Dissipative Dynamics of Two-Level Systems in Low Temperature Glasses

Transcription

1 Aricl pubs.acs.org/jpca Dissipaiv Dynamics of Two-Lvl Sysms in Low Tmpraur Glasss K-Wi Sun, and Yang Zhao*, School of Scinc, Hangzhou Dianzi Univrsiy, Hangzhou 3118, China Division of Marials Scinc, Nanyang Tchnological Univrsiy, 5 Nanyang Avnu, Singapor ASTRACT: An approach basd on a non-marovian im-convoluionlss polaron masr quaion is usd o prob dynamics of a cnral chromophor mbddd in a bah of wo-lvl sysms commonly found in low-mpraur glasss. y raing h Hamilonian in h polaron fram, w can accoun for iniial nonquilibrium bah sas as wll as h spaially corrlad nvironmnal ffc. Rlvan ralisic siuaions ar xplord by adoping paramrs from prvious xprimns. is found ha h mpraur of h boson bah has a subsanial ffc on h populaion rlaxaion and h dcohrnc procss, and a highr mpraur also rsuls in a highr sauraion valu of h nanglmn nropy, whil h coupling bwn h chromophor and h TLS has an ffc ha gos counr o ha of h mpraur.. NTRODUCTON Obsrvaions of glasss a low mpraurs poin o h prsnc of addiional dgrs of frdom, as compard o crysals wih similar composiions. This addiional frdom lads o highr spcific ha, nhancd ulrasound anuaion, and various ohr anomalis. 1, was soon ralizd ha hos dgrs of frdom saura a high microwav or acousic powr, and ar no associad wih vibraions. For vry low mpraurs, only h wo lows nrgy lvls of a doubl minimum ponial nd b considrd. Consqunly, h complx dynamics of glasss can b rducd o a random array of wo-lvl sysms (TLSs). 3 1 As a simpl, phnomnological dscripion, h modl of TLSs was vry succssful in hlping undrsand various anomalis in glasss. A numbr of low-mpraur singl molcul spcroscopy (SMS) sudis on chromophors mbddd in organic glasss hav ld o h obsrvaions of a wid rang of spcral bhaviors Th prsnc of spcral diffusion, a phnomnon of pa absorpion frquncy movmns in succssiv masurmns on chromophors, is dcd by hol-burning spcroscopy (H) and hr-puls phoon cho (3PE) xprimns n addiion, h absorpion spcrum of a dilu collcion of chromophors in glass is usually inhomognously broadnd, 5, rflcing h xn o which h solvn nvironmns diffr, and hnc is rlad o h local disordr around h chromophor. TLS flips du o unnling will also modula h ransiion frquncy of h chromophor. Gnrally, lin shaps show a surprising variaion and, in som circumsancs, singl chromophors ar nown o produc mulipls of lins. 5 is commonly blivd ha h flucuaions of a lowmpraur amorphous sysm can b dscribd by h dynamics of h TLSs ha ar coupld o h chromophors via phonon filds. n addiion, h TLS concnraion in glasss is almos indpndn of h chmical naur of h glass. A a fw Klvins, i urns ou ha a singl chromophor has lss han on TLS in is immdia viciniy wih coupling sufficinly srong o giv ris o spcral diffusion of h chromophor. 11 Thrfor, for simpliciy, w sar wih a modl dscribing a singl TLS and a chromophor undr h influnc of a boson bah. As in h aformniond chromophor TLS sysm, nrgy scals for vibronic rlaxaion and spin phonon coupling ar comparabl, and h sysm can b considrd in an inrmdia coupling rgim, whr h radiional scond-ordr prurbaion approach is inaccura. 