aa r X i v : . [ qu a n t - ph ] F e b Lyapunov Decoherence Rate in Classically Chaotic Systems
Marcus V. S. Bonan¸ca ∗ Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, GermanyCentro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, 09210-170, Santo Andr´e, SP, Brazil † (Dated: November 13, 2018)We provide a path integral treatment of the decoherence process induced by a heat bath on asingle particle whose dynamics is classically chaotic and show that the decoherence rate is given bythe Lyapunov exponent. The loss of coherence is charaterized by the purity, which is calculatedsemiclassically within diagonal approximation, when the particle initial state is a single gaussianwave packet. The calculation is performed for weak dissipation and in the high temperature limit.This situation allows us to simplify the heat bath description to a single random potential. Althoughthe dissipative term is neglected in such approach, the fluctuating one can be treated phenomeno-logically to fit with the above regime. Our results are therefore valid for times shorter than theinverse of the dissipation rate. PACS numbers: 03.65.Yz, 05.45.Mt, 05.40.-a, 85.25.Cp
I. INTRODUCTION
A well-established way of making classical features toappear in quantum systems is to study them when theactions involved are much greater than Planck’s con-stant. This semiclassical approach has allowed us to ex-plore the quantum-classical transition in a very interest-ing class of systems namely those which are classicallychaotic. In different contexts, several results have shownhow the classically chaotic dynamics has an influence onthe quantum behavior [1]. However, it has been shownexperimentally that strictly quantum features such assuperpositions between macroscopically different statesare still present even when actions are bigger than ~ .In Ref.[2], interference patterns measured in Josephsonjunctions show the presence of superpositions betweencurrent states, each of those corresponding to the mo-tion of millions of Cooper pairs. Therefore, besides thesemiclassical regime, a decoherence mechanism must beresponsible for the emergence of pure classical behaviorwhere such superpositions are absent [3].The decoherence process is one of the effects inducedby the coupling to an environment and the path integraldescription of linear systems interacting with a heat bathis very well established [4–7]. Nevertheless, for nonlinearsystems there is no general path integral approach to suchproblem and specific kinds of approximations have beendeveloped [6]. At first glance, it can be argued that de-coherence happens so fast that does not matter whetherthe dynamics is linear or not [8]. On the other hand,calculating a decoherence rate in such regime containsthe assumption that the isolated dynamics of the sys-tem of interest is not relevant at all. That is true onlyif the coupling between system and heat bath is weak[7] and this is not the regime we are interested in here.Once the nonlinearity is relevant, different semiclassical ∗ [email protected] † permanent address approaches are available [9–11] either within the path in-tegral or the master equation framework.In this context, a very interesting result was conjec-tured for the first time in Ref.[12]. There, the authorsproposed theoretically that a classically chaotic systemcould lose coherence at a rate given by the Lyapunovexponent, i.e. a rate completely independent of the envi-ronment parameters and related to the classically chaoticdynamics. A series of works appeared afterwards tryingto implement that proposal in solid basis [13–17] (focus-ing also on the quantum-classical correspondence aspectof the problem). A first approach was developed basedon the resolution of a master equation for the Wignerfunction of the classically chaotic system. It was shownanalytically [14] and numerically [16] that the entropyproduction rate obtained from that approach is propor-tional to the Lyapunov exponent. The analytical calcu-lation was however obtained under the following approx-imations: the chaotic dynamics is taken into account upto its linear regime, the