Phase shifts of atomic de Broglie waves at an evanescent wave mirror

8 Pages·0.19 MB·English

by  C. Henkel

LaserPhysics,Vol.4,No.5,1994,pp.1042-1049
OriginalTextCopyright(cid:13)c1994byAstroLtd.
Copyright(cid:13)c1994byMAHKHayka/InterperiodicaPublishing(Russia). ATOMOPTICS
SPECIALISSUE
“LASER COOLINGAND TRAPPING”
Phase Shifts of Atomic de Broglie Waves
at an Evanescent Wave Mirror
C.Henkel∗,J.-Y.Courtois,R.Kaiser,C.Westbrook,andA.Aspect
Institutd’OptiqueThe´oriqueetApplique´e,B.P.147,F-91403Orsay,France
(ReceivedOctober21,1993)
3
Abstract–AdetailedtheoreticalinvestigationofthereflectionofanatomicdeBrogliewaveatanevanescent
0
wavemirrorispresented.Theclassicalandthesemiclassicaldescriptionsofthereflectionprocessarereviewed,
0
andafullwave-mechanicalapproachbasedontheanalyticalsoutionofthecorrespondingSchro¨dingerequation
2
ispresented.Thephaseshiftatreflectioniscalculatedexactlyandinterpretedintermsofinstantaneousreflection

n of the atom at an effective mirror. Besides the semiclassical regime of reflection describable by the WKB
a
method,apurequantumregimeofreflectionisidentifiedinthelimitwheretheincidentdeBrogliewavelength
J
islargecomparedtotheevanescentwavedecaylength.
6
1
1.INTRODUCTION cal and a quantum regime of reflection. In the semiclassical
1
regime,realizedathighincidentenergy,theSchro¨dingerand
v
the WKB approach coincide, and the evanescent wave mir-
0 Atomicmirrorsareoneofthekeycomponentsinthefield
8 of atom optics[1]. In orderto realize such a device, the use ror behavesas a dephasingdispersivemirror, analogousto a
0 ofevanescentwavesappearsverypromising[2,3].Inviewof dielectric mirror in conventional optics. By contrast, in the
1 quantum regime of reflection where the incident atomic de
thefutureapplicationsfoevanescentwavemirrors,adetailed
0 Brogliewavelengthislargecomparedtotheevanescentpoten-
investigationoftheiropticalpropertiesseemsappropriate.For
3
tialdecaylength,theevanescentwavemirroractsasanondis-
somebasicpurposes(forexample,thedeflectionofanatomic
0
persiveinfinitelysteepbarrier,analogoustoametallicmirror
/ beam), the characterization of the geometric optical proper-
h in conventional optics. Finally, the reflection process of an
ties of the evanescentwave mirror is sufficient. This can be
p atomicwavepacketincidentonanevanescentwavemirroris
achieved by treating the incident atom as classical particles
-
discussed.
t andbyderivingtheirclassicaltrajectories(thisisanalogousto
n
thecalculationoflightraysinconventionaloptics). Formore
a
u elaboratepurposes(atominterferometers[4],atomiccavities
2.CLASSICALANDSEMICLASSICALDESCRIPTIONSOF
q [5, 6]), knowledgeof the wave-mechanicalpropertiesof the
: evanescent wave mirror is also required. One then needs to THEREFLECTIONPROCESS
v
i describe the atom by a de Broglie wave in order to estimate Before turning to the full wave-mechanical treatment of
X
thephaseshiftexperiencedbytheatomduringreflectionatthe
atomic reflection at an evanescent wave mirror, we start by
r mirror.Asemiclassicalderivationofthisphaseshift,basedon
a reviewingtheclassicaldynamicsaswellastheWKBdescrip-
theevaluationoftheactionintegralalongtheclassicalatomic
tion of the reflection process. This will allow us to make a
trajctories(WKBmethod),hasbeengivenbyOpatetal. [7].
cleardistinctionbetweenthesemiclassicalandthepurequan-
Wepresentinthispaperamorecompleteapproachbasedon
tumfeaturesofatomicreflection.
theanalyticalsolutionoftheSchro¨dingerequationdescribing
theinteractionbetweentheatomandtheevanescentwavemir-
rorin theregimeofcoherentatomoptics(limitofnegligible
2.1.PresentationoftheModel
spontaneous emission). We interpret the atomic phase shift
derived from the atomic wave function in terms of instanta- We considerthesimplecase ofatwo-levelatomnormally
neousreflectionatan effectivemirror,whichgeneralizesthe incidentof the surface (z = 0 plane)of an evanescentwave
one introducedin [7]. We distinguish between a semiclassi- mirror.1 Becauseweareinterestedintheregimeofcoherent
∗Presently at Institut fu¨r Physik, Universita¨t Potsdam, 14469 Potsdam, Germany, email [email protected]
RetypingcourtesyofM.PathandR.Donner
1Becauseofthetranslationalsymmetryinthedirectionsparalleltothemirrorsurface,theproblemcanbereducedtoonedimension. This
simplificationholdsforboththeclassicalandthequantumviewpoints.
