IL NUOVO CIMENTO

VOL. 105 B, N. 5

Maggio 1990

Damping Effects on the Coherence Term and Wave Packet Reduction. Y. MORIKAWA

Department of Physics, Waseda University - Tokyo 169, Japan Y. OTAKE

Department of Electronic and Computer Engineering Ibaraki National College of Technology - Katsuta, Ibaraki 312, Japan (ricevuto il 25 Luglio 1989)

Summary. - - We analyse the damping effects of devices such as an rf-coil and Ge or Bi crystals on the interference fringes in the interference-type experiments. We investigate them in terms of the interaction between the device and the wave packets. We also consider the fluctuations deriving from the macroscopic characteristics of devices, and analyse the damping effects. The visibilities of the interference fringes are calculated and compared with the results of the neutron interference experiment. We get reliable results which indicate the tendency of the results measured by the MissouriColumbia group and it is certain that our method is more suitable in order to investigate experiments such as those which make use of wave packets. The intensity of the wave packet is also discussed. The difference between the intensity of a beam made up of several single plane waves and the intensity of a real wave packet such as the one describing pulsed neutrons is pointed out.

PACS 03.65 - Quantum theory; quantum mechanics. PACS 03.65.Bz - Foundations, theory of measurement, miscellaneous theories.

1. - I n t r o d u c t i o n .

Q u a n t u m mechanics has had serious difficulties since the beginning, its theoretical predictions h a v e coincided closely with the r e s u l t s of e x p e r i m e n t s . W h e t h e r the ,(wave p a c k e t r e d u c t i o n , of t h e m e a s u r e m e n t can be explained within the f r a m e w o r k of q u a n t u m mechanics itself or

though various process not h a s 507

508

Y. MORIKAWA

and

Y. OTAKE

been one of these difficulties. Wave packet reduction can be understood as a sort of change from pure states into mixed states. And it can be understood as a disappearance of the coherence or the phase correlation between some eigenstates, when the object system is described by a superposed state of eigenstates of some observables. The wave packet reduction has been explained by means of the interactions between the apparatus system A and the object system Q, in other words, the macroscopic system with fluctuations and the microscopic system[I-3]. In ,,Many Hilbert Spaces Theory, [1, 2] Machida-Namiki pointed out that the wave packet reduction takes place through the interaction of the object system with one local system with macroscopic characteristics of the apparatus system A but not with the whole A. There are some fluctuations in this local system deriving from its macroscopic characteristics, for example the number of the constituent particles of it SN and/or the strengths of interactions between each constituent particle and the object system ~V and/or the distances of these particles ~a and so on. These fluctuations emerge in size parameter l as ~l. In their ~,Many Hilbert Spaces Theory,,, they are able to account for wave packet reduction through a procedure for averaging this fluctuation of the apparatus system in terms of size parameter 1. This theory formulates the criterion of the wave packet reduction as follows. When the fluctuation of the size parameter 1 is given as a distribution width ~l, the simplest form of the criterion for wave packet reduction can be formulated as K~l >> 1, where K stands for an effective wave number. On the other hand, under the condition of K~l ~< 1 wave packet reduction does not occur and the coherence remains. ,,Many Hilbert Spaces Theory,) makes the analysis of actual measurement processes possible by using the criterion above, and enables us to analyse whether a dynamical change, that occurs not only during the measurement process but during the interaction between the macroscopic system and the microscopic system, causes wave packet reduction or not. During earlier research [2] in which one of the authors was involved, the neutron interference experiment with spin-flipper by the Vienna group [4] was analysed producing a conclusion contrary to Vigier's interpretation [5]. We pointed out that K~l ~<1 was satisfied under the conditions of this experiment and that therefore although the spin flipped, wave packet reduction relative to the determination of passage of the neutron did not occur. We made it clear that the phenomena with dynamical change, for instance the energy exchange between the neutron and the oscillating field, does not always cause wave packet reduction. Of course, when the physical condition of the interaction between the neutron and the oscillating field in the g-coil satisfies the criterion for the wave packet reduction, the wave packet reduction will occur, and so we cannot observe the interference pattern. Here, the rf-coil is not the detector in a usual sense, but the argument discussed above is based on the concept that the rf-coil might work as the detector which reveals the passage of the neutron in the interferometer. This

DAMPING EFFECTS ON THE COHERENCE TERM ETC.

