Quantum phase gate for optical qubits with cavity quantum optomechanics

0.13 MB·English

by  Muhammad Asjad

Quantum phase gate for optical qubits
with cavity quantum optomechanics
MuhammadAsjad,PaoloTombesi,andDavidVitali∗
SchoolofScienceandTechnology,PhysicsDivision,UniversityofCamerino,62032Camerino
(MC),Italy
5 ∗[email protected]
1
0
2
Abstract: We show that a cavity optomechanical system formed by a

r mechanical resonator simultaneously coupled to two modes of an optical
a
M cavitycanbeusedfortheimplementationofadeterministicquantumphase
gatebetweenopticalqubitsassociatedwiththetwointracavitymodes.The

7 scheme is realizable for sufficiently strong single-photon optomechanical
1 couplingintheresolvedsidebandregime,andisrobustagainstcavitylosses.

] © 2015 OpticalSocietyofAmerica
h
p OCIS codes: (270.5585) Quantum information and processing; (190.3270) Kerr effect;
- (270.0270)QuantumOptics.
t
n
a
u Referencesandlinks
q 1. M.H.DevoretandR.J.Schoelkopf,“Superconductingcircuitsforquantuminformation:Anoutlook,”Science
[ 339,1169–1174(2013).

2. M.A.Nielsen,andI.L.ChuangQuantumComputationandQuantumInformation(Springer,2000).
3
3. D.Vitali,M.Fortunato,P.Tombesi,“CompletequantumteleportationwithaKerrnonlinearity,”Phys.Rev.Lett.
v
85,445–448(2000).
8
4. D.Bouwmeester,J.-W.Pan,K.Mattle,M.Eibl,H.Weinfurter,A.Zeilinger,“Experimentalquantumteleporta-
2
tion,”Nature390,575–578(1997).
0 5. S.Lloyd,“Almostanyquantumlogicgateisuniversal,”Phys.Rev.Lett.75,346–349(1995).
4 6. Q.A.Turchette,C.J.Hood,W.Lange,H.Mabuchi,H.J.Kimble,“Measurementofconditionalphaseshiftsfor
0 quantumlogic,”Phys.Rev.Lett.75,4710–4713(1995)
. 7. E.Knill,R.Laflamme,G.J.Milburn,“Aschemeforefficientquantumcomputationwithlinearoptics,”Nature
1
409,46–52(2001)
0
8. E.Arimondo,“Coherentpopulationtrappinginlaserspectroscopy,”ProgressinOptics,XXXV,257–354(1996).
5
9. S.E.Harris,“Electromagneticallyinducedtransparency,”Phys.Today50(7),36–42(1997).
1
10. L.V.Hau, S.E.Harris, Z.Dutton, and C.H. Behroozi, “Light speed reduction to 17metres per second in an
:
v ultracoldatomicgas,”Nature397,594–598(1999).
i 11. M.D.Lukin,andA.Imamog˘lu,“Nonlinearopticsandquantumentanglementofultraslowsinglephotons,”Phys.
X Rev.Lett.84,1419–1422(2000).
12. M.D.Lukin,andA.Imamog˘lu,“Controllingphotonsusingelectromagnetically inducedtransparency,”Nature
r
a 413,273–276(2001).
13. C.Ottaviani,S.Rebic,D.Vitali,andP.Tombesi,“Quantumphase-gateoperationbasedonnonlinearoptics:Full
quantumanalysis,”Phys.Rev.A73,010301(R)(2006).
14. C.Ottaviani,S.Rebic,D.Vitali,andP.Tombesi,“Crossphasemodulationinafive-levelatomicmedium:semi-
classicaltheory,”Eur.Phys.J.D40,281–296(2006).
15. S.Rebic,C.Ottaviani,G.DiGiuseppe,D.Vitali,andP.Tombesi,“Assessmentofaquantumphasegateoperation
basedonnonlinearoptics,”PhysRevA74,032301(2006).
16. A.V.Gorshkov,J.Otterbach,M.Fleischhauer,T.Pohl,andM.D.Lukin,“Photon-photoninteractionsviaRyd-
bergblockade,”Phys.Rev.Lett.107,133602(2011).
17. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg
atoms,”Phys.Rev.Lett.107,213601(2011).
18. T.Peyronel, O.Firstenberg, Q.Liang,S.Hofferberth, A.V.Gorshkov,T.Pohl,M.D.Lukin,andV.Vuletic,
“Quantumnonlinearopticswithsinglephotonsenabledbystronglyinteractingatoms,”Nature488,57-60(2012).
