Unraveling quantum Brownian motion: Pointer states and their classical trajectories

Transcription

1 PHYSICAL REVIEW A 9, ) Unraveling quantum Brownian motion: Pointer states an their classical trajectories Lutz Sörgel an Klaus Hornberger Faculty of Physics, University of Duisburg-Essen, Lotharstraße 1, Duisburg, Germany Receive 8 September 015; publishe 9 December 015) We characterize the pointer states generate by the master equation of quantum Brownian motion an erive stochastic equations for the ynamics of their trajectories in phase space. Our metho is base on a Poissonian unraveling of the master equation whose eterministic part exhibits soliton-like solutions that can be ientifie with the pointer states. In the semiclassical limit, their equations of motion in phase space turn into those of classical iffusion, yieling a clear picture of the quantum-classical transition inuce by environmental ecoherence. DOI: /PhysRevA PACS numbers): Yz, Jc, Ab I. INTRODUCTION Decoherence through environmental coupling is a crucial ingreient for unerstaning the quantum-to-classical transition [1 3]. As a prototypical problem, one may consier the motion of a quantum particle interacting with an ieal gas environment. The interaction with the environment selects a preferre set of states, the so-calle pointer states [4 6]. These special states of the particle remain pure for a relatively long time, whereas their superpositions ecohere very fast into a mixture. For simple master equations the pointer states are known [5 9] to be localize wave packets moving in phase space. In the semiclassical limit they turn into arbitrarily sharp peaks moving along classical phase space trajectories. As such, pointer states are ieally suite to explain the emergence of classicality within a quantum framework. The name pointer states erives from the fact that their characteristics matches the behavior of a pointer of a measuring apparatus. Namely, if a quantum system is in an eigenstate of the observable we measure with an apparatus its pointer will turn to a specific value on the ial an stay there. On the other han, if the system is in a superposition of two nonegenerate eigenstates one will not fin the pointer of the apparatus to be in a superposition of the two corresponing values, as prescribe by unitarity, but pointing to either one or the other position with probabilities etermine by Born s rule. In this article, we ientify the pointer states of the master equation of quantum Brownian motion QBM) an erive their equations of motion. Specifically, we unerstan as QBM the issipative ynamics of a free particle resulting from many small environmental collisions as escribe by a Linbla-type master equation [10,11]. One possibility to obtain this master equation is to consier a particle linearly couple to a bath of harmonic oscillators [1 15] an to take the high-temperature limit of the reuce particle ynamics; one arrives at the master equation of the Caleira Leggett moel [5,1,16]. To get a Linbla master equation an aitional term has to be appene [17 19] making the equation completely positive. Alternatively, the Linbla master equation of QBM can be obtaine by consiering a issipative limit of the general particle motion in a gaseous environment as governe by the so-calle quantum linear Boltzmann equation [19 ]. By consiering quantum trajectories using a particular stochastic process in Hilbert space, the so-calle pointer state unraveling, we fin the pointer states of QBM to be rotate Gaussians in phase space, in line with the encounter of Gaussian pointer states in other moels of linear environmental coupling [6,7]. Moreover, we show how superpositions of separate wave packets ecohere into mixtures of pointer states with weights given by the absolute value of the initial amplitue in the superposition i.e., Born s rule). Finally, we erive the stochastic equations of motion for the location of the pointer states in phase space an we show that they turn into the equation of classical iffusion in the semiclassical limit, thus proviing a comprehensive picture of the quantum-to-classical transition for the important issipative process of QBM. For times of evolution much greater than the characteristic ecoherence time, it is then justifie to approximate the ynamics by a mixture of pointer states moving on trajectories in phase space, instea of calculating the whole evolution of the ensity matrix. Our metho of using the pointer-state unraveling to ientify the pointer basis an to provie a perspective on the quantumto-classical transition for particle motion has alreay been emonstrate for the nonissipative case of pure collisional ecoherence [9,3,4]. In that case, the pointer states are exponentially localize, rather than Gaussian, an the emerging classical equations of motion are Newton s equations. In the present case of issipative ynamics we erive a stochastic equation of motion for the pointer states in orer to get the Langevin equation corresponing to classical iffusion. This requires a subtle analysis of the ynamics of the quantum trajectories. The paper is organize as follows: At first, in Sec. II,wegive a general efinition of pointer states. In the following Sec. III we introuce an briefly iscuss the Linbla master equation of QBM an introuce its pointer state unraveling. The careful examination of this particular quantum trajectory approach is the topic of the remainer of the paper: The eterministic part of the unraveling is analyze in Sec. IV proviing us with the Gaussian pointer states of QBM as well as with the eterministic part of the classical equations of motion. By looking at the full unraveling, i.e., the eterministic an stochastic part, we show in Sec. V that any superposition of separate wave packets turns into a mixture of pointer states with the expecte probabilities. In aition, we ientify the trajectories of the pointer states; they exhibit a stochastic motion in phase space an yiel iffusion in the semiclassical limit. Section VI contains our conclusions /015/96)/061110) American Physical Society

