Possible observation of the quantum Zeno effect by means of neutron spin-flipping

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Vistas in Astronomy, Vol. 37, pp. 273-276, 1993 Printed in Great Britain. All rights reserved.

POSSIBLE OBSERVATION OF THE QUANTUM ZENO EFFECT BY MEANS OF NEUTRON SPIN,FLIPPING Seizo l n a g a k i , M i k i o N a m i k i a n d T o m o h i r o Tajiri Department of Physics, Waseda University, Tokyo 169, Japan

In a recent paper, Itano, Heinzen, Bollinger and Wineland (IHBW) claimed that they observed experimentally the quantum Zeno effect by making use of atomic transitions. Their conclusion was opposed by Petrosky, Tasaki and Prigogine who proved, via a detailed theoretical calculation, that the experimental result in the IHBW's paper is just the consequence of a succession of dynamical changes of the wave function, and therefore does •not necessarily be ascribed to any collapse of the wave function. The same conclusion was also drawn by Ballentine on the basis of a rather general discussion on the meaning of wave function collapse by quantum measurement. In the above-mentioned experiment, 9Be+ ions were put in a rf cavity. The ion energy level configuration was such that E1 < E2 < E3, and the resonating if-field frequency w2 = (E2 - E1)/h created a coherent snperposition state of the two lower levels. If we denote the probability of finding the atom in level 1(2) at time ~ by Pl(t) (P2(t)), then we have Pl(t) + P2(t) = 1. If the initial condition PI(0) = 1 (P2(0) = 0) is chosen, it is possible, under appropriate conditions, to find a time T for which the situation is completely reversed, so that PI(T) = 0 (P~(T) = 1). Remember that Pl(t) gradually decreases from 1 to 0 during the time interval (0, T). In order to observe the state of the atom, IHBW irradiated it with very short optical pulses of frequency w3 = (E3 - El)/h, and chose the level configuration in such a way that the spontaneous decay 3 ---* 1 was strongly favoured, while the transition 3 *-* 2 was forbidden. In this way, the atom is known to be in the first level if a photon with frequency w3 is observed, while it is in the second level if no photon is observed. According to the quantum measurement theory, the wave function collapse takes place as a consequence of observation, and consequently the density matrix of the atom loses its off-diagonal components (with respect to the first and second states). If n observations are performed during the time interval (0,T), we can prove that the probabilities of finding the atom in the first and second level, respectively, are given by

P~(T)

=

P~2(T) =

l(l+cos"~), (1 - cos"

5/ n

(1)

Note that P~(T) = PI(T) = 0 and P~(T) = P2(T) = 1. The fact that P~(T) > PI(T) = 0 for n > 2 and lirr~_~ P~,(T) --- 1 disolavs the ouantum Zeno effect: Freauent observations

S. Inagaki et al.

274

on the system '~freeze" the system in its initial state, by inhibiting and finally hindering transitions to other states. In general, a quantum measurement consists of two steps, the first provoking a spectral decomposition, and the second being the detection itself. For instance, in a Stern-Gerlach experiment, the inhomogeneous magnetic field yields the spectral decomposition relative to the up and down spin states, and this is followed by the detection of one of the two states. We consider that the concept of the spectral decomposition plays an important role on the understanding of the IHBW's experiment. At first, let us introduce now the setup of Fig.l, where C (~) stands for a spin flipper(its magnetic field strength B1 along the x direction), R (~) for a reflector for a neutron wave, M (~) for a decomposer in order to split a neutron wave with indefinite spin into two branch waves, S (~) for a superposer of two branch waves with definite spins into a superposed spin state, and D for a detector. The whole system is put in a static magnetic field of strength B0 along the z direction, in order to make the initial spin-state stable. These equipments, such as R, M, S, axe recently invented, mainly effective for neutrons with a rather long wavelength(A -~ 20/~). We denote the length of C (°) by l(~), the distance between C (~) and C (~+1) by I~(~), and the corresponding travel times by ~(~) = l('~)/v and t ~ ) = l'(~')/v, where v is the neutron speed, respectively. We dispose this setup so that l '('~)

=

tO ) =

(No integer)

2~rhvNo/[~lBo,

t(2) . . . . .

=

t(,)__, lr ?20)

T

=

(2)

~ + t ' O ) + t ( 2 ) + . . . + t ~'), 0)

where 0) = 2[/~[B/h, B = ~ + B~, and T is the final time, when the neutron is observed in the detector D. When the initial spin state is the down one (P~(0) = 1, P t ( 0 ) = 0 ), the probabilities for the down and up states at time T, respectively, are give by

Bg

P~(T) = -~,

PT(T)= ~-~,

(3)

PT(T) = 1,

(4)

which yield

PI(T) = O, in the B1 >> B0 limit. I~,(2)

C (')

M"'

S °'

C '2'

M ¢:)

R.,-~)

S ~''

S ~"')

C ~"

M (")

Fig, [ . The reference experimenl.