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atomoptics(limitofnegligiblespontaneousemission),were- ∓p∞/M alongtheasymptotesoftheclassicaltrajectory(see
strict ourselves to the limit of low saturation of the atomic Fig. 1). Thesestraightasymptotesintersectattheposition
transition, where the reactive part of the atom-evanescent
wavecoupling(lightshift)ispredominantoverthedissipative κ z
part. We also assume that the detuning between the evanes- 3
centwaveandtheatomicfrequencyisproperlychosensothat
the atom ca be consideredto followadiabaticallythe optical
potential associated with the light-shifted ground-statelevel.
In thisregime,all the physicalphenomenacan be accounted
forbymeansoftheHamiltonian[3,7]: 2
p2 p2
H = + max exp(−2κz), (1)
2M 2M
whichcontainsthe atomickineticenergy(firstterm)andthe
1
reactive part of the atom-field coupling (second term). In
equation(1),pandz ≥ 0arethemomentumandpositionof κ z
0
theatomiccenterofmass,M istheatomicmass,p >0is
max κ ζ
themaximummomentumthatcanbereflectedbytheoptical cl
potentialbarrier,andκ−1isthecharacteristicdecaylengthof 0
theevanescentwave,oftheorderofthelaserwavelength2(for −2 −1 0 1 2
t / τ
adiscussionoftypicalexperimentalparameters,seeAppendix refl
FIG. 1. Classical trajectory of an atom undergoing specular re-
A).Notethatwhenquantizingtheatomicexternaldegreesof
glection at an evanescent wave mirror. The dimensionless atomic
freedom,onehastosubstitutethemomentumandpositionop-
eratorsP andZ forpandz inequation(1). Thepotentialin positionκz isrepresentedvs.t/τrefl fortheparametersp∞ = 3¯hκ
equation(1) growsexponentiallyas z → −∞. We thusne- manidnipmmuamx =dist1a0n¯hcκe.z0Theat1o.m2κa−p1praonadchiessrethfleecmteirdroornsuarftaimceeastctahlee
glectanyeffectsduetotunnelingthroughthepotentialbarrier ≈
of the order of τrefl. In the asymptotic region (κz 1) of van-
tothephysicalmirrorsurfce. ≫
ishing optical potential, the atom propagates freely at the velocity
p∞/M,andtheclassicaltrajectorycorrespondstostraightasymp-
±totes(dashedlines),whichintersectatthepositionζcl 0.5κ−1.
2.2.ClassicalDynamicsoftheReflectionProcess ≈
ζcl(p∞)=z0(p∞)−κ−1ln2, (5)
Letusfirstconsidertheincidentatomasaclassicalparticle
which is shifted moredeeplyinto the potentialrelative to z
with asymptotic momentum −p∞(0 < p∞ < pmax). With 0
byaquantityindependentoftheincidentmomentum(seeFig.
an appropriate choice of time origin, the classical trajectory
1). Asfarastheasymptoticclassicaldynamicsoftheatomis
oftheatomcanbewritten[7](Fig.1):
concerned,theevanescentwavemirrorthusbehavesasafic-
z(t)=z0+κ−1lncosh(t/τrefl), (2) titiousinfinitelysteepbarrierlocatedatζcl(p∞),atwhichthe
atomwouldexperienceaninstantaneousreflection. Wename
where thisbarriertheclassicaleffectivemirrorafter[7].
z0 =κ−1ln(pmax/p∞) (3)
2.3.TheWKBSolutionfortheEvanescentWaveMirror
isthepositionoftheturningpointoftheturningpointofthe
trajectory(reachedatt=0),locatedaboutκ−1infrontofthe Letusnowconsiderthesemiclassicaldescriptionofthere-
flectionprocess. Inthiscase,theatomisdescribedbymeans
mirrorsurface,andwhere
ofawavefunctionderivedusingtheWKBapproximation.In
τrefl =M/κp∞ (4) theclassicalallowedregionz >z0,thisWKBwavefunction
isgivenby[8]:
isthetimescaleforthereflectionprocessandcorrespondsto
thetimetakentocrossthethicknessκ−1oftheopticalpoten- 4M π 1 z ′ ′
ψ (z)= sin + p(z )dz (6)
WKB
tial at the asymptotic velocity p∞/M (see Appendix A, the sp(z) (cid:18)4 ¯hZz0 (cid:19)
tablefortypicalexperimentalvalues).
In the asymptotic region z ≫ κ−1 of vanishing opti- 4M π p∞
= sin + (artanh(p(z)/p∞)−p(z)/p∞) ,
cal potential, the atom is moving freely at constant velocity sp(z) 4 ¯hκ
h i
2Notethat,thisestimatebreaksdownnearthecriticalanglefortheevanescentlaserwave.Inthiscase,thelengthscaleκ−1tendstoinfinity.