2 LUTZ SÖRGEL AND KLAUS HORNBERGER PHYSICAL REVIEW A 9, ) II. POINTER STATES The evolution of a Markovian open quantum system can be escribe by a Linbla master equation [10,11,5]. For this equation to exhibit pointer states [4] we expect it to isplay a timescale separation into a short ecoherence time t ec an a much longer ynamical timescale which has also a classical meaning [5,6]. The set of pointer states is characterize by the property that superpositions of mutually orthogonal pointer states ecay into a mixture on the ecoherence timescale whereas a single pointer state remains relatively stable an changes on the longer timescale only. Specifically, for a superposition of pointer states ψ 0 = c α 0) π α 0), 1) α=1 the solution of the master equation for times much greater than t ec is well approximate by the mixture of pointer states [5] ρ 0 = ψ 0 ψ 0 ρ t Probα ρ 0 ) π α t) π α t). ) α=1 A proper set of pointer states forms a basis, is inepenent of the initial state, an epens only on the environment an the coupling operators. As a consequence an accoring to Born s rule, the only epenence of the mixture ) on the initial state ρ 0 is via the probabilities Probα ρ 0 ) = Trρ 0 π α 0) π α 0) ) = c α 0). 3) The time epenence of the pointer states π α t) occurs on the longer timescale only. Once the pointer states an their time epenence are characterize, the ynamics of the master equation is fully capture by Eq. ) for times greater than t ec. There are ifferent possibilities to ientify the pointer basis. A well-known one requires that the pointer states shoul be the least-entropy-proucing states of the system; this is the so-calle preictability sieve [6,7]. It can be implemente by consiering the robustness of quantum states uner continuous observation [8], which has been recently extene to feeback stabilization [9]. Here, we follow another approach, in which the pointer states are ientifie as soliton-like solutions of a nonlinear equation associate with the master equation. This nonlinear pointer state equation NLPSE) can be obtaine heuristically by ientifying the pure state evolution closest to the master equation evolution [30,31]. The associate pointer state unraveling is a piecewise eterministic stochastic process in Hilbert space. It connects the solution of the master equation with the NLPSE, since the latter escribes the eterministic part of the unraveling. In the next section, we present the master equation of QBM together with its pointer state unraveling. III. QUANTUM BROWNIAN MOTION A. Master equation of quantum Brownian motion The master equation of QBM escribes the quantum state of a massive marker particle linearly couple to a hightemperature gas environment an experiencing ecoherence as well as issipation. For the sake of simplicity, we restrict ourselves here to the one-imensional case an use imensionless variables by introucing time, length, an momentum scales T = 1 γ, L = 1 γ kb T env m, P = mk B T env. 4) They involve the mass of the particle m, the temperature of the surrouning environment T env, an the friction constant γ. These scales o not epen on an are therefore convenient in escribing the semiclassical limit of QBM. In these units the canonical commutator takes the form [ˆx,ˆp] = i/κ an the momentum operator in position representation is x ˆp = i/κ x x), where we introuce the imensionless parameter κ = k BT env γ. 5) The semiclassical limit is then obtaine by sening κ. The Linbla master equation of QBM is obtaine by using a single Linbla operator ˆL = κ ˆx + i4 ) ˆp, 6) an the Hamiltonian ˆp Ĥ = κ + 1 ) 4 {ˆx,ˆp}. 7) The anticommutator {ˆx, ˆp}/4 escribes a rescaling of the particle energy ue to the environmental coupling. Inserting Eqs. 6) an 7) into the general Linbla master equation [10,5] t ρ = 1 i [Ĥ,ρ] + ˆLρ ˆL 1 ) ˆL 1 ˆLρ ρ ˆL ˆL, 8) we arrive at the Linbla master equation of QBM [19,,5] t ρ = i κ [ˆp,ρ] i κ [ˆx,{ˆp,ρ}] κ [ˆx,[ˆx,ρ]] 1 [ˆp,[ˆp,ρ]]. 9) 16 Equation 9) is the Linbla generalization of the famous Caleira Leggett master equation [1,17,5], having in common the first three terms on the right-han sie. They escribe free motion, friction, an momentum iffusion, respectively. The last term of Eq. 9), which ensures complete positivity, can be ientifie with a position iffusion, see also Ref. [17].This position iffusion term is not present in the Caleira Leggett master equation, an its influence ecreases with growing κ. When writing Eq. 9) in the Wigner phase space representation one can raw the semiclassical limit, κ, to see the formal analogy between the evolution equation for the Wigner function an the Fokker Planck equation for the classical phase space istribution of classical iffusion [1,19]. Further analogies to the classical case of Brownian motion can be seen if one consiers the mean values an variances of the position an momentum operators [5]. Most prominently, one fins that the mean of the momentum operator is exponentially ampe an that the variance of the position operator grows linearly for asymptotically large times; both properties are characteristics of classical Brownian motion

