Nonlinear dynamics and the two-slit delayed experiment

J.S. Nicolis a, G. Nicolis b,*, C. Nicolis c

a Department of Electrical Engineering, University of Patras, Patras 26500, Greeceb Center for Nonlinear Phenomena and Complex Systems, Universit�e Libre de Bruxelles, Campus Plaine, C.P. 231,

Brussels 1050, Belgiumc Institut Royal M�et�eorologique de Belgique, Avenue Circulaire, 3, Brussels 1180, Belgium

Accepted 4 August 2000

Abstract

The two-slit delayed experiment is re-examined from the standpoint of nonlinear dynamics in the presence of multiple attractors

and fractal basin boundaries. It is suggested that the results may be interpreted as the response of the underlying system to a temporary

switch of one control parameter, rather than as a retroaction between this system and the observer. Ó 2000 Elsevier Science Ltd. All

rights reserved.

1. Introduction

The evolution of a quantum system during the process of measurement has a special status in physics. Asa result of the interaction of the system with the measuring apparatus, or ``the environment'' (which formany purposes can be assimilated to a macroscopic system), the plurality of the possible values of anobservable of interest is abolished and the system ends up in a state in which there is a well-de®ned value ofthis observable. It is therefore not surprising that the quantum theory of measurement is intimately relatedto some of the most fundamental questions pertaining to the interpretation of quantum mechanics [1,2].

One of the most famous experiments of relevance in the above context is the two-slit delayed experiment.It has been interpreted by Wheeler [2,3] as providing evidence of a puzzling ``retroaction'' between theobserver and the system under investigation ± referred to by the author as the ``observer participancy'' ±thereby apparently demolishing the view that the universe holds an independent existence.

In the present paper it is proposed to view the results of the two-slit delayed experiment as the responseof the underlying quantum system to a change of an external control parameter. We provide a classicalexample of a simple nonlinear deterministic dissipative system possessing two-point attractors ± the analogsof the eigenstates of a quantum system ± in which the dynamics generated by the underlying evolution lawsleads to a behavior sharing some aspects of the exotic behavior of the two-slit delayed experiment as in-terpreted by Wheeler. Speci®cally, we show that when the process of ``categorizing'' an initial condition ±the analog of the preparation of a quantum state ± to one or the other attractor is still en route we may shiftbefore the collapse on the expected attractor takes place a control parameter during an appropriate lapse oftime and then restore it to its original value, thereby making the system switch over to the other attractor.

The ontological disparity between quantum and classical systems notwithstanding [4], we provide furtherevidence that a nonlinear deterministic system possessing two coexisting (mutually exclusive) attractors

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Chaos, Solitons and Fractals 12 (2001) 407±416

* Corresponding author.

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 1 9 0 - 9

with riddled basin boundaries [5] displays under a similar switching of a control parameter non-comput-ability, non-repeatability and undecidability about the ®nal destination of an initial condition. The latter willfall ultimately on an attractor not known in advance after a transient time that may be uncomputably long,despite the fact that the invariant measures of the two basins of attraction are well de®ned and may even beequal. Statistical considerations are thus not su�cient, but need to be complemented with informationpertaining to the dynamical evolution at the level of individual trajectories. The point is that in the aboveclassical settings no retroaction needs to be invoked to understand results that are strongly reminiscent ofthose of the two-slit delayed experiment in its purely quantum setting.

The paper is organized as follows. In Section 2, the two-slit delayed experiment is presented in somedetail for the purpose of later reference. The response of a nonlinear deterministic system possessing two-point attractors and two attractors separated by fractal boundaries to a switching of a control parameter isanalyzed in Sections 3 and 4. The main conclusions are ®nally compiled in Section 5.

