energy conservation and complementarity in neutron single crystal interferometry

Volume 104A, number 6,7 PHYSICS LETTERS 10 September 1984

ENERGY CONSERVATION AND COMPLEMENTARITY

IN NEUTRON SINGLE CRYSTAL INTERFEROMETRY

C. DEWDNEY 1 Institute Henri Poincar~, Paris, France

A. GARUCCIO Istituto di Fiswa dell'UniversiM, Bari, Italy and INFN Sezzone di Ban, Ban, Italy

and

A. KYPRIANIDIS 2 and J.P. VIGIER Institut Henri Poincard, Paris, France

Received 5 July 1984

The complementarity principle is shown to conflict with the energy conservation laws in neutron single crystal interferometry. Its shortcomings are revealed in specific performed or proposed neutron mterferometry experiments.

The quantum formalism provides the correct pre- dictions for the experiments performed in neutron interferometry [1 ] but nevertheless fundamental ques- tions, open since the Bohr-Einstein debate 50 years ago, are raised again concerning their possible inter- pretations. In a recent letter [2] we discussed the problems related with time dependent neutron interferometry [3] and the possibility to consider the radio- frequency spin flipper as a measuring device, a fact that, if established, should enable the "impossible" simultaneous path/interference detection. In this letter we are mainly concerned with aspects of the complementarity principle and its contradiction with the fundamental energy conservation law.

To this purpose we consider the experimental ar- rangement of fig. 1 with both spin flippers turned off. A simple calculation shows [4] that if an originally spin up polarized beam ff = It z) enters the interferometer, it is subdivided in two partial beams ~1 = eiXltz )

I European Exchange Fellowship. 2 On leave from the University of Crete, Physics Departement,

Herachon, Crete, Greece.

0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Fig. 1.

and ~II --" I t z ) that successively recombine and yield an intensity interference behind the interferometer modulated with the phase shift factor X.

l = (~I + ~II)+(~bI + ~I I ) = 2(1 + cos X), ( I )

while the polarization remains the z-direction.

P = (0, O, 1). (2)

This "double-slit" like situation offers two possible explanations:

(a) Either we say the neutron actually travels along path I o r II only, but is influenced by the physical conditions along both;

(b) Or we say the neutron does not exist as a particle in the interferometer.

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According to the Copenhagen interpretat ion of quantum mechanics (CIQM) if the pamcle actually travels along one path the existence of the other is therefore Irrelevant and interference cannot occur. Interference arises not from our lack of knowledge of the path but from the fact that the neutron does not have one. Thus certain experimental apparatus make the non-localized or wave nature o f the neutron manifest whereas other mutually exclusive apparatus make ~ts particle nature manifest. Tins is the complementarity of wave and partMe.

The first option (a) was that favoured by Einstein [5] and de Broglie [6] in the causal stochastic interpretat ion o f quantum mechanics (SIQM). The neutron always travels along one path whilst its real guiding wave travels along both. In the region of superpositlon the waves combine informatxon about both paths and grade the particle accordingly. Determinate Individual particle trajectories can be calculated in this model in interference experiments (for a detailed calculatmn of the particle trajectories in a two slit expenment see ref. [7]).

In the original Bohr-Eins te in debate Bohr was able to defend the complementar i ty principle by showing that a t tempts by Einstein, to use detailed energy or momentum conservation in individual processes to determine particle trajectories and gwe a fuller description, required a change of the experimental arrange- ment which resulted in a loss o f the wave aspect [8].

The argument is often put in the following way m CIQM. The in t roductmn of a device capable of determining the particle trajectory induces a collapse of the wave functmn in the rest o f the apparatus and a con- sequent loss of interference. Such a collapse is a consequence o f the purely probabihstic interpretat ion of the wave function and follows from the reqmrement that wave and particle pictures are complementary. In fact one should note that the collapse concept (proJec- t ion postulate) need never be used in quantum calcula- tions; whether or not observable interference persists depends on the actual interaction that has taken place with the apparatus, as shown by Cim [9]. For example consider the wave function of an apparatus introduced in one path to be q~i initially and ~f finally, then we have.

