DECOHERENCE IN OPEN QUANTUM SYSTEMS

Transcription

1 DECOHERENCE IN OPEN QUANTUM SYSTEMS A. ISAR Deparmen of Theoreical Physics, Insiue of Physics and Nuclear Engineering, Buchares-Mãgurele, Romania Received December 0, 00 In he framework of he Lindblad heory for open quanum sysems we deermine he degree of quanum decoherence of a harmonic oscillaor ineracing wih a hermal bah.. INTRODUCTION The ransiion from quanum o classical physics and classicaliy of quanum sysems coninue o be among he mos ineresing problems in many fields of physics, for boh concepual and experimenal reasons [,, 3]. Two condiions are essenial for he classicaliy of a quanum sysem [, 5]: a) quanum decoherence (QD), ha means he irreversible, unconrollable and persisen formaion of quanum correlaion (enanglemen) of he sysem wih is environmen [6], expressed by he damping of he coherences presen in he quanum sae of he sysem, when he off-diagonal elemens of he densiy marix of he sysem decay below a cerain level, so ha his densiy marix becomes approximaely diagonal and b) classical correlaions, expressed by he fac ha he Wigner funcion of he quanum sysem has a peak which follows he classical equaions of moion in phase space, ha is he quanum sae becomes peaked along a classical rajecory. In he las wo decades i has became more and more clear ha he classicaliy is an emergen propery of open quanum sysems, since boh main feaures of his process quanum decoherence and classical correlaions srongly depend on he ineracion beween he sysem and is exernal environmen [7]. A remarkable aspec of he curren research helping in undersanding he naure of he quanum o classical ransiion is ha for he firs ime here have recenly been carried on experimens probing he boundary beween he quanum and he classical domains in a conrolled way [3]. In mos of lieraure, he quanum decoherence has been sudied for a sysem coupled o an environmen or hermal bah wih many degrees of freedom. The main purpose of his paper is o sudy QD for a harmonic oscillaor Rom. Journ. Phys., Vol. 50, Nos., P. 7 56, Buchares, 005

2 8 A. Isar ineracing wih an environmen in he framework of he Lindblad heory for open quanum sysems. More concreely we deermine he degree of QD for a sysem consising of a harmonic oscillaor in a hermal bah. For ha purpose, we find he evoluion of he densiy marix and of he Wigner funcion of he considered sysem and hen we apply he crierion of QD. I is found ha he sysem manifess a QD which is more and more significan in ime. The organizing of he paper is as follows. In Sec. we wrie he Lindblad maser equaion for he damped harmonic oscillaor and in Sec. 3 we derive he maser equaion in coordinae represenaion and he corresponding Fokker- Planck equaion in he Wigner represenaion and find he densiy marix and Wigner funcion of he considered sysem. Then in Sec. we invesigae QD and analyze i quaniaively. A summary and concluding remarks are given in Sec. 5.. LINDBLAD MASTER EQUATION FOR THE HARMONIC OSCILLATOR Here we review he Lindblad s axiomaic formalism based on quanum dynamical semigroups. The irreversible ime evoluion of an open sysem is described by he following general quanum Markovian maser equaion for he densiy operaor ρ () [8, 9, 0]: d ρ( ) = i [ H,ρ ( )] + ([ Vjρ ( ), V ] [ j ( ) ]) d j + V,ρ V j. H is he Hamilonian operaor of he sysem and j V j, () V j are operaors on he Hilber space of he Hamilonian, which model he environmen. In order o obain, for he damped quanum harmonic oscillaor, equaions of moion as close as possible o he classical ones, he wo possible operaors V and V are aken as linear polynomials in coordinae q and momenum p [,, 3] and he harmonic oscillaor Hamilonian H is chosen of he general quadraic form µ H = H 0 + ( qp + pq), H m 0 = p + ω q. () m Wih hese choices he maser equaion () akes he following form [, 3]: dρ = i [ H0,ρ ] i ( λ+µ )[ q,ρ p + p ρ ] + i ( λ µ )[ p,ρ q + q ρ ] d Dpp Dqq Dpq [ q, [ q,ρ]] [ p, [ p,ρ ]] + ([ q, [ p,ρ ]] + [ p, [ q,ρ ]]). The quanum diffusion coefficiens D pp, D qq, D pq and he dissipaion consan λ saisfy he following fundamenal consrains [, 3]: D pp > 0, D > 0 and qq (3)

