Decoherence of Schrodinger cat states in a Luttinger liquid

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Quantum decoherence of the damped harmonic oscillator

a r X i v :q u a n t -p h /0606222v 1 27 J u n 2006QUANTUM DECOHERENCE OF THE DAMPED HARMONIC OSCILLATORA.IsarInstitute of Physics and Nuclear Engineering,Bucharest-Magurele,Romaniae-mail:isar@theory.nipne.roAbstract In the framework of the Lindblad theory for open quantum systems,we de-termine the degree of quantum decoherence of a harmonic oscillator interacting with a thermal bath.It is found that the system manifests a quantum de-coherence which is more and more signiﬁcant in time.We also calculate the decoherence time and show that it has the same scale as the time after which thermal ﬂuctuations become comparable with quantum ﬂuctuations.PACS numbers:03.65.Yz,05.30.-d 1Introduction The quantum to classical transition and classicality of quantum systems continue to be among the most interesting problems in many ﬁelds of physics,for both conceptual and experimental reasons [1,2].Two conditions are essential for the classicality of a quantum system [3]:a)quantum decoherence (QD),that means the irreversible,uncontrollable and persistent formation of a quantum correlation (entanglement)of the system with its environment [4],expressed by the damping of the coherences present in the quantum state of the system,when the oﬀ-diagonal elements of the density matrix decay below a certain level,so that this density matrix becomes approximatelydiagonal and b)classical correlations,expressed by the fact that the Wigner function of the quantum system has a peak which follows the classical equations of motion in phase space with a good degree of approximation,that is the quantum state becomes peaked along a classical trajectory.Classicality is an emergent property of open quantum systems,since both main features of this process –QD and classical correlations –strongly depend on the interaction between the system and its external environment[1,2].The role of QD became relevant in many interesting physical problems.In manycases one is interested in understanding QD because one wants to prevent decoherence from damaging quantum states and to protect the information stored in quantum states from the degrading eﬀect of the interaction with the environment.Decoherence is also responsible for washing out the quantum interference eﬀects which are desirableto be seen as signals in experiments.QD has a negative inﬂuence on many areas relying upon quantum coherence eﬀects,in particular it is a major problem in quantum optics and physics of quantum information and computation[5].In this work we study QD of a harmonic oscillator interacting with an environ-ment,in particular with a thermal bath,in the framework of the Lindblad theory for open quantum systems.We determine the degree of QD and then we apply the criterion of QD.We consider diﬀerent regimes of the temperature of environment and it is found that the system manifests a QD which in general increases with time and temperature.The organizing of the paper is as follows.In Sec.2we review the Lindblad master equation for the damped harmonic oscillator and solve the master equation in coordinate representation.Then in Sec.3we investigate QD and in Sec.4we calculate the decoherence time of the system.We show that this time has the same scale as the time after which thermalﬂuctuations become comparable with quantum ﬂuctuations.A summary and concluding remarks are given in Sec.5.2Master equation and density matrixIn the Lindblad axiomatic formalism based on quantum dynamical semigroups,the irreversible time evolution of an open system is described by the following general quantum Markovian master equation for the density operatorρ(t)[6]:dρ(t)¯h [H,ρ(t)]+12(qp+pq),H0=12q2(2)and the operators V j,V†j,which model the environment,are taken as linear polynomials in coordinate q and momentum p.Then the master equation(1)takes the following form[7]:dρ¯h [H0,ρ]−i2¯h(λ−µ)[p,ρq+qρ]−D pp¯h2[p,[p,ρ]]+D pqcase when the asymptotic state is a Gibbs stateρG(∞)=e−H0kT,these coeﬃ-cients becomeD pp=λ+µ2kT,D qq=λ−µmωcoth¯hω2kT≥λ2(5)and the asymptotic valuesσqq(∞),σpp(∞),σpq(∞)of the dispersion(variance),re-spectively correlation(covariance),of the coordinate and momentum,reduce to[7]σqq(∞)=¯h2kT,σpp(∞)=¯h mω2kT,σpq(∞)=0.(6)We consider a harmonic oscillator with an initial Gaussian wave function(σq(0) andσp(0)are the initial averaged position and momentum of the wave packet)Ψ(q)=(14exp[−1¯hσpq(0))(q−σq(0))2+i 2mω,σpp(0)=¯h mω2√∂t=i¯h∂q2−∂22¯h(q2−q′2)ρ−1∂q−∂2(λ−µ)[(q+q′)(∂∂q′)+2]ρ−D pp∂q+∂∂q+∂a damping eﬀect(exchange of energy with environment).The last three are noise (diﬀusive)terms and produceﬂuctuation eﬀects in the evolution of the system.D pp promotes diﬀusion in momentum and generates decoherence in coordinate q–it re-duces the oﬀ-diagonal terms,responsible for correlations between spatially separated pieces of the wave packet.Similarly D qq promotes diﬀusion in coordinate and gener-ates decoherence in momentum p.The D pq term is the so-called”anomalous diﬀusion”term and it does not generate decoherence.The density matrix solution of Eq.(9)has the general Gaussian form<q|ρ(t)|q′>=(12exp[−12−σq(t))2−σ(t)¯hσqq (t)(q+q′¯hσp(t)(q−q′)],(10)whereσ(t)≡σqq(t)σpp(t)−σ2pq(t)is the Schr¨o dinger generalized uncertainty function. In the case of a thermal bath we obtain the following steady state solution for t→∞(ǫ≡¯hω/2kT):<q|ρ(∞)|q′>=(mω2exp{−mωcothǫ+(q−q′)2cothǫ]}.(11)3Quantum decoherenceAn isolated system has an unitary evolution and the coherence of the state is not lost–pure states evolve in time only to pure states.The QD phenomenon,that is the loss of coherence or the destruction of oﬀ-diagonal elements representing coherences between quantum states in the density matrix,can be achieved by introducing an interaction between the system and environment:an initial pure state with a density matrix which contains nonzero oﬀ-diagonal terms can non-unitarily evolve into aﬁnal mixed state with a diagonal density matrix.Using new variablesΣ=(q+q′)/2and∆=q−q′,the density matrix(10) becomesρ(Σ,∆,t)= πexp[−αΣ2−γ∆2+iβΣ∆+2ασq(t)Σ+i(σp(t)2σqq(t),γ=σ(t)¯hσqq(t).(13)The representation-independent measure of the degree of QD[3]is given by the ratio of the dispersion1/√2/αof the diagonal elementρ(Σ,0,t):δQD(t)=1α24{e−4λt[1−(δ+1δ(1−r2)−2cothǫ)ω2−µ2cos(2Ωt)δ(1−r2))µsin(2Ωt)Ω2√2kT,(16)which for high T becomesδQD(∞)=¯hω/2kT.We see thatδQD decreases,and therefore QD increases,with time and temperature,i.e.the density matrix becomes more and more diagonal at higher T and the contributions of the oﬀ-diagonal elements get smaller and smaller.At the same time the degree of purity decreases and the degree of mixedness increases with T.For T=0the asymptotic(ﬁnal)state is pure andδQD reaches its initial maximum value1.δQD=0when the quantum coherence is completely lost,and ifδQD=1there is no QD.Only ifδQD<1we can say that the considered system interacting with the thermal bath manifests QD,when the magnitude of the elements of the density matrix in the position basis are peaked preferentially along the diagonal q=q′.Dissipation promotes quantum coherences, whereasﬂuctuation(diﬀusion)reduces coherences and promotes QD.The balance of dissipation andﬂuctuation determines theﬁnal equilibrium value ofδQD.The initial pure state evolves approximately following the classical trajectory in phase space and becomes a quantum mixed state during the irreversible process of QD.4Decoherence timeIn order to obtain the expression of the decoherence time,we consider the coeﬃcientγ(13),which measures the contribution of non-diagonal elements in the density matrix(12).For short times(λt≪1,Ωt≪1),we have:γ(t)=−mωδ(1−r2))cothǫ+µ(δ−r2δ√2[λ(δ+r2δ(1−r2))cothǫ−λ−µ−ωr1−r2].(18)The decoherence time depends on the temperature T and the couplingλ(dissipation coeﬃcient)between the system and environment,the squeezing parameterδand the initial correlation coeﬃcient r.We notice that the decoherence time is decreasing with increasing dissipation,temperature and squeezing.For r=0we obtain:t deco=12λ(δ−1).(20) We see that when the initial state is the usual coherent state(δ=1),then the deco-herence time tends to inﬁnity.This corresponds to the fact that for T=0andδ=1 the coeﬃcientγis constant in time,so that the decoherence process does not occur in this case.At high temperature,expression(18)becomes(τ≡2kT/¯hω)t deco=1δ(1−r2))+µ(δ−r24(λ+µ)δkT.(22) The generalized uncertainty functionσ(t)(15)has the following behaviour for short times:σ(t)=¯h2δ(1−r2))cothǫ+µ(δ−1This expression shows explicitly the contribution for small time of uncertainty that is intrinsic to quantum mechanics,expressed through the Heisenberg uncertainty prin-ciple and uncertainty due to the coupling to the thermal environment.From Eq.(23)we can determine the time t d when thermalﬂuctuations become comparable with quantumﬂuctuations.At high temperature we obtain1t d=)+µ(δ−1δ(1−r2)squeezing and correlation,it depends on temperature only.QD is expressed by theloss of quantum coherences in the case of a thermal bath atﬁnite temperature.(2)We determined the general expression of the decoherence time,which showsthat it is decreasing with increasing dissipation,temperature and squeezing.We havealso shown that the decoherence time has the same scale as the time after which thermalﬂuctuations become comparable with quantumﬂuctuations and the valuesof these scales become closer with increasing temperature and squeezing.After the decoherence time,the decohered system is not necessarily in a classical regime.There exists a quantum statistical regime in between.Only at a suﬃciently high temperaturethe system can be considered in a classical regime.The Lindblad theory provides a self-consistent treatment of damping as a general extension of quantum mechanics to open systems and gives the possibility to extendthe model of quantum Brownian motion.The results obtained in the framework ofthis theory are a useful basis for the description of the connection between uncertainty, decoherence and correlations(entanglement)of open quantum systems with their en-vironment,in particular in the study of Gaussian states of continuous variable systemsused in quantum information processing to quantify the similarity or distinguishabilityof quantum states using distance measures,like trace distance and quantumﬁdelity. AcknowledgmentsThe author acknowledges theﬁnancial support received within the Project PN06350101/2006. References[1]E.Joos,H.D.Zeh,C.Kiefer,D.Giulini,J.Kupsch,I.O.Stamatescu,Decoherenceand the Appearance of a Classical World in Quantum Theory(Springer,Berlin,2003).[2]W.H.Zurek,Rev.Mod.Phys.75,715,2003.[3]M.Morikawa,Phys.Rev.D42,2929(1990).[4]R.Alicki,Open Sys.and Information Dyn.11,53(2004).[5]M.A.Nielsen and I.L.Chuang,Quantum Computation and Quantum Information(Cambridge Univ.Press,Cambridge,2000).[6]G.Lindblad,Commun.Math.Phys.48,119(1976).[7]A.Isar,A.Sandulescu,H.Scutaru,E.Stefanescu,W.Scheid,Int.J.Mod.Phys.E3,635(1994).[8]A.Isar,W.Scheid,Phys.Rev.A66,042117(2002).[9]B.L.Hu,Y.Zhang,Int.J.Mod.Phys.A10,4537(1995).[10]A.Isar,W.Scheid,Physica A,in press(2006).。

Superconducting qubits II Decoherence

present address: Physics Department and Insitute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; fwilhelm@iqc.ca ‡ mgeller@
The transition from quantum to classical physics, now known as decoherence, has intrigued physicists since the formulation of quantum mechanics (Giulini et al., 1996; Leggett, 2002; Peres, 1993; Feynman and Vernon, 1963; Zurek, 1993). It has been put into the poignant Schr¨ odinger cat paradox (Schr¨ odinger, 1935) and was considered an open fundamental question for a long time.
and compare it to the corresponding classical mixture leading to the same expectation value of σz 1 1 0 ρmix = (2) 2 0 1 we can see that the von-Neumann entropy ρ = −kB Tr [ρ log ρ] rises from Spure = 0 to Smix = kB ln 2. Hence, decoherence taking ρpure to ρmix creates entropy and is irreversible. Quantum mechanics, on the other hand, is always reversible. It can be shown, that any isolated quantum system is described by the Liouville von-Neumann equation i¯ hρ ˙ = [H, ρ] (3)

