Quantum Theory of Open Systems Based on Stochastic Differential Equations of Generalized Langevin (non-wiener) Type

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1 ISSN Joural of Exprmal ad Thorcal Physcs 0 Vol. 5 No. 3 pp Plads Publshg Ic. 0. Orgal Russa Tx A.M. Basharov 0 publshd Zhural Esprmal o Torchso Fz 0 Vol. 4 No. 3 pp ATOMS MOLECULES OPTICS Quaum Thory of Op Sysms Basd o Sochasc Dffral Equaos of Gralzd Lagv (o-wr) Typ A. M. Basharov Naoal Rsarch Cr Kurchaov Isu pl. Aadma Kurchaova Moscow 38 Russa -mal: basharov@gmal.com Rcvd Novmbr 8 0 Absrac I s show ha h ffcv Hamloa rprsao as s formulad auhor s paprs srvs as a bass for dsgushg a broadbad vrom of a op quaum sysm dpd os sourcs ha drm rms of h saoary quaum Wr ad Posso procsss h Marov approxmao h ffcv Hamloa ad h quao for h voluo opraor of h op sysm ad s vrom. Gral sochasc dffral quaos of gralzd Lagv (o-wr) yp for h voluo opraor ad h c quao for h dsy marx of a op sysm ar obad whch allow o o aalyz h dyamcs of a wd class of localzd op sysms h Marov approxmao. Th ma dscv faurs of h dyamcs of op quaum sysms dscrbd hs way ar h sablzao of xcd sas wh rspc o collcv procsss ad a addoal frqucy shf of h spcrum of h op sysm. As a llusrao of h gral approach dvlopd h phoo dyamcs a sglmod cavy whou losss o h mrrors s cosdrd whch coas dcal racavy aoms coupld o h xral vacuum lcromagc fld. For som aomc dss h phoos of h cavy mod ar locd sd h cavy hus xhbg a w phomo of radao rappg ad o-wr dyamcs. DOI: 0.34/S INTRODUCTION I [4] I sablshd w o-wr faurs of h dyamcs of aomc sysms for varous raco modls of a smbl of dcal aoms wh h surroudg phoo-fr vacuum lcromagc fld. Dpdg o h al sa of h smbl of dcal aoms s raso o a sa wh lss (by o) xcao h aomc sysm xhbs xpoal bhavor dscrbd by h fuco xp{γγ W N a } wh h addoal o-wr facor γ W whch may vash for a cra umbr N a of aoms. Hr γ s h cosa of ordary radao dcay of a aom ad h sas of h aomc smbl ar assumd o b symmrc wh rspc o h prmuao of aoms. I h cas of a sgly xcd aomc smbl (h so-calld W sa [5 6] whch s a xampl of a arfcal parcl wh a srog Sar raco [ 3]) h paramr γ W γ W s dfd by [ ( ) 3] ( ) ( ) ( cos( N a γ S )) ( ) ( N a η S ) γ W ( ) η S Π ( Ω ) ---- d () whr Ω s h frqucy of aomc raso wh marx lm d of h dpol mom opraor ad Π (Ω ) s a sadard paramr dfg h Sar shf of h lowr aomc lvl [7]. Th o-wr dyamcs s characrzd by wo ma w phoma. Th frs s h supprsso of collcv spoaous msso ad h sablzao of h xcd sa wh rspc o collcv rlaxao procsss. Ths occurs agas h bacgroud of Dc s suprradao phomo [8 9] whch s characrzd h cas of a smbl of aoms cocrad a small volum by h facor N a h ( ) γγ W xpo of xp{ N a }. Formula () shows ha for h crcal umbr of aoms h smbl whch s dfd by h codo N cr ( ) a η S π h raso ra from h xcd sa bcoms zro. Th raso bhd h sablzao of h xcd sa s h rfrc bw a ral raso (wh h msso of a phoo) o a quaum sa of h aomc smbl characrzd by a smallr umbr of xcaos ad vrual rasos whou chagg h quaum sa ad whou msso of phoos [0]. Vrual rasos ar drmd by h Sar raco of aoms wh h vacuum lcromagc fld. Bg a small paramr (of h scod ordr wh rspc o h couplg cosa o h vacuum fld) h Sar raco crass wh h umbr of parcls h smbl of dcal aoms [3 4] ad sars o play h abovmod rol for cra aomc dss. To oba h oscllaory dpdc () aalycally o should sum up h xpo all h dagrams ovr h Sar raco wh a phoo-fr 37

2 37 BASHAROV vacuum lcromagc fld by sadard mhods of quaum lcrodyamcs of rsoa mda ha ar basd o h KosaovPrlKldysh dagram chqu for Gr s fucos [3]. Ths procdur sms o b xrmly dffcul jus as h chqu volvg h BogolyubovBorGrKrwoodYvo chas [4 5] whch o should ruca h hrarchy of chas a a cra sp. Th mhod of [6 7] complly glcs h Sar raco whl h applcaos of h Zwazg mhod [8] wh smlar (o h oscllaory dpdc h xpo) rsuls ar uow o h auhor. I h paprs o h collcv msso of a sphrcally symmrc smbl of dcal aoms [9 0] h auhors dmosra by umrcal calculaos spcfc cass whr o ca spa of a paral supprsso of msso. Th scod w phomo ha characrzs h o-wr dyamcs s a addoal frqucy shf ha s du o rlaxao dyamcs ad has b complly arbud o h Sar raco rprsd as a quaum Posso procss [4 0]; hrfor hs phomo s closly rlad o h supprsso of collcv spoaous msso. Th addoal frqucy shf as a gralzd Lamb shf has b dscussd h rc publcaos [ ]; howvr hs phomo was o way assocad wh collcv spoaous msso. I [4 0] formulas smlar o hos gv abov wh xpo () ar qu asly obad aalycally by h auomac summao of f srs of prurbao hory h Marov approxmao wh h us of h chqu basd o o-wr-yp quaum sochasc dffral quaos (QSDEs). From h vwpo of h hory of quaum sochasc procsss [37] h Sar raco [4 0] s rprsd by a quaum Posso procss whos algbrac proprs xprssd rms of h HudsoParhasarahy algbra [3] allow o o rjc ay rprsaos of h dagram chqu ad o oba may rsg cass aalycal rsuls udr h ma assumpo ha h procsss ar Marova ad ha h aomc subsysm s cocrad a small volum. I hs papr w gralz h approach of [4 0] ad formula h prcpls of h gral dscrpo h Marov approxmao of varous localzd op sysms whos parcular cas s gv by h modls of [4 0] basd o o-wr yp QSDEs formulad h ffcv Hamloa rprsao. W drv a gral QSDE for h voluo opraor of a op sysm ad s broadbad vrom rprsd by a boso-yp quazd fld wh zro boso dsy. Usg h QSDE obad w df Ldblad opraors ha characrz h c quao for h dsy marx of a op sysm. W sablsh cosras o h applcao of h c quaos obad wh som hory; parcular w show ha s corrc o cosdr varous dsprso lms h c quaos. W apply h hory dvlopd o h dscrpo of h phoo dyamcs of a sgl-mod cavy whou losss o h mrrors ad wh racavy aoms coupld o h xral phoo-fr vacuum lcromagc fld. W show ha h phoo dyamcs of h cavy mod s also of o-wr yp for suffcly larg dsy of racavy aoms ad s characrsc faurs ar h supprsso of msso of phoos from h cavy (rappg of phoos) ad a addoal frqucy shf. Thus w dmosra ha h o-wr faurs of h dyamcs ar of qu gral aur ad may mafs hmslvs op subsysms of dffr yps for xampl subsysms of boso or frmo yp. Th papr s orgazd as follows. I Sco w dscuss h cocp of a op sysm. I Sco 3 w cosdr h mahmacal faurs of h dscrpo of op sysms h Marov approxmao. Sco 4 s dvod o h dscusso of h ffcv Hamloa rprsao. W mphasz h dffrc bw our dfo of h ffcv Hamloa ad h ow dfos ad cosdr corollars o our dfo ha ar mpora for cosrucg QSDEs. I Sco 5 w aalyz h srucur of h rms of h ffcv Hamloa whch ar rprsd by saoary quaum sochasc procsss Sco 6. I Sco 7 w oba h gral form of a QSDE of o-wr yp. Basd o hs gral form Sco 8 w drv c quaos of o- Wr yp for h dsy marx of a op sysm. I Sco 9 w prs a xampl of applyg h hory dvlopd o h sudy of h phoo dyamcs of a sgl-mod cavy wh orsoa racavy aoms. Sco 0 s dvod o h dscusso of h basc faurs of h o-wr dyamcs of op sysms by h xampl of a sgl-mod cavy. I Coclusos w cosdr h applcao lms of h hory ad s possbl gralzaos byod h assumpos mad whr som of h o-wr faurs of h hory may prss.. THE CONCEPT OF AN OPEN QUANTUM SYSTEM Th problms of quaum opcs such as h spoaous msso of a aom or of a smbl of dcal aoms h dyamcs of hgh-q mcrocavs wh losss o mrrors ad h raco of h phoos of a hgh-q cavy mod ad h xral broadbad lcromagc fld wh racavy aoms provd xampls of op quaum sysms ha rac wh h vrom [834]. Op sysms ar characrzd by a rahr wa raco wh h vrom so ha h da of a solad sysm ( h absc of raco wh h vrom) srvs as a good zro approxmao for a op sysm. I addo h vrom s assumd o b mulmod (broadbad) ad a ss homog- JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

