LETTER]~ AL NUOVO CIMENT0 VOL. 40, N. 16 18 Agosto 1984

Time-Dependent Neutron Interferometry: Evidence in Favour of de Broglie Waves.

C. DEWI)~EY (*), A. KYPRIANIDI8 (**) and J. P. VIGIER

Ius t i tu t Henri Poincard - Paris

A . GARUCCI0 (***)

Ist i tuto di Eisiea dell' U:~dversith - Bar i

Ist i tuto ~azionale di Fisica Nueleare - Sezione di Bar i

P m GUE~ET

Ins t i tu t de Mathematique Pure et Appliqude, Universitd P . et M. Curie - Paris

(ricevuto il 2 Maggie 1984)

PACS. 03.65. - Quantum theory; quantum mechanics.

S u m m a r y . - Time-dependent spinor superposition in neutron interferometry by means of radio frequency spin flippers enables a possible simultaneous path and inter- forencc detection and provides evidence for the real physical existence of de Broglie (( pilot )> waves.

The research of the Vienna group (1) on neutron interferometry opens new exciting experimental possibilities to discuss the different interpretations of quantum statistics and to answer the age-old question of whether the neutrons (or any other massive particle) really travel along a path in space-time between their source and the observer (as believed by EINSTEIN and DE BROGLIE) or if such a space-time co-ordination does not exist (as believed by BOlt~ and HEISENBERG). For this reason we intend to discuss the most recent experiment of the Vienna group (~) on the t ime-dependent neutron interferometry.

(*) European Physical Society Fellow. (**) On leave from the University of Crete, Physics Department, Heraclian, Crete, Grecee. (***) Work partially supported by M.P.I. and LN.F.N. (1) J . SUMMHAMMER, G. BADUlgEK, H. I~AUCH a n d O. liTISCHKO: ]Phys. Let t . .d , 90, 110 (1982); G. ]]AI)UI~EK, 1~. RAIYCH, J . SUMMHAMMER, 0. KISCIiKO a n d A. ZEILINGER: J. Phys..d, 16, 1133 (1983); J. SUI~MHAMMER, G. BADUREK, ]=[. RAUCH, O. ]~ISCHKO a n d A. ZEILINGER" Phys, Rev. A, 27, 2523 (1983). (2) G. BADUREK, H. :RAUCH and J . SUMMHAMMER g Phys. Rev. Lett., 51, 1015 (1983).

481

~ 2 C. DEWDNEY, A. KYPRIANIDIS, J . P. VIGIER ]~TC.

The exper imenta l a r rangement can be schemat ica l ly represented as follows: an inc ident neut ron beam conta in ing one neu t ron at a t ime (fig. 1) is subsequent ly divided into beams I and I I . On beam I there acts a nuclear phase shif ter represented by the act ion of a un i t a ry opera tor exp [iz] on V wi th Z = - - :Y" )." be. D, where be is the coherent sca t te r ing length, ). the neut ron wave- length , D the thickness of the phase shif ter and ) r the number of la t t i ce e l emen l s /vo lume element . Beam l I is subjected to the fol lowing combina t ion of magnet ic fields: a) a s ta t ic magnet ic field iu the -}- z direct ion B = (0, 0, B0); b) a rad iof requency t ime-dependen t magnet ic field B~.~.= (Bz'cose)r.f. ' t , Bl'sine)~.~.'t, 0) ro ta t ing in the xy-plane wi th a f requency Wr.f., obeying the resonance condi t ion ?i(o~.~. = 2ffBo, where # is the magnet ic momen t of the neutron, i.e. i t yields exac t ly the Zeeman energy difference be tween the two-spin eigenstates of the neut ron wi th in the s ta t ic field. Neut rons passing through such a device (a spin flipper) reverse the i r in i t ia l + z polar iza t ion into the - - z direct ion, by t ransfer r ing an energy AE = 2fiB o to the coil, whilst ma in ta in ing the i r ini t ia l m o m e n t u m (2).