19. V.Parigi,E.Bimbard,J.Stanojevic,A.J.Hilliard,F.Nogrette,R.Tualle-Brouri,A.Ourjoumtsev,andP.Grangier,
“Observationandmeasurementofinteraction-induced dispersiveopticalnonlinearities inanensembleofcold
Rydbergatoms,”Phys.Rev.Lett.109,233602(2012).
20. O.Firstenberg, T.Peyronel,Q.Liang,A.V.Gorshkov,M.D.Lukin,andV.Vuletic, “Attractive photonsina
quantumnonlinearmedium,”Nature502,71–75(2013).
21. B.He,A.V.Sharypov,J.Sheng,C.Simon,andM.Xiao,“Two-photondynamicsincoherentRydbergatomic
ensemble,”Phys.Rev.Lett.112,133606(2014).
22. J.Volz,M.Scheucher,C.Junge,andA.Rauschenbeutel,“Nonlinearp phaseshiftforsinglefibre-guidedphotons
interactingwithasingleresonator-enhancedatom,”Nat.Photonics.8,965–970(2014).
23. G.S.AgarwalandS.Huang,“Electromagnetically inducedtransparencyinmechanicaleffectsoflight,”Phys.
Rev.A81,041803(2010).
24. S.Weis,R.Rivie`re,S.Dele´glise,E.Gavartin,O.Arcizet,A.Schliesser,andT.J.Kippenberg,“Optomechanically
InducedTransparency,”Science,330,1520–1523(2010).
25. J.D.Teufel,D.Li,M.S.Allman,K.Cicak,A.J.Sirois,J.D.Whittaker,andR.W.Simmonds,“Circuitcavity
electromechanicsinthestrong-couplingregime,”Nature471,204–208(2011).
26. A.H.Safavi-Naeini, T.P.M.Alegre,J.Chan,M.Eichenfield,M.Winger,Q.Lin,J.T.Hill,D.E.Chang,and
OPainter,“Electromagnetically inducedtransparencyandslowlightwithoptomechanics,”Nature472,69–73
(2011).
27. M.Karuza,C.Biancofiore,M.Bawaj,C.Molinelli,M.Galassi,R.Natali,P.Tombesi,G.DiGiuseppe,andD.
Vitali,“Optomechanicallyinducedtransparencyinamembrane-in-the-middlesetupatroomtemperature,”Phys.
Rev.A88,013804(2013).
28. K.Hammerer,C.Genes,D.Vitali, P.Tombesi,G.J.Milburn,C.Simon,andD.Bouwmeester, “Nonclassical
statesoflightandmechanics”,in“CavityOptomechanics:Nano-andMicromechanicalResonatorsInteracting
withLight,”(SpringerBerlinHeidelberg),editedbyM.Aspelmeyer,T.J.Kippenberg,F.Marquardt,pag.25–56,
(2014).
29. S. Mancini, and P. Tombesi “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065
(1994).
30. C.Fabre,M.Pinard,S.Bourzeix,A.Heidmann,E.GiacobinoandS.Reynaud,“Quantum-noisereductionusing
acavitywithamovablemirror,”Phys.Rev.A49,1337–1343(1994).
31. D.W.C.Brooks,T.Botter,S.Schreppler,T.P.Purdy,N.Brahms,andD.M.Stamper-Kurn,“Non-classicallight
generatedbyquantumnoisedrivencavityoptomechanics,”Nature,448,476–480(2012).
32. A.H.Safavi-Naeini,S.Gro¨blacher,J.T.Hill,J.Chan,M.Aspelmeyer,O,Painter“Squeezedlightfromasilicon
micromechanicalresonator,”Nature500,185–189(2013).
33. T.P.Purdy,P-L.Yu,R.W.Peterson,N.S.Kampel,andC.A.Regal“Strongoptomechanicalsqueezingoflight,”
Phys.Rev.X3,031012(2013).
34. M.Ludwig,A.H.Safavi-Naeini,O.Painter,andF.Marquardt,“Enhancedquantumnonlinearitiesinatwo-mode
optomechanicalsystem,”Phys.Ref.Lett.109,063601(2012).
35. K.Stannigel,P.Komar,S.J.M.Habraken,S.D.Bennett,M.D.Lukin,P.Zoller,andP.Rabl,“Optomechanical
quantuminformationprocessingwithphotonsandphonons,”Phys.Rev.Lett.109,013603(2012).
36. W.-Z.Zhang,J.Cheng,andL.Zhou,“Quantumcontrolgateincavityoptomechanicalsystem,”J.Phys.B48,
015502(2015).