3 QUANTUM THEORY OF OPEN SYSTEMS 373 ous so ha h ffc of a op sysm o h vrom ca b glcd. From h spcral po of vw s cov o rpr a op quaum sysm as a sysm ha cosss of svral subsysms wh a small umbr of dgrs of frdom ad dscr spcrum whras h vrom of h op sysm s rprsd by a much largr sysm or by a s of sysms wh a fly larg umbr of dgrs of frdom ad couous spcra (Fg. ). Th dffrc bw h dscrpos of o op quaum sysm ad aohr ls h opraor algbras ha rprs h subsysms of h op sysm ad s vrom. For sac h aomc subsysm s rprsd by h opraors E E j of projco o h aomc sas E characrzd by rgs E ad h orhogoal dcomposo of uy E E E j δ j. I may ypcal problms of olar ad quaum opcs h algbra of projco opraors s rducd o h agular momum algbra su() [30 35]. Th phooc subsysm of a op sysm s usually characrzd by h opraors of brh ( c c ) ad ahlao (c c ) of phoos of h cavy mod whch oby h HsbrgWyl oscllaor algbra: [ c c c c ]. I som problms h aomc ad phooc subsysms may ac as a sgl objc [36] dscrbd by h Karasv polyomal algbras [37 38]. Quaum dos wlls wrs ad ohr lcroc sysms as wll as mulmod phoo sysms for xampl local phoos aocrysals aomchacal rsoaors c. [394] may also srv as h subsysms of a op sysm. Of h vacuum lcromagc fld rprsd by h opraors of brh b q ad ahlao b q of phoos wh momum q rgy ω h dsprso law ω qc (hr c s h vlocy of lgh) ad h commuao rlaos [ b q b q' ] δ qq' plays h rol of h mulmod vrom h xampls of h hory of op quaum sysms a from quaum opcs. Th phooc vrom s also dscrbd by h algbra of boso opraors gv abov. I ohr xampls of op quaum sysms o ca cosdr as h mulmod vrom wh rspc o a gv aom ohr aoms of h sam or dffr d; hus h algbra of h opraors ha rprs h mulmod vrom may also b qu dvrs. Spcra of a op sysm S of h vrom Ev of a cavy aom 3. SPECIFIC FEATURES OF THE QUANTUM THEORY OF OPEN SYSTEMS Th dscrpo of a op quaum sysm mpls h drvao of a c quao (masr quao) for h dsy marx ρ S of h op sysm wh h ffcv Hamloa H EffS ad h rlaxao opraor Γˆ : dρ S [ ρ S H EffS ] Γˆ ρ S. () d Th c quao () udrls h furhr aalyss of physcal ffcs ad phoma. I [43] Ldblad showd ha udr qu gral assumpos h rlaxao opraor Γˆ has h form + Ω E E Ω c Γˆ ρ S - [ ρs H ShfS -L S L S ρ S -ρs L S S L L S ρ S S + L (3) whch s calld h Ldblad form; hr ar h socalld Ldblad opraors ad H ShfS s a addoal rm o h Hamloa of h sysm ha s assocad wh h dyamcs dscrbd by a couous quaum dyamc smgroup [47 43]. S I hs cas h rms coag h opraors L dscrb rlaxao rasos h op quaum sysm ad h opraor H ShfS drms h rgy lvl shfs du o h rlaxao rasos so S ha f L 0 for all h H ShfS 0. Sadard mhods [8 30] for drvg h c quao () volv a a cra sp of rasformaos assumpos of h yp of dcouplg of corrlaors ad glc of mmory ffcs. Appld a h arls sps of rasformao hs assumpos also characrz h so-calld Marov approxmao [303]. Howvr f w formula h Marov approxmao h orgal quaos for h oal ω L S ω a ω c Fg.. Schmac vw of h spcrum of a op sysm ad s vrom. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

4 374 BASHAROV Ψ S + Ev wav fuco of h op quaum sysm ad h vrom h h Schrödgr quaos for h wav fuco ad h voluo opraor bcom mahmacally drma [3 44]. L us brfly xpla hs mpora fac by a xampl of spoaous msso of a aom du o h raso from h xcd lvl E o h groud lvl E. I h lcrc-dpol approxmao [30 45] h Hamloa of h raco of a aom wh h vacuum lcromagc fld h raco rprsao ca b xprssd as usual as H I () E() d() whr E() Γ ωq b qλ qλ ω q q r + H.c. qλ s h lcrc fld srgh opraor of h vacuum fld obad by h sadard quazao of h lcromagc fld [45] d() d j Ω j E j E j s h opraor of h dpol mom of h aom ad qλ dos h u vcor of polarzao. I h cas of ordary hr-dmsoal spac Γ q (π ω q / 3 ) / whr 3 s h quazao volum ad ω q qc s h dsprso law. For h wav fuco Ψ S + Ev () ad h voluo opraor U() of h op sysm ad s vrom w hav sadard Schrödgr quaos --- d Ψ S + Ev () H I ()Ψ S + Ev () d Ψ S + Ev () U()Ψ S + Ev ( 0) ---U d () H I ()U () U( 0) S + Ev. d A soluo o Eqs. (4) ca b rprsd as a srs U() I - I + H ( ' ) d' - + ' 0 H I ( ' )H I ( '' ) d' d'' T - H I xp ( ' ) d' 0 (4) (5) whr s h opraor of ordrg m. As a rul srs (5) s calculad dffr approxmaos. For xampl sad of h orgal Hamloa H I () H I () o as h raco Hamloa H I () H RF () h roag-wav T approxmao [30 46] for a spoaous raso E E bw wo lvls of h aom. For a aom locad a h po r 0 glcg polarzao ffcs ad smplfyg h mod srucur of h fld [3 44] w ca xprss h raco opraor h roag-wav approxmao ( ohr rms a roag fram of rfrc) as H RF () dωγω ( )( b ω d ( ω Ω ) E E ( ) + b ω d ω Ω E E ). Th opraor H RF () coas oly hos rms of h orgal opraor H I () ha dscrb h raso of h aom from h xcd sa E E o h groud sa wh h msso of a phoo of frqucy ω (h rm wh h opraor E E b ω ) ad h rvrs procss volvg h absorpo of a phoo wh h raso of h aom o h xcd sa (h rm wh h opraor b ω E E ). Hcforh w apply h Marov approxmao. I h cas of H I () H RF () h Marov approxmao corrspods o h followg hr codos [ ].. A h al sa of m hr s o corrlao bw h sas of h aom Ψ S ( 0) ad h vacuum lcromagc fld Ψ Ev ( 0) ;.. h wav fuco of h op sysm ad vrom s facorzd Ψ S + Ev ( 0) Ψ S ( 0) Ψ Ev ( 0) whl h mods of h lcromagc fld ar sascally dpd of ach ohr (h propry of dla corrlao of mods): Ψ Ev ( 0) b ω b ω' Ψ Ev ( 0) ( ω)δ( ω ω' ) (6) Ψ Ev ( 0) b ω b ω' Ψ Ev ( 0) ( + ( ω) )δ( ω ω' ).. Th couplg paramr bw h op sysm ad h vrom whch s dfd hs cas by h quay Γ(ω) ad h phoo dsy (ω) h vrom do o dpd o h frqucy ad ar cosa: Γω ( ) Γ( Ω ) cos (7) ( ω) ( Ω ) cos. 3. Th lms of grao wh rspc o frqucy ar xdd from o +. Th frs codo maly guaras h dcouplg of corrlaors h scod h glc of mmory ffcs ad h hrd (oghr wh ohr codos) h rprsao of h opraors of raco bw h op sysm ad h broadbad vrom rms of saoary quaum sochasc procsss. I h abov-lsd codos h grals (5) bcom mahmacally drma: h valu of a JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

