Physical Review A. Atomic, Molecular, and Optical Physics ...565610/FULLTEXT01.pdf · In this paper we present an analysis of how matter waves, guided as propagating modes in potential

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Citation for the original published paper (version of record):

Jääskeläinen, M., Stenholm, S. (2003)

Localization in splitting of matter waves.

Physical Review A. Atomic, Molecular, and Optical Physics, 68(3): 033607

http://dx.doi.org/10.1103/PhysRevA.68.033607

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PHYSICAL REVIEW A 68, 033607 ~2003!

Localization in splitting of matter waves

Markku Jaaskelainen* and Stig StenholmRoyal Institute of Technology, Roslagstullsbacken 21, SE-10691 Stockholm, Sweden

~Received 20 March 2003; published 19 September 2003!

In this paper we present an analysis of how matter waves, guided as propagating modes in potentialstructures, are split under adiabatic conditions. The description is formulated in terms of localized statesobtained through a unitary transformation acting on the mode functions. The mathematical framework resultsin coupled propagation equations that are decoupled in the asymptotic regions as well before as after the split.The resulting states have the advantage of describing propagation in situations, for instance matter-waveinterferometers, where local perturbations make the transverse modes of the guiding potential unsuitable as abasis. The different regimes of validity of adiabatic propagation schemes based on localized versus delocalizedbasis states are also outlined. Nontrivial dynamics for superposition states propagating through split potentialstructures is investigated through numerical simulations. For superposition states the influence of longitudinalwave-packet extension on the localization is investigated and shown to be accurately described in quantitativeterms using the adiabatic formulations presented here.

DOI: 10.1103/PhysRevA.68.033607 PACS number~s!: 03.75.Be, 03.65.Ge, 05.60.Gg, 03.65.Nk

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I. INTRODUCTION

During the past few years, manipulation of confined mter waves has experienced a remarkable progress, both dthe development of laser cooling and trapping of neutraloms@1,2# and progress in lithographical fabrication of nanstructures for ballistic electrons@3#. In the channeling regimeof these experiments, the dynamics will be propagationmatter waves in, possibly multiple, tightly confined waguides.

Several techniques have been utilized@4# for guiding neu-tral atoms, among them magnetic confinement both inshollow glass tubes@5# and above current carrying wires@6#,permanent micromagnets@7#, light force trapping@8#, withmicrofabricated optics@9# and various other schemes bason the interaction of electromagnetic radiation with atomsRef. @10# the splitting of a thermal atomic cloud above curent carrying wires of millimeter size was demonstrated ain Ref. @11# a similar experiment was performed using mcrofabricated conductors. The creation of Bose-Einstein cdensates~BECs! in microfabricated structures was reportalmost simultaneously by two groups@12,13# and has sincebeen reported by several groups@14,15#. In addition, coher-ent transfer of a BEC into waveguide structures has baccomplished@16,17#. In Ref. @18# the splitting and propagation of a BEC in a microfabricated optical trap with interfeometer structures was demonstrated. This experimeprogress has created an interest in the phenomenologlocalized quantum states propagating through various tyof potential structures.

Interferometer experiments depend directly on the wnature of quantum mechanics and in such configurationsdominated by the nonclassical properties of material pticles. The metrological advantage of using matter wacomes mainly from the increased sensitivity offered byvery short de Broglie wavelengths achievable using la

*Electronic address: [email protected]

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cooled atoms@19,20#. Several different setups for atom optcal analog of well-known interferometric devices have beproposed and investigated in previous theoretical studiesRef. @21,22#, waveguide interferometers based on magnetrapping and a time-dependent bias field were described.use of time-dependent currents to change the potentialfiguration was considered in Ref.@23#. The physics of anatomic interferometer in the multimode regime was invesgated in Ref.@24#. Other works have investigated the nonliearities due to many-body effects in atom optics, see, eRef. @25# and references therein. In Ref.@26# the behavior ofa Tonks gas in an interferometer was investigated theocally. The splitting-recombination regions were here treaby assuming that the physics of the splitting could be incporated in boundary conditions of the quantum dynamicsRef. @27#, the influence of weak nonlinearity on the merginof split atomic gases was investigated and shown to causinstability of the transfer of population in the guided modeRecently it was suggested that optical lattices could be uto transport atoms through split potential structures in orto achieve entanglement of neutral atoms@28#.

Propagation of local excitation is also of interest in othphysical systems. In communication technology, the progrof an optical pulse through guiding structures serves asmodel system for both communication and computationplications. In femtosecond laser-induced dynamics of mlecular processes, we have a well established area ofsearch, where vibrational states propagate along electrpotential surfaces.

In confinement to atomic waveguide geometries wtransverse dimensions around or below micrometer scathe quantum nature of the atoms starts to dominate thenamics. Propagation in such potentials will take place inform of matter waves, we are in the regime of atom optiand the phenomena can be used to explore new featurevolving the fundamental properties of quantum dynamiWork is in progress to create the atom optical analogstandard optical components relying on waveguide strtures. This development is partially spurred by the possib

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M. JAASKELAINEN AND S. STENHOLM PHYSICAL REVIEW A68, 033607 ~2003!

ties to utilize matter-wave guides for quantum logic@29# us-ing controlled collisions of neutral atoms@30#.

