Dynamics of two externally driven coupled quantum oscillators interacting with separate baths based on path integrals

Transcription

1 Dynamcs of wo exernally drven coupled quanum oscllaors neracng wh separae bahs based on pah negrals Illaron Dorofeyev * Insue for Physcs of Mcrosrucures, Russan Academy of Scences, 6395, GSP-5 Nzhny Novgorod, Russa Absrac The paper deals wh he problem of dynamcs of exernally drven open quanum sysems. Usng he pah negral mehods we found an analycal expresson for me-dependen densy marx of wo exernally drven coupled quanum oscllaors neracng wh dfferen bahs of oscllaors. I s shown ha a he zerong of exernal forces he densy marx becomes dencal o he prevously obaned one for freely developng coupled oscllaors. Mean values of observables are compued by usng he Herman par of he marx. All elemens of he covarance marx composed by coordnaes and momena of wo drven coupled oscllaors are calculaed. The me-dependen mean values, dspersons and covarances of coordnaes of coupled oscllaors a gven exernal forces are numercally suded. I s shown ha he larger he couplng consan he larger s he dsurbances of he second oscllaor due o exernal acon on he frs oscllaor. Coupled dynamcs of forced oscllaors a relavely large couplng consan s demonsraed a dfferen hermodynamc condons. PACS : 5..Jc;.5.Lc; 5..Gg;

2 I. Inroducon The physcs of open sysems covers varous processes rangng from elemenary-parcle level o asrophyscal scales. Also, such sysems exce a grea neres n echnology and socal scences. An ofen-used model of open quanum sysems consss of a quanum oscllaor (se of oscllaors) coupled o a hea reservor (se of shared or separae reservors) of harmonc oscllaors. Ths mechancal hamlonan sysem allows nvesgang of dsspaon, decoherence, correlaon, connuous quanum measuremen, quanum-o-classcal ranson and oher mporan phenomena n naure. I s well-known ha he model has been successful n descrbng he Brownan dynamcs of seleced parcles coupled o a bah, see, for example, [- 9]. Sudy of he pure Hamlonan compose sysem yelds a reason of rreversbly n he dynamcs of a seleced quanum sysem neracng wh surroundngs afer reducon wh respec o reservors varables. I always deserves consderable aenon because of ranson from pure mechancal dynamcs o hermodynamcal laws due o nevable effecs of he envronmen on objecs under sudy. Thus, he dealed descrpon of relaxaon processes of open sysems o saonary or o quas-saonary saes, and o equlbrum or o quasequlbrum hermodynamc saes s obvously very mporan. For example, n he case of a harmonc oscllaor wh arbrary dampng and a arbrary emperaure an explc expresson for he me evoluon of he densy marx when he sysem sars n a parcular knd of pure sae was derved and nvesgaed n a semnal work [] based on he pah negral echnque. I was shown ha he spaal dsperson n he nfne me lm agrees wh he flucuaon-dsspaon heorem (FDT). To sudy he approach o equlbrum or o some ransen saonary sae of he sysem, a problem for coupled oscllaors neracng wh hea bahs characerzed by s own emperaures was consdered n [-6]. I was concluded ha an arbrary nal sae of a harmonc oscllaor sae decayes owards a saonary sae. A sudy of a harmonc chan o whose ends ndependen hea bahs are aached whch kep a dfferen emperaures was performed n [7]. I was found he chan approaches a saonary sae regme. An analyze of he nonequlbrum seady saes of a one-dmensonal harmonc chan of aoms wh alernang masses conneced o hea reservors a unequal emperaures and a hea ransfer across an arbrary classcal harmonc nework conneced o wo hea bahs a dfferen emperaures were done n [8, 9]. The evoluon of quanum saes of neworks of quanum oscllaors coupled wh arbrary exernal envronmens was analyzed n [3]. The

3 emergence of hermodynamcal laws n he long me regme and some consrans on he low frequency behavor of he envronmenal specral denses were demonsraed. Much aenon has been focused on he bpare connuous varable sysems composed of wo neracng oscllaors. Based on he non-markovan maser equaons, he enanglemen evoluon of wo harmonc oscllaors under he nfluence of non-markovan hermal envronmens was suded n [3]. I was demonsraed ha he dynamcs of he quanum enanglemen s depended on he nal saes, he couplng beween oscllaors and he couplng o shared bahs or o separaed bahs. A sudy of he me-dependen enanglemen and quanum dscord beween wo oscllaors coupled o a common envronmen was provded n papers [3-3]. Dfferen sages of evoluon ncludng sudden dsappearance and appearance of he enanglemen were descrbed provdng a characerzaon of hs process for dfferen reservors ncludng Ohmc, sub-ohmc, and super- Ohmc models. Two coupled oscllaors n a common envronmen a arbrary emperaure and he quanum decoherence of her saes were nvesgaed n [35]. I was shown ha he problem can be mapped no ha of a sngle harmonc oscllaor n a general envronmen plus a free harmonc oscllaor. Besdes, smples cases of he enanglemen dynamcs were consdered analycally and an analycal creron for he fne-me dsenanglemen was derved a he Markovan approxmaon. The me-dependen enanglemen beween wo coupled dfferen oscllaors whn a common bah and whn wo separae bahs was suded n [36] based on a maser equaon. I was found ha n he case of separae bahs a no very low emperaures he nal wo-mode squeezed sae becomes separable accompanyng wh a seres of feaures. For nsance, f he wo oscllaors share a common bah, he observaon of asympoc enanglemen a relevan emperaures becomes possble. The evoluon of quanum correlaons of enangled wo-mode saes n a sngle-reservor and n a wo-reservor model was suded n [37]. I was shown ha n he wo-reservor model he nal enanglemen s compleely los, and boh modes are fnally uncorrelaed, bu n a common reservor boh modes nerac ndrecly va he same bah. In [38] a sysem of wo coupled oscllaors whn separae reservors was nvesgaed. I was shown ha f he bahs are a dfferen emperaures, hen he neracon beween he parcles mus be srong enough n order o reach a seady sae enanglemen. No hermal enanglemen beween wo coupled oscllaors s found n he hgh-emperaure regme and weak couplng lms n [39]. The exsence of a nonequlbrum sae for wo coupled, paramercally drven, dsspave harmonc oscllaors whch has saonary enanglemen a hgh emperaures was repored n []. Based on exac resuls for he non-markovan dynamcs of wo paramercally coupled oscllaors n conac o ndependen hermal bahs, he ou-ofequlbrum quanum lm derved n [] s generalzed o he non-markovan regme n []. I 3

