Physicallyfeasiblethree-level transitionlessquantum driving withmultipleSchro¨dinger dynamics ∗ Xue-Ke Song, Qing Ai, Jing Qiu, and Fu-Guo Deng † DepartmentofPhysics,AppliedOpticsBeijingAreaMajorLaboratory,BeijingNormalUniversity,Beijing100875,China (Dated:July11,2016) Three-levelquantumsystems,whichpossesssomeuniquecharacteristicsbeyondtwo-levelones,suchaselec- tromagneticallyinducedtransparency,coherenttrapping,andRamanscatting,playimportantrolesinsolid-state quantuminformationprocessing. Here,weintroduceanapproachtoimplementthephysicallyfeasiblethree- level transitionless quantum driving with multiple Schro¨dinger dynamics (MSDs). It can be used to control accuratelypopulationtransferandentanglementgenerationforthree-levelquantumsystemsinanonadiabatic way. Moreover,weproposeanexperimentallyrealizablehybridarchitecture,basedontwonitrogen-vacancy- centerensemblescoupledtoatransmissionlineresonator,torealizeourtransitionlessschemewhichrequires fewerphysical resourcesandsimpleprocedures, anditismorerobust againstenvironmental noisesandcon- 6 trolparametervariationsthanconventionaladiabaticpassagetechniques. Allthesefeaturesinspirethefurther 1 applicationofMSDsonrobustquantuminformationprocessinginexperiment. 0 2 PACSnumbers:03.67.Lx,32.80.Qk,76.30.Mi l u J I. INTRODUCTION statesareinvolved. Thistechniquehasgainedtheoreticaland 8 experimentalstudiesinatomicandmolecular[21]andsuper- conductingquantumsystems [22]. When TQD is applied to ] Accuratelycontrollingaquantumsystemwithhighfidelity h speed up the adiabatic operation in three-levelquantumsys- p isafundamentalprerequisiteinquantuminformationprocess- tems,thesituationbecomesmorecomplicated[15,23–26].In - ing[1],highprecisionmeasurements[2],andcoherentcontrol 2010, Chen et al. [15] employed the TQD to speed up adi- t n ofatomicandmolecularsystems[3]. Tothisend,rapidadia- abatic passage techniques in three-level atoms extending to a batic passage[4], which leadstwo-levelquantumsystems to theshort-timedomaintheirrobustnesswithrespecttoparam- u evolveslowlyenoughalongaspecificpath,canproducenear- eter variations. In 2014, Mart´ınez-Garaotet al. [23] studied q perfectpopulationtransferbetweentwoquantumstatesof(ar- [ shortcuts to adiabaticity in three-level systems by means of tificial)atomsormolecules. Theadiabaticevolutionrequires Lie transforms. Alternatively, in 2012, Chen and Muga [27] 4 long runtime, which will generate the extra loss of coher- designedtheresonantlaserpulsestoperformthefastpopula- v ence and spontaneousemission of quantum systems. Short- tiontransferinthree-levelsystemsbyinvariant-basedinverse 0 cutstoadiabaticityarealternativefastprocessestoreproduce 5 engineering.In2014,KielyandRuschhaupt[28]constructed thesamephysicalprocessesinafinite shortertime, whichis 0 fastandstablecontrolschemesfortwo-andthree-levelquan- only limited by the energy-timecomplementarity[5]. There 0 tumsystems. aretwopotentiallyequivalentshortcutstospeedupadiabatic 0 . process in a nonadiabatic route: Lewis-Riesenfeld invariant- Interestingly,multipleSchro¨dingerdynamics(MSDs) [29, 2 based inverseengineering[6–10] and transitionlessquantum 30] were presented to adopt iterative interaction pictures to 0 driving(TQD)[11–17]. Interestingly,TQDhasattractedcon- getphysicallyfeasibleinteractionsordynamicsfortwo-level 6 1 siderableattentioninexperiment[18,19]. In2012,Basonet quantumsystemsrecently.Meanwhile,itenablesthedesigned : al. [18] demonstrated the quantum system following the in- interactionpicturetoreproducethesamefinalpopulation(or