When the essential physical phenomena are of shortration, they are best understood in the time domain@31#. Thisis true even when the Hamiltonian contains no time depdence. To simulate realistic systems, one needs to conmany degrees of freedom, which makes the numerical wdifficult and in many cases well beyond the resourceseven the most powerful modern computers. Due to the exnential growth of Hilbert space with the number of degreof freedom, well known in quantum information theory, thamount of data to be processed increases exponentiallythe number of dimensions. Any method which reducesnumber is extremely interesting and will find uses in diveareas. The use of separated channels or modes offers suapproach, but this requires certain adiabaticity assumptto hold. In the lowest approximation, the center of the wapacket may be assumed to progress along the classical mmum of the potential valley. The transverse curvaturetaken to provide a confining potential around this.

In our previous works, the limitations due to mode copling were examined both for single waveguides@32# andsplit potentials@33#, and it was shown that the breakdownadiabatic following of the transverse modes was connecto the intrinsic diffraction of matter waves. The connectiwith diffraction was shown also for the transition from adibatic propagation to free expansion in Ref.@34#. The occur-rence of reflection of guided matter waves was investigaboth for single guides@35# and for split potentials@33#. Inboth cases considerable backscattering was found to occthe adiabatic limit and in the case of single guides also intransition to free expansion. In this work, we considerchanneled splitting of wave packets on two-dimensiopotential-energy surfaces, the highest dimensionality cofortably available for numerical work at present. The propgating state is assumed initially to be localized closely aloa single minimal path and subsequently propagated throusplit of the guiding structure into two identical waveguideAn adiabatic basis of discrete eigenstates is used to rethe complexity of the problem by reducing the timdependent Schro¨dinger equation to a set of equations fclosely coupled channels. When the physical parameterthe problem allow adiabatic propagation, the discrete bstates decouple, making it possible to model the systemnumber of independent one-dimensional wave packpropagating in their respective potential structures. As sthe model could apply to as well chemical reaction pathwand optical fiber communication@36# as mesoscopic structures@37#, but we essentially have in mind the recently dveloped microstructures for confined atomic wave packe

Interferometry depends on the splitting and recombinatof matter waves together with local manipulations of tpropagating states. It is thus important to understandphysics of matter-wave splitting in terms of localized statHere we describe the dynamics using a unitary transfortion acting on the propagating modes in order to creatbasis with suitable localization properties. The formal framwork used here is in many respects similar to hybridizatin molecular physics@38# which is used to create localize

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electronic states. Propagation on potential surfaces with mtiple minimal valleys also occurs in reactive scattering amany similarities between interferometric devices abranching in chemical reactions exist.

The question whether or not the longitudinal extent opropagating quantum wave packet will give rise to any oservable consequences has caused debate at times. Boperimental results utilizing neutrons@39# and theoretical ar-guments @40# in favor as well as criticism pointing toalternative interpretations@41# have appeared in the literature. We consider here the influence of longitudinal localiztion on the physics of wave packets propagating through spotential structures. This is in contrast to our earlier wo@33# where the splitting was investigated solely in termsthe transverse eigenstates in order to derive adiabaticityteria for propagating states. Here we wish to extend the abatic propagation schemes to include situations describtypical matter-wave interferometers. Due to the differencethe effective propagation equations, the physics aroundsplitting point is nontrivial also for coherent superpositioof different propagating modes.

The structure of the paper is as follows: Section II dscribes the system investigated and the separation ofwave function into distinct modes using transverse eigstates of the potential-energy surface. In Sec. III, we desca framework for adiabatic propagation in terms of localizmodes obtained using a unitary transformation of the traverse eigenstates. Section IV discusses the case of proption of coherent superpositions of transverse eigenstatesthe influence of finite longitudinal wave-packet size. FinalSec. V summarizes the results of the paper and discusseconsequences for interferometric experiments and simtions thereof.

II. ADIABATIC PROPAGATION

Matter waves in potential microstructures evolve accoing to the time-dependent Schro¨dinger equation. In this workwe study a two-dimensional system with stationary poten

i]C~x,y,t !

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Units are taken to be dimensionless unless otherwise stafor further details on the consequences of this choice se.g., Refs.@32,34#. In the cases of interest here, the propagtion will be along the minimal valleys of the potential-energsurface, which we can take to foliate into a continuumtransverse, bound one-dimensional potentials. We thensume that the time-independent Schro¨dinger equation can besolved in the transverse direction at each point alongminimal curve in order to obtain a set of eigenfunctions:

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The transverse Schro¨dinger equation~2!, which constitutes aSturm-Liouville system, has a complete set of eigenfunctio$hn(x,y)%. These can be used in an ansatz together with a

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LOCALIZATION IN SPLITTING OF MATTER WAVES PHYSICAL REVIEW A 68, 033607 ~2003!

of longitudinal wave functions$wn(x,t)% yet to be deter-mined. As these are taken to depend also on time, theytermine the time evolution, both propagation along the mmal valley and the confined dynamics in the transvedirection. The total wave function is given by the expans

