reduced operator approximation for modelling open quantum systems [osid, 22, 1550008 (2015)]

7/21/2019 Reduced Operator Approximation for Modelling Open Quantum Systems [OSID, 22, 1550008 (2015)]

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Qdcuedc Ipdr`tir @pprixgk`tgin hir Kicdoognf

Ipdn Pu`ntuk Uystdks

@fngdszb` Vdrp`emiwsb

Cdp`rtkdnt ih \mysges `nc @striniky, Tngvdrsgty Eioodfd Oincin, Fiwdr Utrddt,Oincin VE0D 8AS, Tngtdc Bgnfcik

@prgo 13, 1


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3/26

` kicdr`tdoy o`rfd sgzd (e` 4< a`sgs st`tds in ` st`nc`rc \E), gntdrètgnf wgtm

nin-K`rbivg`n qu`ntuk m`rkinge isegoo`tir a`tms (hrik sgnfod kicds ti wgcdeintgnuius spdetr`) wgtm ` wgcd r`nfd ih eiupognf strdnftms, wmgod m`vgnf kic-dr`td eikput`tginò rdqugrdkdnts. Gt usds tmd Mdgsdnadrf pgeturd, wmgem k`bdsgt p`rtgeuo`roy d`sy ti hieus tmd `ttdntgin in tmd systdk cdfrdds ih hrddcikwmgod prdsdrvgnf tmd cdeimdrdned ddets cud ti tmd eiupognf ti tmd a`tm, `nceinsdrvgnf tmd titò dndrfy. Gn tmgs nin-K`rbivg`n sdttgnf, iur kdtmic fidsadyinc tmd Biss`biwsbgOgncao`c hirkògsk, `s tmd fdndr`tir ih tmd dvioutginih systdk ipdr`tirs gn tmd Mdgsdnadrf pgeturd cdpdncs in tmdgr mgstiry.

Gn tmd hiooiwgnf sdetgin wd wgoo rdkgnc smirtoy tmd tmdirdtgeò aèbfriunc ihiur wirb `nc o`y iut tmd dkpoiydc hirkògsk (Udes 1.0 `nc 1.1). Ndxt wd wgooprdsdnt tmd cdrgv tgin ih QI@ (Ude. 1.5) `nc pripisd gts twi v`rg nts: oiw `ncmgfm-ircdr gn tmd systdk-a tm gntdrètgin ipdr`tirs. Smdy wgoo ad iptgkgsdc hirtypgeò e`sds ih eintgnuius `nc Oirdntzg`n a`tms gn Ude. 1.3. Gn Ude. 5 wd wgoo

`ppoy iur kdtmic ti ipdn qu`ntuk systdks gn cgdrdnt systdk-a`tm eiupognfstrdnftm `nc a`tm kdkiry rdfgkds, `nc eikp`rd gt ti itmdr pipuo`r kicdoognftdemngquds, suem `s tmd psducikicd kdtmic `nc tmd qu`ntuk st`td cgusgin.Udetgin 3 eint`gns ` smirt sukk ry ih iur wirb.

1 Smdirdtgeò `pprièm

Kist fdndròoy, `n ipdn qu`ntuk systdk gs ` suasystdk ih ` o`rfdr, gntdrètgnfqu`ntuk systdk, d.f. ind ih tmd pmitins gn `n D\Q p`gr, `n `tik gn ` rdsin`nte`vgty, ` qu`ntuk cit gntdrètgnf wgtm pminins gn tmd erystò ir `ny rdò iajdetadeikgnf eo`ssgeò tmriufm se`ttdrgnf ih ` v`st nukadr ih `gr kiodeuods `ncpmitins in gt. Vd einsgcdr tmd e`sd ih ` ffngtd-cgkdnsginò qu`ntuk systdk

eiupodc ti `n gnffngtd-cgkdnsginò qu`ntuk a`tm, eikpisdc ih ` pissgaoy gnffngtdnukadr ih kicds. Gn suem `n `sykkdtrge o sdtup gt gs n`tur o ti gfnird tmdcdt`gos ih tmd a`tm cyn`kges `nc hieus in tmd cyn`kges ih tmd rdcuedc cdnsgtyk`trgx ih tmd systdk. Gn tmgs em`ptdr wd cdrgvd gt usgnf tmd pripisdc QI@`pprièm.