/ \

/ 0 B~ \ / \

/~-'] phcLse sh/ff.en ~ ]~\\

~ R F S,I_~F /

Fig. 1. - Ske tch of the spin superpos i t ion e x p e r i m e n t wi th a r a d i o f r e q u e n c y spin-flip device.

Thus, while ,~ w a v e funct ion of a neut ron in beam I af ter passing th rough the nuclear phase shi t ter is represented by (a)

the corresponding wave funct ion in beam I I af ter a spin-flip should be wr i t t en as

) ' / ~ n = e x P L h J l L } = e x P i T t .

(") G. EDER and A. ZEILINGER: NUOVO Cimen fo ZJ, 34, 76 (1976).

T I M E - D E P E N D E N T NEUTRON INTERFEROMETRY ETC. ~ 8 3

If we assume a 100% efficient radio-frequency spin flipper (RFSP) (4) the polari- zation behind the interferometer lies entirely in the xy-plane of fig. 1 and has the following pat tern (2):

1 p = ~ ~/I~aVf : (eos(og~.f..t-- Z), sin (Wr.,.'t-- Z), 0) ,

while the final intensi ty is constant. Since P is time dependent, a stroboscopic registra- tion is needed to convert the pattern stationary, and, by performing such a detection, one obtains the pat tern of fig. 2 as a function of the nuclear phase shift Z.

200

+

0

'~ -200 AA/V

-3 o-260-16o 16o 260

Fig. 2. - Stroboscopic picture of~ the interference pattern when the polarization component along x-axis is measured. Observed intensity difference between two phase-looked subintervals separated in time by half a period of r.f. field v s . the path difference AD of the interfering beams.

In the actual experiment it is not possible to measure the individual neutrons energy transfer, but only a cumulative energy transfer which correspond to a well-known (but different from one) number of (( particles )~ travelling along the second path and passing through the RFSF.

The interaction of a neutron with the RFSF constitutes a quasi-classical microscopic measurement process because

a) it involves energy (signal) exchange,

b) it modifies in a predictable way the state of the measured and of measuring device,

e) the exchanged energy AE is small compared to the neutron rest energy.

The importance of these kinds of experiments is now evident, because they reproduce the well-known situation of Young's double slit set-up with additional information derived from the interaction between the neutron and the RI~SF. Beeuuse of this additional fact it is of interest to test the explanatory abili ty of the different interpreta- tions of quantum mechanics on the grounds of this recent experiment of the Vienna

( ') H . RAUCH: Z. Phys., 197, 373 (1966); W . G. WILLIAMS a n d J . PENFOZD: Measurement o] the e]]iciencies of Mezei thermal neulron spin ]lippers, iN'BRU, J a n . 1973; B. ALEFELD, (]-. BADUREK and. l:[. RAUCII: Z. Phys. B, 41, 231 (1981).

4 ~ c. D:EWD2N:EY, A. KIPRIA.NIDIS, ,1. t:'. VIGI:ER ETC.

group. We distinguish here three such approaches: the Copenhagen (5), the statistical (s) and the Einstein-de Broglie model (7).

a) The Copenhage.~ interpretat ion. Adherents of the Copenhagen interpretation argue that what happens between source and detection when intereference is observed cannot be conceived in quantum description, which only concerns the statistical pre- diction of results in well-defined experiments. The fundamental uni t for description in these terms is the whole <~ phenomenon ~>, constituted by the system and the experi- mental apparatus which together form an indivisible and unanalysable whole.

Thus in spinor supcrposition of neutrons either we design an apparatus to observe interference and forgo a description in terms of space-time co-ordination or we design an incompatible arrangement to determine the space-time motion and forgo the pos- sibility of observing interference. Any at tempt to subdivide these phenomena leads to ambiguities. The two are complementary phenomena. Complementary is to be understood in this manner, not according to the dictum that matter never reveals its particle and wave aspects together.

