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Syn hronous Quantum Memories with Time-symmetri Pulses ∗ Q. Y. He, M. D. Reid, and P. D. Drummond Centre for Atom Opti s and Ultrafast Spe tros opy, Swinburne Un∗iversity, Melbourne VIC 3122, Australia and Email: pdrummondswin.edu.au We propose a dynami al approa h to quantum memories using a syn hronous os illator- avity model, in whi h the oupling is shaped in time to provide the optimum interfa e to a symmetri inputpulse. This over omes theknown di(cid:30) ulties of a hieving high quantuminput-output(cid:28)delity with storage times long ompared to the input signal duration. Our generi model is appli able to any linear storage medium ranging from a super ondu ting devi e to an atomi medium. We 9 show that with temporal modulation of oupling and/or detuning, it is possible to mode-mat hto 0 time-symmetri pulses that have identi al pulse shapes oninputand output. 0 2 Quantum memories (QM) are key devi es both for ferent shapes on input and output n a quantum information and fundamental tests of quantum An essential feature of our treatment is that we show J me hani s. A QM will write, store then retrieve a quan- how a smooth, time-symmetri se h-pulse an be stored TI 9 tum state after an arbitrary length of time. QM devi es for times longer than , and re alled with high quan- 1 are onsidered vital for the implementation of quantum tum (cid:28)delity. This means that the output pulse-shape networks, quantum ryptographyand quantum omput- repli ates the input pulse. Hen e, this type of quantum ] h ing. At a morefundamental level, they ould enable one logi an be as aded, with inter hangeable inputs and p to generate an entangled quantum state in one devi e, outputs. Thisisavitalfeatureofalllogi devi es. Impor- - then test its de oheren e properties in a di(cid:27)erent lo a- tantly, we do not use a slowly-varying pulse approxima- t n tion. This would allow one to test the equivalen e of tion[5,6,7,8,9,10℄,aswasrequiredinearlierproposals. a the quantum state des ription for more than one physi- Thisisessential,toallowtheuseoffastpulseswhi h an u q al environment. For example, there are proposals that bestoredfortimes mu hlongerthanthepulse-duration. [ gravitational de oheren e may o ur beyond the stan- Our theoreti al al ulations are arried out with sim- dard model of quantum measurement theory [1℄. This ple non-saturating linear os illator models that are an- 1 v would be testable with ontrolled ways to input, store, alyti ally soluble. Cru ially, this allows us to al ulate 6 then readout a quantum state in di(cid:27)ering environments. pulse-shapes that are dynami ally mat hed in time to 5 Theben hmarksforaQMarestoragetimeandinput- the avity-os illator system. This strategy an be om- T 9 output(cid:28)delity. Thememorytime mustbelongerthan bined withavarietyofotherte hnologies. Thesein lude 2 TI T >TI the duration of the input signal: . Otherwise, quantum nondemolition (QND) [11, 12, 13, 14℄ inter- . 