de Broglie wave-front engineering

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  • PHYSICAL REVIEW A, VOLUME 62, 033612de Broglie wave-front engineering

    M. Olshanii,1,2,* N. Dekker,1 C. Herzog,1 and M. Prentiss11Lyman Laboratory, Harvard University, Cambridge, Massachusetts 02138

    2Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089-0484~Received 15 March 2000; published 18 August 2000!

    We propose a simple method for the deterministic generation of anarbitrary continuous quantum state ofthe center-of-mass of an atom. The methods spatial resolution gradually increases with the interaction timewith no apparent fundamental limitations. Such de Broglie wave-front engineering of the atomic density canfind applications in Atom Lithography, and we discuss possible implementations of our scheme in atomic beamexperiments.

    PACS number~s!: 03.75.Be, 42.82.Crdictuataph
















    disat-Engineering of quantum states has been a widelycussed topic for the last decade. Apart from a purely ademic interest, there exist numerous applications of quanstate engineering including preparation of nonclassical stof a cavity electromagnetic field, programming oftrapped-ion-based quantum computer, and atom lithograInitial theoretical suggestions@14# for the preparation of aprechosenquantum state of a cavity field were based onso-called conditional measurement method, where the tastate is reached after a successful sequence of quanmeasurements, while the unsuccessful measuremevents are discarded. In the schemes found in Refs.@7,8,10#,applicable to both cavity light and external motion oftrapped ion, a two-level atom, coupled to the quantum fiof interest as well as to a controllable external laser ligplays a role of a bus, which transfers, in a prescribed wapopulation and coherence between the discrete eigenstatthe quantum field. Similar ideas were used to generatearbitrary internal state of a multilevel atom@9#. According toRefs.@5,6#, the adiabatic population transfer process allowone-to-one mapping between a quantum state of a Zeemultiplet and a cavity field.

    The quantum state engineering methods listed abovewith systems of a discrete spectrum. In our paper we suga simple method to create an arbitrarycontinuousmotionalstate of a free atom starting from a plane wave as an incondition. The role of a bus, transferring the coherencetween the initial and target states, is played by an externuniform force field: the target motional state is encodedthe time dependence of the amplitude of applied laser lig

    The general idea of the setup for the realization of ourBroglie wave-front engineering scheme was inspired byprecision position measurement technique suggested anperimentally realized by Thomas@11#. Let us consider a two-level atom, interacting with a magnetic fieldH(z)52azwhose amplitude varies linearly in space. Suppose thatinternal atomic stateu1& does not interact with the magnetfield ~the corresponding Lande factor equals zero:g150),whereas the Lande factor for the stateu2& has a finite valueg25g. The energy difference between the statesu2& andu1&will, thus, depend linearly on the position of the atom:

    *Email address: olshanii@usc.edu1050-2947/2000/62~3!/033612~4!/$15.00 62 0336s-a-mes



    dt,,s ofan






    \v2,1~z!5\v2,1~0!2Fz, ~1!

    where\v2,1(0)5E22E1 is the energy difference betweeu2& and u1& in the absence of the magnetic field, and tgradient force acting on atom in stateu2& is F5amBohrg. Inwhat follows we will assume, without loss of generality, thboth a andg are positive numbers.

    To describe atomic motion in a superposition of the manetic and laser fields we will use the following timedependent Schrodinger equation:


    ]t S c2c1D 5S p22M 1\v2,1~z! i V~ t !2 i V* ~ t !



    D S c2c1D , ~2!wherep52 i\(]/]z) is the atomic momentum, andV(t) isthe time-dependent Rabi coupling strength. We will furthsuppose that initially all the atoms are in the internal stateu1&and have a momentump0:

    S c2c1D t505S0

    Ar ineip0z/\1 ip02T/2M\D , ~3!

    wherer in is the initial spatial density of atoms,T is the timewhen the distribution is going to be detected, and the iniphase of theu1& state is chosen in such a way that this phabecomes zero at timeT. Note that the equation of motion~2!does not contain any spontaneous emission terms: inimplementation we discuss in the conclusion bothu1& andu2& states are supposed to be ground or metastable atstates coupled by a Raman laser field. As a source forenergy shift~1! either a real magnetic field or far detunedStark-shifting field can be considered.

    Creation of a narrow peak at a desired position: Fixefrequency technique.Suppose for a moment that our goalto create the narrowest possible position distribution ofoms in the stateu2& centered at a positionz5z!, and that theinitial condition corresponds to a stateu1& atom of somemomentump0 ~whose value we can adjust at will!. The sim-plest ~but as we will see below not the optimal! way to ap-2000 The American Physical Society12-1

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    M. OLSHANII, N. DEKKER, C. HERZOG, AND M. PRENTISS PHYSICAL REVIEW A62 033612proach the above goal is to apply a monochromatic spatiuniform laser field of a frequencyv5v2,1(z


    V~ t !5V e2 iv2,1(z!)t ~4!

    for a period of timeT ~solid vertical arrow, inset for the Fig1!. Indeed at the end of the interaction (t5T) a peak of state-u2& atoms centered at the resonant pointz5z! will becreated. In Ref.@11# it is shown though, that for a givenvalue of the forceF the spatial width of this peak is limitedfrom below by a value

    d5S \22MFD1/3


    ~the so-called diffraction limit! no matter how long the interaction timeT is. This limit is reached at a time of the order

    t5S 2\MF 2 D1/3

    . ~6!

