QUANTUM TO CLASSICAL TRANSITION IN THE LINDBLAD THEORY OF OPEN QUANTUM SYSTEMS
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1 Romanan Repors n Physcs, Vol. 57, No., P , 005 QUANTUM TO CLASSICAL TRANSITION IN THE LINDBLAD THEORY OF OPEN QUANTUM SYSTEMS A. ISAR Deparmen of Theorecal Physcs, Insue of Physcs and Nuclear Engneerng, Buchares-Mãgurele, Romana e-mal address: sar@heory.npne.ro (Receved July 5, 005) Absrac. 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 and analyze s ranson from quanum o classcal behavour. Key words: Lndblad heory, open quanum sysems, quanum decoherence, classcal correlaons.. 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 []: 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, ha s he quanum sae becomes peaked along a classcal rajecory. 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]. A remarkable aspec of he curren research helpng n undersandng he naure of he quanum o classcal ranson s ha for he frs me here have recenly been carred on expermens probng he boundary beween he quanum and he classcal domans n a conrolled way [8, 9].
2 57 A. Isar In mos of he leraure, he quanum decoherence has been suded for a sysem coupled o an envronmen or hermal bah wh many degrees of freedom. In hs work we sudy QD and CC for a harmonc oscllaor neracng wh an envronmen n he framework of he Lndblad heory for open quanum sysems. More concreely 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 s more and more sgnfcan n me, whereas CC are less and less srong as he sysem evolves n me. The organzng of he paper s as follows. In Secon we revew he Lndblad maser equaon for he damped harmonc oscllaor and n Secon 3 we derve he maser equaon n coordnae represenaon and he correspondng Fokker-Planck equaon n he Wgner represenaon and deermne he densy marx and Wgner funcon of he consdered sysem. Then n Secon we nvesgae QD and CC and analyze hem quanavely. A summary s gven n Secon 5.. MASTER EQUATION FOR THE HARMONIC OSCILLATOR The rreversble me evoluon of an open sysem s descrbed by he followng general quanum Markovan maser equaon for he densy operaor ρ () [0,, ]: 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 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 [3,, 5] and he harmonc oscllaor Hamlonan H s chosen of he general quadrac form μ H = H + +, = + ω 0 ( qp pq) H m 0 p q. m Wh hese choces he maser equaon () akes he followng form [, 5]: dρ = [ H0,ρ ] ( λ+μ )[ q,ρ p + p ρ ] + ( λ μ )[ p,ρ q + q ρ ] d Dpp Dqq Dpq [ q, [ q,ρ]] [ p, [ p,ρ ]] + ([ q, [ p,ρ ]] + [ p, [ q,ρ ]]). () (3)
3 3 Quanum o classcal ranson n he Lndblad heory 575 The quanum dffuson coeffcens D pp, D qq, D pq and he dsspaon consan λ sasfy he followng fundamenal consrans [, 5]: D > 0, D > 0 and D ppdqq D λ pq. () In he parcular case when he asympoc sae s a Gbbs sae ρ ( ) = H H 0 0 = e kt / Tr e kt, hese coeffcens become [, 5] λ+μ ω λ μ D = ω, = ω pp m coh Dqq coh, Dpq = 0, (5) kt mω kt where T s he emperaure of he hermal bah. From he maser equaon (3) we can oban he equaons of moon for he expecaon values of coordnae and momenum and n he underdamped case ( ω>μ ) consdered n hs paper, wh he noaon Ω ω μ, he soluons have he followng form [, 5]: pp qq G λ μ σ = Ω + Ω σ + q( ) e cos sn q(0) sn Ωσ p(0), Ω mω (6) λ ω μ σ () = e p m sn Ωσ q(0) + cosω sn Ω σp(0) Ω Ω (7) and σq( ) =σp( ) = 0. The relaon () s a necessary condon for he generalzed uncerany nequaly qq pp pq σ () σ () σ () (8) o be fulflled, where 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. The equaly n relaon (8) s realzed for a specal class of pure saes, called correlaed coheren saes [6] or squeezed coheren saes. The asympoc values σqq ( ), σpp ( ), σpq ( ) 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 (5), hey reduce o [, 5] σ ( ) = coh ω, σ ( ) = mω ω qq pp coh, σpq( ) = 0. mω kt kt (9)
4 576 A. Isar 3. DENSITY MATRIX AND WIGNER DISTRIBUTION FUNCTION We consder a harmonc oscllaor wh an nal Gaussan wave funcon ( ) Ψ ( q) = exp σpq(0) ( q σ (0)) + q σ p(0) q, (0) πσqq(0) 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 (0) represens a correlaed coheren sae [6] wh he varances and covarance of coordnae and momenum σ (0) = δ, σ (0) = mω, σ (0) = r qq pp. ω δ( pq () 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 () correspond o a mnmum uncerany sae, snce hey fulfl he generalzed uncerany relaon σ (0) (0) qq σpp σ pq(0) =. () 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 ( )( q q) λ+μ ( ) ( q q) q q ρ+ λ μ + + q q + ρ D pp ( ) ( ) q q ρ+ D qq D pq q q + ρ + ρ q q q q and Wgner dsrbuon funcon W(q, p, ) sasfes he followng Fokker-Planckype equaon: (3) W p = W + mω q W + ( λ+μ ) ( pw) + ( λ μ) ( qw) m q p p q + D W + + pp D W. qq D W pq p q pq ()