3 UNRAVELING QUANTUM BROWNIAN MOTION: POINTER... PHYSICAL REVIEW A 9, ) B. Unraveling the master equation In orer to erive the pointer states of QBM an their motion in phase space, we replace the eterministic master equation evolution for the ensity matrix 9) by an equivalent stochastic pure state evolution in Hilbert space, the pointer state unraveling. The equivalence of an unraveling to a master equation is provie by the fact that the ensemble average over the quantum trajectories yiels the solution of the master equation [9,30 35]. Since the ecomposition of a ensity matrix into a mixture of pure states is not unique, many ensembles of quantum trajectories will recover the same solution of the master equation, hence the choice for an unraveling is ambiguous. One istinguishes between continuous unravelings base on a Wiener process [16,36 39], where quantum state iffusion is the most prominent, an piecewise eterministic unravelings [8,9,30,31,33,34] base on a Poisson process. They have been use to escribe associate ynamics in classical phase space motivate by continuous quantum measurements [40,41] an by the wish to emonstrate regular an chaotic ynamics [4]. However, to our knowlege, there have been no attempts to unravel the QBM master equation 9) with the help of localize pointer states. We use here a piecewise eterministic unraveling which was alreay employe in the case of collisional ecoherence to etermine the pointer states an their ynamics [8,9,31,35]. Its eterministic part is escribe by a nonlinear operator [37]. For QBM it takes the form ˆN[ψ] ψ = i κ ˆp p ) + κ [V x ˆx x) ] [V p ˆp p) ] + κ ic xp 1 ) 4 κ i κ ˆx x)ˆp + p) ) ψ, 10) with expectation values enote as A = Â = ψ Â ψ, variances V A = A A, an the covariance C xp = {x, p} x p. Due to the time epenence of the state ψ, all expectation values are time epenent. The time variable is omitte here for the sake of reaability. The evolution equation t ψ = ˆN[ψ] ψ, 11) will be calle the nonlinear pointer state equation NLPSE) because it exhibits soliton-like solutions which we will ientify as the pointer states. Soliton like is unerstoo here in the sense that the shape of the wave function is time inepenent in phase space. As mentione above, the NLPSE 11) can also be obtaine by looking for the particular pure state equation that mimics the master equation most closely [6,31]. To obtain the whole unraveling a stochastic jump part must be ae to the eterministic NLPSE 11) yieling ) ψ = ˆN[ψ] ψ t Ĵ[ψ] + Ĵ[ψ] ψ 1 ψ N, 1) with the nonlinear jump operator Ĵ[ψ] = κˆx x) + i4 ) ˆp p). 13) The stochastic Poisson increment N can take values N {0,1} an satisfies the Poisson increment rule N = N. 14) The nonlinearity of the jump operator 13) arises because of its epenence on the position an momentum expectation values which epen on the state ψ. To fully efine the above Poisson process we have to specify the ensemble average of the Poisson increment. It is etermine by the rate r = ψ Ĵ [ψ]ĵ[ψ] ψ of the process, given in our case by E[N] = r = κ V x + 1 t 8 V p 1, 15) with E[ ] enoting the ensemble average. A straightforwar calculation shows that this unraveling inee recovers the master equation 9) in the ensemble average [31]. It is also straightforwar to verify that the jump rate in this unraveling is proportional to the change of purity of the state r = 1 t Tr1 ψ ψ ) ). 16) This fining fits with the expecte behavior of a pointer state unraveling: Once the pointer state has been reache it minimizes the rate of purity change accoring to the preictability sieve criterion [7]. Accoring to Eq. 16) the jump rate of the unraveling is then also minimize. On the other han, one can arrive at the pointer state unraveling by minimizing the jump rate of a piecewise eterministic unraveling [43]. IV. POINTER STATES OF QUANTUM BROWNIAN MOTION In this section we present the pointer states of QBM as soliton-like solutions of the NLPSE 11). After that we analyze the action of the NLPSE on a superposition of separate wave packets, which will help us to obtain Born s rule in Sec. V. A. Single wave packets A lengthy but straightforwar calculation confirms that the NLPSE 11) is solve by a Gaussian state of the form 1 x x) ψx) = exp 1 iκc πv x ) 1/4 xp ) 4V x ) + iκx x)p + iφ, 17) where φ is a time epenent global phase. This requires that the first moments an the variances satisfy the following close set of ifferential equations, x = p, 18) t p = p, 19) t

