Vacuum-Induced Coherence in Ultracold Photoassociative Ro-Vibrational Excitations

0.31 MB·English

by  Sumanta Das

Vacuum-Induced Coherence inUltracoldPhotoassociativeRo-Vibrational Excitations
Sumanta Das1,∗, Arpita Rakshit2 and Bimalendu Deb2,3
11RC 58/6 Uttar Raghunathpur, Teghoria Kolkata 700059, India
2Department ofMaterialsScience, and4RamanCenterforAtomic, MolecularandOpticalSciences,
2 Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India.
1
0 Weshowthatcoherencebetweentwoexcitedro-vibrationalstatesbelongingtothesamemolecularelectronic
2
configurationarisesquitenaturallyduetotheirinteractionwithelectromagneticvacuum.Forinitialpreparation

n ofamoleculeinthedesiredro-vibrationalstates,weproposetoemploythemethodofultracoldphotoassoci-
a ation. Spontaneous decay of theexcited moleculethen gives risetovacuum induced coherence between the
J excitedro-vibrationalstates. Wedemonstratetheoreticallyaninterestinginterplayofeffectsduetovacuumin-
1 ducedcoherenceandphotoassociation. WeapplyourtheorytophotoassociationofbosonicYtterbium(174Yb)
1 atomswhichappeartobeapromisingsystemforexploringsuchinterplay. Theeffectsdiscussedherecanbe
importantforcontrollingdecoherenceanddissipationinmolecularsystems.

]
h
PACSnumbers:32.80.Qk,34.50.Cx,34.80.Pa
p
-
m
Quantum coherences and interference form the basis of
o
many fascinating phenomena in atomic and molecular sys-
t J
a tems[1,2]. Itsapplicationsinspectroscopy,metrology,pho- V (r) 2
e
s. tovoltaics and information sciences have been most remark- J1
c able. Oneofthefamousexamplesofquantuminterferenceis
i |φ2!
s theFanoeffect[3,4]thatarisesfrominterferenceoftwocom- |φ1!
hy ptientuinugmosptatitceasl.tArannostihtieornnpoattahbwleaycoshinevreonlvcienpghdeinscormeteenaonndiscothne- ωL2 ωL1 γ2 γ1
p
[ vacuum-inducedcoherence(VIC) which arises due to quan-
tuminterferencebetweentwopathwaysofspontaneousemis- Vg(r)
3
v sions[5]. Itisknownthatunderappropriateconditions,VIC |E!b
can lead to populationtrappingin excitedstates [2, 5]. This
8
Continuum
8 can be utilized in manipulating environment induced relax-
5 ation processes in a wide variety of systems such as atoms,
0 ions, molecules, quantum dots [5–12], and has also been
.
9 found to be effective against decoherence in quantum infor-
FIG.1: (Color online) A schematic diagram for creationof coher-
0 mationprocessing[13]. OneofthekeyconditionsforVICto
ence in two excited ro-vibrational levels by photoassociation. The
1
1 occur is the nonorthogonalityof the dipole moments of two two-atom continuum |Eib is coupled to two ro-vibrational levels
: spontaneoustransitions. For atomic systems nonorthogonal- |v,J1i(|φ1i) and |v,J2i(|φ2i) of an excited molecular electronic
v ityisastringentconditiontoachieve. Possiblerealizationof stateviatwolasersoffrequenciesωL1andωL2,respectively.γ1and
Xi VICfor anexcitedatom interactingwith an anisotropicvac- γ2 arethespontaneousdecayratesatwhich|φ1iand|φ2idecayto
thecontinuum. Coherence arisesbetween thetwoexcitedmolecu-
uum[14–16]andutilizingthej =1/2→j =1/2transition
r lar statesfromthequantum interference of twospontaneous decay
a in198Hg+and139Ba+ionshavebeensuggested[17,18].Re-
pathways|φ1i→|Eiband|φ2i→|Eib.
cently,aproof-of-principleexperimentverifyingitspresence
has been performed in quantum dots [19]. In-spite of these
attempts,aclearsignatureofVICinatomicsystemsisyetto
molecular electronic configuration and differ only in rota-
beobtained.
