Physically feasible three-level transitionless quantum driving with multiple Schr\"{o}dinger dynamics

0.22 MB·

by  Xue-Ke Song

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.
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