friction from the heat bath andthe quantum corrections to the classical evolution of theWigner function are neglected. Despite of this, such re-sults were suported by more rigourous calculations con-sidering a hyperbolic linear system [18].A second approach was developed using path integralmethods and it is known in the literature by Loschmidtecho (LE) [19–21]. The LE is the probability to recovera certain initial state when the Hamiltonian used in theforward time evolution is slightly perturbed for the timereversed propagation of the final state. Considered asan indirect measure of decoherence [22–25], the LE wasshown to decay with the Lyapunov exponent when thesystem is classically chaotic. On the other hand, the LEdeals with isolated systems only and therefore it has beenunderstood as a stability measure of them, as proposedoriginally by Peres [26]. Thus, a path integral formu-lation of a classically chaotic system interacting with aheat bath still is a very challenging problem.The main goal of the present work is to provide a pathintegral derivation of the result conjectured in [12] avoid-ing the LE problems mentioned above. In Sec. II thetreatment of the heat bath is described and the regimeconsidered is presented. In Sec. III we consider the purityas a measure of decoherence and it is calculated throughsemiclassical methods for arbitrary initial states. In Sec.IV it is shown the role played by correlations betweenpairs of classical trajectories in the decay of the purityand how they give rise to a decoherence rate given by theLyapunov exponent when the initial state is a gaussianwave packet of arbitrary width. Finally, conclusions arepresented in Sec. V. In the appendix we present a briefcomparision between LE and purity calculations pointingout their differences. II. DESCRIPTION OF THE HEAT BATH
We consider here a single particle whose dynamics isclassically chaotic as, for example, electrons in a cleanballistic conductor [27]. Our goal is to describe its dy-namics when it is coupled to a heat bath, focusing onthe decoherence process. We start from the path integralformulation in terms of Feynman-Vernon influence func-tionals [28] with a Lagrangian given by L = L S + L I + L B .The L S term describes the isolated dynamics of the par-ticle which we assume to be a classically chaotic billiard-type of confinement. The terms L I and L B , interactionand heat bath Lagrangians respectively, are assumed tobe given by Caldeira-Leggett model [4] with a linear cou-pling between particle and bath coordinates. For initialstates of the form ρ = ρ S ⊗ ρ B , where ρ S and ρ B arethe initial density matrices of the particle and the bathrespectively, the time evolution of the reduced densitymatrix ρ S is given by [28] ρ S ( x f , y f ; t ) = Z d x i d y i J ( x f , y f , x i , y i ; t ) ρ S ( x i , y i ; 0) , (1)where the propagator J , obtained after tracing out theheat bath degrees of freedom with ρ B chosen as a thermalstate, is given by the following path integral J ( x f , y f , x i , y i ; t ) = Z D x D y e i ~ ˜ S eff [ x , y ] , (2)For an Ohmic spectral density and in the high temper-ature limit, ˜ S eff is given by [4–6]˜ S eff [ x , y ] =˜ S o [ x , y ] + mγ Φ d [ x , y ] + i mγk B T ~ Φ f [ x , y ] , (3)˜ S o [ x , y ] = S o [ x ] − S o [ y ], S o [ q ] = R t ds L S [ q ( s ) , ˙ q ( s )] givesthe isolated particle dynamics and the other two terms,Φ d and Φ f , lead to dissipation and decoherence respec-tively. They are local quadratic functionals of the paths x and y without any other prefactors involving the par-ticle mass m , the friction constant γ , ~ or the product k B T of the Boltzmann constant and the heat bath tem-perature [4–6]. In a regime of very weak dissipation andsufficiently high temperature, the term Φ d is negligibleat least for time scales much shorter than 1 /γ . In thissituation, the effective influence of the heat bath can bemodeled in a much simpler way.Let us consider for a moment the following Lagrangian L = L S − q · f ( t ) where q is the position vector of thesystem L S and f ( t ) is a stochastic force. One can alsoconstruct the time