IndirectexcitonqubitmanipulationviatheopticalStarkeffectinquantumdotmolecules J. E. Rolon∗ and J. E. Drut† DepartmentofPhysicsandAstronomy,UniversityofNorthCarolina,ChapelHill,NorthCarolina,27599-3255,USA (Dated:January20,2016) We propose a coherent control scheme based on the optical Stark effect in optically generated excitons in quantumdotmolecules(QDMs). Weshowthat,bythecombinedactionofvoltagebiasdetuningsweepsand Rosen-Zener pulsed interactions, it is possible to dynamically generate and modify an anticrossing gap that emerges between the dressed energy levels of long-lived, spatially indirect excitons. We perform numerical andanalyticnon-perturbativecalculationsbasedontheBloch-Feshbachformalism,whichdemonstratethatthis effectinducesamechanismofcoherentpopulationtrappingofindirectexcitonsinQDMs. Ourresultsshow 6 thatitispossibletoperformanall-opticalimplementationofindirect-excitonicqubitoperations, suchasthe 1 Pauli-XandHadamardquantumgates,acrosstwodefinedaxisoftheBlochSphere. 0 2 PACSnumbers:73.21.La,71.35.Gg,03.67.-a n a J I. INTRODUCTION acotunneling-inducedanticrossingbetweenspatiallyindirect 9 excitons. We demonstrate that this mechanism enables the 1 coherentpopulationtrappingofeitheroftheavoidedcrossed Quantum dot molecules (QDMs) are promising building ] blocksforsemiconductor-basedapproachestoscalablequan- excitons,andfurtherallowscoherentcontrolofindirectexci- l tum information technologies1. They possess remarkable tonicqubitsabouttwoaxisoftheBlochsphere. l a properties,amongwhicharelong-livedchargeandspinstates This paper is organized as follows. Section II introduces h ofconfinedcarriersandexcitons2,3,tunableexcitonrelaxation arealisticphenomenologicalmodeloftheQDMexcitonlevel - s rates4,sustainedcoherentRabioscillations5,6,andtheability structure,whichtakesintoaccounttheparametricdependence e to couple spin and charge to photonic cavity modes7. More- ofsinglechargeconfinementenergiesandinterdotcouplings m over, the tunability of the QDM exciton spectrum, and the onthestructuralparametersofthesystem.Wealsodiscussthe t. selective excitation of the spin and charge degrees of free- numericalmethodsusedtocomputethepopulationdynamics a dom,makeexciton-basedcoherentcontrolprotocolsaviable of the exciton states. In Sec. III, we present the reconstruc- m routefortheimplementationofuniversalquantumgates8. In tionoftheQDMexcitonlevelanticrossingspectrum(LACS) - this context, having controllable exciton states that are re- using level population bias maps. These maps help us iden- d silientagainsttheeffectsofdecoherenceisafundamentalre- tifythedifferentmolecularresonancesandopticalsignatures n o quirement for the construction of the corresponding qubits. asafunctionoftheappliedbiasvoltage, laserexcitationfre- c In particular, spatially indirect (neutral) excitons in QDMs quency,andintensity. SectionIVdiscussestheroleoftheop- [ havebeenproposedassuitableexcitonicqubitsowingtotheir ticalStarkeffectonthespectralcharacteristicsanddynamics extended lifetimes and robust electrically-controlled optical of a qubit manifold comprised by spatially indirect excitons. 1 v properties9,10, and are the object of active experimental and Bymeansofanon-perturbativecalculation,wepresentcom- 0 theoreticalresearcheffortsfortheengineeringofquantumin- prehensive analytical expressions that show the dependence 6 formationschemesinQDMs4,11–13. of the indirect-excitonic dressed energies, avoided crossings, 0 Apartfromthepurelyelectric(ormagnetic)meansofcon- andinteractions,onthebiasdetuningandthematrixelements 5 that couple the qubit manifold to the driving fields. In Sec. trolling exciton states, ultrafast optical excitation can induce 0 V,wediscussthebehavioroftheexcitonpopulationdynam- profoundmodificationstotheexcitoniclevelstructureofsin- . 1 glequantumdots(QDs)andQDMs. Aprominentexampleis ics resulting from the implementation of bias detuning and 0 theopticalStarkeffect14,whichismanifestedasaquasi-static RZ pulses which modulate the intensity of the optical Stark 6 shift. We show that optical Stark effect generates a coherent shiftofexcitonlineshapeswhenthesystemissubjectedtoin- 1 populationtrappingmechanismwhichcanbeusedtoperform tenseultrafastlaserexcitationpulses.InQDs,transientreflec- : v tivitymeasurementswithpulsednon-resonantexcitationdis- indirect-excitonicqubitrotationsabouttwoaxisoftheBloch i sphere. X playaspectralenvelopethatdependsonthestrengthoftheex- citationfield15. Therefore,thecombinedactionoftheoptical r a Starkeffectandtheapplicationofexternalelectricfieldscould serveasausefulmechanismfortheimplementationofcoher- II. MODEL entcontrolofspinandexcitonstatesinQDsandQDMs16–20. Inthiswork,westudytheinfluenceoftheopticalStarkef- The system under consideration consists of two vertically fect on the spectra and dynamics of indirect excitonic qubits