3 3 Decoherence in open quanum sysems 9 D ppdqq D λ pq. () In he paricular case when he asympoic sae is a Gibbs sae H H 0 0 ρ ( ) = e kt / Tre kt, hese coefficiens become [, 3] G λ+µ λ µ Dpp = mω coh ω, Dqq = coh ω, Dpq = 0, (5) kt mω kt where T is he emperaure of he hermal bah. In his case, he fundamenal consrains are saisfied only if λ>µ and ( λ µ )coh ω λ. kt (6) Lindblad has proven [] ha in he Markovian regime he harmonic oscillaor maser equaion which saisfies he complee posiiviy condiion canno saisfy simulaneously he ranslaional invariance and he deailed balance (which assures an asympoic approach o he canonical hermal equilibrium sae). The necessary and sufficien condiion for ranslaional invariance is λ = µ [,, 3]. If λ µ, hen we violae ranslaional invariance, bu we keep he canonical equilibrium sae. The relaion () is a necessary condiion for he generalized uncerainy inequaliy qq pp pq σ () σ () σ () (7) o be fulfilled, where σ qq and σ pp denoe he dispersion (variance) of he coordinae and momenum, respecively, and σ pq denoes he correlaion (covariance) of he coordinae and momenum. The equaliy in relaion (7) is realized for a special class of pure saes, called correlaed coheren saes [] or squeezed coheren saes. The asympoic values σqq( ), σpp( ), σpq( ) do no depend on he iniial values σ qq(0), σ pp(0), σ pq(0) and in he case of a hermal bah wih coefficiens (5), hey reduce o [, 3] σqq( ) = coh ω, σpp( ) = mω coh ω, σpq( ) = 0. mω kt kt (8) 3. DENSITY MATRIX AND WIGNER DISTRIBUTION FUNCTION We consider a harmonic oscillaor wih an iniial Gaussian wave funcion

4 50 A. Isar Ψ ( q) = (0) πσqq exp ( i σ (0))( (0)) i pq q σ q + σ p(0) q, σqq(0) where σ qq(0) is he iniial spread, σ pq(0) he iniial covariance, and σ q(0) and σ p (0) are he iniial averaged posiion and momenum of he wave packe. The iniial sae (9) represens a correlaed coheren sae [] wih he variances and covariance of coordinae and momenum σ qq(0) = δ, σ pp(0) = mω, σ (0) r ( pq =. (0) mω δ r ) r Here, δ is he squeezing parameer which measures he spread in he iniial Gaussian packe and r, r < is he correlaion coefficien a ime = 0. The iniial values (0) correspond o a minimum uncerainy sae, since hey fulfil he generalized uncerainy relaion σ (0) (0) qq σpp σ pq(0) =. () For δ = and r = 0 he correlaed coheren sae becomes a Glauber coheren sae. For a given emperaure T of he bah and for any parameers δ and r he inequaliy (6) alone deermines he range of values of he parameers λ and µ [5]. From Eq. (3) we derive he evoluion equaion in coordinae represenaion: ρ = i ρ imω ( q q ) ρ m q q ( λ+µ )( q q ) ρ+ ( λ µ ) ( q+ q ) + + ρ ( q q ) ( q q ) qq ( ) pq q q ( q q ) Dpp q q D id q q ( ) ρ+ + ρ ( ) + ρ (9) () and in Refs. [6, 7, 8] we ransformed he maser equaion (3) for he densiy operaor ino he following Fokker-Planck-ype equaion saisfied by he Wigner disribuion funcion W(q, p, ): W p = W + mω q W + ( λ+µ ) ( pw) + ( λ µ ) ( qw) + m q p p q (3) W W + D pp + D W qq + D pq. p q pq