evolution of the reduced density ma-trix ρ S in this case but it would depend on f ( t ). Thus,assuming that f ( t ) is gaussian distributed with an av-erage value h f ( t ) i = 0, we can calculate the followingaveraged propagator [29] J ( x f , y f , x i , y i ; t ) = Z D f P [ f ] Z D x D y e i ~ ˜ S [ x , y ; f ] , (4)where P [ f ] is a gaussian functional and˜ S [ x , y ; f ] = ˜ S o [ x , y ] − Z t ds f ( s ) · (cid:16) x ( s ) − y ( s ) (cid:17) . (5)The average over f ( t ) leads to the effective action˜ S eff = ˜ S o [ x , y ] + i ˜ S f [ x , y ] in which [29]˜ S f [ x , y ] = Z t ds Z s ds ′ (cid:16) x ( s ) − y ( s ) (cid:17) ¯M ( s − s ′ ) (cid:16) x ( s ′ ) − y ( s ′ ) (cid:17) (6)and ¯M ( s − s ′ ) is 2 × M i,j ( s − s ′ ) = 1 ~ h f i ( s ) f j ( s ′ ) i , (7)with f i ( t ), i = 1 ,
2, being the f ( t ) components. Since weare free to choose the spectra of h f i ( s ) f j ( τ ) i , we take itas 1 ~ h f i ( s ) f j ( s ′ ) i = 4 mγk B T ~ δ ij δ ( s − s ′ ) . (8)where δ ij is the Kronecker delta. This choice leads to˜ S eff [ x , y ] = ˜ S o [ x , y ] + i mγk B T ~ Φ f [ x , y ] . (9)Eq.(9) is identical to (3) except for the dissipative term.Therefore, we can model the influence of the heat bath inthe regime of very weak dissipation and high temperaturejust by the fluctuating time dependent potential shownin (5) and by replacing the propagator (2) by the one in(4). III. SEMICLASSICAL PURITY
The next step then is to write a semiclassical expres-sion for (4), which can be written in terms of Feynman’spropagators J ( x f , y f , x i , y i ; t ) = Z D f P [ f ] K ( x f , x i , f ; t ) K ∗ ( y f , y i , f ; t ) (10)where K ( x f , x i , f ; t ) = R D x e iS [ x , f ] / ~ , K ∗ is its complexconjugate and S [ x , f ] = S o [ x ] − R t ds f ( s ) · x ( s ).We assume that the coupling to f is classically small inorder that only the actions, i.e. the phases, are affectedwhile the trajectories given by L S remain unchanged.Under this assumption, we replace K by the semiclassical Van Vleck formula [1] for two dimensional systems K sc ( x f , x i , f ; t ) = (cid:16) πi ~ (cid:17) X ˜ α ( x i → x f ,t ) D ˜ α exp (cid:18) i ~ S ˜ α ( x f , x i , f ; t ) (cid:19) (11)where S ˜ α is the classical action S of the trajectory˜ α running from x i to x f in time t , and D ˜ α = | det( ∂ S ˜ α /∂ x i ∂ x f ) | / exp ( − iπ µ ˜ α ) is the Van Vleck de-terminant including the Maslov index.To characterize the decoherence process, we calculatethe purity Tr (cid:0) ρ S ( t ) (cid:1) , where the trace is performed overthe particle degrees of freedom since ρ S is already a re-duced density matrix. If ρ S is initially pure, the puritystarts from one and decays as time evolves due to heatbath influence. The semiclassical expression for the pu-rity is therefore given by the trace of the product of two ρ S ( t ) each one evolved by the semiclassical version of J calculated from (10) when K is replaced by (11). Thatyields the following expression which recalls the LE one[23, 30] (we refer to Sec. V for a brief discussion aboutthis issue)Tr (cid:0) ρ S ( t ) (cid:1) = (cid:18) π ~ (cid:19) Z d x f d y f d x i d y i d x ′ i d y ′ i Z D f D g P [ f ] P [ g ] ρ S ( x i , y i ; 0) ρ S ( y ′ i , x ′ i ; 0) × X ˜ α ( x i → x f ,t ) , ˜ α ′ ( y i → y f ,t ) X ˜ η ( y ′ i → y f ,t ) , ˜ η ′ ( x ′ i → x f ,t ) D ˜ α D ∗ ˜ α ′ D ˜ η D ∗ ˜ η ′ (12) × exp (cid:20) i ~ (cid:16) S ˜ α ( x f , x i , f ; t ) − S ˜ α ′ ( y f , y i , f ; t ) + S ˜ η ( y f , y ′ i , g ; t ) − S ˜ η ′ ( x f , x ′ i , g ; t ) (cid:17)(cid:21) Due to the rapidly oscillatory phase factor contain-ing the action differences, most of the contributions willcancel out except for the semiclassically small action dif-ferences originating from pairs of trajectories which areclose to each other in configuration space. We can thususe a linear approximation in order to relate the actions S ˜ α , S ˜ η ′ along the trajectories ˜ α, ˜ η ′ to the actions S α , S η ′ along nearby trajectories α, η ′ connecting the midpoint r i = ( x i + x ′ i ) / x f . In the same way we relate S ˜ α ′ , S ˜ η to the actions S α ′ , S η along the nearby