Decoherence and Dephasing in a Quantum Measurement Process

PHYSICAL REVIEW A VOLUME 54, NUMBER 2 AUGUST 1996

Decoherence and dephasing in a quantum measurement process

Naoyuki Kono, 1 Ken Machida, 1 Mikio Namiki, 1 and Saverio Pascazio 2 1Department of Physics, Waseda University, Tokyo 169, Japan 2Dipartimento di Fisica, Universita` di Bari, and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy ͑Received 19 April 1995; revised manuscript received 22 January 1996͒ We numerically simulate the quantum measurement process by modeling the measuring apparatus as a one-dimensional Dirac comb that interacts with an incoming object particle. The global effect of the apparatus can be well schematized in terms of the total transmission probability and the decoherence parameter, which quantitatively characterizes the loss of quantum-mechanical coherence and the wave-function collapse by measurement. These two quantities alone enable one to judge whether the apparatus works well or not as a detection system. We derive simple theoretical formulas that are in excellent agreement with the numerical results, and can be very useful in order to make a ‘‘design theory’’ of a measuring system ͑detector͒. We also discuss some important characteristics of the wave-function collapse. ͓S1050-2947͑96͒07507-5͔

PACS number͑s͒: 03.65.Bz

I. INTRODUCTION giving a quantitative estimate of the dephasing occurring in a quantum measurement process. A similar philosophy has Quantum mechanics is the most fundamental physical been followed by several authors at different times. Among theory developed during the last seventy years. Nevertheless, others, quantitative measures for decoherence were also pro- physicists still debate the quantum measurement problem posed by Caldeira and Leggett ͓6͔ and Paz, Habib, and Zurek ͓1,2͔, which has caused them to ponder over some very fun- ͓7͔. It should be emphasized that the present paper follows damental questions. While the original formulation, essen- an approach originally put forward in 1980 ͓3͔. tially due to von Neumann, introduced the so-called ‘‘col- Strictly speaking, the decoherence parameter should be lapse’’ of the wave function as a postulate, currently most estimated by means of a dynamical-statistical analysis of the physicists think that it is necessary to analyze and derive the above-mentioned complicated interactions taking place in- quantum measurement process by starting from fundamental side the apparatus. This kind of theoretical work is, of laws. A quantum measurement must be analyzed as a con- course, very hard to carry out in a satisfactory way. For this crete physical process, by means of quantum mechanics it- reason, in the present paper, we shall study the interaction self, and must not be dealt with by postulating a ‘‘collapse’’ between an object particle Q and a linear array A of a` la von Neumann ͓3–5͔. ␦-shaped potentials ͑a Dirac comb͒, representing the elemen- The seminal formulation, due to von Neumann, is very tary constituents of the macroscopic apparatus. The collision unsatisfactory from the point of view of the internal consis- centers (␦ potentials͒ are regarded as heavy and structureless tency of quantum theory, because the act of ‘‘observation,’’ in the present model, so that every elementary collision is namely the very possibility of obtaining information about elastic and we cannot expect genuine dissipation effects, the quantum properties of the system under investigation, such as absorption, directly stemming from the elementary requires the existence of ‘‘classical’’ devices that, by defini- interactions. Nevertheless, since reflections in the one- tion, do not follow the quantum mechanical laws. It would dimensional case correspond to leakage provoked by elastic be baffling, in our opinion, if such a fundamental theory such deflections in a three-dimensional apparatus, we can natu- as quantum mechanics would need classical ͑i.e., nonquan- rally expect to have some kind of ‘‘dissipation’’ for Q, pro- tum͒ objects in order to ascertain the very value of quantum voked by successive reflections from the constituents of A. observables. In fact, we shall observe a certain kind of irreversibility by A quantum measurement is a very complicated physical taking into account the randomness of the macroscopic sys- process, because it involves the interaction of an object system A, schematized by the fluctuations of the parameters that tem Q with a macroscopic apparatus A, made up of a huge characterize the Dirac comb. This is only a caricature of a number of elementary constituents. The interactions taking measurement process, which can only roughly represent the place between Q and the constituents of A will provoke complicated interactions taking place in a real measurement ‘‘dephasing’’ or ‘‘decoherence’’ on the former. This loss of process. Nonetheless, we shall see that even such a rough quantum-mechanical coherence is the physical process that example can yield, via the evaluation of the decoherence underlines, in our opinion, the so-called ‘‘collapse’’ of the parameter, useful insights into the phenomenon of decoher- wave function. ence, mainly through numerical simulations. At the same In order to analyze the above-mentioned loss of quantum- time, we shall see that our analysis can yield a possible and mechanical coherence, we introduced a ‘‘decoherence pa- preliminary ‘‘design theory’’ in order to ‘‘tailor’’ an appara- rameter’’ ͓4,5͔, which estimates the ‘‘degree of collapse’’ of tus to be used as a detector in a quantum measurement pro- the wave function. This can be regarded as an attempt at cess.

II. DEPHASING AND INTERACTION IN THE APPARATUS in general. For this reason, we shall simply model the phenomenon of decoherence by concentrating only on the fun- In the von Neumann–Wigner approach ͓1,8͔ the quantum damental role played by the many fine interactions taking measurement process is described as place between Q and the elementary constituents of A.To this end, we shall suppress the apparatus states in the follow- ͉⌽ I͘ϭ͉␺͘ ͉AI͘ ͉⌽F͘ϭ͚ ck͉␺k͘ ͉Ak͘, ͑2.1͒ ing description. This is obviously a drastic simplification. In → k order to avoid any misunderstandings, we stress that this simplification aims only at performing a heuristic treatment where , , A are the wave functions of the total ͉⌽͘ ͉␺͘ ͉ ͘ of decoherence. (QϩA) Q and A systems, respectively, the subscripts I and F stand for initial and final, respectively, ͉␺͘ϭ͚kck͉␺k͘, ͕͉␺k͖͘k is a complete set of eigenfunctions of Q, and ͉Ak͘ III. THE DECOHERENCE PARAMETER denotes the apparatus state displaying the kth result. This is often called a von Neumann measurement process of the first A. Definition kind. Let us briefly summarize some of our previous results ͓4͔ Many physicists start their discussion on the measurement and reintroduce the decoherence parameter, in terms of problem from the von Neumann process ͑2.1͒. However, as a which a quantitative definition of wave-function collapse can matter of fact, Eq. ͑2.1͒ does not represent any collapse of be given. Consider a typical Young-type experiment, in the wave function. It simply displays a spectral decomposi- which an incoming Q ‘‘particle’’ ͑namely, a Schro¨dinger tion ͓3–5,8͔, namely, a physical process in which the states wave function relative to a single detection event͒ is split in of the apparatus A become entangled with those of the object two states ␺1 and ␺2 , corresponding to the two possible system Q. No measurement has occurred in ͑2.1͒, because routes in the interferometer. We place the apparatus A along the phase correlation is still perfectly kept in ͉⌽F͘: Indeed, the second path, so that the wave function of Q reads the final density matrix ␺ϭ␺1ϩT␺2 , ͑3.1͒ 2 ␳Fϵ͉⌽F͗͘⌽F͉ϭ͚ ͉ck͉ ͉␺k͗͘␺k͉͉Ak͗͘Ak͉ k where T is the transmission coefficient. Following our discussion in the preceding section, the apparatus state has been ϩ͚ ͚ cic*j ͉␺i͗͘␺j͉͉Ai͗͘Aj͉ ͑2.2͒ suppressed, although the action of the apparatus on the Q i jÞi particle has been properly taken into account via T. For sim- plicity, we assume that and are very close to a plane contains all its off-diagonal terms, which can give rise to any ␺1 ␺2 sort of coherent quantum-mechanical effects, such as inter- wave with wave number k. If we want to take into account ference. wave-packet effects, we have to average our results, as, for instance, Eqs. ͑3.2͒ or ͑3.3͒, with the k-weight factor It should also be emphasized that in order to view the 2 above process as a ‘‘measurement,’’ in some sense, one has ͉a(k)͉ characterizing the wave packet. to require the orthogonality condition among the apparatus Equation ͑3.1͒ holds for every single incoming particle. states: Moreover, every incoming particle is described by the same wave function ␺ϭ␺1ϩ␺2 immediately before interacting with the apparatus. However, after the interaction, the appa- ͗A j͉Ak͘ϭ␦ jk , ᭙j,k. ͑2.3͒ ratus transmission coefficient T will depend on the particular The above requirement is of paramount importance both in apparatus state at the very instant of the passage of the par- the von Neumann–Wigner approach and in the so-called en- ticle. Since the apparatus undergoes random fluctuations vironment theories ͓9͔. It also plays a fundamental role in the ͑which reflect the internal motion of its elementary quantum ‘‘many-worlds interpretation’’ ͓10,11͔. However, it is a pos- constituents͒ we label the incoming particle with j tulate that cannot be proved and is subject to many criticisms ( jϭ1,...,Np, where N p is the total number of particles in ͓4,5,12͔. an experimental run͒, and rewrite the transmission coeffi- By contrast with the approach outlined above, We ana- cient as T j ( jϭ1,...,Np). Notice that the same macro- lyzed the measurement process by describing the apparatus scopic state of the apparatus will correspond to many differ- in terms of density matrices ͑which is physically more sen- ent microscopic states. Consequently, different incoming sible͒, estimated the decoherence parameter ⑀, and discussed particles will be affected differently by the interaction with under which conditions the off-diagonal terms of the total the apparatus, and will be described by slightly different val- density matrix can be shown to vanish: What we need is a ues of T. Accordingly, the probability of detecting the jth ͑macroscopic͒ detector that is able to act as a dephaser, particle after recombination reads namely, to erase the phase correlations between different states of A ͓4,5͔. ͑ j͒ ͑ j͒ 2 2 2 In this paper, we shall concentrate on a very particular P ϵ͉␺ ͉ ϭ͉␺1ϩT j␺2͉ ϭ͉␺1͉ aspect of this problem. Strictly speaking, the wave-function 2 2 ϩ͉T j͉ ͉␺2͉ ϩ2Re͑␺*Tj␺2͒. ͑3.2͒ collapse should be described in terms of the apparatus states, 1 by showing explicitly the disappearance of the off-diagonal part of the total density matrix, as in Eq. ͑3.9͒ of the first After many particles have been detected, the average prob- paper in ͓5͔ or Eq. ͑38͒ of ͓4͔. This problem is very involved, ability will be given by 1066 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