5 5 Quanum o classcal ranson n he Lndblad heory 577 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 and forh erms are he dsspave erms and have a dampng effec (exchange of energy wh envronmen). The las hree are nose (dffusve) erms and produce flucuaon effecs n he evoluon of he sysem. D pp promoes dffuson n momenum and generaes decoherence n coordnae q: reduces he off-dagonal erms, responsble for correlaons beween spaally separaed peces of he wave packe. Smlarly D qq promoes dffuson n coordnae and generaes decoherence n momenum p. The D pq erm s he so-called anomalous dffuson erm. I promoes dffuson n he varable qp + pq, jus lke boh he oher dffuson erms, bu does no generae decoherence. The densy marx soluon of Eq. (3) has he general form of Gaussan densy marces () exp q+ q < q ρ q >= q()) qq( ) σ qq( ) πσ σ σ() σ () ( ) pq q+ q () ( ) q q + ()( ) q q q p q q ( ) qq() σ + σ, σqq σ (5) where σ() σ () () qq σpp σ pq() s he Schrödnger generalzed uncerany funcon [7]. For an nal Gaussan Wgner funcon (correspondng o a correlaed coheren sae (0)) he soluon of Eq. () s { Wqp (,, ) = exp σ ( )( q σ ( )) +σ ( )( p σ ( )) π σ( ) σ( ) σpq( )( q σq( ))( p σ p( )) }. pp q qq p In he case of a hermal bah we oban he followng seady sae soluon for (we denoe ε ω ): kt mω ( ) ω ( q+ q ) < q ρ( ) q >= exp m + ( q q ) cohε. πcohε cohε In he long me lm we have also (6) (7), = ω p W ( q p) exp m q +. πcohε cohε mω (8)
6 578 A. Isar 6. QUANTUM DECOHERENCE AND CLASSICAL CORRELATIONS As we saed n he Inroducon, here are wo condons ha 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, called by Zurek preferred saes n he envronmen nduced superselecon descrpon [, 3]) 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. An solaed sysem has an unary evoluon and he coherence of he sae s no los pure saes evolve n me only o pure saes. 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. On he oher sde, 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), so ha he lack of srong correlaons beween he coordnae and s canoncal momenum does no necessarly mean ha he sysem s quanum. 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,.e. s srongly peaked abou q= q and very small for q far from q. Usng new varables Σ= ( q+ q )/ and Δ= q q, he densy marx (5) can be rewren as ρσ,δ, ( ) = α σ = ασ γδ + βσδ+ ασ Σ+ βσ Δ ασ, π p() exp () () (9) q q q() wh he abbrevaons σ σ α= () pq(), γ=, β= σ ( ) σ ( ) σ ( ) qq qq qq (0)
7 7 Quanum o classcal ranson n he Lndblad heory 579 and he Wgner ransform of he densy marx (9) s α β σ σ,, = [ ( q q()) ( p p())] Wqp ( ) exp α( q σ ( )). π γ q () γ a) Degree of quanum decoherence (QD) The represenaon-ndependen measure of he degree of QD [] 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, ) : whch n our case gves δ α QD =, γ δ QD () =. σ( ) () (3) The fne emperaure Schrödnger generalzed uncerany funcon, calculaed n Ref. [7], has he expresson () e λ coh coh σ = δ+ + ( r δ ) ε ε ω μ Ω ε cos( ) e + λcohε δ+ coh δ( r ) Ω μsn( Ω) rμω( cos( Ω)) coh + δ ( r ) + + ε. δ Ω Ω r In he lm of long mes Eq. () yelds () so ha we oban whch for hgh T becomes σ ( ) = coh ε, (5) δqd ( ) = anh ω, kt δqd ( ) = ω. kt We see ha QD δ 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. (6) (7)
8 580 A. Isar 8 For T = 0 he asympoc (fnal) sae s pure and δ QD reaches s nal maxmum value. A pure sae undergong unary evoluon s hghly coheren: does no lose s coherence,.e., off-dagonal coherences never vansh. 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 marx n he poson bass are peaked preferenally along he dagonal q= q. When δ, we have a srong QD. QD b) Degree of classcal correlaons (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 () of he Wgner funcon becomes α β,, = ( q p) Wqp ( ) exp α q. (8) π γ γ A rdge of he Wgner funcon (8) n phase space s a p= β q, showng he correlaon beween q and p. 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. (8) and he magnude of he average of p ( p = βq ) []: 0 αγ δ CC =, (9) β 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 []: n he sae (8), more squeezed s he Wgner funcon, more srongly esablshed are CC. For our case, we oban σ() δ CC () =, (30) σ () where σ ( ) s gven by Eq. () and σ pq( ) can be calculaed usng formulas gven n Refs. [, 5]: λ ω r ε pq σ pq() = e μω(coh δ ) cos( Ω) Ω δ( r ) r (3) sn( ) μ r +ωω δ coh ( r) Ω + μω δ + ε +. ( r δ δ ) r