v stantaneousadiabatic groundstate nearly perfectly on Bose- state) as those in the original Schro¨dinger picture by appro- i X Einstein condensates in optical lattices. In 2013, Zhang et priate boundaryconditions. In 2012, Iba´n˜ez et al. [29] first r al. [19]implementedtheassistedadiabaticpassagesthrough employed several Schro¨dinger pictures and dynamics to de- a TQD in a two-level quantum system by controlling a single sign alternative and feasible experimentalroutesfor trap ex- spininannitrogen-vacancycenterindiamond. pansions and compressions, and for harmonic transport. In 2013,Iba´n˜ezetal. [30]alsoexaminedthelimitationsandca- For three-level quantum systems, the stimulated Raman pabilitiesofsuperadiabaticiterationstoproduceasequenceof adiabatic passage (STIRAP) technique [20] uses partially shortcutstoadiabaticitybyiterativeinteractionpictures. This overlappingpulses(Stokesandpumppulses)to perfectlyre- raises a significative question: whether one can find an ef- alizethepopulationtransferbetweentwoquantumstateswith fectivewayforthree-levelTQDinexperimentalapplications. thesameparity,inwhichsingle-photontransitionsareforbid- Three-level quantum systems play important roles in solid- denbyelectric dipoleradiation. TheSTIRAPoverthe rapid state quantum information processing as they possess some adiabaticpassageisitsrobustnessagainstsubstantialfluctua- uniquecharacteristicsbeyondtwo-levelones,suchaselectro- tionsofpulseparameters,sincetheevolutionofthequantum magneticallyinducedtransparency,coherenttrapping,Raman system is in the dark state space and only the two quantum scatting, and so on. Therefore, manipulating such quantum systemsinanaccurateandrobustmannerisespeciallyimpor- tant. Inspired by the two-level TQD with MSDs [29, 30], here ∗PublishedinPhys.Rev.A93,052324(2016) Correspondingauthor:[email protected] we employ the iteration process to obtain physically feasi- † 2 ble TQD in three-level quantum systems. More interest- Definingatime-dependentunitaryoperator ingly, we present a physical implementation for the transi- t t tionless scheme with the hybrid quantum system composed U(t)= exp i dt′En(t′) dt′ n0(t′)n˙0(t′) n0(t) n0(0), ofnitrogen-vacancy-centerensembles(NVEs)andthesuper- (−Z −Z h | i)| ih | Xn 0 0 conductingtransmissionlineresonator(TLR).Ithassomead- (4) vantages.First,itcanaccuratelycontrolquantumsystemsina shortertime,asadiabaticquantumevolutioncanbeefficiently which obeys H(t)U(t) = iU˙(t). By analytically solving the accelerated by TQD. Second, the MSDs-based Hamiltonian equationH(t)=iU˙(t)U (t),wehave † requiredforthree-levelTQDisphysicallyfeasible,whichcan beusedtoimplementaccurateandrobustpopulationtransfer H(t) = E n n +i (n˙ n n n˙ n n ) n 0 0 0 0 0 0 0 0 andentanglementgenerationwithhighfidelityinasingle-shot Xn | ih | Xn | ih |−h | i| ih | operation. Third, it is morerobustagainstcontrolparameter H (t)+Hcd(t), (5) fluctuationsanddissipationsthanconventionaladiabaticpas- ≡ 0 0 sage technique. Fourth, the transitionless scheme presented whereallketsaretime-dependent. FromEq.