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wherep(z)istheclassicalmomentumcalculatedfromenergy z →+∞:
conservation:
4M 1 1
p(z)2+p2maxexp(−2κz)=p2∞, (7) ψWKB(z)∼=sp∞ sin(cid:18)2δϕWKB+ ¯hp∞(z−ζWKB)(cid:19),
(11)
and where the phase π/4 results from the WKB connection
formula,whichmatchestheoscillatingpartofthewavefunc-
which correspondsto a plane standing wave whose phase is
tion(6)tothedecayingpartintheclassicallyforbiddenregion
fixed to the value 1δϕ at z = ζ . As far as the
z <z . ThenormalizationoftheWKBwavefunction(6)has 2 WKB WKB
0 asymptotic WKB wave function is concerned, the evanes-
been chosen such that the incident and reflected waves both
centwavemirroristhusequivalenttoaneffectivedephasing
haveunitfluxindependentoftheasymptoticmomentum.
mirrorlocatedat the positionζ , wherethe atomic wave
In the asymptotic region, the atomic wave function is a WKB
is instantaneously reflected and phase shifted by the amount
superposition of two plane waves with wave vectors k∞ =
δϕ (as light on a mirror). By analogy with the classi-
∓p∞/¯h, which correspond to the incident and reflected calWcaKsBe(Section2.2),wecallthismirrortheWKBeffective
waves. The phaseshiftthatwe are interestedin at reflection
mirror. The dephasing character of this mirror results from
∆ϕ is related to the relative phase between these two
WKB theWKBphasefactorπ/4andthushasanonclassicalorigin.
plane waves. We define ∆ϕ by writing the asymptotic
WKB It holds as long as the WKB approximation remains valid,
WKBwavefunctionintheform:
i.e., as long as the incident de Broglie wavelength is small
compared to the decay length κ−1 of the evanescent optical
4M 1 1
z →+∞:ψWKB(z)∼= sin p∞z+ ∆ϕWKB . potential[8]. Furthermore,theevanescentwavemirrorisdis-
sp∞ (cid:18)¯h 2 (cid:19) persive because of the dependence of ζWKB on p∞. More
(8) precisely,theconditionforthemirrortobedispersiveisthat
∂2∆ϕ /∂p2 6= 0: a linear dependence of ∆ϕ on
WKB ∞ WKB
This definition of the phase shift takes as reference p∞ can always be removed by an appropriate choice of ab-
(∆ϕ = 0)astandingwaveinfrontofaninfinitelysteep solute phase reference. Finally, it is interesting to note that
WKB
barrier located at the origin z = 0. It is important to note the position ζWKB [equation (10b)] of the WKB effective
thatthisdefinitionissomewhatarbitrary. In[7],forexample, mirror differs from the classical effective mirror position ζ
cl
thereferenceisastandingwaveinfromtofaninfinitelysteep [equation(5)]bythequantityκ−1independentoftheincident
barrierlocatedatthe positionz = ζ ofthe effectiveclassi- atomicmomentum.Asaresult,theclassicalandtheWKBde-
cl
calmirror[equation(5)].Becauseζ dependsontheincident scription of the reflection processyield comparablephysical
cl
atomic momentum, however, this phase reference is not ab- pictures.
solute. Withourdefinitionofthephaseshift,∆ϕ isthe
WKB
phasecorrectioninthesituationwheretheevanescentoptical
potentialisapproximatedbyaninfinitelysteepbarrierlocated 3.SCHRO¨DINGERWAVEFUNCTIONAPPROACH
atthemirrorsurface[6].Bywritingtheasymptoticexpansion
oftheWKBwavefunction(6)intheform(8),oneobtains[7]
We now turn to the full wave-mechanical description of
atomicreflectionattheevanescentwavemirror.Thisdescrip-
∆ϕWKB(p∞)= π −2p∞ 1+ln pmax (9) tion is based on the analytical solution of the corresponding
2 ¯hκ (cid:20) (cid:18)2p∞(cid:19)(cid:21) Schro¨dinger equation, which allows an exact calculation of
=δϕWKB−2p∞ζWKB/¯h thephaseshiftatreflection.
with
δϕ =π/2 (10a) 3.1.SolutionoftheStationarySchro¨dingerEquation
WKB
ζWKB(p∞)=ζcl(p∞)+κ−1. (10b)
The full quantumdescriptionof atomic reflection consists
The orderof magnitudeof the WKB phase shift(9) is given insolvingexactlythestationarySchro¨dingerequationforthe
bythe ratiop∞/¯hκ, whichrepresentsthe phaseshiftassoci- atomicwavefunctionψ(z):
atedwiththefreepropagationofanatomofmomentump∞,
throughthespatialextentκ−1oftheevanescentopticalpoten- −¯h2 d2 +p2 exp(−2κz) ψ(z)=p2 ψ(z). (12)
tial. dz2 max ∞
(cid:18) (cid:19)
The physicalinterpretation of (10) becomes transparentif
wewritetheasymptoticWKBwavefunction(8)as3 Weusethechangeofvariable
3Thisdecompositionseparatesthephaseshiftintoaconstantandanessentiallylinearterm. Notethatsuchaninterpretationisnotalways
unambiguousbecauseonehastodecompose∆ϕ(p∞)=δϕ 2p∞ζ/¯h,whereδϕ [0,2π]andζareweaklydependentonp∞.