    At times shorter than the timet, a wave packet of a mini-mum position-momentum uncertainty relation (dzdp;\) iscreated. The external forceF broadens the momentum distribution according todp;FT, and, therefore, the spatiawidth of the stateu2& distribution decreases with time adz;\/FT. For long interaction times though, the peak widstarts increasing quadratically as the interaction time

    FIG. 1. Spatial distribution of atoms in the stateu2& ~ab initioquantum-mechanical simulation!. The desired pattern correspondto a d-functional peak centered atz50. The dashed line is a resuof an application of a monochromatic field at resonance atz50.Solid line shows the de Broglie wave-front engineering resuwhere the frequency was chirped according to classical ans~7!, ~8! taken atp050. Dashed-dotted line shows the initial distrbution of the stateu1& atoms. Interaction time isT54.8t, fieldamplitude isV51.0(\/t). The characteristic lengthd and timet ofthe problem are defined by the expressionsd5(\2/2MF)1/3 andt5(2\M /F 2)1/3 respectively. The inset illustrates the basic ideathe de Broglie wave-front engineering method: the trajectorythe resonant point is designed in such a way that stateu2& atomscreated at different stages of the process come to the target asame timet5T.03361ly


    creases~see for example the spatial distribution calculatfor the caseT54.8t shown at the Fig. 1!. Such a broadeningis caused by both the quantum-mechanical diffraction ofwave packet being prepared and the acceleration of the wpacket. In what follows we will show that it is possible tsuppress this broadeningusing a simple modification of thetime dependence of the laser field amplitude.

    Creation of a narrow peak at a desired position: Optimstrategy.Let us write the field amplitudeV(t) in the form

    V~ t !5V e2 i *Tt v2,1(zs(t8,p0 ,z

    !)) dt82 ip0z!/\, ~7!

    where the trajectory of the resonant pointzs(t,p0 ,z!)

    should be optimized in such a way that at the final timeT thestateu2& atoms will form a narrow peak, centered atz5z!.@As we will see below the overall time independent phasethe field~7! is chosen in such a way that at the timet5T theresulting state-u2& wave function will be real at the targetpoint z5z!.#

    Notice now that at a given timet the field~7! plays a roleof a localized at the pointz5zs(t8,p0 ,z

    !) source of atoms ininternal stateu2& and with momentump0. Consider then aclassical analog of our problem: Find a trajectoryzs(t,p0 ,z

    !)of a classical source of atoms of an initial momentump0,such that all the atoms emitted will reach the targetz5z! at a preselected timeT. Atoms are supposed to be afected by a forceF. Such a trajectory does exist: it is giveby

    zs~ t,p0 ,z!!5z!2

    p0~T2t !


    F~T2t !2

    2M. ~8!

    Let us now insert the ansatz~8! to the expression for the fieldamplitude~7! and evaluate the equations of motion~2! usingthis amplitude. To the first order in the field strengthV thestateu2& component of the atomic wave function at timeT~solid line at Fig. 1! will be given by

    @c2~z!# t5T5S VT\ DAr inei [( p01FT/2)(z2z!)/\]3sinc F z2z!dz~T!G1O~V3!, ~9!

    where the spatial width of the distribution


    FT ~10!

    monotonically decreases with the interaction time. Note tthe sinc shape is ultimately the best approximation fod-functional peakd(z2z!) one can create using a@p0 ;p01FT# window in momentum space.

    Preparation of an arbitrary quantum state: de Broglwave-front engineering.The ansatz~7!, ~8! motivates ourstrategy for de Broglie wave-front engineering of motional quantum states. Imagine that ones goal is to prepan atom in a motional statef(z) ~normalized to unity:




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    n of

    de BROGLIE WAVE-FRONT ENGINEERING PHYSICAL REVIEW A62 033612*dzufu251). Let us represent the target state as a cotinuous superposition of thed peaks:



    dz!d~z2dz!!f~z!!. ~11!

    The state engineering process will involve then the followsteps.

    ~1! Prepare the atom in the internal stateu1& and in anexternal state corresponding to thep5p0 eigenstate of theatomic momentum

    @c1~z!# t505Ar ine1 ip0z/\, ~12!

    where the initial momentump0 is chosen in such a way thathe momentum window@p0 ;p01FT# covers entirely themomentum distribution



    dz e2 ipz/\f~z! ~13!

    of the target state,F and T being the typical magneticfield gradient and typical interaction time available in tgiven implementation;

    ~2! Apply for a timeT a laser field

    V~ t !5E dz!v e2 i *Tt v2,1[zs(t8,p0 ,z!)]dt82 ip0z!/\f~z!!~14!