3 5 This lads o dynamics sudis using nonprurbaiv approachs, 6 9 such as numrically xac iraiv pah ingral mhods, 3 35 sophisicad sochasic ramns of h sysm bah modls, 36 and hirarchical quaion of moion approach Howvr, compuaionally innsiv nonprurbaiv mhods always bcom infficin as h sysm siz is larg or whn hr ar mulipl xciaions. Rcnly, h non-marovian imconvoluionlss (TCL) polaron masr quaion, capabl of inrpolaing bwn wa and srong spin phonon coupling rgims and handling iniial nonquilibrium bah sas and spaially corrlad nvironmns, has bn mployd o dscrib xciaion dynamics in mulichromophoric sysms, 5,41 44 and in paricular, his approach is usd o ra an xnsion of h convnional spin boson modl o includ an addiional spin bah. 8 n his wor, w invsiga h dynamics of wo coupld psudospins in conac wih a non-marovian dissipaiv bah. Th mpraur ffc on h von Numann nropy of h Rcivd: January 3, 14 Rvisd: Fbruary 4, 14 Publishd: Fbruary 8, Amrican Chmical Sociy dx.doi.org/1.11/jp5717 J. Phys. Chm. A 14, 118, 7

2 Th Journal of Physical Chmisry A sysm (chromophor+tls) is also sudid. Th linar absorpion spcrum of h chromophor is calculad basd on h xprimnal paramrs s for ryln in polysyrn. Th rmaindr of his papr is organizd as follows. n Scion, h modl and h mhod ar prsnd. Th dynamics of h sysm and h absorpion spcra ar sudid in Scion. Conclusions ar drawn in Scion V.. MODEL SYSTEM AND METHODOLOGY W considr a modl dscribing a chromophor and a singl TLS boh coupld o a common srain fild of h glassy mdium. Th Hamilonian can b wrin as 3,5,45 ω ε = σ + σ σ z z1 x1 + H + f ( b + b ) σ z + g ( b + b ) σ z1 (1) Hr, h chromophor Hamilonian has bn runcad o ha of a psudospin wih s = 1/, assuming ha h xprimns ar carrid nar rsonanc wih h opical ransiion of frquncy ω. Th subindx rfrs o h chromophor, and h 1 subindx o h TLS wih asymmry ε and unnling marix lmn. H = Σ ω b b rprsns h Hamilonian for h boson bah, whr b (b ) corrsponds o h craion (annihilaion) opraor of h h mod of h phonon bah, wih frquncy ω. W hav also inroducd cofficins f (g ) o rprsn h coupling of h chromophor (TLS) o h bah mod. Transformaion of h Hamilonian (1) via a uniary opraor f = U xp ( b b) σz ω q yilds a drssd Hamilonian H = U U ω ε = σ + σ σ z z1 x1 + H + g ( b + b ) σ z1 a + σzσz1 (3) Afr h applicaion of h abov polaron ransformaion, h chromophor phonon coupling is rmovd from h Hamilonian a h xpns of inroducing an xplici form of h chromophor TLS coupling, which ariss from an indirc, phonon-mdiad inracion. This dscripion allows for a convnin, prurbaiv ramn. 5 W shall a h drssd Hamilonian (q 3) as our saring poin. Throughou his papr, h spcral dnsiy J(ω) =Σ g δ(ω ω )= κω ph ω l ( ω/ωc) is assumd wih l bing h spcral xponn. Th cuoff frquncy is ω c = 4, and ω ph is h characrisic phonon frquncy ha will b usd as h nrgy uni. κ is a dimnsionlss consan rprsning h TLS phonon coupling srngh. W adop l = 3 (supr-ohmic form) in all h calculaions. Th paramr a =(γη)/(r 3 ) has h radial and angular dpndnc of a dipol dipol inracion, whr γ is h TLS chromophor coupling consan, and r is h disanc bwn hm. Th orinaion paramr, η, which in principl could assum a coninuum of valus corrsponding o roaions of h TLS in spac, is s o η = ±1 for simpliciy. n h () following calculaions, w simply considr a as h coupling paramr bwn h TLS and h chromophor. 