QUANTUM TO CLASSICAL TRANSITION IN THE THEORY OF OPEN SYSTEMS

QUANTUM THEORY QUANTUM TO CLASSICAL TRANSITION IN THE THEORY OF OPEN SYSTEMS A ISAR Deparmen of Theorecal Physcs, Insue of Physcs and Nuclear Engneerng, Buchares-Magurele, Romana e-mal address: sar@heorynpnero Receved January 3, 006 In he framework of he Lndblad heory for open quanum sysems we deermne he degree of quanum decoherence and classcal correlaons of a harmonc oscllaor neracng wh a hermal bah The ranson from quanum o classcal behavour of he consdered sysem s analyzed and s shown ha he classcaly akes place durng a fne nerval of me We calculae also he decoherence me and show ha has he same scale as he me afer whch hermal flucuaons become comparable wh quanum flucuaons INTRODUCTION The ranson from quanum o classcal physcs and classcaly of quanum sysems connue o be among he mos neresng problems n many felds of physcs, for boh concepual and expermenal reasons [,, 3] Two condons are essenal for he classcaly of a quanum sysem [4]: a) quanum decoherence (QD), ha means he rreversble, unconrollable and perssen formaon of a quanum correlaon (enanglemen) of he sysem wh s envronmen [5], expressed by he dampng of he coherences presen n he quanum sae of he sysem, when he off-dagonal elemens of he densy marx of he sysem decay below a ceran level, so ha hs densy marx becomes approxmaely dagonal and b) classcal correlaons (CC), expressed by he fac ha he Wgner funcon of he quanum sysem has a peak whch follows he classcal equaons of moon n phase space wh a good degree of approxmaon, ha s he quanum sae becomes peaked along a classcal rajecory The necessy and suffcency of boh QD and CC as condons of classcaly are sll a subjec of debae Boh hese condons do no have an unversal characer, so ha hey are no necessary for all physcal models An mporan role n hs dscusson plays he emperaure of he envronmen and Paper presened a he Naonal Conference of Physcs, 3 7 Sepember, 005, Buchares, Romana Rom Journ Phys, Vol 5, Nos 5 6, P 503 53, Buchares, 006

504 A Isar herefore s worh o ake no accoun he dfferences beween low and hgh emperaure regmes For example, purely classcal sysems a very hgh emperaures are descrbed by a classcal Fokker-Planck equaon whch does no follow any rajecory n phase space (for very small knec energy, compared o he hermal energy, when he probably dsrbuon becomes essenally ndependen of momenum), so ha n hs case CC are no necessary Lkewse, one can have a classcal behavour f he coherences are neglgble, whou havng srong CC (for example, n he case of a classcal gas a fne emperaure) and he lack of srong correlaons beween he coordnae and s canoncal momenum does no necessarly mean ha he sysem s quanum On he oher hand, he condon of CC s no suffcen for a sysem o become classcal alhough he Wgner funcon can show a sharp correlaon n phase space, he quanum coherence never vanshes for a closed sysem whch has a unary evoluon Lkewse, n he low emperaure quanum regme one can observe srong CC For example, n he case of a purely damped quanum harmonc oscllaor (a zero emperaure), he nal coheren saes reman coheren and perfecly follow classcal rajecores of a damped oscllaor, bu CC are no suffcen for classcaly In he las wo decades has became more and more clear ha he classcaly s an emergen propery of open quanum sysems, snce boh man feaures of hs process QD and CC srongly depend on he neracon beween he sysem and s exernal envronmen [3, 6, 7] The man purpose of hs work s o sudy QD and CC for a harmonc oscllaor neracng wh an envronmen n he framework of he Lndblad heory for open quanum sysems We deermne he degree of QD and CC and he possbly of smulaneous realzaon of QD and CC for a sysem conssng of a harmonc oscllaor n a hermal bah I s found ha he sysem manfess a QD whch ncreases wh me and emperaure, whereas CC are less and less srong wh ncreasng me and emperaure LINDBLAD MASTER EQUATION FOR THE HARMONIC OSCILLATOR IN COORDINATE AND WIGNER REPRESENTATION The rreversble me evoluon of an open sysem s descrbed by he followng general quanum Markovan maser equaon for he densy operaor ρ () [8]: d ρ( ) = [ H,ρ ( )] ([ Vjρ ( ), V ] [ j,ρ ( ) ]) d j V V j () j H s he Hamlonan of he sysem and V j, V j are operaors on he Hlber space of H, whch model he envronmen In order o oban, for he damped