tionalorvibrationalquantumnumbers. Withthetremendous
In this Rapid Communication we show that unlike atoms progressinhighprecisionphotoassociation(PA)spectroscopy
orions,VICarisesquitenaturallyinmolecularsystems. This [20, 21], low lying rotational levels can be selectively pop-
canbeattributedto thequantuminterferenceofspontaneous ulated in a molecular excited state. This occurs due to PA
emission pathways from two ro-vibrational levels of an ex- transitionsfromthecollisionalcontinuumoftwogroundstate
cited molecular electronic state to a ground molecular state. coldatoms. VICwillbesignificantinsuchanatom-molecule
In this case, the required nonorthogonality condition is sat- interfacesystemprovided(i)thereisnohyperfineinteraction
isfied naturally as the two excited states belong to the same
in the atoms that are photoassociated, (ii) there is no bound
stateclosetothedissociationcontinuumofthegroundmolec-
ularstateand(iii)excitedmolecularlevelshavelonglifetime.
However,toourknowledge,thepossibilityofVICinsuch
∗
[0]Current address: Max-Planck-Institut f´’ur Kernphysic, Heidelberg, Ger- PA systems hasnotbeenaddressedso far. Assuch, we pro-
many pose here a novel PA scheme for realization of VIC in an
2
atom-molecule system. We present results on the interplay where aˆ is the annihilation operator of the vacuum field
κ,σ
ofVICandtheeffectsinducedbyPAlaserssuchasdressing E~ and g (E′,κ) = −hφ | D~ · E~ (κ) | E′i is
vac n,σ n i vac b
ostfattehseccaonntbineucuomn.troWlleedshboywatphpartotphreialtiefemtaimniepuolfattihoeneoxfcittheids the dipole coupling with E~vac(κ) = ~ωκ/2ǫ0V ~εσ, κ
interplay. Moreover, our results predict that for an optimum being the wave number, σ the polariz(cid:16)atpion of the fie(cid:17)ld and
detuningandintensityofPAlasers,theinterplaybetweenVIC ~ωκ/2ǫ0V theamplitudeofthevacuumfield. Letthejoint
stateofthesystem-reservoiratatimetbeexpressedas,
andPAcanleadtocoherentpopulationtrappingintheexcited p
states.
|Ψ(t)i = a (t)|φ ,{0}i
n n
The basic idea of our scheme is depicted in Fig. 1. We
n
considerasourmodela systemoftwoexcitedmolecularro- X
vibrational levels |v,J1i(|φ1i) and |v,J2i(|φ2i) coupled to + dE′ bE′,κ,σ(t)|E′ib |{1κ,σ}i (4)
the two-atomgroundcontinuum|Ei vialasers. Initiallyei- Z κ,σ
b X
ther |φ1i or |φ2iis populatedor partiallybothare populated wherean andbE′,κ,σ aretheamplitudesofn-thexcitedstate
via photoassociation of cold atoms using two lasers L1 and andgroundcontinuum,respectively. Thestate|φ ,{0}icor-
n
L2offrequenciesωL1andωL2,tunednear|Eib →|φ1iand respondstomolecularexcitedstatewithfieldinvacuumand
|Eib →|φ2itransitions,respectively.Boththeexcitedlevels |E′ib|{1κσ}irefersto(ground)barecontinuumstatewithen-
|φ1iand|φ2idecayspontaneouslytothesamegroundcontin- ergyE′andonephotoninmodeκofpolarizationσ.Usingthe
uum with decay rates γ1 and γ2, respectively. The Hamilto- standardWigner-Weisskopfapproach[2],afteralongalgebra
niangoverningthedynamicsofthissystemcanbewrittenas weobtain,
H=H +H ,whereH isthecoherentpartinvolvingPA
c inc c
couplingsandisgivenby, a˜˙1 = (−G1−iδ12)a˜1−G12a˜2 (5)
2 a˜˙2 = −G2a˜2−G21a˜1 (6)
Hc = ~ωbn |φnihφn |+ E′ |E′ib bhE′ |dE′ wherea˜n isthemodifiedamplituderelatedtoan bytransfor-
nX=1 Z mations,a˜1 = a1exp[i(ω˜1−δ12)t] anda˜2 = a2exp[iω˜2t].