Decoherence in a system of many two--level atoms

a r X i v :q u a n t -p h /0511262v 2 19 S e p 2006Decoherence in a system of many two–level atomsDaniel BraunLaboratoire de Physique Th´e orique,IRSAMC,UMR 5152du CNRS,Universit´e Paul Sabatier,118,route de Narbonne,31062Toulouse,FRANCEI show that the decoherence in a system of N degenerate two–level atoms interacting with abosonic heat bath is for any number of atoms N governed by a generalized Hamming distance (called “decoherence metric”)between the superposed quantum states,with a time–dependent metric tensor that is speciﬁc for the heat bath.The decoherence metric allows for the complete characterization of the decoherence of all possible superpositions of many-particle states,and can be applied to minimize the over-all decoherence in a quantum memory.For qubits which are far apart,the decoherence is given by a function describing single-qubit decoherence times the standard Hamming distance.I apply the theory to cold atoms in an optical lattice interacting with black body radiation.Decoherence is considered one of the main ingredients in the transition from quantum mechanics to classical mechanics [1].According to experimental evidence (see for example [2,3])“decoherence”can be well described within the standard framework of quantum mechanics [4].The loss of coherence of quantum mechanical super-positions becomes increasingly rapid when the “distance”between the components of the superpositions gets large [5,6],such that quantum mechanical behavior like in-terference of wave functions is never observed for macro-scopic objects.With few notable exceptions [7,8,9],most work on decoherence has focused so far on systems with only one or a few eﬀective degrees of freedom cou-pled to a heat bath.Even when large numbers of particles come into play,like in superconducting qubits based on Josephson junctions [10]or superradiance [11,12],only a single eﬀective degree of freedom is normally retained.With the rise of quantum information science the funda-mental question whether for large particle (qubit)num-bers there might be new self-limitations of quantum me-chanics [13,14,15],based e.g.on the amount of entan-glement in the quantum states [16,17],has also become of huge practical importance.Indeed,understanding and possibly suppressing decoherence of quantum superposi-tions in the exponentially large Hilbert space of possibly thousands of qubits is considered the most diﬃcult prob-lem on the way to a large scale quantum computer.In the following I show that a generalized Hamming distance with a time–dependent metric tensor given by the heat bath and the couplings to the heat bath com-pletely determines the decoherence process of a an arbi-trarily large number of qubits.The standard Hamming distance,which counts the number of bits by which two code words diﬀer,plays a central role in both classical and quantum error correction [18,19],and determines in particular what code words remain distinguishable in a given error model.It is therefore very satisfying that the heat bath itself determines the metric relevant for the de-coherence process,as decoherence reﬂects to what extent the heat bath distinguishes the superposed states [5].Starting point of the analysis are N two–level atoms at arbitrary but ﬁxed positions R i (i =0,...,N −1)interacting with a common bosonic heat bath.All atoms are assumed identical with level spacing Ω0and energy eigenstates |−1 and |1 ,with σz |±1 =±|±1 .The total Hamiltonian readsH =kωk a †k a k +1ωk /(2ε0V ),where ε0,c ,and V are thedielectric constant of the vacuum,speed of light,and the quantization volume,respectively.The coupling constantof atom i to mode k is denoted by g (i )k =−ed Ek2It is interesting to consider the decoherence process of n selected atoms (i =0,...,n −1)out of the N atoms.This oﬀers the additional information how decoherence scales as function of the size of a sub-cluster in the quan-tum memory.The common heat bath can lead to entan-glement between the atoms [22,23],such that unobserved atoms can become an important source of “indirect”de-coherence.Besides its conceptual importance,this can be relevant practically,if additional atoms are trapped in an optical lattice,or when information is stored in an atomic gas in a large number of atoms [24].We therefore ﬁrst trace out the bosonic modes and secondly the ad-ditional atoms n...N −1.The reduced density matrix ρof the remaining atoms will be expressed in the eigen-basis of the σxi ,the natural basis (also called pointer basis)for studying the decoherence process [5].The ma-trix elements of ρare ρ˜s ˜s ′(t ),where ˜s =(s 0,...,s n −1)is a subset of the code word s =(s 0,s 1,...s N −1)(with σxi |s i x =s i |s i x ,s i =±1,and similarly for ˜s ′,i.e.˜s and ˜s ′are binary representations of the matrix indices).We assume that the heat bath is initially in thermal equilib-rium at temperature T ,and all atoms are prepared in a pure state.Using the method of shifted harmonic oscillators [25]one shows that the time evolution of ρis given by ρ˜s ˜s ′(t )=ρ˜s ˜s ′(0)×(2)exp −n −1i,j =0(s i −s ′i )(s j −s ′j )f ij (t,R i −R j )+in −1 j =1j −1 i =0(s i s j −s ′i s ′j )ϕij (t,R i −R j )N −1 l =ncos n −1i =0(s i −s ′i )ϕil (t,R i −R l ).The functions f ij and ϕij are given by f ij (t,R )=kg (i )kg (j )k2ϕij (t,R )=2kg (i )k g (j )k|ρ˜s ˜s ′(0)|≡1−d ˜s ˜s ′(t ).(4)Expanding eq.(2)to lowest order in f ij and ϕij we ﬁndd ˜s ˜s ′(t )≃123and (|100 +|011 )/√3 metric has the form of a Mahalanobis distance,frequentlyused in statistical analysis[30].Theﬁrst part of the DMT,4f ij(t,R i−R j)in eq.(6),depends on the n selected atoms only.It describes thedirect decoherence of the atoms due to their exposure tothe heat bath.The second part,2Φij(t,R i,R j),however,depends on all atoms,even the non–selected ones,suchthat this contribution is not translationally invariant.Φijdescribes the“indirect decoherence”which arises due tothe eﬀective interaction and subsequent correlations oreven entanglement that the heat bath induces betweenthe atoms[22,23].Remarkably,the total decoherenced tot(t),as deﬁned by s=s′|ρss′(t)|/ s=s′|ρss′(0)|≡1−d tot(t),involves only the trace of the DMT,d tot(t)=12π11−(t/r)2 cos(κr)cos(κt)+t r 2+2 sinκr r ln t−rr Ci(κ|t−r|)−Ci(κ|t+r|)−lnt−rκr sin(κr)(cos(κt)−1),(10)whereα=e2/(4πε0 c)≃1/137.06is theﬁne-structureconstant,andκ=k max d a UV cut–oﬀof the integralover k.A cut–oﬀis only needed for the diagonal partf ii which corresponds to r ij=0,but in order to assurethe positivity of the decoherence metric,the same cut–oﬀshould be used for both diagonal and oﬀ-diagonal parts.The oﬀ-diagonal terms f ij arise due to interference eﬀectsbetween two qubits.They decay as functions of r on alength scale of the order1/κ(see FIG.1).There can alsobe regions in the t,r plane where f ij becomes negative(not withstanding the fact that the decoherence metric isalways non–negative).One such zone is a ridge along thelight-cone t=r.Another one exists forﬁxed r almostindependently of t,for distances of the order1/κ.The functionˆf(t,r,ϑ)is continuous in the limit r→0,and leads to diagonal matrix elements f ii independent ofϑand i,f ii(t,0)=ˆf(t,0,ϑ)≡f ii(t),which describesingle–qubit decoherence(i.e.n=N=1).Explicitly,f ii(t)=22+1−cos(κt)−κt sin(κt)45·101·10fijFIG.1:(Color online)The function ˆf(t,r ij ,ϑ),for κ=0.01and ϑ=π/2at T =0.The limit r →0represents the single–qubit decoherence,f ii (t ).FIG.2:(Color online)Decoherences d s ,s ′(t )of 9qubits in a 3×3square optical lattice at t =200.0as function of the Hamming distances D H (s ,s ′).Small red squares are for a =580d ,large blue circles for a =1000d .Same parameters as in FIG.1.FIG.3:(Color online)Statistical analysis of all 223indepen-dent decoherences of 3×4qubits in a 2D square lattice for a =100d...1100d (parameters as in FIG.2).The average de-coherence,given by the trace of the DMT,is independent of the lattice spacings (central green curve,error bars represent ±one standard deviation).The minimum decoherence (bot-tom curve,blue)indicates a DFS at small spacings;the top curve (red)is the maximum decoherence.The inset shows a histogram of all decoherences for a =580d .does not contain suﬃciently short waves for distinguish-ing the positions of the atoms).The decoherence metric allows to generalize the DFS concept in the sense that even for non–symmetric coupling one may optimize now the performance of the quantum computer by encoding in the states with smallest decoherence,which can be explicitly found using the decoherence metric.In [31]a method was introduced that allows to quantum compute within a DFS,based on a strong coupling to the environ-ment and corresponding broadening of non–DFS states.It is expected that this method can be generalized to the current approach.As a summary,I have derived a “decoherence metric”that induces a natural distance between binary quan-tum code words.The decoherence metric generalizes the well-known Hamming distance and determines directly the time–dependent decoherence in an arbitrarily large system of qubits with degenerate energy levels coupled to a heat bath of harmonic osciallators.Acknowledgments:I wish to thank Olivier Giraud for valuable discussions.This work was supported by the Agence National de la Recherche (ANR),project INFOS-YSQQ.[1]D.Giulini, E.Joos, C.Kiefer,J.Kupsch,I.-O.Stamtescu,and H.Zeh,Decoherence and the Appear-ance of a Classical World in Quantum Theory (Springer,Berlin,Heidelberg,1996).[2]M.Brune,E.Hagley,J.Dreyer,X.Ma ˆıtre,A.Maali,C.Wunderlich,J.M.Raimond,and S.Haroche,Phys.Rev.Lett.77,4887(1996).[3]D.Vion,A.Aassime,A.Cottet,P.Joyez,H.Pothier,C.Urbina,D.Esteve,and M.Devoret,Science 296,886(2002).[4]U.Weiss,Quantum Dissipative Systems (World Scien-tiﬁc,Singapore,1993).[5]W.H.Zurek,Phys.Rev.D 24,1516(1981).[6]W.T.Strunz,F.Haake,and D.Braun,Phys.Rev.A 67,022101(2003).[7]W.D¨u r,C.Simon,and J.I.Cirac,Phys.Rev.Lett.89,210402(2002).[8]H.G.Krojanski and D.Suter,Phys.Rev.Lett.93,090501(2004).[9]F.Mintert,A.R.R.Carvalho,M.Kus,and A.Buchleit-ner,Phys.Rep.415,207(2005).[10]Y.Makhlin,G.Sch¨o n,and A.Shnirman,Rev.Mod.Phys.73,357(2001).[11]R.Bonifacio,P.Schwendiman,and F.Haake,Phys.Rev.A 4,302(1971).[12]P.A.Braun,D.Braun,F.Haake,and J.Weber,Eur.Phys.J.D 2,165(1998).[13]F.DeMartini,F.Sciarrino,and V.Secondi,Phys.Rev.Lett.95,240401(2005).[14]A.Auﬀeves,P.Maioli,T.Meunier,S.Gleyzes,G.Nogues,M.Brune,J.M.Raimond,and S.Haroche,Phys.Rev.Lett.91,230405(2003).[15]mine,R.Herv´e ,mbrecht,and S.Reynaud,5Phys.Rev.Lett.96,050405(2006).[16]S.Bandyopadhyay and D.A.Lidar,Phys.Rev.A72,042339(2005).[17]J.-M.Cai,Z.-W.Zhou,and G.-C.Guo,Phys.Rev.A72,022312(2005).[18]P.W.Shor,Phys.Rev.A52,R2493(1995).[19]A.M.Steane,Phys.Rev.Lett.77,793(1996).[20]M.Scully and M.Zubairy,Quantum Optics(CambridgeUniversity Press,Cambridge,UK,1997).[21]E.V.Goldstein and P.Meystre,Phys.Rev.A53,3573(1996).[22]D.Braun,Phys.Rev.Lett.89,277901(2002).[23]D.Braun,Phys.Rev.A72,062324(2005).[24]B.Juulsgaard,J.Sherson,J.Fiur´aˇs ek,and E.S.Polzik,Nature432,482(2004).[25]G. D.Mahan,Many-particle physics(Kluwer Aca-demic/Plenum Publishers,New York,1990).[26]P.Zanardi and M.Rasetti,Phys.Rev.Lett.79,3306(1997).[27]L.M.Duan and G.C.Guo,Phys.Rev.A57,2399(1998).[28]D.A.Lidar,I.L.Chuang,and K.B.Whaley,Phys.Rev.Lett.81,2594(1998).[29]D.Braun,P.A.Braun,and F.Haake,Proceedings of the1998Bielefeld Conference on”Decoherence:Theoretical, Experimental,and Conceptual Problems”,Lect.Notes Phys.538,55(2000).[30]P.C.Mahalanobis,Proc.Natl.Inst.of Science of India2,49(1936).[31]A.Beige,D.Braun,B.Tregenna,and P.L.Knight,Phys.Rev.Lett.85,1762(2000).。

ETH固体物理英文习题解答12

The ﬁrst fraction only depends on the Fermi surface which is hardly aﬀected by the impurities and the second fraction does not change much because the doping increases the number of free charge carriers only slightly (very small impurity density). Therefore we can conclude that ρ ∝ Z 2. (9) As it was mentioned in the hint we assume that the optimal electronic conﬁguration of the impurity atom valence shell is identical with the valence shell of the surrounding atoms. In this spirit we deﬁne the eﬀective charge of the impurity as the number of electrons which have to be removed from the impurity in order to have the same valence shell conﬁguration as the surrounding atoms, because those electrons are fully delocalized and can therefore no longer screen the nucleus of the impurity atom (apart from Thomas-Fermi screening). A look at the periodic table gives us the electronic conﬁguration, e.g. copper has the valence band conﬁguration 4s1 . By calculating the diﬀerence of the valence electrons between copper and the impurity we ﬁnd the following table: Impurity Be Mg B Al In Si Ge Sn As Sb Electronic conﬁguration 2s2 3s2 2s2 2p1 3s2 3p1 5s2 5p1 3s2 3p2 4s2 4p2 5s2 5p2 4s2 4p3 5s2 5p3 Eﬀ. charge Z 1 1 2 2 2 3 3 3 4 4

自发辐射诱导原子质心运动退相干的理论研究

自发辐射诱导原子质心运动退相干的理论研究姜文英;赵志敏;刘玉洁;张文举;郑丽;郑泰玉【摘要】The decoherence dynamics of a two-level atom coupled into a vacuum electromagnetic field are studied .The analytic solution of the Schrodinger equation has been given , and then the reduced den-sity matrix for the motion of the atomic center of mass has been derived .It is shown that spontaneous e-mission induces the entanglement between the atomic spatial motion and the atomic electronic states , which induces the spatial quantum decoherence of the atom .The decoherence factor is given , and its properties , such as dynamical evolution with time and evolution for different atomic positions , are demon-strated by some figures .%研究单个二能级原子与真空电磁场的相互作用，给出了总系统的薛定谔方程的解析波函数，并分析得出原子质心运动的约化密度矩阵。

结果表明原子内部电子态与真空场的相互作用会诱导原子的质心运动与原子的内部电子态发生纠缠，从而会导致原子的质心运动发生量子退相干。

文中还给出了退相干因子的解析表达式，并通过图形演示了原子质心运动退相干的动力学演化情况，以及与原子的空间尺度和原子-场耦合强度的相关性。

半导体一些术语的中英文对照

半导体一些术语的中英文对照离子注入机ion implanterLSS理论Lindhand Scharff and Schiott theory 又称“林汉德—斯卡夫—斯高特理论”。

沟道效应channeling effect射程分布range distribution深度分布depth distribution投影射程projected range阻止距离stopping distance阻止本领stopping power标准阻止截面standard stopping cross section退火annealing激活能activation energy等温退火isothermal annealing激光退火laser annealing应力感生缺陷stress-induced defect择优取向preferred orientation制版工艺mask-making technology图形畸变pattern distortion初缩first minification精缩final minification母版master mask铬版chromium plate干版dry plate乳胶版emulsion plate透明版see-through plate高分辨率版high resolution plate，HRP超微粒干版plate for ultra—microminiaturization 掩模mask掩模对准mask alignment对准精度alignment precision光刻胶photoresist又称“光致抗蚀剂”。