− ∈
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p
max
z →u= exp(−κz) (13)
¯hκ
which takes advantage of the invariance of the Hamiltonian
(1)underthetransformation
z →z+a
∀a, . (14)
p →e2κap
max max
(cid:26)
Equation(12)transformsintoaBessel-typeequation:
d2 d
u2du2 +udu −(u2−α2) ψ(u)=0, (15) FIG. 2. Comparison between the WKB (ψWKB) and the
(cid:18) (cid:19) Schro¨dinger (ψSchr) wave functions for the same parameters as in
Fig.1. These wave functions coincide both in the asyptotic region
(κz 1) and far inside the optical potential (κz 1) but sig-
whichonlydependsononedimensionlessparameter: ≫ ≪
nificantlydifferaroundtheclassicalturningpointz0,whereψWKB
diverges.
α=p∞/¯hκ. (16)
3.2.PhaseShiftoftheSchro¨dingerWaveFunction
Thesolutionsof(15)arelinearcombinationsoftheBessel
Theexactsolution(17)oftheSchro¨dingerequationallows
functionsI±iα(u). Twoboundaryconditionsimposeaunique ustoderiveexactlythephaseshiftexperiencedbytheatomic
solution:
wavefunctionatreflectionontheevanescentwavemirror.
(i) The wave function must vanish in the limit z → −∞ Followingthe definition(8)ofthe WKB phaseshiftatre-
(theprobabilityoftheatomsbeingintheregionz ≤z0inside flection, we define the Schro¨dinger phase shift ∆ϕSchr by
thepotentialbeingsmallcomparedtotheprobabilityofbeing writingtheasymptoticSchro¨dingerwavefunction(17)inthe
intheclassicallyallowedregionz ≥z ). form:
0
(ii) In the asymptoticregionz → +∞, the wave function
4M 1 1
isnormalizedinthesamewayastheWKBsolution[equation z →+∞:ψSchr(z)∼= sin p∞z+ ∆ϕSchr .
(6)]. sp∞ (cid:18)¯h 2 (cid:19)
As shown in Appendix B, these conditions lead to the (18)
solution4
By using (17) and the asymptotic expansion(u → 0) of the
BesselfunctionsI±iα(u),oneobtains(seeAppendixB)
4M πα 1
ψSchr(z)=sp∞ sinh(πα)2i(I−iα(u(z))−Iiα(u(z))). ∆ϕSchr(p∞)=−2αln p2mh¯aκx +2argΓ(1+iα), (19)
(cid:16) (cid:17)
(17) where Γ is the Euler gamma (factorial) function [9], and
whereargΓ(1+iα)istheargumentofthecomplexnumber
Γ(1+iα)definedasacontinuousfunctionofα.
TheSchro¨dingerwavefunction(17)andthecorresponding Theexact(∆ϕSchr)andthesemiclassical(∆ϕWKB)phase
WKBwavefunction(6)arerepresentedinFig.2asafunction shifts at reflection are represented in Fig. 3 as a function of
ofthedimensionlessparameterκz,inthecasep∞ =3h¯κand the dimensionless parameter α = p∞/¯hκ. One can clearly
p = 10h¯κ. Oneseesthatthewavefunctionsareingood distinguishbetweentwolimitingcases.
max
agreementinboththeasymptoticregion(κz ≫1)andfarin- In the limit α ≫ 1 (high incidentmomentum),where the
sidetheopticalpotential(κz ≪1),butthattheysignificantly incidentatomic de Broglie wavelength is small compared to
differaroundtheclassicalturningpointκz0 ∼=1.2(wherethe thedecaylengthκ−1oftheopticalpotential,theWKBandthe
WKBwavefunctionactuallydiverges). Schro¨dingerapproachesyieldcomparablephaseshifts[8].
4Wehaveneglectedanylossresultingfromatomictunnelingtothemirrorsurface. However,thetunnelingprobabilitycanbeestimatedby
thefluxofthewavefunctionatz=0.Noteaddedafterpublication:Toourknowledge,theexactsolution(17)fortheexponentialbarrierhas
beenfirstderivedbyJ.M.JacksonandN.F.Mottin1932,Proc.Roy.Soc.(London)Ser.A137,703.