4 s shown ha non-markovan dynamcs allows for he survval of saonary enanglemen a hgher emperaures. A saonary regme of wo coupled oscllaors connecng wh ndependen reservors of harmonc oscllaors was suded n [], and analycal formulas for he mean energy of neracon of he seleced oscllaors and her mean energes n hs case were derved. Tme-dependen behavor of varances and covarances of wo coupled oscllaors whn separae bahs n he weak-couplng lm was nvesgaed n [3]. I was demonsraed ha hese characerscs of wo weakly coupled oscllaors n he nfne me lm agrees wh he FDT despe of nal varances. The case of arbrary couplng of dencal oscllaors was consdered n [], and was shown ha he larger a dfference n emperaures of hermal bahs, he larger s a dfference of he saonary values of varances of coupled dencal oscllaors as compared o values gven by he FDT. The general case of wo arbrary coupled oscllaors of arbrary properes neracng wh separae reservors s suded n [5]. As well as n prevous cases he emporal dynamcs of spaal varances and covarances of oscllaors from any gven me up o quas-equlbrum seady saes s suded based on pah negraon. I s shown for arbrary oscllaors ha he spaal varances and covarances acheve saonary values n he long-me lm. I s demonsraed ha he larger he dfference n masses and egenfrequences of coupled oscllaors, he smaller are he devaons of saonary characerscs from hose gven by he FDT a fxed couplng srengh and fxed dfference n emperaures beween hermal bahs. The man goal of hs paper s o derve an analycal expresson for a emporary dependen densy marx of wo seleced oscllaors subjeced by wo ndependen exernal forces a any arbrary mes. The reduced densy marces allow calculang whole se of elemens of a covarance marx. Temporal behavor of some mean values of observables s provded. The paper s organzed as follows. In Sec.II we descrbe a heorecal bass usng a pah negraon mehod o calculae he reduced densy marx of wo neracng quanum oscllaors n dfferen reservors of harmonc oscllaors. Temporal dynamcs and saonary saes of mean values of coordnaes are gven n Sec.III. Our conclusons are gven n Sec.IV. II. Problem saemen and soluon We consder wo blnear coupled oscllaors. In s urn, each of hese oscllaors s blnear coupled wh separae reservors of oscllaors and subjeced by exernal forces. Correspondng me-dependen Hamlonan s wren as follows

5 H ( ) p / M M x / p / M M x / x x x f ( ) x f ( ) N N p j m j m jj q j x pk mk mkk qk x j k / ( ) / / ( ) /, () where x,, p,, M,,, are he coordnaes, momena, masses and egenfrequences of he seleced oscllaors, s he couplng consan, q j, p,, m and j j j q k, p,, m are he coordnaes, momena, egenfrequences and masses of bah s oscllaors. Furher we use he vecors R { q j} { q,..., qn } and R { qk} { q,..., qn } for brevy. k k k We suppose ha, n he me all neracons among oscllaors are swched on and mananed durng arbrary me nerval up o nfny. Then, he arbrary exernal forces begn acng a arbrary me momens,. The problem s o fnd a me-dependen densy marx of wo drven coupled oscllaors n any momen of me followed by calculang all elemens of a covarance marx n hs case. Fgure llusraes he above descrbed scenaro. 5

6 I s well known [6-5] ha he evoluon of he oal Hamlonan sysem seleced wo neracng oscllaors plus wo reservors plus exernal forces s descrbed by he equaon for he densy marx W() of he oal sysem W ( ) W ˆexp ( / ) ( ) () ˆ T H s dsw T exp ( / ) H( s) ds, () where W() W s he nally prepared densy marx of he oal sysem, and we ake no accoun he me dependence of he oal Hamlonan n Eq.() due o exernal forces, ˆ T s he me-orderng operaor. Furher, we use he desgnaons x { x, x} and R { R, R} for shoren noaon. Usng hese desgnaons we wre he compleeness propery of he poson egenfuncons n he coordnae represenaon as usual dxdxdr dr xx RR xx RR dxdr xr xr, (3) where he lms of he mulple negraon exended from mnus o plus nfny. Usng Eq.(3), he Eq. () can be wren n he marx form as follows xr W ˆ(,) ˆ yq dxdrdydq xr U xr xr W yq yq U (,) yq where he unary operaor of evoluon, () Uˆ(,) Tˆexp ( / ) H ( s) ds, (5) The ranson ampludes n Eq.() are expressed va he pah negrals [6-5] n dfferen ways, for example, n he coordnae form xr Uˆ (,) xr K ( x, R, ; x, R,), (6) D x D x D R D R exp{( / ) S[ x ( ), x( ), R ( ), R ( )]} where he negraon along all pahs s carred ou from x() x o x() x, from x() x o x() x, and from R () R o R() R, from R() R o R() R. For convenence, we desgnae he negraon varables by sragh lnes beneah and over he leers for he bah s and oscllaor s coordnaes, correspondngly. The backward amplude n Eq. () s * yq Uˆ (,) yq K ( y, Q, ; y, Q,), (7) D y D y D Q D Q exp{( / ) S[ y( ), y( ), Q ( ), Q ( )]} 6

7 where he negraon along all pahs s carred ou from y() y o y() y, from y() y o y() y, and from Q () Q o Q() Q, from Q() Q o Q() Q. The pah negrals are expressed va he mulple Rehmann negrals n he coordnae space as usual [6-5]. The acon S[ x, x, R, R ] n Eq.(6) s expressed va he Lagrangan correspondng o he Hamlonan n Eq.(). Ths acon n Eqs.(6) and (7) s srucured n accordng wh Eq.() where S S S S S S S S S S. (8) () () ( ) A( ) B( ) I( ) A( ) B( ) I ( ) ( ) ex ( ) ex ( ) A S d M x ( ) / M x ( ) /, (, ), S d [ x ( ) x ( )], N N B j j j j B k k k k j k S d ( m / )[ q ( ) q ( )], S d ( m / )[ q ( ) q ( )], N I j j j j j j S d [ c q ( ) x ( ) ( c / m ) x ( )], N I k k k k k k S d [ c q ( ) x ( ) ( c / m ) x ( )], () ex S d [ x ( ) f ( )], (, ) Pung n hs equaon Hamlonan n Eq. (). c m, c m we oban a sysem exacly correspondng o j j j k k k The reduced densy marx comprsng only varables of seleced oscllaors s obaned by racng he whole densy marx n Eq.() over he varables of he wo bahs ( x, x, y, y, ) dr lm xr W yq QR dxdrdydq dr K x R x R xr W yq K y R y Q * (,, ;,,) (,, ;,,). (9), () where ranson ampludes K and x, R W y, Q W ( x, y; R, Q,) * K from Eq.(6), (7), and s he densy marx of he global sysem a he nal me. III. Resuls and dscusson A. Propagaor for drven coupled oscllaors We choose he nal whole densy marx as follows W ( x, x, y, y; R, R, Q, Q,) ( x, y,) ( x, y,) ( R, Q,) ( R, Q,), () () () () () A A B B 7

