Decoherence-free creation of atom-atom entanglement in cavity via fractional adiabatic passage

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by  Mahdi Amniat-Talab

Decoherence-free creationofatom-atom entanglement incavityvia fractionaladiabaticpassage
Mahdi Amniat-Talab1,2,∗ Ste´phane Gue´rin1,† and Hans-Rudolf Jauslin1‡
1LaboratoiredePhysique,UMRCNRS5027, Universite´ deBourgogne, B.P.47870, F-21078Dijon,France.
2Physics Department, Faculty of Sciences, Urmia University, P.B.165, Urmia, Iran.
(Dated:February1,2008)
We propose a robust and decoherence insensitive scheme to generate controllable entangled states of two
three-levelatomsinteractingwithanopticalcavityandalaserbeam. Lossesduetoatomicspontaneoustransi-
tionsandtocavitydecayareefficientlysuppressedbyemployingfractionaladiabaticpassageandappropriately
designed atom-fieldcouplings. Inthisschemethetwoatomstraversethecavity-mode andthelaserbeamin
opposite directionsas opposed to other entanglement schemes inwhichtheatoms arerequired tohavefixed
locationsinsideacavity.Wealsoshowthatthecoherenceofatravelingatomcanbetransferredtotheotherone
withoutpopulatingthecavity-mode.
5
0 PACSnumbers:42.50.Dv,03.65.Ud,03.67.Mn,32.80.Qk
0
2
Thephysicsofentanglementprovidesthebasisofapplica- gates[6,7,8,9,10]usingopticalcavitieshavebeenproposed.
n
tions such as quantum information processing and quantum Toavoiddecoherenceeffects,itismostconvenienttodesign
a
communications. Particles can then be viewed as carriersof transferstrategiesthatdonotpopulatethedecoherencechan-
J
quantumbitsofinformationandtherealizationofengineered nelsduringthetime-evolutionofthesystem.
6
2 entanglementisanessentialingredientoftheimplementation Inthispaperweproposeanalternativewaytoentangletwo
ofquantumgates[1], cryptography[2] andteleportation[3]. travelingatomsinteractingwith an opticalcavity anda laser

1 Thecreationoflong-livedentangledpairsofatomsmaypro- beam,basedon3-levelinteractionsinaΛ-configuration.This
v videreliablequantuminformationstorage. Theideaistoap- methodisbasedonthe coherentcreationof superpositionof
4 plyasetofcontrolledcoherentinteractionstotheatomsofthe atom-atom-cavitystatesviafractionalstimulatedRamanadi-
5
system inordertobringthemintoa tailoredentangledstate. abatic passage (f-STIRAP) [11], that keeps the cavity-mode
1
Theproblemofcontrollingentanglementisthusdirectlycon- and the excited atomic states unpopulated during the whole
1
0 nectedtotheproblemofcoherentcontrolofpopulationtrans- interaction. In Ref. [12], using f-STIRAP and a one-atom
5 ferinmultilevelsystems. dark state, a robust scheme to generate atom-atomentangle-
0 InthecontextofcavityQED,oneofthemainobstaclesto mentwasproposed,wherethetwotravelingatomsencounter
h/ realize atom-atomentanglementis the decoherenceresulting thecavity-modeonebyone.Here,thetwoatomsentersimul-
p fromthecavitydecay. Additionally,thecavitycouplestoan taneouslyintothecavityinsuchawaythatthesystemfollows
- excited state of the atom that undergoes spontaneous emis- a two-atomdark state that allowsusto keepadditionallythe
t
n sion. Regardingthese considerations,in recentyearsseveral cavity-modeempty.
a schemesto entangleatoms[4, 5] andtoimplementquantum WeconsiderthesituationdescribedinFig.1,wherethetwo
u atomsmoveinplanesorthogonaltothezaxisasfollows:
q
: z = z , x = x +v tcosθ , y = y +v tsinθ ,
v atom1 1 0 1 − 0 1 1 1 − 0 1 1
i z = 0, x =x +v (t τ)cosθ ,
X z0 2 2 0 2 − 2
y = y +v (t τ)sinθ , (1)
ar x0 2 − 0 2 − 2
cavity field where(x ,y ,z ),i=1,2arethecoordinatesofthei-thatom
i i i
(x ,y > 0), τ is the time-delay of the second atom with
0 0
d z
laser beam respect to the first one, θ is the angle that the i-th atom
i
x constructs with the positive direction of the x axis (θ1
0 ∈
[0,π/2[, θ ]π/2,π]),andv isthevelocityofthei-thatom.