3 Quanum o classcal ranson n he heory of open sysems 505 quanum harmonc oscllaor, equaons of moon as close as possble o he classcal ones, he wo possble operaors V and V are aken as lnear polynomals n coordnae q and momenum p [9, 0] and he harmonc oscllaor Hamlonan H s chosen of he general quadrac form μ H = H 0 ( qp pq), H m 0 = p ω q () m Wh hese choces he maser equaon () akes he followng form [9, 0]: dρ = [ H0,ρ ] ( λμ )[ q,ρ p p ρ ] ( λ μ )[ p,ρ q q ρ ] d (3) Dpp Dqq Dpq [ q, [ q,ρ]] [ p, [ p,ρ ]] ([ q, [ p,ρ ]] [ p, [ q,ρ ]]) In he parcular case when he asympoc sae s a Gbbs sae ρ ( ) = H H 0 0 = e kt / Tr e kt, he quanum dffuson coeffcens D pp, D qq, D pq and he dsspaon consan λ sasfy he relaons [9, 0] λμ ω λ μ D = ω, = ω pp m coh D coh, = 0, qq D kt mω kt pq (4) where T s he emperaure of he hermal bah In he Markovan regme he harmonc oscllaor maser equaon whch sasfes he complee posvy condon canno sasfy smulaneously he ranslaonal nvarance and he dealed balance (whch assures an asympoc approach o he canoncal hermal equlbrum sae) The necessary and suffcen condon for ranslaonal nvarance s λ=μ [9, 0] In hs case he equaons of moon for he expecaon values of coordnae and momenum are exacly he same as he classcal ones If λ μ, hen we volae ranslaonal nvarance, bu we keep he canoncal equlbrum sae The asympoc values σqq (, ) σ pp (, ) σ pq ( ) do no depend on he nal values σ qq (0), σ pp (0), σ pq (0) and n he case of a hermal bah wh coeffcens (4), hey reduce o [9, 0] σ ( ) = coh ω, σ ( ) = mω ω qq coh, σ ( ) = 0 mω kt pp kt pq (5) In he followng, we consder a general emperaure T, bu we should sress ha he Lndblad heory s obaned n he Markov approxmaon, whch holds for hgh emperaures of he envronmen A he same me, he semgroup dynamcs of he densy operaor whch mus hold for a quanum Markovan process s vald only for he weak-couplng regme, wh he dampng λ obeyng he nequaly λ ω G

506 A Isar 4 We consder a harmonc oscllaor wh an nal Gaussan wave funcon ( ) 4 Ψ ( q) = exp σ σ pq(0) ( q (0)) q σ p(0) q, (6) πσqq(0) 4 σqq(0) where σ qq(0) s he nal spread, σ pq(0) he nal covarance, and σ q(0) and σ p (0) are he nal averaged poson and momenum of he wave packe The nal sae (6) represens a correlaed coheren sae (squeezed coheren sae) [] wh he varances and covarance of coordnae and momenum σ (0) = δ, σ (0) = mω, σ (0) = r qq pp ω δ( pq (7) m r ) r Here, δ s he squeezng parameer whch measures he spread n he nal Gaussan packe and r, wh r < s he correlaon coeffcen a me = 0 The nal values (7) correspond o a mnmum uncerany sae, snce hey fulfl he generalzed uncerany relaon σ (0) (0) qq σpp σ pq(0) = (8) 4 For δ= and r = 0 he correlaed coheren sae becomes a Glauber coheren sae From Eq (3) we derve he evoluon equaon n coordnae represenaon: ρ = ρ mω ( q q ) ρ m q q ( λμ)( q q ) ρ ( λ μ ) ( q q ) ] ρ q q q q D pp ( ) ρ ρ ( ) ρ q q D qq D pq q q q q q q For he case of a hermal bah wh coeffcens (4) he Wgner dsrbuon funcon Wqp (,, ) sasfes he followng Fokker-Planck-ype equaon: W p = W mω q W ( λμ ) ( pw) ( λ μ) ( qw) m q p p q ω λ μ coh ( λμ) mω W W kt p mω q (9) (0) The frs wo erms on he rgh-hand sde of boh hese equaons generae a purely unary evoluon They gve he usual Louvllan evoluon The hrd