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Couplings~5! introduce velocity-dependent interactions. Whave here assumed that the coordinate line given byy50,i.e., the origin of the transverse coordinate, does not curvthis will introduce further couplings between the differemodes. Propagation of the longitudinal modes is governedEqs. ~4!, a system of coupled one-dimensional Schro¨dingerequations; these can in principle be solved directly, bupractice this procedure becomes cumbersome as it reliesolving the transverse problem and using the solutionsobtained to calculate the coupling coefficients~5! and~6!. Insimplified treatments of matter waves in guiding potenstructures, frequently only the ground state is includedthe couplings to higher modes are neglected. Here we csider the propagation of wave packets guided close tominima of a symmetric double-well~SDW! potential

VDW„y,d~x!…5 12 v0

2$y21d~x!222yd~x!tanh@v0yd~x!#%,~7!

where d(x) is a parameter which determines the distanbetween the two minima, here taken to depend on the lotudinal coordinatex. This potential allows us to determinthe ground state exactly. At constant longitudinal coordinathe transverse ground state for this potential is given bsum of two identical Gaussians, each centered close tominima of the two wells:

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and the ground-state energy is constant,E0„d(x)…5 12 v0. In

order to investigate the behavior of wave packets in SDpotentials the distance function was chosen as

d~x!51

2dmaxF11tanhS x

XD G , ~9!

where dmax is half the asymptotic distance between tminima andX is a parameter which controls the extensionthe region where the transition from a single guide todouble guide takes place. Twice the functiond(x) strictlyspeaking only equals the distance between the minimad50 and in the asymptotic limit of large separations. In F1 the potential given by Eqs.~7! and~9! with the parametersX510, v051, and dmax55 is shown together with thetransverse binding energies for the two lowest states. Thtwo states become degenerate with respect to energylarge interwell distance functiond(x). The behavior of thetransverse eigenstates as functions of longitudinal distacan be seen in Fig. 2, where the same parameters as in Fwere used.

In Fig. 3 the magnitudes of couplings~5! and ~6! areshown for the two lowest-energy eigenstates of the SDpotential used here. For a symmetric potential, the transveigenstates have definite parity. It is easily seen that takthe derivative with respect to the longitudinal coordinate p

FIG. 1. In ~a! a contour line of the potential given by Eqs.~7!and ~9! with the parameters given byX510, v051, dmax55 isshown. The chosen equipotential level corresponds to distatwice the ground-state width away from the bottom of the wells.~b! the transverse binding energies for the first two states are shas functions of longitudinal distance. We see that the two transveigenenergies become nearly degenerate shortly beforex50, re-sulting in a markedly increased tunneling time between thewells. Units are dimensionless.

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serves parity of the transverse states. As a consequencecoupling matrix elementsAi j (x) andBi j (x) vanish for tran-sitions mixing odd and even states.

In our previous work@33#, adiabatic propagation in thSDW potential given by Eqs.~7! and ~9! was investigated.The magnitude of the nonadiabatic excitation, measuredprobability transferred from the ground state to excited staduring propagation through the splitting region, was founddepend on a single dimensionless parameter, the Frenumber, named after the analogous quantity in classicaltics @42#. Small Fresnel numbers correspond to the far-fizone, where the transverse distribution is determined bycontribution from a single Fresnel zone only. For a wapacket propagating in a splitting potential, the equiprobaity lines traced out during the evolution will diverge angiven that the widening is not too rapid, adjustment tonew width will take place. Taking the potential to be chaning over distances of the order ofL, the Fresnel number fothis situation is given as

N5zR

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wherel denotes the de Broglie wavelength correspondingthe longitudinal motion. Adiabatic propagation of wavpackets in potential structures will take place whenchanges are such that the Fresnel parameter is small.the question is whether or not the changes occur ovelength scale comparable to or larger than the Raylelength, which is the length a suddenly released beam requin order to evolve into diverging at a constant angle demined by its momentum distribution. The length scaleL inthe expression for the Fresnel parameter is, in classical,tics the distance over which a beam diffracts in width

FIG. 2. Mode structure of the two lowest-energy eigenstates~a!h0„y,d(x)… and ~b! h1„y,d(x)…. Shown is the probability densityfor the two transverse modes as function of both transverselongitudinal coordinates. The two modes are seen to split intodistinct pieces each centered in one outgoing guide. Axes ardimensionless units.

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about a factor ofA2. Here a suitable choice is the distanover which the interwell distance separates to about

d~xu!5A2Dy5A2v0, ~11!

whereDy denotes the width of transverse ground state. Tis also the point, in the longitudinal direction, at which thcentral barrier height equals the ground-state energy. Aresult, an assembly of classical particles with the sametribution of initial positions and momenta would here seebifurcation in phase space. The value given by Eq.~11! alonedoes not suffice to estimate the length scale and a secpoint must be pinned down using some criterion of the ty

d~xl !*Dy. ~12!

It turns out that the choice of lower end point is fairly insesitive to the exact details of criterion~12!, probably due tothe exponential character of the distance function. A suitavalue is given by the point where distance between the wwas larger than about 1.001 times the ground-state wiThe distance for potential changes is thus taken to be

L5xu2xl , ~13!

wherexu and xl are defined above. With this choice of thlength scale, the nonadiabatic excitation probability has bfound to be a function of the Fresnel parameter only, regaless of the values of other physical parameters@33#.

