Quantum trajectories and quantum measurement theory in solid-state mesoscopics

Transcription

1 Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis HsiSng Gon Cntr or Quntum Computr nology, Univrsity o Nw Sout ls, Sydny, NS 202 Austrli Abstrt. Strting rom t ull Srödingr qution or systm nd n nvironmnt ( dttor), w prsnt uristi drivtion o t stosti Srödingr (mstr) qution or msosopi msurmnt modl to illustrt t ssntil pysis o t quntum trjtory tory. is msosopi modl dsribs twostt quntum systm, n ltron orntly tunnling btwn two oupld quntum dots, intrting wit n nvironmnt ( dttor), low trnsprny point ontt or tunnl juntion. n w provid onntion nd uniid pitur or t quntum trjtory ppro nd t mstr qution ppro. sow tt t mstr qution or t rdud or prtilly rdud dnsity mtrix n b simply obtind wn n vrg or prtil vrg is tkn on t onditionl, stosti dnsity mtrix (or quntum trjtoris) ovr t possibl outoms o t msurmnts.. Introdution tory o quntum trjtoris or stosti Sr ödingr qutions s bn dvlopd in lst tn yrs minly in t quntum optis ommunity to dsrib opn quntum systm subjt to ontinuous quntum msurmnts. But it ws introdud to t ontxt o solidstt msosopis only rntly [, 2, 3, 4. Dirnt utors, owvr, gv somwt dirnt drivtions or t stosti qutions. In s. [, 2 t prsriptions or t quntum trjtoris (or sltiv, onditionl, stosti stt volution) wr drivd bsd on t Bysin ormlism, wil ty wr drivd in s. [3, 4 strting rom unonditionl mstr qution. In ddition, t intrprttions long wit t drivtions do not sm trnsprnt noug. Hn, mny spts o t tory r still poorly undrstood in t ondnsd mttr pysis ommunity. min purposs o tis ppr r (i) to prsnt simpl, uristi drivtion or t sm msosopi modl to illustrt t ssntil pysis o t quntum trjtoris. (ii) to provid uniid pitur or t quntum trjtory ppro nd t mstr qution ppro o rdud or prtilly rdud dnsity mtrix. Hr w rr to t mstr qution ppro o t prtilly rdud dnsity mtrix s t ppro rntly dvlopd in s. [, 6, 7, minly or t purpos o rding out n initil stt o quntum bit (qubit). sow tt t mstr qution o t rdud or prtilly rdud dnsity mtrix n b obtind s rsult o nsmbl vrg or prtil vrg on t onditionl, stosti mstr qution o t dnsity mtrix (onstrutd rom t onditionl systm stt vtor) ovr ll possibl dttion rords. 2. Msosopi msurmnt modl nd quntum trjtoris 2.. Coupldquntumdot nd pointontt modl msosopi msurmnt modl onsidrd in s. [, 2, 3, 4, dsribs quntum systm, n ltron orntly tunnling btwn two oupld quntum dots (CQD s),

2 ; D ; ; ; D F Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis 2 intrting wit n nvironmnt ( dttor), low trnsprny point ontt (PC) or tunnl juntion. Hmiltonin dsribing t wol systm n b writtn s [, 2, 3, 4, () wr! # $ &' # ( *), # ( # $.0/& <; BA D; : > 3#C DE6!F4G D ; $; D0 ; D ; AK H 3#CD # $ 6!I G I rprsnts t tiv tunnling Hmiltonin or t msurd twostt CQD systm (msosopi L rg qubit). 