1.0 Ipdn qu`ntuk systdk

Vd einsgcdr ` qu`ntuk systdk rdprdsdntdc gn `n N-cgkdnsginò Mgoadrt spèdMs sp`nndc ay a`sgs st`tds{|n} `nc `ssieg`tdc wgtm tmd spèd ih ognd`r ip-dr`tirs inMs, cdnitdc ayO(Ms). Gts gntdrnò cyn`kges gs cdsergadc ay tmdM`kgoting`nMs O(Ms),

Ms;N

k,n;0

Xkntkn ,

wmdrdO(Ms) tkn :;|kn| `rd tr`nsgtgin ipdr`tirs adtwddn tmd st`tds|n`nc|k `nc Xkn E `rd tmd gntdrst`td eiupognfs, Xkn ; Xnk. Gn ` kird

5


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einegsd nit`tgin, Ms gs ` trèd ih ` k`trgx pricuet gnO(Ms)NN, tmd spèd ihN N k`trgeds wmisd dodkdnts `rd ipdr`tirs hrikO(Ms):

Ms; SrXSt:;

Nn;0

(XSt)nn .

Gn tmd dqu`tgin àivd,t O(Ms)NN gs `n NNk`trgx ih systdk ipdr`tirs,(t)kn :;tkn, `nc X ENN. Ugned z E e`n ad dkadccdc gnO(Ms) `s zGs(wmdrd Gs gs tmd gcdntgty ipdr`tir inMs), tmd k`trgx pricuet adtwddn ENN`ncO(Ms)NN e`n ad cdffndc, wgtm vòuds gnO(Ms)NN. Sr cdnitds tmd trèdgnO(Ms)NN,

O(Ms)Sr I:;N

n;0

Inn , I O(Ms)NN .

Ay ` n`turò dxtdnsgin ih tmd seò`r pricuet hrikMs,

|MNs :;Nn;0

n|nMs,

`nc MNs :; (Ms)N gs ` Mgoadrt spèd `ssieg`tdc wgtm tmd spèd ih ognd`r ipdr`tirsO(MNs ) O(Ms)NN. Mdrkgtg`n einjuf tgin gnO(Ms)NN e`n ad tmus cdffndc`s (I)kn :; Ink. Gn tmd e`sd ih t, tmgs od`cs tit

; t, sgned tkn ; tkn.Smd systdk gs eiupodc ti ` qu`ntuk a`tm eikpisdc ih ` eioodetgin ih B

gncdpdncdnt m`rkinge isegoo`tirs cdsergadc ay tmd M`kgoting`n ètgnf in `ngnffngtd-cgkdnsginò Mgoadrt spèdMa,

O(Ma)Ma ;Bb;0

b`b`b,

wmdrd`b gs tmd `nngmgo`tgin ipdr`tir ih tmd b-tm kicd ( 0). Smd eiupognfadtwddn tmd systdk `nc tmd a`tm gs cdsergadc ay tmd ipdr`tir

O(M)Mg ;Bb;0

Sr fb t

`b+ m.e. ,

wmdrdM ;Ms Ma `nc{fb} `rd N N cg`finò k`trgeds cdsergagnf tmdeiupognf ih tmd b-tm a`tm kicd wgtm tmd systdk, (fb)kn; knfbn.

Smd hèt tm`t dèm fbk`trgx gs cg`finò kd`ns tm`t tmd a`tm cids nit gncued

tr`nsgtgins adtwddn systdk a`sgs st`tds. Miwdvdr, tmd k`trgx nit`tgin òoiwshir `n d`sy fdndrògs`tgin ih tmd kicdo ti gneoucd suem a`tm-gncuedc tr`nsgtgins.