5 Quanum o classcal ranson n he heory of open sysems 507 and forh erms are he dsspave erms and have a dampng effec (exchange of energy wh envronmen) The las wo are nose (dffusve) erms and produce flucuaon effecs n he evoluon of he sysem They promoe dffuson n momenum p and coordnae q and generae decoherence n coordnae and momenum, respecvely In he hgh emperaure lm, quanum Fokker-Planck equaon (0) wh coeffcens (4) becomes classcal Kramers equaon ( Dpp mλ kt for λ=μ) The densy marx soluon of Eq (9) has he general form of Gaussan densy marces q q ( q ) q ρ () q = exp () qq( ) σ πσ σqq( ) σ() σ () q q ( q q) ( q( ) )( q q) σ σ ( )( q q), σ ( ) qq pq p σqq() () where σ() σ () () qq σpp σ pq() s he Schrödnger generalzed uncerany funcon [] ( σ qq and σ pp denoe he dsperson (varance) of he coordnae and momenum, respecvely, and σ pq denoes he correlaon (covarance) of he coordnae and momenum) For an nal Gaussan Wgner funcon (correspondng o a correlaed coheren sae (6)) he soluon of Eq (0) s { Wqp (,, ) = exp [ σ ()( ()) ()( ()) pp q σ q σqq p σp π σ( ) σ( ) () σ ( )( q σ ( ))( p σ ( ))] pq q p In he case of a hermal bah we oban he followng seady sae soluon for (we denoe ε ω ): kt } ( ) ( q q) mω mω ( ) exp ( ) q ρ q = q q cohε π cohε 4 cohε In he long me lm we have also (3) ( ) exp p W q, p = mω q π cohε cohε mω (4) Saonary soluons o he evoluon equaons obaned n he long me lm are possble as a resul of a balance beween he wave packe spreadng nduced by he Hamlonan and he localzng effec of he Lndblad operaors

508 A Isar 6 3 QUANTUM DECOHERENCE AND CLASSICAL CORRELATIONS As we already saed, one consders ha wo condons have o be sasfed n order ha a sysem could be consdered as classcal The frs condon requres ha he sysem should be n one of relavely permanen saes saes ha are leas affeced by he neracon of he sysem wh he envronmen and he nerference beween dfferen saes should be neglgble Ths mples he desrucon of off-dagonal elemens represenng coherences beween quanum saes n he densy marx, whch s he QD phenomenon The loss of coherence can be acheved by nroducng an neracon beween he sysem and envronmen: an nal pure sae wh a densy marx whch conans nonzero off-dagonal erms can non-unarly evolve no a fnal mxed sae wh a dagonal densy marx durng he neracon wh he envronmen, lke n classcal sascal mechancs The second condon requres ha he sysem should have, wh a good approxmaon, an evoluon accordng o classcal laws Ths mples ha he Wgner dsrbuon funcon has a peak along a classcal rajecory, ha means here exs CC beween he canoncal varables of coordnae and momenum Of course, he correlaon beween he canoncal varables, necessary o oban a classcal lm, should no volae Hesenberg uncerany prncple, e he poson and momenum should ake reasonably sharp values, o a degree n concordance wh he uncerany prncple Ths s possble, because he densy marx does no dagonalze exacly n poson, bu wh a non-zero wdh Usng new varables Σ= ( q q )/ and Δ= q q, he densy marx () can be rewren as ρσ,δ, ( ) = α exp ασ γδ βσδ ασ Σ π q( ) σ p() βσ () Δ ασ q q(), wh he abbrevaons σ σ α= () pq(), γ=, β= σ ( ) σ ( ) σ ( ) qq qq qq (5) (6) and he Wgner ransform of he densy marx (5) s α β σ σ,, = [ ( q q()) ( p p())] Wqp ( ) exp α( q σ ( )) π γ 4 q (7) γ