trajecto-ries α ′ , η connecting the midpoint r ′ i = ( y i + y ′ i ) / y f . The expansion will contain terms up to zero orderin the D ’s and terms up to first order in the exponential. For S ˜ α and S ˜ α ′ , the linearization yields [30] S ˜ α ( x f , x i , f ; t ) ≈ S o,α ( x f , r i ; t ) − Z t ds q α ( s ) · f ( s ) − u · p αi ,S ˜ η ( y f , y ′ i , g ; t ) ≈ S o,η ( y f , r ′ i ; t ) − Z t ds q η ( s ) · g ( s ) + 12 u ′ · p ηi (13)where S o,α is S o [ x ] along the trajectory α , u = ( x i − x ′ i ), u ′ = ( y i − y ′ i ) and p αi , p ηi are the initial momenta oftrajectories α, η . Analagous expressions are obtained for S ˜ α ′ and S ˜ η ′ . As mentioned before, for the prefactors wehave D ˜ α ≈ D α , D ˜ η ≈ D η and analogously for D ˜ α ′ , D ˜ η ′ .In order to evaluate the sums over paths in (12) weconsider only the diagonal contribution obtained by thepairings α = η ′ and α ′ = η . Besides that, we performthe gaussian averages over f and g as described in (6).Using (7) and (8) we obtainTr (cid:0) ρ S ( t ) (cid:1) = Z d x f d y f d r i d r ′ i ρ WS ( r i , p ηi ) ρ WS ( r ′ i , p αi ) × X α ( r i → x f ,t ) ,η ( r ′ i → y f ,t ) | D α | | D η | (14) × exp (cid:20) − κ ~ Z t ds (cid:16) q η ( s ) − q α ( s ) (cid:17) (cid:21) where κ = 4 mγk B T and ρ WS ( r , p ) = (cid:18) π ~ (cid:19) Z d u ρ S (cid:16) r + u , r − u (cid:17) e − i ~ u · p (15)is the Wigner function of the initial ρ S . IV. DECOHERENCE RATES
We consider two kinds of contributions in order to eval-uate (14). First, if α and η are close enough to each otherin phase space, it is possible to linearize the motion ofone trajectory around the other to obtain q η ( s ) − q α ( s ) ≈ (cid:18) ( r ′ i − r i ) + 1 mλ ( p ηi − p αi ) (cid:19) e λs (cid:18) ( r ′ i − r i ) − mλ ( p ηi − p αi ) (cid:19) e − λs λ is the Lyapunov exponent. On the other hand,if they are not close α and η can be first considered asfree particle trajectories for time scales shorter than anaverage free flight time t o . In this case, q η ( s ) − q α ( s ) ≈ ( r ′ i − r i ) + sm ( p ηi − p αi ) . (17)After t o , we can only estimate ( q η ( s ) − q α ( s )) by its av-erage value over the billiard area A . Assuming ergodicity,this average can be calculated as D ( q η ( s ) − q α ( s )) E A = Z A d q d q ′ ( q − q ′ ) A . (18)It can be verified for simple geometries that (18) yields h ( q η ( s ) − q α ( s )) i A ∝ A .The prefactors | D α | and | D η | in (14) can be regardedas Jacobians when transforming the integrals over thefinal positions x f and y f into integrals over the initialmomenta p ηi ≡ p i and p αi ≡ p ′ i [19, 21] leading toTr (cid:0) ρ S ( t ) (cid:1) = Z d r i d r ′ i d p i d p ′ i ρ WS ( r i , p i ) ρ WS ( r ′ i , p ′ i ) × exp (cid:20) − κ ~ Z t ds Q ( r i , r ′ i , p i , p ′ i ; s ) (cid:21) (19) where the function Q is taken either by the square of(16) or (17) or simply by (18) replacing p ηi by p i and p αi by p ′ i . Thus, for initial states given by single gaussianwave packets as ρ S (cid:16) r + u , r − u (cid:17) =1( πσ ) exp (cid:18) − ( r − r o ) σ − u σ + i ~ p o · u (cid:19) , (20)Tr (cid:0) ρ S ( t ) (cid:1) can be calculated for the different situationsmentioned above.Defining τ ≡ γt and using (17), the result for timesbetween 0 and τ o ≡ γt o isTr (cid:0) ρ S ( τ ) (cid:1) = (cid:20) a τ + 23 a τ (cid:16) a τ (cid:17)(cid:21) − (21)where a = k B T / ¯ E , ¯ E = ~ mσ and a = D/γσ with D = k B Tmγ . Since our model is valid for times muchshorter than 1 /γ , τ is certainly smaller than 1. The val-ues of a and a define relations between the free parame-ters of the problem. It is important to mentioned that wehave not assumed at any point a highly localized initialstate. Therefore, a ∼ σ is comparable to D/γ and a ≫ τ approaches τ o .When τ > τ o , there are two contributions: either thetrajectories are close to each other in the sense mentionedbefore or they are not. In the latter case, since (18) ismainly A , (14) together with (20) yieldTr (cid:0) ρ S ( τ ) (cid:1) = exp (cid:18) − π k B T ∆ ( τ − τ o ) (cid:19) , (22)where ∆ = π ~ mA is the mean level spacing for a billiardof area A . In the regime considered here, k B T ≫ ∆ and(22) decays very fast.When the trajectories are correlated by the Lyapunovspreading, (14), (16) and (20) yieldTr (cid:0) ρ S ( τ ) (cid:1) = (cid:26) b [(1 + b ) sinh (2Λ τ ) + (1 − b )2Λ τ ]+ b (cid:20) cosh (2Λ τ ) − (2Λ τ ) − (cid:21)(cid:27) − , (23)where b = k B T ¯ E , b = (2 ¯ E/ ~ λ ) , b = 32 (cid:0) k B T ~ λ (cid:1) andΛ = λ/γ . For Λ τ >