5 is asy o s ha h opraor σ z is a consrvaiv quaniy. Th dfiniion of h psudospin opraors can b wrin as σz = g g σx = + g g (4) whr g () is h ground (xcid) sa of h chromophor, and σz 1 = σx 1 = + (5) whr ( ) is h uppr (lowr) sa of h TLS. Nx, w will adop h rcnly proposd TCL polaron masr quaion approach. 8,5,41 Th polaron ransformaion is gnrad by S = Σ (g /ω )(b b )(σ z1 )/, rsuling in h ransformd Hamilonian S S H = H = H + H 1 + H (6) H = + ε g a Θ + σz σz 4 ω σ σ σ z z1 x1 ω H = [ σx (cosh Θ ) + iσy sinh ] 1 (7) 1 (8) whr = (g /ω )(b b )andθ = cosh =xp[ (1/) (g /ω ) coh(βω /)]. Hr, w assum ha h phonon bah is in hrmal quilibrium a mpraur T =( β) 1 wih olzmann consan. Th TCL quaion up o h scond ordr in h inracion picur is nown as 8,5,47 ρ () = () ρ () + d s () () s ρ () + d s ( ) ( s) ρ ( ) Th firs wo rms in h righ-hand sid (r.h.s.) of q 9 ar inhomognous rms. Thy ar nonzro only whn h iniial bah sa diffrs from h rfrnc sa ρ chosn o b h hrmal quilibrium sa. n our cas, w shall s ha his corrsponds o a nonquilibrium prparaion of h iniial nvironmnal sa wihin h polaron fram. ρ () is h nir dnsiy marix in h polaron fram. Th supropraor is dfind by ( ) =Tr ( ) ρ and =1. Furhrmor,h supropraor is dfind by ()() = i[ H 1(), ]. H 1( ) is givn by + H ( + H) 1 ( ) H () = H 1 = σ+ D + σ D [ 1 ( ) ( ) 1 ( ) ( )] () D () = Θ Aricl (9) (1) whr σ± 1() = σ± 1 and () =Σ (g )/(ω )(b iω b iω ). σ +1 (σ 1 )isanffciv raising (lowring) opraor of h TLS. is convnin o xprss hm in h ign-sa spac in h inracion picur. Firs, w diagonaliz h Hamilonian in q 7. Th four nrgy lvls and h corrsponding ignvcors ar xprssd as 1 dx.doi.org/1.11/jp5717 J. Phys. Chm. A 14, 118, 7

3 Th Journal of Physical Chmisry A E ± g ω ε a = λ ± + Θ ± ω ε + a = λ ± + Θ E θ θ + g () g () ψ = cos + sin g () g () θ θ g () g () ψ = sin + cos g () g () (11) whr λ = Σ (g )/(4ω ), and an θ g =( Θ)/(ε a) (anθ = ( Θ)/(ε + a)). According o q 11, in h ign-sa rprsnaion w can asily obain h xprssions of h opraors in h inracion picur. L ρ s() () and σ s() () dno h Schro dingr (inracion) picur dnsiy opraors of h nir sysm and h sum of h chromophor and h TLS, rspcivly, whil hir counrpars in h polaron fram ar labld as ρ () s () and σ s () (). n h Schro dingr picur and h polaron fram, quanum masr quaion (QME) is givn by s s 1 s σ = () i[ H, σ()] itr{ [ H (), ρ ()] } dstr { [ H ( ), [ H ( s), ρ ()]] } dstr {[ H (), [ H ( s ), σ( ) ρ ]]} (1) whr H = H + H. Th facorizd iniial sa ρ() is assumd in h calculaion of h dynamics C ρ = TLS () ρ = σ() ρ (13) whr ρ = βh /Z and h boson bah is in hrmal quilibrium wih pariion funcions givn by Z. Th dnsiy opraors of h chromophor and h TLS ar prpard iniially in a pur sa. n h polaron fram, w obain ρ () = σ() ρ, whr ρ = 1/ ρ 1/. Furhr, on has ρ () = ρ() Tr { ρ()} ρ = σ() ( ρ ρ ). Th drivaion of h xplici forms of h QME and h woim corrlaion funcions is shown in