stacked QDs embedded in a n-i Schottky junction photodi- inelectricallygatedandopticallydrivenQDMs9. Wepropose ode (see Fig. 1), as typically used in photoluminescence and that, through the combined action of ultrafast laser pumping pump-probespectroscopyexperiments2,22. TheQDsaresep- with a time-dependent Rosen-Zener (RZ) pulse envelope21 aratedbyabarrierofthicknessdandsubjectedtoanapplied and the application of a bias detuning of the indirect energy axialelectricfieldF.Inourmodel,singleneutralexcitonsare levels,itispossibletodynamicallyopenandclosethegapof pumpedbyabroadsquarelaserpulseoffrequency,ω ,with L 2 Theapplicabilityofourmodeldoesnotrelyonanyresonant  conditionbetweenthebareexcitonenergylevelsatzerobias. L GaAs F Thesimulationparametersutilizedinourmodelaregivenin Ref.[34]andalevelconfigurationdiagramoftheHamiltonian z T QD isshowninFig.2. InAs In order to extract the exciton dynamics, molecular reso- GaAs B QD nances, and the response of the system to the effect of the InAs control pulses, we solve for the density matrix, whose time evolutionisgovernedbytheMarkovianmasterequation y GaAs x ∂ρ i =− [H,ρ]+Lρ. (8) ∂t (cid:126) FIG.1. (Coloronline)Schematicrepresentationofthequantumdot In our formalism, the Liouvillian Lρ has the Lindblad form moleculediscussedinthiswork.SeeSec.IIfordetails. givenby atime-dependentelectricfieldenvelopeE(t). Theexcitonic Lρ=(cid:88)1Γ (cid:16)2α ρα†−α†α ρ+ρα†α (cid:17) (9) barestatesaredenotedaccordingtosinglechargeoccupation 2 i i i i i i i i in each QD, i.e. eBeT X, where e ,h = {0,1} are hBhT B(T) B(T) theelectronandholeoccupationnumbersinthe“bottom(B)” where α† (α ) and Γ (cid:39) 1ns−1 are the exciton creation (an- i i i and “top” (T) QDs, respectively. These states comprise the nihilation)operatorsandeffectiverelaxationratesforchannel followingbasis23: thevacuumstate i,respectively. ForappropriatelychosenQDMstructuralpa- rameters and excitation conditions, the characteristic periods |1(cid:105)≡|00X(cid:105); (1) 00 τ (correspondingtocoherentoscillationsoftheexcitonpop- c ulationsρ (t))couldbemuchshorterthantheexcitonrecom- twoneutralspatiallydirectexcitonstates: ii binationtime,i.e. τ (cid:28)τ ;therefore,theexcitonpopulation c X |2(cid:105)≡|1100X(cid:105) and |5(cid:105)≡|0011X(cid:105); (2) dynamicsisnearlycoherentfortimesτc <t(cid:28)τX35–37. andtwoneutralspatiallyindirectexcitonstates: 10X 01 |3(cid:105)≡|0110X(cid:105) and |4(cid:105)≡|1001X(cid:105). (3) 01X 10 t Inthisbasis,theHamiltonianis t th t e h  edF(t) e 01X (cid:88)5 (cid:88) 4 01 H= Ei|i(cid:105)(cid:104)i|+ ((cid:126)Ωj(t)e−iωLt|1(cid:105)(cid:104)j|)+VF|2(cid:105)(cid:104)5| 10X VF i=1 j=2,5  10 X(t) 3edF(t) +t (|2(cid:105)(cid:104)3|+|5(cid:105)(cid:104)4|)+t (|2(cid:105)(cid:104)4|+|3(cid:105)(cid:104)5|)+H.c., 2 e h (4) X(t)   5 X X where the E ’s represent the exciton bare energies24–27 and i t ,t theelectronandholetunnelingmatrixelements28.Here, 00X e h V = µBµT accounts for interdot exciton hopping pro- 00 F 4π(cid:15)0(cid:15)rd3 cessesmediatedbydipole-dipoleinteractions29,30,with(cid:15) be- r FIG. 2. (Color online) Single exciton level configuration diagram. ingthedielectricconstant,andµ theinterbandtransition T(B) Dashed arrows indicate couplings mediated by different processes: dipolemoments31–33. electronandholetunneling,t ,t ;opticalcoupling,Ω (t),andex- e h X The Hamiltonian dynamics is controlled via two parame- citonhopping,V . Relaxationchannelsareindicatedbyreddashed F ters. The first one is achieved via pulsing the applied bias arrows, while solid arrows in gray indicate the respective laser- voltageF(t),whichdrivesthedetuningofthespatiallyindi- excitontransitionenergydetuningsδ =E −(cid:126)ω . i i L recttransitionenergies, E →E −edF(t), (5) 3 3 E →E +edF(t). (6) 4 4 III. LEVELANTICROSSINGSPECTRUM ThesecondparameterisachievedthroughtheopticalStarkef- fectresultingfrompulsingtheshapeofthelaserelectricfield In order to obtain the behavior of the different exciton envelope,i.e. thetime-dependenceoftheexcitoncouplingto molecularresonances,anditsdependenceonthecouplingpa- thelaserfieldgivenby rameters,appliedelectricfieldandexcitationpower,wecon- structalevelanticrossingmap.Wedothisintwodifferentbut Ω (t)=(cid:104)1|µ(cid:126) ·E(cid:126)(t)|j(cid:105). (7) equivalentways,asweexplainnext. j j 3 First,bydiagonalizationoftheHamiltonianEq.4inthero- in Fig. 4). Indeed, as shown in our previous work, see Ref. tating wave approximation (RWA)38, for fixed excitation en- [9],aspectrallyisolatedtwo-levelsystemspannedbytheindi- ergy (cid:126)ω , while varying only the bias voltage F. We thus rectexcitonsshowsananticrossingsignaturewhichcanform L obtainaglobalpictureoftheenergyeigenvalues(dresseden- thebasisforconstructingindirectexcitonicqubitsresilientto ergies), eigenvector components, and the electrically-tunable spontaneousrecombination,andwhosecoherentdynamicsat spatialcharacterofthemolecularexcitons,whichisgoverned lowexcitationpowers(Ω (cid:28)1meV)canbecontrolledbythe X mainlybychargetunnelingandtheopticalStarkeffect. tuningtheelectricfieldF.However,asshownhere,bymeans In the second approach, we reconstruct an averaged level of the optical Stark effect it is possible to take advantage of occupationmapcorrespondingtoeachindividualexcitonstate laser-pulseshapingtechniquestocontrolthecoherentdynam- by direct integration of the diagonal elements of the density ics of the spatially indirect excitons from low to moderately matrix, highexcitationpowers. 