1 Np to the statistical ensemble average over all the possible mi- ¯ ͑ j͒ 2 2 2 Pϵ P ϭ͉␺1͉ ϩ͉T͉ ͉␺2͉ crostates of the apparatus. If we denote the latter with N j͚ϭ1 p ͗•••͘, our assumption reads ϩ2Re͑␺*¯T␺ ͒, ͑3.3͒ 1 2 •••ϭ͗•••͘. ͑3.7͒ where the overbar denotes an event average over all events The significance of the ergodic assumption in quantum phys- in one experimental run ͑which is made up of N events͒. ics is not as clear as it is in classical physics, where the p ergodic theorem states that the long-time average of an ob- Note that ¯T 2р T 2, and that a necessary and sufficient con- ͉ ͉ ͉ ͉ servable can be replaced with an ensemble average over the dition for observing no interference ͑collapse of the wave ¯ phase space. Ergodicity is not easy to prove mathematically function͒ is Tϭ0. We define the decoherence parameter as and is therefore often postulated on physical grounds. In the present context, Eq. ͑3.7͒ must also be regarded as a physical ͉¯T͉2 postulate. ⑀ϭ1Ϫ ,0р⑀р1, ͑3.4͒ 2 In the following, the two above-mentioned averages will ͉T͉ be used interchangeably by virtue of Eq. ͑3.7͒. In particular, 2 and rewrite Eq. ͑3.3͒ as ͉͗T͉͘ ⑀ϭ1Ϫ . ͑3.8͒ ͉͗T͉2͘ ¯ 2 ¯ 2 ¯ i␤ Pϭ͉␺1͉ ϩt ͉␺2͉ ϩ2ͱ t ͱ1Ϫ⑀ Re͑␺1*e ␺2͒, ͑3.5͒ On the other hand, one should keep in mind that, strictly speaking, the above two averages cannot always be com- where ¯Tϵ͉¯T͉ei␤ and¯tϵ͉T͉2. Notice that¯t is the experimen- pletely identified. The decoherence parameter, if defined via tally measured value of the transmission probability. A non- the event average, as in ͑3.4͒͑e.g., in a numerical simula- vanishing ⑀ is a consequence of the statistical fluctuations in tion͒, in general cannot vanish. Indeed, the apparatus. Interference is lost, and hence the wave- N N 1 p 1 p 1 function collapse takes place, in the limit ⑀ϭ1. All effects ¯ 2 2 ͉T͉ ϭ 2 ͚ T jTk*ϭ 2 ͚ ͉T j͉ ϩ 2 ͚ T jTk* , provoked by A on Q are properly taken into account Np j,kϭ1 Np jϭ1 Np jÞk by ¯tϭ͉T͉2 and ⑀. ͑3.9͒ One can also consider the visibility of the interference so that even for a completely random sequence of T j , such pattern that ͚ jÞkT jTk*Ӎ0, one obtains ¯ 2 ¯P Ϫ¯P 2ͱ¯t͑1Ϫ⑀ ͒ ͉T͉ 1 max min ⑀ϭ1Ϫ 1Ϫ . ͑3.10͒ Vϭ ϭ V0 ͱ͑1Ϫ⑀ ͒ ϭ , 2 Ӎ ¯ ¯ T N p P max ϩP min 1ϩ¯t ͉ ͉ ͑3.6͒ Therefore, one needs many events (N pӷ1) in order to be able to ascertain whether the quantum-mechanical coherence where V0ϭ2ͱt/(1ϩt) is the value in the absence of fluc- is lost. If the latter requirement is not satisfied, in general one tuations ͑observe that in such a case tϭ¯t). Once again we is not able to give an operationally meaningful definition of see that coherence between the two branch waves is totally ͑loss of͒ coherence, as shown by Eq. ͑3.10͒. Observe that this lost when ⑀ϭ1, in which case the visibility is zero. condition is independent of the physical requirement that the It is worth stressing that this approach incorporates in a macroscopic apparatus be characterized by a huge number of natural way the possibility of investigating those situations in degrees of freedom and be able to act as a dephaser. which coherence is partially lost or, stated differently, the This observation has interesting spinoffs: The very con- wave function is partially collapsed. These intermediate cept of quantum-mechanical coherence appears to be statis- cases correspond to the values 0Ͻ⑀Ͻ1. tical. One needs the accumulation process of many indi- The loss of quantum coherence is the result of a huge vidual events in order to define ⑀ and quantitatively estimate number of rather ‘‘dirty’’ interactions, acting randomly for a the degree of dephasing. This is true not only when the de- certain amount of time. We are therefore interested in the gree of decoherence is maximum, as in Eq. ͑3.10͒, but even effect on Q of a huge number of dirty (random) interactions. for perfectly coherent systems: Consider, for example, a Each interaction will take place between Q and one ͑or a double-slit experiment yielding a perfect interference pattern. bunch of͒ elementary constituent͑s͒ of A. The global effect Nothing can be said about the coherence properties of the on the wave function of Q will be, as we shall see, a loss of quantum system ͑the beam of particles entering the interfer- phase coherence. This is the idea we shall pursue in the ometer͒ if one does not accumulate many particles in order present study. The task of the decoherence parameter ⑀ will to build up the interference pattern ͓13͔. be to judge whether the global interaction is ‘‘clean’’ The above considerations might make the reader think (⑀Ӎ1) or ‘‘dirty’’ (⑀Ӷ1). that no meaning can be ascribed to single detection events. This would not be correct. One can give a sensible definition of ‘‘collapse’’ for a single particle via Eq. ͑3.7͒. In this way, B. Ergodic assumption the coherence properties of a single quantum system can be We can now formulate an ergodic hypothesis: The defined via the ensemble average of the macroscopic system ‘‘event’’ average over many particles in one experimental it interacted with. Individual detection events were discussed run ͑denoted hitherto with an overbar͒ will be assumed equal in Ref. ͓5͔. 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1067