Appndix A.. NON-MARKOVAN DYNAMCS N THE LA FRAME Gnrally, h xpcaion valu of h sysm (chromophor+ TLS) obsrvabl A in h lab fram is givn by 8,5,41 A = Tr { Aρ( )} S+ s S S S+ s S+ = Tr { A ρ ( )} + Tr { A ρ( )} = A + A rl irrl S S s (14) whr A = rl Tr S{ Aσs ( )} and S S A = Tr { A ρ }. S S A irrl = Tr S+ { A ρ ( ) }. To h zroh ordr h irrlvan conribuion bcoms A Tr + { S A S irrl S ρ ()}. This approximaion is applid o valua h irrlvan xpcaion valus in h calculaion of h dynamical propris. No ha h irrlvan par of h xpcaion valu vanishs if [A, S] =. Th dynamics of h chromophor and h TLS can b obaind basd on qs 13 and 14, such as σ = Tr { σ( ) σ } xy ()rl S s xy () σz1rl = Tr S{ σs( ) σz1} σ = ΘTr { σ( ) σ } xy ()1rl S s xy ()1 σ = xy ( ) irrl σ = z1 irrl σ = Θ Tr { σ() σ ( )} d( ) + c. c. x1 irrl S + 1 σ = iθ Tr { σ() σ ( )} d( ) + c. c. y1 irrl S + 1 (15) whr d() = xp[i (g /ω )sin(ω )] 1. Furhrmor, in ordr o quanify h nanglmn bwn h sysm (i.., h chromophor and h TLS) and h bah, w valua h voluion of h von Numann nropy. f σ s () is h rducd dnsiy marix of h sysm in h original basis, h von Numann nropy of h sysm, which changs wih im du o dphasing and rlaxaion procsss inducd by h bah, is givn by ( σ( )) = Tr( σ( )ln σ( )) (16) s s s n addiion, w can also calcula h linar absorpion linshap funcion 5 1 ω ( ω) = R i μ ( ) μ () d π (17) whr μ() = σ x is h iniial dipol momn opraor in h Condon approximaion, and μ()= μ() is h sam opraor a im in h Hisnbrg picur. Th dipol auocorrlaion funcion in h polaron fram is givn by S+ 1 x x (18) μ( ) μ() = Tr { ρ () σ σ } No ha h avrag valu of h irrlvan par in q 18 is zro. n h calculaion of h linar absorpion spcrum, w hav assumd ε a, ε, and ω a so ha h quilibrium dnsiy marix for h chromophor (h TLS) is givn by 3 q C(TLS) ρ P( ) ( ) ( ) + Pg( ) g( ) g( ) (19) whr P = [ βω/ cosh (βω /)] 1, P g = [ βω/ cosh (βω /)] 1, P = [ βϵ/ cosh (βϵ/)] 1, and P = [ βε/ cosh (βϵ/)] 1. Th paramr rgim of inrs hr corrsponds o a low-mpraur ralisic glassy sysm. n h lab fram, h facorizd iniial condiion for h oal sysm is assumd o b q q Aricl ρ () = ρ ρ ρ 1 C TLS () Furhrmor, in h polaron fram, on has h iniial dnsiy marix dx.doi.org/1.11/jp5717 J. Phys. Chm. A 14, 118, 7

4 Th Journal of Physical Chmisry A q 1/ 1/ 1 C 1/ 1/ + P ρ ) ρ () = ρ ( P ρ (1) V. RESULTS AND DSCUSSON Th ypical paramr s drivd xprimnally for rryln in polysyrn 5 consiss of ε max =17K, = Ko 17 K, a max = 4.5 K, and T =.1 o 1 K. n all h calculaions, w s ω ph = K as h nrgy uni. W now xamin h dissipaiv dynamics of h chromophor and h TLS basd on qs 15 and 16, as shown in Figur 1. Aricl xampl, σ z1 approachs zro du o qual occupaion of h wo lvls. n gnral, i is found ha σ z1 undrgos dampd oscillaions, and h oscillaion frquncy is rducd as h mpraur is incrasd. n addiion, plod in Figur 1d is h von Numann nropy of h nir sysm of h chromophor and h TLS. Th linarizd form of h von Numann nropy has bn prviously usd o quanify xcion phonon nanglmn in h Holsin Hamilonian. 