'M M tunnling N?M Hmiltonin or t PC bt (dttor) is rprsntd by. Hr ( ) nd rprsnt t ltron nniiltion (rtion) oprtor nd nrgy or singl ) ltron stt in dot N rsptivly. N oupling btwn t two dots is givn by. Similrly,, nd, r rsptivly t ltron nniiltion oprtors L nd nrgis or t lt nd rigt rsrvoir stts t wv numbr O. Eqution (4),, dsribs t intrtion btwn t bt (dttor) nd t CQD qubit systm, dpnding on wi dot is oupid. n t ltron in t CQD s is lotd in dot P F D:Q D D I, t tiv tunnling mplitud o t PC dttor ngs rom Continuous msurmnts nd quntum trjtory will driv, strting rom t ull Sr ödingr qution or t systm nd nvironmnt (bt), t stosti Sr ödingr qution wi modls t volution o t CQD systm onditiond on ontinuous in tim msurmnts o t PC bt by dttion dvi. ollow losly t drivtion [8,, 0 o t stosti Sr ödingr qution or n tom, drivn by lssil rdition ilds nd intrting wit t vuum rdition ild, or t CQD/PC modl. bsi id is s ollows. Suppos tt t ombind stt S, o t systm nd t bt is initilly disntngld, S VU VU X U. Hr t initil vuum stt UZ o t PC bt is t stt wr t nrgy lvls in t sour (t lt PC rsrvoir) nd drin (t rigt PC rsrvoir) r illd up to t Frmi nrgis (mil potntil) [ wit ltron numbr ^, nd [ wit ltron numbr ^, rsptivly. rprsnt t Hilbrt sp o t PC bt stts s t numbr or $; Fok stts $; o t lt nd rigt PC rsrvoirs. Sin t oprtor _, ommuts wit t totl Hmiltonin UZ Eq. (), t totl ltron numbr o t PC is onsrvd. n tus writ ` X ` nd `L bx ` b, bing t stts simultnously ving ddition ltrons in t drin ontinuum, nd ols in t sour ontinuum. t o mks t stt o t CQD qubit systm nd t stt o t PC rsrvoirs bom ntngld so tt t totl stt S, no longr torizs s it did t du. For t CQD/PC msosopi modl, t obsrvd quntity or pysil obsrvbl is t numbr o ltrons tunnling troug PC. Hn w n xpnd t totl stt on t ortonorml Fok (numbr) stt bsis o t PC bt s S 38g X Hr w v osn t CQD qubit systm stt to b normlizd, but ty r, in gnrl, not ortonorml. is is rltd to t t tt t indirt msurmnt on t CQD qubit systm, disussd ltr, is not projtiv. onsidr t s tt t msurmnt is prormd nd rptd witin tim intrvl Z mu smllr tn t B; F D0 ; D'; BA (2) (3) (4) ()

3 P Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis 3 typil systm volution or rspons tim. Hn, t systm is tivly ontinuously monitord. Aording to t ortodox quntum tory o msurmnts, t possibl dttor outoms r t intgr ignvlus o t ltron numbr oprtor in t rigt rsrvoir o t PC t tim Z. Morovr, subsqunt to t dttion, t bt prt o t totl wv untion S Z ollpss to t orrsponding ignstt. t is to sy wn t msurmnt idntiis t stt o t PC bt to b in t prtiulr ignstt, t CQD systm is in t orrsponding pur stt Z Z. probbility or tis to ppn is qul to Z. normlizd systm stt, onditiond on t msurmnt rsult just obtind, tn srvs s t initil stt or t nxt volution nd msurmnt tim intrvl Z. us, ording to t msurmnt rord o xprimnt run, w obtin on prtiulr volution o systm stt s rsult o ontinuous projtion o t totl stt S on on o t ignstts. Su n volution is lld quntum trjtory nd its ntur is gnrlly stosti. stosti lmnt in t quntum trjtory orrsponds xtly to t onsqun o t rndom outoms o t msurmnt rord. It is importnt to rliz tt t xprimntr nvr mks dirt msurmnt on t systm o intrst. tr, t xprimntr obsrvs t numbr o ltrons tunnling troug t PC. In t msosopi modl, t CQD qubit systm intrts wit t PC rsrvoirs nd tir quntum stts r ntngld. projtiv msurmnt md on t PC bt, owvr, nbls us to disntngl t systm nd t bt stts. my tink t t on t qubit systm stt is n indirt rsult o t projtiv msurmnt on t PC bt, nd w my modl it in trms o msurmnt oprtors Z ting on t qubit systm stt lon. Howvr, su n tiv msurmnt on t qubit systm stt is, in prinipl, not projtiv. I t initil normlizd stt