37. C.Baker,W.Hease,D.-T.Nguyen,A.Andronico,S.Ducci,G.Leo,andI.Favero,“Photoelastic couplingin
galliumarsenideoptomechanicaldiskresonators,”Opt.Express22,14072–14086(2014)
38. K.C.Balram,M.Davanc¸o,J.Y.Lim,J.D.Song,andK.Srinivasan,“Movingboundaryandphotoelasticcoupling
inGaAsoptomechanicalresonators,”Optica,1,414–420(2014).
39. S.M.Meenehan,J.D.Cohen,S.Gro¨blacher, J.T.Hill,A.H.Safavi-Naeini, M.Aspelmeyer,andO.Painter,
“Siliconoptomechanicalcrystalresonatoratmillikelvintemperatures,”Phys.Rev.A90,011803(R)(2014).
40. D.J.Wilson,V.Sudhir,N.Piro,R.Schilling,A.Ghadimi,andT.J.Kippenberg,“Measurementandcontrolofa
mechanicaloscillatoratitsthermaldecoherencerate,”arXiv:1410.6191[quant-ph].
41. J.F.Poyatos,J.I.Cirac,andP.Zoller,“Completecharacterization ofaquantumprocess:Thetwo-bitquantum
gate,”Phys.Rev.Lett.78,390-393(1997).
42. D.F.Walls,andG.J.MilburnQuantumOptics(Springer,Berlin1994).
1. Introduction
Simple quantum information protocols and quantum gates have been recently implemented
withhighfidelityintrappedionsandincircuitcavityQED(seee.g.,Ref.[1]forareview).In
an efficientquantumnetwork,the informationelaboratedbya solid state processorat a node
shouldbethenrobustlyencodedinsingle-photonqubitsforlong-distancecommunicationand
distributionofquantuminformation.Thepossibilitytoimplementhigh-fidelitytwo-qubitgates
betweensingle-photonswouldgreatlyfacilitatesuchquantuminformationrouting;anexample
is provided by perfect Bell-state discrimination for quantum teleportation and entanglement
swapping [2], which could be implementeddeterministically if a quantumphase gate (QPG)
betweensinglephotonqubitswouldbeavailable[3].ItiswellknownthattoobtainsuchaQPG
one needsnonlinearitiesimplicit in post-processingmeasurementsor a nonlinearsystem [4].
A particularlyconvenientQPG is obtainedwhenthe conditionalphase shiftbetween the two
photonic qubit is equal to p , because in this latter case the QPG is equivalent, up to local
unitarytransformations,to a C-NOTgate[2,5].Using all-opticalsolutionsthisisnoteasy to
achievebecausetoprocesstheinformationoneneedsstrongphoton-photoninteraction.Infact,
to implement quantum information with photons, a nonlinear interaction is needed either to
buildatwo-photongateoperation[6](butthecommonlyachievablec 3 susceptibilityfactoris
typicallytoosmall),orthenonlinearityimplicitatthedetectionstageinlinearopticsquantum
computation[7].
Then, one has to think differentlyas for example using electromagneticallyinducedtrans-
parency (EIT) [8,9]. Indeed the possibility of reducing the speed of light in atomic media
with L -type levels was proposed and experimentally obtained [10]. Although the two weak
fields wave functionstravelingwith slow groupvelocitycan have a veryhigh nonlinearcou-
pling when they propagate in an atomic media with L -like level configurations [11,12], we
need, however,to obtainthe same result atthe single photonleveland thisis extremelyhard
to achieveor evenimpossible with L -likeconfigurations.A fullquantumanalysishasshown
thatalargenonlinearcross-phaseshiftisachievableusinganatomicMlevelstructure[13–15],
but it was also shown that a trade-offbetween the size of the conditionalphase shift and the
fidelityofthegateexists.Thiscanbeavoidedinthetransientregime,whichishoweverexper-
imentallychallenging.Morerecently,importantresultshavebeenachievedbyexploitingtwo
differentsolutionsabletoprovidetherequiredeffectivenonlinearities:i)thestrongdispersive
couplingof lightto stronglyinteractingatomsin highlyexcitedRydbergstates [16–21];ii) a
fiber-integratedcavity QED system employing a whispering gallery mode resonator strongly
coupledtoasingleRubidiumatom[22].