1, only those terms with e τ arerelevant and (23) can be written asTr (cid:0) ρ S ( τ ) (cid:1) ≈ (cid:0) β e τ (cid:1) − , (24)where β = b b )2 + b . Eq. (23) shows that the classi-cally chaotic dynamics induces a decoherence rate inde-pendent of the heat bath parameters. V. CONCLUSIONS AND DISCUSSIONS
Summarizing, we have shown how to describe the de-coherence process of a classically chaotic system coupledto a heat bath within a path integral framework. Ourapproach leads to a decoherence rate given by the Lya-punov exponent. Although the results above were de-rived only for single gaussian wave packets with arbi-trary widths as initial states, the semiclassical approachpresented also allows the treatment of their superposi-tions. This is an important difference compared to pre-vious works where only localized wave packets have beenconsidered [12, 16, 19, 23]. The results for superpositionswill be presented elsewhere and will be compared to thepredictions of Refs.[14, 18] which have claimed that theLyapunov regime does not depend on the initial state.Concerning the description of the stable and unstable di-rections in (16), that is not the most general one butit certainly captures the qualitative behavior one shouldfind in a specific case. Different time scales of the classicalchaotic dynamics were taken into account in the presentcalculation differently from those in Refs.[14, 18] whereit is considered up to its linear regime only.It is possible to extend the present results beyond thehigh-temperature limit as long as the correlation time ofthermal fluctuations is much shorter than the time scalesof the system of interest dynamics [7]. Corrections tothe diagonal approximation performed here could alsobe studied [27] for action differences of the order of ~ .As in the LE case [30], they would take quantum effectsinto account systematically. Dissipation was neglected inthe present calculation making the heat bath treatmentcomparable to previous ones using master equations [14,16]. However when the coupling to the heat bath is strongenough, dissipative terms cannot be neglected and it isstill an open question whether the Lyapunov decoherencerate would be robust to that.Concerning the experimental observation of our theo-retical results, it was predicted recently in Ref.[31] thatJosephson junctions devices could also be used to ob-serve the Lyapunov regime in the time evolution of thefidelity. However, those devices are almost isolated fromthe heat bath in that context. We believe that the sameregime and initial state preparation described there couldbe used to study the decay of interference fringes and toobserve the Lyapunov decoherence rate as long as theheat bath temperature is increased.We briefly compare now the semiclassical calculation ofthe purity presented here and those of the LE [19, 23, 30].At first glance, eq.(12) is identical to eq.(5.7) in Ref.[23] or to eq.(73) in Ref.[30]. They certainly have one thing incommon: all of them are given in terms of four sums overchaotic trajectories which are considered unperturbed ei-ther by the heat bath influence (in the purity case) or bythe extra potential (in the LE case). In other words,those equations result from the same perturbative ap-proach that allows us to use the well-known properties ofthose trajectories. One obtains then from those equationsthe leading order result after performing diagonal ap-proximation, which is another common point they have.The first subtle (though important) difference concernsthe initial state. In Ref.[23] for example, as most of thesemiclassical analytical works on LE, a further approxi-mation is performed over eq.(5.7) since a highly localizedgaussian wave packet is considered as initial state lead-ing to eq.