FIG. 3. Absolute values of the couplings given by Eqs.~5! and~6! for the ground and first excited eigenstates. The relevant enscale is set by the transverse oscillator frequencyv051, i.e., thevalues are to be compared with unity. The velocity-dependent cplingsA00(x) andA11(x) are seen to be very small for the mometum values needed for adiabatic propagation, whereas the coupB00(x) and B11(x) are a few percent of the ground-state enerUnits are dimensionless.

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III. LOCALIZED PROPAGATION

In our earlier works, the investigation of adiabatic propgation was mainly focused around changes related todiffractive spreading of matter-wave beams, this beequivalent to the case of wave packets propagating in potial structures of changing width@32#. For a potential-energysurface with a single minimal path, the idea of separationthe wave function is in principle unproblematic as the lontudinal modes propagate close to the minima.

For the case of splitting, considered here, such anproach neglects the process of dividing the transverse sinto two separate halves by the raising of a central barrierthe incoming transverse oscillator frequency in the potenused here coincides with the ones in the two outgoing guidan alternative picture could be envisioned. It is possibleconsider the incoming wave packet as being a superposof two almost identical ones each to be guided along onethe two degenerate minima. In this case there are twoproximately independent wave packets propagating inwaveguides curving in different directions away from eaother. Nonadiabatic excitations would then be regardedarising due to the longitudinal curvature of each guide.

For the case of symmetric potentials with split minima ttransversally localized propagating modes are the transvenergy eigenstates of the binding potential in the regionfore the splitting. When the potential minima are far apathe localized states are known to be given by linear comnations of pairs of energy eigenstates according to

hL/R„y,d~x!…51

A2@hn„y,d~x!…6hn11„y,d~x!…#, ~14!

where the transverse quantum numbern50,2,4, . . . . Anadiabatic propagation scheme thus has to deal with theparent conflict of continuously guiding energy eigenstatwhich have definite parity and are localized in a single mimum, into states that are delocalized, when it is known twave equations have localized propagating solutions. Forcase of large constant separation between the waveguthe propagation, Eq.~4!, becomes separated into independequations since the couplings depend on the spatial vations of the transverse eigenstates. As, in addition, the enlevels become pairwise degenerate, the propagation etions remain invariant under unitary transformations amothe pairs of eigenstates. Two natural choices for basis stare transverse energy eigenstates or localized states. Thter will be the preferable choice whenever the individual ptential wells are disturbed in a local fashion, for instancean interferometric setup, as the influence from the other wis negligible.

The condition for adiabaticity in quantum dynamicsoften formulated as a requirement that the internal tiscale, in this case the transverse oscillation time

Tosc;1/v ~15!

should be shorter than the time scale under which the Hatonian undergoes considerable changes, here denoted bDt.For wave-packet dynamics in potential structures, the ex

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nal time scale has to be estimated with great care or elsecriterion might become a poor indicator of adiabatic evotion. For wave-packet dynamics in spatially changing pottial structures, the diffraction criterion can be used asmeans to estimate the allowable changes in a lessad hocmanner. In addition to the two time scales mentioned,system under consideration here contains a third, the trverse tunneling time given by the energy split betweground and first excited states,

Ttunn;1/DE, ~16!

a function of the distance between the wells and the traverse oscillator frequency. As the ground-state energy is cstant in the double-well potential chosen, the tunneling tiis always larger than the transverse oscillation time:

Tosc<Ttunn . ~17!

Depending on the relative sizes of the three time scales,can distinguish between different regimes. First, if the extnal time scale is the shortest,

Dt!Tosc<Ttunn , ~18!

the dynamics will be nonadiabatic resulting in sizable extation of higher transverse modes. The potential changescur fast enough to result in the wave packet propagating wconstant shape through changes. On times scales of theof the oscillation time, the resulting changes start to beticeable when transverse oscillations induced by the nonabatic transitions become visible. If both the oscillation atunneling times are shorter than the external time scale,

Tosc<Ttunn!Dt, ~19!

the potential changes over length scales larger than the Rleigh length and the dynamics evolves adiabatically withspect to the transverse eigenstates. Propagating wave pathus behave as if composed of a single mode, althoughmight be split into two partially overlapping halves. For largseparations, the tunneling time approaches infinity rapidlythe two lowest states become degenerate in energy, anmight become larger than the external time scale:

Tosc!Dt!Ttunn . ~20!

In this case, transverse energy eigenstates will not be gadiabatic states that are followed, but the localized stawhich to a very good approximation are single-well groustates, propagate adiabatically and couple weakly toneighboring well only through the overlap which decays eponentially with the distance between the wells. To relthis with the adiabaticity criterion~10! from our earlier work@32,33# we first note that the time scale for the potentchange can be estimated as

Dt'L

kx}Ll, ~21!

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where the lengthL, both for single guides and potential splitings, is related to the transverse change of the guided sand has to be chosen as described in Sec. II. To estimattunneling time~16!, we use the fact that the energy diffeence between the first two transverse states approximawas found to be

DE'v0exp~22v0d2!. ~22!

