Spontaneous localization, environment-induced decoherence and individual-system observations

a& ._-

18 December

1995

‘53

PHYSICS

LETTERS

A

@ Physics Letters A 209 (1995) 129-136

ELSETVIER

Spontaneous localization, environment-induced decoherence and individual-system observations Luca Bonci a, Paolo Grigolini a,b,c, Giuseppina

Morabito a, Luca Tessieri a, David Vitali d

a Dipartimento di Fisica, Universitd di Piss, Piazza Torricelli 2, 56100 Pisa, Italy b Istituto di Biofisica del Consiglio Nazionalc delle Ricerche, Via San Lorenzo 28, 56127 Pisa, Italy ’ Department of Physics, University @North Texas, P.O. Box 5638, Denton, 7X 76203, USA * Dipartimenta di Fisica, Universitci di Camerina, Via Madonna de& Carceri, 62032 Camerino, Italy Received

1 September

1995; revised manuscript received 23 October 1995; accepted for publication 26 October I995 Communicated by J.P. Vigier

Abstract We show that the modified version of quantum mechanics proposed by Ghirardi et al. [Phys. Rev. A 40 ( 1990) 781 is virtually equivalent to ordinary quantum mechanics, if a statistical perspective is adopted. However, this does not conflict with the possibility of an experimental assessment via individual-system observations.

The recent achievements in the field of optical quantum jumps [ I] challenge the traditional statistical picture of quantum mechanics and suggest the adoption of an individual-system picture. A notable example of this new trend is given in the works of Gisin [ 21 and Gisin and Percival [ 31. These authors proposed an individual-system picture, namely a nonlinear and stochastic Schrijdinger equation, driving the evolution of the state vector of a system interacting with its environment. This theory, called quantum state diffusion (QSD) theory, rests on the assumption that the wave function describes the actua1 state of the open system and that the process of environment-induced decoherence [ 41 is equivalent to an environment-induced process of wave function collapse. More precisely, the QSD theory affords an intuitive illustration of the decoherent histories interpretation of quantum mechanics [5] in which, roughly speaking, an event occurs when the density matrix satisfies the decoherence condition. In fact a given solution of the QSD stochastic

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Schrodinger equation is nothing but a decoherent history of the open system [ 61. This theory is now becoming more and more popular as a theoretical tool to describe the processes of optical quantum jumps [ 71. Nevertheless, in our opinion, this theory is not totally satisfactory. According to the QSD prescriptions, the individual-system representation is explicitly constructed in such a way as to be equivalent, when the average on a Gibbs ensemble is made, to the Markovian master equation describing the evolution of the density matrix of the open system, a reduced density matrix. However, a decoherence process resulting from the contraction over the “irrelevant” degrees of freedom cannot be genuinely irreversible, as a genuine wave function collapse would be, since the Schrodinger equation leading the time evolution of system plus environment is reversible. Consequently, the wave function collapse is as “apparent” as the corresponding “irreversibility”. Furthermore, if we study the time evolution of the wave function of the whole

130

L. Bonci et al./ Physics Letters A 209 (1995) 129-136

universe (open system and its environment), we see [ 81 that the decoherence of the system of interest corresponds to establishing quantum correlations between the system of interest and the external world. We think that moving from the global to the reduced representation, this process of growing quantum correlations, or a portion of it, might be perceived as an environment-induced enhancement of the quantum fluctuation of the system of interest, with no conflict with the constraint of statistical equivalence with the Markovian master equation. It is known [ 9,101 that the individual-system picture’s exact equivalence to the statistical prescriptions of ordinary quantum mechanics is not uniquely determined, and that linear rather than nonlinear stochastic Schrodinger equations can be derived so as to fit these statistical constraints without resulting in wave function collapses [ 101. The arbitrary character of the choice of the structure of the stochastic equation of motion for the state vector makes it difficult to establish if the interaction with the environment yields either stochastic phase fluctuations or genuine wave function collapses, or both. Consequently the constraint of exact statistical equivalence with the master equation for the density matrix does not define without ambiguity the rate of wave function collapses but only sets an upper limit to their values. In fact the collapse rate cannot be greater than the decoherence rate predicted by the master equation. Within the context of the individual-system pictures another one is emerging, closely related to an approach originally proposed from a statistical perspective [ 111 and usually referred to as continuous spontaneous localization (CSL) theory [ 121. According to this picture, the dynamics of the matter constituents is already affected by “objective” processes of collapse, with a resulting “actual” rather than “apparent” irreversibility. The processes of wave function collapses, albeit almost silent at the microscopic level of isolated systems, might be triggered and enhanced when dealing with the dynamics of macroscopic objects [ 121. However, even when the macroscopic or mesoscopic level is explored [ 131, the detection of the CSL effects is made difficult by the fact that the statistical effects produced by environmental decoherence often predominate over those of the spontaneous collapses. It has been shown [ 141 that the fluctuation-dissipation processes are only very slightly modified by the adoption

