Coherence width, localization length and classical behavior

Physics Letters A 179 (1993) 249-254 North-Holland

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Coherence width, localization length and classical behavior Luc Dagens Commissariat b I’Energie Atomique, Centre d’Etudes de Lime& Valenton, 94190 Villeneuve St. Georges, France Received 8 April 1993; revised manuscript received 28 May 1993; accepted for publication 14 June 1993 Communicated by J.P. Vigier

The environment-induced transition from quantum to classical behavior is studied according to the Zurek theory. Since the Zurek superselection rule cannot be used for this problem, a more general “strong decoherence” reduction rule is stated. A new discrete environment model, involving a characteristic length L,, is used to discuss the transition to classical behavior. Such a transition is demonstrated when L, is finite. The transition duration is found to scale as L:, and decoherence without localization is shown to occur when the environment is continuous (Le = co )

1. Introduction

An important problem is to decide whether the emergence of classical properties (localization of a pointer after a measurement, for example) can be predicted in terms of quantum concepts only, i.e. under the assumption that quantum mechanics (QM) is a complete theory of what can be observed or predicted about any (possibly macroscopic) system. A possible approach is the decoherence theory, which aims at explaining the emergence of classical properties for an object S :x (x: particle or collective coordinate) in terms of its interaction with an enviwhich induces irreversible ronment d [1,2], correlations between S and 8. A precise theory is due to Zurek [2]. Zurek was able to propose an operational reduction rule, the so-called environmentinduced superselection rule (EISR), which determines, in terms of the S + 6 system state !P( t) only, into which definite (calculable) mixture the pure state is possibly reduced at a given time t. The Zurek theory requires that the pointer observable /1, to which the collapse will give a definite value, commutes with the S+ 6 total Hamiltonian. Let A=I:,;1E, be its spectral representation. Then each !& ( t ) = En PAevolves separately according to the same Schrodinger equation (SE). The reduced density matrix (RDM) is p= C,, pa, with phz, 0375-9601/93/$

= Tr, 1u;l,) ( lu,. 1. There is decoherence at t (for an appropriate environment) if all of the pU, ( t’) vanish for A +A’ and t’> t. Then the RDM p is equivalent to the mixture {PA}mixof the improper RDM pn =p;u, as far as only the observables for S are concerned. The reduction rule p+ {Pn}mixis operationally valid, since no experiment (involving the system S only) is feasible which can distinguish between p and (Pn}mix. But the EISR cannot be applied to the localization process (transition to a well-localized state). Here the role of A is played by the position observable X, which is not, for real systems, a constant of motion. The Zurek rule requires (i) that each E,!P(t) satisfies the SE and (ii) the decoherence of pa, = p(x, x’). Condition (i) is not satisfied and the rule is not effective. Indeed recent model calculations [ 36 ] demonstrate the rapid environment-induced decoherence of the RDM p (x, x’, t) but fail to show the localization process without which classical behavior [7,8] is not obtained. The main purpose of this Letter is to demonstrate that such a theory of the environment-induced localization process can be given, in terms of quantum concepts only, provided that the Zurek EISR is replaced by the closely related, more general, “strong decoherence” reduction rule which is proposed below. The EISR is not strong enough because the state lu, and the mixture { ‘yn},ix may well be empirically distinguishable if some observable for 8 is mea-

06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

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sured. That is why the related concept of strong decoherence (SD) is introduced (section 2). SD is relative to the family 0 of observables for S+ & for which a measurement is assumed to be feasible. Loosely speaking, the state Y(t) decoheres to Yr, !PZ with respect to Loif no measurement of any OE 0 can distinguish between Y and the mixture {Y,, Y~}mix, for example. The corresponding reduction rule is stated in section 2: it is consistent with the assumption that the QM superposition principle is (empirically) valid without restriction, since Y and { Yr, Yz)mixgive the same predictions for any experiment which is feasible. The SD reduction rule is next applied to the localization problem. The amount of localization for the variable x is defined as the x uncertainty LX(Y) when S + 6 is in state Y, see (2) below. x is the S position variable. A localization process occurs if Y is reduced to a mixture of well-localized states (section 2). From this definition and the Omnbs classical theory [ 7,8 1, it is possible in principle to calculate the environment-induced emergence of classical behavior. An illustrative model of environment is given in section 3. It is a spatially discrete version of the continuous environment model of ref. [ 6 1. It illustrates the fact that the d spatial structure is significant for the localization process, and that decoherence (of the RDM) does not always imply classical behavior (section 4).