负性光刻胶negative photoresist正性光刻胶positive photoresist无机光刻胶inorganic resist多层光刻胶multilevel resist电子束光刻胶electron beam resistX射线光刻胶X—ray resist刷洗scrubbing甩胶spinning涂胶photoresist coating后烘postbaking光刻photolithographyX射线光刻X-ray lithography电子束光刻electron beam lithography离子束光刻ion beam lithography深紫外光刻deep—UV lithography光刻机mask aligner投影光刻机projection mask aligner曝光exposure接触式曝光法contact exposure method接近式曝光法proximity exposure method光学投影曝光法optical projection exposure method 电子束曝光系统electron beam exposure system分步重复系统step-and—repeat system显影development线宽linewidth去胶stripping of photoresist氧化去胶removing of photoresist by oxidation等离子［体]去胶removing of photoresist by plasma 刻蚀etching干法刻蚀dry etching反应离子刻蚀reactive ion etching，RIE各向同性刻蚀isotropic etching各向异性刻蚀anisotropic etching反应溅射刻蚀reactive sputter etching离子铣ion beam milling又称“离子磨削”。

$Decay Constants $f_{D_s^}$ and $f_{D_s}$ from ${bar{B}}^0to D^+ l^- {bar{nu}}$ and ${bar{B}$

Decay Constants $f_{D_s^}$ and $f_{D_s}$ from ${bar{B}}^0to D^+ l^- {bar{nu}}$ and ${bar{B}

∗ = 336 ± 79 M eV , and fD ∗ /fD we predict fDs s = 1.41 ± 0.41 for (pole/constant)-type s
form factor.
PACS index : 12.15.-y, 13.20.-v, 13.25.Hw, 14.40.Nd, 14.65.Fy Keywards : Factorization, Non-leptonic Decays, Decay Constant, Penguin Eﬀects
∗ experimentally from leptonic B and Ds decays. For instance, determine fB , fBs fDs and fDs
+ the decay rate for Ds is given by [1]
+ Γ(Ds
m2 G2 2 2 l 1 − m M → ℓ ν ) = F fD D s 2 8π s ℓ MD s
1/2
(4)
.
(5)
In the zero lepton-mass limit, 0 ≤ q 2 ≤ (mB − mD )2 .
2
For the q 2 dependence of the form factors, Wirbel et al. [8] assumed a simple pole formula for both F1 (q 2 ) and F0 (q 2 ) (we designate this scenario ’pole/pole’): q2 F1 (q ) = F1 (0) /(1 − 2 ), mF1
∗ amount to about 11 % for B → DDs and 5 % for B → DDs , which have been mentioned in

OF SOLUTIONS OF SCHR ODINGER'S EQUATION

k
0
(x)k2 r
H =
Z
j(1 ? )r=2 0(x)j2dx ;
r r
then for all t, (x; t) 2 H r (R n ), and k (x; t)kH = k 0(x)kH . (iv) When 0 (x) is in S , the Schwartz class, then for all t, (x; t) is in S . Statement (i) is fundamental in the interpretation of quantum mechanics, where the measure j 0(x)j2dx = dP0 (x) describes the initial spatial probability distribution of a quantum particle, j (x; t)j2dx = dPt(x) is the spatial distribution that quantum mechanics permits us to deduce at another time t through solution of the Schrodinger equation, and (6) is the fact that the Schrodinger equation (5) preserves probability. Based on this interpretation, it is natural to impose the moment condition (7) on the initial distribution, and the result of Theorem 1(ii) is that the solution is instantaneously smooth, however not necessarily localised, for any nonzero time t. It is distinctly not true that the niteness of the initial moments alone will guarantee that they are nite at later time; this will be discussed further below. On the other hand, a nite Sobolev norm of 0 (x) corresponds directly to nite moments of the momentum density j c ( )j2d , and 0 the assertion of Theorem 1(iii) is that these moments are preserved by the Schrodinger evolution. The goal is to study analogous results for solutions of equation (1), and to describe the connection between the dispersive smoothing of solutions and the global behavior of its bicharacteristics from (3). For this purpose we will assume the following estimates on the coe cients of (1):

薛定谔的猫

Roger Penrose criticises:
• "I wish to make it clear that, as it stands, this is far from a resolution of the cat paradox. For there is nothing in the formalism of quantum mechanics that demands that a state of consciousness cannot involve the simultaneous perception of a live and a dead cat“.
Relational interpretation
• All are quantum systems governed by the same rules of wave function evolution, and all may be considered "servers." • Different observers can give different accounts of the same series of events, depending on the information they have about the system.
Many-worlds interpretation and consistent histories • Hugh Everett Both alive and dead states of the cat persist after the box is opened, but are decoherent from each other.
Objective collapse theories

固相直接转变学说和溶剂析出说的英语表达

固相直接转变学说和溶剂析出说是化学领域中的两个重要概念。

固相直接转变学说（Solid State Direct Transformation）指的是在固态条件下，直接由原料转变成产物的化学反应过程。

而溶剂析出说（Solute Precipitation Theory）则是指在溶液中，溶质从溶液中析出形成固体颗粒的过程。

这两种理论对于化学反应和材料合成具有重要的指导意义，因此我们将在本文中对这两个概念展开深入探讨。

1. 固相直接转变学说的英语表达在化学领域中，固相直接转变学说通常被称为Solid State Direct Transformation。

这一理论认为，在固态条件下，原料分子直接转变成产物，而无需经过溶液中的溶解过程。

这种转变的过程可以涉及固体物质的晶体结构变化，也可以包括非晶态材料的相变过程。

固相直接转变学说的重要性在于，它能够解释一些在传统溶液化学中无法得到合理解释的化学反应过程，同时也为高温固相合成材料提供了理论支持。

2. 溶剂析出说的英语表达溶剂析出说在英文中的表达是Solute Precipitation Theory。

这一理论主要描述溶质在溶剂中溶解后，由于某种条件的改变（例如温度、浓度等），而从溶液中析出形成固体颗粒的过程。

在化学合成、晶体生长以及生物化学等领域中，溶剂析出说都具有重要的应用价值。

通过理解溶剂析出过程的特点和规律，我们能够更好地控制化学反应过程，从而实现对材料结构和性质的调控。

回顾本文，我们对固相直接转变学说和溶剂析出说这两个重要概念进行了全面的探讨。

通过深入分析这两种理论，我们不仅可以了解化学反应和材料合成的原理，还可以学习到如何在实际应用中进行有效操作。

相信通过对这些重要概念的理解和掌握，我们能够在化学领域中取得更大的成就。

个人观点：固相直接转变学说和溶剂析出说作为化学领域中的重要理论，对于我们深入理解化学反应和材料合成过程具有重要意义。

在我的研究和实践中，我发现这些理论不仅为实验设计提供了指导，还为我们解释实验现象提供了重要线索。

Decoherence of charge states in double quantum dots due to cotunneling

a r X i v :c o n d -m a t /0208220v 2 [c o n d -m a t .m e s -h a l l ] 10 O c t 2002Decoherence of charge states in double quantum dots due tocotunnelingU.Hartmann ∗and F.K.Wilhelm Sektion Physik and CeNS,Ludwig-Maximilians-Universit¨a t,Theresienstr.37,80333M¨u nchen,Germany Abstract Solid state quantum bits are a promising candidate for the realization of a scalable quantum computer,however,they are usually strongly limited by decoherence.We consider a double quan-tum dot charge qubit,whose basis states are deﬁned by the position of an additional electron in the system of two laterally coupled quantum dots.The coupling of these two states can be controlled externally by a quantum point contact between the two dots.We discuss the decoherence through coupling to the electronic leads due to cotunneling processes.We focus on a simple Gedanken ex-periment,where the system is initially brought into a superposition and then the inter-dot coupling is removed nonadiabatically.We treat the system by invoking the Schrieﬀer-Wolﬀtransformation in order to obtain a transformed Hamiltonian describing the cotunneling,and then obtain the dynamics of the density matrix using the Bloch-Redﬁeld theory.As a main result,we show that there is energy relaxation even in the absence of inter-dot coupling.This is in contrast to what would be expected from the Spin-Boson model and is due to the fact that a quantum dot is coupled to two distinct baths.PACS numbers:03.67.Lx,72.10.-d,73.23.Hk,73.63.Kvweak coupling to the leads (tunable)FIG.1:Sketch of the double dot system.The coupling of the double dot to the leads is assumed to be weak,whereas the coupling between the dots can be strong.The leads are biased such that sequential tunneling is supressed.Quantum dots(“artiﬁcial atoms”)are prototype systems for studying the properties of discrete levels embedded in a solid-state environment[1].In particular,various schemes for realizing quantum bits,fully controlled quantum coherent two-state systems,using quantum dots have been brought forward.Next to using optically excited charge states in quantum dots[2]and electronic quantum dots used for spin manipulation[3],it has been proposed[4] to use the charge states of a double quantum dot as a computational basis.The proposed setup is sketched in Figure1.In order to minimize the inevitable decoherence through coupling to the electronic leads,the system can be brought into the Coulomb blockade regime where sequential tunneling is supressed.We are going to discuss in this article, how the inevitable cotunneling still decoheres the system in this regime.The calculation is carried out for one speciﬁc Gedanken experiment which should capture the most generic features,the decay of a superposition state when the coupling between the dots is switched oﬀ.A more complete treatment of this setup is in preparation[5].We restrict our analysis on spin-polarized electrons.The relevant Hilbert space is char-acterized by four basis states,written as|i,j ,which denotes i additional electrons on the left dot,j additional electrons on the right dot.The two states|1,0 and|0,1 deﬁne the computational basis[6].In order to describe cotunneling,we use the closest energetically forbidden states as virtual intermediate states.These are|v0 =|0,0 and|v2 =|1,1 .Zero-and two electron states with internal polarization are energetically even less favorable due to the high charging energy of the individual dots.The Hamiltonian of this system can be written asH=H0+H1(1)H0=ǫas(ˆn l−ˆn r)−ǫαˆn v0+ǫβˆn v2+γ n(a L†n a R n+a R†n a L n)+ kǫL k b L† k b L k+ k′ǫR k′b R† k′b R k′(2) H1=t c k,n(a L†n b L k+a L n b L† k)+t c k′,m(a R†m b R k′+a R m b R† k′)(3) Note,that the sum over dot states only runs over the restricted Hilbert space described above.H0describes the energy spectrum of the uncoupled system,whereas the tunneling part H1describes the coupling of each dot to its lead and will be treated as a perturbation.ˆn l/r are the number operators for the additional electrons on either dot.The asymmetry energyǫas describes the diﬀerence between the energy level for the additional electron in left dot and the corresponding energy level in the right dot.It can be tuned through via the gate voltages which are applied at each dot.ǫβandǫαare the energy diﬀerences towards the higher level|v2 and the lower level|v0 respectively.γis the tunable inter-dot coupling. The a(†)s and b(†)s denote the creation/destruction operators in the dots and leads.In H1 the symbol t c represents the coupling constant concerning the coupling of the dots to the leads,which should be small compared to the asymmetry energy.Note,that we have chosen a slightly asymmetric notation in order to highlight the physical model:For the actual calculation,H1is also expressed in the eigenstate basis of the dot.For our Gedanken experiment,we assume thatﬁrst the inter-dot couplingγis large (γ≫ǫas,V)such that the system relaxes into the ground state,which is a molecular√superposition state of the form|g =(|0,1 −|1,0 )/that order.In order to treat cotunneling with this formalism,we perform a generalized Schrieﬀer-Wolf transformation[7,8].This transformation maps our original Hamiltonian H1,which is zero in the computational basis but couples the computational states to the |v0/2 onto a Hamiltonian which does not have this coupling to higher states but which hasnonzero matrix elements in the computational basis.The new terms in the Hamiltonian describe the amplitude of transitions between the basis states via the intermediate states. We perform this transformation perturbatively up to second order,i.e.all processes involving at most one intermediate state are taken into account.The new Hamiltomian H I in our special case then can be written asH I,++=A(R†,R,++)b R†m b R n+A(L,L†,++)b L l b L†k(4)H I,−−=A(L†,L,−−)b L†k b L l+A(R,R†,−−)b R n b R†m(5)H I,+−=A(R†,L,+−)b R†m b L l+A(L,R†,+−)b L l b R†m(6)H I,−+=A(L†,R,−+)b L†k b R n+A(R,L†,−+)b R n b L†k.(7) The+and−signs are indizes for the states|1,0 resp.|0,1 .We call the A s Schrieﬀer-Wolﬀcoeﬃcients,they are calculated along the lines of[8]using mainly second order perturbation theory.For example,A(R†,R,++)isA(R†,R,++)=t2cǫR m−(−ǫas+ǫβ)−1The Redﬁeld tensor has the formR nmkl=δlm rΓ(+)nrrk+δnk rΓ(−)lrrm−Γ(+)lmnk−Γ(−)lmnk.(10) The rates entering the Redﬁeld tensor elements are given by the following Golden-Rule expressionsΓ(+)lmnk=¯h−2∞dt e−iωnk t ˜H I,lm(t)˜H I,nk(0) (11)Γ(−)lmnk=¯h−2∞dt e−iωlm t ˜H I,lm(0)˜H I,nk(t) (12)where H I appears in the interaction representation(written as˜H I).In our formalism,it is of crucial importance that the expectation values over H I vanish,i.e.that the bath produces only noise.As a number of expectation values of H I turns out to beﬁnite,we tacitly replace H I by H I− H I in eqs.(11)and(12)and use theﬁnite expectation values to renormalizethe diagonalized,unperturbed Hamiltonian H0→H0+ H I .In our case,the eﬀect of this renormalization is of the order of0.1%of the original matrix elements of H0.After a straightforward calculation of the above Golden-Rule rates,one gets in the general case a(large)sum over terms with the generic formΓ(+)=c iπǫb−ǫa∓2ǫasψ 12π ǫb∓2ǫas−µ1 −ψ 12π ǫa−µ1 −−ψ 12π ǫb−µ2 +ψ 12π ǫa±2ǫas−µ2 (13)Γ(−)=c iπǫb−ǫa∓2ǫas−ψ 12π ǫb∓2ǫas−µ1 +ψ 12π ǫa−µ1 + +ψ 12π ǫb−µ2 −ψ 12π ǫa±2ǫas−µ2 (14)where c=t4cπV2m2∗gn,where g is a conductance in terms of the quantum conductance,E F is the Fermi energy of the leads and n is the number of electrons in the leads.Consequently,c is then changed to c=t2c gmultitude of possibilities for virtual transitions,each element of the Redﬁeld tensor contains a number of terms of this generic structure.In the above equations,the terms containing the Fermi function f(ǫ)only play a role close to resonance and can be neglected in the Coulomb blockade[12].The n l/r’s represent Bose functions for the electron-hole pairs(excitons)that are generated during the virtual processes.Theψ’s denote Digamma functions and hence diverge logarithmically at low temperatures.By solving equation(9),oneﬁnds that the oﬀ-diagonal elements decay towards zero on a time scaleτφ(dephasing time)whereas the diagonal density matrix elements equilibrate on a time scaleτr(relaxation time).Using the above expressions,weﬁnd the rates asΓr=2(Γ(+)+−−++Γ(+)−++−)(15)ΓrΓφ=Y1(19)β1Γ(+)−−−−=Γ(−)−−−−=cY1,−1.(21)βZ is a function containing severalψ-functions(or logarithms).Y1,Y−1and Y1,−1are diﬀerent functions of severalψ’-(Trigamma-)functions(or reciprocals),however,these functions only have a very weak temperature dependence.The most important part of the temperature dependence comes in through100.20.40.60.81Temperature T [K]00.050.10.150.20.25T i m e τ [s ]Dephasing time τφ for εas =0.1 K and ∆µ=0.06 K Relaxation time τr for εas =0.1 K and ∆µ=0.06 KFIG.2:Relaxation and dephasing time (τr and τφ)as a function of temperature T ,with µl =0.85K,µr =0.91K,ǫas =0.1K,ǫβ=11K,ǫα=9K,g =0.1,V =10−12m 2,E F =5meV and n/V =1.7·1015m −2.case,where there is only one lead,the situation corresponding to our Gedanken-experiment (no tunneling between the classical states)would correspond to pure dephasing,whereas in our system relaxation is always possible by extracting an electron on one side and adding one on the other side from the other lead.The numerical values for the relaxation and dephasing times are comparedly huge,on the order of 100milliseconds as compared to the experimentally measured times,which are in the order of nanoseconds.Other possibilities to explain the small decoherence time are phononic and/or photonic baths [14,15,16],or the inﬂuence of the whole electronic circuitry.We analyzed relaxation and dephasing processes in a system of two laterally coupled quantum dots which is coupled to two electronic (i.e.fermionic)baths.We showed that even in the case of vanishing inter-dot coupling,the system’s energy can relax,unlike in the Spin-Boson model.On top of that,the temperature dependence of the rates resembles that of the Spin-Boson model.We identify,that this originates in the fact that the cotunneling rates are mostly sensitive to the distribution function of excitons.As a next step,the case where the inter-dot coupling γhas ﬁnite values will be considered[5].We thank J.von Delft,L.Borda,J.K¨o nig,R.H.Blick,A.W.Holleitner,A.K.H¨u ttel and E.M.H¨o hberger for clarifying discussions.We acknowledgeﬁnancial support from ARO, Contract-Nr.P-43385-PH-QC.[1]R.C.Ashoori,Nature379,413(1996).[2] F.Rossi et al.,phys.stat.sol.(b)233,No.3(2002);P.Zanardi and F.Rossi,Phys.Rev.Lett.81,4752(1998).[3] D.Loss and D.DiVincenzo,Phys.Rev.A57,pp.120-126(1998).[4]R.H.Blick and H.Lorenz,ISCAS2000proceedings,pp.II245-II248.[5]U.Hartmann and F.K.Wilhelm,in preparation.[6]see e.g.M.A.Nielsen and I.L.Chuang,Quantum Computation and Quantum Information,(CUP,Cambridge,2000).[7]J.R.Schrieﬀer and P.A.Wolﬀ,Phys.Rev.149,491(1966).[8] C.Cohen-Tannoudji,J.Dupont-Roc and G.Grynberg,Atom-Photon Interactions:BasicProcesses and Applications,(Wiley,New York,1992).[9]U.Weiss,Quantum dissipative systems,2nd ed.,(World Scientiﬁc,Singapore,1999).[10]K.Blum,Density Matrix Theory and Applications,(Plenum Press,New York,1981).[11]L.Hartmann,I.Goychuk,M.Grifoni and P.H¨a nggi,Phys.Rev.E61,R4687(2000).[12]J.K¨o nig,H.Schoeller and G.Sch¨o n,Phys.Rev.Lett.78,4482(1997).[13]M.Grifoni,E.Paladino,U.Weiss,Eur.Phys.J.B10,719(1999).[14]T.Brandes and B.Kramer,Phys.Rev.Lett.83,3021(1999).[15]S.Debald,T.Brandes and B.Kramer,Phys.Rev.B66,R041301(2002).[16]H.Qin,A.W.Holleitner,K.Eberl and R.H.Blick,Phys.Rev.B64,R241302(2001).。

