Degree of information accessibility: a precursor of the quantum–classical transition J. G. Oliveira Junior1,2∗, J. G. Peixoto de Faria3†, M. C. Nemes2‡ 1Centro de Forma¸c˜ao de Professores, Universidade Federal do Recˆoncavo da Bahia, 45.300-000, Amargosa, BA, Brazil 2Departamento de F´ısica - CP 702 - Universidade Federal de Minas Gerais - 30123-970 - Belo Horizonte - MG - Brazil 3Departamento de F´ısica e Matema´tica - Centro Federal de Educa¸c˜ao Tecnolo´gica de Minas Gerais - 30510-000 - Belo Horizonte - MG - Brazil (Dated: January 26, 2011) One of the fundamental problems with the interpreta- U+ tionofQuantumMechanics,accordingtoBohr[1],isthe in factthat“ourusualdescription ofphysical phenomena is based entirely on the idea that the phenomena concerned © may be observed without disturbing them appreciably”. Furthermore in his articles [1, 2] discussing the subject Bohr argues that the actionof the probe will be affected 1 by the system and inasmuch the system will be affected 1 0 by the probe. Specifically in Gedanken experiments he U_ 2 tests the wave–particleduality of the systemandimplic- out itly assumes thatthe probeis alsoa quantumsystem. A n a universal character can only be attributed to Quantum J Mechanics provided a complementarity relation is also FIG. 1: Mach–Zehnder interferometer with perfect beam 5 valid for the probe. As a consequence the state system– splitters (BS) and mirrors (M). The vector 1 (0 ) repre- 2 probe becomes entangled as extensively discussed by N. | i | i sent the situation where the particle goes through the upper Bohr, W. Heisenberg, A. Einstein, E. Schr¨odinger, de (lower) arm, respectively. After the first beam splitter the ] Broglie and W. Pauli among others starting in the fa- h particle interacts with the probe. It is then reflected by a p mousSolvayconference[3]. Soonafterentanglementhas mirror and acquires a relative phase (Φ). In the sequel it is - been brought to discussion [4, 5] and since then it has detectedinD orD andwillexhibitaninterferencepattern t 1 2 n always been present in several contexts [6]. In the past with sinusoidal modulation as a function of Φ. The action of a fourty years complementarity tests have been proposed the BS, of the probe and of the acumulated relative phase u as,e.g.,usingdoubleslitexperiments[7],quantumeraser between in thearms will berespectively modelled bythefol- q [8–11], cavity interferometry [12, 13], and experiments lowing unitary operators UB = (1l iσy) 1l/√2, UP = 1 [ iHnovwolevvienrg,mwuhcichhl–ewssayatdteenteticotniohna[s1b4–ee1n7]p,atiodqtouottheessotumdey. U|1±ih|1m|⊗i=U+|m+±|0i,ihU0±|⊗U±U†−=e1lUaFnd=σ(yei=Φ|−1(cid:0)i−h1i||1+ih|⊗00|i+h0|i)|0⊗ih11l|w(cid:1).here v of an arbitrary probe system. In the present contribu- 4 tion, we fill in this gap and show that the key ingredi- 9 space (1) (2), where the subsystem in (1) has two entforthequantum–classicaltransitionisnotnecessarily H ⊗H H 6 levels and is entangled to another system in (2). In 4 the information generated by the system–probe interac- general one can describe in the Schmidt base [2H1] . tionbutratherbyitsaccessibility. Ourresultshavebeen 1 successfully tested in the interferometric experiment by 2 0 [13]. Our results also allow for a simple physical inter- Ψ = λ v(1) v(2) (1) 1 | i i| i i| i i 1 pretation of the physics of Ramsey Zones [18] where one Xi=1p : photon (average) interacts with a two level atom in a Xiv claLsseitcaulsmcaonnnsiedre,ri.ea.,Mnaocehn–tZaenhgnledmereninttiesrgfeernoemraetteedr.(see (wfiHchi(ee2nr)et)s,{.r|evTsi(1ph)eeic}stti(va{et|levyi(,o2)afint}dh)etahsryees√toerλmthioiannroermt(h1a)elwSbcialhlsmebsiedintρcHoe(=1f)- r [19] for a revision) with a probe system capable of ob- H (1) a taining and storing the which–way information provided 2i=1λi|vi(1)ihvi(1)|. Since λ1+λ2 =1 it follows immedi- byaparticlesentthroughit,asshowninFigure1. Inthe Patelythat(λ1 λ2)2+4detρ(1) =1,where4detρ(1)isthe − processofobtaining andstoringwhich–wayinformation, concurrence C squared [22] between particle–probe. (1,2) the probe and particle will entangle. This results in a Thesystemin (1)isatwolevelsystemandthereforecan