoms toground state.

IV. OUTLOOK

We have demonstrated that sidebands of atomic mawaves can be produced by diffraction at intensity-frequency-modulated light crystals, either by directly modlating real and imaginary parts of the light potential ormodulating its amplitude and complex phase. The effectbe characterized as a coupled spatial and temporal Bdiffraction process. Specific non-Hermitian potential modlations can have the shape of a time-dependent complexlix. These modulations transfer energy only in one directii.e., either to lower or higher values, and thus violateusual symmetry between absorption and stimulated emisof energy quanta. This is an interesting analog to a recedemonstrated asymmetricmomentumtransfer which occurswhen atoms are diffracted at complex potentials withspatialhelicity @17#.

The investigated temporal diffraction effects imply mainteresting applications. For example, the asymmetric enetransfer produced by non-Hermitian potential modulatiomight be used for cooling atoms to the ground state ofoptical lattice, or to gain complete control over the atompopulation of the energy bands. In this case it is necessachose an atomic system which allows for a repumping ofground-state atoms to the metastable state.

Furthermore, the sideband modulation techniques nowable many applications, which have been advantageoused in light optics and in radio-frequency technology, totransferred into an analogous matter-wave technology. C

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trolled modulation, or frequency shift of matter waves canused in atomic interferometry, in order to build active MacZehnder-type atom interferometers@18,19#, where the coher-ent wave splitters and recombiners consist of modulated lgratings. These are promising tools for high-precision rotion or gravitation sensors@39#. For example, using the described modulation techniques, it is possible to impresprecisely controlled de Broglie frequency difference at tatoms in the two paths of an atom interferometer. Thislows highly increased sensitivity by lock-in detection of thoscillating interference fringes, or by actively tracking thinterference phase with a changing frequency differencemore advanced experiments this might also enable the inferometric detection of mass- or velocity-dependent dispsion phenomena, which are expected in certain atomic stering and collision processes, in magnetic fields, orgravitation physics.

Another application in sophisticated measurement tenology is the use of modulated standing light fields for pcise mass spectroscopy of atomic or molecular beams. Ta particle beam crosses a thick modulated standing lwave at exactly perpendicular incidence. By scanningintensity modulation frequency of the light crystal, diffration peaks will appear only at specific modulation frequecies ~the two-photon recoil frequency!, which depend onlyon the particle mass and the precisely measurable lightquency, but not at hardly controllable parameters likeparticle velocities. Using an intense laser the optical fquency can be selected over a wide range around atommolecular transition lines, since the refractive potential oflight field falls off only slowly with increasing frequencydifference from resonance.

In fundamental research, the control over the sidebstructure of atoms can be used to investigate the quanmechanics of atomic wave packets, an exciting field of qutum optics. Advanced technologies in laser optics, like chiing and compression of laser pulses, which are importtechniques for achieving ultrashort pulses, might alsoemulated advantageously with modulated matter wavThese experiments have now become feasible, since deerate quantum gases of atoms are readily obtainable asherent sources of atomic matter waves.

ACKNOWLEDGMENTS

This work was supported by the Austrian Science Fodation ~FWF!, Project No. S6504, and by European UnioTMR Grant No. ERB FMRX-CT96-0002.

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