Secondly, we propose the experiment of Fig.2 where the spectral decomposers is inserted between the spin-flippers of one step and the ones of next step. We design again the experimental setup so that the conditions in eq.(2) hold ( here, l (°) is the length of C!~) (equal Vi) and 1'(°) is the distance between C~=) and ~ - 1 I°r "~2~ j ). After a lengthy but straightforward manipulation we get the following probabilities of detecting the neutron in

The Quantum Zeno Effect down and up state, respectively, at time T, P~(T)

=

P~T(T) =

275

~ 1+

cos-+2n

sin2

'

~1 1 ,

c o s ~r - + 2~_~2 sins n ~n

'

(5)

where we have supposed that the wave packets do not overlap after any spectral decomposition. If B1 )~ B0, eq.(5) becomes P~I(T)=~

l+cos ~

,

P~T(T) = ~1 ( 1 - cos~ ~ )

(6)

C_,~'~

M2~ i

.

C~"'

M~t"

C:':'

.

.

.

.

M., ~:~

Fill.~. A series o f sl,emral

Cr,t '~

Mr,~.~

dL~.'omposilions.

This change of probabilities at time T, from eqs.(4) to eqs.(6), itself is the same one which the optical pulses in IttBW's experiment yield. From this point, the IHBW's experiment has much analogy with the one of neutron spin-flipping which we propose in Fig.2. The irradiation of a short optical pulse in their experiment can be regarded as a spectral decomposition, because level 1 is "separated" from level 2 by exciting the third level. However, in order to know that the atom is in the first level, we have to observe an wa photon: This would be the detection step. Consequently, the fact that IHBW observed the photon only after n pulses and not at every step means that they performed only a series of spectral decompositions followed by one final detection. This is our understanding of their experiment, from the measurement-theoretical point of view. In the experiment of the neutron spin-flipping, by the setup of Fig.3, we can show the occurrence of the true quantum Zeno effect which stems from the collapse of the wave function. Note that unlike the previous case, the coherence of the spin states has lost at every step by a series of detectors D(a). It is then easy to check that eqs.(5) and (6) for the probabilities of detecting a down or up spin at time T, hold true even in the case of Fig.3.

276

S. Inagaki et al. "R(U

C"~

M~t~

R(2)

R(..~I

SO~ C ~-", M e, S~2~ S~..t~ C~ Fig. 3. Quantum Zeno effect by means of neutron spin-flipping.

M ~.,

Let us now discuss the meaning of our results. As previously stated, the experiment performed by IttBW is immediately identified with the one described in Fig.2, if the atomic levels 1, 2 and a photon emitted in the 3 ~ 1 transition are identified with the spin down, spin up and the spatial component of the neutron wave function. In the light of our discussion therefore IHBW's experiment is completely equivalent to a series of spectral decompositions, and not to the wave function collapse. As far as no detector is placed at any intermediate step, the experiment described in Figs. 1 and 2 are conceptually similar, and imply no loss of coherence. The situation is different in the case outlined in Fig.3, in which coherence is lost many times, due to the presence of the detectors, but the final expression for the detection probabilities for both Figs. 2 and 3 is identical to the previous cases. This is indeed a very peculiar property of the above complete spectral decomposition, and sheds new light on the meaning of the IHBW's experiment. Finally we have to remark that we cannot realize any experiment schematized in Fig.3 unless we use nondestructive detector for neutrons. It is worth discussing the possibility of making nondestructive detectors for neutrons. This is an open problem. Rigorously speaking, we may have to invent a neutron detector sensitive to the neutron magnetic moment. REFERENCES Inagaki, S., Namiki, M., and Tajiri, T.(1992) Phys. Lett. A166 5. Itano, W.M., Heinzen, D.J., Bollinger, J.J., and Wineland, D.J.(1990) Phys. Rev. A41 2295. Petrosky, T., Tasaki, S., and Prigogine, I.(1990) Phys. Lett. A151 109. Ballentine, L.E.(1991) Phys. Rev. A43 5165. Machida, S. and Namiki, M.(1980) Prog. Theor. Phys.63 1457; 63 1833. Namiki, M.(1988) Found. Phys. 18 29. Namiki, M. and Pascazio, S.(1991) Phys. Rev. A44 39; (1992) Found. Phys. 22 452. Yamada, S., Ebisawa, T., Achiwa, N., Al~iyoshi, T. and Okamoto, S.(1978) Ann. Rep. Res. Reactor Inst. Kyoto Vniv.ll (1978) 8. Ebisawa, T., Achiwa, N., Yamada, S., Akiyoshi, T. and Okamoto, S.(1979) J. Nucl. Sci. Tech. 16 647; (1991) Report of the 2th Meeting of the Workshop on "the Developments and Applications of Cold and Ultracold Neutrons" in Kyoto University Research Reactor Institute (in Japanese) 130.