Using Eqs.~21! and ~22! in Eq. ~20!, we find

exp~22v0d2!!N!1, ~23!

whereN is the Fresnel parameter as defined by Eq.~10!. Ifboth conditions in Eq.~23! are met, then the error due tnonadiabatic excitations will be limited by the intrinsic difraction of matter waves as described in Ref.@32#. As thewell separated guides may each go through very differevolution, it is not meaningful to talk about the transvereigenstates of the total potential as being the adiabatic sthat are followed. Problematic situations emerge, forstance, in the case of differing path lengths betweenguides in a Mach-Zender interferometer or propagationright angles or even backwards after a 180° bend.

In order to construct a localized propagation schemethe two lowest transverse states, we take our modes tlinear combinations depending on the longitudinal coornate. In order to preserve orthonormality, we choosestates to be transformed to the new localized basis throuunitary transformation:

FhL~x,y!

hR~x,y!G5Fcosu 2sinu

sinu cosu GFh0„y,d~x!…

h1„y,d~x!…G , ~24!

FIG. 4. Spatial variation of the transformation angleu(x) for theSDW potential given by Eqs.~7! and ~9! with the same parametevalues as for Fig. 1. Shown dashed is the first derivative anddashed the second derivative with respect to the longitudinal cdinate. The potential parameters are the same as in Fig. 1. Axein dimensionless units.

03360

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where the transformation angleu(x) is chosen in such a waythat the new basis states at all positions along the wavegare localized in each well in the sense that the expectavalues of the transverse coordinate taken with respect tother state will equal the positions of the two Gaussians whconstitute the ground state

^hL/RuyuhL/R&57d~x!, ~25!

where the signs were chosen in order to localizehL in theleft minimum, i.e., at negative transverse coordinate valuThis corresponds to the lower well in Figs. 1 and 2. UsiEq. ~24! in Eq. ~25! we arrive at

u~x!561

2arcsinS d~x!

y01D , ~26!

wherey01 denotes the dipole matrix element betweenh0 andh1. The spatial variation of the transformation angleu(x)can be seen in Fig. 4. For small values of separationdmax theangleu will be close to zero whereas for large separationwill approachp/2 as the localized states approximately wbe given by Eq.~14!. The structure of the localized modecan be seen in Fig. 5 for the split potential used in this woFor the mode which equals the transverse ground statelarge negative values of the longitudinal coordinate therelittle change in the transverse size or other properties althe propagation direction. For the other mode, however,node aroundy50 seems to disappear as the transverse shafter the splitting is close to that of a Gaussian localized

t-r-are

FIG. 5. Mode structure of the localized mode functions~a!hL(x,y) and~b! hR(x,y), here shown as the respective probabildensities. Each mode function is seen to be centered on the sguide before the split. After the split each mode has only a smprobability of being in one of the outgoing guides, thus being pdominantly localized to the other. For the modehR(x,y), whichbefore the split is equal to the first excited state, the node seemdisappear around the splitting region. This is not true, howeverthe mixing of the two energy eigenstates causes each locamode to necessarily have a node in the outgoing distributions. Aare in dimensionless units.

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the upper minimum. This change occurs just beforex50 asthe upper half of the eigenstate is seen to grow into donating the mode. The total wave function is now given b

C~x,y,t !5hLwL1hRwR5h0@wL cosu1wR sinu#

1h1@2wL sinu1wR cosu#. ~27!

We thus see that writing the total wave function as a lincombination of two localized transverse modes is equivawith transforming the longitudinal modes according to

Fw0~x,t !

w1~x,t !G→FwL~x,t !

wR~x,t !G5F cosu sinu

2sinu cosuGFw0~x,t !

w1~x,t !G .~28!

If we now use ansatz~28! in the time-dependent Schro¨dingerequation and project away the dependence on the transvcoordinate, we arrive at the propagation equations forlocalized modes

i]wL~x,t !

]t52

1

2

]2wL~x,t !

]x21EL~x!wL~x,t !1WwR~x,t !,

i]wR~x,t !

]t52

1

2

]2wR~x,t !

]x21ER~x!wR~x,t !1WwL~x,t !,

~29!

where the potentials and the off-diagonal coupling are giby

EL~x!5cos2u@E0~x!1B00~x!#1sin2u@E1~x!1B11~x!#,

ER~x!5sin2u@E0~x!1B00~x!#1cos2u@E1~x!1B11~x!#,

W~x!5sinu cosu@E0~x!1B00~x!2E1~x!2B11~x!#.~30!

We have here neglected the spatial variation of the transmation angleu(x) in that its derivatives with respect to thlongitudinal coordinatex have been dropped. The dominaing off-diagonal term is determined byW(x) in Eq. ~30! andthis can be considered to give the coupling matrix elemeto lowest order in the spatial derivatives ofu(x). In Fig. 4the spatial variation of the transformation angle and its fiand second derivatives are shown. The propagation equafor localized modes in a split potential thus have the forma two-level system with spatially dependent energies, cplings, and velocity-dependent interactions. For large septions of the two minima, the level splitting rapidly goeszero and all couplings can be neglected. When the distafunction d(x) is close to zero, the couplings are also neggible and the propagation is in the form of two independmodes. The framework defined here thus models the splitof matter waves using basis states which are localized bothe single-guide region and in the double-guide region. Insplitting region localized propagation necessitates coupbetween the two modes. For a quantitatively more accutreatment, the terms dependent on the derivatives oftransformation angle must also be taken into account. In

03360

i-

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n

r-

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g.