分子水平科赫原则是什么

分子水平科赫原则是什么分子水平科赫原则是一种用于解释和预测分子行为的基本原则。

它是由物理学家、诺贝尔奖得主彼得·格鲁贝（Peter Grünberg）和阿尔伯特·费尔特（Albert Fert）于2007年发现的。

分子水平科赫原则指出，当自旋磁矩相互作用时，电子会选择在相邻原子之间通过自旋耦合来组织自己的自旋状态。

这种自旋耦合导致自旋状态的集体行为，形成所谓的自旋-自旋耦合态。

这一原则在磁性材料中尤为显著，对我们理解和应用于磁性存储器、磁性传感器等领域具有重要意义。

从分子水平科赫原则的角度出发，我们可以更好地理解磁性材料的性质和行为。

它为我们提供了一种全新的思考方式，使我们能够从分子尺度上研究磁性系统的自旋结构、自旋动力学和磁性相互作用。

通过对这些磁性现象的深入研究，我们可以设计出更先进、更高效的磁性材料和器件。

对于实际应用而言，分子水平科赫原则也为我们提供了指导。

通过深入理解分子自旋耦合行为，我们可以在设计磁性材料和磁性器件时更加精确地控制其性能。

这不仅在磁存储器领域有着重要应用，也可以在其他领域如磁共振成像、磁力传感器等方面发挥作用。

此外，分子水平科赫原则还为我们揭示了分子系统的新奇行为。

例如，在纳米尺度下，分子水平科赫原则对磁性纳米颗粒的研究有着重要作用。

磁性纳米颗粒在医学、能源存储等领域具有广泛的应用前景。

通过对分子水平科赫原则的探索，我们可以更好地了解磁性纳米颗粒的自旋行为，并为其应用提供更多的可能性。

综上所述，分子水平科赫原则为我们理解和应用于磁性材料和器件提供了宝贵的指导。

通过深入探索分子自旋耦合行为，我们可以更好地理解磁性系统的性质和行为，并为设计新型磁性材料和器件提供指导。

这一原则的发现开辟了一个新的研究领域，并有望在各个应用领域产生重要的推动作用。

Decoherence induced by a phase-damping reservoir

∞
ρ(t) =
n,n′ =0
ρnn′ e−ı (En −En′ )t/ ×
′ 2 γ (t) (En − En ) 2
exp −
|En En′ |The purity, P = where ρnn′ = En |ρ(0)|En Tr[ρ2 ], then reads ∞ ′ 2 2 γ (t) (En − En ) 2
P (t) =
n,n′ =0
|ρnn′ |2 exp −
.
(3)
When τ is constant, for example, one can observe an exponential decay with time for the oﬀ-diagonal terms of ρ, implying in a decrease of the purity (P (0) > P (∞)), even though the energy of the system remain unchanged, i.e., Tr[H ρ(t)] = Tr[H ρ(0)]. Clearly, the master equation (1) suitably describes nondissipative phase decoherence. Recently, however, the relevance of SID as a physically meaningful process has been questioned [10]. In this work we show that it is also possible to obtain phase decoherence within the EID scenario, without any additional assumption to the standard quantum formalism if we consider a diagonal interaction. Our approach is based on a many-boson model in which the interaction between the components is given by the well† known cross-Kerr coupling a† i ai aj aj . This assumption is grounded in the fact that this kind of interaction is