7 Quanum o classcal ranson n he heory of open sysems 509 a) DEGREE OF QUANTUM DECOHERENCE (QD) The represenaon-ndependen measure of he degree of QD [4] s gven by he rao of he dsperson / γ of he off-dagonal elemen ρ (0,Δ, ) o he dsperson /α of he dagonal elemen ρσ, ( 0, ) : δ α QD =, (8) γ whch n our case gves δ QD () = (9) σ( ) The fne emperaure Schrödnger generalzed uncerany funcon, calculaed n Ref [], has he expresson λ σ ( ) = e 4 δ cohε coh ε] 4 δ( r ) λ ω μ Ω δ cos( ) e cohε coh ε) δ( r ) Ω μ Ω μω Ω δ sn( ) r ( cos( )) coh ε δ( r ) Ω Ω r In he lm of long mes Eq (0) yelds (0) σ ( ) = coh ε, () 4 so ha we oban δqd ( ) = anh ω, () kt whch for hgh T becomes δqd ( ) = ω (3) kt We see ha δ QD decreases, and herefore QD ncreases, wh emperaure, e he densy marx becomes more and more dagonal a hgher T and he conrbuons of he off-dagonal elemens ge smaller and smaller A he same me he degree of pury decreases and he degree of mxedness ncreases wh T δ QD < for T 0, whle for T = 0 he asympoc (fnal) sae s pure and δ QD reaches s nal maxmum value δ QD = 0 when he quanum coherence s compleely los So, when δ QD = here s no QD and only f δ QD <, here s a sgnfcan degree of QD, when he magnude of he elemens of he densy

50 A Isar 8 marx n he poson bass are peaked preferenally along he dagonal q= q When δ, we have a srong QD QD b) DEGREE OF CLASSICAL CORRELATIONS (CC) In defnng he degree of CC, he form of he Wgner funcon s essenal, bu no s poson around σ q( ) and σ p ( ) Consequenly, for smplcy we consder zero values for he nal expecaons values of he coordnae and momenum and he expresson (7) of he Wgner funcon becomes α β,, = ( q p) Wqp ( ) exp α q (4) π γ 4 γ As a measure of he degree of CC we ake he relave sharpness of hs peak n he phase space deermned from he dsperson γ n p n Eq (4) and he magnude of he average of p ( p = βq ) [4]: 0 αγ δ CC =, β (5) where we denfed q as he dsperson / α of q δ CC s a good measure of he squeezng of he Wgner funcon n phase space [4]: n he sae (4), more squeezed s he Wgner funcon, more srongly esablshed are CC For our case, we oban σ() δ CC () =, (6) σ () pq where σ ( ) s gven by Eq (0) and σ pq( ) can be calculaed usng formulas gven n Refs [9, 0]: λ ω ε r σ = μω δ pq( ) e coh cos( Ω) 4 Ω δ( r ) r (7) μ ωω δ Ω μω δ r sn( ) cohε δ( r) δ( r) r When δ CC s of order of uny, we have a sgnfcan degree of classcal correlaons The condon of srong CC s δ CC, whch assures a very sharp peak n phase space Snce σpq( ) = 0, n he case of an asympoc Gbbs sae, we ge δ ( ), so ha our expresson shows no CC a CC