6, snapshots of a wave packet initially in modehL(x,y)which for large negative distances equals the ground stare shown during propagation through a splitting regioWithout any interaction between the two modes, the wapacket will be guided completely into the lower outgoinguide, completely unphysical result as the ground state mis split equally between the two degenerate wells, as canseen from Fig. 2. At the splitting, the interaction leads to tcreation of a component in modehR(x,y), which is chan-neled into the other guide. The dynamics for both modeshown in Fig. 7. It is seen here that although the wave pacis initially completely in the modehL(x,y), the interactionaroundx50 causes a significant growth of probability in thother modehR(x,y). For ground-state propagation, the epected result would be an equal split between the outgomodes resulting in a probability transfer ofDP50.5 regard-less of the longitudinal momentumkx . Any deviations fromthis result come from numerical inaccuracies in the wapacket propagation, failure to include enough couplings,limited accuracy in the calculation of the coupling coefcients~5! and ~6!.

IV. COHERENT SUPERPOSITIONS

A fundamental law of quantum mechanics is the supersition principle, which can give rise to interesting new dnamics through the interference of quantum states. The lincombination given by Eq.~14!, i.e., an equal superposition otwo energy eigenstates, corresponds for the case of far srated potential wells to a localized state. Here we wishinvestigate the dynamics of superposition states with limispatial extent in the longitudinal direction, and take the i

FIG. 6. Snapshots of a wave packet initially in the ground-stmodeh0(x,y). Pictures~a!–~e! show the probability density for awave packet initially in the ground state at instants separatedequal amounts of time. As time progresses the wave packet prgates through the split and enters the asymptotic region beyondsplit. From the final snapshot~e! it is seen that the probability isdistributed in a symmetric manner between the outgoing guidThe initial wave packet was a Gaussian of widthDx520 and mo-mentumkx51. Units are dimensionless.

7-7

u

s,tya

rs

thhpo

inorte

attwotumgtworval

-

er-thisiodect

thatned.intoatingorpa-ceare

in

-av

een


tial state to be a general coherent superposition of eqamounts of ground and first excited states,

C51

A2~h01eiDuh1!, ~31!

where Du denotes the relative phase between the statequantity that will evolve with time causing the probabilidistribution to shift in space. The states with extreme locization are given by

CL/R51

A2~h06h1!, ~32!

i.e., by taking the phase angle to be eitherDu50 or Du5p. If we calculate the expectation value of the transvecoordinate we have

^y&5^CuyuC&

5K 1

A2~h01eiuh1!uyu

1

A2~h01e2 iDuh1!L

5y01cos~Du!, ~33!

where y01 denotes the dipole matrix element betweenground and first excited states, a quantity that approachalf the interwell distance for the case of well separatedtential minima.

For a superposition wave packet of negligible extentthe longitudinal direction, the probability to go to the leftto the right will be given by the projection of the initial staonto transverse states localized in either well,

FIG. 7. Probability density of the localized modes propagatthrough the potential shown in Fig. 1. In~a! uwL(x,t)u2 is shown,and in~b! uwR(x,t)u2. The wave packet was initially in the groundstate mode. In the asymptotic region before the split the wpacket is thus almost completely inwL(x,t), which is guided to thelower well in Fig. 5. It is seen that considerable interaction betwthe localized modes takes place, and that the final distributiosymmetric as expected. Axes are in dimensionless units.

03360

al

a

l-

e

ees-

pL/R5u^CuCL/R&u2

5U K 1

A2~h01eiuh1!U 1

A2~h06h1!L U2

51

2@16cos~Du!#. ~34!

We thus see that the probability to go to the left or rightthe split is dependent on the relative phase between theenergy eigenstates. In the center of mass frame of a quanparticle, well localized in the longitudinal direction, travelinat constant momentum, the phase difference between thestates can be assumed to change over a small time inteaccording to

du5DEdt'DE

]x

]t

dx. ~35!

This gives us for the total phase shift during propagation

Du5Exi

xf DE

]x

]t

dx5C

kx, ~36!

wherexi andxf denote initial and final positions in the longitudinal direction and the proportionality constantC is in-

FIG. 8. The adiabatic splitting of a wave packet being transvsally a coherent superposition of the lowest two eigenstates. Asstate oscillates from side to side in the waveguide, with a pergiven by the transverse binding potential, one would naively expthe outgoing state to be localized in either outgoing guide giventhe initial phase of the components has been appropriately tuOn the contrary it is seen that the wave packet is fractionedsmaller portions each centered in one of the guides and propagoutwards. This behavior is due to the differing arrival times fdifferent longitudinal sections of extended wave packets. The stial period of the slicing is approximately given by the distantraveled longitudinally during one transverse oscillation. Unitsdimensionless.

g

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its

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-

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ares

llys, as

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troduced for convenience. As the energy splitDE goes tozero rapidly after the split, the phase shift slows down, incating that the oscillation time between the wells, i.e.,tunneling time, becomes much larger than the oscillationriod in each well. The dependence on the final positionxf inEq. ~36! will thus be negligible given that the propagatiohas been taken beyond the splitting.