46,5 As is wll-nown, h von Numann nropy is boundd from abov (σ s ()) ln, whr is h dimnsion of h Hilbr spac of h sysm. Thrfor, h nropy may b biggr han 1 in our cas. (σ s ()) is qual o zro du o h facorizd iniial condiion, and incrass wih h im voluion. A abou = 5, i rachs a plaau. Highr mpraur givs ris o a highr plaau. n h limi of infini mpraur, (σ s ()) achivs h largs valu possibl, ln 4, ha is, arriving a h complly mixd sa. Figur 1. Dynamics of h chromophor and h TLS for wo bah mpraurs, T =.5 and T =.. Ohr paramrs ar ω =1,ε =1, =.8, a =.8, κ =.5. (a) Evoluion of h cohrnc of h chromophor. (b) Evoluion of h cohrnc of h TLS. (c) Evoluion of h populaion diffrnc of h TLS. (d) Evoluion of h von Numann nropy of h chromophor TLS sysm. Th opraor σ z commus wih h Hamilonian 3, and σ z is a consan of moion, i.., h populaion of h chromophor dos no chang wih h im. Howvr, h pur dphasing mchanism sill xiss bcaus h chromophor is indircly coupld wih h bah, as shown in Figur 1a. Th dcohrnc procss is fasr a a highr mpraur and σ x rachs h sady sa valu. Th voluion of h cohrnc of h TLS is plod in Figur 1b. Th populaion rlaxaion and h pur dphasing procsss occur in h TLS. A long ims h cohrnc of h TLS rachs a plaau, h high of which dcrass wih an incras in h mpraur. Th populaion diffrnc of h TLS, σ z1, is also invsigad, and h rsuls ar shown in Figur 1c. Th sady sa valu of h populaion diffrnc dmonsras ha h TLS is hrmally quilibrad. n h high-mpraur limi, for 3 Figur. Dynamics of h chromophor and h TLS for wo coupling srnghs, a =.8 and a = 1.6. Ohr paramrs ar ω =1,ε =1, =.8, T = 1, κ =.5. (a) Evoluion of h cohrnc of h chromophor. (b) Evoluion of h cohrnc of h TLS. (c) Evoluion of h populaion diffrnc of h TLS. (d) Evoluion of h von Numann nropy of h chromophor TLS sysm. Th ffc of h coupling bwn h chromophor and h TLS on h dynamics is also inrsing, as shown in Figur. For h chromophor, h oscillaion frquncy of h cohrnc incrass wih incrasing coupling paramr a. From Figur a, i is obsrvd ha h dphasing im is no vidnly conncd wih h coupling srngh bwn h chromophor and h TLS. niially, h voluion of h TLS cohrnc is snsiiv o h coupling srngh a, and a largr a brings abou a largr σ x1, dx.doi.org/1.11/jp5717 J. Phys. Chm. A 14, 118, 7

5 Th Journal of Physical Chmisry A whil in h sady sa, h impac of a on h cohrnc is minimal, as shown in Figur b. n addiion, i is found in Figur c ha h sady-sa populaion diffrnc of h TLS, σ z1, incrass wih an incras in h coupling paramr a. n h limi of a, from q 11 h nrgy lvls approach E ± g() = ± wih only h ground sas E g() occupid, and on also has θ g andθ π, which can b subsiud ino h wav funcions of q 11. is hn found ha h wav funcions of h wo dgnra ground sas saisfy h asympoical rlaions ψ g() ( ) g(), and h wo sas in h TLS ( and ) ar qually occupid, i.., σ z1 ( ) inh limi of a. On h ohr hand, h von Numann nropy of h sysm has a biggr sauraion valu for war coupling bwn h chromophor and h TLS, as shown in Figur d. is inrsing ha h mpraur ffc on h nanglmn nropy is found o b opposi o ha of h coupling paramr a upon comparing Figur 1d wih Figur d. To facilia h invsigaion of h linar absorpion spcrum of h TLS chromophor sysm, a fini radiaiv lifim for h chromophor, γ rad, is inroducd in our