~a = ~bl~I + q~l~0II ~ ~Itf = ~bl~I + ~bf~ll • (3)

If through its functioning the states ¢~ and ¢f become

orthogonal then interference IS destroyed

+ + qz~-xlzf = ~b+~b,~'~I + ~bf~bf'~ll~ii , (4)

and the neutron acts as a particle that goes either on path I or path II. Observation of the measuring instru- ment merely tells us which alternative took place and thus we replace ~ f by ~blt~I or ~bf~ii. This is a collapse of the wave function winch simply represents a change of our knowledge and does not correspond to any real physical changes in the state of the neutron. If ~b 1 and ~bf are not orthogonal then Interference persists

~t~ air f + + + + + + = q~l $1ffI ~ I ÷ ~bf ~bfffll~II + ~bl~I~bf ~I I + +

+ ~bf~llq~l ~ I , (5)

and the neutron acts as a wave in both paths.

If by observing the apparatus we could still In fact determine the path of the neutron then the act of observation in CIQM would have to cause real physical changes in the neutron's state as a consequence of a wave packet collapse. Since if neutrons are conceived as particles that go one way or the other, eq. (5) should reduce to eq. (4).

Thus CIQM concludes that all measurements capable of determining the neutron's path Imply ortho- gonahty of the apparatus wave functions initially and finally. In SIQM determination of pamcle path need not Imply orthogonali ty of apparatus wave functions in order to exclude the in tervent ion o f consciousness in physical processes. What appears as a "pseudo-collapse" is the action of a macroscopic measuring device which makes the interference terms negligible as is con- slstently shown by Cinl [9]. Thus, there IS no a priori Impossibility of path determination and persisting interference, one has only to find an appropriate measuring device that during an interaction with the micro- system does not undergo a change to an orthogonal state, i.e. preserves the Interference terms, and still offers a possibility to decode this small quantum num-

ber change (e.g. one could envisage the possxblhty of using a "quantum non-demolit ion measurement" pro- cess for such a purpose [10,11]).

Consider now the set up with the spin flip device in path II operating (at 100% efficiency). The intensity modulat ion In the emerging beams disappears, there is no spatial interference. Tins lack of Interference need not however imply that the coil acts as measuring device locahsing the neutron xn one beam since interfer-

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ence persists in the spin superposition yielding a final polarization [3]

P = (cos(eOrft - X), sin(6Orft - X), 0 ) .

Each neutron emerging from the interferometer IS polarized in the x - y plane. The spin up I tz> and spin down [ ,L z) states are superposed and hence it is argued along CIQM lines that the neutron actually does not exist as a particle with spin in either beam, if it did a mixture o f spin states would result. Now in order to explain the change of spin state when the polarized neutron acts with the rf coil it is argued [3] that the neutron emits a photon of energy E = ~rWrf to the tame dependent field and hence the energy of the neutron and the cod are altered. No description using the purely wave-like aspect can explain these results. If the neutron actually is only a wave (or in neither beam) during the experiment no such energy exchange could be described: How can the coil exchange a photon with a neutron that does not exist! In order to explain the change of spin states produced by the coil we require a localized particle, but a description in terms of locahzed neutrons in one or other beam cannot explain the superposition. It would seem that we must use here wave and particle aspects o f neutrons simultaneously. Alternatwely it may be argued that when the neutron as gaid to behave like a wave we should not imagine a physical wave, all we see in this case is an interference of the probability amplitudes for an event to happen in indistinguishable ways. The amplitude to travel path I without spin flip "inter- feres" with the amplitude to travel path II with spin flip since the two are said to be indistinguishable. While this pattern seems to be consistent it still maintains a fundamental ambiguity: the neutron must exist in one beam or another in order to exchange energy, while this statement must be denied in order to preserve interference.

One should note the implications of this contradiction: If this scheme is extended from a recipe for pre- dictions to an explanatory pattern, its failure becomes evident. In order to preserve interference when having energy transfer the rf-coil wave functions Cx and ~f must be indistinguishable from the interference point of view and distinguishable concerning the energy transfer omitting the case of orthogonality.

Now consider the apparatus with both spin flippers operational. Since now a spin flip takes place in both

beams spatial interference is recovered. A measurement of the polarization of the neutron behind the interferometer reveals that each neutron has suffered a spin flip. Each emerging neutron has lost an amount of energy AE where AE = 2/tB, the Zeeman splitting. If energy is to be conserved this energy must have gone to one or other of the coils, this is only possible if the neutron passes as a particle through one or other and gives an indivisible photon o f E = hWrf = ZkE to the if-field. The spatial interference can only be explained by assuming that the neutron does not pass through one or other of the coils. The change of energy can only be explained by the particle aspect.