1 (cid:90) tL oaRmbatbpaiilnioteusdcdiebllypautsiloosnleusdtiouofrnpatshtiieto=oneExtthcqLai.tto8in0.sHlpooenρprgieui,(elattnL)toidoustntg,ashn;tdionscpfaorpartcautircceoesn,esavt(aef1enr0wat-)l w (meV)(cid:1)L1111222224680000 ( a11000011)XX 1001X 011001X10X 1001X 0110X11000011XX 11111 222226667703704.....05050 ( b ) 10010110XX 01101001XX W=0.75mEV X RftoiofvarAebeth,ialteemoearsvpcncalhaiiclttleuuiavxduteicemoliytnpo,sisn(otaaFnprte,eeu(cid:126)mce(ωRanpLnoeW,ducΩAgoihnXmptot)pho.uothrtteeoelnistahybfiesletyelledmcv)oew|ml1io(cid:105)pllcuec≡txeuhppia|bi0t.i0iotXTnah(cid:105)rm,eerlasaepo-- w (meV)(cid:1)L1111222224680000 1001X 0110X 1001X 0110X 11111 222226667703704.....05050 10010110XX 01101001XX W=3.69mEV X 00 1280 1274.0 icta.hiletla.otcnotb.hoyeIrndtcriopannmaarstptefielsceru(rteliFand,rg,(cid:126)retiωhtssLesep,ldaΩotpLtXeuAr)laCawtpiSohpnesrporeteaoccththeriausecmnshatawbctloeielrslirsebusdespemtoponoadeppsiupntliegamdteeabxdtye-, w (meV)(cid:1)L111222246000 1001X 0110X 1001X 0110X 1111 222266670370....0505 10010110XX 10010110XX W=6.0mEV X theexcitationpowerdependenceofthedifferentspectralsig- -60 -40 -20 0 20 40 60 -2 0 2 4 6 8 F (k V /c m ) F (k V /c m ) natures resembling those obtained by photoluminiscence or pump-probespectroscopy. FIG. 3. (Color online) Level anticrossing map of |00X(cid:105) as func- 00 Figure 3 (a) shows the level anticrossing spectrum corre- tion of the applied electric field, F, and laser excitation energy, sponding to the energy level diagram in Fig. 2. This was E = (cid:126)ω ,andfordifferentvaluesoftheopticalcoupling,Ω . laser L X reconstructed from the level occupation map of the vacuum (a)Fullspectrumshowingelectrontunnelinganticrossingsignatures state|0000X(cid:105)asafunctionofappliedelectricfieldF andlaser at(cid:126)ωL (cid:39) 1252meV,F (cid:39) (−18.7,23.4)kV/cm, togetherwiththe excitation energy (cid:126)ω , for three values of the optical cou- asymptoticbehaviorofmolecularsatestowardsthespatiallydirect, L pling strength ΩX = 0.75,3.69,6.0meV. The spectral pat- |1100X(cid:105),|0011X(cid:105)andindirectexcitonstates,|0110X(cid:105),|1001X(cid:105).(b)Thespa- tiallyindirectexcitonmanifoldshowsarobustanticrossingsignature tern shows anticrossing signatures between spatially direct atF (cid:39) 3.17kV/cm,whichpersistsunderstrongpowerbroadening and indirect excitonic molecular states at F (cid:39) −18.7 and effectscausedbylargevaluesofΩ . F (cid:39) 23.4kV/cm, each having a gap of (cid:39) 6.23meV; these X are mainly the result of electron tunneling. Clearly, with increasing optical coupling, the direct states |10X(cid:105), |01X(cid:105) 10 01 become prominently affected by power broadening effects IV. THEOPTICALSTARKEFFECT such that, for the highest value of the coupling, the electron tunneling anticrossing intersecting the horizontal line shape (cid:126)ω (cid:39) 1252meVbecomesalmostentirely“washedout”. In Inordertoelucidatetheoriginandbehavioroftheoptical L contrast, the spatially indirect exciton spectral lines, |01X(cid:105) StarkeffectsignaturesshowninFig.3(b),fordifferentvalues 10 and |1001X(cid:105), are more robust to the effects of increasing ΩX, oftheopticalcouplingstrengthΩX,wefirstcalculatenumer- astheirbroadeningweakensas|F|isincreasedrespecttothe ically the excitonic dressed energy spectrum as function of position of the tunneling resonances. This can be easily un- the applied electric field F, for a fixed non-resonant excita- derstood: the optical coupling ΩI of the indirect excitons is tionenergy(cid:126)ωL = 1277meV((cid:126)ωL isdetunedfromboththe mediated by the optical pumping of electrons and holes and direct and indirect exciton transition energies). Figure 4(a) their tunneling rates, which become weaker with increasing showstheanticrossingsignatureresultingfromthemixingof |F|awayfromtheavoidedcrossings. thespatiallyindirectstates|0110X(cid:105)and|1001X(cid:105). Remarkably,as Ontheotherhand,theopticalsignatureinFig.3(b)showsa the optical coupling increases from ΩX = 0.75meV (solid robustanticrossingatF (cid:39)3.17kV/cm((cid:126)ω (cid:39)1269meV)re- lineinblack)thegapshowsanon-monotonicbehavior, van- L flectingthemixingofthetwoindirectstates|01X(cid:105)and|10X(cid:105), ishing