The adiabatic monitoring of quantum objects

PhysicsLettersAl7O(1992)273—279 North-Holland

PHYSICS LETTERS A

The adiabatic monitoring of quantum objects T.P. Spiller, T.D. Clark, R.J. Prance and H. Prance School ofMathematical and Physical Sciences, Physics andAstronomy Division, Universityof Sussex, Falmer, Brighton, Sussex BNI 9QH, UK Received 26 November 1991; revised manuscript received 19 August 1992; accepted for publication 8 September 1992 Communicated by J.P. Vigier

We discuss the behaviour of a coupled system, consisting of a macroscopic classical system and a single quantum object. We argue that, forthe case of a macroscopic quantum object, detectable modifications to an adiabatic classical system’s motion can occur.

In this Letter we discuss a novel approach for monitoring the behaviour of a quantum mechanical object. This involves following the behaviour of a classical system, which is coupled adiabatically to the quantum object of interest. We restrict ourselves to “macroscopic” objects, and give operational definitions of this term in the classical and quantum contexts. We also explain our choice of the term “monitor”, as opposed to “measure” or “observe”. We work in the non-relativistic Schrodinger picture. Our starting point is the quantum object of interest (which has a coordinate x, momentum p = mi, and is described by a potential V(x)) coupled linearly to a single oscillator (coordinate X~, momentum Pa=MXa, and potential 1MQ2X~).The Hamiltonian is

number K it is only valid to take the ground state (K= 0) and some finite number of the excited states (K> 0) in this subset. We assume that, for some nontrivial subset of eigenstates, the adiabatic theorem [1,2] holds. That is, we assume the eigenfrequency differences are greatly in excess of the oscillator frequency Q, or any frequencies of variation in the applied sources F and J The solution to the time dependent Schrodinger equation with the Hamiltonian (1) can then be approximated by

p2a 22 P2 H=~+~MQXaFXQ—/LXQx+~

and f~~f+ pXQ and x~u,~satisfy / ~2~2 \

+V(x)—f.x.

W(XQ,x,t)~x(XQ,1)uK(x)UK.

(2)

Here /

.

U,~ exp~ ~ —

j dt’ EK(JOt)

(1)

The objects are separately coupled to the external driving sources F( t) and f( t), and p is the mutual coupling parameter. Our aim here isto show how the quantum object, whilst remaining in an energy eigenstate, reactively modifies the behaviour of the object to which it is coupled. This may in fact only occur for a subset of the eigenstates of the quantum object (labelled as uK(x) for ic=0, 1, 2, ...); for example if the levels close up with increasing quantum

~ix(X~,1) = ih

(_

(3a)

,

v~+ ~

-~-~-

_jtot .x)u(x)

2m =EK(iO~)uK(x) (3b) and the X~derivatives of UK(x) are taken to be negligible [3, p. 77]. We make two further assumptions. These do not

0375-9601 /92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

273

Volume 170, number 4

PHYSICS LETTERS A

necessarily hold for all systems described by (1) and in the adiabatic limit; however if they do not hold our method simply does not work. (i) Firstly, we assume that the quantum object of interest has energy levels Ek(JO1), found by solving (3b), which depend non-trivially on ft~and K. Clearly we need some dependence on so that ~ does not commute with the bare oscillator Hamiltonian, and thus modifies the momentum in (3a). If the aim is to identify the state u,,, through this effect, then the modification to the momentum should not be identical for any two uk in our subset. (ii) Secondly, we assume that the uncertainty i~XQ is small enough so as not to hide the f50~dependence of EK(ftOt). In principle t~X~ follows from solving (3a). However if it is small so the coupling between the objects is weak we can estimate it from the ~=O uncertainty in XQ. To be precise, then, we assume that I Ith.XQ I << Ifo , where J~is the typical size (in the pararneterf101) of the feature(s) in EK(ftot) we seek to identify. We note that decreasing the coupling p for a given system reduces the effect of the uncertainty; however at the same time it reduces the size of the modification to the momentum in (3a). An example later will show that these two assumptions are not unphysically restrictive, Given that our assumption (ii) holds, we approximate XQ in the argument of EK (f05) by its expectation value. It then follows that this expectation value (evaluated for the solution of (3a), x(X0, t)) satisfies tl dt