5 5 Decoherence in open quanum sysems 5 The firs wo erms on he righ-hand side of boh hese equaions generae a purely uniary evoluion. They give he usual Liouvillian evoluion. The hird and forh erms are he dissipaive erms and have a damping effec (exchange of energy wih environmen). The las hree are noise (diffusive) erms and produce flucuaion effecs in he evoluion of he sysem. D pp promoes diffusion in momenum and generaes decoherence in coordinae q: i reduces he offdiagonal erms, responsible for correlaion beween spaially separaed pieces of he wave packe. Similarly D qq promoes diffusion in coordinae and generaes decoherence in momenum p. The D pq erm is he so-called anomalous diffusion erm. I promoes diffusion in he variable qp + pq, jus like boh he oher diffusion erms, bu i does no generae decoherence. The densiy marix soluion of Eq. () has he general form of Gaussian densiy marices where < q ρ () q >= qq( ) πσ q+ q σ() exp ( σq( ) ) ( q q ) + ( ) σqq σqq( ) iσ pq() q+ q ( q()( ) q q) i + σ + σp()( q q ), σqq() qq pp pq () σ() σ () σ () σ () (5) is he deerminan of he dispersion (correlaion) marix σqq() σpq()(6) M () = σpq() σpp()(7) (6) and represens also he Schrödinger generalized uncerainy funcion [5]. For an iniial Gaussian Wigner funcion (corresponding o a correlaed coheren sae (9)) he soluion of Eq. (3) is { Wqp (,, ) = exp [ σ ( )( ( )) pp q σ q + π σ( ) σ( ) qq p pq q p +σ ( )( p σ ( )) σ ( )( q σ ( ))( p σ ( ))]. } (7) In he case of a hermal bah we obain he following seady sae soluion for (we denoe ε ω ): kt

6 5 A. Isar 6 mω ( ) < q ρ( ) q >= π coh ε ( q q) exp m + ( q q) ω + coh. ε cohε In he long ime limi we have also ( ) exp p W q, p = mω q +. π coh coh mω (9) ε ε Saionary soluions o he evoluion equaions obained in he long ime limi are possible as a resul of a balance beween he wave packe spreading induced by he Hamilonian and he localizing effec of he Lindblad operaors. (8). QUANTUM DECOHERENCE As we saed in he Inroducion, QD is a condiion ha has o be saisfied in order ha a sysem could be considered as classical. This condiion requires ha he sysem should be in one of relaively permanen saes (saes ha are leas affeced by he ineracion of he sysem wih he environmen, called by Zurek preferred saes in he environmen induced superselecion descripion [, 3]) and he inerference beween differen saes should be negligible. This implies he desrucion of off-diagonal elemens represening coherences beween quanum saes in he densiy marix, which is he QD phenomenon. This does no imply ha he knowledge of he sae of he sysem is necessarily precise, by conrary, in he general case we may have acually only a probabilisic descripion. For example, an iniial pure sae wih a densiy marix which conains nonzero off-diagonal erms can non-uniarily evolve ino a final mixed sae wih a diagonal densiy marix during he ineracion wih he environmen, like in classical saisical mechanics. An isolaed sysem has an uniary evoluion and he coherence of he sae is no los: pure saes evolve in ime only o pure saes. The loss of coherence can be achieved by inroducing an ineracion beween he sysem and environmen. The densiy marix does no diagonalize exacly in posiion, bu wih a non-zero widh, i.e. i is srongly peaked abou q= q and very small for q far from q. Using new variables Σ= ( q+ q )/ and = q q, he densiy marix () can be rewrien as ρσ,, ( ) = α exp ασ γ +βσ + i π σ p() + ασ ( ) ( ) q Σ+ i βσq ασ q( ), (0)

7 7 Decoherence in open quanum sysems 53 wih he abbreviaions σ() σ α=, γ=, β= () pq σqq( ) σqq( ) σqq( ) and he Wigner ransform of he densiy marix (0) is () Wqp (,, ) = α π γ [ β( q σ ()) ( ())] q p σp exp ( q ( )) α σ q. γ () The represenaion-independen measure of he degree of QD [] is given by he raio of he dispersion / γ of he off-diagonal elemen ρ (0,, ) o he dispersion /α of he diagonal elemen ρσ, ( 0, ) which in our case gives δ α QD =, γ δ QD () =. σ( ) (3) () The finie emperaure Schrödinger generalized uncerainy funcion (5), calculaed in Ref. [5], has he expression σ () = e λ coh coh δ+ + + δ( r ) ε ε ω µ Ω ε cos( ) e + λcohε δ+ coh + δ( r ) Ω µ sin( Ω ) rµω( cos( Ω)) coh + δ ( r ) + +. r ε δ Ω Ω In he limi of long imes Eq. (5) yields (5) so ha we obain For high T we ge σ ( ) = coh ε, δ ( ) QD =. coh ω kt (6) (7)