1045
FIG.3. DependenceoftheWKB(∆ϕWKB)andoftheSchro¨dinger(∆ϕSchr)phaseshiftsvs.thedimensionlessincidentatomicmomen-
tumα=p∞/¯hκforpmax =10¯hκ. (a)∆ϕWKBand∆ϕSchrcoincideinthelimitofhighincidentatomicmomentum(α 1),whichthus
≫
correspondstoasemiclassicalregimeofreflection. (b)Inthelimitoflowincidentatomicmomentum(α 1),thesemiclassicaldescription
≪
oftheevanescentwavemirrordoesnotyieldthecorrectphaseshiftatreflection.Thiscorrespondstoapurequantumregimeofreflection.
Thiscorrespondstothesemiclassicalregimeofatomicreflec- [equation (22b)], where the atomic wave is instantaneously
tionconsideredinSection2.3. Usingequation(19),itispos- reflected (as light on a metallic mirror). The nondephasing
sible to derive the first correction to the semiclassical phase characterofthe Scho¨dingereffectivemirror[equation(22a)]
shift. Onethusfinds[9] resultsfromthefactthat,onthespatialscaleoftheincidentde
Broglie wavelength, the evanescentopticalpotential appears
1
α≫1:∆ϕSchr(p∞)=∆ϕWKB− +O(α−3). (20) tobeaninfinitelysteepbarrier. Inthequantumregimeofre-
6α
flection, it is thereforelegitimate to approximatethe evanes-
In the limit α ≪ 1 (low incident momentum), where the cent wave mirror by a hard barrier located at the position of
incident atomic de Broglie wavelength is larger than the de- the classical effective mirror for an asymptotic momentum
cay length of the evanescentopticalpotential, the WKB and p∞ ≈ ¯hκ. It is also interesting to note that, contrary to the
the Schro¨dinger approaches yield different phase shifts. In semiclassicalcase,thepositionζ oftheSchro¨dingereffec-
Schr
particular, in the limit α → 0+, ∆ϕWKB tendstoward π/2, tivemirrorisessentiallyindependentofp∞. Asa result, the
whereas ∆ϕ tends toward 0 (see Fig. 3b). The limit of evanescentwavemirrorisnolongerdispersiveinthequantum
Schr
lowincidentmomentumthuscorrespondstoapurequantum regime.5
regimeofreflection,whichcannotbeappropriatelydescribed
insemiclassicalterms. Inordertoobtaininthisregimearep-
resentation of the evanescent wave mirror in terms of an ef- 4.REFLECTIONOFANATOMICWAVEPACKET
fectivemirror,wewriteequation(19)inaformanalogousto
(9).Onethusfinds[9] In the experiments using effusive beams as a source of
atoms,itispossibletodescribetheincidentparticlesinterms
α≪1: of statistical mixtures of de Broglie waves having a well-
p
∆ϕSchr ∼=−2α γ+ln max (21) definedmomentum(planewaves). Inthatcase,thereflection
2h¯κ process at the evanescent wave mirror can be directly char-
=δϕSch(cid:16)r−2p∞(cid:16)ζSchr/¯h(cid:17)(cid:17) acterized using the results of the preceding sections. How-
withtheEulerconstantγ ∼=0.577and ever,insomeothersituations(forexample,whentheincident
atoms originate from an optical molasses where atom local-
δϕSchr ∼=0 (22a) ization takes place [10]), it is more appropriate to describe
ζSchr(p∞ ≪¯hκ)∼=ζcl(h¯κ)+γκ−1. (22b) theparticlesintermsofwavepackets(superpositionsofplane
waves). In such a case, each partial plane wave of incident
Equations(21)and(22)showthat, in thequantumregime momentum p∞ experiences a different phase shift at reflec-
ofreflection,theevanescentwavemirrorbehavesasanonde- tion ∆ϕSchr(p∞) (as given in Section 3.2), which shows up
phasingeffectivemirrorlocatedatthepositionz ≈ζ (h¯κ) inaspatialshiftofthecenterofthewavepacket.
cl
5Itisperhaps surprisingthat theevanescent wavemirrorisnot despersive whiletheSchro¨dinger phase shift(21) depends linearlyonthe
incidentmomentum. Infact,thisdependence isrelatedtothechoiceofreferenceforthephaseshift. Thus, bytakingasphasereferencea
standingwaveinfrontofaninfinitelysteepbarrierlocatedinz ∼=ζcl(h¯κ)+γκ−1,thephaseshiftatreflectioninthequantumregimewould
beindependentoftheincidentmomentum.