for Ω(c) (cid:39) 3.69meV (solid line in orange). Figure 4 10 01 X which persists even under strong power broadening effects shows the value of the anticrossing gap as function of Ω , X caused by large values of Ω (the signature shifts to lower withthepositionoftheminimumindicatedbythereddashed X energies as a result of the optical Stark effect, as explained line. As shown in Fig. 4(c), the position of the gap minima 4 alongthebiasdetuningaxisslightlyshiftsaswevarytheop- all orders) the resulting perturbative corrections to the ma- ticalcoupling,reachingacriticalvalue,F ,whenthegapvan- trix elements of the projected Hamiltonian. To this end, we c ishes. Theobservedshiftofthegapminimaisaconsequence employ a standard non-perturbative procedure based on the of the of the parametrical dependence of the dressed exciton Bloch-Feshbachprojectionoperatorformalism39–41. energies on the values of the interdot couplings and energy Tocalculatetheeffectiveopticalcouplingsofthespatially detuningsoftheindirectexcitontransitions. indirect excitons to the radiation field, namely Ω˜ and Ω˜ , 13 14 In the following subsections, by means of a non- wedefinethetargetsubspaceforourprojectiontobespanned perturbative calculation, we present comprehensive analytic by {|1(cid:105)≡|00X(cid:105),|3(cid:105)≡|01X(cid:105),|4(cid:105)≡|10X(cid:105)}, see Fig. 5(a). 00 10 01 expressionswhichrevealtheoriginoftheaforementionedef- Theprojectionprocedureyields fectintermsoftheinterplaybetweentheenergyandintensity (cid:16) (cid:17)(cid:16) (cid:17) oanfdtheenderrgivyindgetfiuenlidn,gasn.dWaellaolfsothdeisrceulesvsahnotwinttehredtoutncaobuilpiltiyngosf Ω˜13 = zth−ΩδX + ΩX + VzF−ΩδX5 tVe2+ tzh−VδF5 , (11) theindirectexcitonicgapviatheopticalStarkeffectenables 5 z−δ2− z−Fδ5 adynamicallycontrolledcoherentpopulationtrappingmech- (cid:16) (cid:17)(cid:16) (cid:17) ainndisirmec,tweixtchitpoontiecnqtiuabliatspipnliQcaDtiMonss.to the coherent control of Ω˜ = teΩX + ΩX + VzF−ΩδX5 th+ zte−VδF5 , (12) 14 z−δ V2 ( a ) ( b ) 5 z−δ2− z−Fδ5 -7 .9 0 1 X 1 0 X 0.20 wherez =E±i(cid:15)arethecomplexeigenvaluesofHeff,and E(meV)----8888 ....3210 1 0 0 1 WWWWWWWWXXXXXXXX(=======m0233467e.......7006505V59) U(meV)I(30000.c1....001170505)60 1 2 W3X(m4 eV5) 6 7 8 a|tf11hrro00eeXmltah(cid:105)rpegareondscdetetc|us00osn11eniXsntrgii(cid:105)bsn,uvrotoeiflosvpntδδihen25tecogt==istvehpexEEealcty25eii.taf−−oflAlneyc(cid:126)(cid:126)sptωωdisuviLLhmero,,eopwcpitnnteigcixnafcloiEtcloqloonsuw.pt1erladi1nngabsniyatdir(otiu11snen32ss-),, )3.172 -8 .4 1 0 X 0 1 X F(kV/cmUI33..116648 ntieolninignvporlovceesstshees.coSmimbiilnaerdly,acthtieonneoxftelxecaidtoinngpourmdeprincgo,nstirnibgule- -8 .5 0 1 1 0 3.1600 1 2 3 4 5 6 7 8 charge tunneling andexciton hopping processes. Notice that 2 .8 3 .0 3 .2 3 .4 3 .6 W X(m eV ) both couplings are different, a fact that is reflected from the F (kV /cm ) antisymmetric and non-resonant nature of the direct exciton FIG. 4. (Color online)(a) Bias-dependent exciton dressed energy transitions,i.e. Ω˜13 →Ω˜14asδ2 →δ5. spectrumatfixedexcitationenergy,(cid:126)ω = 1277meV,showingthe Ontheotherhand,theresultingeffectivecouplingbetween L anticrossingoftheindirectexcitonstatesfordifferentvaluesofthe bothindirectstatesis opticalcouplingΩX. AsaresultoftheopticalStarkshift, thean- (cid:16) (cid:17)(cid:16) (cid:17) ticrossinggapvanishesforΩ(Xc) (cid:39) 3.69meV(solidlineinorange) U˜ = teth + te+ tzh−VδF5 th+ zte−VδF5 , (14) (tibo)nAfnotricthroessshinifgtingagpomftihneimmainaismfuanacltoinogntohfeΩbXias(ivnoclltuadgiengaxaisc,oFrr,eacs- 34 z−δ5 z−δ2− zV−F2δ5 shownin(c). (a) (b) 10X 01 ~ IV.1. Non-perturbativecalculationoftheopticalStarkeffect U14 10X forindirectexcitons 01X 01 10 ~~ ~ (t)  (F(t), (t)) ~ 1414 34 X  edF(t) ~ ssiegtAnoasftudarisencobunes-tswterdieveainablosqpvuaeta,inathtluleymaipncpdoeihraeercraetnnectxecinoittfoeanraslceptviooeilnn.atsnMttiocortroheseosvoinenrg-, 4 ~13(t) ~13 14 ~3edF(t) UI(X(t)) 1001X thedependenceofthisinteractionontheopticalcouplingΩ X cannotbestraightforwardlyexplainedbytheoff-diagonalma- 00X 00 trixelementsoftheHamiltonianinEq.4, norfromthelevel diagram shown in Fig. 2. However, the physics can be re- FIG. 5. (Color online) (a) Effective Hamiltonian level configura- vealed by an effective Hamiltonian Heff resulting from the tionafteradiabaticallyeliminatingthedirectexcitonstates. Thedi- projection of the full Hamiltonian onto a reduced sector of agram indicates the effective optical coupling of indirect excitons the Hilbert space containing eigenvectors relevant to the an- to the laser field, Ω˜ ,Ω˜ . Dashed red arrows indicate relaxation 13 14 ticrossingregion,witheigenvaluesmatchingexactlythoseof channels. (b)Effectivetwo-levelsystem(qubit)configurationafter thefullHamiltonian. Inotherwords,weaimtoadiabatically projection of the vacuum