r~2 — 2 +~~ —

(r—J4 ,, M\ ôf~J

y-f)

where now f~=1+ p < X0>. This result (4) illustrates the basic idea behind our approach. The differential equation which describes the behaviour of the oscillator is modified in a manner dependent upon the energy eigenstate of the quantum object uK(x). We note that (4) is also valid if the expectation value is for a mixed oscillator state (with each term in the mixture satisfying (3a)), provided that assumption (ii) still holds for the mixture so f,,,, can be replaced by f+ p < X~>. We now turn our attention to macroscopic objects. The aim of this work is to show how a quantum object can affect the detected motion of a classical sys274

9 November 1992

tern in a manner illustrated by eq. (4). However we believe that it is an oversimplification to merely assume that can be taken as the coordinate of a classical system, given the assumed role we require the classical system to play. (iii) We require the classical system to be describable by a coordinate X~1,which obeys a classical differential equation of motion, and whose trajectory can be monitored with some sort of apparatus. The degree of accuracy of this monitoring has to be sufficient to distinguish the classical paths which arise due to different states of the quantum object, but insufficient to significantly modify these paths. Clearly, the classical system’s role is an intrinsically irreversible one, and this irreversibility must be considered. We know that the appropriate quantum expectation values for simple isolated (microscopic) quanturn objects obey classical differential equations (see refs. [4] and [3, p. 41]), when the evolution is given by the relevant Schrodinger equation. However, if we are to use such an expectation value to represent the motion of the classical system [5] ~‘ then we must explicitly consider the irreversible monitoring of its path, in order to justify that it can play the role (iii) that we require. Models for describing this monitoring do exist [6—13].Clearly one cannot consider infinitely frequent quantum measurements, or the watched-pot theorem [14,15], the lack of change of the initial state, applies. The monitoring models therefore stop short of infinitely frequent measurements, and keep a finite interval between them [6— 12], or else consider infinitely frequent imprecise measurements [13]. Here, though, we choose to introduce the irreversibility in a different manner. We consider a macroscopic classical system. As Bohr stressed [16], the results of all experiments on quantum phenomena have to be expressed in classical terms. Here we restrict ourselves to a class of experiments which focus on the behaviour of a macroscopic classical systern coupled to a quantum object. We define a macroscopic classical system as one containing a great many quantum degrees of freedom, sufficient for the concept of a temperature to be meaningful. We as~ Landau and Lifshitz [5] give as example the path of an electron through a cloud chamber.

Volume 170, number 4

PHYSICS LETTERS A

sume that the system is at a sufficient temperature for its quantum energy level spectrum to be unresolvable. We also assume that the system is dissipative, and thus contains fluctuations [17—20]due to its finite temperature. The important point about such a macroscopic system is that the motion of its appropriately chosen coordinate X~ 1can be monitored with insignificant disturbance using some suitable apparatus, to an accuracy level given by the system’s intrinsic thermal noise. Here, then, the additional irreversibility by externally monitoring the trajectory introduced of X,~ 1 is negligible compared to that which is already there, intrinsic to the classical system. In defining our classical system as such we are clearly specifying the location of the “quantum—classical boundary”. Given the conventional definition of a quantum measurement, as “any process of interaction between classical and quantum objects” [5, p. 3], we obviously have to look very carefully at the nature of the adiabatic interaction between our classical system and our quantum object. This will be considered after our discussions of a model macroscopic classical system, macroscopic quantum objects, and a physical example of the coupled classical—quantum system. Given our definition of a macroscopic classical system, it is clear that in eq. (4) will not do as a classical coordinate X~1.We can, however, consider an infinite distribution of quantum oscillators [20] as a model macroscopic classical system. If we sum independent oscillators with a normalized frequency density W(Q) (where f~° cLQ W(Q) =1), and define the collective coordinate XT~f~° dQ W(Q)X0 (so the coupling and source terms now read UXTX and —F•XT respectively), we obtain

9 November 1992

2w~Q2

w(Q)=

(7)

which peaks sharply around WR (if QR>> 1), and choosing ~0R to be much smaller than any eigenfrequency differences for our subset of uK(x), gives a very simple model classical system. Defining as the classical coordinate X~1,it obeys the equation

~ +

R 1ci

+~

=

~

8EK\ M\(F it~—).