8 where ( x, y,), ( x, y,) are nally prepared densy marces of he wo seleced () () A A oscllaors, ( R, Q,), ( R, Q,) are nal densy marces of he separae reservors. () () B B In hs case he reduced densy marx looks lke () () ( x, x, y, y, ) dx dx dy dy J ( x, x, y, y, ; x, x, y, y,) A ( x, y,) A ( x, y,), () where he propagaor J ( x, x, y, y, ; x, x, y, y,) D x D x D y D y exp ( / ) S A[ x ] S A[ y] S [ x ] S [ y ] S [ x, x ] S [ y, y ] S [ x ] S [ y ] () () A A ex ex S x S y S x x S y y S x S y x,,, ] () () A[ ] A[ ] [, ] [, ] ex [ ] ex [ ] FFV [ x y y, (3) where all acons are defned by Eq.(9) and he negraon along all pahs s carred ou from x() x o x() x, from x() x o x() x, and from y () y o y() y, from y() y o y() y, correspondngly. The Feynman-Vernon nfluence funconal FFV n Eq.(3) s he same as n [3]. In our case we have where F [ x, x, y, y ] exp [ x, y ] [ x, y ], () () () FV EV EV () () * EV EV [ x, y ] [ x, y ] d d x ( ) y ( ) ( ) x ( ) ( ) y ( ) ( / ) d[ x ( ) y ( )] ( / ) d [ x ( ) y ( )] where ( ) ( ) ( ) ( ) ( ) ( ) * d d x y x y c ( ) ( ),, N N j J c k J d d j mj j k mk k where J ( ), are he specral denses of nose of wo bahs., (5) (6) The funcons, can be represened n he followng convenen form, ( ), ( ), ( ) d J, ( )[coh(, / )cos( ) sn( )], (7) see, for example [5]. We use he Ohmc case when J, ( ),, whch gves, ( ), ( ) /, where () s he max dela-funcon, /, where max s he maxmal frequency of excaons whn,,,, 8

9 bahs. Takng no accoun Eq. (7) he Eq. (3) for propagaor can be rewren as follows J ( x, x, y, y, ; x, x, y, y,) Dx Dy exp S [ x ] S [ y ] S [ x ] S [ y ] () () A A ex ex d d{ x ( ) y ( )} ( ){ x ( ) y ( )} exp d d{ x ( ) y( )} ( ){ x( ) y( )}exp [ ( ) ( )] d x y. (8) Dx Dy exp S [ x ] S [ y ] S [ x ] S [ y ] S [ x, x ] S [ y, y ] () () A A ex ex d d{ x ( ) y ( )} ( ){ x ( ) y ( )} exp d d{ x( ) y( )} ( ){ x( ) y( )}exp d [ x ( ) y ( )] The expresson n Eq. (8) can be ransformed n o J ( x, x, y, y, ; x, x, y, y,) D x D y D x D y exp S A[ x ] S A[ y] () () () () S A[ x] S A[ y] S[ x, x] S[ y, y] Sex[ x ] Sex[ y] Sex [ x] Sex [ y], (9) M d[ x ( ) x ( ) y ( ) y ( ) x ( ) y ( ) y ( ) x ( )] M d[ x ( ) x ( ) y ( ) y ( ) x ( ) y ( ) y ( ) x ( )] exp[ ( )] where we desgaed,, /M,, and [ x, y] are equal max M d Coh d d { x ( ) y ( )}cos[ ( )]{ x ( ) y ( )} kt B where,., () The Eq. (9) for propagaor can be rewren n he compac form as follows where J ( x, x, y, y, ; x, x, y, y,) D x D y D x D y exp S[ x, x, y, y]exp[ ( )], () S[ x, x, y, y ] dl ( x, x, x, x, y, y, y, y ). () Then, we wre he Lagrangan n Eq. () omng he sragh lnes over he leers for brevy The Lagrange equaons of moon L M x / M y / M x / M y / Mx / M y / M x / M y /. (3) M [ x x y y x y y x ] M [ x x y y x y y x ] x x y y x f ( ) y f ( ) x f ( ) y f ( ) d L L d L L,, (,), () d x x d y y 9

10 as appled o he Lagrangan n Eq. (3) gve he followng sysem of coupled equaons for drven oscllaors x y x ( / M) x f( ) / M x y x ( / M ) x f ( ) / M y x y ( / M) y f( ) / M y x y ( / M ) y f ( ) / M. (5) Soluon of he sysem s done n Appendx A. The classcal soluon s obaned n new varables X x y, x y. In hese new varables he Lagrangan n Eq. (3) reads,,,,,, L M X / M X / M X M X / M X / M X. (6) ( / )( X X ) f ( ) f ( ) Now, we represen he pahs expressed n new varables as he sums X X X,, explcly selecng classcal pahs n Eqs.(A5, A6)-(A8,,,,,,, A9) and flucuang pars X,,, wh boundary condons X, () X, ( ), () ( ). The Lagrangan n Eq.(6) becomes,, L L + L M X / M X / M X M X / M X / M X ( / )( X X ) f ( ) f ( ) M X / M X / M X M X / M X / M X ( / )( X X ) f ( ) f ( ). (7) Wh he above descrbed assumpons and akng no accoun Eq.(7) he acon n Eq.() calculaed n new varables S Scl S s expressed as follows where () () () () () cl L ( ), cl cl cl ex ex S d S S S S S S L ( ) d S S S S S () () ex ex, (8) S S S S D ( X ) [ D D]( X ) [ D D]( X ) () () () () cl cl ex ex f f 5 5 f f 6 6 f f D( X ) D ( X ) D ( X ) D ( X ) D( X ) D( X ) f f f 3 f f 3 f ]( ) D ( X ) [ D7 D7 ]( X f ) [ D8 D8 ]( X f ) [ D9 D 9 X f [ D D ]( X ) [ D D ]( X ) [ D D ]( X ) f d ( ) f ( ) d ( ) f ( ). (9)

11 Then, we subsue no he nhomogeneous par he oal classcal rajecores for, and separae consans and varables of negraon where () d ( ) f ( ) d ( ) f ( ), (3) d ( ) f ( ) d ( ) f ( ) ( ) ( ) ( ) p p, (), () are wren n Appendx C. Then we have aken no accoun he neracons n () Scl beween wo oscllaors S ( X ) ( X ) ( X ) ( X ) () cl f f f f 3 f f f f 5( X f ) 6( X f ) 7( X f ) 8( X f ), (3) ( X ) ( X ) ( X ) ( X ) 9 f f f f ( X ) ( X ) ( X ) ( X ) wh emporal funcons k k(), k,...,6 from [5]. Fnally, we have for he classcal acons n Eq.(8) S S S S S [ D ]( X ) [ D D ]( X ) () () () () () cl cl cl ex ex f f f f [ D D ]( X ) [ D ]( X ) {( D ) 6 6 f f f f 9 f ( D D ) U ( )} X {( D ) ( D D ) U ( )} X 7 7 f f 8 8 f {( D3 5 ) X f ( D D 7 ) X f ( )} {( D 3 8 ) X f ( D D ) X ( )} [ D ]( X ) [ D D ]( X ) f 3 [ D D ]( X ) [ D ]( X ) d ( ) f ( ) d ( ) f ( ) 5 6 p p ( ) U ( ) where he erms U( ), U( ), U( ), ake no accoun nhomogeneous addonal erms n acons (3) () () () cl cl cl paral p p p [ S ( ) S ( ) S ( )] d { M X / M X / M X M X / M X / M X ( / )[ X X ]} p p p p p U ( ) U ( ) X U ( ) X. (33) Then we should calculae of he funcons, n Eq.() akng no acooun he form of he classcal soluon wh exernal forces max M d Coh d d { x ( ) y ( )}cos[ ( )]{ x ( ) y ( )} kt B M M max d Coh d d{ ( )}cos[ ( )]{ ( )} kt B dcoh kt B max where,. d d{ ( )}cos[ ( )]{ ( )} p p, (3)