1 the memory is more like a phase-shifter than a memory. a tions, Raman and ele tromagneti ally indu ed trans- 0 The(cid:28)nalquantumstatemustalsobea loserepli aofthe paren y(EIT)[15,16,17,18,19℄,inhomogeneousbroad- 9 original. In quantitative terms, the mean state overlap ening (CRIB) [20, 21, 22, 23, 24, 25℄, super ondu ting 0 v: [2℄ between the intF¯ended and a hieF¯ve>d Fq¯uCantum sF¯tCates transmission lines and squids [26, 27, 28℄, magneti on- (the mean (cid:28)delity ) must satisfy . Here is trolwithatwo-levelatom[29,30℄,nanome hani alos il- i X thebestmean(cid:28)delityobtainablewitha lassi almeasure latorstorage[31, 32℄, and even intra- avityBEC devi es r andregeneratestrategy[3℄. Furthertothis,anidealQM [33℄. Thisopenssomeex itingexperimentalpossibilities, a proto olmustenablenumeroussequentialquantumlogi in luding omparisonsof(cid:28)delity in QMdevi eswith dif- operationstobeperformed, meaningmanyinput-output ferente(cid:27)e tivemasses,asafundamentaltest ofde oher- (cid:16)quantum states(cid:17), arried on ingoing and outgoing pulse en e in quantum me hani s. waveforms. This means that the output pulse envelope Previous QM experiments were frequently limited by should be identi al to that of the input. relativelyshortstoragetimes[34℄. Otherdemonstrations In this paper we proposea new QM proto ol(Fig. 1), fo us on retrieval e(cid:30) ien y at very high photon number satisfying all of these onstraints, in whi h the (cid:16)state(cid:17) [35℄. However,theseusuallyhaveaverylow(cid:28)delity,sin e is stored in a dynami ally swit hed avity-os illatorsys- the (cid:28)delity at a (cid:28)xed e(cid:30) ien y de reases exponentially tem. The avity a ts as an input-output bu(cid:27)er whi h withphotonnumber. Asarule,previousproposalseither syn hronouslymode-mat hesthe external input pulse to ignore (cid:28)delity, or use riteria only appli able to spe ial a long-lived internal quantum linear os illator. We de- knownstates,like oherentorsqueezedstates[34,36,37℄. rivea onditiononthetime-dependen eoftheos illator- It is more useful in both appli ations and fundamental avity oupling required to mat h to any external pulse- tests to allow for arbitrary input states. Our analysis is shape, in luding time-symmetri pulses. This ontrasts not restri ted to any lass of states, ex ept for an upper with previous work, in whi h the oupling was a step bound on the input photon number. fun tion[4℄,resultinginnon-symmetri pulseshavingdif- Model. Thequantuminformationinatemporalmode 2 voirsareassumedheretobeintheva uumstate,without ex ess phase or thermal noise. √ηM =aoout/aion Hen e, we an solve Eq (1) to obtain Aout byintegratingaRoverthepositive-Poutput(cid:28)eld . This isvalidsin e orrespondstoabosoni operatorwhi h onlya tsonazero-temperaturereservoir,andistherefore equal to zero for the va uum state in the positive P- representation. Figure 1: (Color online) Proposed dynami al atom- avity We will analysethe mode-mat hing onditionsfortwo κ QM. Tuhi0en avituyo0u touples to only one ingoing and outgoing di(cid:27)erent dynami almodels with (cid:28)xed avity damping . mode, and ,anditisthequantumstateofthismode In order to obtain dynami al mode-mat hing we require that is stored. The pulse shape is optimized for e(cid:30) ient t < T an outgoing va uum state for . In the positive P- writing and reading of the state onto and from the os illator medium inside the avity. A symmetri pulse shape is represAeonuttat=ion√th2κisatranAsilnate=s t0o the simplAeirneq=uir√em2κenat used, so thetime-reversedoutputis identi al to the input. that − , so that . Thetwomodels usestrategiesofeithervariable oupling or variable detuning to swit h on and o(cid:27) the ouping oAˆfinth(te)propagating single transverse-mode operator (cid:28)eld between the os illator and the intra- avity (cid:28)eld. For is(cid:28)rsttransferredtoaninternal avitymodewith simpli ity, we treat the ase of zero internal damping aˆ(t) γT <<1 operator , thenwˆbr(itt)teninto