(8)

(6)

Our classical system is thus a lossy oscillator, with a resonant frequency cox, and an (assumed large, but finite) quality factor QR. In order to take f~= 1+ pX~1,our assumption (ii) needs modification. In = X~1,the expectation value for the bath of oscillators is implicitly for a finite temperature mixed state. (In the adiabatic limit we still factor this behaviour off from that of the quantum object state u~(x).)We require the uncertainty in X,~1for this state to satisfy I ,th.XC1 I <
We need to choose W(Q) to satisfy the adiabatic constraint. Taking the function

given on the right hand side of (8). We have deliberately restricted ourselves to macroscopic classical systems, which contain sufficient intrinsic irrevers-

—

=— ~Jdt’

G(t_t’)(F(t’)_it~5~ 8ftot/ (5)

for the analogue of (4). Here G( t— t’) is the Green function for the distribution ofoscillators, which has a Fourier transform -

~(w)

=

J 0

2—Q2+i~ 1 dQ W(Q) w

275

Volume 170, number 4

PHYSICS LETTERS A

ibility so as to avoid the need for a discussion of how X~1(t)is monitored. We regard this as a definition of a macroscopic classical system. However, whilst solving one problem this generates another. We clearly need the “force” in (8), ~uôEK/of,,,,, to be of sufficient size to generate a significant change in the motion of the macroscopic system, given by X~1(t).We believe that this “force”, if it is to be significant and to arise from a single quantum object, has to be due to a macroscopic quantum object. The term “macroscopic” has of course been criticized for its vagueness [26]. We have already given an operational definition of a macroscopic classical systern, which appears to be in accord with our everyday motion of much a system. So, given that definition, we operationally define a macroscopic quantum object as one which can produce a “force”, POEK/ OftOt, sufficient to modify the adiabatic motion of a macroscopic classical system. It is crucial that this “force” arises from just one object. To our knowledge, familiar (microscopic) quanturn objects such as atoms cannot play the role of a macroscopic quantum object as we have defined it. That is not to say experiments cannot be performed on a single atom, or ion. Indeed, single ions held in a Penning tap have been used to investigate, and verify [27], the watched-pot (or quantum Zeno) effect [14,15]. However, we are not aware of any experiments where a single atom (or ion) has demonstrated an effect on an adiabatic macroscopic classical system, such as that described by eq. (8). It is clearly tempting, for the case of an atom, to amplify the “force” POEK/Of,0, by considering an ensemble, a large number of identical atoms. This is fine for spectroscopic experiments; however it is not acceptable for our adiabatic type of experiment. In a real ensemble of atoms there are always interactions which lift the degeneracies of the single atom spectrum; spectral lines are always broadened. We can therefore term an ensemble of atoms a macroscopic quantum system. It possesses a very large number of energy levels, densely spaced in energy. Indeed, if it is at a sufficient temperature so that the discrete energy level structure, at the frequencies at which it is probed, is unresolvable, it can then be regarded as a macroscopic classical system. For example, we are all familiar with the macroscopic classical concepts of complex permittivity and —