(5),onecansee here is quite universal, and it is broadly applicable in other thatthe transitionlessHamiltonian H(t)consistsofthe origi- quantumsystems,suchasatomcavity,superconducting-qubit nal Hamiltonian H (t) for adiabatic evolutionand a counter- 0 TLR,andsoon.Alltheseadvantagesprovidethegoodappli- diabatic driving Hamiltonian Hcd(t) [11, 12, 14, 15]. TQD 0 cations of MSDs on robust quantum information processing offers an effective accurate route for the controlled system inexperimentinthefuture. followingperfectlythe instantaneousgroundstate ofa given This paper is organized as follows: In Sec. II, we show Hamiltonian in theory and experiment. Nevertheless, it is thebasicprincipleofourschemeforobtainingthephysically foundthatthetransitionlessHamiltonianisdifficulttoimple- feasibleTQDinthree-levelquantumsystemsbyusingMSDs. ment for example in three-levelquantum systems [15] since InSec.III,wegivespecificcomparisonsofpopulationtransfer thecounterdiabaticdrivingHamiltonianhastobreakdownthe andsuperpositionstate generationbasedontheconventional energystructureoftheoriginalHamiltonianorbringextrade- STIRAP and MSDs, respectively. In Sec. IV, we present a tunings. physical implementation of the transitionless scheme on an Superadiabaticiterations as an extension of the usual adi- NVEs-TLR system, and analyze the fidelity in the presence abaticapproximationhavebeenintroducedinRef.[31]. The ofdecoherence. Adiscussionandasummaryareenclosedin processofsuperadiabaticiterationcanbesummarizedinTa- Sec. V. ble I, where H (t) donates the j th Hamiltonian by a uni- j − tary transformation A (t) on the (j 1) th Hamiltonian j 1 H (t), A (t) = n (t−) n (0) (j = 1−,2, −), n (t) are the j 1 j j j j − n | ih | ··· | i II. PHYSICALLYFEASIBLEHAMILTONIANWITH eigenstates of thePHamiltonian H (t), and j is the numberof j MSDSONTHREE-LEVELQUANTUMSYSTEMS superadiabaticiteration. A. ThetransitionlessHamiltonianwithmultipleSchro¨dinger dynamics TABLEI:SchemeforsuperadiabaticiterationtorealizeMSDsasa resultofHamiltonianH (t). 0 First,wegiveabriefreviewofTQD.Consideringanarbi- Iteration Hamiltonian Eigenstates Unitaryoperator twrahriychtimhaes-dtheepennodnednetgHenaemrailtteoinnisatnanHta0(nte)ooufsaeqigueannstutamtessy|nst0e(mt)i, 01−tsht HH0((tt)) |nn0((tt))i AA0((tt))==Pn |nn0((tt))ihnn0((00))| withcorrespondingeigenvaluesEn(t),weget − 1 | 1 i 1 n | 1 ih 1 | P H0(t)|n0(t)i= En(t)|n0(t)i. (1) ·j·−··th·· Hj(t) |nj(t)i Aj(t)= n |nj(t)ihnj(0)| P Intheadiabaticapproximation,thestateevolutionofthesys- ······ temdrivenbyH (t)canbewrittenas(~=1) 0 Here,ourgoalistouseMSDstoobtainphysicallyfeasible t t transitionlessHamiltonianforthree-levelquantumsystemsin |Ψn(t)i=exp(−iZ dt′En(t′)−Z dt′hn0(t′)|n˙0(t′)i)|n0(t)i. (2) TQD.Inwhatfollowswewillpresentanexplicitexplanation 0 0 aboutit,reviewingtheideasfromRefs.[29,30].Fortheinitial Asaconsequence,theevolutionoperatorforthisgivenquan- HamiltonianH0(t)witheigenstates n0(t) ,thecorresponding tumsystemisspecified. Alternatively,onecanseekatransi- transitionlessHamiltonianHT(t)for|0 thiiterationreads 0 − tionless Hamiltonian H(t) that can accurately drive evolving state|ψn(t)iinashortestpossibletime,whichguaranteesthat H0T(t)= H0(t)+H0cd(t)= En|n0ihn0|+iA˙0(t)A†0(t), (6) therearenotransitionsbetweentheeigenstatesofH0(t). That Xn is,itshouldsatisfy where A (t) = n (t) n (0) is definedasthe unitaryoper- 0 0 0 n | ih | H(t)ψ (t) =iψ˙ (t) . (3) atorbasedonthPeeigenstates n (t) oftheinitialHamiltonian n n 0 | i | i | i 3 H0(t),and|n0(0)iisthebareadiabaticbasis. Here,theeigen- where η = η21+η22, θ = arctan(η1/η2), and η, η1, and η2 statesarechosentofulfilltheparalleltransportcondition,i.e., are time-depqendenteffective couplingstrengths. The instan- n (t)n˙ (t) =0. h 0 | 0 i taneouseigenvaluesandthecorrespondingnormalizedeigen- Inthefirstinteractionpicture(the1 thiteration),byauni- − statesare tarytransformationA (t),theinteractionpictureHamiltonian 0 H (t)becomes E = η, E =0, 1 0 ∓ ∓ 1 H1(t)= A†0(t)[H0(t)−K0(t)]A0(t). (7) |E−i= √2(cosθ|φ1i+sinθ|φ2i−|φ3i), (12) 1 wHhamerieltoKn0i(atn) i=sdeiAs˙c0r(itb)Aed†0(bt)y. In this case, the transitionless |E+i= √2(cosθ|φ1i+sinθ|φ2i+|φ3i), E =sinθφ cosθφ . 