509

made it clear to us that we should investigate carefully the interaction between the microscopic particle and not only the detector but the macroscopic system, for example devices such as the rf-coil. Especially, when the neutron beam is not precisely monochromatic, we must investigate the interaction between the device and the wave packet carefully [6]. Neutron interference experiments where attention was paid to the fact that the neutron beam is not precisely monochromatic, for instance the experiment by the Missouri-Columbia group [7] and by the Vienna group [8], have produced some interesting results. The Missouri-Columbia group measured the damping of an interference pattern relative to the spatial difference hx by putting a thick slab of material in the interferometer. They measured the longitudinal coherence length from the visibility of the interference fringes. Recently the experiment of pulsed neutron interferometry, where the wave number deviation ~k is not so small that we must treat the incident particle as a wave packet, was done by the Vienna group [8]. They observed interference patterns for neutron pulses. The damping of the interference pattern for the whole pulse and for a time interval of 2 ~s are also measured in this experiment. In both experiments the damping of the interference pattern of the wave packet was measured. Actually, we can only judge whether the wave packet reduction occurs or not from the visibility of the interference pattern in the experiments. In other words, if the interference pattern disappears, we judge that the wave packet reduction occurs. From this point of view, the wave packet reduction is explained by regarding the object system as a plane wave. But, when the object particle is considered as a wave packet, is it certain that the disappearance of the interference pattern is equivalent to the wave packet reduction? We doubt whether this is always the case even when the object particle is considered as a wave packet. Here we investigate the visibility of the interference pattern of the wave packet. Our interest is focused upon the effects of such devices as the spin flipper and Ge or Bi crystals [9] on the coherence of the object particle which is really described not by the wave function of the plane wave but by that of the wave packet. As mentioned before it is known that the fluctuation of a macroscopic system, such as the spin-flipper or the crystal plate, plays an essential role in the wave packet reduction when the object system is described by the wave function of the plane wave. The wave packet has the wave number deviation ~k itself. W e propose to show how ~k affects the visibility of the interference fringes in this paper. In sect. 2 we discuss the damping effects of the interference pattern on the wave packet. We show that the damping effect, which is due to the optical-path difference, is not the cause of the wave packet reduction. In sect. 3 we take into account fluctuations, such as macroscopic characteristics of the devices, and

510

Y. MORIKAWAand Y. OTAKE

discuss the damping e f f e c t s . In sect. 4 we present our results in terms of visibility. The difference between the intensity of an assembly of the plane wave and the intensity of the wave packet is also analysed. Section 5 constitutes the conclusion of this paper.

2. - The damping effects o f the interference pattern on the wave packet.

We will now discuss the interference-type experiment shown in fig. 1, where the incident beam is described by the wave function of the wave packet. We put a device such as the rf-coil or a kind of crystal plate in one path of the interferometer. This device is the macroscopic system and the object system

exp[/I] Fig. i. - The interference expe~ment of double slit type. G is a device in path I I and there

is also an auxiliary phase shifter in path I.

interacts with it. We investigate its effect on the visibility of the interference pattern of the superposed state F. J u s t for simplicity, we now assume that the incident wave packet is described by the Gaussian distribution function with its mean value of wave number ko and its deviation Sk. This wave packet is described as follows: (1)

in , t) F~o(X

-

1 ~dkexp 2(n2(~k)2)l/4

( k - ko)27 . 2~k-~ J e x p [ ~ ( k x - c o t ) ] ,

where ~o= E / h and E is the incident energy. After interaction with the device, the effect of the interaction emerges as the function of the wave number F ( k ) in the phase of the state of the wave packet. This formulation is a sort of generalized form of the Glauber approximation or the optical approximation[10, 11], as follows:

(2)

~ : t ( x , t)

(k - ko)____ 2 + iF(k) + i(kx - oJt)l. 2(~k)2

J

DAMPING EFFECTS ON THE COHERENCE TERM ETC,

511

When F(k) is the quadratic function of k, the shape of the wave packet changes, while it does not change with the linear function [12]. We consider F(k) to be the complex function of k in order to investigate the absorption of the object particle by the device during the interaction. As noted above, the experiment by the Missouri-Columbia group was a very interesting one. Though they mentioned that the neutron beam is not really a plane wave, they treated the effect of the interaction between the potentials of the device and t h e neutron by neglecting its wave number deviation ~k[7, 13, 14]. As a result, their spatial shift A x = - ~ D b c N / 2 r : becomes a constant phase shift, i.e. it is independent of Sk, where D is the thickness of the device, b~ is the scattering length and N is the particle density of aluminium. In this paper, we take into account the k-dependence of F(k), which implies considering the k-dependence of this spatial shift 5x. This aspect of the problem will be explained in more detail later in the paper. When ~k/ko is not very large, it is sufficient to take into account the second order of ~k/ko as, for instance, when the deviation of the wave packet is rather large as $k/ko = 20%, but the amplitude of the second order is still small ($k/ko) 2= 4%. We study two instances where F(k) is the linear function and the quadratic function of k as follows:

F(k') = a + ~o k' ,

(3) (4)

F(k') = a +

k•

~" k,2 ' k' + k-~o

where k = k0 + k' and - Sk ~
(5)

F(k) = ~(~) = - )~bcND.

We expand this in terms of k': (6)

F(k)=~()~)=

2r:b~ND k -

2=b~ND ~1 _ k ' [k'~2~ ko [ ko + ~-ko] J"

When we investigate the experiment with very small $k/ko, it is sufficient to consider the first-order dependence of k'. We adopt eq. (3) putting a = - / ~ = = - 2=bcND/ko. For the experiment with rather large ~k/ko, we adopt eq. (4) (*) In case of the phase shifter in the photon experiment as in the delayed choice experiment [15] or others, the interaction between the photon and the crystal or the phase shifter is represented by the index of the refraction. F(k) is described as F(k) o: k. With the condition of a = ], = 0 we can analyse such an interaction by F(k).

512

Y. MORIKAWA

and

Y. OTAKE

thus

27:beND (7)

~ = - ~ = Y=

k0

To investigate the effect of the device on the damping of the visibility of the interference pattern of the wave packet, we assume that the phase shifter inserted in path I only causes a constant phase shift Z, but, of course, this usually also has wave number dependence as it does when the atuminium phase shifter is used. If we want to take account of this dependence emerging in WI explicitly, we exchange X for F(k). Then we can easily see the effects of both the device and the phase shifter on the damping of the interference pattern. F(k) is derived from the interaction between the device and the object particle, and appears in the state of the object particle. So, the physical conditions of the experiment, i.e. the energy of the incident particle and/or the material of the device and so on, determine the form of F(k), and we can investigate the generalized case by using eqs. (3), (4) without specifying conditions of interaction in detail.

2" 1. The case of a linear function. - When F(k) is the linear function of k, the wave function in the interferometer is described as follows: (S)

1 exp [i(ko x - hk~ t/2m)]. ~gi(x,t) = 2(=8(~k)2),4

+i(x-vgt)k'l],

(9)

1 exp [i(k0 x - hk~ t/2m)]. Tii(x, t) = 2(=8($k)2)1/4

"[exp[ia]fdk'exp[-{2~k)2+i

2m Sht]k'2+i(x-va t+ k~) k ' ] ] 9

The constant term a is the same as • but we can find that fl/ko appears in the coefficient of k'. The coefficient of k' shows the position of the wave packet. This means that the optical path of the wave packet passing through path II is different from that of the wave packet passing through path I. The superposed state W(x, t ) = T~ + TH is described as follows: (10)

?F(x,t) =

exp [i (ko x - hk~ t/2m )]

1

2(=8($k)e)l/4

~/1/2(~k) 2 + iht/2m

9 exp [iz] exp - 4(1/2(~k)~ + iht/2m)

+ exp [i~1 exp - 4(1/2(~k)2 +

im/em)JJ"

513

DAMPING E F F E C T S ON T H E C O H E R E N C E T E R M ETC.

Then the observed intensity I becomes (11)

I-

1/~k\ ~o]

/ T * ( x , t)~F(x, t ) d x = 1 + exp - "4-[k--oo) /~J cos (X - a).

Here we find that the interference pattern is damped down to a degree proportional to the following term: (12)

1

Sk 2

exp [ _ ~ (~0) f12].