o t systm is immditly bor t msurmnt, t unnormlizd stt o t msurd rsult bing t t nd o t tim intrvl Z o t msurmnt n b writtn s Z Z. orrsponding probbility nd t normlizd inl stt o t systm r rsptivly (6) Z Z Z Z Z Z Z _ Hr t msurmnt oprtor stisis t ompltnss ondition wi is simply sttmnt o onsrvtion o probbility. Z or t msurmnt oprtors Z is known t tim umpr volution VU VU X U (7) Z Z P, qustion now is to ind or t CQD/PC modl givn tt First, lt us onsidr t s tt S, nd t msurmnt outom tt no ltron tunnling troug PC is dttd up to tim. trt t sum o t tunnling Hmiltonin prts in nd s t intrtion Hmiltonin. n t dynmis o t ntir systm in t intrtion pitur, is dtrmind by t timdpndnt Hmiltonin [3 D 3#C D 6!F4G D I G A ; $ B; D M! $#& (' wr H.. stnds or Hrmitin onjugt o t ntir prvious trm. totl wv untion o t ntir systm in t intrtion pitur stisis Z S * X S VU X, X Z/. /. S /. () ) (8)

4 Q F P D F Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis 4 ltron troug PC is dttd: Z * P G $ G s rom Eq. (3) tt wn t msurmnt rsult is null (no ltron dttd), t systm ngs ininitsimlly, but not unitrily. nonunitry ntur o t volution xprsss t t tt t projtiv msurmnt on t bt wit rsults o no ltrons dttd still v ts on t systm. In ddition, Eq. (3) is nonlinr bus o t prsn o t xpttion vlu trm, wi nsurs tt t systm stt rmins normlizd.. $ Eqution () is xt nd n b drivd rom t Sr ödingr qution. onsidr t msurmnt sm o ounting t numbr o ltrons troug t PC. totl wv untion in t intrtion pitur n b xpndd in trms o t Fok stts o t bt s in Eq. () wit. t o zroount msurmnt rsults is tt t totl stt vtor S is rptdly projtd into U nd is rnormlizd. Hn during t zroount intrvl, t stt vtor o t systm will b in. Using Eq. () nd rrying out t projtion onto U, w ind or t volution o t unnormlizd stt s ollows: Z, D0 D0 D D 3#CD F G I?G F I. Z/. M [ N: N> [! #! #. (0) wr $, nd P or U nd 7U or U. irst trm in Eq. () dos not ontribut bus U X UZ U d. Undr t ssumption tt t orrltion tim o t bt, [ [, is mu sortr tn t typil tim onstnt xptd rom t systm, nd providd tt, w n rpl. by in t intgrnd o Eq. (0) nd xtnd t lowr limit o t tim intgrtion to ininity tr t ng o vribl.. is ssumption nd rsulting simpliitions to t intgrnd o Eq. (0) r known s t BornMrkov pproximtion. Crrying out t intgrtion nd going bk to t S ödingr pitur, w ind tt Eq. (0) boms Z * P G G. () wr t prmtrs nd r givn by, nd. I. Hr nd. r vrg ltron tunnling rts troug t PC brrir witout nd wit t prsn o t ltron in dot rsptivly, nd [ [ is t xtrnl bis pplid ross F t PC. o rriv t Eq. (), nrgyindpndnt tunnling mplituds rprsntd by nd I, nd nrgyindpndnt dnsity o stts o t rsrvoirs rprsntd by nd r ssumd. quiring tt t stt rmins normlizd so tt Z U, nd wit t lp o Eq. () nd t rltion, w obtin Z 8 (2) G G. wr nd t xpttion vlu!#!#! is tkn wit rspt to t normlizd stt. is tn yilds, rom Eq. (), t ollowing volution qution or t normlizd stt o t CQD qubit systm undr t ondition tt no. HP (3)