kind of Einstein–Podolsky–Rosen [5] pair. The configu- be written in tHerms of Pauli matrices σ ,σ†,σ . The rationisthatofabipartite systemdefinedinthe Hilbert { − − z} equationabovebecomes σ(1) 2+4 σ(1) 2+C2 =1, h z i |h − i| (1,2) where σ(1) = tr(ρ σ ). Note that as we identify h k i (1) k the σ(1) eigenstates with the arms of the interferometer, ∗Electronicaddress: [email protected] z †Electronicaddress: [email protected] σz(1) represents the probability of finding the parti- |h i| ‡Electronicaddress: carolina@fisica.ufmg.br cle in one of the arms, usually called predictability (1) P 2 [14, 23]. On the other hand, the term σ(1) represents will be ρ =tr U U U U ρ U†U†U†U† and the |h − i| (1) (2){ B F P B 0 B P F B} the magnitude of the coherence between the interferom- intensity I will be proportional to I 0ρ 0 = (1) ∝ h | | i eters arms. The interference fringes will exhibit a vis- 1+(z +iy ) m m e−iΦ+(z iy ) m m eiΦ. This 0 0 + − 0 0 − + ibility = 2 σ(1) [23]. Having these two results it yields the vhisibil|ity i = −√1 h 2,| whiere = fwohlleorwesVt12hat=S(|21h)2−++Ci|(212,2) r=ep1r,esreenstusltwfiarvset–fpoaurntidcliend[u2a3l]-, pdirye02ct+lyzd02eipsetnhdesaopnrtiohVre(i1dv)iisstibinilgViut0yis[h2a5b−].ilDiOtybsoefrtvheethinaftoVr0Vm(1a)- S(1) P(1) V(1) ity. Given the symmetry of the Schmidt decomposition tioncontainedintheprobe. Thismeansthatwhich–way the wave–particleduality of the other system will be the informationgenerationisanecessaryconditiontodestroy same, the interferencepatternofthe particle. Itis howevernot enough. In order to destroy interference it is necessary 2 = 2 . (2) not only to generate information but also to make it ac- S(2) S(1) cessible. The concurrence [22] between the particle and In our problem the second system is the probe. Let the probe is given by us consider that the particle has been prepared in the state (1 0 )/√2andsentthroughthe interferometer. C(1,2) = 0 (3) | i±| i V D The probe, prepared in m , evolves to m . We may | i | ∓i and it is easy to the verify that 2 +C2 =1. at this point ask for the probability that a measurement S(1) (1,2) of the state of the probe will yield m . This probabil- The probe: From the preceding discussion its easy to ± ity will be = m m 2, for bo|th ciases and may be see that the complementarity relation for the probe sys- − + interpretedOas th|he im|perfeic|tion of the probe. As a con- tem entangled with a particle has the same form as that sequence we may define Q=1−|hm−|m+i|2 a quantity for the first system S(22) +C(21,2) = 1, where the duality which represents the quality of the probe. The quantity must respect equation (2). In order to have sep- (2) S =√ isaquantitativemeasureofthedistinguishabil- arated expressions to characterize the particle or wave D Q ity of the probe states [14], i.e., of the which–way infor- likenatureoftheprobesystemwewillhavetoconstruct, mationavailableintheprobesystem. Inthisnotationwe e.g., a quantity P which reflect the particle character of cansaythataperfectprobewith =1isthe onewhich the probe. Its wave like character can then be extracted makes the which–way informationDcompletely available. from the equality 2 = 2 . S(2) S(1) Thusaperfectprobe( =1)willhavecompletelydistin- Physical conditions for constructing P: After the probe Q guishable probe states (|hm−|m+i|=0) and the particle interactswiththeparticleitwillunitarilyevolveto|m±i. state will be a statistical mixture showing thus no inter- The operators Π = m m are projectors and their ± ± ± | ih | ference pattern. On the other hand, when the probe is averages Π =tr ρ Π aretheprobabilitiestofind ± (2) ± icnodmisptlientgeulyishimabpleerf(ecmt −(Qm=+ 0=) t1h)eapnrdobdeiffsetraatetsmwoisltl bbye ttihcelepprorebde(cid:10)icitna(cid:11)b|milit±yi,(cid:0)wreeswpoecutl(cid:1)idvelliyk.e tSoimhialavrelyPto=th0ewphaern- phase. This phase|his n|ot cio|ntain which–way informa- Π = 1/2 and P = 1 when Π = 1. Another inter- ± ± tion. Asaconsequencethe globalstateisfactorized. For (cid:10)estin(cid:11)g condition is that P has(cid:10)the(cid:11)interpretation of the aprobewith0< <1wewillhaveinformationgenera- predictability when the