As mentioned above, the difference between these coefficients of k in the case of WI and WI~is equal to the optical-path difference between them, as is well known. This damping factor of the coherence term of the intensity equation (12) originates in the difference between the coefficients of k, in other words, it originates in the optical-path difference. If one compensates for this path difference [3/ko, i.e. if one arranges the length of path II to make it x - vgt, the intensity changes into eq. (13) (13)

I = 1 + cos (X - ~)-

Of course, this compensation is impossible to achieve in an actual neutron interferometer of Si crystal at present but, it may be possible in the case of ultracold neutron interferometry using the multilayer film of Ni-Ti [16] and a phase-grating [17]. If one inserts a device identical to that used in path II into path I, the optical paths of both wave functions become x - v~ t + ~/ko, and it therefore becomes possible to extinguish the optical-path difference. The magnitude of path difference can be estimated for some devices. For Ge, using the experimental value of the Vienna group ~0 = 1.8/~, D = 1 cm, we obtain [~/ko = 193 A. From this result, we can extinguish this damping factor (eq. (12)) by compensating for the optical path difference between T~ and T~. The interference fringes are restored. It might be thought that an effect identical to this compensation can be achieved by using a phase shifter which yields a constant phase. But, as we explained above, the compensation must be applied to the optical path which appears in the term describing the position of the wave packet, i.e. the coefficient of k in the wave function. The constant phase appears in the same term as ~ in the wave function. So, it is meaningless to insert a special phase shifter to yield a constant phase. The spatial phase shift Ax calculated by the Missouri-Columbia group was a constant phase shift. We confirm that the Ax calculated by this group cannot cause spatial shift as the coefficient of k in the phase of the wave function as they expected. We must emphasize that the results of the Missouri-Columbia group

514

Y. MORIKAWAand Y. OTAKE

show this damping phenomenon, but they did not investigate it in terms of the wave function of the wave packet as we have done here. Our study investigates the mechanism of the damping phenomena arising from their experiment. From our results, we can estimate the optical-path difference by eq. (7), i.e. ~/ko with the value of their experiment. Using the following values, ;to = 1.26/~, D = 1 cm, we obtain the value/3 = 267 and so [gko = 53.9/~. This is equivalent to their 5x, and it indicates that our procedure is the case for their experiment and that 5x is the term which appears in the term of the coefficient of k in F ( k ) . But it is not sufficient only to take into account the case where F ( k ) is a linear function when investigating the damping effect of their results. This problem is further discussed later in this section. The interference fringes are restored by compensating for the optical path and changing eq. (11) to eq. (13). Since this damping factor (eq. (12)) is removed by rearranging the optical path, this damping effect on the coherence term is not the same as the disappearance of the phase correlation. The wave packet reduction is a sort of change of states from pure states into mixed states, and this change is observed as the disappearance of the phase correlation. The disappearence of the phase correlation is not a restorable phenomenon. We can assert that this damping effect occurs as the disappearance of interference fringes in the experiment with the wave packet but that this effect is not the wave packet reduction. 2"2. T h e case o f a q u a d r a t i c f u n c t i o n . - We also study the damping effect when F ( k ) is the quadratic function of, k such as in eq. (4). Each wave function is described as follows: (14)

1 }Fi(x, t) -- 2(7:8(~k)~)1/4 exp [i (ko x - hk2ot/2m)] .

(15)

WH(x,t) - 2(~:a(~k)2)l/4 1 exp [i (ko x - hk~ t / 2 m ) ] .

9

The observed intensity is described as follows: (16)

I= 1 + (1 + b2)-li4exp - T r i o ]

1---~| 9cos x - ~ + O / 2 -

4~,ko] " l + b 2 J '

DAMPING EFFECTS ON THE COHERENCE TERM ETC.

515

where

[~k\ z b=-[-~o ) V,

(17)

O=tg -lb.

In this quadratic case there is also the damping factor as in eq. (12), but there is another term derived from the coefficient of the second order of k in F(k). As previously, the length of the optical path of path II is arranged to equal x - vg t, then the intensity is written as follows: (18)

I = 1 +

1 +

cos (z -

+ 0/2),

where (19)

~

[

9

This result shows that the damping factor of eq. (12) is also extinguished but there remains another damping factor which contains the coefficient of k '2 in F(k). The interference pattern cannot be restored completely in the case where F(k) is the quadratic function of k. From these two results, it is observable that when F(k) is the linear function of k, the damping of the visibility of the coherence pattern can be completely restored by compensating for the optical-path difference, but that when it is a quadratic function, the damping is retained and the coherence pattern cannot be completely restored by this compensation. The damping effect, which is restorable, can be considered as the reversible phenomenon, but the damping effect which is not restorable is the irreversible phenomenon. The wave packet reduction is the disappearance of the phase correlation and it is the change of states from pure states into mixed states. So the change of states cannot be reversed. We can conclude that the result of the interference experiment using the wave packet includes two types of damping, one is extinguishable and it is not related to wave packet reduction and the other is related to wave packet reduction.