4 LUTZ SÖRGEL AND KLAUS HORNBERGER PHYSICAL REVIEW A 9, ) t V x = C xp V x 4κ V x 1) + 1 κ Cxp, 0) 16κ t C xp = 1 + κ Cxp 8 C xp 4κ C 16κ xp V x. 1) V x To see this, one inserts the ansatz 17) into the NLPSE 11) an compares coefficients in powers of x. A ifferential equation for V p is reaily erive by noting that the variances of the Gaussian state 17) are relate via 4V x V p = 1 κ + C xp. ) The equations of motion 18) an 19) for the first moments exhibit an exponential momentum amping, which reflects the friction experience by the Brownian particle through the interaction with the gas environment. This movement oes not affect the form of the wave packet but only its position in phase space. The equations of motion 0) an 1) for the variances, however, rive the with an orientation of the wave packet in phase space towars the stable fixe values V x,ps = κ + 1) 16κ + 1 8κ, 3) 16κ + 1 C xp,ps = 4κ 16κ κ. 4) 16κ + 1 The label ps enotes the pointer state, an V p,ps can be calculate with the help of Eq. ). These fixe values are obtaine by solving the nonlinear equations which arise by setting V x /t = C xp /t = 0; their stability can be confirme by a linear stability analysis of Eqs. 0) an 1) aroun V x,ps an C xp,ps. A wave packet with the fixe values V x,ps an V p,ps is a soliton-like solution in the sense that the form of the wave function in phase space oes not change with time. We will call this solution a pointer state of quantum Brownian motion an enote it by πx,p), where x an p parametrize the position an momentum coorinates of the pointer state. As Gaussian wave functions they form an overcomplete basis set in Hilbert space. From Eqs. ) 4) one euces the asymptotic withs of the pointer states as κ : V x,ps 1 κ, V 3/ p,ps 1 κ, C 1/ xp,ps 1 κ. 5) The pointer states thus get more an more localize in phase space as one goes eeper into the semiclassical regime 0) characterize by κ ;seeeq.5). This is illustrate in Fig. 1. We note that similar Gaussian states are obtaine for pure spatial ecoherence with ˆL = ˆx [5 7], iffering in their covariance C xp. B. Superposition of wave packets We now analyze the action of the NLPSE 11) on a superposition of separate wave packets. We will see that the NLPSE suppresses such superpositions until only a single V, C V x,ps V p,ps C xp,ps FIG. 1. Color online) Pointer state variances V x,ps in position soli line) an V p,ps momentum space ashe line) an covariance C xp,ps otte line) as a function of the semiclassical parameter κ.for κ 1, i.e., in the semiclassical limit, all curves follow a potential law. The imensionless units are efine via Eq. 4). localize wave packet remains, which eventually turns into a pointer state πx,p) introuce before. We consier an initial superposition state of the form ψ = j κ c j 0) ψ j, 6) where the wave packets ψ j are well separate in phase space, orthogonal, an normalize with weights w j 0) = c j 0), 7) summing up to unity. We rop the time argument of the weights an amplitues for better reaability, an we enote expectation values of the wave packet components as A j = ψ j  ψ j, an the variances an covariance accoringly. The ψ j are require to be well separate wave packets in the sense that the orthogonality conition ψ j  ψ k =δ jk ψ j  ψ j j,k, 8) hols for the operators  {Î,ˆx,ˆx,ˆp,ˆp,ˆxˆp + ˆpˆx}. The shape of the ψ j can be arbitrary, but their withs in phase space shall be close to that of a pointer state. In particular, we require that the variances V x,j an V p,j of the wave packet components ψ j exhibit the same epenence 5) onκ as the pointer state. The orthogonality conition 8) can then be fulfille arbitrarily well by increasing the parameter κ, which ecreases the pointer state with in phase space accoring to Eq. 5), an thus further orthogonalizes the ψ j. For the following, it is useful to express the expectation values of the superposition state ψ with the help of those of the constituting wave packets ψ j : x = j V x = j w j x j, p = w j p j, j w j V x,j + 1 w j w k x j x k ), j,k

5 UNRAVELING QUANTUM BROWNIAN MOTION: POINTER... PHYSICAL REVIEW A 9, ) V p = j w j V p,j + 1 w j w k p j p k ), j,k C xp = w j C xp,j + w j w k x j x k )p j p k ). 9) j j,k By inserting the superposition state 6) into the NLPSE 11) we get the expression t ψ = t c j ) ψ j +c j t ψ j j = c j ˆN[ψ] ψj, 30) j where the nonlinear operator on the right-han sie contains expectation values with respect to ψ. From this equation we euce a set of couple equations for the normalize states ψ j an their weights w j = c j : t ψ j =ˆN[ψ j ] ψ j + κˆx x j ) + i ) 4 ˆp p j ) κx x j ) i ) 4 p p j ) ψ j, 31) { t w j = w j κ [V x V x,j x x j ) ] + 1 } 8 [V p V p,j p p j ) ]. 3) Note, that the nonlinear operator appearing in 31) now only contains expectation values with respect to ψ j. Moreover, if there is only one constituent in the superposition, Eqs. 31) an 3) turn consistently into the NLPSE 11) an the trivial evolution w j /t = 0 for the weight. This is seen most clearly by looking at the ifferences of the expectation values, which vanish in that case. The couple weight equations 3) exhibit a fixe point if a single w j is equal to one, while all others are zero. It can be checke by a linear stability analysis that this fixe point is a stable one. Thus, Eq. 3) expresses the fact that the NLPSE suppresses superpositions of pointer states, since the ynamics of the weights always ens up in the stable fixe point. To illustrate the suppression of a superposition state, we consier a superposition of two pointer states. The weight of the first component ψ 1 then follows the ifferential equation t w 1 = w 1 1)w 1 w κ x x 1 ) + 18 ) p p 1 ). 