Decoherence induced by a fluctuating Aharonov-Casher phase

a rXiv:q uant-ph/058157v122A ug25Decoherence induced by a ﬂuctuating Aharonov-Casher phase Fernando C.Lombardo ∗,Francisco D.Mazzitelli †,and Paula I.Villar ‡Departamento de F´ısica Juan Jos´e Giambiagi,FCEyN UBA,Facultad de Ciencias Exactas y Naturales,Ciudad Universitaria,Pabell´o n I,1428Buenos Aires,Argentina (Dated:today)Dipoles interference is studied when atomic systems are coupled to classical electromagnetic ﬁelds.The interaction between the dipoles and the classical ﬁelds induces a time-varying Aharonov-Casher phase.Averaging over the phase generates a suppression of fringe visibility in the interference pattern.We show that,for suitable experimental conditions,the loss of contrast for dipoles can be observable and almost as large as the corresponding one for coherent electrons.We analyze diﬀerent trajectories in order to show the dependence of the decoherence factor with the velocity of the particles.PACS numbers:03.75.-b,03.75.Dg,03.65.Yz I.INTRODUCTION Interference eﬀects are the most distinguished characteristic of quantum mechanics.The double-slit ex-periment is often used as the starting point to quantum description of Nature.There,when the path of the interfering particles is measured,the interference pattern disappears and the classical picture is recovered.The interaction between the quantum system with its environment (measuring apparatus)is responsible for the process of decoherence,which is one of the main ingredients in order to reach the quantum-to-classical transition.The Aharonov-Bohm (AB)[1]interference experiment can be a realistic probe for the predictions of decoherence.This experiment starts by the preparation of two electron wave packets ϕ1( x )and ϕ2( x )in a coherent superposition,assuming each of the charged particles follows a well deﬁned classical path (C 1and C 2,respectively).The complete wave function involves the presence of the environment,given an state ψ(t =0)=[ϕ1( x )+ϕ2( x )]⊗χ0( y ),(1)where χ0( y )represents the initial quantum state of the environment (whose set of coordinates is denoted by y ).As time follows,the electron’s coherent state entangles with the environment,and the total wavefunction can be described asψ(t )=ϕ1( x ,t )⊗χ1( y ,t )+ϕ2( x ,t )⊗χ2( y ,t ).(2)Therefore,the two delocalized electron states ϕ1and ϕ2become correlated with two diﬀerent states of the environment.The probability of ﬁnding a particle at a given position at time t (for example when interference pattern is examined)is,Prob( x ,t )=|ϕ1( x ,t )|2+|ϕ2( x ,t )|2+2Re ϕ1( x ,t )ϕ∗2( x ,t ) d 3y χ∗1( y ,t )χ2( y ,t ) .(3)The overlap factor F = d 3y χ∗1( y ,t )χ2( y ,t )is responsible for two separate eﬀects.Its phase generates a shift of the interference fringes,and its absolute value is responsible for the decay in the interference fringe2 contrast.Of course,in absence of environment the overlap factor is not present in the interference term. When the two environmental states do not overlap at all,theﬁnal state of the bath identiﬁes the path the electron followed.There is no uncertainty respect to the path.Decoherence appears as soon as the two interfering partial waves shift the environment into states orthogonal to each st statement suggests that the environment saves(in some way)the information about the path the electron takes.The loss of quantum coherence can alternatively be explained by the eﬀect of the environment over the partial waves,rather than how the waves aﬀect the environment.As has been noted,when a static potential V(x)is exerted on one of the partial waves,this wave acquires a phase,φ=− V[x(t)]dt,(4)and therefore,the interference term appears multiplied by a factor e iφ.This is a possible agent of decoherence. The eﬀect can be directly related to the statistical character ofφ,in particular in situations where the potential is not static.Even more,any source of stochastic noise would create a decaying coeﬃcient.For a general case,φis not totally deﬁned,i.e.it is described by means of a distribution function P(φ).From this statistical point of view,the phase can be written ase iφ = e iφP(φ)dφ.(5)In this way,the uncertainty in the phase produces a decaying term that tends to eliminate the interference pattern.This dephasing is due to the presence of a noisy environment coupled to the system and can be also represented by the Feynmann-Vernon inﬂuence functional formalism.It is easy to prove that Eq.(5) is the inﬂuence functional generated after integrating out the environmental degrees of freedom of an open quantum system.Therefore,in Ref.[2]was shown the formal equivalence between the two ways of studying dephasinge iφ =F= d3yχ∗1( y,t)χ2( y,t).(6)The overlap factor F encodes the information about the statistical nature of noise.Therefore,noise(classical or quantum)makes F less than one,and the idea is to quantify how it slightly destroys particle interference pattern.In many cases,the interaction with the environment cannot be switched oﬀ.Thus,for charged particles or neutral atoms with dipole moment,the interaction with the electromagneticﬁeld is crucial.This interaction induces a reduction of fringe visibility.For example,vacuumﬂuctuations of electromagneticﬁelds have been considered as a decoherent agent[3].Double-slit-like experiments were studied in the presence of conductors, which change the structure of vacuum and modify predictions about decoherence eﬀects.The changes on the fringe visibility induced by the position and relative orientation of the conductors are rather small. Electron interference in mesoscopic devices irradiated by external nonclassical microwaves was considered in Ref.[4].The eﬀect of quantum noise on electron interference for several types of microwaves has been analyzed,and it was shown that entangled electromagneticﬁelds interacting with electrons produce entangled photons[5].Despite the quantum nature of noise(vacuumﬂuctuations for example),classical noise is also present when considering time dependentﬁelds,or some random variables that parametrize the environment. The destruction of electron interference by external classical and quantum noise has been studied in Ref.[6]. In Ref.[7],the overlap factor F was evaluated from a diﬀerent point of view.They studied the eﬀect of time-varying electromagneticﬁelds on electron coherence,including the statistical origin of the AB phase φ.However,they didn’t considerφneither as coming from quantumﬂuctuations nor a time dependent ﬁeld.They included a random variable t0,which was deﬁned as the electron emission time.This variable produces aﬂuctuating phaseφ,and an average over it is needed in order to obtain the result of the double-slit interference experiment.In this simple version of decoherence,the role of a quantum environment is replaced by a time-dependent externalﬁeld which gives a time-varying AB phase.The authors have considered the3 case of a linearly polarized,monochromatic electromagnetic wave,propagating in a direction orthogonal to the plane containing two electron beams.The eﬀect on the fringes seems to be suﬃciently large to be observable.In the present article we follow last idea.We evaluate the overlap factor F for coherent neutral particles with permanent(electric and magnetic)dipoles that are aﬀected by time-varying externalﬁelds.We consider two diﬀerent cases with possible experimental interest:ﬁrstly,the case of dipoles interacting with a linearly polarized electromagnetic plane wave.This will generalize results of Ref.[7]to the case of Aharonov-Casher (AC)[8]case in atomic systems[9],where coherent dipoles follow a closed path around an externalﬁeld. Secondly,the case of dipoles travelling through a waveguide with rectangular section.As we will discuss, not all these eﬀects foresee an observable displacement on the interference pattern related to the phase shift of the wave function of the system.Dependence on the velocity of the interfering particle is crucial.We will show diﬀerent conﬁgurations where the decoherence factor depends on the particle’s speed in a diﬀerent way. In principle,for neutral particles with the same mass and velocity than the charged ones,one should expect the loss of contrast for dipoles to be smaller than the analogous one in experiments performed using charged particles.Although this is basically true,the interaction between electromagneticﬁelds and dipoles is also weaker,allowing to increase the intensity of external classicalﬁelds in the same amount than the dipole eﬀect is smaller,still neglecting the dipole scattering in the externalﬁeld.Therefore,for suﬃciently strong externalﬁelds,the eﬀect on fringe visibility for dipoles will be of comparable magnitude with respect to the case of charged particles.In order to prove this,we will evaluate the scattering cross section for dipoles in interaction with both a plane wave and theﬁelds inside a waveguide.The paper is organized as follows.In Section II we present the evaluation of the phase shift and the deco-herence factor for coherent dipoles(electric and magnetic)in the presence of a time varying electromagnetic plane wave.Symmetric and non-symmetric trajectories are considered in order to analize the dependence on the velocity of the interfering particles.We also include in this section some new results for charged particles.Section III shows the results when the coherent trajectories are inside a waveguide.In Section IV we estimate the decoherence factor for realistic values of the diﬀerent parameters,and compare the loss of contrast for charged particles and dipoles.Section V contains ourﬁnal remarks.We also include an Appendix where we compute the Thomson cross section for dipoles.II.COHERENT DIPOLES AND A PLANE W A VEThe AB phase[1],known to arise when two coherent electrons traverse two diﬀerent paths C1and C2in the presence of an electromagneticﬁeld,is(c= =1)φ=−e δΩdxνAν(x),(7)whereδΩ=C1−C2is a closed spacetime path.If the electromagneticﬁeld’sﬂuctuations happen in a time scale shorter than the total time of the experiment,this shift of phase results in a loss of contrast in the interference fringes.Then,the overlap factor(or decoherence factor)is given by Eq.(6)[2,7].In that expression,angular brackets denote either an ensemble of quantum noise or a time average over a random variable.The classical and quantum interaction of a dipole with an arbitrary electromagneticﬁeld has been described in detail in Ref.[10],where a uniﬁed and fully relativistic treatment of this interaction has been presented. The Lorentz-invariant,classical interaction Lagrangian is14 [8]and is deﬁned byφ=− δΩaν(x)dxν,(8)where aν(x)=(−m·B−d·E,d×B−m×E)plays the role of Aνin the AB case,δΩ=C1−C2,and C1 and C2are the two paths followed by the particles that interfere.In order to evaluate the integral in Eq.(8),we consider the case of a linearly polarized monochromatic wave of frequencyωpropagating in theˆy direction,with an electric and magneticﬁeld in theˆz andˆx direction respectively.We will also assume that the particles’path is conﬁned to theˆx−ˆz plane(see Fig.1).We can write the plane wave as E(x)=E0sin(wt−ky)ˆz,B(x)=E0sin(wt−ky)ˆx and compute aν,which is given byaν(x)=(−d z E z−m x B x,m y E z,d z B x−m x E z,−d y B x)=E0(−d z−m x,m y,d z−m x,−d y)sin(ωt−ky)≡˜aνsin(ωt−ky).(9) Following Ref.[7],we will assume that the phaseφdepends on a random variableξ=ωt0given by the emission time of the dipoles.It is the time t0at which the center of a localized wave packet is emitted. When the measuring time takes longer than theﬂight time,we will observe a result which is the temporal average over t0.Thus,t0is a random variable by whichφhas to be averaged.We can write the AC phase φasφ(t0)=− δΩ˜aνsin(ωt−ky+ωt0)dxν,(10) before taking the time average,we rewrite the phase asφ(t0)=A cos(ωt0)+B sin(ωt0),(11) whereA=− δΩ˜aν(x)sin(ωt−ky)dxν,B=− δΩ˜aνcos(ωt−ky)dxν.(12) The average over the random phase(generating a classical noise)produces a decoherence factorF= e iφ =lim T→∞15A.Symmetric trajectoriesIn this subsection,we will estimate the phase acquired by two neutral particles with electricand magneticdipole moments when they follow symmetric trajectories as the ones depicted in Fig.1.To begin with,we will consider the same trajectory used by authors in [7]to analyze the AB phase factor between charged particles.To perform integration in Eq.(12)along the trajectory of Fig.1(a),we must calculateφ= δΩa ν(x )dx ν=6 i =1 10a ν(σνi (u ))·dσνi ωθsin ωθ2 ,(17)where 2αis the maximum distance between the particles,d y is the electric dipole’s moment in the ˆy direction and T,θare characteristic times of the trajectory.The subindex d indicates that these are the values of A,B and C for dipoles.6For non relativistic particles,we expectωθ,ωT≫1.Therefore,in order to have an estimation for C d√we can replace the sine functions by a typical value1/α2+ing this,we rewrite Eq.(17)as|C d|≈2s λv≈2s λLv.(18)Here L is the characteristic length of an atom with electric dipole d=eL(L≈10−9m),andλis the wavelength of the plane wave.The analogous result for charged particles is given by|C e|≈1s)v [7].Assuming that the charged and neutral particles have the same speed and trajectory,for the same externalﬁeld we obtainL|C d|≈|C e|2παE0d yπαeE0L vλv. It is interesting to check whether the same diﬀerence in behaviours applies to the case of the AB phase for charged particles or not.If the charged particles travel across the paths shown in Fig.1(b),the quantity |C e ellip|can be computed from Eq.