Nevertheless, in recent years it has been proposed [23], and then experimentally real-
ized [24–27], that EIT-like effects could be obtained also within cavity optomechanical sys-
tems. Furthermore,it is well knownthat the ponderomotiveaction of light, togetherwith the
backactionofthemechanicaloscillatorinteractingwithit,isresponsibleforaneffectiveoptical
Kerrnonlinearity[28],whichinturn,maygiverisetointerestingquantumphenomena,suchas
squeezingofthecavityoutputlight,aspredictedacoupleofdecadesago[29,30]andrecently
experimentallyachieved[31–33].
Inthispaperwe willshowthataQPG forsimplephotonicqubitswith aconditionalphase
shift equalto p is achievableby employinga cavity optomechanicalsystem with sufficiently
large single-photon optomechanical coupling, and relatively high mechanical Q-factor Q .
m
Refs. [34,35] first suggested that multi-mode optomechanical systems in the single-photon
strong couplingregime could be exploitedfor quantuminformationprocessing with photons
and phonons, and a first example of optomechanicalimplementation of a QPG has been re-
centlyprovidedinRef.[36].Herewefurtherdeveloptheseideas,byproposingamuchsimpler
scheme,which requiresthe controlofonlytwo cavity modesandof a single mechanicalres-
onator, rather than four optical cavity modes and two mechanical resonators as in Ref. [36].
Recent progressin the realization of strongly couplednano-optomechanicalsystems [37–39]
suggeststhattheQPGschemeproposedherecouldbeimplementedinthenearfuture.
2. TheModel
We consideran optomechanicalsystem consistingof a mechanicalresonatorinteractingwith
twoopticalmodes,whichisdescribedbythefollowingHamiltonian
Hˆ =h¯w aˆ†aˆ +h¯w aˆ†aˆ +h¯w bˆ†bˆ+h¯ g aˆ†aˆ +g aˆ†aˆ bˆ+bˆ† , (1)
1 1 1 2 2 2 m 1 1 1 2 2 2
(cid:16) (cid:17)(cid:0) (cid:1)
where, aˆ(bˆ) and aˆ†(bˆ†) are the annihilation and creation operators for the optical cav-
i i
ity (mechanical) modes, with frequency w /2p and w /2p respectively, and with [aˆ,aˆ†]=
i m i i
[bˆ,bˆ†] = 1; g = (dw /dx)x is the i-th single-photon optomechanical coupling rate, with
i i zpf
x = h¯/2mw thespatialsizeofthezero-pointfluctuationofthemechanicaloscillator.
zpf m
We focusonthesimplestchoiceforanopticalqubit,thestatespacespannedbythelowest
p
Fockstatesofanopticalmode,|0iand|1i;tobemorespecificwewanttoimplementaQPG
between the optical qubits associated with two optical cavity modes of the optomechanical
systemunderstudy.Thegenericinitial(pure)stateofthetwoopticalqubitsisgivenby
|y i =a |0i |0i +a |0i |1i +a |1i |0i +a |1i |1i , (2)
in 00 1 2 01 1 2 10 1 2 11 1 2
corresponding in general to an entangled state of the two modes with up to two photons. A
simple proof-of-principledemonstration could be achieved by restricting to factorized input
statesofthetwocavitymodes,whichcouldbeprovidedbytwoweaklaserpulsesdrivingthe
two selected cavitymodesatfrequenciesw and w , similar to the preliminaryexperimen-
L1 L2
tal demonstrationof a QPG givenin Ref. [6]. In thiscase the two cavity modesare prepared
in a product of two coherent states with amplitudes a and a , |y (0)i=|a i |a i ≃
d1 d2 d1 1 d2 2
[|0i +a |1i ][|0i +a |1i ], where the latter expression is valid for |a |,|a |≪1. For
1 d1 1 2 d2 2 d1 d2
inputlaserpowersP,cavitydetuningsD =w −w ,anddecayratesk , j=1,2,theampli-
j j j Lj j
tudesaregivenbya = 2Pk / h¯w k 2+D 2 .
dj j j Lj j j
r
It is convenient to move to a frahme ro(cid:16)tating at(cid:17)thie corresponding driving laser frequency
for each cavity mode, providing therefore the phase reference for each optical qubit; this is
equivalenttomovetotheinteractionpicturewithrespecttothefreeopticalHamiltonianH =
0
h¯w aˆ†aˆ +h¯w aˆ†aˆ ,inwhichthesystemHamiltonianbecomes
L1 1 1 L2 2 2
Hˆ =h¯D nˆ +h¯D nˆ +h¯w bˆ†bˆ+h¯w fˆ bˆ+bˆ† , (3)
1 1 2 2 m m nˆ
wherewehaveusedthecavitymodephotonnumberoperato(cid:0)rsnˆ =(cid:1)aˆ†aˆ ,andwehavedefined
j j j
fˆ =[g nˆ +g nˆ ]/w .