IV. NUMERICAL SIMULATION detector͒, or an ‘‘absorber’’ ͑that yields a nonvanishing absorption coefficient͒. We also showed that, in contrast to A. Dirac comb widespread belief, absorption effects stemming from imagi- We can now investigate the decoherence process from a nary potentials are not necessarily significant for the collapse numerical point of view. The particle interacts with the ap- of the wave function, because the loss of coherence ͑of paratus ͑a macroscopic object made up of a huge number of which ⑀ is an estimate͒ stems only from the noise and the elementary constituents͒ according to the laws of quantum number N of elementary interactions in the apparatus. In the mechanics ͑Schro¨dinger equation͒. We will describe every present paper, unlike in ͓4͔, we shall introduce fluctuations interacting constituent ͑or every bunch of constituents͒ with of N and shall analyze in detail the dependence of ⑀ on a ␦ potential, and the whole apparatus with a one- several numerical constants involved. In the case of large dimensional array of ␦ potentials ͑a Dirac comb͒. This is the absorptionlike effects due to reflections, we shall see that well-known Kronig-Penney model ͓14͔, which has been in- fluctuations become large enough to yield the wave-function vestigated from several points of view ͓15͔. Previous nu- collapse. merical simulations of the measurement process by means of a Dirac comb were performed in Refs. ͓4,16͔. In the present B. Adjusting the parameters paper, however, we shall focus our attention on the role played by some particular combination of the numerical con- Equation ͑4.2͒ must be solved for many different incom- stants characterizing the array of potentials. We shall see that ing particles ͑‘‘events’’͒ and according to the analysis of Sec. the apparatus acts as a ‘‘dephaser’’ when some factors ap- III, the average transmission-reflection coefficient and prob- pearing in the expression of the decoherence parameter ex- ability read ceed certain ‘‘critical’’ values. 1 Np The total barrier will be written as ¯ ͗T͘ϭTϭ ͚ T j , ͑4.4͒ N Np jϭ1 V͑x͒ϭ ͚ ⌳␦͑xϪbl ͒, ͑4.1͒ l ϭ1 1 Np ¯ ͗R͘ϭRϭ ͚ R j, ͑4.5͒ where N is the total number of ␦ potentials, which play the Np jϭ1 role of ‘‘elementary interactions,’’ ⌳ the strength of the in- N teractions, and bl their locations. 1 p The transmission and reflection coefficients T and R of ͗t͘ϭ͉͗T͉2͘ϭ͉T͉2ϭ ͉T ͉2, ͑4.6͒ N ͚ j the whole barrier are computed in Appendix A. One obtains p jϭ1

NϪ1 T 1 1 Np ϪikbN ikdl ␶3 2 2 2 ϭe Z ͟ e Z , ͑4.2͒ ͗r͘ϭ͉͗R͉ ͘ϭ͉R͉ ϭ ͉R j͉ . ͑4.7͒ ͩ 0 ͪ l ϭ1 ͩRͪ N j͚ϭ1 p where ␶3 is the third Pauli matrix, k the wave number of All quantities will always be computed for N pϭ1000 and the Q, dl ϭbl ϩ1Ϫbl , and ensemble average ͗͘ will be taken over a Gaussian distribution of d and N. 1Ϫi⍀ Ϫi⍀ l Zϭ , ͑4.3͒ We must carefully avoid the case of resonance reflection ͩ i⍀ 1ϩi⍀ ͪ by the latticelike Dirac-comb structure, which may lead to a total reflection probability of order unity. This occurs when where ⍀ϭ⌳/បv, v being the Q particle speed. We shall set k͗d͘ (k being the neutron wave number and ͗d͘ the average ϪikbN b1ϭ0. ͓As explained in Appendix A, the factor e , ap- spacing between scatterers͒ is close to an integer multiple of pearing in ͑4.2͒, should also be put in front of ͑58͒ of Ref. ␲. Such a situation has been observed and investigated dur- ͓4͔ and ͑8.24͒ of the first paper in Ref. ͓5͔.͔ ing the initial part of our numerical situation and is uninter- So far, the internal motions of the elementary constituents esting from the point of view of ‘‘decoherence’’ effects. In of the apparatus have not been taken into account. These our simulation, in order to reduce unwanted spurious effects, internal motions will give rise to an intrinsic stochasticity of we shall always set the parameters describing the constituents themselves. In terms of our Dirac-comb model, this stochasticity will be k͗d͘ϭ4.5␲. ͑4.8͒ modeled as follows: since the interactions between the apparatus’s constituents and the incoming particles will take We shall focus our attention on thermal neutrons interacting place in different parts of the apparatus, the positions b and l with atoms, so that the relative spacings dl ϭbl ϩ1Ϫbl will be subject to statistical fluctuations. Moreover, the total number N of interac- 2␲ tions will also vary for different incoming particles. ␭ϭ ϭ2Å, ͗d͘ϭ4.5 Å . ͑4.9͒ In a previous numerical simulation ͓4͔ we showed that k this simple Dirac-comb model is able to reproduce correctly many different physical devices, such as a ‘‘phase shifter’’ The statistical distributions of the spacings and of the num- ͑that can preserve the quantum coherence͒ or a ‘‘dephaser’’ ber of ␦ potentials will be taken to be Gaussian, with aver- ͑that is able to provoke complete dephasing and work as a ages ͗d͘ and ͗N͘, respectively, and standard deviations 1068 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

␦d that it does not provoke decoherence on the incoming par- Yϵ , ␦dϵͱ ͑dϪ͗d͒͘2 , ͑4.10͒ ͗d͘ Š ‹ ticle. This point will be considered and discussed in Sec. IV F, in particular in connection with neutron interferometry. N Notice also that many detectors make use of thermodynami- ␦ 2 ⌬ϵ , ␦NϵͱŠ͑NϪ͗N͒͘ ‹. ͑4.11͒ cally unstable states, such as gases or liquids in critical con- ͱ͗N͘ ditions, whose physical states are characterized by very large statistical ﬂuctuations. In our simulation, the value of these two parameters will be It is not very easy to understand what are the physically varied between most interesting values of the parameter ⍀ϭ⌳/បv appear- 0рYр.5, 0р⌬р1. ͑4.12͒ ing in Eq. ͑4.3͒. Such values must be determined according to the physical problem investigated. One can guess that The maximum value of Y does not seem appropriate to de- there are situations in which there is a profound link between scribe a rigid lattice. Indeed, as we shall see, there are situ- ⍀ and ͗N͘, which may lead, under some conditions, to a ations in which coherence is lost for YϾ10Ϫ1. In such a precise relation between ⍀ and ͗N͘: This is closely related case, the apparatus cannot be viewed as a solid: A rigid to van Hove’s ‘‘␭2T’’ limit and to the occurrence of a dis- lattice cannot work properly as a ‘‘detector,’’ in the sense sipativelike behavior in quantum mechanics, and has been

FIG. 1. Dependence of ⑀ on ͗N͘ and ⌬, for some values of Y. Observe the sharp transition region from ⑀ϭ0 ͑coherence͒ to ⑀ϭ1 ͑decoherence͒ when Yϭ0. See text. 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1069