2. The two-slit delayed experiment

Consider the specially arranged array of mirrors and photon detectors of Fig. 1. An incoming beam ofphotons ®rst strikes a half-silvered mirror H-S set at a 45° angle to the beam. Such a mirror re¯ects half ofthe photons striking it and lets the other half pass through. The re¯ected beam in turn strikes a regularmirror A lying parallel to the splitter, and the transmitted one hits likewise another regular mirror B alsolying parallel to the beam splitter. The two beams then cross each other at a 90° angle at a ®nal point andenter two separate photon detectors. Both detectors will signal the arrival of photons traveling along theindividual paths, each beam with half the intensity of the original beam.

Now suppose we place a second half-silvered mirror H-S2 exactly at the point where the two beams crossbefore hitting the detectors. It follows that only one detector now registers the arrival of the beam. Ev-erything happens as if one side of the second half-silvered mirror brings the beams into destructive in-terference while the other side of the mirror brings them into constructive interference, since the light wavesreinforce each other. The result is the fully reconstructed original beam of photons which hits the seconddetector at its original intensity.

The action gets really interesting when we start looking at one photon at a time. If we remove thesecond half-silvered mirror, then we will see one detector or the other registering the arrival of aphoton, as each successive photon arrives either by path (1) or path (2). Next we put the second half-silvered mirror back into the apparatus. Now only the second of the two detectors will register at eachtime a photon arriving at the end of the apparatus. The only explanation is that each single photontravels both paths simultaneously. Destructive interference keeps the ®rst detector from ever registeringa photon, while constructive interference makes possible the photon to always register in the seconddetector.

Actually we are here dealing with two di�erent experiments (two di�erent ``preparations''). When thesecond half-silvered mirror is removed we have an experiment which tells us which route the photon takes.When the second mirror is put in its location just in front of the two detectors we have an experiment thattells us the photon traveled both routes at once. But it is impossible to do both experiments at once: one canobserve one feature of nature or the complementary feature but not both features simultaneously.

Now the ``puzzling thing'' which has mysti®ed Wheeler [2,3] so much: instead of placing the ®nal half-silvered mirror in the apparatus before we start the experiment, we put it in or take it out ``at the very lastpicosecond'' after the photon has completed its journey (but of course before the ®nal registration). AsWheeler comments: ``we, now by moving the mirror in or out have an unavoidable e�ect on what we have aright to say about the already past history of that photon'' [3, p. 118]. So deciding whether to put in thehalf-silvered mirror or take it out at the very last moment has the e�ect of in¯uencing what we can sayabout what has already happened to the photon: in a sense we a�ect the photons' past history. In short, ouractions in the ``here and now'' can have an e�ect on what one can say about nature prior to that moment. Inthe next and other sections, we provide evidence that rather than argue in terms of retroaction, the phe-nomenon of two-slit delayed experiment could usefully be viewed as the response of the state of a dynamicalsystem toward the switching of an external control parameter.

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3. A classical, nonlinear deterministic system possessing two coexisting mutually exclusive point attractors

A quantum system does not need an ``observer'' of any kind to collapse quickly and spontaneously todecoherence, i.e., a set of complementary and mutually exclusive eigenstates. It does so by employing in-teractions between the degrees of freedom of the system itself and the enviroment [1]. A quantum mea-surement is thus an irreversible process involving essentially the partitioning of a set of undi�erentiated``raw'' states onto such ``categories''. On the other hand, in a classical nonlinear dissipative system envi-ronmental stimuli are ``captured'' as initial conditions and then ``processed'' as solutions of its dynamics.Eventually, they are compressed irreversibly onto a set of complementary and mutually exclusive states,the system's attractors, each one of which possesses its own basin of initial conditions collapsing onto theattractor. In this respect, therefore, the dynamics of both quantum and classical measurements sharethe common feature that the ®nal destination is ``decoherence'' in the form of the collapse onto one amonga set of coexisting states ± the eigenstates of the quantum system or the attractors of the classical system.

Fig. 1. (a) A layout of the gedanken two-slit delayed experiment. (b) An actual experimental setting (after Ref. [2]).