nˆ 1 1 2 2 m
3. Hamiltoniandynamics
Inordertohaveaphysicaldescriptionofhowtheeffectiveopticalnonlinearityprovidedbythe
optomechanicalinteractionallowstoimplementtheQPG,wefirststudytheidealcasewithno
opticalandmechanicallosses,inwhichthedynamicsisdeterminedsolelybytheHamiltonian
ofEq.(3),i.e.,bytheunitaryoperatorUˆ(t)=e−iHˆt/h¯.Insuchacasethedynamicscanbeex-
actlysolved:infact,profitingfromthefactthatbothphotonnumberoperatorsnˆ areconserved,
j
andmovingtoaphoton-number-dependentdisplacedframeforthemechanicalresonator,one
canrewritetheunitaryevolutionoperatorinaforminwhichtheopticalandmechanicalevo-
lutionoperatorsareconvenientlyfactorized.Infact,actingwiththephotonnumberconserving
mechanicaldisplacementoperator
Dˆ fˆ =exp bˆ†−bˆ fˆ , (4)
nˆ nˆ
(cid:0) (cid:1) (cid:2)(cid:0) (cid:1) (cid:3)
whichseparatestheHamiltonianaccordingto
Dˆ fˆ HˆDˆ† fˆ =Hˆ +Hˆ , (5)
nˆ nˆ opt b
where (cid:0) (cid:1) (cid:0) (cid:1)
(g nˆ +g nˆ )2
Hˆ = h¯D nˆ +h¯D nˆ −h¯w fˆ2=h¯D nˆ +h¯D nˆ −h¯ 1 1 2 2 , (6)
opt 1 1 2 2 m nˆ 1 1 2 2 w
m
Hˆ = h¯w bˆ†bˆ, (7)
b m
(8)
onegets
Uˆ(t)=Uˆ (t)Dˆ† fˆ Uˆ (t)Dˆ fˆ , (9)
opt nˆ b nˆ
where Uˆopt(t)=e−iHˆoptt/h¯, and Uˆb(t)=e−iHˆbt/(cid:0)h¯. (cid:1)Therefor(cid:0)e th(cid:1)e dynamics of the two optical
modesismostlydeterminedbytheeffectiveunitaryoperatorUˆ (t),possessingeitherself-Kerr
opt
terms (cid:181) nˆ2 and the cross-Kerr term −2h¯g g tnˆ nˆ /w ; the other factor Dˆ† fˆ Uˆ (t)Dˆ fˆ
j 1 2 1 2 m nˆ b nˆ
however also affects the optical mode dynamicssince it entanglesthem with the mechanical
(cid:0) (cid:1) (cid:0) (cid:1)
resonator,bycorrelatingtheresonatorpositionwiththephotonnumbers.
The photonnumberconservingdynamicsallowsto stay within thelogicalspacedescribed
above, i.e., the one spanned by optical Fock states with no more than one photon (and this
will remain true even when we will include optical losses). In this case one can always fix
the two detunings in order to eliminate completely the effect of the self-Kerr terms. In fact,
withinthissubspace,nˆ2=nˆ ,andtherefore,takingD =g2/w , j=1,2,onehastheeffective
j j j j m
unitary operatorUˆ (t)=exp[2ig g tnˆ nˆ /w ], yielding a nonlinear conditionalphase shift
opt 1 2 1 2 m
f (t)=2g g t/w onlywheneachcavitymodehasonephoton,i.e.,
nl 1 2 m
|0i1|0i2→|0i1|0i2; |0i1|1i2→|0i1|1i2; |1i1|0i2→|1i1|0i2; |1i1|1i2→ei2gw1mg2t|1i1|1i2.
(10)
Thereforewe expectto geta conditionalphaseshiftequaltop whentheinteractiontimet is
equalto
pw
m
tp = . (11)
2g g
1 2
We now evaluate the exact Hamiltonian evolution in order to see to what extent the in-
teraction with the mechanical resonator affects and modifies the ideal QPG dynamics de-
fined by Eqs. (10) and (11). The natural choice for the initial state is the factorized state
rˆ(0)=|y i hy |⊗rˆth,where|y i isthegenericinitialstateofEq.(2),andrˆth isthethermal
in b in b
equilibriumstateofthemechanicalresonator,withn¯ meanthermalphonons.Byrenumbering
|0i |0i →|0i, |1i |0i →|1i, |0i |1i →|2i, |1i |1i →|3i, we can write the state of the
1 2 1 2 1 2 1 2
wholesystemattimet as
rˆ(t)=Uˆ(t)|y i hy |⊗rˆthUˆ†(t) (12)
in b
3
= (cid:229) a a ∗Uˆ (t)|kihl|Uˆ† (t)⊗Dˆ†(f )Uˆ (t)Dˆ(f )rˆthDˆ†(f )Uˆ†(t)Dˆ(f ),
k l opt opt k b k b l b l
k,l=0
where
Dˆ(f )=exp f bˆ†−bˆ (13)
k k
is now a displacement operator acting only on(cid:2)th(cid:0)e mech(cid:1)a(cid:3)nical resonator degree of freedom,
witha c-numberdisplacement f , with f =0, f =g /w , f =g /w , f =(g +g )/w .
