Driver Phase Correlated Fluctuations in the Rotation of a Strongly Driven Quantum Bit

Transcription

1 [acceped fr PRA Rapid Cmm; quan-ph/] Driver Phase Crrelaed Flucuains in he Rain f a Srngly Driven Quanum Bi M.S. Shahriar,, P. Pradhan,, and J. Mrzinski Dep. f Elecrical and Cmpuer Engineering, Nrhwesern Universiy, Evansn, IL 68 Research Labrary f Elecrnics, Massachuses Insiue f Technlgy, Cambridge, MA 39 Absrac The need maximize he number f perains f a quanum bi wihin is decherence ime may require he rai f he Rabi frequency he ransiin frequency be big enugh invalidae he raing wave apprximain. The sae f he quanum bi under any iniial cndiin hen depends explicily n he phase f he driving field, resuling in driver phase crrelaed flucuains, and a vilain f he rule ha he degree f exciain depends nly n he pulse-area. This is due he inerference f he exciains caused by he c- and cunerraing fields, and is a significan surce f errr, crrecable nly by cnrlling he driver phase. We presen a scheme fr bserving his effec under currenly realizable parameers. PACS Number(s): a, 3.67.Hk, 3.67.Lx, 3.8.Qk

2 In rder minimize he decherence rae f a w-sae quanum bi (qubi embdied in a massive paricle, ne fen chses use lw energy ransiins. In general, ne is ineresed in perfrming hese ransiins as fas as pssible [-5], which demands a srng Rabi frequency. The rai f he Rabi frequency he qubi ransiin frequency is herefre n necessarily very small, hus invalidaing he s-called Raing Wave Apprximain (RWA). A key effec due he vilain f he RWA (VRWA) is he s-called Blch-Sieger shif [6-9], which is negligible in pical ransiins, bu is manifesed in nuclear magneic resnance []. Here, we shw ha VRWA leads anher impran effec, which can lead cnrllable errrs ha are significan n he scale f precisins envisined fr a funcining quanum cmpuer []. Specifically, we shw ha under VRWA he ppulain difference beween he w levels f he quanum bi, wih any iniial cndiin, depends explicily n he phase f he driving field a he nse f an exciain pulse, which is a vilain f he rule [6] ha ha fr a w-level sysem saring in he grund sae, he ppulain difference is a funcin f he inegral f he field ampliude ver he pulse durain, and des n depend n he phase f he field. We prvide a physical inerpreain f his effec in erms f an inerference f he exciains caused by he c- and cuner-raing fields, and presen a scheme fr bserving his effec under currenly realizable parameers. T see he implicain f his resul, cnsider a scenari where ne has a qubi, iniialized he grund sae, and wuld like prepare i be in an equal super-psiin f he grund and excied saes. T his end, ne wuld apply a resnan pulse wih an area f π/ saring a a ime: =. Under RWA, ne des n have knw wha he abslue phase f he field, φ P, is a, and he ppulain difference fr he qubi wuld be zer. Under VRWA, hwever, he desired exciain wuld nly ccur if φ P =. Oherwise, he ppulain difference wuld have a cmpnen varying as ηsin(φ P ), where η is a parameer ha is prprinal he rai f he Rabi frequency he ransiin frequency. Suppse ne were apply his pulse many such qubis, wih a penially differen φ P fr each (e.g., because he pulses are applied a differen imes, r he qubis are spaially separaed), bu wih idenical pulse areas. The ppulain difference fr he qubis will hen exhibi a flucuain, crrelaed heir respecive values f φ P. Fr a quanum cmpuer, his variain wuld represen a surce f errr. Fr sme experimens [e.g., ref 5], he value f η is already clse., s ha he magniude f his errr is much larger han he ulimae accuracy ( -6 ) desirable fr a large scale quanum cmpuer [], and mus be cnrlled. T illusrae his effec, we cnsider an ideal w-level sysem where a grund sae > is cupled a higher energy sae >. We assume ha he - ransiin is magneic diplar, wih a ransiin frequency f ω, and he magneic field is f he frm B=B Cs(ω+φ) [,6]. In he diple apprximain, he Hamilnian can be wrien as: ) H = ε ( σ σ ) / + g( σ z where g( = -g [exp(iω+iφ)+c.c.]/, σ i are he Pauli marices, and ε=ω crrespnding resnan exciain. The sae vecr is wrien as: x ()