probe is perfect. All these con- tionhoweveritsaQccessibilityisonlypartial,proportional ditions will be satisfied when we define P= 1 2 Π ± − to Q. In this case the particle will present an intermedi- (and±x0 −→|x0|). Theprobestateforthee(cid:12)(cid:12)valuat(cid:10)iono(cid:11)(cid:12)(cid:12)f atevisibility. Thismeansthatgeneratingthewhich–way Pwillbeρ =tr U U U U ρ U†U†U†U† . From informationisnotenoughtodestroyitsinterferencepat- the above(c2o)nstru(c1t)i{onBweFgePt B 0 B P F B} tern. The particle: A general initial state of the system can P=1 1 2 . (4) (1) be written as ρ0 = 1l +u0 σ(1) m m/2 where, −(cid:2) −P (cid:3)D uσ(1)== {xσ,x(1y),,σzy(1),σisz({1)t}hearBeloP·cahulvie}cmt⊗oart|riocifehst,h||eu0p|ar=tic1le, theThpeatfhaccthoorse(1n−byPt(h1)e)pqauratinctliefi.esOonutrhuenoctehretraihnatynda,s to2 0 0 0 0 D { } measures the distinguishability of the final states of the state and m m the initial state of the probe. The | ih | probe. Foravalueof fixed,thepowertopredictthefi- particle state immediately before the second BS will be D ρ˜ = tr U U U ρ U†U†U† and its predictability nalstateoftheprobedecreasesasthisuncertaintygrows. (1) (2){ F P B 0 B P F} Theproduct(1 ) 2 quantifiesthecontributiondue P(1) =|x0|whichisidenticaltotheaprioripredictability tothewave–like−chPa(1r)acDteroftheprobeand thenon–local [24]. NowiftheparticleisdetectedD andthe phase P0 1 correlationsbetweenthetwosubsystemsproducedbythe differenceΦvarieswewillobtainedaninterferencefringe interaction. with visibility Probe wave like character: As discussed before it is now I I immediate to obtain the wave–like characteristics of the max min V(1) = I +−I probe. One substitutes equation(4)in (2) andassuming max min that 2 =P2+V2, we get S(2) where I (I ) is the maximum (minimum) fringe in- max min tensity, respectively. After secund BS the particle state V=(1 ) 1 2 . (5) (1) −P D −D p 3 Since 0 1, the product √1 2 varies be- 1 ≤ D ≤ D −D tween 0 and 1/2. Two different values of are asso- C ciated to a particular value of this product.DHence, we 0.8 al Qu have V = 0 both for the perfect probe (which we asso- ss Tr na ciatewiththe extremequantumregime)asforimperfect 0.6 i a t c n u one (extreme classical regime). When the probe is per- a s m fect it entangles with the particle and the global state 0.4 l ti R of the system is an authentic Einstein–Podolsky–Rosen R i e e o g pair [5]. In the opposite limit the probe state remains g n i unaltered and therefore no quantum features of any sort 0.2 mi me e will be present. Interesting to note that one will always have V C where the equality is only reached in 0 (1,2) 0 0.2 0.4 0.6 0.8 1 trivial ca≤se P = 1. Of course this is a particular fea- D ture of the problem in question, since there is no local unitary operationon the probe to enhance V. Thus par- FIG. 2: Probe characteristics: the black (blue) curve is V (C ), respectively, and the red curve represents P (for ticlecharacteristicsanditsabilitytoentanglewillassume (1,2) an important role in the complementarity relation. Re- 0 =0). The vertical lines locate the points =1/√2 and P D =2/(√5+1). Wewillassumethattheprobeiswithinthe sorting to the complementarity relation for the probe, D domains of classical physics when particle characteristics be- 2 +C2 =1,andtoequation(4),itisnowclearthat tSh(2e)prod(1u,c2t) (1 (1)) 2 represents the sum of the con- cinotmereesstdionmgitnoannottPe≥thaVt+inCt(h1,i2s)syitiueladtiinogn0th≤epDar<∼tic0l.e42v8is8i.biIltitiys −P D tributionsrelatedtothewave–likebehavioroftheprobe, will undergo a variation of ∆ 0.0965. This variation is quantified by V, and the non–local correlations due to meaningful however in experiVm(1e)nt≈al apparatuses which may the interaction, measured by C . reproducetheseresults. Inrealisticconditionstheexperimen- (1,2) From the quantum to the classical limit: For an arbi- tal error η will exist and variations resulting from measure- tforarmryaptiaornticwleillstbaetem, tihneimauvamilawbhileitnyoSf(2t)he=whPic=h–w1a,yi.ien.