(5.11). A more general calculation however wasdone recently in Ref.[30] allowing any kind of initial stateand which we have followed closely here. The second dif-ference concerns the averages. Eq.(73) in Ref.[30] showsvery clearly that LE is mostly calculated as an average offidelity amplitude squared modulus. On the other hand,the purity is obtained from the trace of the reduced den-sity matrix squared, i.e. of the product of two averagedquantities. Hence the averages over the random potentialthat appear in (12) are always performed independently.In LE case, the perturbation often depends on the tra-jectories which may be correlated or not. Thus averagescannot be performed always independently. Neverthe-less both purity and LE semiclassical expressions havetwo kinds of contributions arising from uncorrelated andcorrelated trajectories, the last one leading to the Lya-punov regime. The uncorrelated trajectories give rise tothe Fermi golden rule regime in the LE case, which some-times dominates its decay. For the purity they lead to avery big decoherence rate (eq.(22)) that kills this contri-bution very fast and makes the Lyapunov rate always thedominant one (this result agrees with the numerical onesin Ref.[16]). Finally, we have shown that the Lyapunovregime can be obtained from the exponents (eq.(15) and(19)) instead of the prefactors [19, 23] avoiding short timeproblems. ACKNOWLEDGMENTS
The author acknowledges support of the Brazilian re-search agency CNPq and DFG (GRK 638). The author isalso grateful to M. Guti´errez, D. Waltner, C. Petitjean,R. A. Jalabert and K. Richter for valuable discussionsand J. Hausinger and J. D. Urbina for their careful read-ing of the manuscript and valuable suggestions. [1] M. Gutzwiller,
Chaos in Classical and Quantum Mechan-ics (Springer-Verlag, New York, 1990).[2] C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm,R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, Science , 773 (2000).[3] W. H. Zurek, Rev. Mod. Phys. , 715 (2003).[4] A. O. Caldeira and A. J. Leggett, Physica A , 587(1983). [5] H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep. , 115 (1988).[6] U. Weiss, Quantum Dissipative Systems (World Scien-tific, Singapore, 2008).[7] H.-P. Breuer and F. Petruccione,
The Theory of OpenQuantum Systems (Oxford University Press, New York,2007).[8] D. Braun, F. Haake, and W. T. Strunz, Phys. Rev. Lett. , 2913 (2001).[9] G. A. Fiete and E. J. Heller, Phys, Rev. A , 02212(2003).[10] W. Koch, F. Grossmann, J. T. Stockburger, andJ. Ankerhold, Phys. Rev. Lett. , 230402 (2008).[11] A. M. Oz´orio de Almeida, P. de M. Rios, and O. Brodier,J. Phys. A , 065306 (2009).[12] W. H. Zurek and J. P. Paz, Phys. Rev. Lett. , 2508(1994).[13] S. Habib, K. Shizume, and W. H. Zurek, Phys. Rev. Lett. , 4361 (1998).[14] A. R. Pattanayak, Phys. Rev. Lett. , 4526 (1999).[15] J. Gong and P. Brumer, Phys. Rev. E , 1643 (1999).[16] D. Monteoliva and J. P. Paz, Phys. Rev. Lett. , 3373(2000).[17] F. Toscano, R. L. de Matos Filho, and L. Davidovich,Phys. Rev. A , 010101(R) (2005).[18] O. Brodier and A. M. Oz´orio de Almeida, Phys. Rev. E , 016204 (2004).[19] R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. , 2490 (2001).[20] T. Gorin, T. Prosen, T. H. Seligman, and M. ˇZnidariˇc,Phys. Rep. , 33 (2006).[21] Ph. Jacquod and C. Petitjean, Adv. in Physics , 67(2009).[22] F. M. Cucchietti, D. A. R. Dalvit, J. P. Paz, and W. H.Zurek, Phys. Rev. Lett. , 210403 (2003).[23] F. M. Cucchietti, H. M. Pastawski, and R. A. Jalabert,Phys. Rev. B , 035311 (2004).[24] B. Casabone, I. Garc´ıa-Mata, and D. A. Wisniacki, Eur.Phys. Lett. , 50009 (2010).[25] L. A. Raviola, G. G. Carlo, and A. M. F. Rivas, Phys.Rev. E , 047201 (2010).[26] A. Peres, Phys. Rev. A , 1610 (1984).[27] K. Richter and M. Sieber, Phys. Rev. Lett. , 206801(2002).[28] R. P. Feynman and F. L. Vernon, Ann. Phys. , 118(1963).[29] R. P. Feynman and A. R. Hibbs, Quantum Mechanics andPath Integrals (Dover Publications, New York, 2010).[30] B. Gutkin, D. Waltner, M. Guti´errez, J. Kuipers, andK. Richter, Phys. Rev. E , 036222 (2010).[31] E. N. Pozzo and D. Dominguez, Phys. Rev. Lett.98