5 * Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis 2.4. Quntum jumps Nxt w onsidr t s tt ltrons pssing troug t PC r dttd. probbility tt dttion o ltron ours in tim intrvl [, Z ) is qul to t dirn btwn t probbility tt no ltron ws dttd up to tim nd tt up to tim Z. is n b dirtly ound rom Eq. (2) wit t rsult: Z Z (4) wr t rltion s bn usd. I t dttion intrvls Z r osn to b suiintly sort or t dttion probbility to b smll, t probbility o two nd mor ltrons ould v bn dttd my b ngltd. Hn in ininitsiml tim intrvl Z, itr msurmnt rsult is null (no ltron dttd) or tr is dttion o n ltron troug t PC brrir. At rndomly dtrmind tims (onditionlly Poisson distributd), wn tr is dttion o n ltron, t systm undrgos init volution (or suddn jump du to ollps), lld quntum jump. Givn tt S X UZ t tim nd dttion rsult o n ltron t tim Z, t totl stt vtor S Z ollpss to t stt Z X P. know tt t probbility or t dttion to our is ` Z P Z Z () is is obtind rom Eq. (4) by stting P sin t initil stt X UZ t tim, s rsult o prvious msurmnt, sould b normlizd. qustion now is simply to ind t normlizd stt Z X P o t systm. It is suiint to obtin tis stt to Z U Z < t irst ordr in intrtion Hmiltonin. ind 7. Eqution (6) n lso b obtind by mns o tiv msurmnt oprtors. In tis s, only two o tm, Z or P, r ndd in quntumjump msurmnt intrvl. Using t rltions, ^ nd Z Z, w ind rom Eq. () tt P G G. Z (7) msurmnt oprtor Z Z P Z Z in Z n b obtind using Eqs. () nd (6) s Z (8) pprn o nsurs tt only init numbr o dttions n our in init tim intrvl, sin t probbility o dttion rsult is proportionl to Z. n rom Eq. (7), t normlizd stt tr dttion Eq. (6) _ rsults. On n k tt t msurmnt oprtors stisy t ompltnss ondition Z Z P, to irst ordr in tim Z. igt tr t dttion, t PC rsrvoirs r immditly rst bk to its vuum stt. t is, t dttd ltron wi s tunnld into t rigt rsrvoir is dstroyd in t ltri iruit o t dttion dvi, nd n ltron rom t outr iruit immditly lows to t lt rsrvoir to ill up t ol in t msurmnt pross. So nrgy lvls in t sour nd drin r gin illd up to t Frmi nrgis (mil potntil) [ nd [, rsptivly. In otr words, t nw stt S Z tr dttion tror boms Z X UZ. is stt tn srvs s t initil stt or nxt volution nd msurmnt intrvl. n t wol squn n b rptd to dtrmin t rndom tim volution o t CQD qubit systm stt. (6)

6 Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis 6 n ombin t volution o t systm stt or t two possibl outoms o t msurmnt s # Z P # Z! # Z ( Z Hr nd rom wt ollows w xpliitly us t subsript to indit tt t quntity to wi it is ttd is onditiond on prvious msurmnt rsults, t ourrns (dttion rords) o t ltrons tunnling troug t PC brrir in t pst. n tink o ( # _ s t inrmnt in t numbr o ltrons ( pssing troug PC brrir in tim Z. In t quntumjump s, ( is qul to itr zro or on, nd n # (. In ddition, sin t ntur o ltrons tunnling troug t PC is stosti, # tus rprsnts lssil rndom pross. Formlly, t urrnt troug t PC n b writtn s # ( Z. It is t vribl, t umultd ltron numbr trnsmittd troug t PC, wi is usd in s. [, 6, 7. Intuitivly, t nsmbl vrg ( o t lssil stosti pross # quls ` ( Z, t probbility (quntum vrg) o dtting ltrons tunnling troug t PC brrir in tim Z. Hn rom Eq. (), w v ( # Z. # Z (20) wr xpttion vlu ( is tkn wit rspt to t normlizd onditionl stt # t tim. Eqution (20) simply stts tt t vrg urrnt is wn dot is mpty, nd is. wn dot is oupid. Using Eqs. (3) nd (6) [or Eqs. (7), (7) nd (8), nd xpnding nd kping t trms o irst ordr in Z, w obtin rom Eq. () t quntumjump stosti Sr ödingr qution, onditiond on t obsrvd vnt in tim Z : # ^ Not tt ( ( Z * # P G G Z r ignord in Eq. (2). o ommodt