FIG. 2. Transmission probability ͉͗T͉2͘ vs ͗N͘ and ⌬. discussed elsewhere ͓17͔. We shall briefly reconsider this ͉͗T͉2͘ϭ͉͗T͉2͑͘⌬,Y,͗N͒͘, ͑4.14͒ point in Sec. V. The quantity ⍀ plays the role of a coupling constant, ⑀ϭ⑀͑⌬,Y,͗N͒͘. ͑4.15͒ which can be neither too small nor too big in order to ensure the occurrence of some dephasing. In our simulation ͗N͘ will reach the value 104, and it has been found that if Our task is to clarify how the wave-function collapse Ϫ1 takes place (⑀ 1) when the numerical values of ⌬, Y, and ⍀Ӷ10 no decoherence takes place. We therefore set in the → following numerical study ͗N͘ change. For the sake of clarity, we shall perform different simulations: In the first one, we shall let ͗N͘ and ⌬ vary, ⍀ϭ10Ϫ1. ͑4.13͒ while keeping Y fixed to a few different values. In the second one, ͗N͘ and Y will be varied while keeping ⌬ fixed to The values of the transmission probability and decoher- a few different values. In the third and last one, all param- ence parameter ͑or, equivalently, the transmission coeffi- eters shall be varied. cient͒ depend on all the physical quantities we have considered so far. Since the numerical values of ͗d͘ and ⍀ are C. Varying N and ⌬ fixed according to Eqs. ͑4.9͒ and ͑4.13͒, respectively, we can state that, in general, only ⌬, Y, and ͗N͘ determine ͉͗T͉2͘ Figure 1 displays the dependence of ⑀ on ͗N͘ and ⌬ for and ⑀, i.e., several values of Y, Fig. 2 displays the behavior of 1070 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

FIG. 3. Dependence of ⑀ on ͗N͘⌬2 for several values of ͗N͘ and for Yϭ0. The dotted line is Eq. ͑4.19͒. The diamond points are the numerical results. See text. ͗t͘ϭ͉͗T͉2͘ as a function of the same parameters, and Fig. 3 from which we easily derive ͑see Appendix B͒ the behavior of ⑀ versus N ⌬2 for Yϭ0. ͗ ͘ ϪiN⍀ 1 2 Ϫ2i⍀ N Observe that if Yϭ0 the transition region from coherent TӍe ͓1Ϫ 4 ⍀ ͕1Ϫ͑Ϫe ͒ ͖͔. ͑4.17͒ behavior (⑀ϭ0) to totally decoherent behavior (⑀ϭ1) is very sharp and occurs along the lines ͗N͘⌬2ϭ const. This By taking the average over N with a Gaussian distribution behavior can be explained by noting that, for small ⍀, the and neglecting small quantities, we obtain transmission coefﬁcient for a single ␦ ͓see Eq. ͑A2͔͒ reads 2 2 T ͉͗T͉2͘Ӎ1, ͉͗T͉͘2ӍeϪ͗N͘⌬ ⍀ , ͑4.18͒

1 1 Ϫ1 2 ϭ ϭ eϪitan ⍀ӍeϪ͑1/2͒⍀ eϪi⍀, so that T 1ϩi⍀ ͱ1ϩ⍀2 2 2 ͑4.16͒ ⑀Ӎ1ϪeϪ͗N͘⌬ ⍀ . ͑4.19͒ 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1071

FIG. 4. Dependence of ⑀ on ͗N͘ and Y. Observe the ‘‘saturation’’ effect when YϾ10Ϫ1. See text.

In Fig. 3, the ‘‘diamond’’ points are the numerical results D. Varying N and Y and the dotted lines are our approximate formula ͑4.19͒. The Figure 4 shows the dependence of ⑀ on ͗N͘ and Y for agreement is excellent. 2 several values of ⌬. The behavior of ͗t͘ϭ͉͗T͉ ͘ as a func- The above reasoning explains the sharp transition along 2 tion of the same parameters is shown in Fig. 5. For large the lines ͗N͘⌬ ϭ const when Yϭ0. This situation is physi- values of Y, this situation corresponds to a neutron beam cally interesting because it corresponds to the case of a neu- interacting with an absorber made up of a liquid or a dilute tron beam interacting with a macroscopic solid object, like a solution or a gas ͓18͔. ͑A solid lattice would ‘‘melt’’ for such crystal: The spacing between different atoms of the macro- large values of Y.) scopic object is ͑almost͒ constant, but different neutrons im- It is interesting to notice that there is a ‘‘saturation’’ effect pinge on different parts of the crystal, at slightly different in this case: If YϾ0.1, the values of ⑀ and ͗t͘ do not depend angles, interacting therefore with a different number of el- significantly on ͗N͘. The easiest way to understand this ef- ementary scatterers. fect is probably to take the ensemble average of both sides of When the condition Yϭ0 ͑fixed spacings between adja- Eq. ͑4.2͒: Since the fluctuations of dl are independent for cent potentials͒ does not hold anymore, additional random- different l , ͗T͘ contains factors of the type ization effects appear and the above approximations break 2 2 2 down. This is shown in the other graphs of Figs. 1 and 2. ͗eikdl ͘Ӎeik͗d͘eϪk Y ͗d͘ /2, ͑4.20͒ 1072 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

FIG. 5. ͉͗T͉2͘ vs ͗N͘ and Y. which vanish ͑washing away the dependence on Y) when Y large Ϫ1 2 → 2 4 ͑4͒ Yӷ2/k͗d͘Ӎ10 ͓see Eq. ͑4.8͔͒. A similar situation occurs ͉͗T͉ ͘ exp͑ϪN͉ ͉ ͒exp͉͑ ͉ N ͒, ͑4.23͒ for ͉͗T͉2͘. Observe that this phenomenon takes place inde- → R R S pendently of the value of ⌬. Y large A more detailed explanation of this effect is given in Ap- → 4 ͑4͒ ⑀ 1Ϫexp͑Ϫ͉ ͉ N ͒. ͑4.24͒ pendix B: One can see that, when ⌬ϭ0, by expanding T in → R S powers of the reflection coefficient ͑see Appendix A͒ of a where single ␦ potential and writing R Y large 4NϪ 4ϩ Ϫ ͑4͒ → ͉ ͉ N͉ ͉ N 1 N T T . ͑4.25͒ Tϭ ͕1ϩ␣͑ ͖͒, ͑4.21͒ SN → ͉͑ ͉4Ϫ1͒2 T R T one gets at second order in This result is essentially due to a complete randomization of R the phases acquired by multiple reflections, which leads to

Y large Y large → → ͉͗T͉͘2 exp͑ϪN͉ ͉2͒, ͑4.22͒ ͗␣͘ 0. ͑4.26͒ → R → 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1073

FIG. 6. ‘‘Saturation effect’’: Comparison of the perturbative formulas ͑4.22͒–͑4.24͒ with the numerical results. The dotted and solid curves are given by the perturbative formulas ͑4.22͒, ͑4.23͒, and ͑4.24͒, and the diamond points are the numerical results. (⌬ϭ0.)