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The ontological disparity between quantum and classical systems will thus, rather, show up in the nature ofthe initial regime and the particular form of the probabilities of its collapsing to one of the ®nal states. Anatural question to be raised in this context is the existence or not of a classical counterpart of quantumnon-locality and entanglement. We will explore this is the next section. For now, we study in a particularlysimple example the process of en route switching between ®nal states through control parameter variations.We emphasize that, at this stage, our results will o�er no more than suggestive analogies with the quantummeasurement problem.

Consider a one-variable classical deterministic dissipative system whose state variable x satis®es theevolution law

dxdt� ÿx3 � kx� l; �1�

where t is the time and k, l are the two control parameters. As well known Eq. (1) is universal, in the sensethat any dynamical system operating in the vicinity of a pitchfork bifurcation ± one of the elementarybifurcations arising at a simple eigenvalue ± and perturbed by an imperfection (like, e.g., an external ®eld)reduces to this form on a suitable change of variables, time scales and parameters [6]. Furthermore, Eq. (1)has been used widely to model materials under stress as well as autocatalytic chemical reactions or electriccircuits out of equilibrium.

A ubiquitous feature of Eq. (1) is to give rise, in suitable ranges of values of k and l, to two coexistingstable states separated by an unstable one. The bifurcation diagram of these solutions (which for thepurpose of comparison with the quantum case can be thought of as the analog of the ``corpuscle-like'' and``wave-like'' picture of light) as a function of l for a value of k ®xed at k � 2 is depicted in Fig. 2. As can beseen, there are two limit point bifurcations between which the system exhibits both bistability and hystereticbehavior.

In the above setting the parameter l is at each time kept at a given ®xed value, which is subsequentlyvaried in order to sweep the bifurcation diagram in the desired range. Put di�erently, sweeping is beingcarried out in®nitely slowly. We now subject l to a temporary variation in the form of a square pulse asshown in Fig. 3, being understood that if subjected permanently to the lower or to the upper l values of thepulse the system would be led, respectively, to the lower or to the upper branch of the bifurcation diagram.

Fig. 4 summarizes the response of system (1) to the above change of l for di�erent values of the ex-tinction time t1 (keeping the ignition time t0 ®xed), starting with the initial condition x0 � ÿ1. We see(Fig. 4(a)) that on increasing t1 up to 1.25 time units the system is ®rst driven further and further away from

Fig. 2. Steady-state solutions of Eq. (1) as parameter l is varied with k � 2. Full and dotted lines stand, respectively, for stable and for

unstable solutions.

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the lower steady state of Fig. 2 to which it would be rapidly attracted had l remained ®xed at the value l0

(here l0 � 0�. Eventually, however, this state takes over. On the other hand, as seen in Fig. 4(b), for a stilllarger value of the extinction time the system is driven to the upper branch of the hysteresis loop of Fig. 2and stays thereafter in this branch, even though l is re-established to its initial value l0 for all t > t1.

The above behavior is reminiscent of the one in the two-slit experiment, when the second half-silveredmirror is placed or removed between the start of the process and the ®nal registration of the photon. On the

Fig. 3. Switching protocol of the control parameter l of the dynamical system described by Eq. (1).

Fig. 4. Time evolution scenarios generated by Eq. (1) with initial condition x0 � ÿ1 and k � 2. The value of l switches from l0 � 0 to

l1 � 1:2 between t0 � 0:25 and t1 � 0:5, 0.75, 1.00, 1.25 (a); and t1 � 5:0 (b).

J.S. Nicolis et al. / Chaos, Solitons and Fractals 12 (2001) 407±416 411

other hand in our classical model there are, clearly, intrinsically imposed limits to the times of ignition andextinction necessary for switching to another attractor, as a result of which these processes cannotbe postponed to the ``very last picosecond'' as Wheeler puts it. To determine these limits we perform asensitivity study of the model of Eq. (1), in which the value of extinction time t1 necessary to switch to theupper branch of the bifurcation diagram of Fig. 2 (keeping the ignition time t0 ®xed at t0 � 0:25 and theplateau value l1 ®xed at l1 � 1:2) is plotted versus the initial values x0 and l0 of the variable x and theparameter l.