of the new physics. This sets the CSL theory on the same firm ground as the QSD theory, that of the statistical equivalence with ordinary quantum mechanics, albeit as a result of an excellent approximation [ 141 rather than as an exact property. The price to pay for this equivalence is, of course, the virtual impossibility of an experimental assessment of the CSL effects if the experimental observation is the result of an average over many identical systems of a Gibbs ensemble. It has been also observed, however, that in spite of this approximate statistical equivalence with ordinary quantum statistical mechanics, the frequency of spontaneous collapses is enhanced by the interaction between the system of interest and its environment [ 151. The case studied in Ref. [ 151 is the same as that discussed by Zurek [ 41, and refers to the decoherence process between two distinct spatial components of the same wave function, which in turn represents the state of a dissipative harmonic oscillator. Although dissipation enhances significantly the process of spontaneous collapses, for plausible values of the distance between the two distinct components the spontaneous collapse of this linear superposition would occur at a time larger than that necessary for the two components to merge into one another by the process of longitudinal relaxation [ 151. In this paper we plan to discuss a model where the effect of the environmentinduced enhancement of the spontaneous collapses, if an individual-system picture is adopted, shows up at the time scale that plausible physical arguments [ 161 assign to the wave function collapses in the case of the optical quantum jumps [ 11. Furthermore we shall study the joint action of this mechanism of collapse enhancement and that resulting from the macroscopic nature of the pointer, namely the process of enhancement pointed out in the original work of Ref. [ 111. The purpose of this paper is twofold: (i) We plan to show that if the CSL perspective is adopted the processes of wave function collapses are defined with no ambiguity. The interaction between the system to measure and its environment has the twofold effect of enhancing the quantum fluctuations of the system to measure and of accelerating the occurrence of real collapses. We want to define a criterion to assess whether an environment-induced enhancement of quantum fluctuations or an environment-induced spontaneous wave function collapse occurs. The joint action of these distinct processes is expressed with no

L. Bonci et al./Physics

131

Letters A 209 (1995) 129-136

ambiguity by a stochastic Schrodinger equation (see Eq. (13) below). (ii) We plan to show that the CSL processes, enhanced by the environmental interaction, or by the macroscopic nature of the detector, although still invisible to a statistical treatment, can be observed by means of individual-system observations. The rate of environment-induced decoherence is larger than the enhanced CSL rate by many orders of magnitude. This should make it possible, in principle, to assess which one is the correct prediction. The model we use to illustrate these ideas is described by the following Hamiltonian,

The former mechanism concerns the center of gravity of macroscopic rigid bodies, discussed by the authors of the CSL theory [ 11 ,I 21. This process is accounted for by us through the increased localization rate Aa N,h, where N, is the number of constituents of the pointer and A = lo-l6 s-r is the localization rate of the CSL theory. The latter mechanism is the frictioninduced process of accumulation recently revealed by the analysis of the authors of Ref. [ 151. This analysis proves that the interaction of the pointer with its bath is a channel to collect the spontaneous collapses with mean rate A; = A (i = 1, . . . , N) affecting the bath particles as well as a channel to dissipate pointer’s en-

k=G&-,&+&n,

ergy. The time evolution

(1)

of a generic

operator

R(t)

within the CSL theory is given by R(t) = (&, + LCSL) A(t), where .&,A = (i/Fi)[fi,d] and