The agreement of the perturbative formulas ͑4.22͒–͑4.24͒ FIG. 7. General case: Comparison of the perturbative formulas with the numerical results is excellent, as can be seen in ͑4.28͒–͑4.30͒ with the numerical results. The solid curves are given Fig. 6. by the perturbative formulas ͑4.28͒–͑4.30͒ and the diamond points On the other hand, when there is no ‘‘saturation’’ effect are the numerical results. (⌬ϭ0.) and ⌬ϭ0, one ﬁnds it convenient to deﬁne the number Neff of potentials the Q particle effectively interacts with Neff . Also in this case, the agreement of the perturbative formulas ͑4.28͒–͑4.30͒ with the numerical results is excel- ͑2͒ NeffϵNϩ2Re N lent, as can be seen in Fig. 7. S s͑sNϪNsϩNϪ1͒ ϵNϩ2Re , ͑4.27͒ ͭ 2͑sϪ1͒2 ͮ T

2 2 where sϵ 2͗e2ikd͘ϭϪ 2eϪ2(k͗d͘) Y . By making use of this newlyT deﬁned parameter,T one obtains

͉͗T͉͘2Ӎexp͑ϪN ͉ ͉2͒, ͑4.28͒ eff R

2 2 4 ͑4͒ ͉͗T͉ ͘Ӎexp͑ϪNeff͉ ͉ ͒exp͉͑ ͉ ͒, ͑4.29͒ R R SNeff

⑀Ӎ1Ϫexp͑Ϫ͉ ͉4 ͑4͒ ͒. ͑4.30͒ R SNeff

These formulas are identical with those obtained for the satu- FIG. 8. Dependence of ⑀ on ͗N͘Y 2 for several values of ͗N͘. rated region ͓Eqs. ͑4.22͒–͑4.24͔͒, when N is replaced by (⌬ϭ0.) 1074 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

FIG. 9. Decoherence parameter ⑀ and transmission probability ͉͗T͉2͘ vs ͗N͘⌬2 and ͗N͘Y 2, for several values of ͗N͘. The solid curves are given by ͑4.36͒ and ͑4.32͒ and the diamond points are the numerical results.

The decoherence parameter versus NY2 when ⌬ϭ0is case discussed in Sec. IV C, when ⌬Þ0 additional random- shown in Fig. 8. It is interesting to notice that, for large N, ization effects appear and the above approximations break ⑀ becomes a function of the single variable NY2.Asinthe down. The resultant effect is shown in Figs. 4 and 5. 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1075

FIG. 9. ͑Continued). 1076 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

FIG. 10. Phase diagrams for T. The transmission coefﬁcients T j of N pϭ150 particles are displayed for Yϭ0 and ͗N͘ϭ2000. From left to right and top to bot- tom: ⌬ϭ 0, 0.02, 0.05, 0.1, 0.2, 0.4, 0.5, and 1.0.

E. Varying N, ⌬, and Y ⑀ϭ⑀͑͗N͘⌬2,͗N͘Y 2͒. ͑4.31͒ We have seen that the decoherence parameter essentially depends on some particular combination of the parameter In the present section we shall study the behavior of ⑀ as a considered. Namely, function of these quantities. This is the most interesting situ- 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1077

FIG. 11. Phase diagrams for T. The transmission coefﬁcients T j of N pϭ150 particles are displayed for ⌬ϭ0 and ͗N͘ϭ2000. From left to right and top to bot- tom: Yϭ 0, 0.001, 0.005, 0.01, 0.02, 0.05, 0.1, and 0.5.

ation, from the physical point of view. It corresponds to the In order to explain these results, it is convenient to ob- case of a neutron interacting with a gas or liquid absorber, in serve that, in our numerical simulation, the transmission which both the positions and the total number of scatterers probability turns out to be largely independent of ⌬. We can change from event to event. therefore assume that 1078 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

FIG. 12. Phase diagrams for T. As in Fig. 10, but Yϭ0.02. 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1079

FIG. 13. Phase diagrams for T. As in Fig. 11, but ⌬ϭ0.5. 1080 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

FIG. 14. ‘‘Summary’’ of different numerical simulations. The points refer to different values of ⌬, Y, and ͗N͘.In͑a͒,⌬ϭ0, 10р͗N͘р104, and 0рYр0.5: All points lie on the same curve. In ͑b͒,0Ͻ⌬р1, 10р͗N͘р104, and 0рYр0.5. In all cases, ⑀ 1 when → ͉͗T͉2͘ 0. Dephasing and decoherence cannot be avoided at very low transmission probability. →

2 2 ϪiN⍀ ͉͗T͉ ͘N,dӍ͉͗T͉ ͘d ͗T͘N,dϭ͗e ͘N͉͗T͘d͉. ͑4.34͒ 2 4 ͑4͒ Ӎexp͑ϪNeff͉ ͉ ͒exp͉͑ ͉ ͒. ͑4.32͒ R R SNeff In this way we obtain

2 ϪiN⍀ 2 2 where we explicitly wrote which variables the averages are ͉͗T͘N,d͉ Ӎ͉͗e ͘N͉ ͉͗T͘d͉ taken over. Moreover, the analysis of Appendix B shows that Ӎexp͑Ϫ͗N͘⍀2⌬2͒exp͑ϪN ͉ ͉2͒, eff R ͗T͘ Ӎ N͕1Ϫ͉ ͉2 ͑2͖͒ӍeϪiN⍀͉͗T͘ ͉. ͑4.33͒ ͑4.35͒ d T R SN d We therefore assume that so that the decoherence parameter turns out to be 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1081