initil nonpur or mixd stts, w xprss t stosti Sr ödingr qutions s stosti mstr qutions o t CQD qubit systm dnsity mtrix. From Eq. (), t onditionl, stosti dnsity mtrix # Z # ( Z stisis #. is o ordr Z. Hn trms proportionl to ( Z wr t rltion # P ( Z! # ( Z # # Z Z ( ( s bn usd. Using Eqs. (3) nd (6) [or Eqs. (7), (7) nd (8), nd kping trms up to ordr Z, w obtin t stosti mstr qutions, onditiond on t obsrvd vnt in tim Z : ( ( Z # P # # wr!!!, nd!!!!! # Z # Z Z # B Z * #. On n lso driv, wit t lp o Eq. (2), t stosti mstr qution (23) using t stosti Itô lulus [3 # ( # ^ ( ( # ( ^ # '. Equtions (2) nd (23) r t sm s Eqs. (3) nd (33) o. [3. But t drivtion prsntd r, strting rom t Sr ödingr qution or t ombind wol systm nd illustrting t ssntil pysis o t quntum trjtoris, is mor trnsprnt nd uristi. # () (2) (22) (23)

7 S Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis 7 3. Conntions to mstr qution ppro sow nxt tt t mstr qution or t rdud or prtilly rdud dnsity mtrix simply rsults wn n vrg or prtil vrg is tkn on t onditionl, stosti dnsity mtrix (onstrutd rom t onditionl, stosti CQD qubit systm stt vtor) ovr t possibl outoms o t msurmnts on t PC bt. is rsult provids uniid pitur or ts smingly dirnt ppros. trditionl, unonditionl mstr qution ppro is to din totl dnsity mtrix Z S Z Z, nd tn tr out t stt o t bt. is lds to t rdud dnsity mtrix ' Z Z 3 Z Z Z (24) or t CQD qubit systm lon. t o intgrting or tring out t nvironmntl (dttor) dgrs o t rdom to obtin t rdud dnsity mtrix is quivlnt to tt o ompltly ignoring or vrging ovr t rsults o ll msurmnt rords #. is n b sn by tking nsmbl vrg on t onditionl, stosti dnsity mtrix Eq. (22), idntiying Z # Z # Z, nd stting ( qul to its xptd vlu Eq. (20). n t rsultnt qution lds to Eq. (24) wit Z Z nd Z P Z or t quntumjump s. Furtrmor, wit t lp o Eqs. (3) nd (6) [or Eqs. (7), (7) nd (8), w ind t unonditionl mstr qution: (2) * Not tt t trm originting rom t onditionl stt Eq. (6) or (8) rprsnts t t, du to dttion o n ltron tunnling troug t PC, on t CQD dnsity mtrix. is is wy somtims is lld jump suproprtor. Eqution (2) n lso b obtind s in. [3 by tking t nsmbl vrg ovr t obsrvd stosti pross on Eq. (23) by stting ( qul to its xptd vlu Eq. (20). In tis ppro o mstr qution o t rdud dnsity mtrix, t inlun o t PC bt, dorn t or xmpl, on t CQD systm n b nlyzd [3. But tis ppro or Eq. (2) dos not tll us nyting bout t xprimntl obsrvd quntity, nmly t ltron ounts or urrnt troug PC. Hn, t PC dttor in tis ppro is trtd s pur nvironmnt or t systm, rtr tn msurmnt dvi, wi n provid inormtion bout t ng o t stt o t systm. An ltrntiv ppro rntly dvlopd in s. [, 6, 7 is to tk tr ovr nvironmntl (dttor) mirosopi dgrs o t rdom but kp trk o t numbr o ltrons,, tt v tunnld troug t PC brrir during tim in t prtilly rdud dnsity mtrix. is llows on to xtrt inormtion bout t quntum stt o t qubit, by msuring t tim vrg urrnt X troug t PC. mstr (rt) qution or tis prtilly rdud dnsity mtrix or t CQD qubit systm is drivd in. [ rom t solld mnybody Sr ödingr qution. il it is drivd in s. [6, 7, by mns o t digrmmti tniqu in t Kldys orwrd nd bkwrd in tim ontour, or Cooprpir rg qubit oupld pitivly to singlltron trnsistor. Hr w sow tt it n b obtind or t CQD/PC modl by tking prtil vrg on t onditionl, stosti CQD qubit systm dnsity mtrix ovr t possibl outoms o t msurmnt on t PC bt. produr to tk t prtil vrg n b dsribd s ollows. First, tking t nsmbl vrg on Eq. (23), w obtin Eq. (2). n to kp trk o t numbr o ltrons tt v tunnld, w nd to idntiy t t o t jump suproprtor trm in Eq. (2). I ltrons v tunnld troug t PC t tim Z, tn t umultd numbr o ltrons in t drin t t rlir tim, du to t