12 The exernal forces are no lead o new resuls n Eq.(3) compare wh prevous calculaons n [3-5], because relaed addonal erm n Eq.(3) has no varables for furher negraon. Calculaons of a classcal acon and funcons, n Eq.() yeld he fnal expresson for he propagaor n case of wo drven coupled oscllaors J X X X X C S S S S S () () () () () ( f, f, f, f, ;,,,,) ( )exp { cl cl cl ex ex } exp { A ( ) B ( ) C ( ) }exp { A ( ) B ( ) C ( ) } f f f f exp { E ( ) E( ) f E3 f E f f ( ) ( ) }, (35) G( X, X,,, ; X, X,,,) f f f f where all me dependen funcons A (), B (), C (), E (),,,,,3, are wren n [5]. The funcon C () consss all rrelevan erms, whch have no varables for furher negraon. The flucuaonal negral n Eq.(35) s G( X f, X f, f, f, ; X, X,,,) exp { } D X D X D D S S S S ex, (36) exp { [, ] [, ]}exp { [, ] [, ]} T T T T where he negraon s carred ou along all closed pahs because X, () X, ( ) and, (), ( ), and funcons T, T can be found n [3]. The flucuaonal negral n Eq.(36) whou of he addonal erm relaed o exernal forces s calculaed n [3], and he addonal erm S ex s no change our fnal resul. B. Reduced densy marx of drven coupled oscllaors Thus, afer calculang of he flucuaonal negral he fnal form of he propagang funcon n Eq.(35) reads J ( X, X,,, ; X, X,,,) C( )exp S () f f f f cl exp S exp { A ( ) B ( ) C ( ) } () cl f f exp S exp { A ( ) B ( ) C ( ) } () cl f f. (37) exp { E ( ) E( ) f E3( ) f E f f ( ) }

13 In order o calculae he reduced densy marx ( x, x, y, y, ) n Eq.() we ough o assgn () he nal densy marxes ( x, y,) and A () A ( x, y,) for wo seleced oscllaors. In new varables we have nsead of Eq.() he followng expresson for he reduced densy marx ( X f, X f, f, f, ) dx dx dd J( X f, X f, f, f, ; X, X,,,), (38) () () ( X,,) ( X,,) A A where we choose he same nal saes of oscllaors as n [3-5] ( k ) / A ( X k, k,) ( ) exp k ( X k k ) / 8 k, ( k, ) (39) where, (,) k k are he nal spaal varances of wo oscllaors. Usng he propagaor from Eq.(37) and Eq.(39) an negraon n Eq.(38) s sraghforward and leads o an explc expresson for he reduced densy marx of wo drven coupled oscllaors, whch s vald for any me, ncludng a seady sae regme. In case of exernal forces he oal non-herman, n general, densy marx s as follows ( X, X, ;,,) C( )exp g ( ) X g ( ) X X g ( ) X f f f f f f f f exp g ( ) f g ( ) f f g ( ) f exp g ( ) X g ( ) X g ( ) X g ( ) X f f f f f f f f Aex X f Bex ( ) X f exp Aex ( ) f Bex ( ) f exp ( ) aex X f bex X f aex f bex f exp ( ) ( ) exp ( ) ( ), where C () s he normalzaon consan, g( ), g( ), g ( ), A ( ), B ( ), a ( ), b ( ) are ex ex ex ex pure real funcons. The non-hermcy of he n Eq.() can be easly proved recallng ha X f x f y f, f xf yf and akng no accoun he realy properes of he jus above menoned funcons. The Herman par of he of he n Eq.() can be easly exraced represenng as usual as a sum of he Herman and an-herman pars. Bu, s occure n hs problem ha A A, B B and a a, b b. Tha s why we can pu A X B X A B A x A y B x B y ex f ex f ex f ex f ex f ex f ex f ex f a X b X a b a x a y b x b y ex f ex f ex f ex f ex f ex f ex f ex f (),, () The Herman densy marx n hs case s as follows ( X, X,,, ) C( )exp g ( ) X g ( ) X X g ( ) X f f f f f f f f exp g ( ) f g ( ) f f g ( ) f exp g ( ) X g ( ) X g ( ) X g ( ) X f f f f f f f f aex X f bex ) X f exp Aex ( ) f Bex ( ) f exp ( ) (, () 3

14 where g( ), g( ), g ( ) and a ( ), b ( ) ex ex, A ( ), B ( ) are wren down n Appendx B. ex ex The Hermcy ( x, x, y, y, ) ( y, y, x, x, ) * f f f f f f f f of he marx n Eq.() s clear now and all mean values of observables are real, see he Appendx C. Also, we can o represen he non-herman marx n Eq.() as follows ( X, X, ;,,) C( ) ( ) ( ), (3) f f f f H N where he Herman par () s he marx from Eq.() and he non-herman par H ( ) exp A ( ) X B ( ) X exp a ( ) b ( ). () N ex f ex f ex f ex f Then, represenng he oal marx as a sum of he Herman and an-herman pars we can o oban C[ ( ) / ( ) / ] C, (5) H N N H N N H whch can be fullflled under resrcons A A, B B and ex ex ex ex a a, b b. These relaons are n agreemen wh approxmaons acceped n ex ex ex ex Eq.(). C. Temporal dynamcs of mean values All elemens of a covarance marx for he drven coupled oscllaors are obaned n Appendx C usng he Herman densy marx from Eq.(). I should be noed ha all dspersons of coordnaes and momena n hs problem are dencal o he problem of coupled quanum oscllaors whou exernal forces. Bu, he mean quadrac values of coordnaes and momena are esseally dfferen n hese problems. Besdes, mean values of coordnaes and momena are no equal o zero n he problem under our sudy, conrary o he problem whou of exernal forces, where he mean values of coordnaes and momena are equal zero. We nvesgaed dspersons n our prevous papers n [3-5] ha s why we pu our man aenon o sudy he emporal dynamcs of mean values of coordnaes of coupled oscllaors under he acon of exernal forces. For our sudy we chose he exponenal-lke exernal forces f ( ) f ( ) Exp( ), (,), (6) whch are swched on a dfferen mes, (,) n general, where f s he force amplude, s he un sep funcon, s he decayng facor. To chose he approprae ampludes of he forces we consder ha he oal ncrease of momena of an oscllaor durng a me nerval from o s as follows p p d f ( ) f exp( ). (7)

15 Then, we use he vral heorem o oban f x exp( ), where x, for example, can be of he dsperson x /M. For numercal calculaons we have chosen he followng parameers of oscllaors: 3 M 3 g, M 5M, rad / s, 3,., relang o sold maerals, and,, s, 3 sfor he exernal forces. Fg. exemplfes he normalzed mean values of he coordnaes x ( ) /, (,) of he frs a) and second b) oscllaors versus a me, where / M, (,) usng Eq.(C, C8). These fgures correspond o he case of no couplng, and o he case of he oal equlbrum, when T T 3K. The profles of exernal forces f (), of un amplude and of dfferen sgns n accordance wh Eq.(6) s shown beween of hese graphs, as well as n fgures 3,. I s clearly seen ha he mean values ends o zero, when boh of he forces f,. Also, we can see absoluely ndependen and dfferen dynamcs of seleced oscllaors, as mus be a and a dfferen parameers of oscllaors and forces. Fg.3 shows he normalzed mean values of he coordnaes x ( ) /, (,) of he frs a) and second b) oscllaors versus a me usng Eq.(C, C8) a relavely srong couplng 5