3. - The damping effects with f l u c t u a t i o n s of the devices.

During the analyses above we did not consider fluctuations, such as macroscopic characteristics of the devices, which cause wave packet reduction in the case of a plane wave [1, 2]. We must take them into account to investigate the wave packet reduction in the case of the wave packet. Here, we take the potential series as a model of the device and use the

516

Y. MORIKAWA a n d Y. OTAKE

Glauber approximation.

F(k) is described as follows: F(k) =

(20) We expand

F(k)[18], using the following coefficients: mVNa ~_ _ gV. h2ko

~= - ~=7=

(21) Then,

mVNa h2k

F(k) is described in terms of k' as follows:

(22)

1-~o+

F(k')=-gV

t~07

j"

This potential series has some fluctuations such as the strength of the potentials ~V, the interval of potentials ~a, the number of the potentials SN, and so on as done in a computer simulation[19]. For simplicity, we consider that the fluctuation of potential strength only exists among the fluctuations above and that it has a Gaussian distribution with its central value 11o and its deviation SV:

(23)

~-'(Vo,3V) :

~ f

dVexp [ (V-

V2.(~v) ~-

L

17o)2]

~6-v7

-J

The states before superposition are described in terms of g and V as follows:

(24)

1 exp [i (ko x - hk~ t/2m)]. Ti(x, t) = 2(rca(Sk)2)1/4

" [ e x p [ i z ] f d k ' e x p [ - ( 2 ~ + i h t ] k~--~m '2 S

(25)

1 exp [i (ko x Tii(x, t) = 2(=3($k)2)i/4

9f d k ' ] d V

1 exp V2~(~V) 2

+fix-vgt)k']],

- hk~ t/2m)].

(v - g0)2~

- ~ - ~- .j.

. [exp[_ igV]exp [_ {2~k)2 + i( h t,~+ gV,,]j, k,2 +

i(x _ v~t + ~o )k,}] "

Optical-path difference also occurs in this case, and, as shown in sect. 2, the damping term which is due to this difference is not the cause of the wave packet reduction. So we compensate for this difference, then the intensity of the

517

DAMPING EFFECTS ON THE COHERENCE TERM ETC.

superposed state F is as follows: (26)

1 1 exp [ _ ($y)2g2 + 244c_ 1 ] + I = ~ + 2(2d - 1)1/2

+ [ d 2 T b 2 ] _ l / 4 e x p [ ( ~ V ) 2 g 2~-~ ]

cos ( ~ + gVo + ~0 +

b.~_~__~),

where _

Sk 2

_ 1

4 ~k 2

(27) __

3

2

~k

_ -lb O=tg-~.

2

d-- 1 + ~(~V) g2(~0) ,

We affirm that compensating for the optical path does not remove all factors related to the interference term. These factors damp the interference fringes down exponentially. These factors are derived from the fluctuation SV. It is convenient to consider the case where F(k') is described in a more generalized form as follows: k , k F(k') = a'V + fl'V ~ + y V

(28)

.

Then the intensity is described as follows: (29)

1

1

[

I = ~ + 2(2d - 1)la exp - (~V)2~,2 .~_ + [d2+ b2]-1/4exp

+

( ~V)2~'2 + dc ] COS (~ 2 d 2 + b2J

bc otrYo +o_+ 2 d 2 + b2] '

where ,T~[~k\ 2

1

, ,2

4 ~k 2

(30)

This representation enables us to see which order of k in F(k) mainly causes the damping of the interference fringes. In sect. 2, we can show that, after the compensation for the optical-path difference, the coefficient of k '2 only maintains in the damping term. But eq. (29)

- Il Nu~voC ~ o B.

518

Y. MORIKAWA and Y. OTAKE

shows that, when we consider the fluctuation of device, there also remains ~' in the damping t e r m with compensation for the optical path. It means that even if the deviation of the wave packet ~k is small enough that we can neglect the dependence of k '2, i.e. even in the case where F(k) is the linear function of k, the damping effect cannot be extinguished completely by compensation. We contend that this unrestorable effect is one of the cause of the wave packet reduction and that we must distinguish the restorable damping effect from other damping effects. This restorable damping effect occurs only in the case of the wave packet.