Since both interference and spin direction can be measured simultaneously, according to CIQM the neutron actually travels path I or II and at the same time does not exist as a particle at all.

In the Bohr-Einstein debate the application of particle momentum conservation in individual events always led to the consistency of CIQM. Here the energy conservation leads to the inconsistency of CIQM since wave/particle aspects appear together. If it is in- sisted in CIQM that neutrons do not travel one way or the other, no energy can be transferred to the coils and then there is no conservation o f energy in individual events. Further if a statistical ensemble of individual neutron passages is considered we see that, even there, there is no conservation of energy in CIQM.

If we wash to consider the mechanism of spin flip and the conservation of energy, then the neutrons must travel on one path and through a coil. If we wish to consider spin superposition then the neutrons must travel along neither.

Is the consequence of the above presentation to renounce complementarity for the CIQM? Probably not, because the Heisenberg uncertainty relations could provide a means to escape the conclusiveness of the presented reasoning. In fact one could argue that since the energy uncertainty 6E introduced to the neutron energy E due to the Bragg scattering o f the crystal is greater than the energy transfer AE to the rf-coil (10 -6 and 10 -8 respectively) no energy conservation could be established. Still a problem remains: Since the final spin polarization (i.e. - z ) is detectable, the Zeeman energy loss of the neutron in B is known with respect to its initial polarization (i.e. +z). Because of this, the magnetic f~eld energy of the neutron spin is accurately known. One can now interpret the uncer-

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talnty in E as inherent in the corresponding operator, in which case the problem of a "hypothetical" energy

transfer to the if-coil inferior to 6E is not legitimate but the simultaneous sharp value of the Zeeman part

remains incomprehensible. One could then still escape to a formulation of the kind. "'the energy transfer cannot be measured because AE < 6E". This opera- tlonal aspect of the uncertainty clearly promotes an existence of sharp instantaneous values which are steadily perturbed. This explains the expenmentai detection of a spin state but in CIQM fails to account for interference because it yields a definite neutron path. In both aspects/versions of the energy uncertainty in- terpretation m CIQM contradictions arise that do not

perhaps affect the complementarity prlnctple as a useful recipe in most of the cases but which do reveal its shortcomings as an explanatory pattern in specific situations as the performed or proposed experiments on neutron interferometry.

The authors want to thank Professor Rauc.h for the useful discussions. One of us (C.D.) wants to thank the British Royal Society for a European Exchange Fel-

lowship, another (A.K.) the French Government for a grant and another (A.G.) the Admmlstration Council

of the University of Barl for a grant which made this collaboration possible.

References

[1] J Summhammer, G Badurek, tf Rauch and O Kxschko, Phys. Lett. 90A (1982) 110, G Badurek, H Rauch, J Summhammer, U Klschko and A Zedlnger, J. Phys. AI6 (1983) 1133, J Summhammer, G. Badurek, H Rauch, U Klschko and A. Zelhnger, Phys. Rev A27 (1983) 2523

[2] C Dewdney, Ph. Gueret, A Kypnanldls and J P Vlgler, Phys. Lett. 102A (1984) 291

[3] G Badurek, H. Rauch and J. Summhammer, Plays Rev. Lett 51 (1979) 15.

[4] G. Eder and A Zethnger, Nuovo Clmento 34B (1976) 76.

[5] A Einstein, Proc. Congres Solvay (1927). [6] L. de Broghe, Une tentative d'lnterpr~tatlon causale et

non-lm~aire de la mgcamque ondulatolre (la th~orle de la double solution) (Gauthler-Vdlars, Pans, 1956)

[7] J C Phlhppldls, C. Dewdney and B J Hlley, Nuovo Cimento 52B (1979) 15

[8] N. Bohr, Atomic physics and human knowledge (Wdey, New York, 1958)

[9] M Cmi, Nuovo Clmento 73B (1983) 27 [10] C W Caves, R W. Drever, V Sandberg, K.S. Thorne and

M. Zlmmermann, Phys. Rev. Lett. 40 (1978) 667, Rev Mod. Phys. 52 (1980) 341.

[11] WG. Unruh, Phys Rev D17(1978) 1180, D18(1978) 1769, D19 (1979) 2888.

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energy conservation and complementarity in neutron single crystal interferometry

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