state onto the indirect exciton subspace. eliminatethespatiallydirectexcitonicsectoroftheHamilto- Both,thecouplingandlevelseparationarefunctionsofthecontrol parameters,Ω (t),F(t). nian, while retaining its dynamical effects by including (to X 5 where the first term shows that electron-hole cotunneling is Therefore, to find the conditions for which the gap vanishes, theleadingprocessthatcouplesthetwoindirectexcitons,with weset∇Λ =0. Thisyields I thesecondtermdescribingsinglechargecotunnelingandex- √ citonhopping.Noticehowever,thatatthisstageoftheprojec- Ω(c) = 2 2 (cid:113)V (t2 +t2)−t t (δ +δ −2z), (21) tion procedure, Eq. 14 does not reveal the optical Stark shift X t −t F h e h e 2 5 e h dependence of the anticrossing features shown in Fig. 4, as its effects are still embedded in the matrix elements involv- 1 (cid:12) ing the RWA vacuum state |00X(cid:105). To make this dependence F = ((δ −δ )+(∆ −∆ ))(cid:12) . (22) more explicit, we further re0d0uce the target subspace of the c 2ed 3 4 t h (cid:12)Ω(Xc) projectionproceduretoatwo-level(qubit)subspacespanned Itisimportanttoremarkthatintheabsenceofopticalexcita- by{|01X(cid:105),|10X(cid:105)},seeFig.5(b). Thisresultsinaneffective 10 01 tionΩX = 0,anon-vanishingindirectexcitonicanticrossing Hamiltonian signature,i.e. Λ (cid:54)= 0,isconditionedbythecouplingU˜ in I 34 (cid:18)δ −∆ +∆ U (cid:19) Eq.14,i.e. onlybyexcitonhoppingprocessesandtunneling. H = 3 S h I , (15) I UI δ4+∆S +∆t ForVF = 0,theanticrossinggapemergesonlybytheaction of electron-hole cotunneling processes (see first term in Eq. wherethediagonaltermscontaincontributionsfromthebias 14), while for V (cid:54)= 0, the gap emerges by the action of ei- F detuning∆S =edF andenergyshiftsgivenby therelectronorholetunneling. Ontheotherhand,underthe influence of the optical Stark effect Ω (cid:54)= 0, the vanishing- (cid:16) (cid:17)2 X ∆h = z−t2hδ + te+ tzh−VδF5V2 + z−Ω˜21∆3 , (16) gexacpitcoonndhiotpiopninsginaEndqss.in2g1leancdar2r2ierarteunanlseolicnogn.dFiotiroVnFed(cid:54)=by0b,oththe 5 z−δ2− z−Fδ5 U condition is fulfilled with the tunneling of either electron or hole, while for V = 0, the condition is fulfilled only when F (cid:16) (cid:17)2 bothelectronandholetunnelingaredifferentfromzero. ∆ = t2e + th+ zte−VδF5 + Ω˜214 . (17) t z−δ V2 z−∆ 5 z−δ2− z−Fδ5 U V. QUBITCOHERENTCONTROLVIATHEOPTICAL Ontheotherhand,theoff-diagonalcouplingisgivenby STARKEFFECT U =U˜ + Ω˜13Ω˜14 , (18) V.1. Proposedcontrolsetup I 34 z−∆ U To illustrate the controllability of the Hamiltonian in Eq. where 15, we interpret the tunability of the indirect excitonic gap (cid:16) (cid:17)2 ∆ = Ω2X + ΩX + VzF−ΩδX5 . (19) ivnectteorrmassosofciitasteedffetocttohne sthtaeteteomfptohrealinedvioreluctti-oenxcoitfotnhiecBqulobcith. U z−δ V2 5 z−δ2− z−Fδ5 Tothisend,werecastEq.15asfollows, TheleadingtermU˜34inEq.18originatesfromchargecotun- HI =ασ0+βσz+UIσx, (23) nelingandexcitonhopping,anddominatesthebehaviorofU I whereσ isthe2×2identitymatrix,σ ,σ arethePaulima- for0 ≤ Ω < 1meV,i.e. theeffectsoftheopticalcoupling 0 z x X trices, with corresponding coefficients given by the coupling arenotsignificantinthisregime,seeFig.4(b). However,the smeecVon.dThteisrmfeaΩ˜zt1−u3r∆Ω˜eU1e4ndaobmleisnaanteasltlh-oepbteichaalvcioorhoerfeUnItcfoonrtΩroXlo≥ve1r UI (definedinEq.(cid:18)18δ)a+ndδ (cid:19) (cid:18)∆ +∆ (cid:19) thespatiallyindirectexcitonicqubitsubspace,startingfroma α= 3 4 + t h , (24) 2 2 regimeinwhichthedynamicsispurelydominatedbycotun- neling to a regime where the dynamics becomes highly con- trollable by the bias detuning F(t), and the optical coupling (cid:18)δ −δ (cid:19) (cid:18)∆ −∆ (cid:19) β = 3 4 − t h +∆ . (25) ΩX. 2 2 S Adistinctiveeffectofthetunabilityoftheopticalcoupling Ω ,istheopeningandclosingoftheindirectexcitonicanti- Assuming fixed values of the variables that depend on the X crossinggapΛ ,achievedbymeansoftheopticalStarkshift. QDM structural and material composition parameters (i.e. I After diagonalization of Eq. 15 we obtain the corresponding single charge confinement energies, single charge tunneling gapequation, and exciton hopping strengths), time-dependent rotations of theBlochvectoraboutthezzzˆˆˆ-axiscanbecontrolledprimarily (cid:113) ΛI = 4UI2+(2∆S +(δ4−δ3)+(∆t−∆h))2. (20) by the value of the bias detuning ∆S, and by ΩX-dependent energyshifts∆ ,∆ ,asshowninEq.25andFig.6. Onthe h t Equation 20 defines an energy surface over the control pa- other hand, rotations about thexxxˆˆˆ-axis are controlled by cou- rameters,withminimaatthecriticalvaluesofthebiasdetun- pling U , whose strength is modulated solely by the optical I ingandopticalcoupling∆(c) = edF andΩ(c),respectively. couplingΩ .Therefore,acoherentcontrolschemeviaEq.23 S c X X 6 𝟎𝟏𝟏𝟎𝑿 δ3+edF(t)andδ4+edF(t)withinthequbitmanifold. This sweep brings the indirect transitions in and out of resonance b