2 i
∈
Sincethelaserfieldpropagatesindirectionofthey axis,the
atom2 time-dependentoptical phase of the laser field seen by each
x atomis
ϕ (t)=ω t kv tsinθ , ϕ (t τ)=ω t kv (t τ)sinθ ,
FIG.1: Geometricalconfigurationoftheatoms-cavity-lasersystem 1 L 1 1 2 L 2 2
− − − −
intheproposedscheme.Thepropagationdirectionofthelaserbeam (2)
isparalleltoyaxisandperpendiculartothepage. where k is the wavevectormagnitudeof the laser field. Fig-
ure 2 represents the linkage pattern of the atom-cavity-laser
system.ThelaserpulseassociatedtotheRabifrequencyΩ(t)
couplesthestates g and e ,andthecavity-modewithRabi
1
| i | i
frequencyG(t)couplesthestates e and g . TheRabifre-
∗Electronicaddress:[email protected] | i | 2i
†Electronicaddress:[email protected] quenciesΩ(t) and G(t) are chosenreal withoutloss of gen-
‡Electronicaddress:[email protected] erality. These two fields interact with the atom with a time
2
|e,0> |e,0> bewrittenas(¯h=1)
Ω G Ω G
1 1 2 2
|ag1 t,o0>m1−cav|git2 ,y1> |ag1 t,o0>m2−cav|gi2 ,t1y> H(t) = iX=1,2(cid:2)ωe|eiiihe|+(cid:0)Gi(t)a|eiiihg2|+H.c.(cid:1)
+ Ω (t)eiϕi(t) g e +H.c. +ω a†a, (4)
|e , g2 , 0> |g2 , e , 0> i | 1iiih | C
(cid:0) (cid:1)(cid:3)
Ω G G Ω
1 1 2 2 wherethesubscriptionthestatesdenotesthetwoatoms,ais
|g1 , g 2, 0> atom1−a|gt2 o, gm2 , 21>−cavity|g2 , g 1, 0> othfethaennaithoimlaitcioenxocpiteerdatsotratoef(tωheca=vitωymo=de0,)ω,eanisdthωeenisertghye
g1 g2 C
frequencyofthecavitymodetakingresonantω =ω =ω
C L e
FIG.2: Linkagepatternofthesystemcorrespondingtotheeffective whichimplieskv sinθ ω . Inthefollowingweconsider
i i L
Hamiltonian. the state of the atom1-a≪tom2-cavity system as A1,A2,n
| i
where A1,A2 = g ,e,g , and n = 0,1 is the number
1 2
{ } { }
ofphotonsinthecavity-mode.
delay, each of the fields is in one-photonresonancewith the
respective transition. The semiclassical Hamiltonian of this Regarding Figure 2, the subspace generated by the
S
systemintheresonantapproximationwhere states g1,g2,0 , e,g2,0 ,g2,g2,1 , g2,e,0 ,g2,g1,0 is
{| i | i| i | i| i}
decoupled under H from the rest of the Hilbert space of
Ω , G ω ,ω , ∂ϕ /∂t, ∂ϕ /∂t, (3) the system. If we consider the initial state of the system as
0 0 e C 1 2
| | | |≪ | | | | g ,g ,0 , the Hamiltonian of the system in the subspace
1 2
| i S
withΩ ,G thepeakvaluesoftheRabifrequencies,canthen willbe
0 0
0 Ω (t)e+iϕ1(t) 0 0 0
1
 Ω (t)e−iϕ1(t) ω G (t) 0 0 
1 e 1
H (t):=PHP = 0 G (t) ω G (t τ) 0 , (5)
P  1 C 2 − 
 0 0 G (t τ) ω Ω (t τ)e−iϕ2(t−τ) 
 2 − e 2 − 
 0 0 0 Ω (t τ)e+iϕ2(t−τ) 0 
 2 − 
whereP istheprojectoronthesubspace . TheeffectiveHamiltonianisthusgivenby
S
0 Ω (t) 0 0 0
1
Ω (t) kv sinθ G (t) 0 0
∂R  1 1 1 1 
Heff :=R†H R iR† = 0 G (t) kv sinθ G (t τ) 0 , (6)
P 1 1 1 2
− ∂t  0 0 G (t τ) kv si−nθ Ω (t τ) 
 2 − 1 1 2 − 
 0 0 0 Ω (t τ) k(v sinθ v sinθ )
 2 − 1 1− 2 2 
wheretheunitarytransformationRis mericsshowsthat,inpractice,thecondition(8)issatisfiedfor
∆ < Ω ,G /100. Assuming thisconditionallowsone to
1 0 0 0 0 con∼sid{er0∆ 0}kv sinθ 0 in Eq. (8), if we additionally
 0 e−iϕ1(t) 0 0 0  assumev ∼v 1 v. In1t∼hiscaseτ isthedelayofarrivalat
R= 0 0 e−iϕ1(t) 0 0 . 1 ∼ 2 ≡
thecenterofthecavity(x = y = 0)ofthesecondatomwith
 
 0 0 0 e−iϕ1(t)  respecttothefirstone.