16 / MM.3beween of oscllaors. These fgures also correspond o he case of he oal equlbrum, whent T 3K. As well as n he prevous fgure s clearly seen ha he mean values ends o zero, when boh of he forces f,. Bu, n hs case we can see ha he frs exernal force also acs on he second oscllaor due o couplng a when f. Ths value of he couplng consan has been chosen because of he model of blnear couplng breaks down a sronger couplngs, see, for nsance, [, 5]. The normalzed mean values of he coordnaes x ( ) /, (,) of he frs a) and second b) oscllaors versus a me usng Eq.(C, C8) a.3 beween of oscllaors are shown n Fg. n he case ou of oal equlbrum n he sysem, when T 3K and T 9K. A comparson of fgures 3 and shows ha he dfference n emperaures yelds n a dfferen dynamcs of coupled oscllaors. I should be noed ha despe of parameers and hermodynamcal condons n he problem under sudy, he mean values of coordnaes ends o zero a he force zerong. Ths corresponds o A ( ), B ( ), a ( ), b ( ) of ex ex ex ex he densy marx n Eq.(). The funcons A ( ), B ( ), a ( ), b ( ) n Eq.() are no zero ex ex ex ex 6

17 only n he case when he exernal forces are no zero, see correspondng formulas n Appendx B. I should be noed ha n calculaons of mean values we can use arbrary forms of exernal forces ncludng sngle mpulses, whch are fne n me, or mpulse sequences wh gven onoff me rao, seady sae forces alernang n me, consan n me and so on, whch can be appled a any me o any of wo coupled oscllaors. Besdes, he seleced oscllaors can be characerzed by arbrary properes a dfferen emperaures of separaed bahs, n general. IV. Concluson Our paper s devoed o sudy of he relaxaon problem of open quanum sysems drven by exernal forces. We consdered wo blnear coupled oscllaors and, n s urn, each of hese oscllaors s coupled wh separae reservors of harmonc oscllaors and subjeced by exernal forces. Correspondng Hamlonan s provded. In he nal me all neracons among oscllaors are swched on and mananed durng arbrary me nerval. Then, he arbrary exernal forces begn acng a arbrary me momens. Usng he pah negral mehods we found and analyzed an analycal expresson for me-dependen densy marx of wo forced coupled quanum oscllaors neracng wh dfferen reservors of oscllaors. We calculaed correspondng propagaor n hs case. All elemens of he covarance marx are calculaed usng 7

18 he known reducon procedure. I s shown ha he mean values of coordnaes and momena of coupled oscllaors are no zero n case of exernally drven oscllaors. Tme-dependen behavor of he mean values a dfferen condons s graphcally llusraed. Coupled dynamcs of seleced oscllaors a relavely large couplng consans s demonsraed a dfferen hermodynamc condons. I s neresng o noe ha he mean quadrac characerscs of oscllaors are dfferen for he case of freely developng par of oscllaors and for he drven par of oscllaors, bu her dspersons are dencally equal. Acknowledgemen References [] V.B. Magalnsk, Zh. Eksp. Teor. Fz, 36, 9 (959) [Sov. Phys. JETP 9, 38 (959)] [] I.R. Senzky, Phys.Rev. 9, 67 (96). [3] G.W. Ford, M. Kac and P. Mazur, J. Mah. Phys. 6, 5 (965). [] P. Ullersma, Physca 3, 7, 56, 7, 9 (966). [5] A.O. Caldera and A. J. Legge, Phys. Rev. Le. 6, (98). [6] V. Hakm and V. Ambegaokar, Phys. Rev. A 3, 3 (985). [7] P.S. Rseborough, P. Hängg, U.Wess, Phys. Rev. A 3, 7 (985). [8] A. J. Legge e al., Rev. Mod. Phys. 59, (987). [9] G.W. Ford, J.T. Lews and R.F. O Connell, Phys. Rev. A 37, 9 (988). [] G.W. Ford, J.T. Lews and R.F. O Connell, Ann. Phys. (N.Y.) 85, 7 (988). [] G.W. Ford, J.T. Lews and R.F. O Connell, J. Sa. Phys. 53, 39 (988). [] X.L. L, G.W. Ford and R.F. O Connell, Am. J. Phys. 6, 9 (993). [3] M.B. Mensky, Connuous Quanum Measuremens and Pah Inegrals, (IOP Publshng Ld, Brsol and Phladelpha, 993) [] T. Drch, P. Hangg, G.-L. Ingold, B. Kramer, G. Schon, W. Zwerger, Quanum Transpor and Dsspaon, (Wley-VCH, Wenhem, 998) [5] U. Wess, Quanum Dsspave sysems, (World Scenfc, Sngapore, 999) [6] M.B. Mensky, Phys. Usp (3). [7] P. Hängg and G-L. Ingold, Chaos 5, 65 (5). [8] G.W. Ford and R.F. O Connell, Phys. Rev. B 75, 33 (7). [9] P. Hängg, G-L Ingold and P. Talkner, New J. Phys., 58 (8). [] A. O. Caldera and A. J. Legge, Physca A, 587 (983). [] J. Rau, Phys. Rev. 9, 88 (963). [] M. Bolserl, M. Rch and W.M. Vsscher, Phys. Rev. A, 86 (969). 8

19 [3] M. Rch and W.M. Vsscher, Phys. Rev. B, 6 (975). [] R. Glauber and V.I. Man ko, Zh. Eksp. Teor. Fz. 87, 79 (98) [Sov. Phys. JETP 6, 5 (98)]. [5] U. Zürcher and P. Talkner, Phys.Rev. A, 378 (99). [6] A. Chmondou and E.C.G. Sudarshan, Phys. Rev. A 77, 3 (8). [7] R. J. Rubn and W. L. Greer, J. Mah.Phys. 97, 686. [8] Venkaeshan Kannan, Abhshek Dhar, and J. L. Lebowz Phys. Rev. E 85, 8. [9] Kej Sao and Abhshek Dhar, Phys. Rev. E 83,. [3] Eseban A. Marnez and Juan Pablo Paz, Phys. Rev. Le., 36 (3). [3]Kuan-Lang Lu and Hs-Sheng Goan, Phys. Rev. A 76, 3 (7). [3]Juan Pablo Paz and Auguso I. Roncagla, Phys. Rev. Le., (8). [33]Juan Pablo Paz and Auguso I. Roncagla, Phys. Rev. A 79, 3 (9). [3]Jose Nahuel Freas and Juan Pablo Paz, Phys. Rev. A 85, 38 (). [35]Chung-Hsen Chou, Tng Yu and B.L. Hu, Phys. Rev. E 77 (8). [36]F. Galve, G.L. Gorgu, and R. Zambrn, Phys. Rev. A 8 () 67. [37]C. Hörhammer and H. Büner, Phys. Rev. A 77 (8) 35. [38]A.Ghesquere, I.Snaysky, F.Peruccone, Phys. Scrpa, 5 () 7. [39]A.Ghesquere, I.Snaysky, F.Peruccone, Phys. Le. A 377 (3) [] Fernando Galve, Leonardo A. Pacho n, and Davd Zueco, Phys. Rev. Le. 5, 85 () [] Andres F. Esrada, Leonardo A. Pachon, [hp://arxv.org/abs/.338]. [] I. A. Dorofeyev, Can. J. Phys. 9, 537 (3). [3] I. A. Dorofeyev, Can. J. Phys., 9(), 8- (). [] I. A. Dorofeyev, Can. J. Phys., (5). arxv:7.9v [quan-ph] [5] I. A. Dorofeyev, arxv:.5 (). [6] R. P. Feynman and A. R. Hbbs, Quanum mechancs and pah negrals (McGraw-Hll Book company, New-York, 965). [7] L. S. Shulmann, Technques and applcaons of pah negraon, (John Wley& Sons Inc., New-York, 98). [5] R. P. Feynman and F. L. Vernon, Annals of Phys. 8, 57 (). [5] H. Klener, Pah Inegrals n Quanum Mechancs, Sascs, Polymer Physcs, and Fnancal Markes (World Scenfc, Sngapore, 99). [5] G-L. Ingold, Coheren Evoluon n Nosy Envronmens Lecure Noes n Physcs 6, (). 9