If the initial wave packet is spread out in the longitudindirection, different parts will reach the split at different timthus having different local values of the relative phaseDu,possibly differing by large amounts. In Fig. 8 we see aherent superposition wave packet propagating througsplit. As the longitudinal momentum is low, a spatially peodic slicing of the wave packet into portions entering diffeent outgoing guides is seen to take place. If several osctions occur during the split, there will be approximateequal probability for the particle to go in either direction.Fig. 9 the same splitting is presented as probability densitthe transverse eigenstates as functions of time and longinal distance. It is seen that both modes propagate forwwithout noticeable disturbances. The only exception to thithe acceleration of the first excited mode aroundx50 due tothe release of additional energy as the transverse binenergies approach degeneracy. In Fig. 10 the splittingshown in terms of localized modes in the same way as in9. Here, however, in contrast to the previous figure, theriodic slicing of the wave packet is clearly visible in bolocalized modes. The localized mode picture thus hasability to reveal information hidden in the relative phasessuperposition states.

If we want a wave packet to go into only one of thoutgoing guides and, in addition propagate adiabaticathere are several conditions that will have to be fulfilleFirst, the wave packet must not conduct a full transve

FIG. 9. The adiabatic splitting of a wave packet initially beingcoherent superposition of the lowest two transverse eigenstatesfigure shows a density plot of the probability in the propagateigenstates,uw0(x,t)u2 in ~a! and uw1(x,t)u2 in ~b!, as functions oflongitudinal coordinate and time. The oscillations seen in Fig. 8absent here as no probability is transferred between the modthis basis. Axes are in dimensionless units.

03360

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ef

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oscillation as it propagates a distance corresponding tolongitudinal spread,

2Dx

kx!

2p

v0. ~37!

Second, the wave packet should not spread out too mduring the split as this will cause the first condition tobroken. We require that the time it takes to propagate throthe split should be much smaller than the time of spreadin the longitudinal direction,

L

kx!Dx2. ~38!

If the atoms are kept in an optical lattice in the longitudindirection, as suggested in Ref.@28#, requirement~37! is nolonger needed as there is no dispersion broadening thetribution. Instead an adiabaticity condition for the longitudnal motion of the optical lattice has to be fulfilled if thatoms are to remain in the localized ground states ofoptical lattice. Finally, the propagation should take plawith negligible nonadiabatic excitation

1

Lv0l!1, ~39!

wherel is the longitudinal de Broglie wavelength. Combining Eqs.~37!–~39! we have

Dx

kx!

1

v0!

L

kx!Dx2. ~40!

he

ein

FIG. 10. The adiabatic splitting of a wave packet being initiaa coherent superposition of the lowest two transverse eigenstatedescribed in terms of the localized modes. The figure shows pability density vs longitudinal coordinate and time~a! for the modewL(x,t) and ~b! for the modewR(x,t). The coherent oscillationsduring the splitting of the wave packet are clearly visible heUnits are dimensionless.

7-9

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ed

ss

ke

gqsnn

seuono

tlytiotioo

vema

rentng

esol

tu-thebethe

Thelo-

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eo-rts,pli-

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If all the conditions included in Eq.~40! are fulfilled, weexpect the wave packet to exit the splitting region predonantly in only one of the outgoing guides. In order to esmate the amount exiting in the left guide, we calculateaverage ofpL(u) taken over the initial wave packet extendin the longitudinal direction as

PL5E2`

`

uw~x,0!u2pL„u~x!…dx, ~41!

where the intial probability distribution is taken to be Gauian,

uw~x,0!u251

A2pDxexpS 2

~x2x0!2

2Dx2 D , ~42!

with an initial width Dx and initial positionx0. If the initialposition is far from the splitting region and the wave pacis sufficiently localized att50, we take

Du~x1X!'Du~x!1Ex

x1XDE

kxdx'Du~x!1

DE

kxX

~43!

as an approximate expression for the shift of relative anwith increasing distance from the splitting region. Using E~34!, ~42!, and~43! in Eq. ~41!, we can calculate the fractioof the wave packet, averaged over the initial distributioexiting through the left guide after propagation as

PL51

2 F12expS 21

4

Dx2v02

kx2 D cosS C

kxD G , ~44!

where the constantC is determined by Eq.~36! with xi5x0, in principle. In practice it turns out that the total phais not given by a simple expression of this kind probably dto the nonuniform acceleration in the longitudinal directiaround the splitting, which was ignored in the derivationEq. ~44!. In the discussion above, adiabaticity was impliciassumed and the longitudinal spreading during propaganeglected. As a result, the maximal transverse localizainto the outgoing guides depends, apart form an oscillatfactor, on the parameter combination

Dxv0

kx!1. ~45!

This condition is also given by Eq.~37!. This shows thatlocalization in either outgoing matter-wave guide for a wapacket of finite longitudinal extent is determined by the nuber of transverse oscillations that are completed during psage through the splitting region.

This was tested by propagating wave packets initially ppared in a coherent superposition state through a potesplitting, varying the longitudinal momentum and observithe fraction of the wave packet entering the left~in Fig. 8 thelower! guide. The result is shown in Fig. 11. Open circlshow the results from wave-packet simulation and the sline is given by Eq.~44! where the constantC was used as a

03360

i--n

-

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le.