3 3 C ξ =. () C( We perfrm a raing wave ransfrmain by peraing n ξ(> wih he uniary perar Q: ) Q = ( σ + σ z ) / + exp( + iω + iφ)( σ σ z ) /. (3) ξ > The Schredinger equain hen akes he frm (seing h=): = ih ξ >, where he effecive Hamilnian is given by: = α σ α H ( ) ( ) σ, (4) wih α(= -g [exp(-iω-iφ)+]/, and he raing frame sae vecr is: + + > ˆ C ( ) >= ( ) ξ Q ξ. (5) C( Nw, ne may chse make he RWA, crrespnding drpping he fas scillaing erm in α(. This crrespnds ignring effecs (such as he Blch-Sieger shif ) f he rder f (g /4ω), which can easily be bservable in an experimen if g is large [6-9]. On he her hand, by chsing g be small enugh, ne can make he RWA fr any value f ω. Here, we cnsider he general resuls wihu he RWA. Frm Eqs.4 and 5, ne ges w cupled differenial equains: g C = i [ + exp( iω iφ )] C (, (6a) g C ( = i [ + exp( + iω + iφ )] C. (6b) Given he peridic naure f he effecive Hamilnian, he general sluin Eq.6 can be wrien in he frm: an ξ >= exp( n( iω iφ )) n b. (7) = n Insering Eqs.6 in Eq.7, and equaing he cefficiens wih he same frequencies, ne ges, fr all n: = n n n a n i nω a + ig ( b + b ) /, (8a) = n n n+ b n i nω b + ig ( a + a ) /. (8b)

4 4 Here, he cupling beween a and b is he cnveninal ne, and he cuplings he neares neighbrs, a ± and b ± are deuned by an amun ω, and s n. T he lwes rder in (g /ω), we can ignre erms wih n >, hus yielding a runcaed se f six equains: a = ig b ( b + )/ (9a) b = ig + a ( a )/ = i a + ig ( b b )/ (9b) a ω + (9c) = i a b a ω b + ig / (9d) + = i ω a ig b / (9e) = i b + ig ( a a )/ b ω + (9f) We cnsider g have a ime-dependence f he frm g = g M [ exp( / τ )], where τ >>ω -, g - M. We can slve hese equains by emplying he mehd f adiabaic eliminain, which is valid firs rder in η (g /4ω). Ne ha η is als a funcin f ime, and can be expressed as η = η[ exp( / τ )], where η (g M /4ω). We cnsider firs eqs.e and f. In rder simplify hese w equains furher, ne needs diagnalize he ineracin beween a - and b -. Define µ - (a - -b - ) and µ + (a - +b - ), which nw can be used re-express hese w equains in a symmeric frm as: = ( µ µ i ω + g / ) ig a / (a) + = ( µ + µ i ω g / ) + ig a / (b) Adiabaic fllwing hen yields (again, lwes rder in η ): µ ηa and µ + ηa which in urn yields a and b ηa. In he same manner, we can slve he equains 9c and 9d, yielding: a ηb and b. Ne ha he ampliudes f a - and b are vanishing (each prprinal η ) lwes rder in η, hereby jusifying ur runcain f he infinie se f relains in Eq.9. I is easy shw nw a = ig b / + i ( a / b = ig a / i ( b / (a) (b) where =g (/4ω is essenially he Blch-Sieger shif. Eqn. can be hugh f as a wlevel sysem excied by a field which is deuned by a frequency. Fr simpliciy, we assume ha his deuning is dynamically cmpensaed fr by adjusing he driving frequency ω. This assumpin des n affec he essence f he resuls fllw, since he resuling crrecin η