The quantum world has no independent real existence. The wave function ~]~ is the most complete description of an individual that can be given. I t is merely a proba- bility amplitude which states the odds on various results and is subject to instantaneous changes on measurement. If sonle preparation device (source, shutter, collimator) is designed to produce a wave packet then all we can say is that the wave packet represents the fact that a single particle has a probabili ty of appearing at a position ~ given by IT (e ) l -~ i/a meas~,reme~t is made. Until such a t ime it is not legitimate even to conceive of a particle, let alone its properties. For the Copenhagen interpretation it is thus impossible to us~ this experiment for a simultaneous detection of both a path of particle and a self-interference pattern. This view is founded also on the grounds of the Heisen- berg uncertainty relations, i.e. iu the present case the so-called fifth, phase-number uncertainty relation AN.A~ ~> 2.~. Indeed, this argumentation claims that the know- ledge of the accurate phase of the radio-frequency field needed for the stroboscopic registration of the oscillating polarization pattern destroys the possibility of the detection of a single-photon transition because AN hecomes indeterminate.

b) The Stat is t ical i~terpretation,. As emphasized by BALLENTIN:E (6) the statistical interpretation is to be distinguished from the Copenhagen interpretation. He asserts that the wave function simply represents an ensemble of similarity systems and does not provide a complete description of an individual system.

(~ In general, quantum theory predicts nothing which is relevant to a single measure- ment ~) (6).

The interpretation of a wave packet is that although each particle has always a definite position ~, each position is realized with relative frequency IT(~)I 2 in an ensemble of similarily prepared experiments. I t follows that each particle has ,~ well-defined trajectory, but its specification is beyond the statistical quantum theory, probabilities arising in the predictions of the theory are to be interpretated as in classical theory.

C) The Eins te in-de Broglie )nodel. In this interpretation it is argued that the quantum-mechanical description, through the wave function, of an individual is incom- plete in the sense of Einstein. The entities of the micro-world are thought of as being particles and waves, in the sense given by de Broglie in his model of oscillators accom- pained by a real physical wave (the de Broglie <( pilot )) wave) beating in phase. Quantum

(5) 1%. Bomb: .Atomic Physics and the Description of Nature, CUP (1934). (~) L . E . BALLANTINE: RaY. 3led. Phys., 42, 358 (1970). (~) L. DE BROGLIE; Nonlinear Wave Mechanics (Elsevier Publ. Co., Houston, Tax., 1960).

TIME-DEPENDENT NEUTRON INTERFEROMETRY ETC. 485

phenomena can be described in this model by means of space-time pictures and the neutron interference experiments can be conceived in a straightforward way: while particles travel along one of the paths in the interferometer the <( pilot )> wave propagatcs on both.

We now claim that the experimental facts established by the Vienna group indeed support this interpretation, namely along the following lines of reasoning.

i) A detection of an energy amount Edot by the RFSF implies tha t this amount of energy has been transferred to the coil by the neutrons involved in the experiment.

ii) A coil absorb energy only at its reasonance frequency %.f., i . e . the energy transfer has occurred as a series of single-energy transfers ~%.f. = A E , i .e . as a series of energy transfers, corresponding to the Zeeman energy splitting. This implies that Edo~ is a sum of equal individual energy transfers corresponding to a spin-flip of each individual neutron, i .e . Eaot = n A E .

iii) Consequently the energy Eao~ corresponds to a sum of ~ spin-flip, hence neutrons have passed through the path containing the RFSF coil.

iv) Therefore, if N neutrons are successively involved in the experiment, 2 V - ~b neutrons have passed through the path without the RFSF coil.

v) By means of this measurement one cannot tell which neutron has gone through which path, but one establishes the following: Out of N neutrons involved in the ex- periment n neutrons pass through path I I and N - ~ through path I. Every neutron has a probabili ty given by the transmission/reflection coefficient of the first incident plane in the interferometer of going in I or I I , but it either goes through path I or through path II .

vi) Since now neutron self-interference persists and shows that (, each neutron in the aIea of interference knows simultaneously what has happened in both paths ~> (s), this implies that something which has a real physical existence independent of the particle travels along both paths and contributes to the forming of the interference.