5 QUANTUM THEORY OF OPEN SYSTEMS 375 gral as h lm of gral sums dpds o h choc of pos h paro rvals of h grao doma a whch h grad s calculad [3 44]. Th corrc formulao of hs quaos s gv rms of QSDEs [3 44] whos formalsm allows o o oba h c quaos h smpls way. Th approach volvg QSDEs urs ou o b qu ffcv o oly for obag c quaos ad a rlaxao opraor problms of quaum opcs [ ] bu also for smulag h rajcory of a op sysm [ ]; morovr hs approach urs ou o b ffcv h hory of couous masurms [ ]. Howvr hr s subsaal drmacy (or complss) hr h radoal formulao [3 44] of h QSDE approach whch lads o a corrc rsul. Ths complss cosss h absc of clar codos udr whch h Marova crra ar formulad ad a QSDE s wr. As a rsul h QSDE wr for h orgal mos gral Hamloa H I () H I () whch dscrbs a op quaum sysm s vrom ad hr raco gvs rs o a zro rlaxao opraor [65] whl corrc rsuls ar obad h roagwav approxmao [3 44]. 4. THE EFFECTIVE HAMILTONIAN REPRESENTATION AS A BASIS FOR THE FORMULATION OF A QUANTUM STOCHASTIC DIFFERENTIAL EQUATION FOR THE EVOLUTION OPERATOR OF AN OPEN SYSTEM I [48] I suggsd ha h QSDE chqu should b appld o h ffcv Hamloa obad by a uary rasformao of h orgal oal wav fuco Ψ S + Ev () Q() Ψ S + Ev () Q() Q () rahr ha o h orgal mos gral Hamloa or o h parcular cas of h Hamloa h roag-wav approxmao. Th Schrödgr quao for h rasformd wav fuco of a op sysm ad s vrom --- d Ψ S + Ev () H ()Ψ S + Ev () d s drmd by h rasformd Hamloa H () Q() H I () Q() Q() --- d Q(). (8) d Nx accordac wh h cocp of a op quaum sysm w xpad h opraor Q() ad h rasformd Hamloa H () a prurbao hory srs h couplg cosas bw h op quaum sysm ad xral flds whos ordr s dcad blow by suprscrps [48]: Q() Q ( 0 ) ()+ Q ( 0 ) () + H () H ( 0 ) () + H ( 0 ) () + (9) + H ( 0 ) () + H ( 0 ) () +. Th uary symmry ha udrls h srs (9) s a hr propry of quaum hory ad has b appld spcfc problms sc h vry bgg of quaum mchacs [667]. Howvr h prcpls how h rms of h srs (9) ar dsgushd hav b dffr from hos proposd [48]. For xampl h auhors of [69] ad may ohr paprs cosdrd h cosrvao of oal rgy or h umbr of xcaos as h ma rqurm o h rms of srs (9). I [48 7] I formulad h ma rqurm o h rms of srs (9) as h absc of rms h raco rprsao ha vary rapdly m. I h cas of a op quaum sysm racg wh broadbad vrom such a smpl modfcao of h codo for dsgushg h rms of srs (9) lads o h cocps of h ffcv Hamloa ad h ffcv Hamloa rprsao whch grally spag ar dffr from h ow ffcv Hamloa hors [ ]. Morovr h codo for dsgushg h rms of srs (9) mpls a umbr of mpora corollars ha udrl h quaum hory of op sysms basd o QSDEs h ffcv Hamloa rprsao ad hghlgh h dffrc bw h ffcv Hamloa cosdrd h prs sudy whch s basd o h prcpls of [48 7] ad ohr ffcv Hamloas [ ]. Blow w ls four mpora corollars for h hory of op quaum sysms ha follow from h cosruco prcpls [48 7] of h ffcv Hamloa. Corollary. By a uary rasformao [48 7] (whou rapdly varyg rms h raco rprsao) h broadbad vrom s auomacally dcomposd o a famly of dpd broadbad quaum os sourcs h cssy of roducg whch was mod v [73]. Ths fac sms obvous h cas of spoaous wo-quaum msso a xral orsoa moochromac fld. Hr dpd os sourcs [48] ars from cosdraos of h wo-quaum rsoac hory [7 7]. Th phoos of hs sourcs a par boh ral wo-quaum rasos wh a quaum of a moochromac wav ad vrual wo-quaum rasos wh rur o h al lvl (Fg. ). Fgurs 3 ad 4 rprs slghly mor complx cass whch a subsysm of h op quaum sysm ha s o drcly coupld o h vrom (h aoms h cas of Fg. 3 ad h cavy phoos h cas of Fg. 4) vrhlss urs ou o b subjcd o h ffc of h vrom h scod ordr of prurbao hory bu a dffr rgo of h vrom spcrum coras o h subsysm ha s drcly ( h frs ordr of prurbao hory) JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

6 376 BASHAROV ω cl ω s ω s E ω cl E (a) (b) (c) racs wh h vrom (h cavy phoos h cas of Fg. 3 ad h aoms h cas of Fg. 4). Corollary. Th Marova codos cludg h dla-corrlao codo (of h yp of codo (6)) of h mods of h os sourcs should b formulad oly h ffcv Hamloa rprsao whch dos o coa rms ha rapdly vary m ( h raco rprsao). Ths prcluds h apparac of paradoxcal rsuls smlar o hos prsd [65] ad s maly coss wh h das of h quaum sochasc lm [74]. ω a ω a ω cl ω ω Fg.. Two-quaum rasos a aom a xral orsoa moochromac fld of frqucy ω cl ad a vacuum lcromagc fld. A uary rasformao [48] dcomposs h vacuum fld o hr dpd os sourcs wh h Sos ω (E E ) s ω cl (a) ad a-sos ω (E E ) a + ω cl (b) cral frqucs ad wh h frqucy ω ω cl (c). Th phoos of hs sourcs a par h opcally forbdd aomc raso E E ad duc Sar shfs of lvls (vrual rasos wh rur o h al lvl). Corollary 3. Th uary rasformao [48] (volvg h rqurm ha rapdly varyg rms b mssg h raco rprsao) mposs srg cosras o h furhr aalyss of h lm cass of h c quao (). I gral s corrc o cosdr () ad (3) varous dsprso lms as s do for xampl [ ]. Fgurs 3 ad 4 show ha a src rsoac h raco bw aoms ad h cavy mod (Δ 0) corrspods o o par of h frqucy spcrum of h vrom of h op sysm whras h dsprso lm (growh of Δ) volvs h mods of h vrom ha ar locad aohr par of h frqucy spcrum o h raco procsss. Th cras h dug Δ from rsoac lads o h fac ha facors of h yp xp(±δ) bcom rapdlyvaryg m facors h Hamloa h raco rprsao ad h accordg o Corollary s corrc o formula h Marov approxmao wh a QSDE ad h c quao dfd by hs Hamloa. Th approach of [48 7] rqurs ha ach dsprso lm should corrspod o s ow ffcv Hamloa wh appropra os sourcs (dffr from h sourcs of h src rsoac cas) ha df hr ow QSDEs ad h c quao. Th c quaos corrspodg o h cass rlad o ach ohr by h dsprso lm ar o rlad h gral cas by ay lm raso. I addo o h fudamal dffrc h aur of os sourcs h approach of [48 7] lads o corrc xprssos for h paramrs of h c quao dscrbg h dsprso lm ha dffr from aalogous paramrs obad udr passag o h dsprso lm h c quao orgally drvd for h cas of src rsoac. Fally h approach of [48 7] jusfs h roag-wav approxmao ad pos ou s applcao doma. I h cass of rsoa raco of a aom wh a cavy mod of frqucy Ω c or wh a wav pac wh carrr frqucy ω cl h dugs Δ Ω c Ω ad Δ Ω from h rsoac should b ω cl E Δ Ω c ω c Ω ω c ω E ω Spcrum of h vrom Fg. 3. Sgl-mod cavy wh losss o h mrrors ad wh racavy aoms ha ar orsoa wh h mod. A uary rasformao [48] dcomposs h vacuum lcromagc fld o wo dpd os sourcs wh cral frqucs Ω c (cocds wh h frqucy of h cavy mod) ad Ω (cocds wh h frqucy of h opcally allowd aomc raso E E ). I h frs ordr of prurbao hory h cavy phoos ar coupld o h os sourc wh cral frqucy Ω c. I h scod ordr of prurbao hory h aoms ar coupld o h os sourc wh cral frqucy Ω [4 5]. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