The main results of this analysis are summarized in Figs. 5(a) and (b), which provide two-dimensionalcuts of the t2 versus x0 and l0 dependence corresponding, respectively, to l0 � 0 and x0 � ÿ1. As seen inFig. 5(a), t1 ®rst exhibits a plateau and then decreases rapidly to zero as x0 tends itself to zero: a naturalresult, since for l0 � 0 the value x0 � 0 is an unstable ®xed point of the dynamical system separating thedomains of attraction of the upper and the lower steady states. As for Fig. 5(b), the decreasing trend in t1

corresponds to the approach of l0 to the rightmost limit point of the bifurcation diagram of Fig. 2, giventhe (®xed) value of )1 chosen for x0: switching on l0 to l1 beyond this point accelerates the evolution to the

Fig. 5. Switching time t1 necessary for the trajectory to reach upper branch of Fig. 2(a) versus initial conditions, x0, with l0 � 0; (b)

versus l0 with x0 � ÿ1. Parameter values k � 2, t0 � 0:25 and l1 � 1:2.

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upper branch which is now the only branch available, and forces the system to a ``point of no return'' evenwhen l is re-established to the value l0.

A di�erent view of the system's sensitivity is provided by Fig. 6. Curves (I) and (II) in this ®gure are thesystem's trajectories for a common initial condition x0 � ÿ1 and a common ignition time t0 � 0:25, but forextinction time values equal, respectively, to t1 � 2:9 and t1 � 3:1. The trajectories split and subsequentlydiverge considerably beyond a time around t1 � 3. To the unaware observer, it would appear that hisdecision to switch o� l at t1 � 2:9 from l1 to l0 had an interference with the system's history. This is, again,reminiscent of the two-slit delayed experiment, yet there is here no retroaction whatsoever.

4. The switching process in the presence of riddled basin boundaries

In the simple dynamical system of the previous section the two basins of attraction of the stable steadystates were delimited by an intermediate unstable ®xed point. In a multi-dimensional analog, the situationwould be qualitatively similar, the di�erence being that the attraction basins would now be limited bysmooth, di�erentiable curves ± the separatrices of the unstable ®xed point.

One should perhaps question the relevance of this classical model in dispelling a puzzle generated by aquantum system. In the present section, we consider a di�erent class of deterministic classical dynamicalsystems which for all phenomenological purposes mimic the behavior of a quantum system under exper-imental scrutiny, namely, undecidability, non-computability, non-repeatability and an inherent probabilisticbehavior. What makes this behavior possible is the fact that a given part, no matter how small, in the basinof attraction of one of the invariant states has, arbitrarily nearby, pieces belonging to the attraction basin ofanother invariant state. We refer to this situation as a case of riddled basins [5].

A prototypical dynamical system possessing riddled basins is the two-dimensional iterative mapping

hn�1 � hn � a sin 2hn ÿ b sin 4hn ÿ xn sin hn; xn�1 � ÿJ0 cos hn: �2�For a � 1:32; b � 0:9 and J0 � 0:3 it possesses two coexisting stable ®xed points corresponding toA� � �h � 0; x � ÿJ0� and Aÿ � �h � p; x � J0�.

Consider a set of two real numbers de®ning an initial condition �h0; x0� (the analog of the preparation ofthe quantum system), represented by a point in the embedding two-dimensional phase space. With prob-ability one such an initial condition, chosen randomly is an irrational number and, as almost all irrationalnumbers, is random that is non-compressible and uncomputable [7]. Hence if the riddleness of the basinsholds for many scales it is even theoretically impossible to know in advance on whose attractor basin thisinitial condition belongs. Under the action of the (dissipative) dynamics of the ``measurement'', Eqs. (2), an

Fig. 6. As in Fig. 4 but with l0 � 1, l1 � 1:2 and t1 � 2:9 (I); t1 � 3:1 (II).