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Let us consider an atomic wave packet incident on an α¯ ≫1:ζwp =ζcl(p¯∞)+O(1/α¯2), (29)
evanescent wave mirror. In the asymptotic region, one may
writetheincidentpartoftheatomicwavefunctionψinc(z,t) theasymptoticwavepacketappearstobeinstantaneouslyre-
as flectedataneffectivemirrorlocatedinz =ζ [equation(5)].
cl
Thereflectionprocessisthusanalogoustothatofaclassical
ψinc(z,t)= dp∞ψ˜inc(p∞)exp −i p2∞t −ip∞z , particleofincidentmomentump¯∞.6 Notethatthepositionof
2M¯h ¯h
Z (cid:18) (cid:19) theeffectivemirrorfortheatomicwavepacketcorrespondsto
(23) ζ ,andnottoζ [equation(10b)].
cl WKB
In the limit α¯ ≪ 1 (quantumregimeof reflection), where
whereψ˜inc(p∞)denotesthe Fouriertransformofψinc. Dur- equation(27)reads
ingthereflectionprocess,eachpartialplanewaveexperiences
adifferentphaseshift∆ϕSchr(p∞)[equation(19)],sothatin α¯ ≪1:ζwp ∼=ζSchr(α¯ ≪1)∼=ζcl(h¯κ)+γκ−1, (30)
the asymptoticregion, the reflectionpartof the atomic wave
functionψ (z,t)reads
ref theevanescentwavemirrorbehavesasaninfinitelysteeppo-
tential barrier located at the position z ≈ ζ (h¯κ) + γκ−1,
ψref(z,t)=− dp∞ψ˜inc(p∞)× which instantaneously reflects the atomic wacvle packet. The
Z fact that this position coincides with that of the effective
×exp −i p2∞t +ip∞z +i∆ϕSchr(p∞) , (24) mirror for an atomic plane wave of incident momentum p¯∞
2M¯h ¯h
(cid:18) (cid:19) [equation (22b)] is not surprising because in the quantum
regime, the position of the effectivemirror ζ is indepen-
wheretheminussigninfrontoftheintegralresultsfromour Schr
dentoftheincidentatomicmomentum(seeSection3.2).Asa
peculiarchoiseofphaseorigin.
By assuming that ψ˜inc(p∞) is peaked around the average result,allpartialplanewavesofthewavepacketarereflected
atthesamebarrier,andthewavepacketbehavesinthesame
momentum p¯∞, it is possible to characterize the position
way.
z (t) of the center of the wave packet (23) or (24) via the
wp
methodofstationaryphase.Onereadilyfinds The positionζwp of the effectivemirrordescribingthe re-
flection process of the wave packet at the evanescent wave
zinc(t)=−p¯∞t, (25a) mirrorisrepresentedinFig. 4togetherwithζcl asafunction
wp M ofthedimensionlessparameterα¯ =p¯∞/¯hκ. Oneclearlydis-
zref(t)=−p¯∞t−¯h ∂∆ϕSchr(p∞) , (25b) tinguishesbetweenthesemiclassicalandthequantumregime
wp M (cid:18) ∂p∞ (cid:19)p¯∞ ofreflectionofthewavepacket.
wherezinc(zref)denotesthepositionofthecenteroftheinci-
wp wp
dent(reflected)wavepacket. Equation(25)showsthat,asfar
astheasymptoticwavepacketsareconcerned,theevanescent
wavemirrorbehavesasaninfinitelysteepeffectivemirrorlo-
catedatthepositionζ givenby
wp
¯h ∂∆ϕ
Schr
ζ =− . (26)
wp
2 (cid:18) ∂p∞ (cid:19)p¯∞
Substitutingfor∆ϕ in(26)using(19)gives
Schr
ζ =κ−1(ln(p /2h¯κ)−ReΨ(1+iα¯)), (27)
wp max
where α¯ = p¯∞/¯hκ is the dimensionlessparametergivenby
equation(16),andwhereΨisthedigammafunctiondefined
as
FIG. 4. Dependence of the positions of the wave packet (ζwp)
Ψ(x)=∂lnΓ(x)/∂x. (28) and the classical (ζcl) effective mirrors vs. the dimensionless av-
erage incident momentum of the wave packet α¯ = p¯∞/¯hκ for
As in the preceding section, we distinguish between two pmax = 10¯hκ. ζcl andζwp coincideinthesemiclassicalregimeof
limitingregimesofreflectionoftheatomicwavepacket. reflection(α¯ 1).Bycontrast,inthequantumregimeofreflection
≫
Inthelimitα¯ ≫ 1(semiclassicalregimeofreflection,see (α¯ 1),ζwp tendstowardaconstant,whereasζcl tendstoinfinity
Section3.2),whereequation(27)reducesto asα¯≪ 0+.
→
6WithintheframeworkoftheWKBapproach,itcanbeshownthatthisresultholdsforanymirrorpotentialvanishingatlargez.