2. Strong decoherence and localization (I) General decoherence theory. A state Y( t ) splits into two incoherent parts Yr, YZ if no experiment can distinguish between Y and the mixture { Yr, Y~}mil. Decoherence cannot occur if any observable, 1Y) ( Yy( for example, can be measured and must thus be defined with respect to a restricted set of observables Co.We state the following.

Definition A. A set of S+ B states { Yi( t)} is a decohering set for t > to, with respect to the set of observables 0, if the following properties hold for all t>to: ( 1) Every Y(t) satisfies the Schrodinger equation. 250

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(2) Any pair Y, Y satisfies the decoherence condition (

YjlOl

E)

=Oidij+Eu(t)

(1)

for any OE 0, with eii negligible for all practical purposes. Lois the set of observables which are considered as measurable in practice. There is simple decoherence if Lo= C!s(the observables for S). Simple decoherence implies decoherence for the RDM p, but is not sufficient for a reduction rule [ 21. Let r!& be the set of observables which couple at most N different S or d particles. Such an observable is a linear combination of products of at most N (absorption and creation) field operators. Of course & 3 CC& and simple decoherence clearly does not imply the impossibility to distinguish empirically between Y and { Yi}mix. On the other hand, precise quantum measurements are possible only if the number of involved particles N is not too large. By choosing N sufficiently large, we may assume safely that strong decoherence, defined as decoherence with respect to c&, implies the operational strict equivalence of the pure state Y and the mixture {Yi}mti. We state thus the “strong decoherence” reduction rule (SD rule).

Rule A. If at time to the S + 8 state vector Y( t ) can be written as Ci !&(t; to), where { .. .. !&(t; to), . ..} is a strongly decohering set for t > to (according to definition A), then ( 1) Y(t) can be replaced by the proper mixture {K::(C

h3)}mix,

(2) the RDM ps( t) can be replaced by the mixwith pi(t)=Tr,] Yi(t; to)) ture (Pitt; to))e, X(Yi(t; tO)l, for any tatto. The probability of a component pi is ]IYi]j2/ll Yll’. This rule is sufficient to solve operationally the socalled “measurement problem”: assuming that Y( 1) is known, we can calculate at every time t if Y(t) is equivalent to some well-defined mixture (Yi( t; tO)}mix. It does not solve it in principle, as shown by Bell [ IO] and d’Espagnat [ 111, since there is some arbitrariness in the choice of UN. We discuss briefly the dynamic reduction theory proposed by Ghirardi et al. [ 121 (GRW). The GRW theory involves a slightly modified, nonlinear evo-

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lution equation. It solves the measurement problem, but, contrary to the SD theory, can lead to predictions which contradict QM. For example, the superposition principle can be empirically contradicted according to the GRW theory but not (by construction) according to the SD rule. Moreover, what is quantum with respect to the SD theory, may well be classical with respect to the GRW theory. For example, an isolated diamond crystal in its ground state has only six degrees of freedom, and is purely quantum according to the decoherence theory; its behavior would be always classical according to the GRW theory, at least if the number of carbon atoms is large enough, more than 1015, say. (II) The localization process and classical behavior. Two characteristic lengths can be ascribed to the quantum state Y or the RDM p(x, x’). First, the localization length LX, which is defined as the spread of the probability density p(x, x),

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UP) =fi/26P, 9

(3)

so that L,>l, is always true. Physically, I,(p) is related to the disappearance of observable interference effects: when I, is small, Sp, is large and the interferences are blurred out. The localization process is not a consequence of the dynamic evolution: if Hs N 0, for example, LX(p) is almost a constant. Nevertheless a transition to a localized state is possible if the environment induces a collapse from p to one of the pi( t; to) according to rule A. The localization length is now one of the LX(pi). We may define the effective localization length as

L*(t, to) =suPiLX(Pi(t; to))