fromthepointofclosestapproachattheanticrossingmixing |01X(cid:105) and |10X(cid:105), see Fig. 3(a) and dashed black box in Fig. 10 01 7. Subsequentlyattheanticrossing,thesystemisdrivenwith a pulsed optical Stark shift Ω (t), that controls the strength X ofU ,andconsequentlythewidthoftheanticrossinggap. I U Inthepresentwork, weconsiderapulsedinteractionwith I a smooth rise profile which controls the strength of the opti- 𝟏𝟎𝟎𝟏𝑿 cal Stark effect within the qubit manifold, i.e UI or equiva- lently Λ . Among the different choices for the pulse shape I profile (e.g. smooth rectangular, Gaussian, hyper Gaussian, FIG. 6. (Color online) Bloch sphere geometrical representation of etc.)44, we employ a Rosen-Zener hyperbolic tangent pro- the two-level system (|1001X(cid:105) and |0110X(cid:105)) and the two rotation axes file21,45. Thischoiceismadefirstlybecauseitconformswell (UUUˆˆˆIII andβββˆˆˆ)allowingtheimplementationcontrolledrotationsofthe with the assumed non-resonant excitation conditions and en- correspondingBlochvector. ergylevelstructureinourmodel,andsecondlybecauseitpro- duces an efficient population transfer between pairs of anti- 1 0 crossedstates,whileminimizingnon-resonantpopulationos- 0 1 X 1 0 X cillations into states outside the controlled target subspace46. 1 0 0 1 0 0 X TheformofourRZpulseisasfollows, 0 0 In itia liz a tio n O p tic a l p u ls e c o n tro l 0 (cid:18) (cid:19) -1 0 000...048 0000X 0110X ~ 2 W 130000X d+edF(t)3F (t) --88..20 0110X U I(W X(10t01)X) ΩX(t)=ΩX(ti)+(ΩX(tf)−ΩX(ti))tanh αtτ , (26) whereτ = t −t definesthetimeoverwhichthepulsedin- eV) --00..84 1001X D etuning control -8.4 1001X 0110X teraction is nfon-stiationary, and α ≥ τ−1 controls the pulse E(m-2 0 0 1-X12 -10 -8 -6 -4 10 01 X 01 10 X 2.8 3.0 3.2 3.4 1 30.X6 rcirseea.seThmeofnoortmonoicfatlhliysfpruolmseacnaiunsietisatlhveaolupetitcoawlacroduspalincgontsotainn-t 0 1 1 0 value and ithas great applicability in thecoherent control of 0 1 X -3 0 1 0 X 0 1 multi-level systems approaching the two-state limit, such as 1 0 thespectrallyisolatedindirectexcitonmanifoldshowninFig. 1 0 X 0 1 X 5(b). 0 1 1 0 -4 0 -6 0 -4 0 -2 0 0 2 0 4 0 6 0 F (k V /c m ) V.2. Resultingpopulationdynamics FIG. 7. (Color online) Level anticrossing spectrum showing schematicsofthecoherentcontrolscheme. Thesystemispumped Figure 8 shows the exciton population dynamics resulting at (cid:126)ωL = 1277meV with ΩX = 0.75meV. The system is ini- fromapplyingthecontrolprocedureshowninFig.7.Thesys- tialized into the state |0110X(cid:105) at the avoided crossing Ω˜13 at F = temisinitializedwithanopticalcouplingofΩX =0.75meV −7.967kV/cm(dashedredbox-leftinset). Subsequently,thesys- attheanticrossinglocatedatF =−7.96kV/cm,whichmixes tem is driven by a detuning control bias sweep δ3 +edF(t) into theRWAvacuum|00X(cid:105)andtheindirectstate|01X(cid:105). Thisan- theavoidedcrossingatF = 3.1759kV/cm(dashedblackbox). At ticrossinghasagap002Ω˜ (cid:39) 0.2meV,suchthat10theRabihalf- theavoidedcrossingtheopticalcouplingissubjectedtoatemporal 13 periodcorrespondingtoaπrotationis (cid:126)π (cid:39)10.2ps.Aftera Rosen-ZenerpulseΩX(t),whichcontrolsdynamicallythewidthof 2Ω˜13 theindirectexcitonicgap(rightinset). 3πrotation,asuddendetuningbiaspulse(seeFig.8(e))traps and brings the spatially indirect exciton into the qubit mani- fold defined by the avoided crossing at F = 3.1759kV/cm, allowstheimplementationofarbitraryrotationsovertwoaxes which has a gap 2U (cid:39) 0.1mev. Here, the π-rotation period I oftheBlochSphere,openingthepossibilitytoimplementuni- correspondingtocoherentoscillationsoftheindirectexcitonic versalindirectexcitonicqubitoperationsinQDMs42,43. qubitbasishasaperiodof (cid:126)π (cid:39) 20.3ps,whichcanbecon- 2UI Toimplementuniversalcoherentoperationsontheindirect trolledbythestrengthoftheopticalStarkeffect.Atanelapsed excitonic qubit subspace, we constructed a control scheme timet=118.03psweapplyaRZpulsewithapulserisetime based on the combination of a bias time-dependent linear τ = 40.3ps and rate constant α = 0.2, which increases the sweepF(t),andapulsedopticalStarkinteractionΩX(t).The opticalcouplingfromΩX =0.75meVuptothecriticalvalue bias sweep serves two purposes: firstly, it is used to initial- Ω(c) (cid:39) 3.69meV; the RZ pulse has the effect of closing the X ize the qubit by selecting a value of the bias detuning such avoidedcrossingandtrappingtheexcitonpopulationswitha that the radiation field (RWA vacuum |00X(cid:105)) becomes reso- weight determined by the pulse duration τ, and the rise rate 00 nant and coupled (with strength Ω˜ ) to one of the indirect constantα,seeEq.26. Fig.8(a)showstheeffectonthepop- 13 excitons |01X(cid:105), see red dashed box and inset in Fig. 7; sec- ulation dynamics of the RZ pulse in (b), which traps the ex- 10 ondly, the bias sweep controls the indirect exciton detunings citon 10X with near unity fidelity. On the other hand, Fig. 01 7 ρ(t)°°1.0 0000X 1001X 1001X 1100X 0011X n o (a) ati0.5 ul p Po0.0 4.0 Ω(t)X23..00 Ω=0.75meV ΩX=3.69(mb)eV 1.0 X ρ(t)°°1.0 n (c) o ati0.5 ul p Po0.0 4.0 Ω(t)X23..00 Ω=0.75meV ΩX=3.69(mde)V 1.0 X m)4.0 c0.0 F=3.17kV/cm kV/-4.0 Initialization Rapiddetuningbiaspulse (e) F(-8.0 F=-7.96kV/cm 0.0 30.0 60.0 90.0 120.0 150.0 180.0 t(ps) FIG.9.