This means a quantum system of two degenerate states j +) and I-), the eigenstates of the Pauli operator 6;) interacting with the coordinate $ of a pointer oscillator. We place the i-spin system in a linear superposition of the eigenstates j +) and I-), and the measurement process is realized, through the coupling between the quantum system to measure and the measurement apparatus, via either an environment-induced decoherence [ 451, or a real collapse [ 11,121. The measurement apparatus is given by the Hamiltonian &n which consists of the macroscopic oscillator, playing the role of a pointer, coupled to its environment, a set of independent harmonic oscillators, through an ohmic spectral density [ 17 J . Following Pathria [ 181 we depict the environmental influence as caused by the interaction between the “pointer” oscillator and a set of distinguishable bath oscillators, thereby making the CSL theory equivalent to the original theory of Ref. [ 111, which we indeed use to define the parameters of the processes of spontaneous localization. Notice that the environment of the pointer, whose influence is the crucial ingredient of the environmental decoherence theories [ 41, also plays an important role when the CSL theory is applied, since spontaneous localizations act on both the microscopic constituents of the pointer and the bath oscillators resulting in two distinct processes of enhanced collapses.

X AeXp[-ia(Gj

-

I

X)2] - A ,

(3)

l/fi = 10d5 cm being the localization length of the CSL theory. The exact time evolution of the mean values of the three i-spin operators starting from the factorized initial condition br(O) = iTO) ~exp(-PEis>/TrB[exp(-pA~>l, where@(O) is an arbitrary initial state of the $-spin system, can be obtained in a straightforward way. Within this class of initial conditions, the contracted i-spin dynamics is exactly reproduced by the following master equation, b(t)

=-[z(t)

+g(f)l[@(f)

-*;p(t)*rl,

(4)

where

(5) and g(t)

= e/&j1 i=a

- exp[--cur?(s)]}.

(6)

s(t) is the symmetrized correlation function of the free motion of the damped oscillator’s coordinate b,

132

L. Bonci et al./Physics

whose general expression is given in Ref. [ 191. The functions ri( t) are defined by

Letters A 209 (1995) 129-136

where B,“(X) = D;(X) -

r-o(t) = -G

ds x(s)

(7)

and

(filfji(~>l+l>5

(10)

312

B,i(X)

=

(> 2

expl -$x(~~

- x)*]

(11)

and ds’sin[tii(s’0

dBi(x)

s)]x(s’),

dBi( X) dBj (2)

0

i Z 0,

(8)

where x(t) is the susceptibility of the damped harmonic oscillator [ 191. The master equation (4) is characterized by a time-dependent rate which is the sum of two distinct contributions. The former, z.(t), describes the environment-induced decoherence and corresponds to the prescriptions of ordinary quantum mechanics. The latter, g(t), represents the influence of the CSL processes and must be regarded as a true correction to ordinary quantum mechanics. However, for any reasonable choice of parameters, it turns out that g(t) < z(t). thereby implying that this new theory does not produce significant corrections to the statistical predictions of ordinary quantum mechanics. This has the twofold and conflicting effect of setting the CSL theory on a firm ground and of leading to the pessimistic conclusion that no experimental check of the CSL processes is possible. It would be essentially so if only statistical treatments were possible. This changes radically if we adopt an individualsystem perspective. How to make an individual-system picture equivalent to (4) practicable? The direct use of the CSL prescription would lead to the following stochastic equation for the state vector of the whole system [ 121,

-

= 0,

[$ (2) -“*;7 dx [d”(x),*dr]

I$,>,

--DC) (9)

= SijS( X - x’) dt.

(12)

The conventional averaging over the quantum state I$) is denoted by ( ) while the ensemble averaging is denoted by the bar. The stochastic equation (9) makes it practically impossible to carry out any theoretical prediction with analytical expressions. However, we notice that writing an effective Schriidinger equation statistically equivalent to the exact result (4) is made easy by remarking that the rate g(t) corresponds to objective collapses while the rate z (t) cannot be mistaken for that of a collapse and rather it corresponds to the blurring caused by environment-induced quantum fluctuations. Thus it is plausible that the process corresponding to (6), and implying the action of spontaneous collapses, is given the fundamental nonlinear structure prescribed by the QSD theory. On the contrary, the traditional decoherence process, corresponding to (5)) with no collapses behind it, can be described by the linear stochastic term adopted by Spiller (see I$. (5) of Ref. [ lo] ) . We are thus led to the much simpler stochastic equation for the i-spin state vector I#), dl$) = [m@d5-