 
 0 0 0 0 e−i[ϕ1(t)−ϕ2(t−τ)]  Thesystemistakentobeinitiallyinthestate g ,g ,0 ,
 (7) | 1 2 i
The dynamicsof the system is governedby the Schro¨dinger Φ(ti) = g1,g2,0 . (9)
| i | i
equationi∂ Φ(t) =Heff(t)Φ(t) .
∂t| i | i The goal is to transform it at the end of interaction into an
An essential condition for the STIRAP and f-STIRAP
atom-atomentangledstate
processes is the four-photon resonance between the states
|g1,g2,0iand|g2,g1,0iwhichmeans: |Φ(tf)i = cosϑ|g1,g2,0i+sinϑ|g2,g1,0i
|∆:=k(v1sinθ1−v2sinθ2)|≪|Ω0|,|G0|. (8) = (cosϑ|g1,g2i+sinϑ|g2,g1i)|0i, (10)
This condition can be achieved by controlof the velocity of where ϑ is a constant mixing angle (0 ϑ π/2), and
≤ ≤
atoms, andof the deflectionanglesof the atomsθ . Nu- the cavity-mode state factorizes and is left in the vacuum
i=1,2
3
state. The qubits are stored in the two degenerate ground
states of the atoms. The decoherence due to atomic sponta- 10
neousemissionisproducedifthestates e,g ,0 , g ,e,0 8 0.4
are populated, and the cavity decay {o|ccur2s iif |th2e stait}e m)6 • 0.3
µ
g2,g2,1 is populated during the adiabatic evolution of the d (4 0.2
| i
system. Therefore we will design the Rabi frequencies 2 0.1
Ω (t),G (t),Ω (t),G (t) in our scheme such that these 0
{ 1 1 1 2 } 0 5 10 z (1µ5 m) 20 25 30
statesarenotpopulatedduringthedynamics. 0
One of the instantaneous eigenstates (the two-atom dark 10 0.6
state) of Heff(t) which corresponds to a zero eigenvalue is 8 0.5
[10] m)6 • 0.4
µd ( 4 00..23
D(t) = C G1Ω2 g1,g2,0 Ω1Ω2 g2,g2,1 2 0.1
| i (cid:16) | i− | i 0
0 5 10 15 20 25 30
+ G2Ω1 g2,g1,0 , (11) z0 (µ m)
| i(cid:17)
where C is a normalization factor. The possibility of FIG. 3: Top panel: contour plot at the final time tf of |12 −
decoherence-free generation of atom-atom entanglement P|g1,g2,0i(tf)| as a function of z0 and d (black areas correspond
to approximately half population transfer) with the pulse parame-
arisesfromthefollowingbehaviorofthedarkstate:
ters as WL = 20µm, WC = 40µm, v = 2 m/s, λ = 780nm,
tl→imti ΩΩ12((tt)) =0, |D(ti)i∼|g1,g2,0i (12a) sPΩu|0mg2=,go2f2,10tihW(evtLffi),na+Gl0pPo=|gp2u,1lea0,0t0iioW(ntvsCf).inwBihonetttreoermmbelpadacinkaetela:rseTtaahsteecssoaPrmr|eees,pgp2ol,no0dti(fttoofr)athp+e-
proximatelyzeropopulation.Thewhitedotshowsspecificvaluesof
Ω (t)
lim 1 =tanϑ, z0anddusedinFig.4toobtainahalf-STIRAPprocess.