Smd titò M`kgoting`n fdndr`tgnf tmd dvioutgin ih tmd systdk `nc tmd a`tmgn tmd Uemricgnfdr pgeturd gs fgvdn ay

M;Ms+Ma+Mg ; SrXSt+

Bb;0

b`b`b+

SrBb;0

`bfb+`bf

b

t . (0)

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Gn tmd tr`cgtginò gncdx nit`tgin

M;N

k,n;0

Xkntkn+Bb;0

b`b`b+

Nk;0

Bb;0

(fbk`b+fbk`b)tkk .

Smd rdcuedc cdnsgty k`trgx ih tmd systdk gs cdffndc `s

s(t) :; Sra(t) ,

wmdrd(t) gs tmd cdnsgty k`trgx ih tmd systdk `nc tmd a`tm `s ` wmiod `nc tmdtrèd Sra :O(M) O(Ms) gs pdrhirkdc ivdr tmd a`tm cdfrdds ih hrddcik inoy.Gn tmd|n a`sgs,

(s(t))kn ; SrR(t)tkn[ , (1)

wmdrd tmd trèd gs ivdr aitm systdk `nc a`tm cdfrdds ih hrddcik. Smd k`gnt`sb ih tmd prdsdntdc kdtmic gs iat`gngnf s(t) wgtmiut eòeuo`tgnf(t).

1.1 Cyn`kges gn tmd Mdgsdnadrf pgeturd

Gn tmd Mdgsdnadrf pgeturd tmd w`vdhunetgin gs tgkd-gncdpdncdnt, (


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wmdrd wd m`vd usdc tmd gcdntgty (vògc òsi gn tmd Uemricgnfdr pgeturd)

tkn(t)tkn(t)tkn (t)nk. (8)Gn tmd kird eikpèt nit`tgin, RSr(@t(t)), t(t)[ ; R@, t[.

Smd àivd gcdntgtgds e`n ad `ppogdc ti (5) ti iat`gn tmd dvioutgin ih t `ncipdr`tirs fdndr`tdc ay tmd M`kgoting`n (0). Hir tmd systdk wd iat`gn

t(t) ;gRXS, t(t)[ +gBb;0

Rfb, s

b(t)[ + Rf

b, sb[

, (6)

wmdrd sb(t) :;t(t)`b(t) `rd systdk-a`tm gntdrètgin ipdr`tirs `nc, sgnedt(t) ;

t(t) `nc a`tm ipdr`tirs eikkutd wgtm systdk ipdr`tirs, sb(t) ;t(t)`b(t). Hir

tmd a`tm, usgnf tmd e`ningeò eikkut tgin rdo`tgins hir aisinge erd`tgin/`nng-

mgo`tgin ipdr`tirs, wd iat`gn

b(t) ;gb`b(t) gSr fb t(t) . (=)Gn tmd gncdx nit`tgin, tmd àivd dqu`tgins m`vd tmd hiooiwgnf hirk (`ssukgnf

(fb)kn; knfbn):

c

cttkn(t) ;g

Nk;0

(Xkktkn(t) Xnktkk (t))

+gtkn(t)Bb;0

(fbk fbn)`b(t) +gtkn(t)Bb;0

(fbk fbn)`b(t)(7)

`ncc

ct`b(t) ;gb`b(t) g

Nk;0

fbktkk(t) .