9 Quanum o classcal ranson n he heory of open sysems 5 c) DISCUSSION WITH GAUSSIAN DENSITY MATRIX AND WIGNER FUNCTION If he nal wave funcon s Gaussan, hen he densy marx () and he Wgner funcon () reman Gaussan for all mes (wh me-dependen parameers whch deermne her amplude and spread) and cenered along he rajecory gven by he soluons of he dsspave equaons of moon Ths rajecory s exacly classcal for λ=μ and only approxmaely classcal for no large λ μ The degree of QD has an evoluon whch shows ha n general QD ncreases wh me and emperaure The degree of CC has a more complcaed evoluon, bu he general endency s ha CC are less and less srong wh ncreasng me and emperaure δ QD < and δ CC s of he order of uny for a long enough nerval of me, so ha we can say ha he consdered sysem neracng wh he hermal bah manfess boh QD and CC and a rue quanum o classcal ranson akes place Dsspaon promoes quanum coherences, whereas flucuaon (dffuson) reduces coherences and promoes QD The balance of dsspaon and flucuaon deermnes he fnal equlbrum value of δ QD The quanum sysem sars as a pure sae, wh a Wgner funcon well localzed n phase space (Gaussan form) Ths sae evolves approxmaely followng he classcal rajecory (Louvlle flow) n phase space and becomes a quanum mxed sae durng he rreversble process of QD In Ref [] we deermned he me d when hermal flucuaons become comparable wh quanum flucuaons: d = λ δ cohεμ δ cohε λ δ δ ( r ) ( r ) A hgh emperaure, nroducng he noaon τ kt, ω ε expresson (8) becomes = d τ λ δ μ δ δ δ ( r ) ( r ) (8) (9) (30) We see ha d s decreasng wh ncreasng of boh emperaure T and dsspaon λ Dffuson n momenum occurs a he rae se by D pp In he macroscopc lm, when s small compared o oher quanes wh dmensons of acon,

5 A Isar 0 such as D < ( q q ) >, he erm n Eq (9) conanng D / domnaes pp and nduces he followng evoluon of he densy marx: pp ρ Dpp = ( q q ) ρ Thus he densy marx loses off-dagonal erms n poson represenaon: (3) q ρ () q = q ρ (0) q exp ( q q ), (3) whle he dagonal ( q= q ) ones reman unouched Quanum coherences decay exponenally a a rae gven by D pp ( q q ) /, so ha he decoherence me scale s of he order of (33) D q q pp ( ) D pp In he case of a hermal bah, we oban (see Eq (4)) = deco, ( λμ) mωσ (0)cohε qq (34) where we have aken ( q q ) of he order of he nal dsperson n coordnae σ qq (0) As expeced, he decoherence me deco has he same scale as he me d afer whch hermal flucuaons become comparable wh quanum flucuaons We can asser ha n he consdered case classcaly s a emporary phenomenon, whch akes place only a some sages of he dynamcal evoluon, durng a defne nerval of me [3] Due o he dsspave naure of evoluon, he approxmaely deermnsc evoluon s no more vald for very large mes, when he localzaon of he sysem s affeced by he spreadng of he wave packe and of he Wgner dsrbuon funcon REFERENCES E Joos, H D Zeh, C Kefer, D Guln, J Kupsch and I O Samaescu, Decoherence and he Appearance of a Classcal World n Quanum Theory, nd Edn, Sprnger, Berln, (003) J P Paz and W H Zurek, n Coheren Aomc Maer Waves, Les Houches Sesson LXXII, ed by R Kaser, C Wesbrook and F Davd, Sprnger, Berln, p 533 (00) 3 W H Zurek, Rev Mod Phys 75, 75 (003) 4 M Morkawa, Phys Rev D4, 99 (990) 5 R Alck, Open Sys and Informaon Dyn, 53 (004)

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