7 QUANTUM THEORY OF OPEN SYSTEMS 377 ω Ω ω c Ω c ω c E Δ Ω Ω c E ω Spcrum of h vrom Fg. 4. Sgl-mod cavy whou losss o h mrrors ad wh racavy aoms ha ar orsoa wh h mod ad ar coupld o h xral vacuum lcromagc fld. A uary rasformao [48] dcomposs h xral vacuum lcromagc fld o wo dpd os sourcs wh cral frqucs Ω c (cocds wh h frqucy of h cavy mod) ad Ω (cocds wh h frqucy of h opcally allowd aomc raso E E ). I h frs ordr of prurbao hory h racavy aoms ar coupld o h os sourc wh cral frqucy Ω. I h scod ordr of prurbao hory h cavy phoos ar coupld o h os sourc wh cral frqucy Ω c. small much lss ha all h characrsc frqucs of h problm. I h cas of raco of a aom wh a broadbad fld wh cral frqucy ω h dug Δ ω Ω from h rsoac should b assumd zro Δ 0 so ha aga s corrc o cosdr varous dsprso lms of h c quaos obad h roag-wav approxmao. Corollary 4. Th uary rasformao [48] uformly roducs o h quaum hory of op sysms QSDEs of gralzd Lagv (o- Wr) yp whch a subsaal rol s playd by h Sar raco bw a op sysm ad s vrom. For h frs m hs was dmosrad [4 0]. I [ ] a raso o h ffcv Hamloa (o h prcpls roducd [48 7] for dsgushg rms srs (9)) s characrzd as a raso o h ffcv Hamloa rprsao. I hs cas o ca spa of h ffcv Hamloa rprsao h Schrödgr ad Drac (raco) scaro whch ar obvously rlad o ach ohr [ ]. 5. THE STRUCTURE OF THE TERMS OF THE EFFECTIVE HAMILTONIAN Cosdr h rms of srs (9). L for dfss h frs (lfmos) dx h group of uppr dcs of h rms H ( j ) () xpaso (9) dca h ordr h couplg paramr of aoms o h broadbad fld of h vrom h scod dx dca h ordr h couplg paramr of a aom o h cavy mod ad h hrd h ordr h couplg paramr of a aom o a cohr pump fld. O may also cosdr usg appropra (ohr) dcs h ordrs h couplg paramr (o h mrror) of h cavy mod o h (grally dffr) broadbad fld of h vrom (for xampl h fourh dx) ad so o. I h lows ordrs of xpaso (9) w hav rms ha ar lar h couplg paramrs o h flds of h problm: H ( 0 0 ) H ( 0 0 ) ()H ( 0 0 ) ()H ( ) () () c.; rms ha ar blar h couplg paramrs: H ( 0 ) ()H ( 0 ) () H ( 0 ) () c.; ad rms ha ar quadrac h couplg paramrs: H ( 0 0 ) H ( 0 0 ) ()H ( 0 0 ) ()H ( ) () () c. Ohr rms ars mulphoo (wh hr or mor phoos) rsoacs [77] orsoa procsss [7 7 78]. Th rms ha ar lar h couplg paramrs dscrb sgl-quaum rsoacs wh h xchag of xcaos of dffr aurs boh wh h op sysm ad bw h op sysm ad h vrom (s Fgs. 59). Th rms of h ffcv Hamloa ha ar blar h couplg paramrs dscrb woquaum rsoacs ad h xchag of xcaos JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

8 378 BASHAROV Ω E E Fg. 5. Rsoa raco bw h aomc subsysm ad h xral broadbad lcromagc fld (aomc rlaxao). I s dscrbd by h rm H ( 0 0 ) () ( ω a Ω ) dω a Γ(ω a )d b ωa E E + H.c. h ffcv Hamloa whch h grao s prformd oly wh rspc o h frqucy of h os sourc wh cral frqucy ω a Ω. I s hs rqurm ha dsgushs H ( 0 0 ) () from h raco Hamloa H RF () h roag-wav approxmao. Ω E E Fg. 6. Rsoa raco bw h aomc ad phooc subsysms of h op quaum sysm (rgy xchag bw h subsysms). I s dscrbd by h rm H ( 0 0 ) ( ω c Ω ) () g c d b ωc E E + H.c. h ffcv Hamloa whch g c s h couplg paramr of a aom o a phoo mod (for xampl h cavy mod) of frqucy Ω c. Ω E E Ω c ω a E () ω ~ cl + H.c. ω cl Fg. 7. Rsoa raco bw h aomc subsysm ad h xral cohr (classcal) lcromagc fld (cohr pumpg). I s dscrbd by h rm H ( 0 0 ) () *()d ( ω cl Ω ) E E + H.c. h ffcv Hamloa whch () s a slowly varyg amplud of h lcrc fld srgh E of frqucy ω cl. of dffr aurs boh wh h op sysm ad bw h op sysm ad h vrom (s Fgs. 35). Hr w do o xplcly prs hs rms (s [ ]) bu cosdr som parcular cass blow hs sco ad Scos 6 ad 8. Th rms ha ar quadrac h couplg paramrs dscrb wo-quaum rsoac rasos h op sysm h Lamb ad Sar shfs of lvls of h subsysms of h op sysm ad h raco bw h subsysms (dcal aoms). Th ypcal form of h opraor of h Sar shf of lvls s gv by h xprssos H ( 0 0 ) () H C Sar () gc b c b c Π ( Ω c ) E E H ( 0 0 ) () () Π ( ν) E E. I hs cas h ypcal form of h opraor of h Sar raco bw h aoms ad a os sourc wh cral frqucy ω ha s slcd from h broadbad vrom by h raso o h ffcv Hamloa rprsao [ 7 36] s xprssd as H S( ) () ω ω ' Γω ( )Γ( ω ' )b ω b ω ' ω ω ' d -( Π ( ω ) + Π ( ω ' )) E E. Hr w usd h sadard oao of opcal rsoac hory [7 7]: Π ( ν) j d j ν ν ω j W spcally srss ha h scod-ordr rms quazd flds for xampl scod-ordr rms h couplg paramrs o h broadbad fld of h vrom coa opraors ha dscrb h Lamb shf H Lamb h raco of aoms wh ach ohr E x ad h Sar raco of aoms H Sar () [ 7 36]. Th opraors ha dscrb scod-ordr procsss ar a ss uvrsal ad characrz a vary of op quaum sysms. Th opraor of h Lamb shf of aomc lvls dos o coa h opraors of h broadbad fld ad s asly lmad from furhr aalyss by a appropra uary rasformao [ 36]. Th opraor of raco bw aoms lads o phas modulao ad of dos o drcly affc h ra of radav rasos bw aomc lvls [] xcp for h cas wh h combao ω j H ( 0 0 ) () H Lamb + H Sar () + H Ex ( ) JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