0 1 2 | i | i− | i H1T(t) = A†0(t)[H0(t)−K0(t)+H0cd(t)]A0(t) Itiseasytoseethat E E˙ =0(m=+, ,0).FromEq.(7), = A†0(t)H0(t)A0(t). (8) onecan obtainthe inhtemra|ctmioinpictureHam−iltonianin the 1 th iteration for the effective Hamiltonian H (t) in the basi−s Here, we employ the relation K (t) = Hcd(t). In the 0 Schro¨dinger picture, the Hamilton0ian for TQ0D is H (t) + {|E−i,|E+i,|E0i}asfollows: 0 H0cd(t). ItisworthnoticingthatH0T(t)andH1T(t)arerelatedby η 0 iθ˙ a unitary transform A (t), and they represent the same com- − −√2 0 H (t) = 0 η iθ˙ , (13) mthoeInnHutahnmedielsrteolcynoiinnagndpHihn1yt(estri)acwcs.tiitohnepigicetnusrteat(etshe|n21(−t)ith,thiteerianttieorna)c,tifoonr 1  √iθ˙2 √iθ˙2 −0√2  pictureHamiltonianH2(t)canbeexpressedas where θ˙ = (η˙1η2 η˙2η1)/η2. The unitary transform matrix − relatedtoH (t)andH (t)is 0 1 H2(t)= A†1(t)[H1(t)−K1(t)]A1(t), (9) 1 cosθ 1 cosθ sinθ √2 √2 wtcthhaheenesHroaebammtAae1iinl(wtto)aanyn=i,oabnthPyneinar|ndtT1dr(aQitnn)Digshintaii1soc(n0oHl)ue0|ns(astten)rHd+daiKamHb11ic(aldttt)(iotcn)=,diarwniivAh˙iHe1nr(g2Tet)(tAtHe)†1r.1cm(dt)(T,.t)ohneI=nne ThenormalAiz0ed=eige√n1−2ves√1icn2toθrso√1f2t√s1h2ienHθam−icl0otosnθian.H1(t)are(14) A0(t)K1(t)A†0(t). Similarly,inthehigh-orderinteractionpicture[the(j+1) λ = η2+θ˙2, λ =0, thiteration],onecanalsogetthecorrespondingHamiltonia−n ∓ ∓q 0 torealizeTQDintheSchro¨dingerpictureas λ = iW E + iQ E+ + √2θ˙ E0 , | −i R | −i R | i R | i (15) H0(t)+Hcjd(t)= H0(t)+iBj(t)A˙j(t)A†j(t)B†j(t), (10) λ+ = iQ E + iW E+ √2θ˙ E0 , | i R | −i R | i− R | i w(jh=ere1,B2j,(t)=)wA0it(ht)An1j((tt))··b·Aeijn−g1(tth)eanedigAenj(stt)a=tesPno|fntjh(te)iHhnamj(0il)-| |λ0i= −iR√2θ˙|E−i+ i√R2θ˙|E+i+ 2Rη|E0i, ··· | i tonianH (t)forthe j thiteration.Notethataphysicallyfea- j sible Hamiltonian is−hard to obtain due to the unpredictable where W = η + η2+θ˙2, Q = η + η2+θ˙2, and R = − number of superadiabatic iterations needed for execution in 2 η2+θ˙2. Itgenepratestheunitaryoperatpor thespecificquantumsystems. p iW iQ i√2θ˙ R R − R A = iQ iW i√2θ˙ , (16) 1 B. Physicallyfeasiblethdrreiev-ilnevgeltransitionlessquantum  √RR2θ˙ −R√R2θ˙ 2RRη  withwhichonecangettheinteractionpictureHamiltonianin In three-levelquantum systems, the effective Hamiltonian the 2 th iteration. Substituting Eq. (14) and Eq. (16) into for achieving adiabatic populationtransfer in the orthogonal Eq.(1−0)when j=1,onecanobtaintheHamiltonianH (t)in M basisof φ1 , φ2 , φ3 takestheformof MSDsforrealizingshortcutstoadiabaticityas {| i | i | i} 0 0 cosθ 0 0 ηcosθ+V 1 H0(t) =ηco0sθ si0nθ si0nθ , (11) HM(t) =ηcosθ0+V1 ηsinθ0−V2 ηsinθ0−V2 , (17) 4 whereV1 = 4sinθ(η˙θ˙ ηθ¨)/R2,V2 =4cosθ(η˙θ˙ ηθ¨)/R2,η˙ = 1 (a) 1 (b) (η η˙ +η η˙)/η,andθ¨−=[(η¨η η η¨)η 2η˙(η˙η− η η˙)]/η3. 1 1 2 2 1 2 1 2 1 2 1 2 IhtitaoisnsatnhloectosduaipmffiliecnugflostrtamondfiansddetthtuhenaitHntgahmse−.iHlTtoahmnuisial,nto−anHsi0aim(nt)pH,leMwa(−ittn)hdoinufetMaasSdibDdlies- opulation0.5 P1 opulation0.5 P1 P P P P control of TQD for three-level systems is physically imple- 2 2 mented with MSDs by flexibly tuning the effective coupling P3 P3 0 0 strengths. 