Fig. 7. Basins of attraction for the two attractors (0, ÿJ0) (dark region) and (p; J0) (bright region) of system (2) as obtained from

256� 256 initial conditions, ÿ0:56 x06 0:5, 06 h06p after an integration time of 100 units. Parameter values a � 1:32; b � 0:9 and

J0 � 0:3.

J.S. Nicolis et al. / Chaos, Solitons and Fractals 12 (2001) 407±416 413

initial preparation starts falling toward one of the coexisting attractors. Now, in view of the above, it re-mains undecidable to which attractor it will eventually register and it is unknown (and uncomputable) howlong the transient will last. We now repeat the experiment, resetting the initial condition with the greatestpossible accuracy but the process is non-repeatable due to the sensitive dependence of the ``pinball''character involved between the preparation and the ®nal categorization [8].

We next consider a whole of �x0; h0� values and register the percentages of its member-preparationswhich land on the individual attractors. In this way we get the probabilities of categorizing a set of rawundi�erentiated stimuli onto the coexisting, complementary and mutually exclusive invariant sets of thenonlinear operator in Eq. (2). These percentages also represent the invariant measures of the sub-basinsinvolved and stand for the probabilities of getting a registration of a given preparation onto a given at-tractor-eigenstate. Fig. 7 depicts the result of this experiment. The dark and bright parts represent, re-spectively, the set of initial conditions ending up in �0;ÿJ0� and in (p; J0) which turn out to be equiprobable.

Further examples of classical, deterministic, nonlinear systems with riddled boundaries separating the®nal outcomes include the dynamics of coin tossing [9] and die tossing [10]. In these cases, the particularoutcome toward the two and six ®nal possibilities, respectively, is not predictable though, statistically, itoccurs with probabilities 1/2 and 1/6, respectively.

The uninformed reader would perhaps infer from the above that we are talking about a protocol of aquantum mechanical measurement but the fact is that if one replaces the term ``eigenstate'' with ``attrac-tor'', the term ``decoherence'' with ``®nal destination'', and the term ``entanglement'' with ``riddleness'' (in asense an initial preparation is ``smeared out'' along all other preparations thereby conferring to the dy-namics a non-local character), nothing phenomenological remains which could di�erentiate between the twoclasses of physical phenomena. The upshot is that we have been perhaps excused in trying to explain awayin Section 3 the Wheelerian ``mystery'' of the two-slit delayed experiment by using a classical, local non-linear dissipative operator possessing two-point attractors and showing that during the transient process wecan shift a control parameter and make the ¯ow switch between attractors even though no retroaction inthe sense proposed by Wheeler is involved.

To establish the analogy in a more convincing manner we now again study, on the dynamical system ofEq. (2), the response to the switching of a control parameter chosen here quite naturally to be J0. Theswitching protocol is similar to Fig. 3, J0 simply replacing the control parameter l. Fixing the initial valueat J00 � 0:3 we perform a series of experiments corresponding to di�erent plateau values J01 and to di�erentvalues of ignition and extinction times t0 and t1. In all cases studied, it turns out that after J0 has been re-established to its initial value there are no detectable di�erences in the picture of Fig. 7. This at ®rst sight is

Fig. 8. Initial conditions that change course when parameter J0 switches from 0.3 to 0.5 during the ®rst 20 time units. Parameter values

as in Fig. 7.

414 J.S. Nicolis et al. / Chaos, Solitons and Fractals 12 (2001) 407±416

disconcerting result merely means that attractors A� and Aÿ remain equiprobable. To detect the role ofswitching ± if any ± on the dynamical behavior one therefore needs to monitor individual trajectories ratherthan statistical properties.