1047
5.CONCLUSION Ω2/2
s= ≪1, (A1)
∆2+Γ2/4
Wehavepresentedadetailedtheoreticalinvestigationofthe
reflectionprocessofanatomicdeBrogliewaveatanevanes-
whereΩ=−dE /¯histheresonantRabifrequencythatchar-
0
centwavemirrorintheregimeofcoherentatomoptics. Our
acterizes the coupling between the atomic dipole d and the
calculation of the atomic phase shift at reflection using the
evanescentfieldofmaximumamplitudeE (attheevanescent
0
exactsolutionofthecorrespondingSchro¨dingerequationhas
wavemirrorsurface),Γisthenaturalwidthoftheatomicex-
allowedustoidentifytwolimitingregimesofreflection. The
cited state, and ∆ = ω − ω is the detuning between the
A
semiclassicalregimecorrespondstoincidentdeBrogliewave-
frequency (ω) of the evanescent wave and the atomic (ω )
A
lengthsmuchsmallerthanthedecaylengthoftheevanescent
frequency.Italsorequiresthattheincidentatomfollowsadia-
opticalpotentialandcanbesatisfactorilyaccountedforbythe
baticallytheopticalpotentialassociatedwiththelight-shifted
WKB method. The evanescentwave mirror then behavesas
ground-statelevel. This is achieved in the limit of large de-
a despersivedephasingmirror. In thequantumregimeofre-
tuningfrom resonance(we assume ∆ > 0, which allows an
flection, where the incidentatomic de Broglie wavelengthis
atom entering the optical potential in the ground-state to be
larger than the decay length of the evanescent potential, the
reflectedattheevanescentwavemirror):
evanescentwave mirrorbehavesas a nondispersivehard po-
tential barrier located in front of the actual evanescent wave
∆≫Γ (A2)
mirrorsurface.
In experiments using either a supersonic beam or laser-
cooledatomsacceleratedbytheearthgravityfield,themini- Intheregimewhere(A1)and(A2)arefulfilled,itispossible
mumachievableatomicincidentmomentumistypicallyofthe todescribetheatomreflectionprocessbymeansoftheHamil-
order of 10¯hκ. Under such conditions, the atomic reflection tonian(1). Note,however,thatforagivenlaserintensity,the
process can always be accounted for in semiclassical terms. maximumreflectibleatomicmomentump decreasesasthe
max
However,theexperimentalobservationofatomicreflectionin frequencydetuningincreases,asshownbytherelation
the quantum regime seems feasible, using, for example, an
evanescent wave mirror located at the summit of an atomic
p2 1
fountainwherethe atomicmomentumapproacheszero. An- max = ¯h∆s (A3)
2M 2
other,morechallengingpossibilitywouldbetoinvestigatethe
lowest bouncing modes of a gravitational cavity [6], which
correspondtoadeBrogliewavelengthoftheorderofthede- In fact, by designing the evanescentwave mirror with the
caylengthoftheevanescentopticalpotential,andthusrealize multilayercoatingtechnique[12],itispossiblefofulfillcon-
thequantumregimeofreflectionforthebouncingatoms. ditions(A1)and(A2)whilesimultaneouslybeingabletore-
Theexactderivationoftheatomicwavefunctionpresented flect atoms having high incident momentum (pmax ≫ ¯hκ).
inthispaper[equation(17)]canserveasastartingpointforthe Typicalexperimentalparametersareindicatedinthe table in
investigationof manyother effects. These include tunneling thecaseoftheD2lineof85Rb.
through the optical potential barrier, the influence of atomic
internalstatesonthereflectionprocess,and,especiallyinter-
esting, dipole-surfaceeffects (such as the Van der Waals in- Typicalexperimentalparametersforatomicreflectionatanevanes-
teraction),whichmaymodifythepotentialwehaveassumed centwavemirrorusingtheD2lineof85Rbatoms
here,andhencealsothephaseshiftatreflection.
Nota- Typical
Physicalparameters
tion value
ACKNOWLEDGEMENTS
Laserwavelength 780nm
Naturallinewidth Γ 6MHz
This work was supported by DRET (under Grant no.