8 Quntum trjtoris nd quntum msurmnt tory in solidstt msosopis 8 ontribution o t jump trm, sould b P or P mtrix s: *. Atr writing out t numbr dpndn xpliitly, w obtin t mstr qution or t prtilly rdud dnsity P Evluting Eq. (26) in t logil qubit rg stt (i.., prt loliztion stt o t rg in dot nd dot 2, rsptivly), w obtin t rt qutions, t sm s Eq. (3.3) o. [. I t sum ovr ll possibl vlus o is tkn [i.. tring out t dttor stts ompltly, _, Eq. (26) tn rdus to Eq. (2). is produr o rduing Eq. (23) or (2) to Eq. (26) nd tn to Eq. (2) by vrging ovr (tring out) mor nd mor vilbl dttor inormtion provids uniid pitur or ts smingly dirnt ppros rportd in t litrtur. is produr is prtiulrly simpl using our ormlism. t is i t (stosti) mstr qution is xprssd in orm in trms o suproprtors nd, nd t t o t jump suproprtor trm is idntiid. o summriz, w v prsntd uristi drivtion o t quntum trjtory (stosti Sr ödingr or mstr) qution strting rom t ull Sr ödingr qution o t systm nd nvironmnt. ous on t msurmnt intrprttions to t quntum trjtoris. n t onpt o quntum trjtoris riss quit nturlly rom n xpnsion o t totl stt vtor in ign bsis o t oprtor tt rprsnts t pysil quntity or obsrvbl o t nvironmnt tt is msurd. In t CQD/PC modl, t obsrvd quntity o t PC nvironmnt is t numbr o ltrons tunnling troug t PC brrir. stostiity in t quntum trjtory n b viw s bing du to t rndomnss in t possibl outoms o t msurmnt rord. v sown tt t quntum trjtory or stosti Sr ödingr qution ppro provids us wit t most (ll) inormtion s r s t systm stt volution is onrnd. In tis ppro, w r propgting in prlll t inormtion o onditiond (stosti) stt volution # nd dttion rord # rsrvoirs is rovrd nd ontind by t msurmnt rords # in ontinuous msurmnt pross. All t inormtion rrid wy rom t systm to t o prt dttion or iint msurmnt. is is wy t systm n b ontinuously dsribd by stt vtor rtr tn rdud or prtilly rdud dnsity mtrix. v lso sown tt t mstr qutions o t rdud or prtilly rdud dnsity mtrix n b obtind s rsult o tking n nsmbl vrg or prtil vrg ovr t possibl msurmnt rords in t quntum trjtory ppro. is provids uniid pitur or ts smingly dirnt ppros. produr to iv tis uniid viw is prtiulrly sy to undrstnd using our ormlism. E quntum trjtory nd orrsponding dttion rord mimis possibl singl run o t ontinuous in tim msurmnt xprimnt. will prsnt lswr t simultion rsults or n initil qubit stt rdout xprimnt using bot o t quntum trjtory nd prtilly rdud dnsity mtrix ppros. rns [ Korotkov A N Pys. v. B [2 Korotkov A N 200 Pys. v. B [3 Gon HS,Milburn G, ismn H M, nd Sun H B 200 Pys. v. B [4 Gon HS nd Milburn G 200 Pys. v. B [ Gurvitz S A 7 Pys. v. B 6 2; quntp/ [6 Snirmn A nd Sön G 8 Pys. v. B [7 Mklin, Sön G nd Snirmn A 200 v. Mod. Pys [8 Dlibrd, Cstin nd Molmr K 2 Pys. v. Ltt [ Hgrldt G C nd Sondrmnn D G 6 Quntum Smilss. Opt [0 vn Dorsslr F E nd Ninuis G Opt. B: Quntum Smilss. Opt. 2, 2. (26)