t→tf Ω2(t)
D(t ) cosϑg ,g ,0 +sinϑg ,g ,0 (12b)
f 1 2 2 1
| i∼ | i | i
velocityv(onthey =0planeatz =z line)throughanopti-
0
calcavityinitiallyinthevacuumstate 0 andthenencounters
ti <t<tf, G1(t)∼G2(t)≫Ω1(t),Ω2(t), the laser beam, whichis parallelto th|eiy axis(orthogonalto
D(t) Ω2(t)g1,g2,0 +Ω1(t)g2,g1,0 . (12c) thecavityaxisandthetrajectoryoftheatom).Thelaserbeam
| i∼ | i | i
is resonant with the e g transition. The distance be-
Equations(12a)and(12b)areknownasf-STIRAPconditions | i ↔ | 1i
tween the center of the cavity and the laser axis is d. The
[11, 12], and the condition (12c) guarantees the absence of
second atom, synchronized with the first one τ = 0, moves
populationinthestate g ,g ,1 duringthetime-evolutionof
| 2 2 i with the same velocity v on the y = 0 plane at z = 0 in
thesystem[13, 14]. Equation(12b)meansthattheRabifre-
theoppositedirectionwithrespecttothefirstatom. Thetrav-
quencies fall off in a constant ratio, during the time interval
eling atoms encounter the time-dependentand delayed Rabi
where they are non-negligible. We remark that this formu-
frequenciesofthecavity-modeandthelaserfieldsasfollows
lation opens up the possibility to implement f-STIRAP with
Gsuacuhssaiapnulpsuelsseesq.ueTnhcee cgaonalbiendtehseigfnoelldowininagcaisvittoysbhyowantahpa-t G1(t) = G0e−(vt)2/WC2 cos 2πz0 , (15a)
(cid:18) λ (cid:19)
propriatechoiceoftheparameters.
In an optical cavity, the spatial variation of the atom-field Ω1(t) = Ω0e−z02/WL2 e−(vt−d)2/WL2, (15b)
coupling for the maximum coupling TEM00 mode, resonant G2(t) = G0e−(vt)2/WC2, (15c)
withthe e g atomictransition,isgivenby
| i↔| 2i Ω2(t) = Ω0e−(vt+d)2/WL2 (15d)
G(x,y,z)=G0e−(x2+y2)/WC2 cos 2πz , (13) wherethetimeoriginisdefinedwhentheatomsmeetthecen-
(cid:18) λ (cid:19)
ter of the cavity at x = y = 0. The appropriate values of
where WL is the waist of the cavity mode, and G0 = z0 and d that lead to the f-STIRAPprocesscan be extracted
µ ωC/(2ǫ0Vmode)withµandVmoderespectivelythedipole from a contour plot of the final population P|g1,g2,0i(tf) :=
−mompentof the atomic transition and the effectivevolume of g1,g2,0Φ(tf) 2 as a function of z0 and d that we calcu-
|h | i|
thecavitymode. Thespatialvariationoftheatom-lasercou- lated numerically (see Fig. 3). The white dot in Fig. 3
plingforthelaserbeamofFig. 1is showsvaluesofz0anddtoobtainanf-STIRAPprocesswith
ϑ π/4(calledhalf-STIRAP).Ithasbeenchosensuchthat
Ω(x,z)=Ω0e−(x2+z2)/WL2, (14) at≃theendofinteractionP|g1,g2,0i(tf)≃P|g2,g1,0i(tf)≃0.5,
andthe populationsofthe otherstates of the subspace are
S
whereW is the waist of the laser beam, and Ω = µ /2 zero.