(7)and the parametrization Eq.(20).The result is|C e ellip|=2παeE0λJ1[ωτ],(23) showing that the dependence on the velocity is similar for both neutral and charged particles.Finally,it is worth noting that,as both trajectories in Figs.1(a)and(b)are symmetric respect to theˆx (ˆt)axis,only the term proportional to A of Eq.(12)contributes to the AC phase,and,consequently,to the decoherence factor.This is due to the parity of the integrand with respect to x and t.B.Asymmetric trajectoryIn this subsection we will consider the case of dipole wave packets travelling across the asymmetric tra-jectory depicted in Fig.2.As we will see,in this case,not only do both terms in Eq.(12)contribute to the7C 2C 1FIG.2:Paths C 1and C 2are shown for the asymmetric trajectory.AC phase but also diﬀerent components of the dipole moments.Consequently,speed dependence will be diﬀerent from the case of symmetric trajectories.We write C d asym =A d asym +iB d asym ,where each of these coeﬃcients is deﬁned in Eq.(12).The parametriza-tion of the asymmetric curve can be read from Eqs.(15)and (20).After performing the corresponding integrations,we obtainA d asym =πd y E 0αJ 1[ωτ]+4E 0d y α2]sin[ωω sin[ωθ2(T +θ)]+sin[ωT ωθsin[ωθ2(T +θ)]+4dE 0m y 2].(24)The ﬁrst term in the B coeﬃcient is the most important in the low velocity limit,because all other terms are O (1/√πE 0Lλ,(25)being independent of the velocity.For charged particles,the situation is very diﬀerent.The decoherence factor can be computed by inte-grating Eq.(7)along the asymmetric trajectory of Fig.2.We obtain C e asym =A e asym +iB e asym withA e asym =2πeE 0αω2θsin[ωθ2(T +θ)],B e asym =0.(26)Therefore,we can approximate|C e asym |≈e 2πE 0α vλ38 III.COHERENT DIPOLES INSIDE A W A VEGUIDENow we consider theﬁeld generated inside a waveguide(along theˆy direction)with rectangular section. For the TE mode,the electromagneticﬁelds inside the pipe are the real part ofB y=B0cos(k x x)cos(k z z)exp(i(k y y−ωt)),B x=−ik y k xγ2B0cos(k x x)sin(k z z)exp(i(k y y−ωt)),E x=iωk zγ2B0sin(k x x)cos(k z z)exp(i(k y y−ωt)),(28)where k x=mπa(with m and l integers and a,b the dimensions of the pipe),γ= a)2+(mπγ2+k2y,and B0is a complex number.We will write B0as B0=|B0|exp(iωt0),where t0is the particle emission time,as was deﬁned in the previous Section.2 d=bFIG.3:The particles interfere in the presence of a time-dependent electromagneticﬁeld.The particles follow the path C1and C2in the x−z plane as shown above for the waveguide.Let us consider the trajectory depicted in Fig.3.The particles are set together at one side of the pipe(x i) (at some initial time t0),released and meet again at the other side(x f),having gone through it along straight paths.As theﬁelds vanish outside the waveguide,the particles will interact with the electromagneticﬁeld only when travelling inside it.Bearing in mind that C d guide=A d guide+iB d guide,we obtain for this caseA d guide=0B d guide=−4γ2sin(lπα2)sin(ωT 2)sin(mπ9For the ﬁrst mode of the waveguide (l =1,m =0),this expression looks simpler:C d TE 10=−4πsin(πα2)(d x ωT −m z k y T −bd y k y ).(30)We can estimate the quantity |C d TE 1,0|in the same way we did in the preceding section,yielding |C d TE 10|≈2√πB 0a sin(πα(2π3)1/2eB 0aλv sin απ(ωT )2−(mπ)2k za ) ωT cos(mπ2)−mπcos(ωT 2) d x ωT −m z k z T −bd y k y sin(ωt 0).(33)It is easy to prove that the result is negligible in the case of odd m and mπ≪ωT .In any other case we get|C d TE |≈B 0a |sin(παωT d y k z )2−2bγ2E 0cos(k x x )cos(k z z )exp(i (k y y −ωt )),B z =−iωk xγ2E 0cos(k x x )sin(k z z )exp(i (k y y −ωt )),E y=E 0sin(k x x )sin(k z z )exp(i (k y y −ωt )),E z =ik y k z (ωT )2−(mπ)2|E 0|sin(lπα2)sin(ωT 2)sin(mπγ2T +m z ωγ2 + ωT cos(ωT 2)−mπcos(mπ2) −d y T −bm z sin(ωt 0).(36)10 Using the same arguments than in the TE case,we can check that there is also a contribution independent of the dipoles’velocity but this time proportional to m z.Therefore,as we are assuming|m|<<|d|all along this work,we conclude that|C d TM|≪|C d TE|.It is worth noting that in the TM case,as there is no component of the magneticﬁeld along the guide axis,the AB phase vanishes.Therefore,the eﬀect for neutral particles is the only one we expect.IV.NUMERICAL ESTIMATIONSWhen we compared the decoherence factors for charged and neutral particles(see for example Eq.(19)), we had assumed that the relevant parameters(velocity,maximum separation)were the same in both cases. Even though it is a valid thread of thought,it is not a realistic one.Therefore,in this section we will estimate the loss of coherence in more realistic situations.In interference experiments with electrons,the wave packets can be moved apart up to100µm[17].A typical non relativistic velocity is v e∼0.1.This yields a relationωT∼10for aﬁeld that has a wavelength of about100µm.On the other hand,in atomic interferometry,two neutral particles can be separated up to 1mm[18,19].Typical speeds are of the order v d∼10−5[20].We will assume the dimensions of the pipe a∼cm.The energy density of the electromagneticﬁeld is deﬁned asρwave=E20/2for a plane wave.The energy density for theﬁeld inside a waveguide isρguide=(aB0)2/λ2.Considering the typical values ofλand a given above,we note that a/λ∼1for dipoles whereas a/λ∼10for electrons.As we we are using Lorentz Heaviside units with =c=1,ρis also the energyﬂux in the electromagnetic wave.We will assume an energyﬂux of10Watts/cm2,approximately.With all these values,we can estimate the C factor for all the cases presented in the previous sections. The results are summarized in Table1.As we can see,all the results for electrons are of order one or bigger, which means that the eﬀect is experimentally observable.Dipoles’results are smaller but not that much as one would naively expect.In electron’s interference experiments,we conclude that the best experimental setup would be either the asymmetric trajectory or the elliptic path.In those conﬁgurations,for adequate parameters of the trajectories it is in principle possible to obtain a complete destruction of the interference pattern(setting the value of|C| equal to a zero of the Bessel function J0).In an interference experiment with dipoles,the best experimental setup would be either the asymmetric trajectory or the waveguide.In these cases,the eﬀect is non negligible thanks to the fact that the C factor is independent of the velocity.Moreover,as shown in the Appendix, one is allowed to increase the intensity of the externalﬁeld,since the scattering cross section for dipoles is much lower than for electrons,being still possible to neglect the direct interaction with the electromagnetic ﬁeld.Trajectories Dipoles|C|10−610|C asym|10−1111V.FINAL REMARKSWe have estimated the loss of contrast produced when interfering particles are shined by a classical electromagneticﬁeld.We considered both a monochromatic,linearly polarized electromagneticﬁeld,and electromagneticﬁelds inside a waveguide.We considered diﬀerent trajectories for both charged particles and dipoles.Symmetric and asymmetric paths have been used in order to illustrate the dependence of theﬂuctuating phase upon the velocity.We have shown examples(asymmetric trajectories,waveguide)in which the loss of contrast for dipoles is independent of the velocity in the low velocity limit.This particular behaviour does not show up for charged particles.However,we have found that the eﬀect for electrons is in general larger than the one computed in Ref.[7]for a particular symmetric trajectory.We also estimated the backreaction eﬀect of theﬁelds over the dipoles,evaluating the scattering cross section.When the mean free path for dipoles is much larger than the characteristic dimensions of their trajectories,the direct interaction between dipoles and photons can be neglected.This condition puts an upper bound over the externalﬁeld intensity.Given an externalﬁeld,the decoherence for dipoles is smaller than the one expected for electrons.However,as the scattering cross section is also smaller,it is in principle possible to increase the intensity of the externalﬁelds in order to partially compensate the diﬀerence,still within the upper bound mentioned above.Finally,we estimated the magnitude of the eﬀect for values of the diﬀerent parameters achievable in the laboratory.In the case of electrons,for an adequate setup the interference fringes can dissappear totally.For dipoles,the independence with the velocity makes the eﬀect much more important than naively expected, and could be observed for suﬃciently strong externalﬁelds.VI.APPENDIXThe loss of contrast computed in the previous sections depends on the intensity of the classical electro-magneticﬁeld.In the usual AB and AC eﬀects,the force on the charges and dipoles vanishes.However, in the time dependent case considered in this paper,the dipoles have a direct interaction with the electro-magneticﬁeld,which we neglected.Therefore the intensity of the electromagneticﬁeld is limited by the scattering cross section:the mean free path for dipoles should be much larger than the characteristic size of its trajectory.In this section we will evaluate the scattering cross section for both dipoles in interaction with a plane wave,and dipoles travelling inside the waveguide,and compare these with the results for charged particles.1.Plane Wave.The force experienced by a non-relativistic neutral atom with arbitrary electric andmagnetic dipole moments in the presence of an electromagneticﬁeld is[16]F=∇ d·E(R)+m·B(R) +∂t d×B(R) ,(37) where R is the position of the center of mass of the atom.For the plane wave of Section II,the force on the particle readsF=−k y E0cos(ωt−k y y)(d yˆz+m xˆy).(38)If the atom’s center of mass is oscillating around the origin of coordinates(y=0),we can approximate the force it feels as the force evaluated at y=0for every time t.In the non-relativistic limit,the acceleration of the particle is¨x=F/m A.Writing the dipole moment as d=e x,we can compute¨d from the force,and use Larmor’s formula to know the angular distribution of radiated power d P12 averaged in time,is thereforeσd=8πm2Ak2y(d2y+m2x).(39)We can compare this result with Thomson’s scattering cross sectionσe=8πm e )2for the case of anon-relativistic electron interacting with a plane wave.If we consider that|d y|>|m x|,then:σd≈σe(L mA)2,(40)where L is the dipole’s characteristic length andλis the wavelength of theﬁeld.Thus,we can see that the mean free path for dipoles l d mfp∼1/σd is bigger than the corresponding l e mfp for electrons ina factor(λ/L)2(m A/m e)2.Based on this simple observation,we conclude that the suppression of fringe visibility in the experiment done with coherent electrons or with coherent dipoles could have a similar order of magnitude.Indeed, although for a given externalﬁeld the suppression is bigger for electrons than for dipoles,in the latter case it is possible to increase the intensity of the externalﬁeld to partially compensate the diﬀerence, still within the limit of negligible direct interaction.2.Waveguide.For the case of a dipole interacting with theﬁelds of the TE mode inside a waveguide theforce is written as:F d=kk zγ2B0 d x k x cos(k x x)cos(k z z)sin(k y y−ωt)+d y k y sin(k x x)cos(k z z)cos(k y y−ωt)−d z k z sin(k x x)sin(k z z)sin(k y y−ωt) ˆz(41)Following the reasoning above for the case of the plane wave,we obtain that the scattering cross section of the TE mode is given byσd TE≈8πm2A kk y=8πm2AL2kk y=σe TE k2y L2(m e3e4k yis the total cross section for an electron inside the waveguide in the TE mode(forsimplicity we assumed that k x∼k z,and a∼b).Therefore,we obtainσd TE≪σe TE,and l d,TE mfp≫l e,TE mfp For the TM modes,an analogous calculation shows thatσd TM≈σd TE.VII.ACKNOWLEDGMENTSWe thank Juan Pablo Paz for useful discussions.This work was supported by UBA,CONICET,Fundaci´o n Antorchas,and ANPCyT,Argentina.[1]Y.Aharonov and D.Bohm,Phys.Rev.115,485(1959).13[2]A.Stern,Y.Aharonov,and Y.Imry,Phys.Rev.A41,3436(1990).[3]F.D.Mazzitelli,J.J.Paz,and A.Villanueva,Phys.Rev.A68,062106(2003).[4]C.C.Chong,D.I.Tsomokos,and A.Vourdas,Phys.Rev.A66,033813(2002).See also: D.I.Tsomokos,C.C.Chong,and A.Vourdas,quant-ph/0502113.[5]D.I.Tsomokos,C.C.Chong,and A.Vourdas,Phys.Rev.A69,03810(2004).[6]A.Vourdas,Phys.Rev.A64,053814(2001).[7]Jen-Tsung Hsiang and L.H.Ford,Phys.Rev.Lett.25,250402(2004).[8]Y.Aharonov and A.Casher,Phys.Rev.Lett.53,319(1984).[9]K.Sangster,E.Hinds,S.Barnett,E.Riis,and A.Sinclair,Phys.Rev.A51,1776(1995).[10]J.Anandan,Phys.Rev.Lett.85,1354(2000).[11]M.V.Berry,Proc.Soc.London Ser.A392,45(1991).[12]Y.Aharonov,S.A.Pearle,and L.Vaidman,Phys.Rev.A37,4052(1988).[13]G.Spavieri,Phys.Rev.Lett.82,3932(1999).[14]A.Cimmino,G.I.Opat,A.G.Klein,H.Kaiser,S.A.Wermer,M.Arif,and R.Clothier,Phys.Rev.Lett.63,380(1989).[15]K.Sangster,E.Hinds,S.Barnett,and E.Riis,Phys.Rev.Lett.71,3641(1993).[16]J.Schwinger,L.L.DeRaad,ton,and W.Tsai,Classical Electrodynamics,Advanced Book Program,Westview Press,1998.[17]F.Hasselbach,Z.Phys.B:Condens.Matter71,443(1988).[18]D.W.Keith et al.,Phys.Rev.Lett.66,2693(1991).[19]T.Pfau et al.,Phys.Rev.Lett.71,3427(1993).[20]D.M.Greenberger,D.K.Atwood,J.Arthur,C.G.Shull,and M.Schlenker,Phys.Rev.Lett.47,751(1981).。