k 0 1 1 m 2 2 m 3 1 2 m
Weareinterestedinthestateofthetwoopticalqubitsonly,andthereforewehavetotraceover
themechanicalresonator.UsingtheexplicitexpressionofUˆ (t)(withthechoiceofdetuning
opt
specifiedabove),andperformingthetrace,thereducedstateoftheopticalmodesreads
rˆ (t)= (cid:229) 3 c (t)a a ∗exp i2g1g2t d −d |kihl|, (14)
opt k,l k l w k,3 l,3
k,l=0 (cid:20) m (cid:21)
(cid:0) (cid:1)
whered istheKroneckerdelta,whilec (t)isthefactordescribingthedecoherencecaused
k,l k,l
bytheinteractionwiththemechanicalresonatorandwhoseexplicitexpressionisgivenby(see
theappendixforitsderivation)
c (t)=exp −(f −f )2(1−cosw t)(2n¯+1)+i f2−f2 sinw t . (15)
k,l k l m l k m
h (cid:0) (cid:1) i
WequantifytheQPGperformancewiththefidelityrelativetotheidealpuretargetstatecorre-
spondingtoap conditionalphaseshift,thatis
3
|y tgti=a 00|0i1|0i2+a 01|0i1|1i2+a 10|1i1|0i2+eip a 11|1i1|1i2= (cid:229) a keipd k,3|ki. (16)
k=0
ThecorrespondingfidelityF(t)canbewrittenas
F(t)≡hy |rˆ (t)|y i= (cid:229) 3 c (t)|a |2|a |2exp i 2g1g2t −p d −d . (17)
tgt opt tgt k,l k l w k,3 l,3
k,l=0 (cid:20) (cid:18) m (cid:19) (cid:21)
(cid:0) (cid:1)
Actually,theQPGperformancecanbecharacterizedbythegatefidelity,whichistheaverage
of the abovequantityoverall possible inputstates of the two qubits[41]. Since the averages
aregivenby|a |4=1/8,∀k,and|a |2|a |2=1/24,∀k6=l [41],usingtheexplicitexpression
k k l
for c (t) ofEq. (15) implyingin particularthatc (t)=1 ∀k and thatc (t)=c (t)∗, and
k,l k,k k,l l,k
theexplicitvaluesofthec-numbers f ,weget
k
1
F(t)=hy |rˆ (t)|y i= (18)
tgt opt tgt
2
1 g2 g2
+ exp − 1 (1−cosw t)(2n¯+1) cos 1 sinw t
12 w 2 m w 2 m
(cid:26) (cid:20) m (cid:21)(cid:20) (cid:18) m (cid:19)
g2+2g g 2g g t
+cos 1 1 2sinw t+p − 1 2
w 2 m w
(cid:18) m m (cid:19)(cid:21)
g2 g2
+exp − 2 (1−cosw t)(2n¯+1) cos 2 sinw t
w 2 m w 2 m
(cid:20) m (cid:21)(cid:20) (cid:18) m (cid:19)
g2+2g g 2g g t
+cos 2 1 2sinw t+p − 1 2
w 2 m w
(cid:18) m m (cid:19)(cid:21)
g −g 2 g2−g2
+exp − 1 2 (1−cosw t)(2n¯+1) cos 2 1sinw t
w m w 2 m
" (cid:18) m (cid:19) # (cid:18) m (cid:19)
g +g 2 g +g 2 2g g t
+exp − 1 2 (1−cosw t)(2n¯+1) cos 1 2 sinw t+p − 1 2 .