20 [53] L. I. Mandelsam, A Complee Collecon of Works, Ed. by M.A. Leonovch, Vol. (AN SSSR, Moscow, 955). [5] S. P. Srelkov, Inroducon o he Theory of Oscllaons (Nauka, Moscow, 96). [55] D. I. Trubeskov, A. G. Rozhnev, Lnear Oscllaons and Waves (Fzmal, Moscow, ). [56] M. F. Sarry, Sov. Phys. Usp. 3 () 958 (99). Appendx A. Soluon o he sysem n Eq.(5) for classcal pahs wh exernal forces. The oal analyss of he coupled moon has been done on he bass of he exbooks [53-55]. In new varables we obaned from Eq.(5) wo pars of coupled equaons for classcal pahs X X X ( / M) X, (A) X X X ( / M ) X ( / M) f( ) / M ( / M ) f( ) / M. A general soluon of he frs par of homogeneous equaons can be found n [3]. In s urn, he second par of equaons n (A) for he classcal pahs ( ), of backward ampludes mus (A) be solved akng no accoun paral soluons () p and () p of nhomogeneous equaons ( ) B sn( )exp( ) r B sn( )exp( ) ( ), p ( ) r B sn( )exp( ) B sn( )exp( ) ( ), p (A3) where ( ) r ( ) ( ) r( ) B sn, sn, p p p p B ( rr ) ( rr ) ( ) r ( ) ( ) r ( ) B cos exp( ) co( ), f p f p p p ( r r )sn( ) ( r r ) ( ) r ( ) ( ) r( ) B cos exp( ) co( ), f p f p p p ( r r )sn( ) ( r r ) (A) (A5) where we use he bref noaons ( ) and ( ), ( k,) for he paral pk pk pk pk soluons. I should be noed ha he paral soluon can be chosen ha,. Correspondng way o oban such a soluon s descrbed jus below. p p p p

21 Le s desgnae n (A) F ( ) f( )/ M, F ( ) f( ) / M and represen he rgh par n (A) as follows Fk( ) Re{ Fk( )exp( ) d / }, ( k, ) (A6) We seek for he paral soluon of he sysem (A) n he same form pk ( ) Re{ gk ( )exp( ) d / }, ( k, ) (A7) Subsuon (A6),(A7) no he sysem n (A) yelds n ( / M ) f ( ) D ( ) f ( ) g ( ) A ( ) D( ), D ( ) D ( ) / MM ( / M ) f ( ) D ( ) f ( ) g ( ) A ( ) D( ), D ( ) D ( ) / MM (A8) (A9) where D( ) D ( ) D ( ) / M M, D ( ) ( ), ( k,), [ D] s he k k k Drac dela funcon, A ( ), are unknown funcons, whch properes can be found from he analycy of g ( ) and from boundary condons for, (), a,. The delafunconal erms n (A8),(A9) appear n he soluons due o reasonng descrbed, for nsance n [56]. From Eqs.(A8),(A9) we can see ha he vbraons of coupled oscllaors are deermned by boh of exernal forces va her couplng consan. The deermnan equaon D( ) D ( ) D ( ) / M M has dfferen n general smple p p poles as he roos of he equaon 3 a b c d where a, b, c, d are pure real numbers a ( ), b ( ), c ( ), d / M M. (A) The above equaon of he fourh order has four dfferen complex roos, and,3. Furher, we no need he cumbersome explc expressons for,. Takng no accoun (A7)-(A9) and properes of he Kronecker dela-funcon we oban j d A, ( )e j p, p( ) Re C,( )e Re, (A) j D( j ) where D ( j ) s he frs dervaon of he deermnan funcon a he roos j, ( j,...,), and ( / ) ( ) ( ) ( ) ( / ) ( ) ( ) ( ) C ( ) M F D F, C ( ) M F D F. D ( ) D ( ) / MM D ( ) D ( ) / MM (A)

22 I s clear from he second pars n (A) ha he sums mus be pure real. Ths allows pung some useful resrcons on he unknown complex funcons A, ( ). Takng no accoun ha D( ) D( ) and D( ) D( 3) because, and,3, and represenng he unknown funcons as A, A, A, we have where and j A, ( j )e e A, A, j D( j ) D( ) Re Re ( )e ( )e e Re A, ( )e A, ( 3)e, D( ) Re{ A ( )e A ( )e } Im{ A ( )e A ( )e },,,, [ A ( ) A ( )]cos [ A ( ) A ( )]sn,,,, {[ A ( ) A ( )]cos [ A ( ) A ( )]sn },,,, Re{ A ( )e A ( )e } Im{ A ( )e A ( )e },,,, [ A ( ) A ( )]cos [ A ( ) A ( )]sn,, 3,, 3 {[ A ( ) A ( )]cos [ A ( ) A ( )]sn },,, 3, 3, (A3) (A) (A5) Because n Eq.(A),(A5) we need only he pure real pars, he pure magnary pars n hese equaons can be subjeced by some convenen relaons beween he real and magnary pars of he funcons A, A ( ) A ( ), A ( ) A ( ),,,,, A ( ) A ( ), A ( ) A ( ).,, 3,, 3 (A6) Ths perms o rewre he sums n (A3) as follows j A, ( j )e e Re A, ( )cos A, ( )cos D( ) D( ) j j e A, ( )cos A, ( 3)cos, D( ) Then, we can pu addonal resrcons A ( ) A ( ), A ( ) A ( ), and (A7) A ( ) A( ), A ( ) A ( ), whch follow from smlar groundngs. Oher choce of 3 3 relaons leads o rval denes. Usng he obaned relaons we have he rajecores n he followng forms d e e p C D( ) D( ) ( ) Re ( )e ( ) ( ), (A8)

