de Broglie wavefront engineering
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PHYSICAL REVIEW A, VOLUME 62, 033612de Broglie wavefront 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 900890484~Received 15 March 2000; published 18 August 2000!
We propose a simple method for the deterministic generation of anarbitrary continuous quantum state ofthe centerofmass of an atom. The methods spatial resolution gradually increases with the interaction timewith no apparent fundamental limitations. Such de Broglie wavefront 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
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disatEngineering 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 oftrappedionbased quantum computer, and atom lithograInitial theoretical suggestions@14# for the preparation of aprechosenquantum state of a cavity field were based onsocalled 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 twolevel 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 allowonetoone 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 wavefront engineering scheme was inspired byprecision position measurement technique suggested anperimentally realized by Thomas@11#. Let us consider a twolevel 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.edu10502947/2000/62~3!/033612~4!/$15.00 62 0336sames
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\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:
i\]
]t S c2c1D 5S p22M 1\v2,1~z! i V~ t !2 i V* ~ t !
p2
2M
D S c2c1D , ~2!wherep52 i\(]/]z) is the atomic momentum, andV(t) isthe timedependent 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 detunedStarkshifting 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 simplest ~but as we will see below not the optimal! way to ap2000 The American Physical Society121
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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 stateu2& 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
~5!
~the socalled 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 minimum positionmomentum 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 initioquantummechanical simulation!. The desired pattern correspondto a dfunctional peak centered atz50. The dashed line is a resuof an application of a monochromatic field at resonance atz50.Solid line shows the de Broglie wavefront engineering resuwhere the frequency was chirped according to classical ans~7!, ~8! taken atp050. Dasheddotted 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 wavefront 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 quantummechanical 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 stateu2& 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 !
M2
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
dz~T!52\
FT ~10!
monotonically decreases with the interaction time. Note tthe sinc shape is ultimately the best approximation fodfunctional peakd(z2z!) one can create using a@p0 ;p01FT# window in momentum space.
Preparation of an arbitrary quantum state: de Broglwavefront engineering.The ansatz~7!, ~8! motivates ourstrategy for de Broglie wavefront 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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de BROGLIE WAVEFRONT ENGINEERING PHYSICAL REVIEW A62 033612*dzufu251). Let us represent the target state as a cotinuous superposition of thed peaks:
f~z!5E2`
1`
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
f~p!5E2`
1`
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
v5V0FT
2p\f~0!~16!
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 preparation 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
2p\Ep0p01FT
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 overall timeindependent amplitude, a product of two distintimedependent factors. The first one is not specific fo03361
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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 distribution 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 wavefunction 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 wavefronengineering 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 onedimensionaltime dependence of the coupling field to the onedimensiotarget 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 magneticfieinduced~alternatively the Stark shift in a spatially varyinlaser field induced@11#! gradient force F/\52p3109t.
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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 between 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...