Y, eventually leading to a complete loss of quantum- mechanical coherence, can be suitably tailored and exploited in order to yield the desired type of dephasing ͑e.g., by trans- mitting the Q particle with high or low probability͒. Second, this analysis can be important for the study of neutron interferometry experiments at very low transmission probability ͓18,19͔, because it suggests that different physical devices, such as a silicon crystal, a gaseous or a liquid absorber ͓18͔, can modify the coherence properties of an incoming neutron beam in many different ways. For example, one cannot expect a strong dependence on Y (d fluctuations͒ for a crystal. On the other hand, an interesting dependence of ⑀ on Y can certainly be expected when a gaseous ͑say Ne͒ neutron absorber or a water solution of some highly absorbing material ͑say Gd͒ is inserted in an interferometer ͓18͔. Incidentally, notice that a dependence on ⌬ is to be expected both for a crystal or a gaseous device, because geometrical factors play here an important role here. FIG. 15. N eff vs Y and ͗N͘. An exhaustive discussion of these effects, beyond the qualitative estimates given here and in Ref. ͓19͔, requires a different study and is not easy to perform, because it in- ⑀Ӎ1Ϫexp͑Ϫ͗N͘⍀2⌬2͒exp͑Ϫ͉ ͉4 ͑4͒ ͒. ͑4.36͒ volves a careful estimate of the most important characteris- Neff R S tics of the experimental setup. Such a program is at present In spite of the crudeness of the above approximations, the under consideration. agreement of these formulas with the numerical results is However, it is possible to draw some general conclusions excellent, as can be seen from Fig. 9. Equations ͑4.32͒, even from the results of our simulation: A collection of many ͑4.35͒, and ͑4.36͒ are our main results and cover all the par- different numerical simulations is shown in Fig. 14. In all the ticular cases hiterto considered. cases investigated, one observes that when ͉͗T͉2͘ is close to 0, in general ⑀Ӎ1. In other words, it appears that in general, F. Comments decoherence effects cannot be avoided at low transmission probability. In order to clarify the mechanism underlying the loss of Notice that this phenomenon exists for all values of the quantum-mechanical coherence, we have drawn, in parameters ⌬, Y, and ͗N͘. Even though the present numeri- Figs. 10–13, the phase diagrams of the transmission prob- cal simulation neglects many important characteristics of a ability T. Every figure displays the transmission coefficients real experiment, it is difficult to believe that the behavior T j of N pϭ150 particles. displayed in Fig. 14 be just a coincidence. It is worth stress- In Fig. 10, Yϭ0, ͗N͘ϭ2000, and ⌬ is varied between 0.0 ing that the statistical fluctuations of a real macroscopic ap- and 1.0. Observe that although the phases spread, the value paratus can never be neglected, even in principle, because of of the transmission probability is constant and independent 2 finite-temperature effects and of the impossibility of isolating of ⌬. In the particular case displayed, ͉͗T͉ ͘ is close to 1 and completely the apparatus from its environment. the phases are completely randomized for ⌬ϭ1 ͑‘‘collapse’’ of the wave function͒. V. CONCLUDING REMARKS In Fig. 11, ⌬ϭ0, ͗N͘ϭ2000, and Y is varied between 0.0 and 0.5. In this case the effect is different: The transmission We have discussed some important characteristics of the probability ͉͗T͉2͘ is strongly dependent of Y. The quantum quantum measurement process by making use of a schematic coherence is completely lost for Yϭ0.5 ͑‘‘collapse’’ of the representation of a macroscopic device: In our analysis, the wave function͒. complicated dynamical behavior of the elementary constitu- In Figs. 12 and 13, both sources of fluctuation are ents of the apparatus ͑our ‘‘detector’’͒ was represented by switched on: In the first case, ͗N͘ϭ2000, Yϭ0.02 ͑fixed͒, means of an array of potentials undergoing ͑Gaussian͒ ran- and ⌬ is varied as in Fig. 10. The phase coherence is lost dom fluctuations. The ‘‘collapse’’ of the wave function is a already for ⌬ϭ0.5 ͑‘‘collapse’’ of the wave function͒.Inthe consequence of the dephasing effects that provoke a loss of second case, ͗N͘ϭ2000, ⌬ϭ0.5 ͑fixed͒, and Y is varied as quantum-mechanical coherence. in Fig. 11. There is complete dephasing already for Our main conclusions are Eqs. ͑4.32͒, ͑4.35͒, and ͑4.36͒, Yϭ0.05 ͑‘‘collapse’’ of the wave function͒. which give a realistic estimate of the transmission probabil- It is interesting to observe that the parameters ⌬ and Y ity, transmission coefficient, and decoherence parameter, re- provoke decoherence in different ways: The value of the spectively. The values of these parameters alone suffice to transmission probability is practically ⌬ independent; on the outline the essential features of the interaction between the other hand, it is strongly dependent on Y. This is important Q particle and the apparatus. for two independent reasons. First, this analysis sets the pre- The dephasing effects can always be quantitatively ana- liminary basis for a ‘‘design theory’’ of a quantum- lyzed by means of the decoherence parameter ⑀, which plays mechanical detector. The combined action of ͗N͘, ⌬, and the effective role of an ‘‘order parameter’’ for the wave- 1082 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54 function collapse. This is only an analogy, and must be con- We would like to conclude this discussion with a perspec- sidered with great care. Indeed, we are not supplying any tive on the measurement problem and the general issue of evidence that the quantum measurement process can be decoherence. We believe that decoherence should be under- viewed as a phase transition, and are unable, at the present stood as a dissipative phenomenon of some sort ͓17͔. A simi- stage, to corroborate this idea with clear-cut arguments. Nev- lar idea was independently put forward by Leggett ͓26͔ a few ertheless, Figs. 1, 4, and 9 are rather suggestive, in that they years ago. However, we are not thinking of ‘‘dissipation’’ in lead one to ͑naively͒ view the loss of quantum-mechanical the sense of energy loss: Obviously, in the present model ͑a coherence as a phase transition of some sort. The idea which Dirac comb made up of real potentials͒ one can expect no the ‘‘collapse’’ of the wave function can be viewed as a sort energy loss. Rather, we are thinking of an irreversible phe- of phase transition was first proposed, as far as we know, by nomenon stemming from the complicated and disordered Ne’eman ͓20͔, although in a different context. motion of the many elementary constituents of the macro- The whole measurement process has been analyzed scopic detector, eventually leading to an irretrievable loss of within the quantum-mechanical framework. In this sense, we coherence. This problem is at present under investigation. need neither modify quantum mechanics ͓21͔, nor ‘‘complete’’ it by introducing additional parameter ͑hidden vari- ables͓͒22͔. At the same time, we need not invoke meta- ACKNOWLEDGMENTS physical concepts ͓23͔ in order to explain the measurement We thank Hiromichi Nakazato for a very careful reading process. We believe that quantum mechanics alone suffices of the manuscript. S.P. is very grateful to the Physics Depart- to describe all the important features of the loss of coher- ment of Waseda University for their kind hospitality. ence, which eventually lead to the ‘‘collapse’’ of the wave function. To this end, one needs an effectual operational principle that ‘‘works,’’ yielding the desired decoherence. Such an operational principle is introduced in Eq. ͑3.7͒.It APPENDIX A: THE DIRAC COMB enables us both to discuss single events and to give an op- Let the potential V(x)ϭ⌳␦(x) and the incident wave am- erational definition of quantum coherence, valid both for plitude be normalized at 1 for xϭϪ . Then single events and collections of experimental data. As em- ϱ phasized in Sec. IIIB, we simply regard ͑3.7͒ as a postulate, at the present stage, and hope to substantiate it by more eikxϩ eϪikx, xϽ0 fundamental arguments in the future. It goes without saying R ␺ϭ eikx, xϾ0, ͑A1͒ that the prescription ͑3.7͒ does not imply any fundamental ͭ T modification of quantum mechanics. It is worth stressing the analogies and differencies that we where the reflection and transmission coefficients, and think exist between the philosophy underlying the present respectively, read R T work and those approaches that make use of the technique of partial tracing. A detector must be an open system ͓5͔: This ϭϪi⍀͑1ϩi⍀͒Ϫ1, is the only way to circumvent all the well-known no-go theo- R rems ͓8,24͔. The above-mentioned openness of the macro- ͑A2͒ scopic detector makes it similar to a sort of ‘‘environment’’: ϭ͑1ϩi⍀͒Ϫ1, In this sense, our numerical simulation leads to results that T are consistent with the so-called ‘‘environment approach.’’ However, we believe that there is a noteworty difference, with ⍀ϵ⌳/បv (v is the particle speed͒. The reflection and because we make no use of projection operators a` la von transmission probabilities read Neumann and of partial tracing over the environment states. ͉ ͉2ϭ⍀2͑1ϩ⍀2͒Ϫ2, Decoherence is obtained by applying the ensemble average R and the ergodic hypothesis ͑3.7͒. Of course, the computation ͑A3͒ of such an ensemble average ͑if properly performed over the ‘‘many Hilbert spaces,’’ see ͓3,5͔͒ leaves us only with the ͉ ͉2ϭ͑1ϩ⍀2͒Ϫ2. dynamical variables of the Q particle. Needless to say, ap- T plication of partial tracing has the same effect. Nevertheless, Analogously, if the wave impinges on the potential from we feel that the underlying philosophy and the physical in- the right and is normalized at 1 for xϭϩϱ, one gets, by terpretation are different. Partial tracing is only a convenient explicit calculation or by applying space reflection invari- ‘‘working rule’’ ͑see the criticisms put forward against the ance, ϭ Ј and ϭ Ј. environment approach in Ref. ͓25͔͒. By contrast, the en- TheR totalR potentialT T barrier is semble average ͑3.7͒ has its own independent logical status. Admittedly, it is a postulate that should be carefully consid- N ered and, possibly, justified. Work is now in progress in or- V͑x͒ϭ ͚ ⌳␦͑xϪbl ͒, ͑A4͒ der to clarify its role in a more delicate context, such as l ϭ1 ͑quantum and classical͒ chaos. ͑Notice that the ensemble average endeavors to describe the effects that stem from the where N is the total number of ␦ potentials, ⌳ their macroscopicity of the apparatus.͒ These issues will be dis- strengths, and bl their positions. We write the wave function cussed elsewhere. as 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1083

ik͑xϪb1͒ Ϫik͑xϪb1͒ A1e ϩB1e , xϽb1 Ӈ A eik͑xϪbl ϩ1͒ϩB eϪik͑xϪbl ϩ1͒ϭC eik͑xϪbl ͒ϩD eϪik͑xϪbl ͒, b ϽxϽb ␺ϭ l ϩ1 l ϩ1 l l l l ϩ1 ͑A5͒ Ӈ