(4)

for t 2 to. The notation L*( t, to) means that only the reductions up to to are taken into account. An important point is that it is not possible to express L* in terms of the RDM alone: some knowledge of the complete S + 8 state itself is required in order to ver-

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ify the conditions stated in definition A. The localization process is then determined by the time evolution of L*(t, t). The classical behavior is defined with respect to some observational resolution length Y [ 71. 5? is, for a given system, the smallest spatial interval which can be measured or/and controlled by an observer. We state the definition: the object S is said to behave

classically in the time interval [to, tl ] if the result of a position measurement, done at ts [to, t, 1, and accurate to 9, can be predicted with certainty from the sole knowledge of the initial S+ 8’ quantum state Y( to). Thus “classical” is loosely speaking identified to determinism [ 7,s 1. Since p(t) = Tr, ] Y) ( Y] is a deterministic quantity, a sufficient condition for classical behavior is L,(p( t) ) < 3’ for tg [to, t, 1. But this condition is not necessary since a density matrix may be a mixture of classical-like states. The condition is then

L*(t, to)=L*(Y(t), Next, the coherence width (or decoherence length [ 61) Z, (p) which gives the off-diagonal extent of the density matrix. For a Gaussian RDM exp{ f [ (x+ x’) 2/a2+ (x-x’) */b’]} we get L,(p) =Zxp=a, Z,(p) = b and the momentum dispersion Za,=fi/2b. This suggests the definition

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t,)<9,

(5)

for tE [to, tl 1. L* takes into account the possible &induced wave packet collapse at time to as given by rule A. The largest tl consistent with (26) defines the classical behavior duration T,,. An estimate is Tc,-fM=%(b)lfi,

(6)

for some t2e [to, to + T,,], since the velocity dispersion is h/2&7(t). Equation (27) shows that a too small coherence width makes the system less predictable.

3. A discrete probes environment model (I) The model described here is relative to an object S:x in a complex medium 8 made of discrete probes (such as photons, atoms, loosely bound electrons). .Xis a particle coordinate or a (macroscopic) collective coordinate. Each probe i is in as a loosely bound or free state, with fairly well-defined position Rj( t) &L, and momentum hqi4 h/L,; these are both well defined because of the repeated irreversible interactions of the probe with the medium. These properties are realistic for many mediums: a photon in a plasma, for example, has a finite extent between its emission and absorption, and a well-defined momentum since its wavelength is generally much

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smaller than the photon mean free path. As a consequence of these properties, the effect of a collision of the object which interacts locally with the probe is an irreversible (approximate) localization near the probe mean position Rj( t) -t L, and a momentum increase hk?h/L, equal to the one lost by the scattered probe. L, determines the accuracy to which the position is “measured” at each collision. The model is devised in order to include these features in a schematic way, so that the model remains tractable. Let ‘1:be the average time between two successive collisions. The value of Y(t) is considered only at the discrete times 0, r, . ... nr, ... and YE denotes Y( t= nr). We consider the evolution from (n- 1 )r to nr. It is due to free motion and to collision. We assume that there is exactly one irreversible’ collision per time r, so that

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($ vwdi2)=1,

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(10)

for every n. The average is over a statistical ensemble of environments. Equations (7 )- ( 10) are the main equations of the model. It involves three numerical parameters L,, Ak and r, and will be referred to as the ‘
5: C Y&n)

,

(11)

(7) where H, is the object free Hamiltonian and &(n) describes the effect of a collision with i at time nr-. Its effect is to localize S near the probe position Rj(n), to increase the momentum of S by about Ak and to induce correlatively in irreversible change of the environment state. Thus

Qk(n)=

Jdxe’“ix>(xlS,(i,

k,n),

(8)

where S,(i, k, n) is an operator for G which describes the irreversible transform of YE- i due to one collision with S at position x. The wanted features are obtained if S, is assumed to satisfy the identity