compuaion o Figur 3. Linar absorpion spcra for wo mpraurs, T = (solid lin) and T =.5 (dashd lin). Two srnghs of TLS chromophor coupling ar usd: (a) a =.1; (b) a =.. nsur ha hr is a fini widh for spcral pas. Th ffc of mpraur on h spcra is dmonsrad in Figur 3 for wo chromophor TLS coupling srnghs, a =.1,.. Ohr paramrs adopd ar ω =1,ε =,κ =.5, =.1,and γ rad =.3.TworansiionsE g ± E ± occur in our cas. Th oscillaor srngh of E g E is grar a lowr mpraurs. No ha h facorizd iniial condiion is assumd. Th oscillaor srngh is proporional o h occupaion probabiliy of h nrgy lvl E g ±. Thrfor, w can xpc ha h ransiion srngh of E g + E + will incras wih h incras of h mpraur, as shown in Figur 3a,b. From q 11, h coupling srngh a drmins h spliing of h nrgy lvls, and as a rsul, h wo absorpion pas mov away from ach ohr wih an incras in a, as shown in Figur 3b. Only on TLS has bn considrd in our discussion hr. Howvr, i is also inrsing o loo ino h ffc of mulipl TLSs on h lin shap of h absorpion spcrum. n fac, i has bn shown ha srong chromophor TLS coupling du o hir clos proximiy rsuls in non-gaussian faurs in absorpion spcra. 53,54 n a ralisic sysm, TLS TLS inracions inducd by srain filds can also xis, and hir ffc 4 Aricl was invsigad by rown al. 5 who concludd ha h TLS TLS coupling dos no xr any significanly dirc ffc on h chromophor s spcral propris, including h lin widh hisograms, and ha only in rlaivly rar cass dos his coupling affc h singl molcul lin shaps. Furhr discussion of h TLS TLS inracions and hir ffc on h absorpion lin shap, howvr, is byond h scop of his wor. Du o h quivalnc bwn h quanum microscopic modl (q 3) and h sochasic suddn-jump ramns, 3 som discussion on is implicaions o h lin shap is in ordr. Firsly, w assum h individual conribuions from ach TLS o h lin shap ar small, hrby giving ris o h obsrvd cnral limi yp bhavior of h lin shap as h cumulaiv ffc of many small prurbaions. Wihin h sochasic suddn-jump ramn, h ransiion frquncy of h chromophor dpnds upon h insananous sas of h TLS, and ach flipping TLS modifis h chromophor ransiion frquncy. As h wo absorpion pas mov away from ach ohr wih an incras in h chromophor TLS coupling, h lin shap can morph from Gaussian o Lornzian. Accordingly, w hav shown ha lin shap compud in our modl closly rsmbls ha from h uncoupld suddn-jump modl. 5 Th obsrvd shaps of singl molcul lins ar primarily h rsul of spliing of h chromophor s absorpion pa ino many ovrlapping lifim limid Lornzians. Th rlaiv highs of hs Lornzians ar dominad by h occupaion probabiliis of h nrgy lvl E g ±, and h spliings hmslvs ar dpndn upon h chromophor TLS sparaions. n addiion, for h muli-tls nvironmn, h rsulan lin shap is Gaussian in h absnc of disordr bu bcoms Lornzian for complly disordrd TLSs in h long im limi. 54 Du o h low TLS dnsiy in glass, h TLS TLS coupling will hav insignifican influnc on h lin widh hisograms. 