8 5 A. Isar 8 δqd ( ) = ω. (8) kt We see ha δ QD decreases, and herefore QD increases, wih emperaure, i.e., he densiy marix becomes more and more diagonal a higher T and he conribuions of he off-diagonal elemens ge smaller and smaller. A he same ime he degree of puriy decreases and he degree of mixedness increases wih T. For T = 0 he asympoic (final) sae is pure and δ QD reaches is iniial maximum value. A pure sae undergoing uniary evoluion is highly coheren: i does no lose is coherence, i.e. off-diagonal coherences never vanish. δ QD = 0 when he quanum coherence is compleely los. So, when δ QD = here is no QD and only if δ QD < here is a significan degree of QD, when he magniude of he elemens of he densiy marix in he posiion basis are peaked preferenially along he diagonal q= q. When δqd we have a srong QD. For simpliciy we consider zero values for he iniial expecaions values of coordinae and momenum and he expression () of he Wigner funcion becomes ( q p) Wqp ( ) α β exp q,, = α. π γ γ (9) In coordinaes βq p and q, γ and / α are he lenghs of he shorer and longer semi-axes of he σ conour and heir produc gives he area of he σ ellipse. We see from Eq. (3) ha δ QD is inversely proporional o his area. Besides his geomeric inerpreaion, δ QD is also conneced wih he linear enropy [9, 0]. We have seen ha if he iniial wave funcion is Gaussian, hen he densiy marix () and he Wigner funcion (7) remain Gaussian for all imes (wih ime-dependen parameers which deermine heir ampliude and spread) and cenered along he rajecory given by he soluions σ q( ) and σ p ( ) of he dissipaive equaions of moion. The degree of QD has an evoluion which shows ha in general QD increases wih ime and emperaure. δ QD < for long enough ime, so ha we can say ha he considered sysem ineracing wih he hermal bah manifess QD. Dissipaion promoes quanum coherences, whereas flucuaion (diffusion) reduces coherences and promoes QD. The balance of dissipaion and flucuaion deermines he final equilibrium value of δ QD. The quanum sysem sars as a pure sae, wih a Wigner funcion well localized in phase space (Gaussian form).

9 9 Decoherence in open quanum sysems 55 This sae evolves approximaely following he classical rajecory (Liouville flow) in phase space and becomes a quanum mixed sae during he irreversible process of QD. 5. SUMMARY AND CONCLUDING REMARKS In he presen paper we have sudied QD wih he Markovian equaion of Lindblad in order o undersand he quanum o classical ransiion for a sysem consising of an one-dimensional harmonic oscillaor in ineracion wih a hermal bah in he framework of he heory of open quanum sysems based on quanum dynamical semigroups. The role of QD became relevan in many ineresing physical problems from field heory, aomic physics, quanum opics and quanum informaion processing, o which we can add maerial science, heavy ion collisions, quanum graviy and cosmology, condensed maer physics. Jus o menion only a few of hem: o undersand he way in which QD favorizes he quanum o classical ransiion of densiy flucuaions; o sudy sysems of rapped and cold aoms (or ions) which may offer he possibiliy of engineering he environmen, like rapped aoms inside caviies, relaion beween decoherence and oher caviy QED effecs (such as Casimir effec); on mesoscopic scale, decoherence in he conex of Bose-Einsein condensaion [, 3]. In many cases physiciss are ineresed in undersanding he specific causes of QD jus because hey wan o preven decoherence from damaging quanum saes and o proec he informaion sored in quanum saes from he degrading effec of he ineracion wih he environmen. Thus, decoherence is responsible for washing ou he quanum inerference effecs which are desirable o be seen as signals in some experimens. QD has a negaive influence on many areas relying upon quanum coherence effecs, such as quanum compuaion and quanum conrol of aomic and molecular processes. The physics of informaion and compuaion is such a case, where decoherence is an obvious major obsacle in he implemenaion of informaion-processing hardware ha akes advanage of he superposiion principle []. The sudy of classicaliy using QD leads o a deeper undersanding of he quanum origins of he classical world. Much work has sill o be done even o sele he inerpreaional quesions, no o speak abou answering hem. Neverheless, as a resul of he progress made in he las wo decades, he quanum o classical ransiion has become a subjec of experimenal invesigaions, while previously i was mosly a domain of philosophy [, 3]. The issue of quanum o classical ransiion poins o he necessiy of a beer undersanding of open quanum sysems. The Lindblad heory provides a selfconsisen reamen of damping as a general exension of quanum mechanics

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