theots= il0latorormemory ( ) in the equations, while still in luding os illa- with mode operator up to time . Subsequently, tordampinginthegraphstodemonstratethatthise(cid:27)e t theintera tionisturnedo(cid:27) ordetuned fora ontrollable an be made small if ne essary. With no loss of general- T κ=1 storage time . The intera tion is swit hed on again af- ity, we onsider units for whi h . T ter tiAmôeut(t,)allotw>ingTreadout into an outgoing quantum (cid:28)eld at (Fig. 1). We treat quantum infor- 2 b mation en oded into single propagating modes that are 1.5 Ain temporally and spatially mode-mat hed to the memory 1 oudpeouvetri( aiento)[(3rts8),a3re9℄aˆ.o0Hute(rine)t=heRreule0ouvta(nint)⋆in(tp)uAˆtoaunt(dino)(utt)pdut,twmhoedree outA, b 0.5 0 is understood to represent the output (input) inA, 0 temporal mode shape. −0.5 We use theAˆpouots(iitni)v,eaôPut-(rienp)raê,seˆbntation [40℄, in whi h −1 all operators Aout(in),a0out(in)a, bare formally repla ed Aout by -numbers 0 . Using input-output −1−.510 −5 0 κ t 5 10 15 theory[41℄, the resulting dynami al equations are: a˙(t) = (κ+iδ(t))a(t)+g(t)b(t)+√2κAin(t) − Figure 2: (Color online) Case 1: Cavity input (dashed) and b˙(t) = (γ+i∆(t))b(t) g(t)a(t)+ 2γBin(t). output amplitudes (solid). The dotted line gives the os illa- − − p (1) κ t oorupalminpglisthuadpee. inThtiemdeags(hte)d.-Hdoetrteedt0 =ya−n5l,inTe=re5p,reas0e=nts1.the Here is the avity damping (assumed (cid:28)xed), with δ(t). dg(ett)uning The internal avity-os illator oupling is ∆,δ = 0 (assumed variable),γw, ∆hi(let)the damping and detun- 1. Variable oupling ( ). In this approga( th),, ing of the os illatorinare respe tively, with an os- we propose that the avity de ay is (cid:28)xed, and that B illator reservoir . These equations an be applied theintera tionofthe avity(cid:28)eldwiththelinearmeAdiinum=, to a range of experiments ranging from solid-state rys- √is2sκwait hed. Durinag(tth)e input stage, the relation tals or old atoms to super ondu ting avities or nano- meansAthina(tt) is predetermined for any degs(itr)ed os illators. The ompleteness of the representation al- mode-shape . Thisγgives0an expressgio(tn)f=or b˙/a, lows us to treat any quantum state or memory proto ol. sin e from Eq (1), with → , one has − . Sin e the equations are linear, the overall time-delayed Hen e: input-output relationship must be given by: [a˙ gb]/a=a˙/a+(b˙2)/(2a2)=1. − (3) out in R ao =√ηMao +p1−ηMa . (2) a =Inaorsdeecrht(otreatli)ze a time-symmetri input mode with 0 0 ηM − , from Eq (b3)=wea0seete−tt0hsaetchth(te intte0r)- Here an amplitude retrieval e(cid:30) ieanR y is introdu ed nal (cid:28)eld amplitude must be − . forthetime-delayedread-out,and representstheover- The optimal shapge(to)f=the b ˙/aavit=y-os siellcaht(otr otup)ling in 0 all e(cid:27)e ts of the loss reservoirs. For simpli ity, all reser- time is therefore − − − . This 3 a0 F¯n¯ = 1/[1+n(1 √ηM)2] is independent of the amplitude whi h en odes the mean (cid:28)delity is − . These (cid:28)- quantuminformation. Here the ouplingissyn hronized delities may orrespond to quite high e(cid:30) ien ies, sin e to t0, whi h is the pulse arrival time. The quantum ηM >[1−p1/(n+1)]2isneededforaQM.Withn=20, ηM >0.61 memory readout is obtained simply by time-reversal af- QM should be a hieved for , provided there is ter half the memory storage time has elapsed, so that no other de oheren e. g(t)=g(T t) − . The resulting output mode is alsotime- For many quantum information appli ations, a