33) We see immeiately that w 1 gets greater if an only if it is alreay the greater of the two weights, leaing eventually to the suppression of the other component. If there exists a superposition of many wave packets, which, in aition, all have ifferent withs the situation becomes far more complex an cannot be capture intuitively. However, the eventual ecay of the superposition into one of the components is certain, though it is not easily preictable which component of the superposition will survive in the course of the evolution ue to Eq. 3). V. UNRAVELING QUANTUM BROWNIAN MOTION In this section we return to the unraveling of QBM, which consists of the NLPSE as eterministic part, complemente by stochastic jumps. This stochastic part is necessary to prouce the require ensemble of quantum trajectories since the NLPSE alone woul always lea to the same asymptotic pointer state for one particular initial state. We first show how the jump process leas to stochastically selecte asymptotic pointer states, which occur with relative frequencies accoring to Born s rule. After that, in Sec. VB, we iscuss the action of the unraveling on a single pointer state. The NLPSE escribes pointer state trajectories which are exponentially ampe in momentum. We will see how the stochastic part inuces a ranom walk of the pointer state trajectory isplaying the expecte iffusive behavior. In the semiclassical limit this will eventually yiel classical iffusion. A. Stochastic ynamics of a superposition Let us consier the action of a single jump on the superposition state 6) of Sec. IV B, consisting of wave packets ψ j that are well separate in phase space, accoring to the orthogonality relation 8), an with withs comparable to the pointer state with, ψ =Ĵ[ψ] ψ Ĵ Ĵ = j c j ψ j. 34) Here, we introuce the normalization factor Ĵ Ĵ = ψ Ĵ [ψ]ĵ[ψ] ψ an the normalize wave packets ψ j =Ĵ[ψ] ψ j, 35) Ĵ Ĵ j involving the normalization factor Ĵ Ĵ j = ψ j Ĵ [ψ]ĵ[ψ] ψ j = κ [V x,j + x x j ) ] [V p,j + p p j ) ] 1. 36) In the following, only the square mouli w j = c j of the new coefficients will be require. One reaily fins w j = c j Ĵ Ĵ j. 37) Ĵ Ĵ Note that the new wave packet components ψ j can be safely assume to be still separate an localize because the jump operator moifies the shape only linearly in ˆx an ˆp. The superposition state 34) after the jump is therefore again a superposition of separate wave packets, but with ifferent weights. The effect of a jump can thus be approximately accounte for by reshuffling a finite number of weights w j in the superposition. We confirme this by a numerical implementation of the unraveling 1) by combining the Crank Nicolson metho with the split-operator technique [44 46]. From the action of the NLPSE on a superposition state we calculate in Sec. IV B the eterministic evolution 3) ofthe weights w j. Knowing the new weights 37) ue to a jump, we are now in a position to construct a stochastic process for the

6 LUTZ SÖRGEL AND KLAUS HORNBERGER PHYSICAL REVIEW A 9, ) weights of the wave packets where the jumps occur with the same rate as in the original unraveling: Ĵ Ĵ ) w j = w j Ĵ Ĵ Ĵ j Ĵ j )t + w j Ĵ Ĵ 1 N. 38) Here, we have written the eterministic part 3) more conveniently with the help of the jump rate r = Ĵ Ĵ 15) an the normalization factor Ĵ Ĵ j 36). The Poisson increment N has the same ensemble average E[N] = rt as the pointer state unraveling 1). The set of equations 38) represents the evolution of the weights in a quantum trajectory. Each quantum trajectory ens up asymptotically in a pointer state, when all weights but one vanish. From this we euce that the relative frequency of fining a particular pointer state πx j,p j ) is equal to the probability of the associate asymptotic state in the ensemble of quantum trajectories. This latter probability is evaluate easily by carrying out the ensemble average of 38), E[w j ] = 0. 39) This implies E[w j t)] = const. an, in particular, E[w j t )] = E[w j 0)] = c j 0), 40) where c j 0) are the initial amplitues of the wave packets in the superposition state 6). Equation 40) confirms Born s rule 3), as expecte for a superposition of separate wave packets. B. Stochastic ynamics of a single wave packet So far, we investigate the ynamics of the unraveling if the initial state is a superposition of localize wave packets. We confirme in the previous section that the probability to en up asymptotically in a certain pointer state is given by its weight in the initial superposition Born s rule). This process takes place on the fast ecoherence time scale t ec after which the system is well escribe by a pointer state. In terms of the pointer state unraveling, this point is reache when each quantum trajectory in the ensemble has turne into its asymptotic pointer state. Since the unerlying quantum ynamics is iffusive [5] it is natural to escribe the statistics of the position x an the momentum p of the pointer state by the phase space iffusion ) ) ) x p B11 B = t + 1 W1, 41) p p) B 1 B W with the Wiener increments W 1 an W obeying the rules E[W i ] = 0, W i W j = δ ij t. 4) Equation 41) is a special case of the two-imensional Ornstein Uhlenbeck process with constant coefficients [47] an the erivation of its ensemble variances an covariance is presente in the appenix. The ifferent iffusion constants D x, D p, an D xp escribing position, momentum, an covariance iffusion are relate to the B ij by D x = B 11 + B 1, D p = B1 + B, D xp = B 11 B 1 + B 1 B. 43) We procee to show how the stochastic jumps acting on the motion of the pointer state lea to phase space trajectories which turn into classical iffusion in the semiclassical limit κ. To achieve this, we erive a simple moel allowing the escription of the pointer state trajectories by the phase space iffusion 41) an then compare it to a numerical stuy. 1. Analytic moel Consier a pointer state moving on the phase space trajectory governe by the NLPSE see Sec. IV A). This eterministic motion is interrupte by a jump, escribe by the jump operator 13) acting on the pointer state 17), ψ x) = x Ĵ[πx,p)] πx,p) Ĵ[πx,p)] πx,p) [ = κ r ps 1 1 iκc xp,ps 8κ V x,ps ) ] x x) x πx,p). 