5 5 is negligible. Wih he iniial cndiin f all he ppulain in > a =, he nly nn-vanishing ( he lwes rder in η ) erms in he sluin f Eq.9 are: a Cs( / ); b isin( / ) a iη Sin( / ); b Cs( η / ) ( where we have defined: = g ( ') d' = g[ ( ) ( exp( / τ ))] τ We have verified his sluin via numerical inegrain f Eq.7, as discussed laer. Insering his sluin in Eq.6, and reversing he raing wave ransfrmain, we ge he fllwing expressins fr he cmpnens f Eq. : C = Cs( / ) η Σ Sin( / ) (a) C ( i ( ω+ φ ) * = ie [ Sin( / ) + ησ Cs( / )] (b) where we have defined Σ ( i / ) exp[ i(ω + φ )]. T lwes rder in η, his sluin is nrmalized a all imes. Ne ha if ne wans prduce his exciain in an ensemble f ams using a π / pulse and measure he ppulain f he sae > afer he exciain erminaes ( a (τ )τ /= π /), he resul wuld be a signal given by C ( τ ), ) = [+ηsin(φτ )], (3) ( φ where we have defined he phase f he field a =τ be φ τ ωτ+φ. This signal cnains infrmain f bh he ampliude and he phase f he driving field. This resul can be appreciaed bes by cnsidering an experimenal arrangemen f he ype illusraed in Fig.. Cnsider, fr example, a cllecin f 87 Rb ams, caugh in a diple frce rap, where he saes > 5 S / : F=,m=> and > 5 S / : F=,m=> frms he w level sysem. These saes differ in frequencies by GHz. When illuminaed by resnan righcircularly plarized ligh a a frequency f 3.844X 4 Hz, sae > cuples nly he sae > 5 P 3/ : F=3,m=3>, which in urn can decay nly sae >. This cycling ransiin can hus be used pump he sysem in sae >. When a righ-circularly plarized micrwave field a GHz is applied, sae > cuples nly sae >, even when he RWA apprximain breaks dwn. The srng cupling regime (e.g., η f he rder f.) can be reached, fr example by using a supercnducing, high-q ( ) micrwave caviy []. The hereical mdel develped abve is hen a valid descripin f he cupling beween > and >. The srng micrwave field is urned n adiabaically wih a iching ime-cnsan τ SW, saring a =. Afer an ineracin ime f τ, chsen s ha g '(τ)τ=π/, he ppulain f sae > can be deermined by cupling his sae he sae > wih a shr (faser han /ω and /g M ) laser pulse, and mniring he resuling flurescence [3]. We have simulaed his prcess explicily fr he fllwing parameers: ω=π* * 9 sec -, g M =.ω, and τ SW =.τ. These numbers are easily achievable experimenally. The laser pulse widh, τ LP, is chsen be - sec, in rder saisfy he cnsrain ha τ L <</ω and τ L <</g M. In rder pimize he signal, he laser Rabi-frequency, Ω L is chsen be such ha Ω L τ L =π, s ha all he

6 6 ppulains f sae > is excied sae > a he end f he pulse. Fr he cycling ransiin ( ), and a pulse fcused an area f 5 µm, he pwer needed fr achieving his Rabi frequency is. W, which is achievable experimenally. Afer he laser pulse is urned ff, he flurescence is clleced fr a durain lnger han he spnaneus-decay lifeime (3 nsec) f sae >. Under his cndiin, ur simulain verifies ha he deecr signal is essenially prprinal he ppulain f sae >, as given by eqn. 3, wih he prprinaliy cnsan deermined by he efficiency f he deecin sysem. If 6 ams are used (a number easily achievable in a diple rap), he signal--nise rai can be mre han fr he parameers cnsidered here, assuming a deecr slid angle f.π, and a quanum efficiency f.8. In figure (a), we have shwn he evluin f he excied sae ppulain C ( τ ) as a funcin f he ineracin ime τ, using he analyical expressin f eqn.b. Under he RWA, his curve wuld represen he cnveninal Rabi scillain. Hwever, we nice here sme addiinal scillains, which is magnified and shwn separaely in figure (b), prduced by subracing he cnveninal Rabi scillain [Sin (g ' (τ)τ / )] frm figure (a). Tha is, figure (b) crrespnds wha we call he Blch-Sieger Oscillain (BSO), given by ηsin( ( τ ) τ) Sin(φ τ ). The dashed curve (c) shws he ime-dependence f he Rabi frequency. These analyical resuls agree clsely he resuls bained via direc numerical inegrain f Eq.7. Cnsider nex a siuain where he ineracin ime, τ, is fixed s ha we are a he peak f he BSO envelpe. The experimen is nw repeaed many imes, wih a differen value f φ each ime. The crrespnding ppulain f > is given by η Sin(φ τ ), and is pled as a funcin f φ in he inse f figure. This dependence f he ppulain f > n he iniial phase φ (and, herefre, n he final phase φ τ ) makes i pssible measure hese quaniies. Ne, f curse, ha he speed f he deecin sysem is limied fundamenally by he spnaneus decay rae, γ - ( 3 ns in his example), f sae >. As such, i is impssible in his explici scheme mnir he phase f he micrwave field n a ime scale shrer han is perid. If ne were ineresed in mniring he phase f a micrwave field f a lwer frequency (s ha ω - >>γ - ), i wuld be pssible rack he phase n a imescale much shrer han is perid. One pssible se f amic levels ha can be used fr his purpse are he Zeeman sublevels (e.g., hse f he 5 S / :F= hyperfine level f 87 Rb ams), where he energy spacing beween he sublevels can be uned by a dc magneic field mach he micrwave field be measured. Hwever, he number f sublevels ha ge cupled is ypically mre han. A simple exensin f ur hereical analysis shws ha he signaure f he phase f he micrwave field sill appears in he ppulain f any f hese levels, and can be used measure he phase. Mre generally, he phase signaure is likely appear in he ppulain f he amic levels, n maer hw many levels are invlved, as lng as he Rabi frequency is srng enugh fr he RWA break dwn. A recen experimen by Marinis e al. [5] is an example where a qubi is driven very fas. In his experimen, a qubi is made using he w saes f a curren-biased Jsephsn juncin, he resnance frequency is ω = 6.9 GHz, and he Rabi frequency is g=8 MHz. If his experimen is carried u wihu keeping rack f he phase f he driving field, he degree f qubi exciain will flucuae due he BSO, leading an errr which is f he rder f g/ω =., i.e. nearly %. This errr is much larger han he permissible errr rae f -6 fr an errr crrecing quanum cmpuer ha wuld cnsis f 6 qubis []. In rder eliminae he BSO induced errr, ne can design he driving sysem such ha he phase is measured, e.g., by using an auxiliary cluser f bis lcaed clse he qubi f ineres, a he nse f he qubi