+ 2 ΛnE′e−iωLntSˆE†′ +H.C. dE′. (1) Texhperedsrseisosnedarferegqiuveenncybyω˜n~ω˜annd=the d|eAtunnEin|2gEδd12Einanthdeδa1b2ov=e
Z nX=1n o (ωL1−ωL2)+(ω˜1−ω˜2). TheGnisthedecayconstantofthe
R
Here ~ω are the binding energies of the bound states n-thboundstate,givenby
bn
|φni(n= 1,2);|E′ib isthebarecontinuumstateandΛnE′ = 1
hφn | D~n·E~Ln | E′ib isthelasercouplingforthetransition Gn = ~2 dE′ dt′ |gnσ(E′,κ)|2
from the n-th bound state to the bare continuum|E′ib. The Z Z Xκ,σ
vectorsD~n andE~Ln arethedipolemomentandelectricfield exp i ω +ω˜ − E′ −ω (t−t′) (7)
of the laser associated with the n-th transition, respectively. Ln n ~ κ
(cid:20) (cid:18) (cid:19) (cid:21)
The operator Sˆ† = |φ i hE′ | is a raising operator. H is
E′ n b c A key feature of Eqs. (5 − 6) is the coupling between the
exactly diagonalizable[22, 23] in the spirit of Fano’s theory
amplitudesviathecrossdampingterm
[3]. TheeigenstateofH isthedressedcontinuumexpressed
c
as 1
Gnn′ ≃ ~2 dE′ dt′ gnσ(E′,κ)gn∗′σ(E′,κ)
|Eid =A2E|φ2i+A1E|φ1i+ CE′(E)|E′ibdE′ (2) Z Z Xκ,σ E′
Z exp i ω +ω˜ − −ω (t−t′) (8)
Ln n ~ κ
with the normalization condition hE′′|Ei = δ(E − E′′). (cid:20) (cid:18) (cid:19) (cid:21)
HereAnE = ΛnE(qΓn′/2+δn′E)/~[(δ1E +iΓ1/2)(δ2E + where n 6= n′. For simplicity in writing the above equa-
iΓ2/2) − Γ1Γ2(q − i)2/4] with n′ 6= n and CE′(E) = tion we have assumed (ωL1 + ω˜1) ≃ (ωL2 + ω˜2). It is
δ(E −E′)+(A1EΛ1E′ +A2EΛ2E′)/(E −E′). The term important to understand that G12 arises due to quantum in-
Γn(E) = 2π|ΛnE|2/~, is the stimulated line width of the terferenceofthespontaneousemissionpathwaysresultingin
n-thboundstate duetocontinuum-boundlasercouplingand VIC between the excited states amplitudes. Summing over
δnE = E/~ − δn with detuning δn = ωbn − ωLn. Here the vacuum modes and then carrying out the time integral
q = V12/(πΛ1EΛ2E)isanalogoustothewell-knownFano’s under Born Markov approximation, we finally obtain Gn =
iqwnhtpTeearhrraeeacm‘tiiPnoecntoe’orhsfet[ar3vnea]ndcwtsupfuiotamhrrtpVfiore1fi2nldtchi=wepaiHtlPhavmtahRlieultdeso.Eynsi′taΛenm1EHa′iΛnndc2Eids′e/gs(icEvreibn−ebsyEt,h′e) ((G3ω1π2Lǫ10=+~cω˜3G)1−2−11RE=d′/E~(′)3(3ωǫ/0L2~(nωc3+Lπ2ω)˜+−n1ω−˜R2Ed−E′/E′~h′φ)/31~||h)Dφ~31/n2||E.D~′niS|hEEol′′vi|ibD~n|2g2|aφtnh2dei
coupledEqs. (5)and(6)analytically,weobtain
2
H = dE′ g (E′,κ)Sˆ† aˆ e−iωκt+H.c. (3) a˜1(t) = c1−ez−t+c1+ez+t (9)
inc Z "n=1κ,σ n,σ E′ κ,σ # a˜2(t) = c2−ez−t+c2+ez+t (10)
XX
3
1 1
ρnn0000....2468 ρρ1212 ρ1100..68 III111 === 025...100 WWW///cccmmm222
0.4
0
0 2 4 γt 6 8 10
1 0.2
δ = 0 MHz
ρ11000...468 δδ222 == 13 MMHhzz 00 3 6 γt 9 12 15
0.2 1
0
0 2 4 γt 6 8 10 0.8 I = 0.1 W/cm2
1 1
ρ22000...468 δδδ222 === 301 MMMHHhzzz ρ2200..46 II11 == 25..00 WW//ccmm22
0.2
0.2
0 0
0 2 4 6 8 10 0 3 6 9 12 15
γt γt
FIG.2: (Coloronline)UppermostPanelshowstheplotsofρ11 and FIG.3: (Coloronline)SameasinFig. 2,butfordifferentvaluesof
ρ22 inthe absence of laser which isthe case of normal VIC. Both intensityoffirstlaser,I1,keepingtheintensityofsecondlaserfixed
ρ11 and ρ22 go to zero in the long time limit. In the middle and at0.1mW/cm−2,keepingδ1 = δ2 = 0. Thepopulationistrapped
lowerpanel,theexcitedstatepopulationρ11andρ22areplottedasa atexcitedstatesforintensityof2W/cm−2.