0 0.5 1 1.5 0 0.5 1 1.5 t(µs) t(µs) 1.8 1.8 (a) (d) FIG.2: Timeevolutionsofthepopulationsinthesuperpositionstate generationschemefor φ , φ ,and φ basedon(a)theSTIRAP MHz) η1/2π MHz) η1/2π and(b) MSDs, respecti|ve1liy,|w2iithη0/2|π3i= 1.6MHz, t3 = 1.15µs, (π 0.9 η2/2π (π0.9 η2/2π t4=0.25µs,andT =0.408µs. 2 2 /ηj /ηj 0 0 0 0.3 0.6 0.9 1.2 0 0.3 0.6 0.9 1.2 t(µs) t(µs) strengthsareintheGaussianshapesas 1(b) 1(e) ηj =η0e−[(t−tj)/T]2, (18) n n whereη , t , and T are the amplitude, time delay, and width o o 0 j opulati0.5 PPP123 opulati0.5 PPP123 opflatyhveacroiautpiolinnsgosfttrheengtwtho,orepstpimecatliveeffleyc.tiIvneFcoigu.pl1in(ag)s,twreengdtihss- P P withtimet forachievingpopulationtransfer,whereη /2π = 0 00 0.3 0.6 0.9 1.2 00 0.3 0.6 0.9 1.2 1.6MHz,t1 = 0.75µs,t2 = 0.25µs, andT = 0.408µs. Fig- t(µs) t(µs) ures. 1(b)and1(c)presenttime evolutionofthe populations during the transfer process from φ to φ based on STI- 1(c) 1(f) RAP and MSDs, respectively. Th|e2piopu|lat1iion is defined as Population0.5 PPP123 Population0.5 PPP123 loPunkt(itoth)ne=oifnhidφteika|nρls(sittt)ya|φtmekia|φt(r2kiix.=aIfn1t,et2rh,tihs3e)cwapsoietph,ubρlao(tttih)obntheteirnatgnimstfheeeretovipmoeleurateitovionons- governed by the Hamiltonians H (t) and H (t) can achieve 0 M near-perfect population transfer from φ to φ , while the 0 0 2 1 0 0.3 0.6 0.9 1.2 0 0.3 0.6 0.9 1.2 | i | i populationP ofintermediatestate φ showsa slightlydif- t(µs) t(µs) 3 3 | i ferentbehavior. When the time delay of η is changedto be 1 t = 0.9 µs, which reducesoverlapof the two effective cou- FIG. 1: Comparisons of robustness of the population transfer 1 based on the STIRAP and MSDs. (a) The time-dependent effec- pling strengths, we plot variations of the effective coupling tive coupling strengths η and η are in the Gaussian shapes as strengths,timeevolutionofthepopulationsbasedonSTIRAP 1 2 ηj = η0e−[(t−tj)/T]2 (j = 1,2)withη0/2π = 1.6MHz, t1 = 0.75µs, andMSDsinFigs. 1(d),1(e),and1(f),respectively. Onecan t = 0.25µs, andT = 0.408µs. Timeevolutionofthepopulation seethatthepopulationtransferbytheHamiltonianH (t)with 2 M P (t)during thepopulation transfer from φ to φ based on: (b) MSDsisperfectlyrealizedina shortevolutiontime, andthe k | 2i | 1i tSiismTcIehRaeAnvgoPe,ludatnitoodnb(ocef)tt1hMe=SeDff0s.e9,ctrµievssepacenocdutipovlteihlneygr.sptWarerahnmegntehttsehraesndrtiemtmheeaipdnoeilpnauvylaaortiifaonnηts1, fithnealHpaompiulltaotnioiannoHft0h(te)twarigthetSstTaItReA|φP1iccaannnroeta.cMh1o0re0o%v,ewr,hniule- mericalcalculationsrevealthattheHamiltonianH (t)isalso basedonSTIRAPandMSDsareshownin(d),(e),(f),respectively. M validforhigh-fidelitypopulationtransfereventhetimedelay of η becomes much bigger than 0.9 µs, suggesting that our 1 transitionlessschemewithMSDsisveryrobustandcaneffi- cientlyrealizeperfectpopulationtransfer. III. HIGH-FIDELITYPOPULATIONTRANSFERAND SUPERPOSITIONSTATEGENERATION B. Superpositionstategeneration A. Populationtransfer Assumingtheinitialstateofthesystemis φ ,onecaneas- 2 ilygetthesuperpositionstate ψ = 1 (φ |φi)bySTIRAP FromEq.(12),onecanseethatwhenθ = 0attimet = 0, | i √2 | 1i−| 2i i andMSDs. Forthispurpose,thetwotime-dependenteffective thedarkstate E (t) becomes φ withaglobalphasefactor | 0 i | 2i couplingstrengthsaredesignedas π. If the system evolvesadiabaticallyalongthe state E (t) , 0 sthimepfilneaplostpautelaitsio|φn1tirawnhsfeenrθis=coπ2maptlleatteelrytirmeaelitzfe.dAbsyaSrT|esIuRlAt,iPa η1 =η0e−[(t−t3)/T]2, (19) [20]. Forthispurpose,thetime-dependenteffectivecoupling η2 =η0e−[(t−t4)/T]2 +η0e−[(t−t3)/T]2, 5 (a) TLR (b) e trol of ηj(t) by changing Rabi frequency ΩL,j(t) of the mi- crowavepulsewhentheparametersg and∆ areprescribed. g j j j TheHamiltonian H (t) conservesthetotalexcitationnumber I j N = 2j=1σ†jσ−j +nc duringthedynamicalevolutionwithnc gj beingPthephotonnumberintheresonatorandσ†j =(σ−j)†.The NVE NVE j whole system evolves in the one-excited subspace spanned 1 2 by φ = 0ge , φ = 0eg , φ = 1gg , where the 1 2 3 c,1,2 x {| i | i | i | i | i | i} y a subscripts c, 1, and 2 donate the resonator mode, the first NVE, and the second NVE, respectively. In the basis of FIG.3:(a)Schematicdiagramofthehybridquantumsystem,which φ , φ , φ , the interaction Hamiltonian H (t) is equiva- 1 2 3 I {| i | i | i} consists of two NVEs coupled to a high-Q TLR. (b) The V-type lenttotheHamiltonianH (t). Consequently,onecanachieve 0 energy-levelconfigurationforthegroundstateofNVEdrivenbythe therobustandaccuratepopulationtransferandmaximallyen- resonatorandappropriateexternalmagneticfields. tangledstategenerationbetweentwoNVEs,wherethecavity stateisemployedasanancillary. 1 1 whichshouldsatisfytheboundaryconditionsoftheSTIRAP (a) (b) that at the beginning of the operation η /η = 0 and at the 0.8 et4nd=η01.2/η52µs=,a1n.dTGi=ve0n.4t0h8eµpsa,rtahmeepteerrfso1ηrm0/2a2nπce=so1f.t6heMpHopz-, Fidelity00..46 Fidelity00..9926 ulations for φ and φ with variation of t have two con- | 1i | 2i 3 0.2 ditionsas follows: (i) whenthe parametert getsan optimal 3 0 0.88 time 0.75 µs, time evolutions of the populations P1 and P2 0 0.3 0.6 0.9 1.2 0 40 80 120 160 200 with STIRAP andMSDs reachan approximatevalue 1, that t(µs) κ′/κ 2 is, the two approacheseffectively generate the superposition FIG. 4: (a) Fidelity of the population transfer scheme with MSDs state ψ ;(ii)whent increases,thepopulationdynamicswith STIR|APi and MSDs3exhibit significantly differentbehaviors. fIrnomthe|φs2iim=ul|a0teiogni,toκ|φ11i==5|00gµesiu[3n2d]e,rγthe1i=nflu6enmcse,oγfd1is=sip6a0ti0onµss. The equivalent populations with P = P = 1 can be im- − − ϕ− 1 2 2 [33], and otherparametersarethesameasthoseforFig. 1(c). (b) plemented with MSDs, implying that time evolution of the Fidelityof thepopulation transfer schemewithMSDsfor different quantumstategovernedbyHM(t)isinthesuperpositionstate cavitydecayratesκ′(inunitsofκ). ψ , while the Hamiltonian H (t) in STIRAP leads to oscil- 0 | i latory behaviors for P and P , as shown in Figs. 2(a) and 1 2 2(b),respectively.TheseresultsconvinceusthatMSDscould Inthe presenceof dissipations, thedynamicsofthe NVE- paveanefficientwaytoachieveaccurateandrobustquantum TLRhybridsystemisdescribedbytheLindbladmasterequa- informationprocessing. tion: dρ = i[H(t),ρ]+κD[a]ρ+γD[σ ]ρ+γ D[σz]ρ, (21) IV. PHYSICALIMPLEMENTATIONOFTHE dt − − ϕ TRANSITIONLESSSCHEMEONANNVES-TLRSYSTEM whereρisthedensitymatrixoperatorforthehybridsystem, H(t) is the Hamiltonian in the form of Eq. (20), D[L]ρ = To experimentally realize the population transfer and en- (2LρL+ L+Lρ ρL+L)/2, κ is the decay rate of TLR, and tanglementgeneration,we considerthe hybridquantumsys- − − γ and γ are the relaxation and dephasing rates of NVE, re- tem, in which two NVEs are coupled to a high-Q TLR, as ϕ spectively. For the proposed transfer scheme, the fidelity is shown in Fig. 3(a). The NVE can be modeled as a V-style defined as F = φ ρφ with φ being the corresponding three-levelqubitwith g and e beingtwo upper-levels,and h 1| | 1i | 1i | i | i ideally final state under the population transfer on its initial a servingasthelower-level. AsillustratedinFig. 3(b),the q|truaiennsictyiowni|tahic↔oup|leiingissltarergnegltyhdgetuannedddteotutnhiengres∆on,aatonrdftrhee- astsa:teg/|φ22πi.