9 QUANTUM THEORY OF OPEN SYSTEMS 379 phoma [50] l radav collsos [ ] ar a o cosdrao. I may cass h fac whhr or o hs opraor s a o accou ca b cosdrd as a characrsc faur of a modl. Th opraors of h Sar raco bw aomc smbls ad lcromagc flds drm h dscv faur of h gralzd Lagv (o- Wr) yp of QSDEs h hory of op quaum sysm. L us cosdr hm grar dal. Th procsss dscrbd by h Sar raco opraors ca b rprsd as a vrual msso of a phoo followd by s absorpo so ha a aom rmas a h sam rgy lvl (Fg. 0). Ths procsss lad o h shf of h rgy of hs lvl whch s ow olar opcs as h hgh-frqucy Sar ffc [7 30 8] ad producs a sgfca ffc o h raco ad rasmsso of s opcal pulss a mdum [7]. Th ffccy of hs procsss s drmd by h paramr Π (ω) ad dpds o h prsc of a quas-rsoa lvl (h lvl E q Fg. 0) whch may or may o b prs dpdg o frqucy. Thrfor h Sar raco may hav dffr valus for dffr pars of h spcrum of h broadbad vrom of a op sysm. Th Sar raco opraors ar addv wh rspc o all lcromagc flds ad os sourcs prs a problm; addo o h rms H ( 0 0 ) () h Sar raco opraors ar also H ( 0 0 ) rprsd by h rms () (h Sar raco wh a cavy mod) ad h rm H ( ) () (h classcal hgh-frqucy Sar ffc). I a smbl of dcal aoms h Sar raco has h followg mpora spcfc faur: crass as h umbr of aoms h smbl crass [3 4]. Ths s llusrad by h followg gralzaos of h formulas prsd abov. I a classcal lcromagc fld (wh lcrcfld srgh E() () xp( l r ω cl ) + c.c. carrr frqucy ω cl cl c ad a slowly varyg amplud ()) h Sar raco opraor for a aomc smbl has a smpl form [7 7] H ( ) () () Π ( ω cl ) E () E (). If hr ar svral classcal flds wh carrr frqucs ω ad ampluds (s) cl () () s h ω c H ( ) () s () s Π ( ω cl ) E Ω c Fg. 8. Rsoa raco bw h phooc subsysm ad h xral broadbad lcromagc fld (losss o h mrror). I s dscrbd by h rm H ( ) () dω c g(ω c ) b ωcbc ( ω c Ω ) + H.c. whch h grao s prformd oly wh rspc o h frqucy of h os sourc wh cral frqucy ω c Ω c. Ω c ω cl Fg. 9. Rsoa raco bw h phooc subsysm ad h xral cohr (classcal) lcromagc fld (cohr pumpg of h mod). I s dscrbd by h rm H ( ) () g c b c () ( Ω c ω cl ) + H.c. h ffcv Hamloa. E j E q E E l Fg. 0. Vsual rprsao of h Sar raco as a procss of vrual msso ad absorpo of h sam phoo of frqucy ω whou ay ral quaum raso. Th ffccy of vrual rasos from lvl E ad bac dpds o h prsc of a quas-rsoa lvl E q. ω s ω () () () E (). JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

10 380 BASHAROV Hr h uppr dx N a of h quaum sa vcors of a aom umras h aoms h smbl. For h quazd fld of a os sourc wh cral frqucy ω (wh polarzao s glcd ad h mod srucur s smpl) h Sar raco opraor H S() () for a aomc smbl localzd a doma wh dmsos much lss ha h characrsc wavlgh ca b obad from h classcal cas by h subsuos [] ω ' Γω ( ' )b ω ' ω ' so ha (0) If h xral broadbad fld s rprsd by svral os sourcs wh cral frqucs h For h quazd fld of a cavy mod wh frqucy Ω c ad h couplg paramr g c h subsuos yld Π * ( ) ω cl ω Γω ( )b ω ω -( Π ( ω ) + Π ( ω ' )) H S( ) () ω ω ' Γω ( )Γ( ω ' )b ω b ω ' ω ω ' d -( Π ( ω ) + Π ( ω ' )) E () E H Sar () H S( ) (). ( ) For svral mods w also oba a sum of smlar opraors. I hs cas Hr h subscrp dcas h opraors ha ar smlar o hos roducd abov for H ( 0 0 ) () bu rsul from h raco of h aomc subsysm wh h quazd fld of h cavy mod. No also ha h rms () () c. ca also b rprsd as h opraors of h Sar raco udr whch a phoo of o fld s vrually md (absorbd) whl a phoo of aohr fld s absorbd (md). For xampl g c c c Ω c (). ω * g c c c Ω c Π j ( ω cl ) Π j ( Ω c ) H C Sar H ( 0 0 ) g c c c c c Π j ( Ω c ) E j () E j j Lamb () H C (). Sar + H C + H Ex C. H ( 0 ) H ( 0 ) H ( 0 ) H ( ) () ca b obad from () by h subsuos g c c c Ω c * ω c Γω ( c )b ωc ω Π ( ω cl ) -( Π ( ω c ) + Π ( Ω c )) ad by addg h Hrma cojuga rm;.. H ( 0 ) () g c c c ω c Γω ( c )b ωc ω c Ω c -( Π ( ω c ) + Π ( Ω c )) E () E () + H.c. Hr s also clarly s ha hs procsss h ffcv Hamloa rprsao slc a os sourc from h broadbad fld of h vrom. H ( 0 ) I ordr ha h rm () should o coa facors ha rapdly vary m h rm xp(ω c Ω c ) should o rapdly osclla;.. h cral frqucy ω c of h os sourc whos phoos ar mard by a subscrp c should cocd wh h frqucy Ω c of h cavy mod ω c Ω c (s Fg. 4). Thus h Sar raco of h aomc (lcroc) subsysm of h op sysm sablshs a couplg bw flds of dffr aurs. 6. REPRESENTATION OF THE TERMS OF THE EFFECTIVE HAMILTONIAN IN TERMS OF STATIONARY QUANTUM STOCHASTIC PROCESSES O accou of h srucur of h rms ha ar lar h couplg paramr o h broadbad fld of h vrom H ( 0 0 ) () H ( 0 ) () H ( 0 ) () c. s cov o roduc h opraors (h dmsolss m varabls ar spcfd Sco 9) b a () ω a ( ω a Ω ) d b ωa π B a () d'b a ( ' ) 0 () db a () B a ( + d) B a (). Th subscrp a characrzs h opraors roducd as opraors of a os sourc wh cral frqucy ω Ω. I sp of h f lms of grao a wh rspc o frqucy () oly a arrow fr- Ulss w smplfy h mod srucur hs frqucy grals ar fac grals wh rspc o h projcos of h phoo momum [8] so ha w ca aurally xd h dfo of h opraors apparg (9). JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

11 QUANTUM THEORY OF OPEN SYSTEMS 38 qucy doma ar h cral frqucy corbus o physcally obsrvabl phoma [30]. Dfg h grals xpaso (4) as grals h ss of Io [ ] ϕ( ' ) db ( a ' ) lm ϕ( )( B a ( ) B a ( )) 0 N N (hr h lm s ma h ma-squar ss) w oba h followg opraor algbra [3 44] (h GardrColl algbra): db a ()db a () ( + a )d ()db a () a d db a ()db a () db a ()db a () db a ()d db a db a ()d dd 0. () Hr a ( ω a ) s h phoo dsy of a os sourc a s cral frqucy. Th abov rlaos should b udrsood as h qualy of appropra grals wh so-calld oacpag opraors as grads [3]. Opraors () rprs quaum Wr procsss B a () ad B a () (s blow ad [5 6] for mor dal) for a os sourc wh cral frqucy ω a. Th subscrp dcas ha h opraors ar rlad o h grag os sourc dsgushd by h uary rasformao [48 7] of h orgal Hamloa. I h Marov approxmao h ffcv Hamloa rprsao h Wr procsss B () ad B () of all os sourcs obad by h uary rasformao [48 7] df opraors h ffcv Hamloa ha ar lar wh rspc o h couplg cosa o h broadbad flds: ( H ( 0 0 ) () + H ( 0 ) () + H ( 0 ) () + )d ( Y db () + Y db () ). (3) Th sum ovr h subscrp mpls summao ovr all dpd os sourcs roducd by h ffcv Hamloa rprsao; all producs of ay wo dffrals of Wr procsss for dffr os sourcs vash. Hr Y do opraors ha characrz h op quaum sysm h procsss volvg h h os sourc. Ths opraors ar proporoal (lar or blar) o h couplg paramrs o boh h xral broadbad flds ad ohr flds h problm. W assum ha all h quas hr ad h x sco ar proprly rducd o a dmsolss form; howvr hy ar dod by h prvous symbols. A opraor lar h couplg paramr o a broadbad fld was frs rprsd by h dffrals of quaum Wr procsss [44] for h raco Hamloa H I () H RF () h roagwav modl; h Marov approxmao for hs Hamloa s xprssd rms of h codos prsd Sco 3. W srss ha srcly spag a lar approxmao h couplg cosa should corrspod o h opraor H I () H ( 0 0 ) () [ ] wh zro dug from rsoac Δ ωa Ω 0. I [ ] quaum Wr procsss wr roducd o dscrb varous racos wh h vrom ha ar lar wh rspc o h couplg paramrs o broadbad flds ad ca b rprsd by rms of h form H ( 0 ) () H ( 0 ) () c. h ffcv Hamloa. I [ ] for varous modls of couplg bw a op sysm ad s vrom I showd ha h Sar raco s dscrbd h Marov approxmao by a quaum Posso procss whch s roducd as Λ a () d'b a ( ' )b a ( ' ) 0 dλ a () Λ a ( + d) Λ a (). (4) Th ag o accou all os sourcs w oba h followg rprsao: H Sar ()d dλ (). (5) Hr do opraors ha characrz h aomc subsysm of h op quaum sysm scod-ordr procsss volvg h h os sourc. Ths opraors ar proporoal o h squard couplg cosa o h h os sourc ad ar dagoal. Th quas dλ () ar calld h crms (dffrals or h Io dffrals) of h quaum Posso procsss Λ (). Somms hy ar calld smply quaum Posso procsss. Hr o should bar md ha coras o h classcal cas quaum Wr ad Posso procsss ar rlad o ach ohr. Wll-dfd quaum Wr W() ad Posso P() procsss ar dfd by h opraors roducd abov (ha ar appropraly rducd o a dmsolss form ad rlad o h sam os sourc) as [5 6] W() B() + B () P() Λ() + ( B () B() ). JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