functionofdimensionlesstimeγtfordifferentvaluesofthedetun-
ingofthesecond laser, δ2, keeping δ1 = 0. Atδ2 = 1MHz, the
populationistrappedbetweenthetwoexcitedstates. Theintensities
ofL1andL2are50mWcm−2and0.1mWcm−2,respectively. trappingintheexcitedstates. Whenδ12 =0,ρ11(t →∞) =
ρ22(t → ∞) = [ρ11(0)+ρ22(0)−2Re[ρ12(0)]]/4become
exactly same as normal VIC case [2]. It is worthwhile to
where, z± = 21[−iδ12 − G+ ± Ω] and c1± = emphasize that the results given in Eqs. (9), (10) and (12)
[±(−iδ12 − G− ± Ω)a˜1(0) ∓ 2G12a˜2(0)]/(2Ω) and c2± = are general because they are applicable to any PA coupling
[±(iδ12 + G− ± Ω)a˜2(0) ∓ 2G12a˜2(0)]/(2Ω) with Ω = regime.
(G−+iδ12)2+4G122,G− =G1−G2andG+ =G1+G2. ForexperimentalrealizationofVIC,ourmodelcanbeap-
pInthelimitwhenlaserintensitiesgoingtozero(weakcou- pliedtothespinforbiddenintercombinationtransition1S0−3
pling), we find ω˜n → (ωbn −ωLn), thus Gn reduces to the P1 ofbosonic174Yb[24–26]whichhasnohyperfineinterac-
usualdampingconstantγn.Moreover,forlowenergywehave tion. The onlymoleculargroundelectronicstate of174Yb is
G12 ≃γ12 = p(3(πωǫb10~ωcb32))3hφ1 |D~1·D~2 |φ2i. (11) r1TeΣhpegreewsxehcniitctsehdthcsoetarorteensslpy|oφbnnadirsectacono1nbStei0ncu+huo1msSen|0Eaaistbrlooo-nfvgoibusrreapmtaioroandtaeilollnaevnaednlds.
Thus in the absence of lasers, the model reduces to normal inlongrangeOu+ statethatcanbepopulatedbyPA.Forillus-
tration, we specifically consider excited ro-vibrationallevels
VICcaseinV-typesystem[2].Theaboveequationshowsthat
γ12 vanishes if the molecular transition dipole moments D~1 |φ1i = |v = 118,J = 1iand|φ2i = |v = 118,J = 3i[25].
andD~ areorthogonal. InourmodelD~ andD~ arethetran- According to the selection rules of continuum-boundtransi-
2 1 2
sition dipolemomentsbetweenthe same groundand excited tions,theminimumpartialwave(l)thatbecoupledto|φ2iby
PA is d wave (l = 2). Usually at ultracold temperatures, d
electronicstates,thereforetheyareessentiallynonorthogonal.
wavescatteringamplitudebecomesinsignificantduetolarge
theTchoeheexrecnitceedρs1ta2te=pa˜o1pa˜u′2l∗atcioannsbρe1o1b=tai|na˜e1d|2f,roρm22E=qs|.a˜2(9|2)aanndd c1n7ee4nnYttrdbif-uwagraeavleebxsachrerapipeteri.orneBasuoltninagnrtocheuenasdtetnsetsmaetpetehsracatatutitrteeresixnahgsiblpoirwtospaaesrp2tir5eosmµoKif-
(10). Explicitly
[24,27].