1.5 Qdcuedc Ipdr`tir @pprixgk`tgin

1.5.0 Fdndr o cdsergptgin

Smd `gk ih tmd prdsdntdc kdtmic gs ti kicdo tmd dvioutgin ih tmd systdk, gneouc-gnf gts cdeimdrdned eùsdc ay tmd gntdrètgin wgtm tmd a`tm. Smd gnhirk tginàiut tmgs priedss gs eint`gndc gn tmd rdcuedc cdnsgty k`trgx ih tmd systdks(t). @s cdkinstr`tdc gn prdvgius sdetgins, sdd (1) `nc (3), gt e`n ad iat`gndchrik tmd kd`n vòuds ih systdk ipdr`tirs t(t). Smus, ti eòeuo td s(t) gn tmd

Mdgsdnadrf pgeturd, ind m`s ti dviovd t(t) gn tgkd. Ugned tmd dvioutgin dqu`tginhir t(t) (6) gnviovds tmd a`tm ipdr`tirs `b(t), cud ti tmd systdk-a`tm gntdrè-tgin, gt gs ndedss`ry ti dviovd `b(t) `s wdoo. Miwdvdr, ` nukdrgeò cdsergptgin ihaitm typds ih ipdr`tirs gn tmd titò systdk `nc a`tm a`sgs gs gkpissgaod, `sMags gnffngtd-cgkdnsginò.

@eeircgnf ti (3), fgvdn tmd gngtg o systdk st`tds wd inoy nddc ti bniw tmdp`rtgò dxpdet`tgin vòuds ih systdk ipdr`tirs, Sraatkn(t), ti iat`gns(t). Smd

8


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eirrdspincgnf p`rtgò dxpdet`tgins ih tmd a`tm ipdr`tirs, Sraa`b(t), eint`gn

p`rt ih tmd gnhirk`tgin in miw gntdrètgin eirrdo`tdc tmd a`tm `nc tmd systdkgh tmdrd w`s ni suem eirrdo`tgin, Sraa`b(t) wiuoc ad pripirtginò ti `n gcdntgtyipdr`tir gnMs(rdeòo tm`t`b(


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hiroR


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`ppoy tmd s`kd emiged ih m ti tmd `pprixgk`tgin ih KRsb(t)`b(t)[. @nitmdr

priaodk gs tm`tKRsb(t)`b

(t)[ e`n ad `pprixgk`tdc ay dgtmdr KRsb(t)[KR`b

(t)[ir KRsb(t)[KR`b(t)[. Ugkgo roy, wd e`n emiisd adtwddn KR`b(t)[KRsb(t)[ `nc

KRsb(t)[KR`b(t)[ hir KR`b(t)sb(t)[. Si dxpoigt huooy tmd gnhirk`tgin àiut

tmd systdk-a`tm eirrdo`tgins eint`gndc gn KRsb(t)[ k`trgeds, wd usd `n dquòoywdgfmtdc `vdr`fd ih tmd twi `pprixgk`tgins. Gn tmgs w`y wd iat`gn tmd dvioutgindqu`tgin hir tmd rdcuedc rdprdsdnt`tgin ih gntdrètgin ipdr`tir

c

ctKRsb(t)[ ;gbKRsb(t)[ gKRt(t)[fb+gRXS, KRsb(t)[[

+ g

1

Bb;0

Rfb , KRs

b (t)[[ + Rf

b , KRsb(t)[[

KR`b(t)[

+

g

1

Bb;0

KR`b(t)[Rfb

, KRsb(t)[[ +

g

1

Bb;0

Rfb , KRsb(t)[[KR`b

(t)[ .

Sifdtmdr wgtm (0


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Miwdvdr, gh wd usd gt ti eòeuo`td s(t), gts pisgtgvgty gs nit fu`r`ntddc. Si ffx

tmgs priaodk, wd k`bd usd ih tmd gcdntgty (8) ti cdrgvd t(t)

t(t) N

t(t) `ncrdpoèd (3) wgtm ` cgdrdnt `pprixgk`tgin

(s(t))kn ; 0

NSrssKRtkn(t)[

1 . (05)

Ugned tkk(t)tkn(t) ; tkk (t)(tnk(t)) (`nc tmd s`kd hir tmd rdcuedc rdprd-

sdnt`tgins), tmd àivd hirkuo` fu`r`ntdds tm`t s(t) gs pisgtgvd-sdkgcdffngtd. Gneintr`st, tmd trèd ih cdnsgty k`trgx (05) gs nit einsdrvdc, adeùsd KRt(t)[1 ;KRt(t)1[. Smus, wd nirkògsd tmd cdnsgty k`trgx ti iat`gn

(s(t))kn ; SrssKRtkn(t)[1/

N

n;0SrssKRtnn(t)[

1 .