,

e

f

nn

ry

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id

single fitting parameter. In the derivation above, the longidinal momentum was taken to be constant during all ofpropagation. In reality the excited-state component willaccelerated as the energy difference goes to zero aftersplit causing an additional phase shift neglected here.expression given by Eq.~44! agrees well with the numericaresults over the whole range of momenta used. For low mmenta, there is a probability close to one-half of going inthe left guide due to the periodic slicing of the wave packas illustrated in Fig. 8. For increasing momenta, the wapacket starts to get split asymmetrically as fewer oscillatiotake place during the splitting. The probability of going ether way shows an oscillatory behavior with increasing aplitude, exactly as predicted by Eq.~44!. For the limitingcase of infinite longitudinal momentum, which is far beyothe adiabatic approximation, the relative phase betweencomponents of the wave packet does not change at all analso homogenous along the wave packet. The probabilitythe wave packet to exit in either of the guides should thapproach a constant value. This region lies beyond thementum range presented in Fig. 8, and would from a coputational point of view be very demanding to probe. Wthus see that, given suitable initial wave packets are ppared, propagation through a potential splitting can be uto extract information about the longitudinal structurematter waves.

V. DISCUSSION

We have investigated the propagation of guided mawave in split potential structures. For well separat

FIG. 11. Probability of finding a wave packet exiting from thleft guide as a function of longitudinal momentum. For low mmenta, the wave packet is equally split between the output powhereas for higher momenta an oscillation can be seen. The amtude increases with increasing momenta, whereas the oscillafrequency decreases. Open circles denote results obtained frommerical solutions of the time-dependent Schro¨dinger equation~1!for a potential with parametersv051, L5500, andkx varying. Thelongitudinal extent of the wave packet wasDx540. The parameterC in Eq. ~44! has been fitted here.

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waveguides it is intuitively appealing to model the dynamas being in the form of wave packets propagating alongdependent potential minima. Under these circumstanceswave packets are expected to be subjected to local adiabity conditions depending on how rapidly the local potentichange. When the potential foliates into a continuum of odimensional potentials, it is natural to use the transveeigenstates provided by the transverse binding potentThis becomes less intuitive when there are several degeate minima, as the eigenstates in this case are delocahaving equal probability of being found in each. In thecases, although it has been shown that splitting is adequadescribed using the eigenstates@33#, it seems natural to transform to a basis that allows for a description in termslocalized modes in the same way as is common in molecphysics using the method of hybridization to analyze bolocalization@38#. The formalism described in Sec. III intepolates between a basis localized in a single minimum bethe split and localization in each of the outgoing guides athe split. It should be observed that the equivalence betwthe upper~right! and lower~left! guides broken in an arbitrary way is correct, as the sign of the transformation angiven by Eq. ~26!, which can be chosen, is arbitrary bnevertheless gives a unique mapping between the transvand localized states. The choice of mapping the ground sto either the upper or lower guide does not influencedynamics, only the representation of it. We feel that theterion used here is a natural choice if the new modes arbe localized in either well. The framework described in thwork should find uses in simulations of interferometric dvices as the manipulations lead to phase shifts between srated parts of matter wave packets, which act as indepenstates sensitive mainly to changes in the local binding potial valley. This makes the transverse eigenstates unsuitas a basis since they are unable to properly take local pebations into account when the guides are far separated.viously the localized states given by Eq.~28! offer a betterchoice. In this picture, the effective dynamics is formaidentical to that of two weakly coupled single waveguideeach being sensitive only to local perturbations. For th

.

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states adiabatic propagation is limited by the same conditas those that apply to single waveguides. In the present wwe only outline the formalism and show the equivalenwith adiabatic splitting in terms of transverse eigenstates

Traditionally the conditions for validity of the adiabatiapproximation are stated in terms of the time scales owhich the Hamiltonian describing the system changes. Inearlier work we found connections between adiabaticity athe diffraction of matter waves. This made it possible to epress these conditions in geometrical terms borrowed frclassical optics. In Sec. III the regimes of validity of adibatic propagation schemes based on transverse eigensversus ones based on localized states were outlined uheuristic arguments based on the relevant time scales. Cnections were also made to our earlier work in order to rethe different regimes to the adiabaticity conditions givenintrinsic diffraction of matter waves. It would be desirablerather state the validity criteria in geometrical terms inmore stringent way than has been done here. It appears lithat such an approach would yield further insights intofundamental understanding of how matter waves are splpotential structures.

The behavior of coherent superpositions of transveeigenstates in split potential structures is nontrivial. The rson is that it involves the relative phase between the progating states, a quantity possibly varying with spatial codinate. The resulting probability distribution after the splitconveniently described using the localized states insteathe transverse states which are delocalized and thus coninformation in the relative phases. A superposition wapacket of finite extent propagating through a potential swill be divided between the outgoing guides depending onlongitudinal extent relative to the distance propagated durone transverse oscillation. Observation of any asymmetrytween the outgoing guides for wave packets prepared inperposition states can thus reveal information about thegitudinal extent of the state, something not possibleinterferometric studies as there are alternative explanatequally valid@41#.

s,

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Documents

Physical Review A. Atomic, Molecular, and Optical Physics ...565610/FULLTEXT01.pdf · In this paper we present an analysis of how matter waves, guided as propagating modes in potential