=B2y0cMhoHozsi,n∆g/t2hπe =fea2s0ib0leMeHxpze,rκi−m1e=nta5l0paµrsa,mγe−t1er=s transition a g isoff-resonantjdrivenbyatime-djependent 6 ms, γϕ−1 = 600 µs, and ΩL,j(t) = Ω0e−[(t−tj)/T]2 MHz with microwave| ipu↔lse| wi ith Rabi frequency Ω (t) and the same t1 = 0.75µs, t2 = 0.25µs, T = 0.408µs, Ω0/2π = 16MHz, d~e=tu1nifnogr∆thje,hreysbpreidctsivyesltye.mTishegiivnetnerbayctionL,Hj amiltonian with wwihtihchτm=e1e.t2s µthse[2a0d]i,aboanteiccacnonfidnitdiotnhatthtahteRp0τroΩpLo,sj(etd)dstch≫em1e withMSDscanrealizeperfectpopulationtransferwiththefi- 2 delity being 100%, as shown in Fig. 4(a). To illustrate the HI(t)= ηj(t)aσ†j +H.c., (20) robustnessofthepresentscheme,wealsosimulatethedepen- Xj=1 denceofthefidelityF versusthephotondecayrateκ′ inFig. 4(b). Itshows that a highfidelity of 89.60%can still be ob- where σ†j = |eijhg| and ηj(t) = gjΩL,j(t)/∆j is the effective tained even for κ′/κ = 200. The reasons are two-manifold: coupling strength. Obviously, it is easy to realize full con- The cavity state is just used as an ancillary in the present 6 scheme, so it is insensitive to the photon decay in the res- been experimentallyobservedat roomtemperature[33]. An onator; the mean photon number n¯ = a a , in consistency optimized dynamical decoupling microwave pulse has been † h i withthepopulationP forintermediatestate φ inFig. 1,re- demonstrated to increase the dephasing time of NVE from 3 3 | i mainsatrivialvalueduringthetransferprocess,whichcannot 0.7 ms to 30 ms [37]. Moreover, our transitionless scheme achieve the complete occupation of photon states [34]. The withMSDsrequiresfewerresources,oneTLRandtwoNVEs, above results suggest that time evolution of the populations whichgreatlysimplifiestheexperimentalcomplexity. withMSDsismorerobustagainstcontrolparameterfluctua- Insummary,wehavepresentedasimpleschemeforphys- tionsandimperfectionsthanSTIRAP. ically feasible TQD for three-level quantum systems with MSDs, which is used to realize perfect population transfer andentanglementgenerationin a single-shotoperation. Our V. DISCUSSIONANDSUMMARY experimentallyrealizabletransitionlessprotocolbasedonthe NVE-TLR hybrid system requires fewer physical resources We considerthe feasibility with the currentaccessible pa- andsimpleprocedures(one-stepindeed),worksinthedisper- rametersintheNVE-TLRhybridsystem. ForanNVEplaced siveregime,andisrobustagainstdecoherenceandcontrolpa- attheantinodesofthemagneticfieldofthefull-wavemodeof rameterfluctuations. Thesefeaturesmakeourprotocolmore theTLR,thecouplingstrengthg/2π=16MHzbetweenthem accurate for the manipulation of the evolution of three-level is reported experimentally [34, 35]. The amplitude of mi- quantum systems than previous proposals, which may open crowavepulseisavailablewiththecurrentexperimentparam- upfurtherexperimentalrealizationsforrobustquantuminfor- eterΩ /2π=16MHz[36].Thedetuningis∆/2π=160MHz mationprocessingwithMSDs. 0 so that ∆ g and ∆ Ω , which can adiabatically elim- 0 inate the s≫tate a . Fr≫om Eq. (20), the effective coupling strength is η /2|πi= 1.6 MHz. When the coupling strength ACKNOWLEDGMENTS j g and detuning ∆ remain invariant, we have full control of the populationtransfer and entanglementgenerationby con- WewouldliketothankDr. Sof´ıaMart´ınez-GaraotandWei trollingflexiblythetime-dependentRabifrequencyΩ (t)of Xiong forhelpfuldiscussion. Thiswork is supportedby the L,j the microwave pulse with a single-shot operation. The mi- National Natural Science Foundation of China under Grants crowave coplanar waveguide resonators with the decay rate No. 11474026andNo. 11505007,andtheFundamentalRe- of κ 1 = 50 µs can be reached [32]. The dephasing time of search Funds for the Central Universities under Grant No. − T > 600 µs for an NVE in bulk high-purity diamond has 2015KJJCA01. 