Fig. 8 depicts the result of a switching experiment of this kind. The dark spots stand for initial conditionsthat have changed their course. As could be expected these initial conditions lie, essentially, in theboundaries between the dark and light regions of Fig. 7.

Fig. 9 gives a ®rst idea of the sensitivity of the response to the parameters of the switching protocol. Itdescribes the dependence of the number of initial conditions (out of a total of 256� 256, changing coursefor di�erent plateau values of J0, as a function of the extinction time t1 (for an ignition time t0 � 0). Thedependence saturates beyond a value of t1 � 5 which seems to be common to all three J01 values studied.

A di�erent view of sensitivity is provided by Fig. 10. Here the plateau of J0 is kept ®xed at J01 � 0:5 andfour di�erent values of extinction times t1 are studied: 5 (circles), 10 (crosses), 20 (triangles) and 30 (dots).For each of these values the ignition time t0 is varied by steps between t0 � 0 and t0 � t1. The resultingdependencies of the number of initial conditions changing course practically superpose each other and canall be ®tted by a law of exponential decay with exponent equal to about 0.26.

The above behaviors are, again, reminiscent of the e�ect of placing or removing the second half-silveredmirror during the two-slit delayed experiment. As in Section 3, the di�erence is that there are intrinsicallyimposed limits to the timing of switching necessary to have an observable e�ect, an ``event horizon'' as itwere surrounding the attractor. Furthermore, no retroaction is involved as one deals with a deterministicdynamical system obeying local, causal evolution laws. The relevance of fractal structures in quantummeasurement has also been analyzed, from a di�erent standpoint, in Refs. [11±13].

5. Conclusions

In this paper some connections between, on the one side, the two-slit delayed experiment and, on theother side, the nonlinear dynamics of classical systems possessing coexisting point attractors separated bysmooth or fractal boundaries were proposed. The main rationale behind this proposal was the observationthat both quantum and classical measurements share the common feature to lead eventually to decoher-ence, through the collapse of a plurality of initially ordered (or ``entangled'') states onto one among alimited set of ®nal states: the eigenstates of the quantum system, or the attractors of the classical system.

The particular aspect on which the work focused was the possibility to alter the initial disposition of thestate vector and induce it to switch from one ®nal state to another through the switching of one controlparameter. In the quantum two-slit delayed experiment such a switching has long been recognized to exist,

Fig. 9. Ratio of the number, N of initial conditions that change course over the total number of initial conditions, NT versus extinction

time t1 for three values of the plateau of the switching parameter J01 with t0 � 0. Parameter values and numerical procedure as in Fig. 7.

Fig. 10. As in Fig. 9 but versus the ignition time, t0, with J01 � 0:5 and t1 � 5 (circles), t1 � 10 (crosses), t1 � 20 (triangles) and t1 � 30

(dots). Dashed line represents best ®t with an exponential function.

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related or not to the insertion of the second half-silvered mirror in the setting of Fig. 1. The interpretationof the result has led to some far-reaching conclusions, including the system±observer retroaction.By applying a very similar switching protocol to two representative classical models we observed similarbehavior, yet in the setting considered there could be no question of retroaction since the classical systemswere governed by fully local deterministic evolution laws.

We suggest that a possible explanation for the discrepancy is to be sought in the existence of an ``eventhorizon'' surrounding the attractor involved, beyond which intrinsically generated inertia e�ects do notallow the switching over between attractors. Recent experimental results on the two-slit delayed experiment[2] seem to indicate an appreciable lack of inertia of the associated state vector. It would be interesting, inthe light of our results, to re®ne these experiments by more swift switching devices to see whether there isany detectable event horizon, after all. This would put the ``retroactive'' interpretation of the experiment toan altogether di�erent perspective.

Acknowledgements

This research is supported, in part, by the Interuniversity Attraction Poles program of the BelgianFederal Government.

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