Decoherence, einselection, and the quantum origins of the classical

a r X i v :q u a n t -p h /0105127v 3 19 J u n 2003DECOHERENCE,EINSELECTION,AND THE QUANTUM ORIGINS OF THE CLASSICALWojciech Hubert ZurekTheory Division,LANL,Mail StopB288Los Alamos,New Mexico 87545Decoherence is caused by the interaction with the en-vironment which in eﬀect monitors certain observables of the system,destroying coherence between the pointer states corresponding to their eigenvalues.This leads to environment-induced superselection or einselection ,a quantum process associated with selective loss of infor-mation.Einselected pointer states are stable.They can retain correlations with the rest of the Universe in spite of the environment.Einselection enforces classicality by imposing an eﬀective ban on the vast majority of the Hilbert space,eliminating especially the ﬂagrantly non-local “Schr¨o dinger cat”states.Classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit:Combination of einse-lection with dynamics leads to the idealizations of a point and of a classical trajectory.In measurements,einselec-tion replaces quantum entanglement between the appa-ratus and the measured system with the classical corre-lation.Only the preferred pointer observable of the ap-paratus can store information that has predictive power.When the measured quantum system is microscopic and isolated,this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the eﬀective “collapse of the wavepacket”.Existential in-terpretation implied by einselection regards observers as open quantum systems,distinguished only by their abil-ity to acquire,store,and process information.Spreading of the correlations with the eﬀectively classical pointer states throughout the environment allows one to under-stand ‘classical reality’as a property based on the rela-tively objective existence of the einselected states:They can be “found out”without being re-prepared,e.g,by intercepting the information already present in the envi-ronment.The redundancy of the records of pointer states in the environment (which can be thought of as their ‘ﬁt-ness’in the Darwinian sense)is a measure of their clas-sicality.A new symmetry appears in this setting:Envi-ronment -assisted invariance or envariance sheds a new light on the nature of ignorance of the state of the system due to quantum correlations with the environment,and leads to Born’s rules and to the reduced density matri-ces,ultimately justifying basic principles of the program of decoherence and einselection.ContentsI.INTRODUCTION2A.The problem:Hilbert space is big 21.Copenhagen Interpretation 22.Many Worlds Interpretation 3B.Decoherence and einselection 3C.The nature of the resolution and the role of envariance4D.Existential Interpretation and ‘Quantum Darwinism’4II.QUANTUM MEASUREMENTS5A.Quantum conditional dynamics51.Controlled not and a bit-by-bit measurement 62.Measurements and controlled shifts.73.Ampliﬁcationrmation transfer in measurements91.Action per bit9C.“Collapse”analogue in a classical measurement 9III.CHAOS AND LOSS OF CORRESPONDENCE11A.Loss of the quantum-classical correspondence 11B.Moyal bracket and Liouville ﬂow 12C.Symptoms of correspondence loss131.Expectation values 132.Structure saturation13IV.ENVIRONMENT –INDUCED SUPERSELECTION 14A.Models of einselection141.Decoherence of a single qubit152.The classical domain and a quantum halo 163.Einselection and controlled shifts16B.Einselection as the selective loss of information171.Mutual information and discord18C.Decoherence,entanglement,dephasing,and noise 19D.Predictability sieve and einselection20V.EINSELECTION IN PHASE SPACE21A.Quantum Brownian motion21B.Decoherence in quantum Brownian motion231.Decoherence timescale242.Phase space view of decoherence 25C.Predictability sieve in phase space 26D.Classical limit in phase space261.Mathematical approach (¯h →0)272.Physical approach:The macroscopic limit 273.Ignorance inspires conﬁdence in classicality 28E.Decoherence,chaos,and the Second Law281.Restoration of correspondence 282.Entropy production293.Quantum predictability horizon 30VI.EINSELECTION AND MEASUREMENTS30A.Objective existence of einselected states 30B.Measurements and memories31C.Axioms of quantum measurement theory321.Observables are Hermitean –axiom (iiia)322.Eigenvalues as outcomes –axiom (iiib)333.Immediate repeatability,axiom (iv)334.Probabilities,einselection and records342D.Probabilities from Envariance341.Envariance352.Born’s rule from envariance363.Relative frequencies from envariance384.Other approaches to probabilities39 VII.ENVIRONMENT AS A WITNESS40A.Quantum Darwinism401.Consensus and algorithmic simplicity412.Action distance413.Redundancy and mutual information424.Redundancy ratio rate43B.Observers and the Existential Interpretation43C.Events,Records,and Histories441.Relatively Objective Past45 VIII.DECOHERENCE IN THE LABORATORY46A.Decoherence due to entangling interactions46B.Simulating decoherence with classical noise471.Decoherence,Noise,and Quantum Chaos48C.Analogue of decoherence in a classical system48D.Taming decoherence491.Pointer states and noiseless subsystems492.Environment engineering493.Error correction and resilient computing50 IX.CONCLUDING REMARKS51 X.ACKNOWLEDGMENTS52 References52I.INTRODUCTIONThe issue of interpretation is as old as quantum the-ory.It dates back to the discussions of Niels Bohr, Werner Heisenberg,Erwin Schr¨o dinger,(Bohr,1928; 1949;Heisenberg,1927;Schr¨o dinger,1926;1935a,b;see also Jammer,1974;Wheeler and Zurek,1983).Perhaps the most incisive critique of the(then new)theory was due to Albert Einstein,who,searching for inconsisten-cies,distilled the essence of the conceptual diﬃculties of quantum mechanics through ingenious“gedankenexper-iments”.We owe him and Bohr clariﬁcation of the sig-niﬁcance of the quantum indeterminacy in course of the Solvay congress debates(see Bohr,1949)and elucidation of the nature of quantum entanglement(Einstein,Podol-sky,and Rosen,1935;Bohr,1935,Schr¨o dinger,1935a,b). Issues identiﬁed then are still a part of the subject. Within the past two decades the focus of the re-search on the fundamental aspects of quantum theory has shifted from esoteric and philosophical to more“down to earth”as a result of three developments.To begin with, many of the old gedankenexperiments(such as the EPR “paradox”)became compelling demonstrations of quan-tum physics.More or less simultaneously the role of de-coherence begun to be appreciated and einselection was recognized as key in the emergence of st not least,various developments have led to a new view of the role of information in physics.This paper reviews progress with a focus on decoherence,einselection and the emergence of classicality,but also attempts a“pre-view”of the future of this exciting and fundamental area.A.The problem:Hilbert space is bigThe interpretation problem stems from the vastness of the Hilbert space,which,by the principle of superposi-tion,admits arbitrary linear combinations of any states as a possible quantum state.This law,thoroughly tested in the microscopic domain,bears consequences that defy classical intuition:It appears to imply that the familiar classical states should be an exceedingly rare exception. And,naively,one may guess that superposition principle should always apply literally:Everything is ultimately made out of quantum“stuﬀ”.Therefore,there is no a priori reason for macroscopic objects to have deﬁnite position or momentum.As Einstein noted1localization with respect to macrocoordinates is not just independent, but incompatible with quantum theory.How can one then establish correspondence between the quantum and the familiar classical reality?1.Copenhagen InterpretationBohr’s solution was to draw a border between the quantum and the classical and to keep certain objects–especially measuring devices and observers–on the clas-sical side(Bohr,1928;1949).The principle of superposi-tion was suspended“by decree”in the classical domain. The exact location of this border was diﬃcult to pinpoint, but measurements“brought to a close”quantum events. Indeed,in Bohr’s view the classical domain was more fundamental:Its laws were self-contained(they could be conﬁrmed from within)and established the framework necessary to deﬁne the quantum.Theﬁrst breach in the quantum-classical border ap-peared early:In the famous Bohr–Einstein double-slit debate,quantum Heisenberg uncertainty was invoked by Bohr at the macroscopic level to preserve wave-particle duality.Indeed,as the ultimate components of classical objects are quantum,Bohr emphasized that the bound-ary must be moveable,so that even the human nervous system could be regarded as quantum provided that suit-able classical devices to detect its quantum features were available.In the words of John Archibald Wheeler(1978; 1983)who has elucidated Bohr’s position and decisively contributed to the revival of interest in these matters,“No[quantum]phenomenon is a phenomenon until it is a recorded(observed)phenomenon”.3 This is a pithy summary of a point of view–known asthe Copenhagen Interpretation(CI)–that has kept manya physicist out of despair.On the other hand,as long as acompelling reason for the quantum-classical border couldnot be found,the CI Universe would be governed by twosets of laws,with poorly deﬁned domains of jurisdiction.This fact has kept many a student,not to mention theirteachers,in despair(Mermin1990a;b;1994).2.Many Worlds InterpretationThe approach proposed by Hugh Everett(1957a,b)and elucidated by Wheeler(1957),Bryce DeWitt(1970)and others(see DeWitt and Graham,1973;Zeh,1970;1973;Geroch,1984;Deutsch,1985,1997,2001)was toenlarge the quantum domain.Everything is now repre-sented by a unitarily evolving state vector,a gigantic su-perposition splitting to accommodate all the alternativesconsistent with the initial conditions.This is the essenceof the Many Worlds Interpretation(MWI).It does notsuﬀer from the dual nature of CI.However,it also doesnot explain the emergence of classical reality.The diﬃculty many have in accepting MWI stems fromits violation of the intuitively obvious“conservation law”–that there is just one Universe,the one we perceive.But even after this question is dealt with,,many a con-vert from CI(which claims allegiance of a majority ofphysicists)to MWI(which has steadily gained popular-ity;see Tegmark and Wheeler,2001,for an assessment)eventually realizes that the original MWI does not ad-dress the“preferred basis question”posed by Einstein1(see Wheeler,1983;Stein,1984;Bell1981,1987;Kent,1990;for critical assessments of MWI).And as long asit is unclear what singles out preferred states,perceptionof a unique outcome of a measurement and,hence,of asingle Universe cannot be explained either2.In essence,Many Worlds Interpretation does not ad-dress but only postpones the key question.The quantum-classical boundary is pushed all the way towards theobserver,right against the border between the materialUniverse and the“consciousness”,leaving it at a veryuncomfortable place to do physics.MWI is incomplete:It does not explain what is eﬀectively classical and why.Nevertheless,it was a crucial conceptual breakthrough:4ment)can be in principle carried out without disturbing the system.Only in quantum mechanics acquisition of information inevitably alters the state of the system–the fact that becomes apparent in double-slit and related experiments(Wootters and Zurek,1979;Zurek,1983). Quantum nature of decoherence and the absence of classical analogues are a source of misconceptions.For instance,decoherence is sometimes equated with relax-ation or classical noise that can be also introduced by the environment.Indeed,all of these eﬀects often ap-pear together and as a consequence of the“openness”. The distinction between them can be brieﬂy summed up: Relaxation and noise are caused by the environment per-turbing the system,while decoherence and einselection are caused by the system perturbing the environment. Within the past few years decoherence and einselection became familiar to many.This does not mean that their implications are universally accepted(see comments in the April1993issue of Physics Today;d’Espagnat,1989 and1995;Bub,1997;Leggett,1998and2002;Stapp, 2001;exchange of views between Anderson,2001,and Adler,2001).In aﬁeld where controversy reigned for so long this resistance to a new paradigm is no surprise. C.The nature of the resolution and the role of envariance Our aim is to explain why does the quantum Universe appear classical.This question can be motivated only in the context of the Universe divided into systems,and must be phrased in the language of the correlations be-tween them.In the absence of systems Schr¨o dinger equa-tion dictates deterministic evolution;|Ψ(t) =exp(−iHt/¯h)|Ψ(0) ,(1.1) and the problem of interpretation seems to disappear. There is no need for“collapse”in a Universe with no systems.Yet,the division into systems is imperfect.As a consequence,the Universe is a collection of open(in-teracting)quantum systems.As the interpretation prob-lem does not arise in quantum theory unless interacting systems exist,we shall also feel free to assume that an environment exists when looking for a resolution. Decoherence and einselectionﬁt comfortably in the context of the Many Worlds Interpretation where they deﬁne the“branches”of the universal state vector.De-coherence makes MWI complete:It allows one to ana-lyze the Universe as it is seen by an observer,who is also subject to decoherence.Einselection justiﬁes elements of Bohr’s CI by drawing the border between the quan-tum and the classical.This natural boundary can be sometimes shifted:Its eﬀectiveness depends on the de-gree of isolation and on the manner in which the system is probed,but it is a very eﬀective quantum-classical border nevertheless.Einselectionﬁts either MWI or CI framework:It sup-plies a statute of limitations,putting an end to the quantum jurisdiction..It delineates how much of the Universe will appear classical to observers who monitor it from within,using their limited capacity to acquire, store,and process information.It allows one to under-stand classicality as an idealization that holds in the limit of macroscopic open quantum systems.Environment imposes superselection rules by preserv-ing part of the information that resides in the correlations between the system and the measuring apparatus(Zurek, 1981,1982).The observer and the environment compete for the information about the system.Environment–because of its size and its incessant interaction with the system–wins that competition,acquiring information faster and more completely than the observer.Thus,a record useful for the purpose of prediction must be re-stricted to the observables that are already monitored by the environment.In that case,the observer and the en-vironment no longer compete and decoherence becomes unnoticeable.Indeed,typically observers use environ-ment as a“communication channel”,and monitor it to ﬁnd out about the system.Spreading of the information about the system through the environment is ultimately responsible for the emer-gence of the“objective reality”.Objectivity of a state can be quantiﬁed by the redundancy with which it is recorded throughout Universe.Intercepting fragments of the environment allows observers toﬁnd out(pointer) state of the system without perturbing it(Zurek,1993a, 1998a,and2000;see especially section VII of this pa-per for a preview of this new“environment as a witness”approach to the interpretation of quantum theory). When an eﬀect of a transformation acting on a system can be undone by a suitable transformation acting on the environment,so that the joint state of the two remains unchanged,the transformed property of the system is said to exhibit“environment assisted invariance”or en-variance(Zurek,2002b).Observer must be obviously ignorant of the envariant properties of the system.Pure entangled states exhibit envariance.Thus,in quantum physics perfect information about the joint state of the system-environment pair can be used to prove ignorance of the state of the system.Envariance oﬀers a new fundamental view of what is information and what is ignorance in the quantum world. It leads to Born’s rule for the probabilities and justiﬁes the use of reduced density matrices as a description of a part of a larger combined system.Decoherence and ein-selection rely on reduced density matrices.Envariance provides a fundamental resolution of many of the inter-pretational issues.It will be discussed in section VI D.D.Existential Interpretation and‘Quantum Darwinism’What the observer knows is inseparable from what the observer is:The physical state of his memory implies his information about the Universe.Its reliability de-pends on the stability of the correlations with the exter-nal observables.In this very immediate sense decoher-5ence enforces the apparent“collapse of the wavepacket”: After a decoherence timescale,only the einselected mem-ory states will exist and retain useful correlations(Zurek, 1991;1998a,b;Tegmark,2000).The observer described by some speciﬁc einselected state(including a conﬁgu-ration of memory bits)will be able to access(“recall”) only that state.The collapse is a consequence of einse-lection and of the one-to-one correspondence between the state of his memory and of the information encoded in it. Memory is simultaneously a description of the recorded information and a part of the“identity tag”,deﬁning observer as a physical system.It is as inconsistent to imagine observer perceiving something else than what is implied by the stable(einselected)records in his posses-sion as it is impossible to imagine the same person with a diﬀerent DNA:Both cases involve information encoded in a state of a system inextricably linked with the physical identity of an individual.Distinct memory/identity states of the observer(that are also his“states of knowledge”)cannot be superposed: This censorship is strictly enforced by decoherence and the resulting einselection.Distinct memory states label and“inhabit”diﬀerent branches of the Everett’s“Many Worlds”Universe.Persistence of correlations is all that is needed to recover“familiar reality”.In this manner,the distinction between epistemology and ontology is washed away:To put it succinctly(Zurek,1994)there can be no information without representation in physical states. There is usually no need to trace the collapse all the way to observer’s memory.It suﬃces that the states of a decohering system quickly evolve into mix-tures of the preferred(pointer)states.All that can be known in principle about a system(or about an observer, also introspectively,e.g.,by the observer himself)is its decoherence-resistant‘identity tag’–a description of its einselected state.Apart from this essentially negative function of a cen-sor the environment plays also a very diﬀerent role of a“broadcasting agent”,relentlessly cloning the informa-tion about the einselected pointer states.This role of the environment as a witness in determining what exists was not appreciated until now:Throughout the past two decades,study of decoherence focused on the eﬀect of the environment on the system.This has led to a mul-titude of technical advances we shall review,but it has also missed one crucial point of paramount conceptual importance:Observers monitor systems indirectly,by in-tercepting small fractions of their environments(e.g.,a fraction of the photons that have been reﬂected or emit-ted by the object of interest).Thus,if the understand-ing of why we perceive quantum Universe as classical is the principal aim,study of the nature of accessibil-ity of information spread throughout the environment should be the focus of attention.This leads one away from the models of measurement inspired by the“von Neumann chain”(1932)to studies of information trans-fer involving branching out conditional dynamics and the resulting“fan-out”of the information throughout envi-ronment(Zurek,1983,1998a,2000).This new‘quantum Darwinism’view of environment selectively amplifying einselected pointer observables of the systems of interest is complementary to the usual image of the environment as the source of perturbations that destroy quantum co-herence of the system.It suggests the redundancy of the imprint of the system in the environment may be a quantitative measure of relative objectivity and hence of classicality of quantum states.It is introduced in Sec-tions VI and VII of this review.Beneﬁts of recognition of the role of environment in-clude not just operational deﬁnition of the objective exis-tence of the einselected states,but–as is also detailed in Section VI–a clariﬁcation of the connection between the quantum amplitudes and probabilities.Einselection con-verts arbitrary states into mixtures of well deﬁned possi-bilities.Phases are envariant:Appreciation of envariance as a symmetry tied to the ignorance about the state of the system was the missing ingredient in the attempts of ‘no collapse’derivations of Born’s rule and in the prob-ability interpretation.While both envariance and the “environment as a witness”point of view are only begin-ning to be investigated,the extension of the program of einselection they oﬀer allowes one to understand emer-gence of“classical reality”form the quantum substrate as a consequence of quantum laws.II.QUANTUM MEASUREMENTSThe need for a transition from quantum determinism of the global state vector to classical deﬁniteness of states of individual systems is traditionally illustrated by the example of quantum measurements.An outcome of a “generic”measurement of the state of a quantum sys-tem is not deterministic.In the textbook discussions this random element is blamed on the“collapse of the wavepacket”,invoked whenever a quantum system comes into contact with a classical apparatus.In a fully quan-tum discussion this issue still arises,in spite(or rather because)of the overall deterministic quantum evolution of the state vector of the Universe:As pointed out by von Neumann(1932),there is no room for a‘real collapse’in the purely unitary models of measurements.A.Quantum conditional dynamicsTo illustrate the ensuing diﬃculties,consider a quan-tum system S initially in a state|ψ interacting with a quantum apparatus A initially in a state|A0 :|Ψ0 =|ψ |A0 = i a i|s i |A0−→ i a i|s i |A i =|Ψt .(1)Above,{|A i }and{|s i }are states in the Hilbert spaces of the apparatus and of the system,respectively,and a i6are complex coeﬃcients.Conditional dynamics of such premeasurement(as the step achieved by Eq.(2.1)isoften called)can be accomplished by means of a unitary Schr¨o dinger evolution.Yet it is not enough to claim thata measurement has been achieved:Equation(2.1)leads to an uncomfortable conclusion:|Ψt is an EPR-like en-tangled state.Operationally,this EPR nature of the state emerging from the premeasurement can be made moreexplicit by re-writing the sum in a diﬀerent basis:|Ψt = i a i|s i |A i = i b i|r i |B i .(2)This freedom of basis choice–basis ambiguity–is guar-anteed by the principle of superposition.Therefore,if one were to associate states of the apparatus(or the ob-server)with decompositions of|Ψt ,then even before en-quiring about the speciﬁc outcome of the measurement one would have to decide on the decomposition of|Ψt ; the change of the basis redeﬁnes the measured quantity.1.Controlled not and a bit-by-bit measurementThe interaction required to entangle the measured sys-tem and the apparatus,Eq.(2.1),is a generalization of the basic logical operation known as a“controlled not”or a c-not.Classical,c-not changes the state a t of the target when the control is1,and does nothing otherwise: 0c a t−→0c a t;1c a t−→1c¬a t(2.3) Quantum c-not is a straightforward quantum version of Eq.(2.3).It was known as a“bit by bit measurement”(Zurek,1981;1983)and used to elucidate the connection between entanglement and premeasurement already be-fore it acquired its present name and signiﬁcance in the context of quantum computation(see e.g.Nielsen and Chuang,2000).Arbitrary superpositions of the control bit and of the target bit states are allowed:(α|0c +β|1c )|a t−→α|0c |a t +β|1c |¬a t (3) Above“negation”|¬a t of a state is basis dependent;¬(γ|0t +δ|1t )=γ|1t +δ|0t (2.5) With|A0 =|0t ,|A1 =|1t we have an obvious anal-ogy between the c-not and a premeasurement.In the classical controlled not the direction of informa-tion transfer is consistent with the designations of the two bits:The state of the control remains unchanged while it inﬂuences the target,Eq.(2.3).Classical measurement need not inﬂuence the system.Written in the logical ba-sis{|0 ,|1 },the truth table of the quantum c-not is essentially–that is,save for the possibility of superpo-sitions–the same as Eq.(2.3).One might have antici-pated that the direction of information transfer and the designations(“control/system”and“target/apparatus”)of the two qubits will be also unambiguous,as in the clas-sical case.This expectation is incorrect.In the conjugate basis{|+ ,|− }deﬁned by:|± =(|0 ±|1 )/√2|1 1|S⊗(1−(|0 1|+|1 0|))A(5) Above,g is a coupling constant,and the two operators refer to the system(i.e.,to the former control),and to the apparatus pointer(the former target),respectively. It is easy to see that the states{|0 ,|1 }S of the system are unaﬀected by H int,since;[H int,e0|0 0|S+e1|1 1|S]=0(2.10) The measured(control)observableˆǫ=e0|0 0|+e1|1 1| is a constant of motion under H int.c-not requires inter-action time t such that gt=π/2.The states{|+ ,|− }A of the apparatus encode the information about phase between the logical states.They have exactly the same“immunity”:[H int,f+|+ +|A+f−|− −|A]=0(2.11) Hence,when the apparatus is prepared in a deﬁnite phase state(rather than in a deﬁnite pointer/logical state),it will pass on its phase onto the system,as Eqs.(2.7)-(2.8),show.Indeed,H int can be written as:H int=g|1 1|S|− −|A=g7 phases between the possible outcome states of the ap-paratus.This leads to loss of phase coherence:Phasesbecome“shared property”as we shall see in more detailin the discussion of envariance.The question“what measures what?”(decided by thedirection of the informationﬂow)depends on the initialstates.In“the classical practice”this ambiguity does notarise.Einselection limits the set of possible states of theapparatus to a small subset.2.Measurements and controlled shifts.The truth table of a whole class of c-not like trans-formations that includes general premeasurement,Eq.(2.1),can be written as:|s j |A k −→|s j |A k+j (2.13)Equation(2.1)follows when k=0.One can thereforemodel measurements as controlled shifts–c-shift s–generalizations of the c-not.In the bases{|s j }and{|A k },the direction of the informationﬂow appears tobe unambiguous–from the system S to the apparatus A.However,a complementary basis can be readily deﬁned(Ivanovic,1981;Wootters and Fields,1989);|B k =N−1N kl)|A l .(2.14a)Above N is the dimensionality of the Hilbert space.Anal-ogous transformation can be carried out on the basis{|s i }of the system,yielding states{|r j }.Orthogonality of{|A k }implies:B l|B m =δlm.(2.15)|A k =N−1N kl)|B l (2.14b)inverts of the transformtion of Eq.(2.14a).Hence:|ψ = lαl|A l = kβk|B k ,(2.16)where the coeﬃcientsβk are:βk=N−1Nkl)αl.(2.17)Hadamard transform of Eq.(2.6)is a special case of themore general transformation considered here.To implement the truth tables involved in premeasure-ments we deﬁne observableˆA and its conjugate:ˆA=N−1k=0k|A k A k|;ˆB=N−1 l=0l|B l B l|.(2.18a,b)The interaction Hamiltonian:H int=gˆsˆB(2.19)is an obvious generalization of Eqs.(2.9)and(2.12),withg the coupling strength andˆs:ˆs=N−1l=0l|s l s l|(2.20)In the{|A k }basisˆB is a shift operator,ˆB=iN∂ˆA.(2.21)To show how H int works,we compute:exp(−iH int t/¯h)|s j |A k =|s j N−1。