—

—

276

9 November 1992

permeability applied to a gas of atoms. The dense packing of energy levels implies that, even at very low ensemble temperatures, the adiabatic constraint cannot be satisfied for any physically reasonable model macroscopic classical system (such as our lossy oscillator) which is coupled to the ensemble. The frequency W~ would have to be unreasonably small to yield adiabaticity. This is where the distinction between a macroscopic quantum system and a macroscopic quantum object emerges. In order to produce an “ensemblesized force”, —POEK/OfOj, but still maintain the adiabaticity, we require something containing a typical ensemble size number of microscopic quantum objects, but whose energy level spectrum is not everywhere dense. It should at least contain a window at the bottom of its level spectrum which contains a finite number of levels all spaced by gaps enabling the adiabatic limit to be utilized for reasonable values of WR and experimentally attainable temperatures. As long as one operates in this window, the system will satisfy our operational definition for the behaviour of a macroscopic quantum object. Such a single object can modify the equation of motion of a macroscopic classical system, as illustrated by eq. (8). In order to demonstrate that all of our constraints and assumptions are not unphysically restrictive, we turn now to an example. The need for a sparsely levelpopulated energy window in a system containing a very large number of microscopic quantum objects immediately suggests a superconducting, or possibly a superfluid, system [28]. In a BCS-superconductor, for example, there is a finite energy gap 2A at the base of the microscopic excitation spectrum; at least 2A is required to produce two (they are fermionic) quasiparticle excitations [29] ~2• In the case at hand, then, what is required is a superconducting system which contains just a few energy levels in this window which is devoid of microscopic excitations. An example of such a system is a superconducting weak link ring, which consists of a thick (compared to the magnetic penetration depth [29, p. 61) superconducting ring incorporating some form of small capacitance weak link. Within the energy window (for a detailed discussion, see ref. [28]), such a system can be deemed Chapter 2 ofref. [29] deals with the BCS theory and contains references for all the original papers.

Volume 170, number 4

PHYSICS LETTERS A

a macroscopic quantum object, and it is describable

EK( ~j5~t)

=

0, where fi

vcos(2xci’/ci’0)

2A

Q] =ih.

.

(9)

(10)

The inductance of the ring is A, the capacitance of the weak link is C, and the third term in (9) is the Josephson coupling energy [33]; h p/2 is the matrix element for a pair charge to tunnel across the weak link. The superconducting flux quantum is defined by ci~ h/2e. The external source applied to the weak link ring, ~P~‘, is the contribution to the total ring flux ci~which is generated by other (classical) circuits. Clearly ci ~ couples linearly to ci’ in (9). In order for it to be monitored we consider the macroscopic quantum circuit to be coupled to a lossy resonant macroscopic classical circuit. The questions to address are whether our assumptions (i) and (ii) together with adiabaticity, to an equation of motion can for be thesatisfied, classicalleading circuit which is of the form (8). The coordinate space time independent Schrödinger equation (3b) for our example quantum object reads H0 ( ihO /8’,, cTi,: —

(12)

to first order in v/w

Here ci,, the macroscopic coordinate of the object, is the total magnetic flux threading the ring, and Q, the conjugate momentum, is the charge on the small Capacitor. These satisfy the commutation relation [30— 32] [ci,

(ic+ ~)hw0

_~lvcos(2I~’~°’/c~0) exp(—7t

(~ ~tOt)2

2C+

=

2fl)L~(2,t2f3)

by the Hamiltonian [2 1,23,30] Ho(Q,ci’;ci~°’)

Q2

9 November 1992

‘,tOt)

UK( ci’S ,

‘,tOt)

(11) =EK(ci~’)U~(~ ti, with the Hamiltonian (9), and the substitution t0t)