w m w m w
" (cid:18) m (cid:19) # "(cid:18) m (cid:19) m #)
From Eqs. (17) one can see that the gate fidelity F achieves the ideal value of unity when
two conditions are satisfied: i) the conditional phase shift is equal to p , (or more generally
to an odd multiple of p ), 2g g t/w =(2m+1)p (integer m); ii) c (t)=1, ∀k,l. Eq. (18)
1 2 m k,l
shows that the latter conditions are achieved for generic nonzero couplings g and g only
1 2
aftereverymechanicaloscillationperiod,i.e.,whenw t=2pp ,p=1,2,....TheQPGwillbe
m
minimallyaffectedbylossesfortheshortestinteractiontimetp ofEq.(11),andtherefore,the
idealconditionsforaunit-fidelityQPGwithaconditionalphaseshiftequaltop are
2gw1g2tp =p w mtp =2p ⇒g1g2= w4m2. (19)
m
Therefore,whenopticalandmechanicallossesarenegligible,onecanrealizeanidealQPGwith
asimpleoptomechanicalsetupbyfixingtheinteractiontimebetweenthemechanicalresonator
andthecavitymodesaccordingtoEq.(11),andprovidedthatthesingle-photonoptomechanical
couplingscanbetunedto thestrongcouplingconditionofEq.(19).Theinteractiontimecan
becontrolledintunableoptomechanicalsystemsinwhichtheinteractioncanbeturnedonand
off,asitcouldbedoneforexampleintheoptomechanicalsetupofRef.[40],whereavibrating
nanobeam is coupled to the evanescentfield of a whispering gallery mode of a microdisk. It
is also relevant to stress that under these conditionsthe QPG is practically insensitive to the
effectofthermalnoiseactingontheresonator,becauseattheexactgatedurationtp thefidelity
becomescompletelyindependentuponthemeanthermalphononnumbern¯.Eq.(18)showsthat
insteadthegatefidelitysignificantlydropsforincreasingn¯assoonast6=tp .
4. Dissipativedynamics
Letusnowconsidertherealisticsituationinordertoseetowhatextentopticallosses,mechan-
icaldamping,andthermalnoiseaffectthisidealgatebehavior.Inthatcasetheevolutionisno
moreanalyticallytractable,andwewillconsiderthenumericalsolutionofthemasterequation
forthedensitymatrixoftheoptomechanicalsystemunderstudy.
Introducingthecavitymodesdecayk (i=1,2),themechanicaldampingg =w /Q ,and
i m m m
themeanthermalphononnumberassociatedwiththereservoirofthemechanicalresonator,n¯,
themasterequationintheusualBorn-Markovapproximationcanbewrittenas[42]
d 1 k
rˆ(t)= [Hˆ,rˆ(t)]+ 1(2aˆ rˆ(t)aˆ†−aˆ†aˆ rˆ(t)−rˆ(t)aˆ†aˆ )
dt ih¯ 2 1 1 1 1 1 1
k
+ 2(2aˆ rˆ(t)aˆ†−aˆ†aˆ rˆ(t)−rˆ(t)aˆ†aˆ ) (20)
2 2 2 2 2 2 2
g g
+ m(n¯+1)(2bˆrˆ(t)bˆ†−bˆ†bˆrˆ(t)−rˆ(t)bˆ†bˆ)+ mn¯(2bˆ†rˆ(t)bˆ−bˆbˆ†rˆ(t)−rˆ(t)bˆbˆ†),
2 2
whereHˆ istheHamiltonianinEq.(3).
InFig.1wecomparethebehaviorofthegatefidelityF(t)intheabsenceofdampingand
lossesofEq.(18)versusthedimensionlessinteractiontimew t,eitheratn¯=0(reddot-dashed
m
curve),andatn¯=10(fullblackcurve)withthecorrespondingcurvesinthepresenceofoptical
and mechanical damping processes. These latter curves (the dotted blue line corresponds to
n¯=0,whilethegreendashedlineton¯=10)areobtainedfromthenumericalsolutionofthe
master equation Eq. (20) in the case k =k =10−2w ,Q =106, while we have fixed the
1 2 m m
couplingsaccordingtotheidealstrongcouplingconditionofEq.(19)g =g =w /2.
1 2 m
Wesee,asexpected,thatintheabsenceofopticalandmechanicallosses,F(t)=1exactly
attheinteractiontimetp ofEq.(11),regardlessthevalueofthetemperatureofthemechanical
reservoir. Atdifferenttimesthe gate performanceis stronglyaffectedby thermalnoise;what
isrelevantisthatinthepresenceofrealisticvaluesofmechanicaldampingandofopticalloss
rates,thisscenarioisstillmaintained,withalimiteddecreaseofthegatefidelity.