- [im%,dr+

$g(W~)*dW) $(t)dtljlCI),

(13)

where &$ is defined as in ( lo), d,.$is a complex Wiener process and dr an independent real Wiener process, such - that d$ = @ = 0, dcdt* = dt, ;i; = d
L. Bonci et al./Physics

important condition of being statistically equivalent to (4). Let us now consider some quantitative estimates of the environmental decoherence rate l/to and of the collapse rate 1/tC. For the pointer oscillator, with the damping rate r, we assume: M = 1 g, T = 100 K, 0-r = 10 s-i, G/Ma* = 1 cm, ha = lo7 s-i. By adopting the small time approximation z(t) M 2G2kBTtlh4f2*1i2, we get I _=_ tD

G -ksT im d- A4 ’

(14)

implying l/to N lo*’ s-i, which is a remarkably large rate of decoherence. As to the collapse rate, we split Eq. (6) into two independent contributions, namely, the term corresponding to i = 0 and that given by the sum from i = 1 to N. The former contribution refers to the processes of spontaneous localization directly affecting the pointer, and it is denoted by go(t) . The latter, indicated by gs ( t) , refers to the process of accumulation of spontaneous collapses on the bath of oscillators. Adopting the small time approximation, one derives for the first contribution

(15) which is essentially the time that it takes the pointer to move a distance l/G. The determination of the rate constant corresponding to the latter contribution is more involved. We consider only the lowest order contributions in the coupling strengths Ei to the ri and, as usual, we assume that the spectral density of the thermal oscillators is continuous and ohmic [ 17 1. Then, in the small time condition Tt << 1, we get

(16) where m is the mass of the thermal oscillators. this approximated equality we derive 1 -z

tc

From

Letters A 209 (199.5) 129-136

133

this order of magnitude being in the latter case scarcely dependent on whether we identify m with the electron or the proton mass. We thus see that the resulting CSL rate is still much smaller than l/to and this suggests that if an experimental setup corresponding to model ( 1) existed, it would be easy in principle to discriminate between our proposal ( 13) based on the CSL theory and the QSD theory, by performing an appropriate individual-system observation. This would be so because according to the QSD approach, the collapse to an eigenstate of uZ would occur, in a given experimental run, in a time of the order of tD, while according to the theory illustrated in this paper, it would occur with the much more extended time scale tc. The distinction between the environment-induced process of enhancement of spontaneous wave function collapses and the environment-induced enhancement of quantum fluctuations is pointed out by means of the numerical results illustrated in Figs. l-3, which refer to the numerical solution of ( 13). Fig. 1 shows some trajectories of (&-,(t)), corresponding to the initial condition (ah.(O)) = 1, collapsing into the eigenstates I+) and I-) of the measured observable cZr with the rate 1/tC calculated above, and with no significant dependence on the environmental decoherence. Nevertheless, as shown in Fig. 2, the effect of the environmental decoherence becomes predominant if we look at the (i?+(t)) component of a single trajectory. Now it is the turn for the CSL process to result in negligible effects. To further stress the ambiguity of a statistical treatment we compare in Fig. 3 the average of (a,(t)) over 500 trajectories with the corresponding mean value over the reduced density matrix driven by Eq. (4), both denoted by (a,(t)) to stress with the bar their common statistical meaning. The analytical prediction derived from Eq. (4) is obtained using the short-time approximation which yields ___

____

bAt>) = (~x(0))ew M exp

( ’ - 2

J 0

dT [z(.r> + d~)l) (18)

(17)

With the same choice of parameters as that leading to 102’ s-i, we obtain 1/tC N lo4 s-i for both l/tD the direct process ( 15) and the indirect process ( 17),

It is evident from Fig. 2 that the decay of (a,(t)) occurs in the time tD given by Eq. ( 14). The effect of the environment is statistically equivalent to that of a real collapse, but it occurs with a rate seventeen orders of

L. Bonci et d/Physics

134

-l.O0

200

loo

300 t (set)

400

560

I 6&10d

1.0

0.5

A

0.0

9

-0.5 -1

n 0

I

I

2

4

I

I

6

8

t (==)