Frequencydetuningfromresonance ∆ 5 104Γ
91055)andtheEEC(ScienceSC1-CT92-0778). ×
Maximumintensityoftheevanes-
centwave E2 104W/cm2
0
Saturationparameter s 6 10−4
×
APPENDIXA:EXPERIMENTALINVESTIGATIONOFTHE Maximalreflectedmomentum pmax 150h¯κ
REFLECTIONPROCESS
Probabilityofspontaneousemission
perreflectionforp∞ =pmax 2.5 10−3
×
We wouldliketo commentin thisappendixonthetypical Probabilityofnonadiabaticdepar-
experimentalparametersrequiredtoobserveatomicreflection turefromthelight-shiftedground-
onan evanescentwave mirror. Theregimeofcoherentatom statelevelforp∞ =pmax 8 10−15
≤ ×
opticsisrealizedprovidedthattheprobabilityofspontaneous Decaylengthoftheevanescentopti-
calpotential 1/κ−1 100nm
emissionduringreflectionisnegligible. Itcorrespondstothe ≈
limitofsmallsaturationoftheatomictransition: Reflectiontimeforp∞ =pmax τrefl ≈4Γ−1
1048
APPENDIXB:SOLUTIONOFTHESTATIONARY 4M πα 1
SCHRO¨DINGEREQUATION ψSchr(z)= p∞ sinh(πα)2i(I−iα(u(z))−Iiα(u(z))), (B7)
r
ThegeneralsolutionoftheBessel-typeSchro¨dingerequation(15) orequivalently
isalinearcombinationoftheBesselfunctionsI±iα(u).Thesefunc-
tionaredefinedby[11] 4M
ψSchr(z)= sinh(πα)Kiα(u(z)). (B8)
∞ π¯hκ
1 u 2n±iα r
I±iα = , (B1)
n!Γ(n+1 iα) 2
Xn=0 ± (cid:16) (cid:17)
where Γ denotes the Euler gamma function. They both diverge as
z u + accordingto[11]
→−∞⇔ → ∞
1 u
u→+∞:I±iα(u)∼= √2πue (1+O(1/u)). (B2)
[1] Balykin,V.I.andLetokhov,V.S.,1989,Phys.Today;Pritchard,
Asaresult,theonlylinearcombinationofI±iα(u)satisfyingthe D.E.,1991,AtomOptics,ICAP12,Zorn,J.C.andLewis,R.P.,
boundarycondition(i)ofSection3.1correspondstothedifferenceof Eds.(AnnArbor:AmericanInst.Physics),p.165.
Iiα(u)andI−iα(u).ThisdifferenceisproportionaltotheBessel-K [2] Balykin, V.I.,Letokhov, V.S.,Ovchinnikov, Y.B.,etal.,1987,
function: JETPLett.,45, 353; Kasevich, M.A., Weiss,D., andChu, S.,
1990,Opt.Lett.,15,607;Aminoff,C.G.,Bouyer,P.,andDes-
π 1
Kiα = (I−iα(u) Iiα(u)), (B3) biolles,P.,1993, C.R.Acad.Sci.,316,Ser.II,1535; Aminoff,
sinh(πα)2i −
C.G., Steane, A.M., Bouyer, P., et al., 1993, Phys. Rev. Lett.,
whoseasymptoticexpansionis[11]: 71,3083;Sigel,M.,Pfau,T.,Adams,C.S.,etal.,1993,Lecture
NotesinPhysics,Ehlotzky,F.,Ed.,(Heidelberg:Springer),vol.
u→+∞:Kiα(u)∼= 2πue−u(1+O(1/u)). (B4) 420,p.3.
[3] Cook,R.J.andHill,R.K.,1982,Opt.Commun.,43,258.
q
Equation (B4)shows that theatomicwavefunction decays very [4] Appl. Phys. B, 54, Special issue on Atom Interferometry and
rapidly(asexp[ (pmax/¯hκ)e−κz])insidethepotentialbarrier. Atom Optics, Mlynek, J., Balykin, V., and Meyste, P., Eds.,
−
Wefinallyconsidertheboundarycondition(ii)oftheSection3.1. Appl.Phys.B,54.
Intheasymptoticregionz + u 0+, theexpansion of [5] Balykin,V.I.andLetokhov,V.S.,1989,Appl.Phys.B,48,517.
→ ∞ ⇔ →
I±iα(u)isgivenbythefirsttermoftheseriesexpansion(B1): [6] Wallis,H.,Dalibard,J.,andCohen-Tannoudji,C.,1992,Appl.
Phys.B,54,407.
1
z→+∞:I±iα(u(z))∼= Γ(1+iα) exp(∓ip∞z/¯h [7] Opat,G.I.,Wark,S.J.,andCimmino,A.,1992,Appl.Phys.B,
| | 54,396.
iαln pmax iargΓ(1+iα) [8] Messiah, A., Quantum Mechanics (Amsterdam: North Hol-
± 2h¯κ ∓ land),vol.1,p.214.
(cid:16) (cid:17) (cid:17) (B5) [9] 1979, HandbookofMathematicalFunctions,Abramowitz,M.
andStegun,I.A.,Eds.(NewYork:Dover),p.253.
with[9]:
[10] Westbrook, C.I.,Watts,R.N.,Tanner,C.E.,etal.,1990, Phys.
πα Rev.Lett.,65,33;Grynberg,G.,Lounis,B.,Verkerk,P.,etal.,
Γ(1+iα) = . (B6)
| | sinh(πα) 1993,Phys.Rev.Lett.70,2249(andreferencestherein).
r [11] 1979, HandbookofMathematicalFunctions,Abramowitz,M.
Combining (B3), (B5), and (B6), one readly finds that the only andStegun,I.A.,Eds.(NewYork:Dover),p.374.
solutionofequation(15)satisfyingtheboundaryconditions(i)and [12] Kaiser,R.,Le´vy,Y.,Vansteenkiste,N.,etal.,1994,Opt.Com-
(ii)ofSection3.1is: mun.,104,234.
1049