23 where d e e p C D( ) D( ) ( ) Re ( )e ( ) ( ), (A9) ( ) A( )cos A ( )sn, ( ) A( )cos A ( )sn 3 (A) Then, sasfyng he condons () and ( ) we fnd he unknown funcons,, A ( ), A ( ), A ( 3), A ( ) and oban he seekng for paral soluons d e sn( ) sn( ) p( ) Re C( ) e C ( ) C ( ) sn( ) sn( ) e cos( ) e cos( ) cos( ) sn( ) C () cos( ) sn( ) C () sn( ) sn( ) d e sn( ) sn( ) p( ) Re C( ) e C ( ) C ( ) sn( ) sn( ) e cos( ) e cos( ) cos( ) sn( ) C () cos( ) sn( ) C () sn( ) sn( ) where (A) (A3) d d C () Re C ( ) Re C ( ), d d C ( ) Re C ( ) e Re C ( ) e. (A) Fnally, we wre a general soluon of he homogeneous sysem of equaons (A) sasfyng boundary condons X ( ) w ( )sn( )exp( ) w cos( )exp( ) w ( ) r sn( )exp( ) w r cos( )exp( ), 3 (A5) X ( ) w ( ) r sn( )exp( ) w r cos( )exp( ) w ( )sn( )exp( ) w cos( )exp( ), 3 where X f r X f X r X X r X w ( ) exp( ) co( ), w, ( r r )sn( ) ( r r ) ( r r ) X f r X f X r X X r X w3 ( ) exp( ) co( ), w. ( r r )sn( ) ( r r ) ( r r ) A general soluon of he nhomogeneous sysem of equaon (A) sasfyng he condons (A6) (A7), (), and, () f, s as follows 3

24 ( ) v ( )sn( )exp( ) v cos( )exp( ) v ( ) r sn( )exp( ) v r cos( )exp( ) ( ), 3 p ( ) v ( ) r sn( )exp( ) v r cos( )exp( ) v ( )sn( )exp( ) v cos( )exp( ) ( ), 3 p (A8) (A9) where v r r v r f f ( ) exp( ) co( ), ( r r )sn( ) ( r r ) ( r r ) f r f r r v3( ) exp( ) co( ), v, ( r r )sn( ) ( r r ) ( r r ) and paral soluons (), are from (A),(A3). p p (A3) Appendx B. Temporal funcons relaed o Eq.() Some of he emporal funcons n Eq.() can be found n [5]. Here we repersen oher funcons whch are necessary n calculang of a covarance marx relaed o he case of drven coupled oscllaors. Below we keep n mnd ha g( ) g(,, T, T ), g ( ) g (,, T, T), a ( ) a (,, T, T ), bex ( ) bex (,, T, T), Aex ( ) Aex (,, T, T), Bex ( ) Bex (,, T, T) ex ex and, all he funcons n Eq.() are pure real ( D D ) e ( C / a ) Z Y g ( ), g ( C a) a( C a) ( D 6) Z Y ( D )( D D ) e e ( C / a ) Z Z Y Y ( ), ( C a) a( C a) ( D 6) Z Y ( D 3 8) e5 ( C / a) Z3 Y g(), ( C a ) a ( C a ) ( D ) Z Y 6 A E e ( C / a ) Z Y g ( ), 3 ( C a) a( C a) ( D 6) Z Y E B E e e ( C / a ) Z Z Y Y g ( ), ( C a) a( C a) ( D 6) Z Y A B e ( C / a) Z5 Y3 g (), ( C a ) a ( C a ) ( D ) Z Y 6 (B) (B)

25 ( D ) E ( D D ) e e ( C / a ) Z Z Y Y g ( ), ( C a) a( C a) ( D 6) Z Y ( D D ) E ( D ) e e ( C / a ) Z Z YY g () ( C a) a( C a) ( D 6) Z Y ( D D ) B ( D D ) e e ( C / a ) g () ( C a) a( C a) ( D 6) Z Z Y Y Z Y 5 3 5, ( D ) B ( D ) e e ( C / a ) Z Z YY 3 g (). Y ( C a) a( C a) ( D 6) Z The funcon whch ake no accoun of exernal forces are a b ex ex ( D D ) ( e ( D )) () ( C a) [ a( C a) ( D 6) ] Z ( [ On]) ( y [ Ro]) ( y Z ), Z Y Y Z ( D 3 8) ( e5 ( D 6)) () ( C a ) [ a ( C a ) ( D ) ] 6 Z ( [ On]) ( y [ Ro]) ( y Z ), Z Y Y Z M( ) r M ( ) r r M ( ) r M ( ) ( E3) Aex () ( r r )sn( ) exp( ) ( r r )sn( ) exp( ) ( C a) z ( [ On] ) ( e )( D ) ( Y ) Z ( [ Ro ] 6 ), Z 6 Z [ a( C a) ( D 6) ] Y r M ( ) M ( ) r r M ( ) r M( ) ( B ) Bex () ( r r )sn( ) exp( ) ( r r )sn( ) exp( ) ( C a) z ( [ On] ) ( e )( D ) ( Y ) Z ( [ 6 Ro] ), Z Z [ a( C a) ( D 6) ] Y E e3 ( D 6) where On ( C a ) a ( C a ) ( D ) 6, and ( D D ) e ( D ) Z Ro [ On]. 6 6 ( C a) a( C a) ( D 6) Z ( f r f )[ M( ) r M ( )] ( f r f )[ M ( ) r M ( )] ( ), ( r r )sn( )exp( ) ( r r )sn( )exp( ), (B3) (B) (B5) (B6) 5

26 ( ) co( ) M ( ) N ( ) r r co( ) M ( ) r r N ( ) ( rr ) r co( ) M ( ) r N( ) r co( ) M ( ) r N ( ), ( ) r co( ) M ( ) r N ( ) r co( ) M ( ) r N ( ) ( rr ) r r co( ) M ( ) r r N( ) co( ) M ( ) N( ), (B7) where M ( ) d f ( )sn( ) exp( ), k k k N ( ) d f ( )cos( ) exp( ), ( k, ) k k k M ( ) d f ( )sn( ) exp( ), k k k N ( ) d f ( )cos( ) exp( ), ( k, ) k k k (B8) Below he funcons whch are no nvolved n our numercal calculaons, bu hey deermne he non-herman marx n Eq. (). a ex () e V ( C a ) ( z V ) e ( D ) [ a ( C a ) ( D ) ] Z a ( C a ) ( D ) Y ( e e Z / Z )( C a ) [ V V ], ( ) ( ) b ex 3 6 Y a C a D 6 () ev ( C a ) [ a ( C a ) ( D ) ] Z a( C a) ( D 6) Y ( e e Z / Z )( C a ) [ V V ], ( ) ( ) A ex Y a C a D 6 () ( z V ) e ( D ) ( e V ( C a )) ( ZV) e ( D ) [ a ( C a ) ( D ) ] Z a ( C a ) ( D ) ( y ) ( e e Z / Z )( C a ) [ V V ]; ( ) ( ) B ex Y a C a D 6 () ( e5v ( C a)) ( Z3V ) e3 ( D 6) [ a ( C a ) ( D ) ] Z a ( C a ) ( D ) ( y ) ( e e Z / Z )( C a ) [ V V ]; ( ) ( ) 3 6 Y a C a D (B9) (B) Juxaposon of a (), b (), A (), B () wh a (), b (), A (), B () shows ha n ex ex ex ex mos cases we mus evenually compare he funconsv, wh,. Our verfcaon shows ha ex ex ex ex V,,, and we dd no wre ou hem here, because her rrelevance. 6