This at least prooves the incompleteness of the quantum-mechanical Copenhagen description because the persistence of an interference pat tern is combined with the existence of a definite trajectory for each particle, a fact forbidden in Copehangen interpretation.

Of course the experiment only represent an indirect argument in favour of the Ein- stein-de Broglie point of view and one can legitimately feel that a final proof of the existence of such paths requires the individual detection of passage of each neutron in the RFSF coil, i.e. the detection of photons of energies ~ 1 ~ev; this may indeed be possible by using superconducting quantum interference detectors (SQUID).

If every individual neutron energy transfer is measured, the RFSF will behave as a yes-no device and two conflicting results are possible.

The Copenhagen interpretation of quantum mechanics in this case implies the wave packet collapse (9). This also implies the use of a mixture because each particle either follows path I or path I I which yields, for every individual case, either px = 1, PH = 0 or p~ = 0, Pn = 1, with a relative frequency determinated by the reflexion/ transmission coefficient of the first incident plane. This means that, since each particle

(s) H. RAUOH: Seminar given at the Insti lul H. Poincard, Paris, 21.2.198~. (') ~. BOLEV: ,Atomic Physics and Human Knowledge (Wiley and Sons, New York, N.Y., 1958).

4 ~ 6 C. D:EW'DNEY, A. XYPRIANIDIS, J . P. V][GIJ~K ]4TC.

has a def ini te t r a j e c t o r y in the a p p a r a t u s , t he p r o b a b i l i t y wave collapses in the o t h e r p a t h .

Th i s s i t ua t i on is ca lcu la ted as follows. W h e n ene rgy is t r a n s f e r r e d to t h e coil, t h e n e u t r o n h a s passed a n d t h e w a v e is desc r ibed b y t h e s t a t e ~: = exp [i(AE/f~)] I4~}

w i t h no wave co r r e spond ing to p a t h I. W h e n no ene rgy is t r ans fe r red , t he wave is g iven b y V n = exp [i7.] I ~ ) w i t h no wave co r re spond ing to p a t h I I .

The co r r e spond ing p robab i l i t i e s are l )~, pH w i t h p:- l - p n = 1, a n d if t he t r a n s - miss ion/ ref lec t ion coefficient in t he first p l a t e of t he i n t e r f e r o m e t e r is equal , one ha s to a s sume p x = p n = �89 U n d e r t he a s s u m p t i o n we can ca lcu la te th i s i n t e r f e r ence i n t e n s i t y

I ~- pl<vi[%~l) + /)n<q;iilTll ) = iol @ 1)n

a n d co r r e spond ing ly t he p o l a r i z a t i o n

/~ = pI(o, O, 1) + pn(O, O, - - 1) .

One can i m m e d i a t e l y see t h a t , w i t h i n th i s k i n d of Ansa tz , t he osc i l la t ing p a t t e r n of t h e in t e r f e rence i n t e n s i t y van i shes comple te ly a n d the po l a r i z a t i on p a t t e r n consis ts aga in of two c o n s t a n t c o n t r i b u t i o n s d i rec ted a long t h e z-axis (in t he + z a n d - - z d i rect ions) . One should also no t ice t h a t t he t heo re t i c a l l y p red ic ted p a t t e r n exh ib i t s no in t e r f e rence effects (due to t h e lack of ove r l ap be tween t he wave func t i ons on each p a t h ) a n d consis ts of two d i s t i nc t sets of poin ts , t he r e l a t i ve f r equency of e ach set b e i n g specified b y the a s sumed p robab i l i t i e s p~ a n d pU. F i n a l l y i t shou ld be s t ressed t h a t th i s comple t e de s t r uc t i on of a n y in t e r f e r ence is a s t r a i g h t f o r w a r d consequence of a pe r fec t ly work ing m e a s u r i n g device (100~ efficient coil) in B o h r ' s wave p a c k e t col- l apse t heo ry .