44) Here, r ps = κ V x,ps V p,ps 1 45) isthejumprate15) of the pointer state an V x,ps an V p,ps are the withs of the pointer states efine in Eqs. ) 4). The resultingstateisasymmetric ouble-peake wave packet. This symmetry reflects the assume symmetry of the state before the jump. Since in reality the system state is never exactly symmetric even small asymmetries will trigger the emergence of a single wave packet uner the NLPSE; see Fig.. Which of the two subpeaks survives epens on the asymmetry before the jump. We assume equal probabilities an that the single wave packet is restore sufficiently fast by the NLPSE. This results in an effective jump of the pointer state s first moments in phase space by the istances j x = x x = V x,ps, j p = p p = 46) V p,ps. The withs j x an j p are calculate from the positions of the two peaks in Eq. 44). Since the irection of the jump is positive or negative with equal probability, we en up with the two possible jumps in phase space ) ) jx jx j 1 =, j =, 47) j p j p each of them characterize by a Poisson process N k with jump rate r k = E[N k] t = r ps. 48) We are now in the position to write own a stochastic ifferential equation for x an p, ) ) x p = t + j k N k. 49) p p k=1 It consists of the eterministic evolution 18) an 19) erive in Sec. IVA an the stochastic jumps 47) with corresponing jump rates 48). By using the Poisson rule N k N l = δ kl N k

7 UNRAVELING QUANTUM BROWNIAN MOTION: POINTER... PHYSICAL REVIEW A 9, ) FIG.. Color online) Sketch of the simplifie jump process: A jump turns the wave packet at position x into a slightly asymmetric ouble-peake structure. The NLPSE then suppresses one of the peaks resulting in a single wave packet at the position x of one of the ouble-peaks. A similar picture applies in momentum space. an Eq. 48), one can easily erive all moments of the jump part of the stochastic process 49): E[x) n 1 ] = 0, E[x) n ] = jx n r pst, E[p) n 1 ] = 0, 50) E[p) n ] = jp n r pst, E[xp) n ] = j x j p ) n r ps t, with n 1. If one is intereste in calculating moments up to secon orer only, the stochastic jump process 49) can be well approximate by the phase space iffusion 41) upon ientifying the iffusion constants 43) with D x = j x r ps, D p = j p r ps, D xp = j x j p r ps. 51) Inserting the jump withs 46) an the jump rate 45) into Eqs. 51) gives their epenence on κ as κ D x 1 κ, D p, D xp κ. 5) We see that both D x an D xp vanish in the semiclassical limit κ so that only the momentum iffusion D p = contributes an the stochastic process 41) thus reuces to ) ) x p 0 = t + )W. 53) p p Dp One can also arrive irectly at Eq. 53) without efining the phase space iffusion 41) by rawing the semiclassical limit of Eq. 49) an thereby performing a iffusive limit analogous to that of a ranom walker [47]. Writing Eq. 53) in physical imensions by using the scales efine in Eq. 4) finally leas to the Langevin equation of classical Brownian motion [5,48] p = γ pt + 4γmk B T env W, 54) where the imension of W is a square root of time. It shoul be remarke here that the crucial assumption leaing to this moel is the immeiate restoring of the pointer state after a jump. The real ynamics will take a finite time until the ouble-peake wave function returns to a pointer state, an there will be several jumps in between. We show in the following, by numerically simulating trajectories of the pointer state unraveling, that this is inee the case, giving rise to complex ynamics. Nonetheless, the general picture of the analytic moel can be confirme.. Numerical stuy By using a combination of the Crank Nicolson metho an a split-operator technique [44 46] we generate quantum trajectories of the unraveling 1) an calculate their first moments x an p an their variances V x, V p, an C xp. Figure 3 shows exemplarily the temporal evolution of p an V p for two sample trajectories. A similar behavior is foun in position space not shown) a) b) p _ V p t t FIG. 3. Color online) a) Momentum expectation p an b) momentum variance V p of two sample trajectories starting at the same pointer state π0,0). One observes that, in general, more than a single jump occurs in the unraveling before the with of the wave function is restore to the pointer state with. The calculations are mae at κ = 50 an all quantities are imensionless with units efine via Eq. 4)

8 LUTZ SÖRGEL AND KLAUS HORNBERGER PHYSICAL REVIEW A 9, ) 0.0 a) 0.80 b) E N [p] 0.00 Var N [p] t t FIG. 4. Color online) E N [p] anvar N [p] soli) as well as their theoretically expecte eviations ashe) ue to a finite sample size. The fluctuations are well characterize by the ashe lines. Calculations are one at κ = 50 for N = 8000 trajectories; the imensionless units are efine via Eq. 4). Verification of the stochastic moel. In orer to compare the statistics of the phase space trajectories preicte by the phase space iffusion 41) with that of the numerically generate trajectories we consier the expectation values of a finite sample of size N: E N [x] = 1 N E N [p] = 1 N x i, p i, Var N [x] = 1 N 1 Var N [p] = 1 N 1 Cov N [x,p] = 1 N 1 x i E N [x]), 55) p i E N [p]), x i E N [x])p i E N [p]), where x i, p i are the first moments of quantum trajectory i. The sample expectation values 55) are themselves stochastic values ue to the stochasticity of the unerlying moel. The ensemble variances of E N [x] an E N [p] on the one han an