Decoherence of the Unruh Detector

a r X i v :g r -q c /9508016v 2 8 A u g 1995Decoherence of the Unruh DetectorJ EAN-G UY D EMERSDepartment of Physics,McGill University3600University St.,Montreal,PQ,Canada H3A 2T8.jgdemers@hep.physics.mcgill.caAbstractAs it is well known,the Minkowski vacuum appears thermally pop-ulated to a quantum mechanical detector on a uniformly acceler-ating course.We investigate how this thermal radiation may con-tribute to the classical nature of the detector’s trajectory through the criteria of decoherence.An uncertainty-type relation is ob-tained for the detector involving the ﬂuctuation in temperature,the time of ﬂight and the coupling to the bath.1IntroductionIn the Copenhagen interpretation of the measurement process,the existence of classical systems is postulated from the start.In quantum cosmology however,the whole universe receives a quantum mechanical description and classicality must be somehow derived.An important mechanism by which some degrees of freedom may behave classically is the environment-induced decoherence.In that case,partial diagonalization of the density matrix in a given basis is obtained by integrating out some unobserved degrees of freedom[1].A simple illustration is given by a system and its environment having each a single de-gree of freedom described by the actions S sys [x ]and S env [y ]respectively,with interaction described by the action S int [x,y ].Starting with an uncorrelated density matrix at t =0,ρ(x 1,y 1,x 2,y 2;t =0)=ρsys (x 1,x 2;t =0)×ρenv (y 1,y 2;t =0),(1)then at a time t later,the reduced density matrix (where the environment is integrated out)will be given by[2,3]:ρred (x f 1,x f2;t )=dx i 1dx i 2P (x f 1,x f 2,t |x i 1,x i 2,0)ρsys (x i 1,x i2;0),(2)with the kernelP (x f 1,x f 2,t |x i 1,x i2,0)=x f 1x i 1D x 1x f 2x i 2D x 2e i (S sys [x 1]−S sys [x 2])e i Γ[x 1,x 2],(3)where the boundary conditions on the functional integrals are given for times 0and t (our units are such that ¯h =c =1).Γ[x 1,x 2]is the inﬂuence func-tional (IF).It is a property of the environment and the way it is coupled to the system[4].For an environment in a pure state at t =0,it takes the conceptually simple forme i Γ[x 1,x 2]= ψ2(t )|ψ1(t ) .(4)Here,|ψ1(t) is the state that evolved from the initial state at t=0under the dynamics dictated by S env[y1]+S int[x1,y1]in which x1acts as a c-number, time dependent source;likewise|ψ2(t) is governed by S env[y2]+S int[x2,y2]. For paths[x1(t)]and[x2(t)]such that the states in(4)are not adiabatically disturbed,Γwill have a(positive)imaginary partΓI.It is clear that ifΓI is large for pair of paths(x1(t),x2(t))that are far apart from one another in spacetime,then the contribution of these pairs will be suppressed in(3).As a result,ρred(x f1,x f2;t)will be more diagonal.Decoherence can thus be studied throughΓI.In recent times,the study of black holes received renewed interest,in part du to the introduction of more tractable two dimensional models.An impor-tant issue arising in that context is the limit of the semiclassical approximation, in which the Hawking radiation is derived.As part of the groundwork to that problem,we report here on the related issue of the decoherence of a detector in uniform acceleration.In the next section,we review how the IF predicts the heating of the scalarﬁeld forming the environment of a detector on a uniformly accelerating course[5,6].In Section3,the spacetime trajectory is taken to be a decohered spacetime path,with a spread around the uniformly accelerating trajectory,and taking now the detector’s excitation to beﬁxed. We then obtain an uncertainty-type relation involving the spread in acceler-ation involved in constructing the approximately classical path,the time of ﬂight and the coupling to the thermal bath.2Unruh EﬀectConsider an environment made of a massless scalarﬁeld in two spacetime dimensions[3,6].In aﬁnite box of spatial dimension L,the mode expansion readsφ(x,t)= L k[y+k cos kx+y−k sin kx],(5) with k=2πn/L with n=1,2....The kinetic term isL env= dx12 σ=+,− k[(˙yσk)2−k2(yσk)2],(6)where dots denote time derivative.The system is formed by a detector that is linearly and locally sensitive to the matterﬁeld with an interacting Lagrangian densityL int(x)=−εQφ(x,t)δ(x−X(t)),(7) whereεis a coupling constant while Q(t)is the DeWitt monopole moment[7] and plays the role of x(t)in Section1while X(t)is the trajectory of the de-tector.For a uniform acceleration a,X(τ)=1a sinh(aτ)whereτis the proper time and the resulting IF is given by[3]ΓR[Q1,Q2]=− t0ds1 s10ds2[Q1(s1)−Q2(s1)]µ(s1,s2)[Q1(s2)+Q2(s2)],ΓI[Q1,Q2]= t0ds1 s10ds2[Q1(s1)−Q2(s1)]ν(s1,s2)[Q1(s2)−Q2(s2)],(8)whereµ(τ1,τ2)=−∞0dωε22πωcothπωa −2i +t 2with i =1,2.For the scalar ﬁeld in (5)with interaction in (7),the sum over modes can be converted into an integral,and we then ﬁnd [9]ΓI (a 1,a 2,t )=ε2Q 2A 2−F 2D 2−F 2a −21+s 21− a −22+s 21−a −21+s 21−a −22+s 21−8πt R (t,a m ,a d ),(13)with:R (t,a m ,a d )=1dy P (s 1=t,s 2=yt,a 1,a 2)=R (˜t ,˜a d ).(14)24681000.050.10.150.20.25A BCFigure 1:Plot of R (˜t ,˜a d )for ˜a d =0.05(curve A ),˜a d =0.075(curve B )and ˜a d =0.1(curve C ).In (14),we made the change of variable s 2=yt (y is unitless),and workedwith the average a m ≡116πε2 Q 2t2a 1−a 2T≈1given by1。

deprotect-consensus analysis -回复

deprotect-consensus analysis -回复什么是deprotect consensus analysis（DCA）？DCA，全称deprotect consensus analysis，也称为去保护共识分析，是一种用于序列分析和比较的方法。

在生物学研究中，DCA被用于分析序列之间的保守性和变异性，以揭示蛋白质、DNA或RNA序列的功能和进化关系。

该方法可以帮助研究人员理解生物分子在不同物种中的相似性和差异性，并推断它们的结构和生物学功能。

首先，DCA将序列比对结果转化为一个Kolmogorov-Smirnov统计量，该统计量用于揭示序列之间的差异。

Kolmogorov-Smirnov统计量可以衡量两个序列分布之间的差异程度，从而反映了它们之间的保守性和变异性。

接下来，DCA利用一个保守序列模型来解析Kolmogorov-Smirnov 统计量。

保守序列模型将序列分为两个部分：保守部分和变异部分。

保守部分是那些在不同物种中高度保守的序列段，而变异部分则是那些在不同物种中具有较高变异性的序列段。

通过对序列进行这种划分，DCA可以推断出保守性和变异性之间的关系。

然后，DCA使用一种聚集性分析方法，将序列按照它们的保守性和变异性进行聚类。

聚集性分析是一种将数据分组为一些相似的子集的方法，可以帮助研究人员更好地理解序列之间的相似性和差异性。

通过聚类分析，DCA可以揭示序列之间的共识或者特定群体的保守序列模式。

最后，DCA将聚类结果可视化，以便研究人员更好地理解和解释序列的保守性和变异性。

通常，DCA可以生成保守性和变异性的热图或树状图，帮助研究人员在序列之间进行比较和分类。

这些可视化结果可以为进一步研究提供有价值的线索和指导。

综上所述，DCA是一种应用于序列分析和比较的方法，可以揭示序列之间的保守性和变异性。

通过将序列比对结果转化为Kolmogorov-Smirnov统计量，DCA可以研究序列的保守性和变异性，并使用保守序列模型、聚集性分析和可视化方法来进一步解释这些结果。

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Decoherence of Schr˝ odinger cat statesБайду номын сангаасin a Luttinger liquid
P. Degiovanni∗ and S. Peysson†
Laboratoire de Physique de l’Ecole Normale Sup´ erieure de Lyon, Unit´ e de mixte de recherche CNRS et Ecole Normale Sup´ erieure de Lyon (UMR 5672), Groupe de Physique Th´ eorique, 46 all´ ee d’Italie, 69364 Lyon Cedex 07, France.
arXiv:cond-mat/0001452v1 [cond-mat.str-el] 31 Jan 2000
Schr˝ odinger cat states built from quantum superpositions of left or right Luttinger fermions located at diﬀerent positions in a spinless Luttinger liquid are considered. Their decoherence rates are computed within the bosonization approach using as environments the quantum electromagnetic ﬁeld or two or three dimensional acoustic phonon baths. Emphasis is put on the diﬀerences between the electromagnetic and acoustic environments. LPENSL-Th-01/2000
I.
INTRODUCTION AND MOTIVATIONS
Decoherence is a very important issue for mesoscopic systems since it governs the crossover between quantum and quasi-classical transport regimes. Coherence of mesoscopic conductors gives rise to various Aharonov-Bohm interference eﬀects such as permanent currents in a mesoscopic ring, and conductance oscillations as a function of the external magnetic ﬁeld. The problem of electron decoherence in metals at zero temperature is an active area of research both from the theoretical and experimental point of view. Recent experiments claim to observe a saturation of the dephasing time τφ at very low temperatures1 . Strong discussions among theorists arose from these observations2,3,4,5,6 . The heart of the debate, summarized for example in Mohanty’s recent letter7 , is to determine whether the conventional theory of dephasing in Fermi liquids8,9,10 could explain the saturation of τφ (for example as an eﬀect of an external microwave radiation2,11 ) or whereas one should reconsider the theory completely12,13,14 . Although such an excitement is a strong motivation for working on decoherence in disordered mesoscopic conductors, we shall present a model for studying the decay of Schr˝ odinger cat states in 1D ballistic conductors. From a methodological point of view, our line of thought is very close to the one used in atomic physics, for example in theoretical works15,16 on decoherence experiments in cavity QED17 . It relies heavily on the use of simple exactly solvable models of decoherence, such as the Caldeira-Leggett model18 . We think that this point of view brings a diﬀerent light on the question of decoherence by taking into account the electron ﬂuid as a whole and studying the coupling of this many-body system to the external reservoirs. It makes a bridge between atomic physics situations, where decoherence is extremely well controlled and for which simpliﬁed decoherence models apply almost directly and mesoscopic conductors which are complex interacting systems. In particular, this point of view is well suited to the 1D case because of interactions. In a non-Fermi liquid, one cannot keep track of an individual electron because of orthogonality catastrophe eﬀects. Moreover, electromagnetic or acoustic radiation emitted by one part of the system and absorbed by another part may have drastic eﬀect on the strongly correlated electron state. These remarks motivated our method, based on the study of the coupling of an external environment to the low energy excitations of the 1D electron system. Following general ideas of Stern, Aharonov and Imry19 , we have computed the decoherence rate of Schr˝ odinger cat states built from localized excitations (e.g Luttinger fermions) introduced at diﬀerent places in the system or at the same place but moving in diﬀerent directions (left and right moving components). As expected, we have found that such linear superpositions decay into statistical mixtures even at zero temperatures and computed various decoherence scenarii. This is the main result of this paper and it means that a 1D pure ballistic conductor exhibits decoherence at absolute zero in the sense of Schr˝ odinger cat states decay. One dimensional conductors in the ballistic regime are appropriately described by an eﬀective interacting theory: the Luttinger liquid20 . In this eﬀective theory, interactions between electrons are put by means of an electrostatic short-ranged potential. Obviously, within the approximation of a linear dispersion relation for particle-hole excitations, the Luttinger eﬀective theory contains no source for decoherence. A coupling to an external quantum environment is necessary to introduce decoherence in the Luttinger liquid. The quantized electromagnetic ﬁeld and a 2D or 3D bath of longitudinal phonons will be considered in the present paper. The combined use of bosonization and non-equilibrium techniques make it possible to solve this problem. Within the bosonization framework, the pioneering work21 by Martin and Loss investigates the question of equilibrium permanent currents induced by ﬂuctuations of the quantized electromagnetic ﬁeld. They have shown that coupling the Luttinger liquid to QED leads to a renormalization of the Luttinger liquid parameters. Our discussion of dynamical and therefore non-equilibrium aspects of the coupled Luttinger & QED system will show how this renormalization appears dynamically. Let us mention that coupling a Luttinger liquid to one dimensional phonons also renormalizes the Luttinger liquid parameters and drives a 1D Fermi