-, tηminetghnuetiwsvhailirlniabgteitohanecce∆offuV¯en(c1tt)esdmoffaotyrhbebyew∆whiVi¯cth(h1–i)nw=athyηe∆ineVfror(r1om)r.abDtaieropsn,esntthdouirnsegddoiisnn- the probe is completely imperfect and devoid of quan- the probe. This contributes to the idea of classical behavior tum behavior. In this case, the extreme classical regime see [13] . is reached and the probe exhibits corpuscular behavior . Ontheotherhand,itwillbemaximumwhen =P= (2) S x0 whenitsbehaviorwillbecompletelyquantum. Note global state immediately after the interaction will be | | that = C when the particle is prepared in a state (1,2) whichDsatisfies =0. Inthiscasetheprobecomplemen- 1 tarity relationPo0nly depends on its own characteristics. |φi= √2 |ei|α+i+|gi|α−i (cid:0) (cid:1) Wecanthusdefineagoodprobewhen >1/√2,i.e.,its D imperfectionwillbesmallerthan1/2. Theoppositelimit, with α = √2 C cos(Ω√n+1t )n and i.e., <1/√2 is a weak condition since there will be a | +i (cid:20) n n i | i(cid:21) D P region 2/ 1+√5 < < 1/√2 where C P V, as shown(cid:0)in figure(cid:1)2. DThus a better defini(t1i,o2n) ≥for a≥bad |α−i = √2(cid:20) nCnsin(Ω√n+1ti)|n+1i(cid:21), where Cn = probe is P>C(1,2) <2/ √5+1 . e−|α|2/2αn/√Pn!, Ω is the Rabi frequency in vacuum and −→ D (cid:0) (cid:1) t isthenecessarytimefortheatomtosufferaπ/2pulse i The physics of Ramsey Zones: To display the general- defined by C 2cos2(Ω√n+1t ) = 1/2. If we have ityofourfindingswewilldiscussintheirlightanintrigu- α2 = 1 wPe wni|llnh|ave P 0,3271i, V 0,4691 and | | ≈ ≈ ing question. Ramsey Zones are microwavecavities with C 0,8203. So one should expect that the field in (1,2) ≈ lowqualityfactor. Itservesthepurposeofrotatingeffec- the Ramsey Zonewouldhavemeasurablequantumchar- tivelytwolevelRydbergatoms[18]. Theprocessconsists acteristics and behave as a good which–way discrimina- in sending atoms through the cavity where there is a co- tor, since the informationavailable is large 0,8203. D≈ herent field of approximately one average photon main- Why don’t the expected features appear? tainedby anexternalsource. The interestingconceptual Immediately after the atom leaves the cavity the dy- issue is: Why do Ramsey Zones work as if they contain namics of the feeding source along with strong cavity classical fields and no entanglement between atom and dissipationactonthefieldstate. SincetheRamseyZone photon is generated? A full microscopic calculation has possessesarelaxationtimesveryshort,i.e., smallerthan been performed [20], however a convincing transparent T =10ns[13,18],theninatimeintervaloftheorder T r r physical picture has not emerged. For typical values of the photons inside the cavity are renewed and the state the atomic flight velocity the time t of the atom field α is restored resulting = 0 with the information no i | i D interaction is that of a few µs [13]. Let us consider t longer available. It is important to note that this not i sufficiently small so that one can assume a unitary evo- mean that the which–way information disappeared. It lutionduring the interaction. Under this hypothesis,the has become stored in a system with infinite degrees of 4 freedom which renders it inaccessible. Therefore atom– might explain why quantum objects like photons cross field disentangle. In other words the field goes almost macroscopic apparatuses and suffer unitary evolution in instantaneouslyfromaprevailingquantumregimewhere the complete absence of entanglement, e.g.: photon plus it was, a priori,a good probe, to a situation where there beam splitters or photons plus quarter wave plates, to will be only corpuscular characteristics P = 1. This is a quote some. classical regime as we defined. This suggests the follow- ing general picture: systems which do not store neither The present work, simple as it is, led us to believe make the information about its interaction available, it thatfurther exploiting Bohr’scomplementarityprinciple can be by intrinsic properties of the systems or by aux- may lead to very simple and solid physical explanation iliaries dynamics, may be consider classical because the for important conceptual problems which at first sight act on the quantum system without get entangled. This appear to be of high mathematical intricacy. 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