12 38 BASHAROV Th opraors B() B () ad Λ() (or db() db () ad dλ()) ar also calld lmag grag ad gaug sochasc procsss [5 6]. Th crms of hs opraors oby h Hudso Parhasarahy algbra [3]: dλ()dλ () dλ() db()db () d dλ()db () db () db()dλ () db() dλ()db () dλ()d db ()dλ () db ()db () db ()d (6) db()d dd 0. Hr s assumd ha all h opraors ar rlad o h sam os sourc wh zro phoo dsy. Th producs of wo crms of sochasc procsss for dffr os sourcs vash bcaus dffr os sourcs ar dpd of ach ohr. No ha f rprsao (5) coas h crm of h Posso procss of som sourc whl h ffcv Hamloa s dpd of h crm of h Wr procss of h sam sourc h hs Posso procss ad h sourc ca b glcd h dscrpo of h dyamcs of h op quaum sysm []. Tag o accou scod-ordr rms h ffcv Hamloa s dffr from ag o accou oly h frs-ordr rms h couplg paramr o h broadbad flds of h vrom by h algbra of h opraors (() ad (6)) of quaum sochasc procsss. Th rprsao of h ffcv Hamloa by h quaum saoary procsss (3) ad (5) srvs as a bass for obag QSDEs for h voluo opraor ad h c quao for h op sysm. I h cas of algbra () h QSDEs wh quaum Wr procsss ar aurally calld QSDEs of h Wr yp. I s also aural o rfr o h c quaos obad by such QSDEs as c quaos corolld by quaum Wr procsss. A QSDE xprssd wh rgard o a quaum Posso procss ad basd o aohr algbra (6) s aurally calld a QSDE of gralzd (ag o accou Posso procsss) Lagv yp or of o-wr yp h ss ha h QSDE s o dfd solly by quaum Wr procsss. I gral h Sar raco opraors ar small bcaus hy ar scod-ordr quas h couplg cosa. Howvr a smbl of dcal aoms h valus of hs opraors cras as h umbr N a of aoms crass [3 4]. Du o h rlaos of h HudsoParhasarahy algbra h Sar raco (Posso procss) wh a os sourc wh zro phoo dsy sars o mafs slf rrspcv of a smlar cras h smbl of opraors Wr procsss cludg frs-ordr procsss h couplg cosa ad wh hy rma rlavly small compard wh h opraors Wr procsss. Hr a Posso procss xhbs s gaug or accumulao propry dλ()dλ() dλ(). 7. A QUANTUM STOCHASTIC DIFFERENTIAL EQUATION OF NON-WIENER TYPE FOR THE EVOLUTION OPERATOR OF AN OPEN QUANTUM SYSTEM Th ffcv Hamloa H Eff () of a op quaum sysm ad s vrom h raco rprsao ad h Marov approxmao s drmd by h Io crms of quaum Wr ad Posso procsss as H Eff ()d H EffS ()d + ( Y db () + Y db () ) + dλ () (7) whr h ffcv Hamloa H EffS () of h op quaum sysm s dfd by h opraors ha dscrb cohr pumpg ad varous racos bw h subsysms of h op sysm whl h Lamb shfs ar lmad by a addoal uary rasformao [ 36]. I hs cas h quao for h voluo opraor of h op quaum sysm ad s vrom cao b obad by a drc subsuo of (7) o (4) bcaus h dfos of h Io crms (() ad (4)) ad h algbra of opraos ovr hm (() or (6)) ar basd o h dfo of grals h ss of Io. I h absc of o-wr rms ( 0) (7) quaos for h Io crm of h voluo opraor hav b obad by varous mhods [ ]; howvr o ca qu asly a o cosdrao o-wr rms [ ] by applyg h gral rprsao (4) o h calculao of h crm of h voluo opraor du() U( + d) U(). Ths ylds h followg gralzg xprsso [4 0]: du() xp H EffS ()d + ( Y db () + Y db () ) + dλ () U(). (8) JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

13 QUANTUM THEORY OF OPEN SYSTEMS 383 Ths rprsao xplcly dmosras h uary of h voluo opraor ad h fulfllm of h Io dffrao rul: If w s 0 formula (7) h usg h GardrColl algbra (() or a mor gral algbra [44]) w ca asly oba a Wr QSDE o h bass of whch varous op quaum sysms wr dscrbd [ ]. If 0 (8) h a o-wr QSDE s obad by a srs xpaso of h xpoal fuco ad usg h HudsoParhasarahy algbra (6). Sc w assum ha h opraors ar dagoal w oba h followg rsul: Hr facors d( U()U () ) ( du() )U () + U()dU () + ( du() )( du () ). du() H EffS ()du () Y + Y Λ + ( ) Y d Y Y Λ db () -+ Y Λ Y db () + dλ () U(). (9) xp( ). Th o-wr opraor Y Λ (0) ( ) dsgush a o-wr-yp QSDE from a Wr QSDE. I h absc of Posso procsss 0 h o-wr opraor facors ar gv by + Y Λ ( ) - ad h o-wr QSDE (9) mpls a Wr QSDE dfd solly by h crms of Wr procsss: du() H EffS ()du () -Y Y d + Y db () + Y db () U(). Hr h ma dsy of phoos h os sourcs s assumd o b zro bcaus w usd h Hudso Parhasarahy algbra wh drvg Eq. (9). Howvr h absc of Posso procsss w ca apply h GardrColl algbra () or a algbra for comprssd flds [44] ad drv a Wr QSDE for ozro phoo dsy xpadg h xpo (7). W oba a quao ypcal of all Wr QSDEs for h voluo opraor of a op sysm ad s vrom wh h vrom s rprsd by broadbad bosoc flds. 8. KINETIC EQUATIONS OF NON-WIENER TYPE W wll carry ou all h calculaos volvg quas dfd by sochasc procsss for xampl by h dsy marx of a op sysm ad s vrom by calculag h crms of hs quas followd by h applcao of h algbra of crms of radom varabls of yp () or (6). I hs way w ca asly oba a quao for h dsy marx ρ S +Ev (). Th cha of rasformaos volvd loos as follows: All such calculaos ar basd o h QSDE (9). Dpdg o whhr h QSDE s of Wr or o- Wr yp h c quaos obad for h dsy marx ar sad o b of Wr or o-wr yp. Th dyamcs of a op sysm dscrbd by appropra c quaos s also sad o b of Wr or o-wr yp. Th c quao for h dsy marx ρ S () Tr Ev (ρ S +Ev ()) of a op quaum sysm s obad from h quao for ρ S +Ev () afr s avragg (ag h rac Tr Ev ) ovr h sas of h vrom wh rgard o h rlaos W hav + ρ S + Ev dρ S + Ev ρ S + Ev + d () Ψ S + Ev () Ψ S + Ev () () ρ S + Ev ( + d) ρ S + Ev () ( ) Ψ S + Ev ( + d) Ψ S + Ev ( + d) U( + d) Ψ S + Ev ( 0) Ψ S + Ev ( 0) U ( + d) dρ S + Ev () du()ψ S + Ev ( 0 ) Ψ S + Ev ( 0) U () + U()Ψ S + Ev ( 0 ) Ψ S + Ev ( 0) du () + du()ψ S + Ev ( 0 ) Ψ S + Ev ( 0) du (). Tr Ev ( ρ S + Ev ()db () ) Tr Ev ( ρ S + Ev ()db () ) Tr Ev ( ρ S + Ev ()dλ () ) 0. dρ S () [ H EffS ()ρ S () ] d Y Λ Y ρ S Y ()Y Λ Y Y Λ Y Y ρ S () Λ () JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