(Coloronline)Blochsphererepresentationoftheoperations FIG.8. (Coloronline)Excitonpopulationdynamicsresultingfrom resultingfromthecombinedactionoftheopticalRZandbiasF(t) thecontrolprocedureshowninFig.7. (aandb)Thesystemisini- tializedinthestate|01X(cid:105)andbroughtintotheanticrossingmixing pulsesshowninFig.8(b-d-e). (a)Usingarapiddetuningbiaspulse |01X(cid:105)and|10X(cid:105)wit1h0arapiddetuningbiaspulseshownin(e).After theBlochvectorisinitializedinthestate|3(cid:105)=|1001X(cid:105),subsequently 10 01 subjected to a 5π rotation about thexxxˆˆˆ-axis. At the end of the RZ afewRabirotations,apulsedRosen-ZeneropticalStarkshiftΩ (t), X transfersthepopulationintothestate|10X(cid:105),effectivelytrappingthis pulsetheBlochvectoriseffectivelytrappedatthesouthpoleintothe 01 state|4(cid:105) = |01X(cid:105). Notethatthisoperationconstitutesaconcatena- state. (candd)AtemporalshiftappliedtoΩX(t)trapsthesystem 10 intoanequallyweightedsuperpositionof|01X(cid:105)and|10X(cid:105). tionofseveralPauli-Xquantumgates.(b)AtemporalshiftoftheRZ 10 01 pulse(seeFig.8(d)),causestheBlochvectortoprecessaboutthezzzˆˆˆ- axis,creatingacoherentsuperpositionstatesofbothindirectstates; theresultingoperationisequivalenttotheapplicationofaHadamard 8(c)showstheeffectofatemporalshiftδτ = 29.6psforthe quantumgateoperation. sameRZpulse;inthiscasethesystemevolvesintoanequally weightedsuperpositionof|01X(cid:105)and|10X(cid:105). Clearly,thepop- 10 01 ulation dynamics of the indirect excitons corresponds to that of a TLS model, thus corroborating the validity of the an- ing the initialization step, as the system leaves the subspace alytical expressions obtained by the Bloch-Feshbach projec- {|0000X(cid:105),|0110X(cid:105)},andentersthequbitsubspace,theBlochvec- tionprocedure,andtheapplicabilityoftheRZpulseshaping. tortiprisesfromtheoriginoftheBlochsphereuntilreaching Note,thatthereisasmallresidual(fastoscillating)population the north pole at |3(cid:105) = |0110X(cid:105). Within the qubit subspace, ofthespatiallydirectexciton|10X(cid:105)(redline)duetothefast the Bloch vector precesses performing several full rotations 10 dynamicstakingplaceoutsidethequbitmanifold,whichorig- aboutthexxxˆˆˆ-axisundertheinfluenceoftheinteractionUI,see inatesbytheproximityoftheelectron-tunnelinganticrossing; Eq. 18; subsequently, under the action of RZ pulse, the sys- inaddition,theindirectexcitonmanifoldattheavoidedcross- tem is trapped in the state |4(cid:105) = |1001X(cid:105) at the south pole of ingexhibitsresilienceagainstrelaxationprocesses,asshown the Bloch sphere. Interestingly, this operation constitutes a bytheslowriseinthepopulationamplitudeofthestate|00X(cid:105) concatenationofseveralPauli-Xquantumgates,bywhichthe 00 fort>150ps,seeblackdashedlineinFigs.8(a)and(c). qubit basis states are mapped amongst each other42. On the To highlight the operational significance of the controlled otherhand,Fig.9(b)showstheeffectofapplyingatemporal exciton dynamics in Fig. 8 within the Bloch sphere of the shift to the RZ pulse. In this case, the RZ pulse brings the qubit subspace {|01X(cid:105),|10X(cid:105)} (see Fig. 6), we reconstruct system into the superposition state of both indirect excitons, 10 01 thetemporalevolutionoftheBlochvectorviathefullnumer- withtheBlochvectorprecessingaboutzzzˆˆˆ-axis,withapreces- icalsolutionsforthedensitymatrixdynamicsobtainedviaEq. sion frequency dominated by the β interaction, see Eq. 25. 8. Tothisend, weparameterizethecoordinatesoftheBloch Notably,thisoperationrepresentstheactionoftheHadamard vectorrrr(t),suchthat quantum gate, by which the qubit basis states are mapped intotheirsymmetricandantisymmetricsuperpositions42. As r (t)=ρ (t)+ρ (t), shown above, the Bloch vector evolution is fully consistent x 34 43 r (t)=i(ρ (t)−ρ (t)), withthestructureofthequbitHamiltonianinEq.23. y 34 43 r (t)=ρ (t)−ρ (t). (27) As discussed in Sec. IV (Eqs. 11 and 12), indirect exci- z 33 44 tonspossessaneffectiveopticalcouplingtotheradiationfield, Figure 9 shows the Bloch vector evolution corresponding andconsequentlyafinitelifetime. Tohighlighttherelaxation tothedynamicsshowninFig.8. AsshowninFig.9(a),dur- dynamicswithinthequbitsubspace, Fig.10showsthelong- 8 00X 10X 01X 10X 01X 00 10 10 01 01 1.00 0.75 ρ(t)ii0.50 (a) 0.25 0.00 0.03 0.02 ρ(t)ii0.01 (b) 0.00 1.00 ρ(t)ii00..5705 (c) 0.25 0.00 0.04 ρ(t)ii00..0023 (d) 0.01 0.00 0.0 6.4 12.7 19.1 25.5 31.8 t(ns) FIG. 10. (Color online) Long-term relaxation dynamics of the ex- citon population for times beyond which the RZ pulse reaches its constantcriticalvalueΩ(c) =3.69meV.