5 V. CONCLUSON n summary, w hav mployd h im-convoluionlss polaron masr quaion mhod o invsiga h dissipaiv dynamics of a cnral chromophor mbddd in a bah of wo-lvl sysms commonly found in low-mpraur glasss. Our hory includs inhomognous rms which accoun for nonquilibrium iniial prparaion ffcs, and i gos byond h wa rsonanc coupling limi of h Fo rsr and Dxr hory. 55,56 y raing h rlvan dgrs of frdom in a polaron fram, his hory is valid for modra spin-phonon coupling and is capabl of handling iniial nonquilibrium bah sas and spaially corrlad nvironmns. Th ohr advanag is h highr compuaional fficincy of our dynamics calculaions as compard o ohr nonprurbaiv approachs. is found ha h TLS populaion diffrnc undrgos dampd oscillaion, and is oscillaion frquncy is rducd as h mpraur is incrasd, whil h oscillaion frquncy of h chromophor cohrnc incrass wih an incras in h coupling paramr a. Th mpraur ffc on h nanglmn nropy gos counr o ha of h coupling paramr a. n addiion, h oscillaor srngh in h linar absorpion spcra is found o b modifid by h mpraur. As h wo pas in h absorpion spcrum mov away from ach ohr wih an incrasing chromophor TLS coupling srngh, srong coupling adds non-gaussian dviaions o h absorpion lin shap in a muli-tls nvironmn. dx.doi.org/1.11/jp5717 J. Phys. Chm. A 14, 118, 7

6 Th Journal of Physical Chmisry A APPENDX A: THE QME AND THE TWO-TME CORRELATON FUNCTONS Th scond rm of h r.h.s. in q 1 can b wrin as itr { [ H1 ( ), ρ ()] } = σ+ σ i () Tr { D( ρ ρ 1 )} + σ σ i () D Tr { ( ρ ρ 1 )} i σ() σ+ Tr {( ρ ρ ) D} 1 i σ() σ Tr {( ρ ρ ) D} 1 whr g i sin( ω) Tr { D ( ρ = Θ( ω 1) =Θd( ) Tr {( ρ ρ ) D} = Θd( ) Th hird rm of h r.h.s. in q 1 can b wrin as dstr { [ H ( ), [ H ( s), ρ ()]] } = σ+ σ+ ( s ) σ() Tr { DD( s )( ρ σ+ σ ( s ) σ() Tr { DD ( s )( ρ σ σ+ ( s ) σ() Tr { DD( s )( ρ σ σ ( s ) σ() Tr { DD( s )( ρ + σ+ ( s ) σ() 1 σ+ 1 Tr { Ds ( )( ρ ρ ) D} + σ+ ( s ) σ() 1 σ 1 Tr { Ds ( )( ρ ρ ) D} + σ ( s ) σ() 1 σ+ 1 Tr { D( s )( ρ ρ ) D} + σ ( s ) σ() 1 σ 1 Tr { D( s )( ρ ρ ) D}} + h. c. (A1) (A) (A3) whr Tr { DD( s )( ρ () s () = Tr { ρ ( Θ)( Θ)} Tr { ρ DDs ( ) ( )} () + s () 1/[ (), s ()] = Tr { ρ } Θ [ d ( ) + ds ( ) + 1] Tr { ρ DDs ( ) ( )} ϕ() ϕ( s) ϕ( s) = { dds ( ) ( ) + ( 1)[ d ( ) + ds ( )]} Tr { DD ( s )( ρ ϕ() ϕ( s) ϕ( s) = { d( ) d* ( s) + ( 1)[ d( ) + d* ( s)]} Tr { DD( s )( ρ ϕ() ϕ( s) ϕ( s) = { d* ( ) d( s) + ( 1)[ d* ( ) + d( s)]} Tr { DD( s )( ρ ϕ() ϕ( s) ϕ( s) = { d* ( ) d* ( s) + ( 1) [ d* ( ) + d* ( s)]} (A4) n h calculaion of q A4, w hav usd h following woim corrlaion funcions ϕ() ϕ( ) 1 ϕ() ϕ( ) Tr { ρ D ( ) D ( )} = [ 1] 1 Tr { ρ D ( ) D( )} = [ 1] Tr { ρ D( ) D ( )} = Tr { ρ D ( ) D( )} Tr { ρ D( ) D( )} = Tr { ρ D( ) D( )} (A5) whr ϕ()=σ (g /ω ) (cos ω coh(βω /) i sin ω ). Th las rm of h r.h.s. in q 1 can b wrin as dstr {[ H (), [ H ( s ), σ( ) ρ ]]} s = σ σ σ ρ ( s ) s( ) Tr { DD( s )} σ+ 1σ 1( s ) σs ( ) Tr { ρ DD( s )} σ 1σ+ 1( s ) σs ( ) Tr { ρ DD( s )} σ 1σ 1( s ) σs ( ) Tr { ρ DD( s )} + σ+ 1( s ) σs ( ) σ+ 1 Tr { D( s ) ρ D} + σ+ 1( s ) σs ( ) σ 1 Tr { D( s ) ρ D} + σ 1( s ) σs ( ) σ+ 1 Tr { D( s ) ρ D} + σ 1( s ) σs ( ) σ 1 Tr { D( s ) ρ D}} + h. c.. Aricl (A6) Subsiuing h abov qs A1 A6 ino q 1, w can solv h moion quaion of h rducd dnsiy marix. 5 dx.doi.org/1.11/jp5717 J. Phys. Chm. A 14, 118, 7

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