woA/cP~and LK is

the Laguerre polynomial of order.’c [35]. The ,cdependence here is fairly weak (albeit discernable in principle), as all the energy levels have the same single harmonic dependence on ~ ~ merely with different ic-dependent amplitudes. However for w0/2 ~ v ~ 2w0 there is much more dramatic ,c-dependence [23,34]. Higher harmonics contribute to the forms of the E~(‘i f’), giving significant ,c-dependence to the shapes of the levels [23,34]. The upper limit v ~ w0 is given to ensure adiabaticity. With the physically reasonable choices of A 5 x l0’° H and C-~l016 F, taking v 2w0 gives a minimum interlevel spacing of a few hundred GHz (in frequency units). This is to be compared with a typical applied classical resonant (radio) frequency of 20 MHz for the time-dependence in ci, tot Thus our assumption (i), and adiabaticity, can be simultaneously satisfied for our example macroscopic quantum object. So can assumption (ii). If the classical lossy radio frequency circuit has inductance LR, and is inductively coupled to the quantum circuit through a mutual inductance M, then the fraction of the integrated classical flux noise felt this by the quantumexample circuit 2 For particular is ~ dimensionless ~‘ p ( kB TLR)” the coupling is given by P=MILR, and has a typical value of 0.01. With LR 0.2 IIH, and taking the classical circuit to be at liquid helium ternperature (4.2 K) gives~4~’ 0.02cb 0 [23,34]. Since the features in E,~(~ ~) to be identified are at the ci0 scale, we see that assumption (ii) can also typically be well satisfied for our example. (More detailed discussion is given equation in refs. [23,34].) In this case the general (8) takes the specific one-dimensional form ‘~

BEK

Q—~’—ift8/Oci. from (10). The energy levels

\

____

E~(~°’) are even, ~

0-periodic functions of ~ for each level ic; these properties respectively follow from H0 (Q, ci,; ~ ) = H0 ( Q, ‘,tOt) and H0 ( Q, ii,; ‘,tOt) = H0 ( Q,2ci’ +w ci’0 ‘,tOt + ci’0). For example, when (AC)” 0>> v the levels are given by [23,34] ~‘,;

—

~

~

~(IIN(t_P8’,~~).

(13)

This is the equation of motion for the macroscopic classical flux variable ço of a lossy resonant circuit (inductance capacitance CR, quality factor QR, 2RLRCRLR, 1). Such a circuit satisfies our criand CO(iii) for a classical system. The circuit is drivterion 277

Volume 170, number 4

PHYSICS LETTERS A

en with a current IIN(t), and is coupled (through a mutual inductance) to a macroscopic quantum circuit, described by (9). The total flux felt by the quantum circuit is ~ where i,,, is a quasi-static applied flux. Since the energy levels EK( ‘,~“) are all periodic functions of their argument, (13) is a non-linear equation for the motion of ço. In an experiment, the detected classical motion is typically that of the voltage, ço. We should stress that our example coupled classical—quantum system (described by (13)) is not simply a contrived model, constructed to show that our assumptions leading to eq. (8) are not unphysical. It is a genuinely practical example, and amenable to laboratory investigation. Experiments on such systems (placed in otherwiseextremely “quiet” electromagnetic environments) have yielded striking evidence for the ground and lowest few excited states of a superconducting weak link ring [21,36— 38], through their reactive effect on a classical circuit, as laid out in (13). The adiabatic monitoring method is thus a practical method for investigating macroscopic quantum circuit behaviour. The possibility of using the approach for seeking potential macroscopic quantum behaviour of ferromagnetic [24] or ferroelectric grains [39] has also been suggested. We turn finally to our choice of the term “monitor”. As we have already pointed out, the conventional definition of a quantum measurement [6, p.3] involves any form of interaction between a quantum object and a classical system. Since we have been very restrictive about the form of the interaction we consider (it must be adiabatic), we have chosen to emphasize this restriction by changing the terminology. The term “observe” is used to describe the interaction between a metastable quantum object and classical apparatus designed to register the occurrence of the object’s decay when it happens [40]. As this is not the sort of interaction we discuss here, we have chosen the term “monitor”, to try and stress the continuous but non-disruptive nature of the interaction. However that is not to say that the state of the quantum object suffers no irreversible effect whatsoever from this adiabatic interaction. If this were so then the extraction of information from it would appear to be questionable. A conventional quantum measurement involves an irreversible 278

9 November 1992

change (a projection) of the state of the quantum object being measured. This still holds in the adiabatic limit, there is an irreversible change to the state of the quantum object; however it is of a very specific kind. It is a change in the phase of the state. Perhaps the best illustration ofthis can be given by considering a superposition state for the quantum object, ~(x, t) =auk(x) exp(_