7 7 exciain, and he measured value f he phase is used deermine he durain f he exciain pulse, in rder ensure he desired degree f exciain f he qubi[4,5]. Finally, we pin u ha by making use f disan enanglemen, he BSO prcess may enable eleprain f he phase f a field ha is encded in he amic sae ampliude, fr penial applicains reme frequency lcking [6- ]. In cnclusin, we have shwn ha when a w-level amic sysem is driven by a srng peridic field, he Rabi scillain is accmpanied by anher scillain a wice he ransiin frequency, and his scillain carries he infrmain abu he abslue phase f he driving field. One can deec his phase by simply measuring nly he ppulain f he excied sae. This leads a phase-crrelaed flucuain in he exciain f a qubi, and vilain f he rule ha he degree f exciain depends nly n he pulse-area. We have shwn hw he resuling errr may be significan, and mus be cnrlled fr lw-energy fas qubi perains. We hank G. Cards fr useful discussins. We wish acknwledge suppr frm DARPA gran # F under he QUIST prgram, ARO gran # DAAD under he MURI prgram, and NRO gran # NRO---C-58. References:. D. Buwmeeser, A. Eker, and A. Zeilinger, Eds., "The Physics f Quanum Infrmain," Springer,.. A.M. Seane, Appl. Phys. B 64, 63 (997) 3. A. Seane e al., quan-ph/ D. Jnahan, M.B. Pleni, and P.L. Knigh, quan-ph/9. 5. J. M. Marinis e al., Phys. Rev. Le. 89, 79 (). 6. L. Allen and J. Eberly, Opical Resnance and Tw Level Ams, Wiley, F. Blch and A.J.F. Sieger, Phys. Rev. 57, 5(94). 8. J. H. Shirley, Phys. Rev. 38, 8979 (965). 9. S. Senhlm, J. Phys. B 6, (Augus, 973).. R. J. Abraham, J. Fisher and P. Lfus, Inrducin NMR Specrscpy, Wiley, 99.. J. Preskill, quan-ph/ S. Brake, B. T. H. Varce, and H. Walher, Phys. Rev. Les. 86, 3534 (). 3. C. Mnre e al., Phys. Rev. Le. 75, 474 (995). 4. P. Pradhan and S. Shahriar, presened a PIERS, Cambridge, MA (July ). 5. P. Pradhan and M. S. Shahriar, presened a he APS annual meeing, March,. 6. R. Jzsa, D.S. Abrams, J.P. Dwling, and C.P. Williams, Phys. Rev. Les. 85, (). 7. S.Llyd, M.S. Shahriar, J.H. Shapir, and P.R. Hemmer, Phys. Rev. Le. 87, 6793 (). 8. G. S. Levy e al., Aca Asrnau 5, 48(987). 9. M.S. Shahriar, The prceedings f he Cnference n Quanum Opics 8, Rcheser, NY, July.. M. S. Shahriar, quan-ph/964.

8 Fig. : Schemaic illusrain f an experimenal arrangemen fr measuring he phase dependence f he ppulain f he excied sae >: (a) The micrwave field cuples he grund sae ( >) he excied sae ( >). A hird level, >, which can be cupled > pically, is used measure he ppulain f > via flurescence deecin. (b) The micrwave field is urned n adiabaically wih a iching ime-cnsan τ SW, and he flurescence is mnired afer a al ineracin ime f τ. 8

9 Figure. Illusrain f he Blch-Sieger Oscillain(BSO): (a) The ppulain f sae >, as a funcin f he ineracin ime τ, shwing he BSO superimpsed n he cnveninal Rabi scillain. (b) The BSO scillain (amplified scale) by iself, prduced by subracing he Rabi scillain frm he pl in (a). (c) The ime-dependence f he Rabi frequency. Inse: BSO as a funcin f he abslue phase f he field. 9