14 384 BASHAROV Ths c quao ca asly b rwr h Ldblad form () ad (3) wh h Ldblad opraors whr + ρ S Y ()Y Λ Y. L S Y Λ Y H ShfS sy Y Λ Y L S L S cosy Y. () (3) Udr h glc of h Sar racos h Ldblad opraors () ad (3) ar rducd o L S Y L S S L Y Y H ShfS 0. (4) I fac Eqs. (9) () ad () ar gral quaos (of o-wr yp) h Marov approxmao ha dscrb varous op sysms coag a aomc/lcroc subsysm ad locad a broadbad bosoc vrom. A sgfca rol h dyamcs of a op sysm s playd by Sar-yp racos ha mafs hmslvs h prsc of o-wr facors (0) () ad (3) ad ar rsposbl for h spcfc faur of h o-wr dyamcs. Rcall ha Sar-yp racos ar of h scod ordr wh rspc o h couplg cosa o h broadbad fld ad h rol of hs racos crass wh h umbr of parcls h aomc/lcroc subsysm rrspcv of a smlar cras h procsss of h frs ordr wh rspc o h couplg cosa o h broadbad fld. 9. EXAMPLE: NON-WIENER DYNAMICS OF A SINGLE-MODE CAVITY WITH INTRACAVITY NONRESONANT ATOMS L us apply h gral quao () o h sudy of h dyamcs of a sgl-mod cavy whou mrror losss ha coas dcal racavy aoms. Suppos ha h aoms ar cocrad a small volum ar h po r 0 ad rac wh h cavy mod of frqucy Ω c ad wh h xral quazd vacuum lcromagc fld va lcrc-dpol forcs (s Fg. 4). Th Drac scaro s characrzd by h followg orgal Schrödgr quao for h Ψ S + Ev wav fuco () of h phoos of h cavy mod racavy aoms ad h xral lcromagc fld: + Ev d ΨS () ( H A C () + H A F () ) Ψ S + Ev () d j H AC () g c c c Ω c c c Ω c ( + ) d j Ω j E () E j () Ω j H AF E E j () ωγω ( )( b ω ω + b ω ω ) d j Ω j E () E j j Hr H AC () s h opraor of raco bw h aoms ad h cavy mod H AF () s h opraor of raco bw h aoms ad h xral broadbad vacuum lcromagc fld ad g c ad Γ(ω) ar couplg paramrs. W wll assum ha h racavy aoms ca occupy oly a par of lvls E ad E whch ar rlad by a opcally allowd raso ad h phooc mod s orsoa wh h aomc rasos. Th h frs-ordr rms xpaso (9) H ( 0) () ---Q d 0 d ( ) H ( 0 ) () ---Q d 0 d ( ) h quas ha slowly vary m ar coad oly H ( 0) () ad hs quas ar gv by h followg rms of H AF (): Th rm () formally ( s xror) cocds wh h raco opraor H RF () h roagwav approxmao; howvr h slow-m-varao rqurm mafss slf hr ha h frqucy grao () s prformd oly ovr a arrow doma ar h cral frqucy (). () + H AF () () + H AC () H ( 0) () ω a Γω ( a ) b ωa ω a d Ω E () E () b ωa d ωa Ω + H ( 0) E () E (). H 0 ) ωa JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

15 QUANTUM THEORY OF OPEN SYSTEMS 385 whch cocds wh h aomc raso frqucy Ω. Ths s dod by ω a h dfo of h frqucy wh rspc o whch h grao s prformd. Thus ω a Ω ; blow h cas of zro dug s omd bcaus ω Ω. a Thr s o such rqurm h roag-wav approxmao. Ths fac mmdaly forbds h aalyss of dsprso lms h furhr cosdrao. Afr cludg h rms of H AF () h dfo of H ( 0) () h opraor Q ( 0 ) () s uquly drmd from h quao for H ( 0) (). Sc h raco bw h cavy mod ad a aom s of orsoa characr hr ar o rms H AC () ha slowly vary m so ha H ( 0 ) () 0. Ths codo also uquly drms h opraor Q ( 0 ) (). Examg h scod-ordr rm H ( ) () ---Q d d ( ) () -[ Q ( 0 ) ()H AF () ] - [ Q ( 0 ) ()H ( 0) () ] -[ Q ( 0) ()H AC () ] - [ Q ( 0) ()H ( 0 ) () ] w ca s ha h followg slowly varyg rm arss du o h commuaors: H ( ) () ω c Γω ( c )b ωc ω c g c c c Ω c -( Π ( Ω c ) + Π ( ω c )) E () E + ω c Γω ( c )b ωc ω c g c c c Ω c -( Π ( Ω c ) + Π ( ω c )) E () E whr h frqucy grao s oly ovr a arrow doma ar h cral frqucy ω c whch cocds wh h frqucy Ω c of h cavy mod. Ths s dod by ω c h dfo of h frqucy wh rspc o whch h grao s prformd. Thus hr ω c Ω c ; blow w cao dscuss h cas of zro dug ω c Ω c bcaus ω c Ω c. Thus w oba wo quaum os sourcs wh frqucs ω a ad ω c. Th cral frqucy of h Mor prcsly h cral frqucy of a os sourc cocds wh h rormalzd frqucy Ω of h aomc raso whch as o accou h Lamb shfs of h lvls. () () frs sourc cocds wh h frqucy Ω of h aomc raso whl h cral frqucy of h scod sourc cocds wh h frqucy Ω c of h cavy mod (Fg. 4). Ths os sourcs also df h Sar raco opraors mag hr ow Sar Sar corbuos H Γa () ad H Γc () o h scodordr rm wh rspc o h couplg paramr o h xral vacuum lcromagc fld: H ( 0) Lamb () H F H Γc H Γa Sar Sar Sar Sar + H Γa () + H Γc () + H Ex F () ω c ω c ' Γω ( c )Γ( ω c ' )b ωc b ωc ' ω c ω c ' d -( Π ( ω c ) + Π ( ω c ' )) E () E ( ) Th maguds of h Sar raco opraors Sar Sar H Γa () ad H Γc () may apprcably dffr from ach ohr bcaus of dffr valus of h paramrs Π (ω a ) ad Π (ω c ) ha drm h valus of h Sar shfs of aomc lvls [7 7] h flds wh carrr frqucs ω a ad ω c. Th valu of Π (ω) dpds o h prsc of a quas-rsoa lvl ar E ± ω h aomc spcrum. Ohr pars of h frqucy spcrum of h xral vacuum lcromagc fld ca also b cosdrd as dpd os sourcs ha ma hr ow corbuo o h Sar raco opraor. Howvr hy do o affc [] h dyamcs of a op quaum sysm h cas of zro phoo dsy ad h absc of h opraor of raco of h op sysm wh such sourcs rprsd by a Wr procss h appropra rgo of h spcrum. Th rmag rms of h ffcv Hamloa hs xampl ar gv by () () ω a ω a ' Γω ( a )Γ( ω a ' )b ωa b ωa ' ω a ω a ' d H C Lamb -( Π ( ω a ) + Π ( ω a ' )) E () E H ( 0 ) Lamb () H C Sar + H C + H Ex C ' g cd j d j E j H F Lamb ω c Ω j ω ' Γ ( ω)d j d j E () E ω j Ω j ( ) (). () E () () JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

16 386 BASHAROV H C Sar g c c c c c Π ( Ω c ) E () E Ex H C - g c d j d j j () Th opraors roducd abov oby algbra (6) whos quaos h varabl s rplacd by a appropra dmsolss varabl τ s s a c. A QSDE for h voluo opraor s gv by (9) wh h abovmod rplacm h ffcv Hamloa H EffS (τ s ) ad h s of opraors Y ad wh a c of h form H C Sar Ω c Ω j Th prm h sum dcas h absc of rms wh dvrg domaors. W wll dscrb h os sourcs roducd abov by opraors of yps () ad (4) B a B c b a τ a whch h followg dmsolss varabls ar usd: Ω c Ω j E () E j () E j ( ' ) E ' ' Ex H F - ωγ ( ω) ( ) d ' j d j d j d j ω+ ω j Ω j Ω j E () E j () E j ( ' ) E ' ' ( ) g c c c c c Π ( Ω c ) E () E ( ) τ a ν a ν a π ( )τ a b νa ( τ a ) dτ'b a ( τ' ) Λ ( a τ ) a dτ'b a ( τ' )b a ( τ' ) 0 b c τ c ( ) τ c τ a ν c ν c π 0 ( )τ c b νc ( τ c ) dτ'b c ( τ' ) Λ ( c τ ) c dτ'b c ( τ' )b c ( τ' ) 0 τ c 0 (). τ a Ω τ c Ω c ν a ω a /Ω ν c ω c /Ω c Ω b ωa b νa Ω c b ωc b νc. H EffS H Ex Sar ( ) H c + H Ex τ s H Ex C Ω Ex + H F H c Sar Sar H C Ω ( + )N Y a χ a R Y Λa η a ( a η ) a R 3 ( ± ) π η a ( + )N Y c G c c c η a ( c ---- η ) + c R 3 ( ± ) η π c ( + )N Y Λc η a ( c η ) c R 3 χ a Γω ( a )d π Ω ----Γ( Ω )Γ( Ω )( Π ( Ω ) ± Π ( Ω )) G c (5) (6) Th c quao for h dsy marx ρ S of h phoos ad aoms of h cavy s gv by () wh opraors (5) ad (6). Blow w cosdr h smpls cas. Suppos ha hr ar o corrlaos bw h aomc ad phooc subsysms a h al sa of m. Th h c quaos for h dsy marcs of h aomc (ρ A Tr C (ρ S )) ad phooc (ρ C Tr A (ρ S )) subsysms also hav h g c πω c ΓΩ ( c ) ----Γ( Ω c )Γ( Ω c )( Π ( Ω c ) ± Π ( Ω c )) R 3 - E () E () ( E () E () ) R E () E () R E + () E (). JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