9 9 Quanum o classcal ranson n he Lndblad heory 58 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 c) Dscusson wh Gaussan densy marx and Wgner funcon We have seen ha f he nal wave funcon s Gaussan, hen he densy marx (5) and he Wgner funcon (6) reman Gaussan for all mes (wh medependen parameers whch deermne her amplude and spread) and cenered along he rajecory gven by Eqs. (6) and (7), whch are he soluons σ q( ) and σ p() 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 long enough 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. From expressons () and (9) we noce ha he key parameer whch descrbes QD and CC s γ. Ths coeffcen deermnes he spread of he Wgner funcon () around he pah n phase space and measures he conrbuon of non-dagonal erms n he densy marx (9). Therefore, when decoherence ncreases, he correlaons beween he canoncal varables of coordnae and momenum decrease. The exreme lm of QD (γ ) s ncompable wh CC and ha of CC (γ 0) s ncompable wh QD. Ther smulaneous realzaon s no a rval ask: QD requres neracon wh an envronmen, whch nevably suppresses CC and produces flucuaons n he evoluon of he sysem, whereas classcal predcably requres hese flucuaons o be small. Therefore he exsence of he envronmen s crucal for he quanum o classcal ranson and, consequenly, classcaly s an emergen propery of an open quanum sysem. We can say ha a relave compeon appears beween QD and exsence of CC, snce decoherence (dagonalzng or he decreasng of he wdh of he densy marx) mples a spreadng of he Wgner dsrbuon funcon (whch s he Fourer
10 58 A. Isar 0 ransform of he densy marx) along he rajecory n phase space, whereas CC requre he exsence of sharp peaks n he Wgner funcon. Alhough here exss hs compeon, here s a broad compromse regme n whch QD and CC can hold well smulaneously. 5. SUMMARY We have suded QD and CC wh he Markovan equaon of Lndblad n order o undersand he ranson from quanum o classcal mechancs for a sysem conssng of an one-dmensonal harmonc oscllaor n neracon wh a hermal bah n he framework of he heory of open quanum sysems based on quanum dynamcal semgroups. Dependng on he relave magnude beween he measures of QD and CC, he sysem recovers classcaly n a sgnfcan measure. The classcaly s condoned by he CC, expressed by he fac ha he Wgner funcon has a peak whch follows (exacly for λ = μ and approxmaely for λ μ) he classcal rajecory n phase space and also by QD, expressed by he loss of quanum coherence n he case of a hermal bah a fne emperaure. For an nal Gaussan quanum sae, whch s a correlaed coheren sae, he Wgner funcon s posve for all mes, so ha represens a rue classcal probably dsrbuon n phase space and descrbes CC. The sudy of classcaly usng QD and CC leads o a deeper undersandng of he quanum orgns of he classcal world. As a resul of he progress made n he las wo decades, he quanum o classcal ranson has become a subjec of expermenal nvesgaons, whle prevously was mosly a doman of heory [, 3]. The ssue of quanum o classcal ranson pons o he necessy of a beer undersandng of open quanum sysems and he Lndblad heory provdes a selfconssen reamen of dampng as a general exenson of quanum mechancs o open sysems. 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, 00), p W. H. Zurek, Rev. Mod. Phys., 75, 75 (003).. M. Morkawa, Phys. Rev., D, 99 (990). 5. R. Alck, Open Sys. and Informaon Dyn.,, 53 (00). 6. J. P. Paz, S. Habb and W. H. Zurek, Phys. Rev., D 7, 88 (993). 7. W. H. Zurek, Phys. Today, No. 0, 36 (99); 6, No., 8 (993); Prog. Theor. Phys., 89, 8 (993). 8. M. Brune, E. Hagley, J. Dreyer, X. Maîre, A. Maal, C. Wunderlch, J.-M. Ramond and S. Haroche, Phys. Rev. Le., 77, 887 (996).
11 Quanum o classcal ranson n he Lndblad heory D. A. Kokorowsk, A. D. Cronn, T. D. Robers, D. E. Prchard, Phys. Rev. Le., 86, 9 (00). 0. E. B. Daves, Quanum Theory of Open Sysems (Academc Press, New York, 976).. G. Lndblad, Commun. Mah. Phys., 8, 9 (976).. H. Spohn, Rev. Mod. Phys., 5, 569 (980). 3. G. Lndblad, Rep. Mah. Phys., 0, 393 (976).. A. Sandulescu and H. Scuaru, Ann. Phys. (N.Y.) 73, 77 (987). 5. A. Isar, A. Sandulescu, H. Scuaru, E. Sefanescu and W. Sched, In. J. Mod. Phys., E 3, 635 (99). 6. V. V. Dodonov, E. V. Kurmyshev and V. I. Man ko, Phys. Le., A 79, 50 (980). 7. A. Isar and W. Sched, Phys. Rev., A 66, 07 (00).
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