Volume 157, number 4,5 PHYSICS LETTERS A 29 July 1991

Comment

Comment on "Proposed neutron interferometry test of Einstein's "Einweg" assumption in the Bohr-Einstein controversy"

Peter B. Lerner Physics Department, University of Tennessee, 401 A.H. Nielsen Bid., Knoxville, TN 37996-1206 USA

Received 15 March 1991; accepted for publication 11 June 1991 Communicated by J.P. Vigier

The gedankenexperiment recently proposed by Vigier and Rauch is not necessarily supportmg Einstein's Einweg paradigm. It can be more easily comprehended within the traditional framework of quantum mechanics.

The proposition for experiment [ 1 ] made by Vigier and Rauch is remarkable, since all the com- ponents of this gedankenexperiment were tested ex- perimentally by Rauch and his coworkers [2,3 ]. This arouses the hope that the experiment will soon pro- vide the expected results, which are: the simultane- ous observation of the interference pattern and the quantized recoil from the spin-flipping coils inside the arms of the neutron interferometer.

The interpretation given by Unnerstall [4 ] seems quite satisfactory to me, yet, I would like to put it in more transparent form.

Vigier seems to be in favor of the hypothesis sug- gesting that if the (measurable) energy gain is equal to hco, it proves that the neutron have been passed only one coil during its flight. This, in his view, prompts that the trajectory of the neutron inside the interferometer is definite, while quantum mechanics says nothing about it.

This is simply not true! The energy gain AE is the following:

AE= •O)rf.l WL, I Psf, l + hO)rf,2 WL,2Psf,2 , ( 1 )

since the quantum states of the if-field inside the coils are non-interfering. Here, numbers 1 and 2 are as- signed with respect to the coils, tOe.l,2 are the oper- ation frequencies of the spin-flippers, P~f,~,2 are the probabilities of the spin-flip, and WL, I,2 are the prob-

abilities of the localization of the neutron inside the coil 1 or 2. This should be added both in classical and quantum mechanics, since the probabilities P~f,l,2 are calculated in assumption that there is a neutron inside the spin-flipping coil.

The wavefunction of the neutron inside the inter- ferometer with 50-50% beam splitter can be ex- pressed as

I¢/) =2-1/2[11 , ( + - ) ) - i e i Z l 2 , ( + - ) ) ] ,

11)=11)~®10)2 , 1 2 ) = 1 0 ) ~ ® 1 1 ) 2 , (2)

the numbers 1, 2 are now referring to the arms of the interferometer, the numbers 0 and 1 are the pure number states of the neutron on each arm, Z is the phase difference inside the arms of the interferom- eter, and the signatures ( + - ) are referring to the "spin-up" and "spin-down" states of the neutron. This could be omitted, since in the proposition of ref. [ 1 ] the spin states in different arms are always the same, due to 100% efficiency of the flipper. A more detailed description of the construction of the particle wavefunction in interferometers is found in refs. [5,6].

The energy gain in the conditions of experiment [ 1 ] is given, when the frequencies of the rf field are said to be equal and the probabilities of the spin flip are also equal to 100%. The probability localization

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Volume 157, number 4,5 PHYSICS LETTERS A 29 July 1991

inside each coil is expressed by the formulas

WL.,= (11~'), WL.2= ( 2 I y / ) and is equal to ½. It is independent of the phase difference inside the arms. The energy gain according to ( 1 ) is, obviously, equal to

A E = hoax ½ X 1 + h o a x ½ X 1 =boa . (3)

I f the coils are operat ing with slightly different fre- quencies with appropr ia te adjustment of their lengths ( t ime-of-f l ight) , and magnet ic field strength, the wavefunct ion (af ter the event o f the energy transfer) will take the form

I~" ) = 2 - t / 2 { e x p [ - i ( o a - o a r . l ) t ] l l , -- )

+ e x p [ - - i ( o a - - o a r , 2 ) t + i z ] 12, - ) } , (4 )

are not interfering with each other. In the case of a par t icular selection, one should interpret the square modulus of the wavefunction, propor t ional ei ther to condi t ional or absolute probabi l i t ies o f the event. Einstein 's concept of Einweg is, in principle, the in- tui t ive idea that the pre-selected and post-selected ensembles are somehow related by a certain pre- script ion called " t ra jec tory of the par t ic le" inside the interferometer . This is obvious "mul t ip l ica t ion of essences". It applies, because of the absence of in- terference of the quan tum states of the electromag- netic field in the coils in the proposed exper iment ,1, to the quan tum mechanical uncer ta inty as well as to the statistical uncer ta inty and should not be in- cluded in the formal ism of quantum mechanics.

and the beats will proceed with frequency ~ : oar, l - O)r,2 in spite o f the fact that local izat ion o f the neutron does not happen with 100% probabil i ty . This is because we have al ready selected an ensemble of interfering neutrons which have s imilar direct ion of spins, or, what is the same, we al ready registered the events o f the energy transfer in each coil. In other words, the interference pat tern (z-dependent part of ( ~ u ' l ~ ' ) ) is a lready built by the neutrons which undergo t ransi t ion in both coils. As it was correctly stated by Unnerstall , this experiment is not a Welcher Weg experiment . Any a t tempts to make a Welcher Weg exper iment will lead to the al ternat ives de- scribed in ref. [ 7 ].

All this can be put in modern parlance of the "pre- cision QM tests" communi ty (see e.g. ref. [8] ). I f one hopes to interpret I~'1: as a probabi l i ty density, one should select an ensemble o f events. This en- semble could be pre-selected or post-selected with re- spect to the exper iment . For instance, pre-selected ensembles contain the states of neutrons which d id not undergo the spin flip in one o f the coils. How- ever, a post-selected ensemble excludes such states. This is true both in the case o f statist ical or quan tum mechanical uncer ta inty and the choice of the arms by the neutron, since the quan tum state of the coils

I use this possibi l i ty to express my indebtness to the kind hospitali ty of Professor Rauch and his group during my stay in Vienna.

~1 However, it is possible to imagine an experiment in which the energy transfer in each arm of the interferometer is provided by the common field of the single cavity. Interaction of the neutron spin with rf-field is, indeed, too weak for observable effect (P.B. Lerner, private communication to H. Rauch, 1989 ), nevertheless it could be achieved through Ramsey-type schemes involving Rydberg masers [9].

References

[ 1 ] H. Rauch and J.P. Vigier, Phys. Len. A 151 (1990) 269. [2] G. Badurek, H. Rauch and D. Tuppinger, Phys. Rev. A 43

(1990) 2600. [3] H. Weinfurter, G. Badurek, H. Rauch and D. Schwann, Z.

Phys. B 72 (1988) 195. [4] T. Unnerstall, Phys. Lett. A 151 (1990) 263. [ 5 ] T. Hellmuth, H. Walther, A. Zajonc and W. Schleich, Phys.

Rev. A 35 (1987) 2532. [ 6 ] B.C. Sanders and G.J. Milburn, Phys. Rev. A 39 ( 1989 ) 694. [ 7 ] M.O. Scully, B.G. Englert and J. Schwinger, Phys. Rev. A 40

(1989) 1175; M.O. Scully and H. Walther, Phys. Rev. A 39 (1989) 5229.

[ 8 ] D.T. Pegg and P.L. Knight, Phys. Rev. A 37 (1988) 4303. [9] M.O. Scully et al., Phys. Rev. A 34 (1986) 2032.

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