Fig. 1. Numerical solution of the master equation Eq. (20) for the gate fidelity F(t)
versus the dimensionless interaction time w mt. We compare four different cases: i) zero
damping and losses gm = k 1 = k 2 = 0 and n¯ = 10 (full black line); ii) zero damp-
ing and losses and n¯ =0 (red dashed-dotted line); iii) with damping and losses (k 1 =
k 2=10−2w m,Qm =106,) and n¯=10 (green dashed line); iv) with damping and losses
(k 1 =k 2 =10−2w m,Qm =106,) and n¯ =0 (blue dotted line). The numerical solutions
forthezerodampingandlosscaseareindistinguishablefromtheanalyticalexpressionof
Eq.(18)eitheratn¯=0andatn¯=10.Inallcaseswehavefixedthecouplingsaccordingto
theidealstrongcouplingconditionofEq.(19),g1=g2=w m/2.
5. Conclusions
We have proposed a simple optomechanicalsetup which is able to implement an ideal QPG
withaconditionalphaseshiftequaltop betweentwoopticalqubitsassociatedwiththelowest
Fockstates(zeroandonephoton)ofanopticalcavitymode.Theschemeisminimalbecauseit
employsonlytwomodesofahigh-finesseopticalcavityandasinglemechanicalresonatorcou-
pledtothem.Theschemeisrobustinthepresenceofrealisticvaluesofopticalandmechanical
losses and,if theinteractiontime is appropriatelyfixed,it isalmostcompletelyinsensitiveto
thethermalnoiseactingonthemechanicalresonator.Themoststringentandchallengingcon-
dition is the required strong optomechanicalcouplingcondition, given by Eq. (19), in which
the single-photonoptomechanicalcouplingmustbeofthe orderofthe mechanicalresonance
frequency. Such a condition has not been achieved yet in current solid-state nanomechani-
cal setups, for which record values corresponds to g /w ∼10−3 [37–39]. On the contrary,
j m
such a strong coupling situation is normally achieved in ultracold atom realizations of cav-
ityoptomechanics,wherethemechanicalresonatorcorrespondstothecollectivemotionofan
ensembleoftrappedultracoldatoms;forexampleonehasg/w ∼0.3inRef.[31].Thelimita-
m
tionintheselattersystemsisrepresentedbycavitylosses,becauseinthiscaseoneistypically
far from the resolvedsidebandregime k /w ≪1 which is requiredhere in orderthat cavity
m
losses donotaltersignificantlythe effectivecross-Kerrnonlinearinteractionmediatedbythe
resonator,responsiblefortheQPG dynamics.Thereforethepresentproposalcouldbeimple-
mentedinexperimentaloptomechanicalplatformsabletocombineasignificantlylargesingle
photoncouplingg/w ∼0.5witharesolvedsidebandoperationconditionk /w ≪1.
m m
6. Appendix
Wenowderivetheexplicitexpressionforthedecoherencecoefficientsc (t)ofEq.(15).From
k,l
Eqs.(12)-(14)andusingthecyclicpropertyofthetrace,onehas
c (t)=Tr Dˆ†(f )Uˆ†(t)Dˆ(f )Dˆ†(f )Uˆ (t)Dˆ(f )rˆth , (21)
k,l b l b l k b k b
h i
whichisathermalaverageofacombinationofdisplacementoperators.Wenoticethenthatthe
factorUˆ†(t)Dˆ(f )Dˆ†(f )Uˆ (t)withinthetraceisjusttheHeisenbergtimeevolutionforatime
b l k b
t ofadisplacementoperatorofafreemechanicalresonator,sothat
Uˆ†(t)Dˆ(f )Dˆ†(f )Uˆ (t)=exp bˆ†eiw mt−bˆe−iw mt (f −f ) . (22)
b l k b l k
Inserting this solution within Eq. (21), and us(cid:2)in(cid:0)g the property of(cid:1)the displ(cid:3)acement operator
Dˆ(a )Dˆ(b )=Dˆ(a +b )exp[iIm(ab ∗)],onegets
c (t)=Tr Dˆ (f −f )(eiw mt−1) rˆth exp i(f2−f2)sinw t . (23)
k,l b l k b l k m
n (cid:2) (cid:3) o (cid:2) (cid:3)
Performingthethermalaverageofthefinaldisplacementoperator,accordingto
1
hexp a bˆ†−a ∗bˆ i =exp −|a |2 n¯+ , (24)
th
2
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:2) (cid:3)
wefinallygetEq.(15).
Acknowledgments
ThisworkhasbeensupportedbytheEuropeanCommission(ITN-MarieCurieprojectcQOM
andFET-OpenProjectiQUOEMS),andbyMIUR(PRIN2011).