(9)

for any Y. Here 8i,(x)=B(x-R,(n)) is a function related to the probability for S at x to interact with probe i at time nz. Its extent should be about L,. R,(n) is the probe wavefunction center. din(x) is calculable in principle if a complete description of F is available. Here the Gaussian form e(x) = exp( -x2/2Lz) is assumed for simplicity. The model must be completed by some information about the possible values of k and Ri( n). k is assumed to be a Gaussian random variable, with (k) = 0 and ( kz) = Ak’. The Ri( n) are statistically distributed so that 252

i,k

(12)

t/z>

for n = 1. I and K are respectively the n-dimensional index (i i, .. .. i,) and (k, ,..., k,). K,,=k,+... + k,. The product is over iE{fl and kE{K). S1 is a product of n Gaussian factors 8,,(x), in {I}, with t9,,(x)=B(x-R,(m)) and is of the form OI(x) +exp[ -n(x-R1)‘/2L:]. Most of the 0, are very small. Condition ( 10) implies ( Cr [e,(x) I’) = 1, so that (7 e~(x)o,(x’))=exp[-n(x-x’)Zi2L:l.

( ‘VIS$ (i’, k’, n’)S,( i, k, n) I Y) =s,i,skk,s,,,eilz(x)ejn(X’)

x n Wi, k,n) ao,

(13)

The corresponding density matrix is now calculated from (12) and (13). We get ( CIKKI(x)OI(x’) xeX’fx)f*(x’)), and, using (exp(iK,y)) = exp( - fnAk2y2),

(14) with y=x-x’ and n=t/r. Equation (14) reduces to the Joos-Zeh result p. exp ( - y2/2Lz) [ 61 when the object is well localized, i.e. L,(p) <
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4. Discussion (I) The model is first applied to discuss the localization process for a system initially in an extended quantum state p,, =f( x)f * (x’); the initial characteristic lengths are lo -LX(po)%ffo, with Lo very large. We consider first the caseLo > L, and next the opposite case. 1, and Lx calculated for the RDM ( 14) are respectively I&)

= (LF~

+ Ak’) -1’2(r/t)“2,

LAP,) =L(Po)

(15)

=-Le.

(16)

The coherence width becomes smaller than L, after a very short time r, irrespective of its initial value, and decreases steadily with time, in accordance with the models [ 3-61. On the other hand the localization length L,(p,) is seen to remain equal to its (large) initial value: the RDM pn is clearly unable to represent the expected transition to classical behavior. The same comment is appropriate for the numerical results obtained with the help of the solvable models in refs. [ 3,4,6]. The reduction rule A is now applied. Thanks to assumption (9) it is easily shown that the strong decoherence conditions of definition A are satisfied for the !&(n) defined by (12). The object state is thus described as a mixture of improper mixtures PIK(n)=Tr,Ily,(n))(~~(n)I,

xexp[iK,(x-x’)]f(x)f*(x’)

.

(17)

The corresponding physical localization found to be (see (4)) L*(t,

t)=Lx(pl~)=L,(7/t)“2

(l>7)

.

length is

(18)

Of course L* (0) = Lo > L,. There is a sudden localization to the value L, after exactly one decoherence time 7. This value is determined by the spatial structure of the environment, as suggested by Simonius [ 11. On the other hand L*( t, t) decreases as l/t1j2 and the localization length may become much smaller than L,. The coherence width L(t) (for p,) and the true localization length L*(t) = L* (t, t) are related as lx(t)=L*(t)/(l+Ak2L,Z)1’2

(19)

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and lx(t) is always smaller than L*(t). When Ak2Lz >> 1 we have I, -=xL* and the coherence width does not give any useful information upon the localization L* of the object. The reduced density matrix then describes a mixture of well-localized states, each of which with a momentum K,, = k, + . ..+ k,, with ( K2) = nAk2. The physical picture is one of a fairly localized particle experiencing Brownian motion. This illustrates clearly that decoherence of the RDM is not a sufficient condition for classical behavior. Case Lo << L,. Result ( 17 ) is specialized to the case L,=co. The collapsed mixture is now bK( n)},, with pK(n) =exp(iK,y)f(x)f*(x’), y=x-x’. The physical localization length is L*( t, t) = L,,, and is not changed by the environment. The coherence width is estimated as (20) and becomes much smaller than L,, for t sufficiently large [ 6 1. The effect of the environment is only to induce decoherence without localization. The decoherence length describes essentially the environment-induced velocity dispersion. This dispersion prevents the observation of interference effects, but this does not mean that the behavior is classical. (II) Classical behavior. A precise criterion for classical behavior was given above ( 5 ). We consider now the transition to classical behavior from a nonlocalized initial (t =O) state. This occurs for example when a high energy charged particle is produced in an s state through a radioactive decay and hits a ionizable medium, where its track is detected. The “L,, Ak, 7” model explains schematically this transition. Let the initial wavefunction be f(x) N 1. L,(f) = Lo >>L, is assumed. The localization length L*( t, t) has been calculated above as L,( z/t) li2. Let to be the smallest time such that L*( to, to) < $8. A possible collapsed density matrix is then pIK( to) ( 17). Its motion is determined by eq. (7) which gives at time t> to At; to) =exp( - Xv’)