17 QUANTUM THEORY OF OPEN SYSTEMS 387 Ldblad form () ad (3) wh h Ldblad opraors (L A ad H ShfA for h aomc subsysm ad L C ad H ShfC for h phooc subsysm) gv by ( + )N η a ( xp a η ) a R 3 L A χ a R ( + )N η a ( a η ) a R 3 H ShfA χ a R + ( + )N η a ( s a η ) a R 3 η ( + )N a ( a ---- η ) a R R ( + )N η a ( a η ) a R 3 L C ( + )N G c c c xp η a ( c η ) c R 3 H ShfC N G a ( + ) () c s ---- ( η c η c ) N a ( + ) () ---- ( η c η c ) cc c c. (7) (8) Tm formula () s rplacd by h rspcv dmsolss m τ a or τ c ad h ffcv Hamloas ar gv by H EffA Sar ( τ a ) H c + H Ex H EffC ( τ c ) H Sar c Sar g H c c c c c c Π + ( Ω c ) N a Π ( Ω c )R 3 Ω Sar g H c c c c c c Π + ( Ω c ) N a Π ( Ω c )R 3. Ω Hr Π ± (ω) Π (ω) ± Π (ω) ad h agular bracs do avragg: Π + ( Ω c ) N a Π ( Ω c )R 3 Tr ρ A Π + ( Ω c ) N a Π ( Ω c )R 3 c c c c Tr( ρ C c c c c ). 0. THE MAIN FEATURES OF THE NON-WIENER DYNAMICS OF OPEN SYSTEMS W ca dsgush h followg ma characrsc faurs of o-wr dyamcs whch ar dfd by Eqs. (9) ad () h gral cas.. A addoal shf of h lvls of a op sysm whch s dscrbd by h opraor H EffS ad s abs h cas of o-wr dyamcs (s (4)).. Supprsso of collcv rlaxao procsss ad sablzao of xcd sas wh rspc o h collcv dcay for som dss of h aomc/lcroc subsysm. Ths phomo occurs o oly h aomc/lcroc subsysm of a op sysm bu also ohr subsysms for xampl h phooc subsysm. I s oly mpora ha ohr subsysms of h op sysm should somhow rac wh h aomc/lcroc subsysm as h xampl cosdrd abov. Sc h frqucy shfs ca b drmd xprmally o a suffcly hgh dgr of accuracy h addoal frqucy shf may mafs slf ad b a o accou agas h bacgroud of o y clarly mafsd ( fac h absc of) supprsso of collcv rlaxao procsss ad sablzao of xcd sas wh rspc o h collcv dcay. Th supprsso of collcv rlaxao procsss for h aomc ad phooc subsysms of a sglmod cavy follows from Eqs. () (3) (7) ad (8). Frs cosdr a spoaous dcay of h aomc subsysm a xcd smbl of dcal wolvl aoms localzd a sgl-mod cavy ad coupld (by h opraor H AF ) o h xral vacuum lcromagc fld. Suppos ha a h al sa of m 0 h wav fuco of h smbl s symmrc wh rspc o h prmuao of aoms. Th sas of hs smbl ar dscrbd by h gvcors r m of h opraor R 3 R 3 r m m r m o whch a (r + )-dmsoal rprsao of h algbra su() wh graors R 3 ad R ± ad commuao rlaos [R 3 R ± ] ±R ± ad [R + R ] R 3 s mplmd. Hr r s h umbr N a of aoms h smbl N a r. A smbl of oxcd aoms corrspods o h sa r r a smbl wh o xcd aom o h sa r r+ ad wh N a xcd aoms o h sa r r. Th cas cosdrd hr dffrs from h cas of a smbl of dcal aoms localzd a small doma of h ordary hr-dmsoal spac oly by a addoal Sar raco wh h phoos of h orsoa cavy Sar mod (dscrbd by h opraor H c ) ad a slghly dffr valu of h opraor H Ex of raomc raco. Ths dffrcs affc h lcromagc fld md by h aoms bu do o affc h JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0

18 388 BASHAROV raso ras bw h quaum sas of h aomc subsysm. Th W sa r r+ of h aomc smbl h cavy jus as h fully xcd sa r r dcays by h xpoal law xp{γγ W N a } wh a o-wr ( ) facor γ W γ W h cas of h W sa ad γ W for h fully xcd sa ( ) γ W (a) E ω a db a (τ a ) E (b) ω a dλ a (τ a ) Fg.. Irfrc procsss ha drm h supprsso of collcv rlaxao ad sablzao of a xcd sa of h aomc smbl. Ω c (a) (b) ω c db c (τ c ) ω c dλ c (τ c ) E Fg.. Irfrc procsss ha drm h supprsso of collcv rlaxao ad h sablzao of a xcd sa of h phoos of h cavy mod whou losss o mrrors bu wh racavy aoms coupld o h vacuum lcromagc fld of h vrom. E E ω Fg. 3. Dug Δ from h rsoa raco. Is growh s o dsgushd a all: as Δ crass h raco mooocally dcrass. Δ ( ) ( ) ( cos( N a η S )) ( ) ( N a η S ) γ W ( ) η S γπ ( Ω ) Ω d (9) dfd by h paramr Π (Ω ) whch s characrzd by h Sar raco of a aom h xcd cr sa E. Wh h umbr N a of aoms of h cr smbl sasfs h codos N a η S π h sas of xcd aoms ur ou o b sabl wh rspc o collcv rlaxao procsss. Now cosdr h varao ra of h avrag umbr of phoos of h cavy mod h cas of racavy aoms a oxcd sa. Th avrag umbr c () of phoos h mcrocavy dcrass v h absc of mrror losss ad h dcras s also xpoal c () c ( 0) G c N ( c xp{ a γ ) W } wh h Wr dcay cosa G c ad a o-wr facor h xpo ( c) ( ) ( cosn a η S ) γ W c ( c) η π S N a ----Γ ( Ω c )Π ( Ω c ). (30) I hs cas hr also xss a s of crcal umbrs cr N a of aoms N cr ( c) a η S π for whch h phoos ar locd h cavy. Th sablzao of a xcd sa wh rspc o collcv rlaxao procsss cludg collcv spoaous dcay s assocad wh h rfrc of wo procsss (Fgs. ad ). Th frs procss s h phoo msso; drms rlaxao rasos h quaum sysm (Fgs. a ad a). I Sco 6 hs procss s rprsd by a quaum Wr procss wh crms db a (τ a ) ad db c (τ c ). Th scod rfrc procss rprss a vrual msso of a phoo followd by s absorpo (Fgs. b ad b). Ths procss drms h Sar raco ad dos o chag h quaum sa of h sysm. I Sco 6 hs procss s rprsd by a quaum Posso procss wh crms dλ a (τ a ) ad dλ c (τ c ). W should srss ha h supprsso of collcv rlaxao phoma cao b arbud o h cras h frqucy shf ad dug from rsoac whch rsul a dcras h raco wh h vrom (Fg. 3). Frs hr s ow Sar shf of lvls propr a phoo-fr vacuum lcromagc fld ad o dals wh a rlaxao frqucy shf dscrbd by h opraor H EffS. Scod for som dug from rsoac a furhr JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 3 0