ik͑xϪbN͒ Ϫik͑xϪbN͒ Ά CNe ϩDNe , bNϽx

where we used the coefficients A ,B (C ,D ) on the very l l l l N 2 iD 4 iDЈ 6 left ͑right͒ of the l th potential. We get Tϭ 1ϩ ͚ ••• e jϩ ͚ ••• e j ϩO͑ ͒ T ͭ R j T T R j T T R ͮ B ϭA ϩD Ј ϵ N͑1ϩ␣ ϩ␣ ϩ ͒ l l R l T T 2 4 ••• ͑A6͒ ϵ N͑1ϩ␣͒, ͑B2͒ T C ϭA ϩD Ј. l l T l R where each term corresponds to a fixed number of reflections inside the Dirac comb and the number of ’s and the phase By solving for Cl ,Dl , we obtain T factors D j ,DЈj , . . . in each summation depend on where the reflections take place ͑see the following formula͒. Cl 1 ЈϪ Ј Ј Al Al ϭ TT RR R ϭZ . At the second order in , the solution for T reads ͩ D ͪ Ј ͩ Ϫ 1 ͪͩB ͪ ͩB ͪ R l l l T R ͑A7͒ N 2 2ikd 2ikd Tϭ ͕1ϩ ͕͑e 1ϩ•••ϩe NϪ1͒ In our case ϭ Ј, ϭ Ј and T R 2 2ik͑d ϩd ͒ 2ik͑d ϩd ͒ R R T T ϩ ͑e 1 2 ϩ•••ϩe NϪ2 NϪ1 ͒ 2 2 T 1 Ϫ 1Ϫi⍀ Ϫi⍀ 2͑NϪ3͒ 2ik͑d1ϩ•••ϩdNϪ2͒ 2ik͑d2ϩ•••ϩdNϪ1͒ Zϭ T R R ϭ , ͑A8͒ ϩ•••ϩ ͑e ϩe ͒ ͩ Ϫ 1 ͪ ͩ i⍀ 1ϩi⍀ ͪ T T R ϩ 2͑NϪ2͒e2ik͑d1ϩ•••ϩdNϪ1͖͖͒ϩO͑ 4͒, ͑B3͒ T R where we made use of Eq. ͑A2͒. Finally, we define the whole barrier’s transmission (T) and reflection (R) coefficients, by which yields, since 2ӍϪ͉ ͉2, setting R R

ik͑bNϪb1͒ 2 2ikd 2ikd A1ϭ1, B1ϭR, DNϭ0, CNϭTe ͑A9͒ ␣ ϭϪ͉ ͉ ͕͑e 1ϩ ϩe NϪ1͒ 2 R ••• ϩ 2͑e2ik͑d1ϩd2͒ϩ ϩe2ik͑dNϪ2ϩdNϪ1͒͒ In Eq. ͑A9͒ we eliminated an incorrect kinematical factor T ••• appearing in the definition of T, as given in Ref. 4 . Equa- 2͑NϪ3͒ 2ik͑d ϩ ϩd ͒ 2ik͑d ϩ ϩd ͒ ͓ ͔ ϩ ϩ ͑e 1 ••• NϪ2 ϩe 2 ••• NϪ1 ͒ tion ͑4.2͒ is easily obtained from Eqs. ͑A7͒–͑A9͒. ••• T ϩ 2͑NϪ2͒e2ik͑d1ϩ•••ϩdNϪ1͖͒. ͑B4͒ T APPENDIX B: PERTURBATIVE EXPANSION FOR T In this appendix we shall concentrate on the d fluctuations The average transmission coefficient and probability, as of our one-dimensional Dirac comb. The symbol ͗͘ must given by Eqs. ͑4.4͒ and ͑4.6͒, read therefore be understood as an average over the d variables only. The reader should also keep in mind that the barrier’s ͗T͘ϭ N͑1ϩ͗␣͒͘ϭ N͑1ϩ͗␣ ͘ϩ ͒, ͑B5͒ transmission coefficient T depends, among others, on N.Itis T T 2 ••• worth stressing that all the theoretical formulas to be derived in this Appendix hold even in the case of a linear array of ͗t͘ϭ͉͗T͉2͘ϭ͉ ͉2N͑1ϩ2Re͗␣͘ϩ͉͗␣͉2͒͘ arbitrary shaped potentials, provided they are separated by T ϭ͉ ͉2N͑1ϩ2Re͗␣ ͘ϩ͉͗␣ ͉2͘ϩ ͒. ͑B6͒ force-free regions. T 2 2 ••• By Eq. ͑4.2͒, the exact solution for T reads

NϪ1 Notice that ⑀ is proportional to the square of the standard T 1 deviation (␦␣)2: ϭeϪik͑d1ϩ•••ϩdNϪ1͒Z ͟ eikdl ␶3Z , ͑B1͒ ͩ 0 ͪ l ϭ1 ͩRͪ ͉͗T͉͘2 ͑␦␣͒2 ⑀ϭ1Ϫ ϭ , where dl is the distance between adjacent potentials and we ͉͗T͉2͘ ͉͗1ϩ␣͉2͘ set the ﬁrst potential at xϭ0. If is small, the transmission coefﬁcient is expanded as ͑␦␣͒2ϭ͉͗␣͉2͘Ϫ͉͗␣͉͘2ϭ͉͗␣Ϫ͗␣͉͘2͘. ͑B7͒ R 1084 KONO, MACHIDA, NAMIKI, AND PASCAZIO 54

2 2 1. Calculation of ͦŠT‹ͦ : Definition of N eff ͗T͘Ӎ N͕1ϩ͗␣ ͖͘ϵ N͕1Ϫ͉ ͉2 ͑ ͖͒. ͑B8͒ T 2 T R SN We explicitly calculate the average value of the transmission coefficient when the spacings dl undergo random fluctuations. We write where

͑2͒ϵϪ͗␣ ͉͘ ͉Ϫ2ϭ͗͑e2ikd1ϩ ϩe2ikdNϪ1͒ϩ 2͑e2ik͑d1ϩd2͒ϩ ϩe2ik͑dNϪ2ϩdNϪ1͒͒ϩ ϩ 2͑NϪ3͒ SN 2 R ••• T ••• ••• T ϫ͑e2ik͑d1ϩ•••ϩdNϪ2͒ϩe2ik͑d2ϩ•••ϩdNϪ1͒͒ϩ 2͑NϪ2͒e2ik͑d1ϩ•••ϩdNϪ1͒͘ T ϭ͑NϪ1͒͗e2ikd͘ϩ͑NϪ2͒ 2͗e2ikd͘2ϩ T ••• ϩ2 2͑NϪ3͒͗e2ikd͘NϪ2ϩ 2͑NϪ2͒͗e2ikd͘NϪ1 T T 2 NϪ2 NϪ1 ϭ͑NϪ1͒sϩ͑NϪ2͒s ϩ•••ϩ2s ϩ1s s͑sNϪNsϩNϪ1͒ ϭ , ͑B9͒ 2͑sϪ1͒2 T