xexp[iK(to)Y181(x)81(x’),

(21)

where i=Ak2(t-to)/7 and y=x-x’. The localization length, which takes into account the possible re253

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ductions up to time t,,, is now L*( t, to) = L[p(t; to)]=L,(z/to)*/’ (the width of O,(x) at to) and remains smaller than 9’. Moreover, the future position can be predicted with certainty to remain in the region where @r(x) is significant. The behavior is thus proved to be classical. Another result is that the duration of the transition may be quite long if L, > 9, and depends strongly upon the spatial structure of 8: this time is of the order of (L,/9)2z. On the other hand I,(t) decreases with increasing t: the velocity uncertainty increases and the future behavior is made less predictable. The effect of the environment is thus, intuitively, (i) to (inaccurately) “measure” repeatedly the approximate object position and (ii) to perturb dynamically its motion, in a noncontrollable way, by repeated small random momentum transfers. It is worth noting that an environment is not necessary to classical behavior if it is already established (Lo< Y), in agreement with an analysis of Halliwell [ 13 1.

5. Conclusion The behavior of a simple object S, coupled to an environment b, is conditioned by two characteristic lengths, the coherence width l(p) and the localization length L(p), with Z
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tificial but is hoped to include some features of a real environment and how it interacts with the object

C1,61. Transition to classical behavior is obtained when L, is not too large: the time required for the transition Lo+L*< 9 is T*= (L,/9)2z. When the environment is continuous (L, = co ), T * is infinite and there is decoherence (Z(t) -+O) without localization: the interference effects are suppressed but the behavior is not classical. Another result is that L*( t, to)/Z( t) = ( 1+ Ak2L2)“2 may be much larger than 1, if AkL, > 1, and it is then not true that S settles to a Gaussian coherent state (6x-A/Gp), as assumed by Unruh and Zurek [ 41. The motion of S is rather described by a mixture of states with a spread 6x much larger than the de Broglie wavelength. We comment briefly the solvable Gaussian models of refs. [ 3-5 1. Decoherence (for the density matrix) is rigorously demonstrated, but not the tendency to localization. There is no L, length in these models since the range of the interaction Hamiltonian is infinite. Whether these environments are able to localize an initially extended state or not is an open problem. References [ 1 ] M. Simonius, Phys. Rev. Lett. 40 (1978) 980. [2] W.H. Zurek, Phys. Rev. D 26 (1982) 1862. [3] A.O. Caldeira and A.J. Le’ggett, Phys. Rev. A 31 (1985) 1059. [4] W.G. Unruh and W.H. Zurek, Phys. Rev. D 21 (1989) 1698. [ 5 ] J.P. Paz, S. Habib and W.H. Zurek, Phys. Rev. D 47 (1993) 488. [6] E. Joos and H.D. Zeh, Z. Phys. B 59 (1985) 223. [7] R. Omnts, Ann. Phys. 201 (1990) 354. [8] R. Omnes, Rev. Mod. Phys. 64 (1992) 339. [9] W.H. Zurek, New techniques and ideas in quantum measurement theory, ed. D.M. Greenberger (New York Academy of Sciences, New York, 1986) p. 86. [lo] B. d’Espagnat, Found. Phys. 20 (1990) 1147. [ 1 l] J.S. Bell, Helv. Phys. Acta 48 (1975) 93. [ 121 G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34 (1986) 470. [ 131 J.J. Halliwell, Phys. Rev. D 46 (1992) 1610.