( ~$

~

$

dt’

EK(fOl))

))~

dt’ E~(f10,

(14)

+ a u . (x) exp with I a I 2 + a’ 2 Recall that we define our macroscopic classical system to be adiabatic, but to be at a finite temperature and thus to exhibit thermal fluetuations. For our example of a iossy oscillator the frequency spectrum of the noise power is given by I G(w) 12, is proportional to the temperature T, and different frequency components are uncorrelated. The argument of the energies in (14), ~ therefore has a noise contribution from the thermal motion of the classical coordinate Xe,. Since we have insisted that EK and EK have different dependence onf,01, the noise inf,0, will give rise to a noise component in the relative phase between the terms in the superposition (14). This is a very simple and specific example of the decoherence of a macroscopic quantum state [41—441due to its environment. In general if the classical system (the environment) coupled to the quantum object does not satisfy adiabaticity, then one has to consider changes in the amplitudes of superposition coefficients, as well as phases. It is interesting to note that although the relative phase between the terms in (14) acquires a noise component due to the thermal fluctuations in the classical systern, the “force” which (14) generates on the classical system is =

(

—

P

a

2

OE,.

2

‘

+

a

I

OEK

\

~TJ’

independent of this relative phase. In order to investigate the phase relationship in the quantum state (14), something other than an adiabatic monitoring experiment must be performed. One example would be a macroscopic quantum coherent oscillation experiment [43—47}, which involves repeated prepa-

Volume 170, number 4

PHYSICS LETTERS A

rations and conventional projective quantum measurements. We summarize. In this paper we have described a novel method for investigating certain quantum objects, namely those which lie in a defined class of macroscopic quantum objects. The method involves coupling one such object to an adiabatic macroscopic classical system. It is the motion of this system which is detected. This yields information on the state of the quantum object, through a statedependent “force” which acts on the classical systern. The adiabatic interaction can be considered to be a very specific form of quantum measurement (given the general definition of this term), in which the irreversibility simply appears in the phase of the quantum state. We have therefore termed this the monitoring of a quantum object. This approach is a practical method for investigating macroscopic quantum phenomena, and we have illustrated this with the example ofa superconducting weak link ring. We would like to thank the Royal Society and the Science and Engineering Research Council for their generous funding of this research.

[17] [18] [19] [20] [21]

9 November 1992

H. Nyquist, Phys. Rev. 32 (1928) 110. H.B. Callen and J.H. Welton, Phys. Rev. 83 (1951) 34. R. Kubo, J. Phys. Soc. Japan 12 (1957) 570. R.P. 118. Feynman and F.L. Vernon Jr., Ann. Phys. 24 (1963) H. Prance, T.P. Spiller, J.E. Mutton, R.J. Prance, T.D. Clark and R. Nest, Phys. Lett. A 115 (1986)125.

[22] T.P. Spiller, T.D. Clark, R.J. Prance, H. Prance and D.A. Poulton, Nuovo Cimento B 105 (1990) 749. [23] T.P. Spiller, T.D. Clark, R.J.physics PranceXIII, anded. A. D.F. Widom, in: Progress in low temperature Brewer (Elsevier, Amsterdam, 1992) ch. 4, p. 219. [24] A. Widom andY. Srivastava, Nuovo Cimento B 103 (1989) 85. [25] A. Widom andY. Srivastava, Nuovo Cimento B 107 (1992) 71. Bell, Phys. world 3 (1990) 33. [26] J.S. [27] W.M. Itano, D.J. Heinzen, J.J. Bollinger and D.J. Wineland, Phys. Rev. A 41(1990) 2295. [28] T.P. Spiller, T.D. Clark, D.A. Poulton, R.J. Prance and H. prance, Microscopicviews ofsimple macroscopic quantum behaviour, submitted to Nuovo Cimento. [29] M. Tinkham, Introduction to superconductivity (McGrawHill, New York, 1975). [30] A. Widom and Y. Snvastava, Phys. Rep. 148 (1987) 1. [31 ]A. Widom, J. Low Temp. Phys. 37 (1979) 449. [32] A. Widom and T.D. Clark, Phys. Lett. A 90 (1982) 280.