L 0
− E
with theamplitudeofthelaserfield. Figure1showsasitu- Figure4showsforparametervaluesassociatedtothewhite
E
ationwherethefirstatom,initiallyinthestate g ,goeswith dot in Fig. 3, (a) the time dependence of the Rabi frequen-
1
| i
4
(a) is split at 50% among the states g ,g ,0 , g ,g ,0 and is
MHz)5 G almost zero for the other states o|f1 2duriing| 2the1inteiraction
ncies (34 2 G1 tfhoer mGa0xi∼ma3lΩly0a.toTmh-iastocmaseenctoarnrgelsepdoSnstdasteto1/th√e2gegne,rgati,o0n o+f
e 1 2
u | i
Rabi Freq−01250 −40 −30 −20 Ω−210 Ω10 10 20 30 40 50 |pWgr2o,fi/glv1e,sr0efios(cid:1)prebΩcy(titva)edaliyan,bdtahGtiec(stpu)afofisfscaiwgenied.ttchAosnsTsdLuitmi=oinnWgofGLa/a(cid:0)duvisaasbniaadtnicTpiCtuyls=ies
C
(b) Ω T ,G T 1.
0 L 0 C
1 P ≫
|g,g,0〉 Thestate g ,g ,0 isstationarystateofthesystem. Ifthe
s0.8 1 2 | 2 2 i
n secondatomentersinsidethecavitybeforethefirstone(τ <
pulatio00..46 0), we can transfer completely the population from the state
Po0.2 P|g,g,0〉 |g1d,ugr2i,n0gitthoe|dgy2n,ga1m,i0cisw(sietheoFuitgp.o5p)u.laIntinpgartthiceuolathr,ergisvteantetshoatf
0 2 1 S
−50 −40 −30 −20 −10Time0 (µ s)10 20 30 40 50 thefirstatominitiallyispreparedinacoherentsuperposition
of the ground states αg +β g , that the second atom is
1 2
| i | i
FIG.4:(a)Rabifrequenciesofthecavity-modeandthelaserfieldfor initiallyinthestate g2 , andthatthecavity-modeisinitially
| i
twoatoms.(b)Timeevolutionofthepopulationswhichrepresentsa inthevacuumstate,atwo-atomSTIRAPwillcoherentlymap
two-atomhalf-STIRAP. thissuperpositionontothesecondatom:
es (MHz)567 G2 (a) G1 α|g1,g2,0i+β|g2,g2,0i→α|g2,g1,0i+β|g2,g2,0i. (16)
nci4 In summary,we haveproposeda robustand decoherence-
e
bi Frequ123 Ω2 Ω1 ffr-eSeTIsRchAePmetectohngieqnueeraitneΛat-osmys-taetmoms. enTthainsglsecmheemnte, uissinrogbtuhset
Ra−060 −40 −20 0 20 40 60 with respect to variations of the velocity of the atoms v, of
thepeakRabifrequenciesG ,Ω andofthe fielddetunings,
(b) 0 0
1 P|g,g,0〉 but not with respect to the parameters d,z0, describing the
s0.8 1 2 relative positionsof the laser beam and the cavity, shown in
n
atio0.6 Fig. 1. Ourschemecanbeimplementedinanopticalcavity
Popul00..24 P|g,g,0〉 cwaivthityGd0ec∼ohκer∼encΓe.isTGhe0n≫eceΩss0awryhiccohndisitsiaotnistfioedsuipnpprerasscttihcee
−060 −40 2 1−20 Time0 (µ s) 20 40 60 tfhoerGad0ap∼te3dΩv0al(useeseoFfigds.an4d,5z). Finorthgeivfe-nSTvaIRluAesPopfroWcCes,sWcaLn,
0
be determined from a contour plot of the final populations.
FIG.5: (a)Rabi frequencies ofthecavity-modeandthelaserfield
Decoherencechannelsare suppressed duringthe whole evo-
fortwoatomswiththepulseparametersasv=2m/s,WL=20µm, lutionofthesystem.Inthisscheme,asopposedtothescheme
WC = 40µm, λ = 780 nm, Ω0 = 2 MHz, G0 = 6.5 MHz, ofRef. [10],wedonotneedtofixtheatomsinsidetheoptical
z0 = 5.5µm, d = 6µm and τ = −9µs. (b) Time evolution of
cavitynortoapplytwolaserbeamsforeachoftheindividual
thepopulationswhichrepresentsatwo-atomSTIRAP.
atoms.
M.A-T.wishestoacknowledgethefinancialsupportofthe
cies of half-STIRAP for two atoms, and (b) the time evo- MSRTofIranandSFERE.Weacknowledgesupportfromthe
lution of the populations which shows that the population ConseilRe´gionaldeBourgogne.
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