    5 v e2 i *Tt v2,1[zs(t8,p0 ,0)]dt8f@p~ t,p0!#. ~15!

    @see Eqs.~14!, ~7!, and~11!#, where



    is the field amplitude adjusted to the target statp(tcr ,p0)5p01F(T2tcr) is the momentum a stateu2& atomcreated at a timet5tcr acquires by the end of the preparatioproceduret5T, and the source trajectoryzs(t,p0 ,z

    !) isgiven by the expression~8!,

    ~3! At the time T measure the internal state. If the atoremains in the stateu1&, take another atom and repeat tabove steps. If atom is detected in the stateu2&, the prepara-tion procedure is complete. After a lengthy but straightfward calculation one can show that in the course ofpreparation procedure the initial state gets transformedstate

    @c2~z!# t5T51


    dp f~p!e1 ipz/\1O~V02!'f~z!~17!

    close to the target statef, which was, we recall, assumeto be localized in momentum space within a@p0 ;p01FT#interval. This is the central result of our paper.

    Notice that the field amplitude~15! is, apart from an over-all time-independent amplitude, a product of two distintime-dependent factors. The first one is not specific fo03361-



    particular target, but only for a given momentum window@p0 ;p01FT#. This factor can be set once and for all forbroad class of targets. We recall that the purpose offactor is tosuppress the acceleration broadeningof the dis-tribution being generated. The second factor correspondsfinite duration pulse whose amplitude is proportional to ttarget wave functionf(p) in the momentum representationit allows encoding of the target wave-function in thespectrum of the applied laser field.

    Note also that the overall field strength constantV0 isdefined in such a way that prior to the internal state measment step~step 3! the density of theu2& fraction of theatomic distribution is given by uc2(z)u2

    5r in(V0T/\)2uf(z)u2/uf(0)u2.

    In Fig. 2 we show the result of our attempt to reproduthe boa swallowed an elephant pattern@12# using the deBroglie engineering technique.

    Summary and applications.To conclude, we have presented a method for generation of anarbitrary motionalquantum statef of a free atom. Our de Broglie wave-fronengineering method allows one to modify a plane waveatoms in an internal stateu1& and external statep5p0 to astate

    u1&eip0z/\u2&f~z!, ~18!

    wheref can beany state such that its momentum represetation f(p) is localized within an intervalp0

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    M. OLSHANII, N. DEKKER, C. HERZOG, AND M. PRENTISS PHYSICAL REVIEW A62 033612So far we do not see any simple way to generalizestate engineering scheme to more than one dimension.~1D!Whereas a 1D1D mapping from a one-dimensionaltime dependence of the coupling field to the one-dimensiotarget wave function@V(t)f(z)# turned out to be a relatively simple problem, an analogous 1D2D(3D) mappingof a form V(t)f(x,y) @V(t)f(x,y,z)# seems to bemuch harder to design.

    Note that our method allows a modification, suitablelithography with atomic beams. A simplest~but not the onlypossible! lithographic version of the de Broglie engineering technique would involve a replacement of the time dpendent phase factor of a type~7! by a suitably chosen gradient of the magnetic fieldalong the direction of the atomicbeam @in addition to thetransversegradient ~1! describedearlier#. Also in order to ensure a continuous interaction btween the laser field and atoms in the beam the laser p~15! should be replaced by a periodic sequence of pulsethe same shape. Such a lithographic scheme will providmassivelyparallel deposition of anarbitrary atomic pattern.Furthermore, the spatial resolution of the suggested meis not restricted by the diffraction limit d suggestedin Ref. @11#.

    Let us give some realistic estimates for the spatial restion one can achieve using the above lithographic methwe will use Fig. 1 as an example. Recall that this plot shothe narrowest peak, which could be obtained for given valof the field gradientF and interaction timeT. For the ArgonmassM530 amu and a realistic value of the magnetic-fieinduced~alternatively the Stark shift in a spatially varyinlaser field induced@11#! gradient force F/\52p3109t.












    Hz/cm (2p31012 Hz/cm! the natural units of length andtime will be given byd5(\2/2MF)1/35108.0 nm~10.8 nm!andt5(2\M /F 2)1/3514.7ms (0.147ms!, respectively. TheHWHM of the peak shown in Fig. 1 will correspond then63.0 nm ~6.3 nm! obtained for an interaction time ofT54.8t570.8 ms ~0.708 ms!. Our scheme can be implemented, for example, using as12s2 Raman coupling be-tween themJ50 ~as a stateu1&) andmJ512 ~as a stateu2&)Zeeman sublevels of the 4s@3/2#2 metastable level in argonThemJ52 atoms can be then quenched to the...