larger rAeovuetrs=ed a√n2daissecuhn( Tha+ngted ta)part from being inverted: lass of possible quantum inputs is needed. In the most 0 0 − − . Atypi alresult isshown general ase, we an de(cid:28)ne the input state as any state nm in Fig. 2, from integrating Eq (1). withaphotonnumberboundedby . This orresponds to an arbitrary state (cid:12)(cid:12)Ψ~E of 1+nm levels. Fnm is then (cid:12) 2 1.5 Ain b tthhee aovnersatrgaein(cid:28)tdethliatyt(cid:12)(cid:12)oΨ~ve(cid:12)(cid:12)r=al1l.pToshseib(cid:28)lede loitey(cid:30)l imienittsfosra(tiimsfypienrg- (cid:12) (cid:12) 1+nm outA, b 0.15 fisea nFt¯)nb mlmeau≤rllty2ip/gl(eenn melro+antien2)gan[o4yf5,annu4m6a℄.brbeSirtirnoa fre yoap ielassosif aalleqvmuealenmsttouartmye inA, 0 state, it must be onstrained by this (cid:28)delity bound also. −0−.15 Detunings−1100000 ∆ δ δ’ ∆’ a1nWdnem2 a=nn2o,wwh ai lh ualalltoewtshfeo(cid:28)rdaerlbitiytrianrythseta taessewoiftnhmup=to1 −5 0 5 10 15 Aout and photonsrespe tively. Aftertra ingoverthereser- −1.5 −10 −5 0 κ t 5 10 15 voir modes, we obtain the predi ted memory (cid:28)delities in our beam-splitter model of [4℄: ηM +2√ηM +3 Figure 3: (Color online) Case 2: Cavity input (dashed) and F1 = 6 output amplitudes (solid). Other lines and parameters as in Fig (δ2()t.) The inset gives the detuning shapes in time: ∆(t) F2 = ηM2 +2ηM√ηM +3ηM +2√ηM +4. and . 12 (5) 2. Variable detuning. In this approa h, the ou- An arbitrary quantum state (cid:28)delity measure gives a ∆(t) δ(t) pling is hanged by varyingthe detunings and . better indi ation of the power of a QM than a measure g = κ = 1 We onsider the simplest ase with and a onstrained to the oherent states. For example, any a = a0sech(t t0) ηM >0.23 symmetri pulse − . To give a va uum storagedevi ewith anpotentiallybeaquan- 2 output during the writing phase we must have: tum memory for arbitrary states with up to photons. ∆ = i(b˙+a)/b δ = i(a˙ a b)/a 2 b1 − − (4) 1.5 Ain b3 b2 b = b + ib δ = i(a˙ a bW)e/asu+ppbos/eat,h∆at = [(b˙1 + a)2b, so tb˙habt]/b2 + i[−(b˙ −+ b 1 a1)b1 + b˙2b22]/b2. 1Im(∆) 2=−Im2(δ1) |=| 0 1 outA, 0.5 tthhaatt bb12 ==(aa˙0|−e|t−at)0Ssi=enc h−e(at0−et−t0t0)tsaenchh2((tt−−tt00)).,Fa,inndwalehlye,(cid:28)nn tdoe inA, −0.05 γ / κ = 0.05 A3out −1 γ / κ = 0.0125 A2out eore−bat(ltai−zinte0)sftryaomnmhm(tEet−qrit (04)i)n+ptsuhetechar(netqd−uiotr0ue)dt,pδud=tetpueutn−lisnte0gtsasnhoahfp:(te−s∆,tw0=)e. −1−.510 −5 0 κ γt / κ = 05 Ao1ut 10 15 After a ontrollable storage time, the intera tion is sδw′ it hedδba k b∆y′time r∆eversal of the detunings, so that Figure 4: (Color online) Case 1 with losses: Cavity input → − and → − . The readout is obtained as (dashed) and output amplitudes (solid) with various loss ra- γ/κ = 0, 0.0125, 0.05 before, as shown in Fig. 3. tios during the storage time: . Other Memory Fidelity. Coherentstates haveproved use- lines and parameters as in Fig (2). fulinquantumappli ationssu hasteleportation[42℄and quantum state transF¯fecr=fro(m1 +lignh)t/(o2nnto+a1t)oms [11℄. It Os illator Loss: Figure 4 shows the typi al input- is well known that n¯ serves as a output relation for various loss ratios during a storage T = 5 γ/κ = 0, √ηM = 1 ben hmark for a QM with a gaussian ensemble of o- time of duration . For , for γ/κ=0.0125, √ηM =0.84 herent states [43, 44℄ having a mean photon number we(cid:28)ndane(cid:30) ien yof ,while n¯ γ/κ = 0.05, √ηM = 0.50 . For our beam-splitter solution Eq. (2), the output for we obtain . 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