Fig, 2. A single stochastic trajectory of (v.~( t)), showing the details of large fluctuations in the short time scale tD of Eq. ( 14). The values of the parameters are the same as in Fig. 1.

larger than the rate of the genuine collapse. We want to point out that in future experiments that might be made to assess whether the correct physical perspective is given by the the QSD theory or by the CSL theory, the fans of the QSD might possibly argue that also the process expressed by the first term on the r.h.s. of ( 13) might derive from some kind of interaction with the environment. However, we believe that it is difficult to derive from the current decoherence theories processes as slow as those stemming from the CSL theory. Furthermore, we must remark again that only the CSL theory ensures the genuine irreversibility of both contributions to ( 13). It is worth remarking that, with the choice of pamagnitude

-l.o’o”’ 0 1

2

3

4

5

t (set)

Fim _f 1. Time evolution of (frz (t)) starting from the state with cr, = I. We show three different trajectories, all collapsing into one of the two eigenstates in a time of the order of the CSL collapse time tc. The environmental decoherence has no effect in this case. The values of the parameter are M = I g, m = IO-” g, T = 100 K, 0=9s-I, I’= lOs-t, G=81 cmg sP2, Ac=107 s-t.

s

Letters A 209 (1995) 129-136

Fig. 3. Time evolution of the ensemble average (rrr (t)). The result of an average over 500 trajectories, compared with the analytical prediction of Eq. ( 18) (dashed line). The values of the parameter are the same as in Fig. 1.

rameters adopted for our model, the collapse time tC is compatible with that of the existing experiments on optical quantum jumps. Note first of all that ( 1) mimics some aspects of the three-state scheme currently adopted to describe the quantum jumps of a single ion in a trap [ 11. In these experiments, by means of two radiation fields, the ground state IO) is weakly coupled to a metastable state 12) and closely to the unstable state 11) whose life-time is of the order of 1O-* s, so that the Hilbert space spanned by IO) and 11) can be regarded as being a single state I+) [ 161. This state interacts with the photodetector, mimicked by the damped oscillator of (I), through the radiation emitted by the spontaneous decay of state II). State 12) can be identified with state I-) of our two-state model. It must be pointed out that from the literature on the experiment on optical quantum jumps, we cannot derive a clear indication on the value of the collapse time tc. However, we can assess its range of values with a plausible conjecture. As an effect of the interaction with the measurement apparatus initially placed in state I_!?,,), the $-spin system must change from an initial condition (c+ I+) + c- I-)) IEo) into the entangled condition c+ I+) IE+) + c_ I-) (E_), with ]E+) denoting the state with the photodetector having absorbed a photon and amplified it into a macroscopic signal, and IE_) coinciding with the rest state IEo). This entanglement is a necessary condition for the collapse to occur, since an “observation” of one of the two states jE&) yields a collapse of the i-spin state. Therefore, the collapse time tC cannot be shorter than the time it

L. Bonci et al./Physics

takes the two-level system to become entangled with the states I&-), and this, in turn, cannot be shorter than the time it takes the fluorescent state 11) to emit a photon, thereby setting the condition tc > lOAs s. The upper limit for tc can be derived from the time duration of the optical pulses used by Itano et al. [ 201 for their experimental realization of the Zeno effect, i.e., 1.4 x lop3 s, thereby leading to the time range 10V3 > tc > 10P8 s, a condition compatible with the CSL predictions for our model. Unfortunately, we cannot yet use these arguments to rule out the QSD perspective. Neither can we try to convince with indisputable arguments the community of physicists to adopt the CSL perspective. Let us explain why. We must stress that the model discussed in this paper does not take properly into account the important ingredient of the interaction between the system of interest and the radiation field. The interaction between the state II) and the photon vacuum is here imagined, in a sense, as contributing to the interaction between the system of interest and the dissipative oscillator (considered as playing the role of the photodetector in the experiments of optical quantum jumps [ 1 ] ). The weak radiation field coupling between the state IO) and the state 12) is, on the contrary, totally missing and for this reason we cannot reproduce the Zeno-like [ 211 nature of the real experiments. It has to be pointed out indeed that the real experiments, rather than providing a direct measurement of tc, show a stochastic sequence of light and darkness with time distributions which depend indirectly on the wave function collapse time. For this reason, a more conclusive verdict on which the correct theory is, would only be reached by adopting a model closer to the Zeno-like nature of the current experiments, and if the stochastic sequence of light and darkness, the experimental observation, were proved to be sensitive to the widely different values that, as suggested by this paper, the QSD theory and the CSL theory are expected to assign to the time scale of the detector-induced collapses. The current theoretical treatments on optical quantum jumps focus on the dynamics of the system to measure, and we have essentially adopted the same perspective, as illustrated by the reduced equation of motion (4). It would have been more convenient to derive an equation of motion for the pointer oscillator. Notice, however, that the simplified nature of the