where all fluctuations are assumed independent, all spacings N eff ϵ Ϫc2 . ͑B14͒ identically distributed, and sϵ 2͗e2ikd͘. 2 T 2 Moreover, since ͉ ͉ ϭ1Ϫ͉ ͉ , we obtain As a first application of the above formulas, let Yϭ0. Then, T R by Eq. ͑4.8͒, sϭϪ 2 and 2 2N T ͉͗T͉͘ Ӎ͉ ͉ ͕1ϩ2Re͗␣2͖͘ T 1 2 4 2 ͑2͒ 4 ͑2͒ Ϫ2i⍀ N ϭ͕1ϪN͉ ͉ ϩO͑ ͖͕͒1Ϫ2͉ ͉ Re ϩO͑ ͖͒ N Ӎ ͕Ϫ͑Ϫe ͒ Ϫ2Nϩ1͖ ͑B15͒ R R R SN R S 4 ϭ1ϪN͉ ͉2Ϫ2͉ ͉2Re ͑2͒ϩO͑ 4͒ R R SN R so that 2 ͑2͒ 4 ϭ1Ϫ͉ ͉ ͕Nϩ2Re N ͖ϩO͑ ͒ N 2 ͑2͒ R S R TӍ ͕1Ϫ͉ ͉ N ͖ 2 4 T R S ϵ1Ϫ͉ ͉ Neff ϩO͑ ͒ Ϫ ͑1/2͒ ⍀2 Ϫi⍀ N 1 2 Ϫ2i⍀ N R R ϭ͑e e ͒ ͑1Ϫ 4 ͉ ͉ ͕Ϫ͑Ϫe ͒ Ϫ2Nϩ1͖͒ 2 R Ӎexp͑ϪNeff ͉ ͉ ͒, ͑B10͒ ϪiN⍀ 1 2 Ϫ2i⍀ N R ϭe ͓1Ϫ 4 ⍀ ͕1Ϫ͑Ϫe ͒ ͖͔, ͑B16͒ where we roughly took into account the effects of higher- which yields Eq. ͑4.17͒. order reflections, in the last approximate equality. ͑Notice As a second application, consider the large d-fluctuation that this approximation corresponds to the assumption that case. Notice that, by Eq. ͑4.8͒ and the Gaussian property of *4 Ӎ *2 *2 . This formula holds as a Gaussian rule when ͗ ͘ ͗ ͗͘ ͘ the fluctuation, all higher-order moments are discarded.͒ This is Eq. ͑4.28͒. The parameter 2 2 2 2 sϭ 2͗e2ikd͘ϭ 2e2ik͗d͘eϪ2͑k͗d͒͘ Y ϭϪ 2eϪ2͑k͗d͒͘ Y . T T T ͑B17͒ N ϵ Nϩ2Re ͑2͒ ͑B11͒ eff SN By Eq. ͑4.8͒,ifYϾ0.1, we get ͉s͉Nϳ0(NϾ1). From Eqs. plays the role of the number of ␦ potentials the Q particle ͑B9͒ and ͑B17͒, we obtain therefore effectively interacts with. Indeed, suppose that the Dirac ͑2͒ comb is made up of N eff potentials acting incoherently: We N ӶN ͑Y large͒ , ͑B18͒ obtain S → Y large 2 2N 2 → ͉͗T͉͘ ϭ͉ ͉ eff Ӎ ͉͗T͉ ͘ . ͑B12͒ ͉͗ T ͉͘2 exp͑ϪN͉ ͉2͒, ͑B19͒ T → R The last approximate equality is fully justified if NՇ100, which yields Eq. ͑4.22͒ and explains the ‘‘saturation’’ effect but, surprisingly, yields very accurate results even if mentioned in Sec. IV D. Nϳ10 000. It is interesting to observe that in this case A more rigorous definition of N eff can be given by starting from the perturbative expansion of the square of the av- NϭNeff . ͑B20͒ erage transmission coefficient This is in agreement with our physical intuition: when Y is 2 2 4 6 large, the phases are completely randomized, the number of ͉͗T͉͘ ϭ1ϩc2͉ ͉ ϩc4͉ ͉ ϩ ͉͑ ͉ ͒ϩ•••. ͑B13͒ R R O R ‘‘effective’’ interactions is N, and ͉͗T͉͘2Ӎ͉ ͉2N 2 T In this expansion, Ӎexp(ϪN͉ ͉ ). R 54 DECOHERENCE AND DEPHASING IN A QUANTUM . . . 1085

We give here a short list of properties of Neff . The proofs Y large 2 → 2N 4 ͑4͒ are left to the reader. ͑i͒ If Nϭ1, then also Neff ϭ1; ͑ii͒ if ͉͗T͉ ͘ ͉ ͉ ͑1ϩ͉ ͉ SN ͒ 2 → T R Yϭ0 and k͗d͘ϭ(nϩ1/2)␲, then Neff ϭsin (N⍀ ) for even 2 2N 4 ͑4͒ N and Neff ϭcos (N⍀ ) for odd N; ͑iii͒ if Yϭ0 and Ӎ͉ ͉ exp͉͑ ͉ SN ͒Ӎexp 2 T R k͗d͘ϭn␲, then Neff ϭ N ͑‘‘resonance’’ effects͒; ͑iv͒ if 2 4 ͑4͒ ͑ϪN͉ ͉ ͒exp͉͑ ͉ N ͒ YϾ0.1 then NeffϭN. R R S The behavior of Neff as a function of Y and N is shown in ϭ͉͗T͉͘2exp͉͑ ͉4 ͑4͒͒. ͑B24͒ Fig. 15. R SN

2 where we partially took into account the effects of higher- 2. Calculation of ŠͦTͦ ‹ order reflections, such as in Eq. ͑B10͒. This is Eq. ͑4.23͒. Let us compute the average transmission probability. We Let us now turn to the general case in which there is no consider first the ‘‘saturated’’ region. In this case, from Eqs. saturation (YՇ0.1). Such a situation is difficult to tackle in ͑B6͒ and ͑B18͒, the average transmission probability be- full generality. We shall therefore follow a heuristic ap- comes proach, and observe that comparison of Eqs. ͑B10͒ and ͑B19͒ enables us to evince that one can formally obtain the Y large 2 2 → 2N 2 expression for ͉͗T͉͘ in the general case by substituting ͉͗T͉ ͘ ͉ ͉ ͕1ϩ͉͗␣ ͉͘ ͖. ͑B21͒ → T 2 We write ͑2͒ N Nϩ2Re N ϵNeff ͑B25͒ ͉͗␣ ͉͘2ϵ͉ ͉4 ͑4͒ , ͑B22͒ → S 2 R SN (4) into the expression for the ‘‘saturated’’ case. Roughly speak- where the term N stems from multiple reflections and is explicitly given byS ing, this assumption can be justified as follows: In our numerical simulation, both the transmission probability and de- Y large coherence parameter depend with very good approximation → ͑4͒ 2ikd1 2ikdN-1 2 N ͗ ͉ ͑e ϩ ••• ϩ e ͒ on NY . Moreover, in the ‘‘saturated’’ region, the average S → transmission coefficient and its square depend only on N. ϩ 2͑e2ik͑d1ϩd2͒ϩ ϩe2ik͑dNϪ2ϩdNϪ1͒͒ϩ T ••• ••• Therefore a change of the value of N in the nonsaturated case 2 NϪ3 2ik d ϩ ϩd can be thought of as equivalent to a change of the value of ϩ ͑ ͒͑e ͑ 1 ••• NϪ2͒ T Y 2. In this way one obtains, from Eq. ͑B10͒, ϩe2ik͑d2ϩ•••ϩdNϪ1͒͒ϩ 2͑NϪ2͒e2ik͑d1ϩ•••ϩdNϪ1͉͒2͘ T 4 4͑nϪ2͒ 1 NϪ1͒ϩ NϪ2͒ϩ ϩ 1 2 2 4 ͑4͒ Ӎ ͑ ͉ ͉ ͑ ••• ͉ ͉ ͉͗T͉ ͘Ӎ͉͗T͉͘ exp͉͑ ͉ ͒ T T Neff 4N 4 R S ͉ ͉ ϪN͉ ͉ ϩNϪ1 2 4 ͑4͒ Ӎ T T , ͑B23͒ ϭ exp͑Ϫ͉ ͉ Neff ͒exp͉͑ ͉ ͒ , ͑B26͒ ͉͑ _͉4Ϫ1͒2 R R SNeff T In conclusion, by Eq. ͑B19͒ which is Eq. ͑4.29͒.

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