(a)Relaxationdynamicsof FIG. 11. (Color online) Bloch sphere representation of the long thestate|ΨI(cid:105) ∼ √12(cid:0)|0110XX(cid:105)+|1001X(cid:105)(cid:1). Thisstateprimarilyrelaxes term exciton relaxation dynamics shown in Fig. 10. (a) After the intotheRWAvacuum|00X(cid:105),accompaniedbyaslightfillingofthe initialization and RZ pulsing steps, the Bloch vector for the state directexcitonstatesass0h0owningreenandredin(b).(c)Relaxation |ΨI(cid:105)∼ √12(cid:0)|1100X(cid:105)+|0011X(cid:105)(cid:1)evolvesbyprecessingaboutthez-axis dynamicsofthestate|01X(cid:105). BesidestheRWAvacuumsaturation, whileitsmagnitudedecreasesasitcompletesseveral2π rotations, 01 thereisaslightfillingoftheindirect-directpairofstatesconnected eventuallycollapsingattheorigin. (b)ThetipoftheBlochvector byelectronhoppingshowninblueandredin(d). ofthetrappedstate|4(cid:105) = |0011X(cid:105),movesawayfromthesouth-pole alongthez-axis,eventuallycollapsingattheorigin. term temporal evolution of the exciton population for times allycollapsingattheorigin, asthestateofthesystemleaves beyondwhichtheRZpulsereachesitsconstantcriticalvalue thequbitsubspace. Ω(c) (cid:39) 3.69meV. In all cases, the population of the indirect X excitons relaxes to the vacuum state |00X(cid:105), while the small 00 residual population of the spatially direct states, |10X(cid:105) and 10 |01X(cid:105),saturatestowardsastationaryvalue. Fig.10(a)shows VI. SUMMARYANDCONCLUSIONS 01 the decay of the superposition state |Ψ (cid:105), with both indirect I excitonsshowingoppositerelaxationenvelopes;smallampli- We have characterized the signatures of the optical Stark tudeoscillationsareobservedinthepopulationofbothdirect effectonthespectrumanddynamicsofspatiallyindirectex- andindirectstates,whichreflecttheinfluenceofthefasttran- citons in optically driven quantum dot molecules. By recon- sitiondynamicsoutsidethequbitsubspacedrivenbyelectron structing the QDM level anticrossing exciton spectrum, we tunneling. On the other hand, the relaxation of the trapped have found an avoided crossing signature between two spa- state |0011X(cid:105) shown in Fig. 10(c) is accompanied by a slight tiallyindirectexcitonswhichpersistsunderstrongexcitation relaxation of the direct state |0011X(cid:105), and population transfer power broadening effects. Under the influence of the op- preferentiallyintothestates|1100X(cid:105)and|0110X(cid:105). tical Stark effect, the gap of this anticrossing exhibits non- Now,Figure11showsthecorrespondingBlochsphererep- monotonicbehaviorwithavanishingvaluethatdependspri- resentation of the relaxation dynamics presented in Fig. 10. marilyontheinterplayofchargetunnelingandopticalexcita- InFig.11(a),relaxationofthesuperpositionstate|Ψ (cid:105)takes tion.Usinganon-perturbativeBloch-Feshbachprojectionfor- I place long after the completion of the Hadamard gate. As malism, we presented comprehensive analytic results which shown,relaxationinducesadecreaseintheBlochvectormag- explaintheoriginandbehavioroftheaforementionedsigna- nitudeasitevolvesintimeprecessingaboutthezzzˆˆˆ-axisunder turefordifferentconditions. Wehaveshownthatthedynami- theinfluenceofthecouplingβ,seeEq.25. Afteranelapsed calopeningandclosingoftheindirectexcitonicgapenablesa precession time of 31.8ns, the Bloch vector eventually col- coherentpopulationtrappingforindirectexcitons,abehavior lapses into the origin, as the state of the system within the that is akin to the coherent destruction of tunneling mecha- qubit subspace evolves from a pure state into a completely nism. We devised a coherent control scheme based on the mixed state. On the other hand, Fig. 11(b) shows the Bloch opticalStarkeffect,whichreliesonthevariationofthepulse vectortimeevolutionofthetrappedstate|4(cid:105) = |10X(cid:105). Here, intensity envelope of the pumping laser by means of a RZ 01 longafterthecompletionofthePauli-Xgate,theBlochvector hyperbolic tangent pulsed interaction. In particular, within tipmovesawayfromthesouth-polealongthez-axis,eventu- the effective two-level system spanned by the spatially indi- 9 rect excitons, we defined a qubit manifold whose dynamics ACKNOWLEDGMENTS is controlled by the combined action of optical RZ and ap- pliedelectricfieldsweeps,enablingfullcontroloftheBloch vectoracrosstwowelldefinedaxesoftheBlochsphere. Our This material is based upon work supported by the Na- findingspavethewayforfurtherresearchaimedatthedesign tional Science Foundation under Grants No. PHY1306520 and implementation of indirect-excitonic qubit operations in (NuclearTheoryprogram)andNo. PHY1452635(Computa- quantum dot molecules using available ultrafast optical ma- tional Physics program). 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