27 Appendx C. Mean values, varances and covarances of coordnaes and momena of drven coupled oscllaors For furher convenence we nroduce desgnaons 8 g( ), g ( ), 8 g( ), and A () A, B () B, a () a, b () b for brevy. ex ex ex Then he oal Herman densy marx wh exernal forces s ex ( X f, X f, ; f, f,) C( )exp ( ) X f / 8 ( ) X f X f / ( ) X f / 8 exp g ( ) f g ( ) f f g ( ) f exp g ( ) X f f g ( ) X f f g ( ) X f f g ( ) X f f exp Aex ( ) f B ex ( ) f exp a ex ( ) X f b ex ( ) X f, (C) where X f x f y f and f xf yf, (, ). I s known ha n order o oban mean values of observables of one oscllaor we should o reduce he oal densy marx wh respec o he observables of oher oscllaor. For example or wh use of he oher varables ( x, ; y,) lm { ( x, x, ; y, y,)} dx, (C) f f y f f f f f f xf ( X, ;,) lm { ( X, X, ;,,)} dx, (C3) f f f f f f f f and a smlar formulae for he second oscllaor. The reduced densy marxes are as follows 8 ( X, ;,) C exp[ ax ( / 8) X A g X f f f f f f f (b X g ) / 8 g ], f f f 8 ( X, ;,) C exp[ ax ( / 8) X B g X f f f f f f f (a X g ) / 8 g ]. f f f, (C) (C5) The normalsaon consan C () can be found, for nsance from Eq.(C) or Eq.(C5) lm { ( X, ;,)} dx lm { ( X, ;,)} dx f f f f f f f f. (C6) For cross-correlaed values we need anoher normalsaon condon as follows lm { ( X, X, ;,,)} dx dx, (C7) f f For any case he normalsaon consan s f f f f f f 7

28 ( b ab a ) C ( ) exp. (C8) Afer hs we have everyhng o calculae all he mean values, varances and covarances as usual L lm { Lˆ ( x, ; y,)} dx, f f f yf xf L lm { Lˆ ( x, ; y,)} dx, f f f yf xf L lm { Lˆ ( x, x, ; y, y,)} dx dx, ( m, n, ) mn yf xf mn f f f f f f yf xf, (C9) where Lˆ xˆ pˆ xˆ pˆ,(, ), and L ˆ { xˆ xˆ, pˆ pˆ, xˆ pˆ, pˆ xˆ }, ( m, n,), {,,( ),( ) } mn m n m n m n m n where pˆ / x. m m Usng above wren formulas we fnd all he characerscs. For he frs oscllaor he mean value of he coordnae ( b a ) x (), (C) for he mean value of he coordnae n square for he dsperson of he coordnae for he mean value of he momenum b (8 ab ) ( a ) () ( ) x x ( ) x ( ) ( ) ( ) ( ) a g p g b g g for he mean value of he momenum n square, (C), (C) A, (C3) p ( ) ( ) {6 b ( g g ) 6 a ( g g ) 8 aa( g g )( ) 8 b( g g ) [ a( g g ) A( )] ( )[8gg g g A ( )] ( ) g} for he dsperson of he momenum, (C) ( g g g g ) p ( ) p ( ) g, (C5) for he smmersed mean value of he cross covarance for he frs oscllaor 8

29 [ x p ( ) p x ( )]/ ( ) { b ( g g ) a ( g g ) ( g g )( ) ba( ) 3 b[ A ( ) a( g g g )]} for he commuaor of he frs oscllaor, (C6) x p ( ) p x ( ). (C7) For he second oscllaor he mean value of he coordnae ( a b ) x (), (C8) for he mean value of he coordnae n square for he dsperson of he coordnae for he mean value of he momenum x a (8 ab ) ( b ) () ( ) x ( ) x ( ) ( ) ( ) ( ) a g p g b g g for he mean value of he momenum n square, (C9), (C) B, (C) p ( ) ( ) {6 b ( g g ) 6 a ( g g ) 8 ab( g g )( ) 8 b( g g ) [ a( g g ) B( )] ( )[8g g g g B ( )] ( ) g} for he dsperson of he momenum, (C) ( g g g g ) p ( ) p ( ) g, (C3) for he smmersed mean value of he cross covarance for he second oscllaor [ x p ( ) p x ( )]/ ( ) { b ( g g ) a ( g g ) ( g g )( ) ab( ) 3 b[ B ( ) a( g g g )]} for he commuaor of he second oscllaor, (C) x p ( ) p x ( ), (C5) for he coordnae covarance of boh oscllaors 9

30 ab( ) ( a ) b 3 () ( ) x x for he momenum covarance of boh oscllaors, (C6) for he oher covarances of boh oscllaors p p ( ) p p ( ), (C7) x p ( ) p x ( ) x p ( ) p x ( ). (C8) I s easy o oban from above wren formulas all elemens of he covarance marx for he case of zero exernal forces by pung A B a b. In hs case all he mean values are equal zero x ( ) x ( ) p ( ) p ( ). Tha s why he formulas have a smpler vew. Namely, for he dsperson of he coordnae of he frs oscllaor x () for he dsperson of he momenum of he frs oscllaor, (C9) ( g g g g ) p( ) g, (C3) for he smmersed mean value of he cross covarance for he frs oscllaor for he commuaor of he frs oscllaor [ x p ( ) p x ( )]/ for he dsperson of he coordnae of he second oscllaor ( g g ), (C3) x p ( ) p x ( ), (C3) x () for he dsperson of he momenum of he second oscllaor, (C33) ( g g g g ) p( ) g, (C3) for he smmersed mean value of he cross covarance for he second oscllaor [ x p ( ) p x ( )]/ for he coordnae covarance of boh oscllaors x x () for he commuaor of he second oscllaor ( g g ), (C33), (C35) 3

31 x p ( ) p x ( ), (C36) and oher covarances of boh oscllaors are equal zero. Fnally, we have oal se of 6 marx elemens for he covarance marx boh n cases wh and whou exernal forces and mean values of observables. Fgure capons: Fgure. A skech of he problem suded: a) - wo ndependen reservors,, and wo uncoupled seleced oscllaors; b) - he couplngs among all oscllaors are swched on a ; c) he frs exernal force s appled o he frs coupled oscllaor a ; d) he second exernal force s appled o he second coupled oscllaor a. Fgure. Normalzed mean values of he coordnaes x ( ) /, (,) of he frs a) and second b) oscllaors versus a me, where / M, (,) usng Eq.(C, C8). Ths fgures correspond o he case of no couplng, and oal equlbrum, when T T 3K. The emporal dynamcs of exernal forces f (), n accordance wh Eq.(6) s shown beween of hese graphs. Fgure 3. Normalzed mean values of he coordnaes x ( ) /, (,) of he frs a) and second b) oscllaors versus a me usng Eq.(C, C8) a relavely srong couplng / MM.3beween of oscllaors. These fgures also correspond o he case of he oal equlbrum, whent T 3K. Fgure. Normalzed mean values of he coordnaes x ( ) /, (,) of he frs a) and second b) oscllaors versus a me usng Eq.(C, C8) a.3 beween of oscllaors n he case ou of oal equlbrum n he sysem, when T 3K and T 9K. 3