e−2G1t Wenowdiscussournumericalresults. InFig. 2,weshow
ρ (t) = 2 AΩ2−Bδ2 ρ (0)
nn 4Ω2 12 nn thedynamicalbehaviorofpopulationsρ11 andρ22 asafunc-
−8BG12(cid:2)2ρn(cid:8)′n′(0)+8BG(cid:9)12δ12Im[ρ12(0)] tion of scaled time γt. The uppermostpanelshowspopula-
−8G12ΩRe[ρ12(0)]sinh(Ωt)], (12) tionsintheabsenceofthelasersassumingγ1 = γ2 = γ12 =
γ = 2.29MHz[28]. Theshorttimedynamicsclearlyshows
wheren′ 6= n,A = [1+cosh(Ωt)]andB = [1−cosh(Ωt)] exchange of population between |φ1i and |φ2i due to VIC.
and we have considered G1 = G2. At 4G12 = [4G122 −δ122], Inthelowertwo panels, weplotρ11 andρ22 ofEq. (12)for
itfollowsfromaboveequationthatρ11andρ22becometime- different values of δ2, keeping δ1 fixed. We find that, as δ2
independentin long time limit meaning coherentpopulation increasesuptoanoptimumfrequency,thelifetimeofboththe
4
0 0.06 electromagnetic vacuum. We have analyzed the effects of
-0.05 0.03 PA lasers on VIC. Our results show that under certain con-
-0.1 ditionspopulationcanbetrappedinexcitedstates. Notethat
ρ]Re[12-0-0.1.25 ρIm[]12-0.003 VWIeChcaavnebdeisbcuesstseredaVlizICabilneafonraatonmid-emaolilzeecdultehrienetelrefvaeclessyysstteemm.
-0.25 -0.06 where one of the levels is the collisional contunuum of two
-0.09 groundstateatoms.Ourmodelreliesontheconditionthatthe
0 2 4 γt 6 8 10 0 2 4 γt 6 8 10
twoexcitedmolecularlevelsdecaytothesamecontinuum. It
maybeveryhardtofulfillthisconditioninalkalimetalatoms
0 0.06
since they have severalgroundcontinuadue to hyperfinein-
-0.05
ρRe[]12-0-.01.51 ρIm[]120.03 tshetaarsatecnstoioinnhsyg.proeFrufiunnrdtehmeinromtleeorcaruecltaieorxncc,iotientdfihglauesvraeotlnisolynm.oanSyeindgcerecoabuyonsdtoocnboiocnutYinnbd-
-0.2 0
uum. Photoassociatedexcitedrotationallevelswith largevi-
-0.25
brationalnumbersas considered in this work are unlikely to
-0.03
0 2 4 γt 6 8 10 0 2 4 γt 6 8 10 decay to any bound level since their Frank Condon overlap
with the least bound state close to the ground continuum is
FIG.4: (Coloronline)Plotofrealandimaginarypartofρ12against verysmall. Therefore,bosonicYbappearstobeapromising
γtfordifferentdetuningsofsecondlaser(twouppersubplots)and
candidate for exploringVIC. The manipulationof VIC with
differentintensitiesofthefirstlaser(twolowersubplots). Allother
parametersoftheuppersubplotsissameasinFig.2whiletheother PA may be importantfor controlling decoherenceand dissi-
parametersoflowersubplotsarethesameasinFig. 3. Inthetwo pation in cold molecules. It may also be used for coherent
uppersubplots,thedetuningδ2 =0(blacksolidline),δ2 =1MHz controlofatom-moleculeconversionandoptical[29,30]and
(reddottedline)andδ2 = 3MHz(bluedashed lines)whileinthe magneto-opticalFeshbachresonance[23].
lowertwosubplotstheintensityI1=0.1Wcm−2(blacksolidline),
I1 = 2.0 W cm−2 (red dotted line) and I1 = 5.0 W cm−2 (blue AR gratefully acknowledge support from CSIR, Govern-
dashedlines) mentofIndia.
excitedlevelsalso increases. Thenatan optimumfrequency
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