References

[33] B.D. Josephson, Phys. Lett. 1(1962)251. [34] T.P. Spiller, D.A. Poulton, T.D. Clark, R.J. Prance and H. Prance, Int. J. Mod. Phys. B 5 (1991)1437. [35] M. Abramowitz and I. Stegun, Handbook of mathematical

[1] M. Born and V.A. Fock, Z. Phys. 51(1928) 165. [2] A. Messiah, Quantum mechanics, Vol. II (North-Holland, Amsterdam, 1961) p. 747 [3] E. Merzbacher, Quantum mechanics, 2nd Ed. (Wiley, New York, 1970). [4]P. Ehrenfest, Z. Phys. 45 (1927) 455. [5] L.D. Landau and E.M. Lifshitz, Quantum mechanics, 3rd Ed. (Pergamon, Oxford, 1977) p.3. [6] A. Beskow and J. Nilsson, Ark. Phys. 34 (1967) 561. [7] H. Ekstein and A.J.F. Siegert, Ann. Phys. (NY) 68 (1971) 509. [8] E.B. Davies, Helv. Phys. Ada 48 (1975) 365. [9] L. Fonda, G.C. Ghirardi, A. Rimini and T. Weber, Nuovo Cimento A 15 (1973) 689; 18 (1973) 805. [10] L. Fonda, G.C. Ghirardi and A. Rimini, Rep. Prog. Phys. 41(1978) 587. [11] R.A. Han-is and L. Stodolsky, Phys. Lett. B 116 (1982) 464. [12] C.B. Chui, E.C.G. Sudarshan and B. Misra, Phys. Rev. D 16 (1977) 520. [13] A. Barchielli, L. Lanz and G.M. Prosperi, Nuovo Cimento B 72 (1982) 79 [14] C.N. Friedman, Indiana Math. J. 21(1972) 1001. [15] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18 (1977) 756. [16] N. Bohr, in: Albert Einstein, philosopher—scientist, ed. P.A. Schilpp, Library ofLiving Philsophers (Evanston, 1949) ~. 209.

functions (Dover, New York, 1972) ch. 22. [36] R.J. Prance, J.E. Mutton, H. Prance, T.D. Clark, A. Widom and G. Megaloudis, Helv. Phys. Acta 56 (1983) 789. [37] J.E. Mutton, R.J. Prance and T.D. Clark, Phys. Lett. A 104 (1984) 375. [38]R.J. Prance, T.P. Spiller, H. Prance, T.D. Clark, J. Ralph, A. Clippingdale, Y. Srivastava and A. Widom, Nuovo CimentoB 106 (1991) 431. [39] J.E. Mutton, A. Widom, T.D. Clark, H. Prance and R.J. Prance, Nuovo Cimento Lett. 41(1984) 587. [40] A. Sudbery, Ann. Phys. 157 (1984) 512. [41] A.0. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211. [42] A.O. Caldeira and A.J. Leggett, Ann. Phys. 149 (1983) 374. [43] A.J. Leggett, in: Proc. Int. Symp. on Foundations ofquantum mechanics, ed. S. Kamefuchi (Physical Society of Japan, Tokyo, 1983). [44] A.J. leggett, in: Proc. 1983 NATO ASI on Percolation, localization and superconductivity (Pergamon, Oxford, 1984). [45] C.D. Tesche, in: SQUID’85, Proc. 3rd Int. Conf. on Superconductingquantum devices, eds. H.E. Hahlbohm and H. Lubbig (Dc Gruyter, Berlin, 1985) p. 355. [46] C.D. Tesche, Ann. NY Acad. Sci. 480 (1986) 36. [47] C.D. Tesche, in: Macroscopic quantum phenomena, eds. T.D. Clark, H. Prance, R.J. Prance and T.P. Spiller (World Scientific, Singapore, 1991) p. 67.

279