2 [1] J.Stolzeand D.Suter, QuantumComputing: AShort Course [14] M.V.Berry,J.Phys.A:Math.Theor.42,365303(2009). from Theory to Experiment (Wiley-VCH, Berlin, 2008), 2nd [15] X.Chen,I.Lizuain,A.Ruschhaupt,D.Gue´ry-Odelin,andJ.G. ed. Muga,Phys.Rev.Lett.105,123003(2010). [2] T.W.Ha¨nsch,NobelLecture:Passionforprecision.Rev.Mod. [16] A.delCampo,Phys.Rev.Lett.111,100502(2013). Phys.78,1297(2006). [17] M.MolinerandP.Schmitteckert,Phys.Rev.Lett.111,120602 [3] P.Kra´l,I.Thanopulos,andM.Shapiro,Rev.Mod.Phys.79,53 (2013). (2007). [18] M.G.Bason,M.Viteau,N.Malossi,P.Huillery,E.Arimondo, [4] N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Annu.Rev.Phys.Chem.52,763(2001). Morsch,Nat.Phys.8,147(2012). [5] A.C.SantosandM.S.Sarandy,Sci.Rep.5,15775(2015). [19] J. Zhang, J.H. Shim, I. Niemeyer, T. Taniguchi, T.Teraji, H. [6] H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 Abe, S. Onoda, T. Yamamoto, T. Ohshima, J. Isoya, and D. (1969). Suter,Phys.Rev.Lett.110,240501(2013). [7] J. G. Muga, X. Chen, A. Ruschhaupt, E. Torrontegui, and D. [20] K. Bergmann, H. Theuer, and B. Shore, Rev. Mod. Phys. 70, Gue´ry-Odelin,J.Phys.B42,241001(2009). 1003(1998). [8] X. Chen, E. Torrontegui, and J. G. Muga, Phys. Rev. A 83, [21] B. W. Shore, Manipulating Quantum Structures Using Laser 062116(2011). Pulses(CambridgeUniversityPress,NewYork,2011). [9] E.Torrontegui,S.Iba´n˜ez,S.Mart´ınez-Garaot,M.Modugno,A. [22] K.S.Kumar, A.Vepsalainen, S.Danilin,andG.S.Paraoanu, delCampo,D.Gue´ry-Odelin,A.Ruschhaupt, X.Chen,andJ. arXiv:1508.02981. G.Muga,Adv.At.Mol.Opt.Phys.62,117(2013). [23] S.Mart´ınez-Garaot,E.Torrontegui,X.Chen,andJ.G.Muga, [10] Y.H.Chen,Y.Xia,Q.Q.Chen,andJ.Song,Phys.Rev.A89, Phys.Rev.A89,053408(2014). 033856(2014). [24] M.Lu,Y.Xia,L.T.Shen,J.Song,andN.B.An,Phys.Rev.A [11] M. Demirplak and S. A. Rice, J. Phys. Chem. A 107, 9937 89,012326(2014). (2003). [25] L.GiannelliandE.Arimondo,Phys.Rev.A89,033419(2014). [12] M. Demirplak and S. A. Rice, J. Phys. Chem. B 109, 6838 [26] X.ShiandL.F.Wei,LaserPhys.Lett.12,015204(2015). (2005). [27] X.ChenandJ.G.Muga,Phys.Rev.A86,033405(2012). [13] M. Demirplak and S. A. Rice, J. Chem. Phys. 129, 154111 [28] A.KielyandA.Ruschhaupt,J.Phys.B47,115501(2014). (2008). [29] S. Iba´n˜ez, X. Chen, E. Torrontegui, J. G. Muga, and A. 7 Ruschhaupt,Phys.Rev.Lett.109,100403(2012). N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. [30] S.Iba´n˜ez,X.Chen,andJ.G.Muga,Phys.Rev.A87,043402 Dre´au, J. F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, (2013). andP.Bertet,Phys.Rev.Lett.107,220501(2011). [31] M.V.Berry,Proc.R.Soc.A414,31(1987). [35] Y.Kubo,F.R.Ong, P.Bertet,D.Vion,V.Jacques, D.Zheng, [32] A.Megrant,C.Neill,R.Barends,B.Chiaro,Y.Chen,L.Feigl, A. Dre´au, J. F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, J.Kelly,E.Lucero,M.Mariantoni,P.J.J.O’Malley,D.Sank, M.F.Barthe,P.Bergonzo,andD.Esteve,Phys.Rev.Lett.105, A.Vainsencher, J.Wenner, T.C.White,Y.Yin,J.Zhao,C.J. 140502(2010). Palmstrøm,J.M.Martinis,andA.N.Cleland,App.Phys.Lett. [36] G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, 100,113510(2012). andD.D.Awschalom,Science326,1520(2009). [33] P.L.Stanwix,L.M.Pham,J.R.Maze,D.LeSage,T.K.Yeung, [37] D.Farfurnik,A.Jarmola,L.M.Pham,Z.H.Wang,V.V.Do- P.Cappellaro,P.R.Hemmer,A.Yacoby,M.D.Lukin,andR. brovitski, R.L.Walsworth,D.Budker, andN.Bar-Gill,Phys. L.Walsworth,Phys.Rev.B82,201201(2010). Rev.B92,060301(R)(2015). [34] Y.Kubo,C.Grezes,A.Dewes,T.Umeda,J.Isoya,H.Sumiya,