The E in s t e in -d e Brogl ie model excludes t he poss ib i l i ty of collapse of the rea l de Brogl ie <~pilot ,) wave and, therefore , in th i s second case p red ic t s t h e de t ec t ed in te r - ference (i.e. po l a r i z a t i on in t he xy-plane) , and impl ies a s i m u l t a n e o u s de tec t ion of p a t h a n d in t e r f e r ence p a t t e r n (*).

As for t he theo re t i ca l objec t ions , based on t he p h a s e - n u m b e r unce r t a in ly , conce rn ing such a s i m u l t a n e o u s de tec t ion , i t is b y no m e a n s c lear t h a t th i s f i f th u n c e r t a i n t y r e l a t i o n is correct . I n d e e d th i s re la t ion , a l r e a d y con te s t ed b y DE B n 0 G H E (u), is shown to be inco r rec t b y C~taRUTHERS a n d NIETO (12) and m u s t be s u b s t i t u t e d b y a more complex one wh ich does no t reduce to A N . A ~ >~ 2.n in t h e case e x a m i n e d here (:3). F u r t h e r m o r e , fo l lowing a sugges t ion p r e s e n t e d e lsewhere (:4) we can by-pass th i s rela- t ion a n d t h e r e l a t ed p rob lems .

(*) This problem remains in the statistical interpretation, which also admits definite (but unknown) particle trajectories in each individual case, the wave referring only to ensemble probabilities, statistical frequenees. In Young's double-slit experiment the same problem is resolved by refer- ence to Duane's extension (,0) of the Bohr-Sommcrfeld theory. In the present case i t is not clear how a time ensemble of spin-up and spin-down particles results in an ensemble in which the polariza- tion lles in the xy-plaue. (~0) "W. DUANE: Proc. Natl. Acad. Sci. USA, 9. 158 (1923). (*l) L. DE BROGLIE: Wave Mechanics, the First 50 Years (Bathemorths, London, 1973), Chapt. 5. (,2) 1 ~. CAI~RUTHERS and 5L •IETHO: Reg. ~]lod. Phys., 40, 4:11 (1968). (is) C. DEV~'DNEY, A. GARUCCIO, PH. GUERET, A. KYPRIANIDIS and J.-P. ~rIGIER: Time-depe*~denl neutron inter/erometry: Evidence Agai~sf IVave Packet Collapse, to be published. (~a) C. DIXIWDNlgY, PH. GUERIgT, A. ]bXYPRIANIDIS and J.-P. VIGIER: Testing ]Vave-ParlicIe Dualis~ with Time Dependent Neulror~ Inter/erometrg, Phys. Left. A., 102, 291 (198~).

TIME-DEPENDENT NEUTRON INTERFEROMETRY ETC. 4 8 7

We wish to conclude with the following remark. Even if the existing experimental evidence has not direct ly proved the real physical existence of the de Broglie waves, i t has at least shown tha t nei ther the Copenhagen nor the s ta t is t ical in terpreta t ion provide an adequate description of neutron spinor superposition interferometry. On the other hand, the Einstein-de Broglie model can provide a sat isfactory intui t ive description and i t seems tha t further experiments will provide a posit ive confirmation of this model.

The authors wish to thank Prof. H. RAuC~ for many discussion and helpful sug- gestions. One of the authors (AK) wants to thank the French government for a grant and another (CD)wishes to thank the Royal Society for the European Exchange Fel- lowship award which enabled him to do this research.