those of Var N [x], Var N [p], an Cov N [x,p] onthe other are proportional to 1/N an 1/N 1), respectively see appenix). We compare the finite-size variances of the numerically generate trajectories with those preicte by the above moel. Figure 4 compares E N [p] an Var N [p] of the sample with the theoretically expecte eviations. The interval of one stanar eviation aroun the ensemble mean value E [p] shoul contain 68% of the trajectories of the sample, which we confirme at selecte times. The same analysis applies for E N [x], Var N [x], an Cov N [x,p] an yiels the same result. Thus, we conclue that the statistics of the moments of the pointer state unraveling can inee be escribe by the stochastic moel 41). Extraction of the iffusion constants. Having verifie the stochastic moel we are now able to extract the iffusion constants D x, D p, an D xp by fitting the variances shown in Eqs. A5) to the generate trajectories. Since Var[p] only epens on D p, an Cov[x,p] only epens on D p an D xp, fitting is one consecutively by extracting D p from Var[p], D xp from Cov[x,p], an finally D x from Var[x]. The numerically obtaine values for ifferent κ, aswellas the calculations from the analytic moel 5), are shown in Fig. 5. We see that the calculate D p is in goo quantitative agreement with the simulation. The eviation of the calculate D x an D xp from the simulate ones is ue to the crue simplification mae by the assumption that a jump in the unraveling correspons to a phase-space jump of the pointer state without further ynamics; Fig. 3 shows clearly that this is not the case. For large κ both the analytic moel an the numerical simulation exhibit the classically expecte behavior. Specifically, the position iffusion D x an the covariance iffusion D xp ten to zero, whereas the momentum iffusion D p approaches the value. D D x D p D xp κ FIG. 5. Color online) Depenence of the iffusion constants D x soli line), D p ashe line), an D xp otte line) on the semiclassical parameter κ. The lines are calculate from Eqs. 51) of the analytic moel, whereas the symbols represent the results of the numerical simulation. One observes for all three iffusion constants that the analytic moel captures the main features of their epenence on κ. Specifically, D p approaches the classical value of in the semiclassical regime κ, whereas D x an D xp vanish in that limit. The imensionless units are efine via Eq. 4)

9 UNRAVELING QUANTUM BROWNIAN MOTION: POINTER... PHYSICAL REVIEW A 9, ) VI. CONCLUSION We ientifie the pointer states of quantum Brownian motion an erive the stochastic equations of motion for their trajectories in phase space. The pointer states turn out to be Gaussian wave packets with fixe withs an a finite position-momentum covariance. Their localization gets more pronounce for increasingly strong environmental coupling an the phase space trajectory characterizing their motion obeys a stochastic ifferential equation combining momentum amping with position an momentum iffusion. In the semiclassical limit, these trajectories turns into the Langevin equation of classical iffusion. This provies us with a consistent an transparent picture of the quantum-classical transition as inuce by quantum Brownian motion. We saw that any initial state, which can be represente by a superposition of sufficiently separate wave packets, ecays into a mixture of corresponing pointer states, an that the ensuing statistical weights are consistent with Born s rule. By using the pointer states an their trajectories one can thus capture an essential part of the ynamics of quantum Brownian motion. Our analysis epens crucially on linking the nonlinear equation of motion for the pointer states to a particular piecewise eterministic unraveling of the master equation. It allows one to connect the open quantum ynamics to stochastic trajectories in phase space. We thus emonstrate a stringent an unbiase metho to erive equations of motion for trajectories in classical phase space using only the quantum master equation as input. For the present case of Brownian motion we arrive in the semiclassical limit κ at the expecte classical iffusion equation, while the quantum corrections to classical iffusion are capture by the κ epenence of the iffusion constants 51) an 5). The escribe technique of the pointer state unraveling may become particularly useful in cases where even approximate solutions of a master equation are not known. For example, the full Markovian ynamics of a tracer particle in a gaseous environment is escribe by a complicate integroifferential equation not reaily accessible by analytic means []. If pointer states can be ientifie for this equation we expect their trajectories to be escribe by a general Chapman Kolmogorov equation in phase space turning into the linear Boltzmann equation in the semiclassical limit. ACKNOWLEDGMENT This work was supporte by the DFG via SFB/TR 1. APPENDIX: DIFFUSION IN PHASE SPACE We analyze the properties of the classical stochastic process x = pt + B 11 W 1 + B 1 W, p = pt + B 1 W 1 + B W, A1) use in Sec. VBto escribe the ynamics of the first moments x, p of the pointer states in phase space. At first, we iscuss the ensemble statistics, an afterwars the statistics of a sample of finite size. 1. Ensemble statistics In orer to erive the statistical properties of the stochastic process A1), we recall the stochastic properties of the realvalue Wiener increments W 1 an W : E[W i ] = 0, W i W j = δ ij t. A) From