1.3 A`tms wgtm eintgnuius spdetrò cdnsgtgds

Hir `ny typd ih a`tm wd e`n cdffnd tmd spdetrò cdnsgty J() ;

bfbfb(b).

Gn tmd ogkgt ih `n gnffngtd nukadr ih kicds, J() e`n ad ` eintgnuius hunetgin.Ind w`y ti m`ncod tmgs sgtu`tgin gs ti cgserdtgsd J() gnti ` ffngtd, aut o`rfdnukadr ih kicds. @ssukgnf ` einst`nt kicd hrdqudney spègnf , wd cdffnd` eiupognf einst`nt hir ; b ti ad

fb ;

J(b) . (03)

S`bgnf tmd squ`rd riit dnsurds pripdr nirkògs`tgin `s


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`nc tmdn gnsdrt tmd iat`gndc hirkuo` gnti (6):

t(t) ;gRXS, t(t)[ +g

Bb;0

Rfb,u

b(t)[ + Rf

b,ub(t)[

+gBb;0

dgbt`b(


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Ipdr`tirs k(t) `nc k(t) s`tgshy e`ninge`o eikkut`tgin rdo tgins,

Rk(t), n(t)[ ; < `nc Rk(t), n(t)[ ; kn

kn ogkBB

b;0 fbnfbk ;kn.Smdy `rd psducikicd erd`tgin `nc `nngmgo`tgin ipdr`tirs, erd`tgnf ir cdstriy-gnf eioodetgvd dxegt`tgins gn ` sgnfod a`tm R17[. Smdgr cyn`kges gs cdsergadc aytmd dqu`tgin

c

ctk(t) ; ogk

B

Bb;0

fbkbd

gbt`b(


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wmdrd (


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Dvioutgin dqu`tgins hir tmd rdcuedc rdprdtdnt`tgins ih tmd `aivd systdk `nc

gntdr`etgin ipdr`tirs, rdspdetgvdoy, `rd

c

ctKRtkn(t)[ ;g

Nk;0

(XkkKRtkn(t)[ XnkKRtkk(t)[)

+j

kj

KRsknkj(t)[ KRsnkkj(t)[

+j

nj

KRsnknj(t)[ KRsknnj(t)[

(07)

`nc

c

ct

KRsknkj(t)[ ;gN

k;0

(XkkKRsknkj(t)[

XnkKRskkkj(t)[)

+0

1

j

kj(KRsknkj(t)[ KRsnkkj(t)[)KRkj(t)[

+0

1KRsknkj(t)[

j

(

kj KRkj(t)[

njKRnj(t)[)

+0

1

j

nj(KRs

nknj(t)[ KRsknnj(t)[)KRkj(t)[

01

j

(

kjKRkj(t)[

njKR

nj (t)[)KRsknkj(t)[

(gkj +kj)KRsknkj(t)[ + kjnkKRtkn(t)[

(1


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5 Nukdrge`o dx`kpods `nc eikp`rgsin wgtm itmdr

kdtmicsGn tmgs sdetgin wd cdkinstr`td QI@ ay `ppoygnf gt ti ` kiodeuo`r `ffrdf`tdgntdr`etgnf wgtm ` nin-K`rbivg n qu`ntuk a`tm, `nc eikp`rd tmd iat`gndcrdsuots wgtm twi itmdr tdemngquds: tmd psducikicd kdtmic R17, 5


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< 4 0< 04 1

reduced operator approximation for modelling open quantum systems [osid, 22, 1550008 (2015)]

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