Letters A 209 (1995) 129-136

135

model makes it possible to reach unambiguous conclusions on the value of the collapse times: there is no essential difference between the collapse time of the system of interest and the collapse time of the detector, since both imply a collapse of the same entangled form

c+j+)lE+)

+ c-I-)lE-).

In spite of all these yet unsolved difficulties, which require further research work, we think that this paper has some merits: we derive the exact solution of model ( 1) under the action of the CSL collapses and the stochastic Schriidinger equation ( 13) which nicely shows the joint action of the environment-induced enhancement of quantum fluctuations and the environment-induced enhancement of the CSL collapses. Furthermore, the fact that the QSD and the CSL predictions on the collapse time are so widely different ( 17 orders of magnitude!) is a significant property even if it cannot yet be directly assessed with the current experimental quantum jumps techniques.

References [ 1 ] H. Dchmelt, Rev. Mod. Phys. 62 ( 1990) 525. [ 21 N. &in, Phys. Rev. Lett. 52 ( 1984) 1657; Helv. Phys. Acta 62 (1989) 363. [31 N. Gisin and I.C. Percival, J. Phys. A 25 (1992) 5677; 26 (1993) 2233,2245; Phys. L&t. A 167 (1992) 315. r41 E. Joos and H.D. Zeh, 2. Phys. B 59 ( 1985) 223; W.H. Zurek, Prog. Theor. Phys. 89 ( 1993) 281. [51 M. Gell-Mann and J.B. Hartle, in: Santa Fe Institute Studies in the science of complexity, No. 8. Complexity, entropy and the physics of information, ed. W.H. Zurck (AddisonWesley, Reading, MA, 1991); R. Omnks, Rev. Mod. Phys. 64 ( 1992) 339. [61 L. D&i, N. Gisin, J. Halliwell and I.C. Percival, Phys. Rev. Lett. 74 (1995) 203. [71 N. Gisin, P.L. Knight, I.C. Percival, R.C. Thompson and D.C. Wilson, J. Mod. Opt. 40 ( 1992) 1663; B.M. Garraway and P.L. Knight, Phys. Rev. A 49 (1994) 1266. Quantum mechanical irreversibility and 181 P. Grigolini, measurements (World Scientific, Singapore, 1993) p. 103. [91 H.M. Wiseman and G.J. Milbum, Phys. Rev. A 47 (1993) 1652. IlO1 T.P. Spiller, Phys. I&t. A 192 ( 1994) 163. 1111 G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34 ( 1986) 470. [ 121 F! Pearle, Phys. Rev. A 39 ( 1989) 2277; G.C. Ghirardi, P. Pearle and A. Rimini, Phys. Rev. A 42 ( 1990) 78; P. Pearle and E. Squires, Phys. Rev. I&t. 73 ( 1994) 1. [ 131 A.I.M. Rae, J. Phys. A 23 (1990) L57.

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[ 141 D. Vitali, L. Tessieri and P. Grigolini, ( 1994) 967. [ 151 L. Tessieri, D. Vitali and P Grigolini, ( 1995) 4404. [ 161 D.T. Pegg and P.L. Knight, Phys. Rev. A 1171 A.O. Caldeira and A.J. Leggett, Ann. ( 1983) 274.

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[ 181 R.K. Pathria, Statistical mechanics (Pergamon, Oxford, 1972) p. 75. [ 191 H. Grabert and U. Weiss, Z. Phys. B 55 (1984) 87. [20] W.M. Itano, D.J. Heinzen, J.J. Boilinger and D.J. Wineland, Phys. Rev. A 41 ( 1990) 2295. [21] R.J. Cook, Progress in optics XXVIII, ed. E. Wolf (Elsevier, Amsterdam, 1990).