these two relations one can erive a ifferential equation, e.g. for x, by expanig to secon orer in x an keeping only termsuptoorert; this is known as the Ito rule of stochastic calculus. One gets x ) = xx + x) = xp + B11 + 1) B t + xb11 W 1 + B 1 W ). A3) In the same manner one calculates p ) an xp), which then allows the erivation of ifferential equations for the ensemble variances Var[x] = E[x ] E[x] an Var[p] = E[p ] E[p] an the ensemble covariance Cov[x,p] = E[xp] E[x]E[p]: t Var[x] = Cov[x,p] + D x, t Var[p] = Var[p] + D p, A4) t Cov[x,p] = Cov[x,p] + Var[p] + D xp, where efinition 43) of the iffusion constants was use. Solving the ifferential equations A4) for initial conitions Var[x]0) = Var[p]0) = Cov[x,p]0) = 0 yiels Var[x]t) =D x + D p + D xp )t 1 D p1 e t ) D p + D xp )1 e t ), Var[p]t) = 1 D p1 e t ), Cov[x,p]t) = 1 D p1 e t ) + D xp 1 e t ). A5). Statistics with finite sample size We are now intereste in the statistics of the first moments an the variances of a finite sample of size N, an, in particular, their eviations aroun the ensemble values A5). With the help of the stochastic moel A1) an the sample expectation values 55) one can write own, e.g., the stochastic ifferential equation for the sample momentum expectation: E N [p] = E N [p]t + 1 B 1 W 1,i + B W,i ), N A6) with inepenent Wiener increments W 1,i an W,i for all i. In the same manner as in the previous section one can then erive E N [p] ) = E N [p]e N [p] + E N [p]) = E N [p] + B 1 + ) B t N + N E N[p] B 1 W 1,i + B W,i ), A7)

10 LUTZ SÖRGEL AND KLAUS HORNBERGER PHYSICAL REVIEW A 9, ) an finally t Var[E N[p]] = Var[E N [p]] + D p N. A8) Here, the ensemble variance of E N [p] is enote by Var[E N [p]] = E[E N [p] ] E[E N [p]] an the efinition of the iffusion constants 43) was use. Similarly, one erives a stochastic ifferential equation for the sample variance of the momentum: Var N [p] =D p Var N [p])t + N 1 p i E N [p]) B 1 W 1,i + B W,i ), A9) yieling t Var[Var N[p]] = 4Var[Var N [p]] + 4D pvar[p] A10) N 1 for the ensemble variance of Var N [p]. The variances of E N [x], Var N [x], an Cov N [x,p] are obtaine accoringly in a long an teious calculation, giving the solutions Var[E N [x]] = 1 N Var[x], Var[E N [p]] = 1 N Var[p], Var[Var N [x]] = N 1 Var[x], Var[Var N [p]] = N 1 Var[p], Var[Cov N [x,p]] = 1 N 1 Var[x]Var[p] + Cov[x,p] ). A11) [1] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, an I. O. Stamatescu, Decoherence an the Appearance of a Classical Worl in Quantum Theory Springer-Verlag, Berlin, 003). [] W. H. Zurek, Rev. Mo. Phys. 75, ). [3] M. Schlosshauer, Decoherence an the Quantum to Classical Transition Springer-Verlag, Berlin, 007). [4] W. H. Zurek, Phys. Rev. D 4, ). [5] W. H. Zurek, S. Habib, an J. P. Paz, Phys. Rev. Lett. 70, ). [6] L. Diósi an C. Kiefer, Phys. Rev. Lett. 85, ). [7] J. Eisert, Phys.Rev.Lett.9, ). [8] M. Busse an K. Hornberger, J. Phys. A: Math. Theor. 4, ). [9] M. Busse an K. Hornberger, J. Phys. A: Math. Theor. 43, ). [10] G. Linbla, Commun. Math. Phys. 48, ). [11] V. Gorini, A. Kossakowski, an E. C. G. Suarshan, J. Math. Phys. 17, ). [1] A. Caleira an A. Leggett, Phys. A Amsteram, Neth.) 11, ). [13] W. G. Unruh an W. H. Zurek, Phys.Rev.D40, ). [14] W. T. Strunz, F. Haake, an D. Braun, Phys. Rev. A 67, ). [15] P. Hänggi an G.-L. Ingol, Chaos 15, ). [16] W. T. Strunz, L. Diósi, N. Gisin, an T. Yu, Phys.Rev.Lett.83, ). [17] L. Diósi, Europhys. Lett., ). [18] F. Petruccione an B. Vacchini, Phys. Rev. E 71, ). [19] B. Vacchini an K. Hornberger, Eur. Phys. J. Spec. Top. 151, ). [0] B. Vacchini, Phys. Rev. Lett. 84, ). [1] K. Hornberger, Phys. Rev. Lett. 97, ). [] B. Vacchini an K. Hornberger, Phys. Rep. 478, ). [3] M. R. Gallis an G. N. Fleming, Phys.Rev.A4, ). [4] K. Hornberger an J. E. Sipe, Phys.Rev.A68, ). [5] H.-P. Breuer an F. Petruccione, The Theory of Open Quantum Systems Oxfor University Press, New York, 006). [6] W. H. Zurek, Prog. Theor. Phys. 89, ). [7] D. A. R. Dalvit, J. Dziarmaga, an W. H. Zurek, Phys. Rev. A 7, ). [8] D. J. Atkins, Z. Bray, K. Jacobs, an H. M. Wiseman, Europhys. Lett. 69, ). [9]L.Li,A.Chia,anH.M.Wiseman,New J. Phys. 16, ). [30] L. Diósi, Phys. Lett. A 114, ). [31] N. Gisin an M. Rigo, J. Phys. A: Math. Gen. 8, ). [3] C. W. Gariner, A. S. Parkins, an P. Zoller, Phys. Rev. A 46, ). [33] J. Dalibar, Y. Castin, an K. Mølmer, Phys.Rev.Lett.68, ). [34] K. Mølmer, Y. Castin, an J. Dalibar, J. Opt. Soc. Am. B 10, ). [35] M. Rigo an N. Gisin, Quantum Semiclassical Opt. 8, ). [36] N. Gisin, Phys.Rev.Lett.5, ). [37] L. Diosi, J. Phys. A: Math. Gen. 1, ). [38] G. C. Ghirari, P. Pearle, an A. Rimini, Phys.Rev.A4, ). [39] N. Gisin an I. C. Percival, J. Phys. A: Math. Gen. 5, ). [40] T. Bhattacharya, S. Habib, an K. Jacobs, Phys. Rev. Lett. 85, ). [41] T. Bhattacharya, S. Habib, an K. Jacobs, Phys. Rev. A 67, ). [4] T. Brun, N. Gisin, P. O Mahony, an M. Rigo, Phys. Lett. A 9, ). [43] F. Lucas an K. Hornberger, Phys.Rev.A89, ). [44] J. J. Fleck, J. Morris, an M. Feit, Appl. Phys. 10, ). [45] M. Feit, J. F. Jr., an A. Steiger, J. Comput. Phys. 47, ). [46] W. H. Press, S. A. Teukolsky, W. T. Vetterling, an B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3re. Cambrige University Press, New York, 007). [47] C. W. Gariner, Hanbook of Stochastic Methos for Physics, Chemistry an the Natural Sciences, 4the.Springer-Verlag, Berlin, 009). [48] H. Risken, The Fokker-Planck Equation: Methos of Solutions an Applications, n e. Springer-Verlag, Berlin, 1989)