4. - The visibility and intensity of the wave packet. We present our result in terms of visibility [20], as is widely done. First of all, we consider visibility in the case where F(k) is a quadratic function. When the optical path difference is not compensated for, it is displayed as follows:

(31)

v(~k, fl/ko)=(1 + b

) "exp [

1[~k~2~2 )

1

1

where

(32)

b---

v($k,~/ko) stands for the visibility of the wave packet with its wave number deviation ~k and without any compensation for the optical path length, i.e. with the optical-path difference ~/ko. And when this difference is compensated for, the visibility changes into the following form: (33)

v(~k, O) = [1 "4- [ ~k ~4 ~21-1/4 .

L

k0j J

We can reduce v i s i b i l i t y in both eases to that of a plane wave by p u t t i n g ~k = O:

(34)

v(O,fl/ko)=l,

(35)

v(0, 0 ) = 1 .

There is no damping factor in either of the above results, and no difference between above two results originating from the optical-path difference. The visibility is not damped down in the case of a plane wave, so such damping effects occur only in experiments relating to the wave packet. The visibility of the case where the strength of the potential series has the

519

DAMPING EFFECTS ON THE COHERENCE TERM ETC.

fluctuation ~V and F(k) is described by eq. (28), when the optical path has been compensated for, is described as follows: (36)

v~v($k, 0) = [d 2 + b2]-1/4exp [

[

dc

(~V)2s + 2

]

d 2 + b2J '

where

(37)

b---y

vOlVo),

,

,./~k\ 2 d = l + 2 (~V)2(fl'~ + 2 ~ ' , )t~-~o) . This result can be directly reduced to eq. (33) in the case where ~V-* 0. As noted above, though the Missouri-Columbia group calculated the spatial ' phase shift 5x as a constant phase shift, they measured the intensities in the case where the optical path is compensated for and in the case where it is not compensated for. They measured the intensity putting the same aluminum crystals into both paths in the interferometer in order to compensate for the optical path difference. This compensation makes the optical length of both paths x - Vgt + [~/ko. Their result with 2 aluminium crystals indicates the tendency of our prediction of the visibility equation (36). As they explained, this result indicates that even if this path difference is compensated for there remains the damping effect of the interference fringes. The damping factor comprises the fluctuation of the potential which represents the fluctuation of the device, and the wave number deviation Sk of the wave packet. And eq. (36) shows that the interference term damps exponentially down in terms of g, i.e. as D in their results. In practice, we can estimate the magnitude of the damping, that is to say, the visibility, using their experimental value. These results are shown in table I. The magnitude of the fluctuation is assumed to have the following values, ~V/Vo = 1% TABLE I. - The magnitude of ~he visibility in terms of the thickness of aluminum

crystal. Visibilities

v(~k, fl/ko) v(~k, O) v~v(~k, O) (~V/Vo-- 1%) v~v(~k, O) (~V/V~: 0.1%)

Thickness of A1 D 0.5cm

lcm

2cm

0.256 0.999 0.409 0.990

0.0617 0.998 0.0282 0.964

1.54- 10-5 0.993 6.49- 10-7 0.865

520

Y. MORIKAWA

and Y. OTAKE

and SV/V0=0.1% in order to estimate the visibility v~v(~k,0). The visibility without compensation for the optical-path difference for 0.5 cm aluminium crystal damps down to 25.6% and this value changes into 99.9% by compensating for the path difference. Such remarkable damping effect is due to $k/ko. But we can find that even with such compensation the visibility is not restored as 99.9% in the results of the experiment. This suggests that the fluctuations of the device are significant for the damping of the visibility. The visibility with a fluctuation of the order of SV/V0 -- 1% damps down very quickly in terms of the thickness of aluminium crystal, i.e. thickness of the device. When we assume SV/V0 = 0.1% then the visibility becomes 86.5% for 2 cm. From this result, we find that the optical path difference affects the damping of the visibility as much as the fluctuation does. These results, when compared with the experimental results of the Missouri-Columbia group, reveal that the magnitude of the fluctuation must be larger than 0.1% and/or smaller than 1%. The device has some kinds of fluctuation in practice, which should be estimated more precisely, and taken into account. In our analysis we derived the intensity of the wave packet from the wave function of the wave packet as in eq. (1). But the following formulation is familiar [21] and usually used to discuss the intensity of the wave packet: (38)

I = ~ dAW(Z ) I Q ) ,

where W(~) is a distribution function of the wavelength of this wave packet, and /(2) is the intensity of the plane wave whose wavelength is ~ [9, 22]. The intensity is derived as follows based on eq. (38) with Gaussian distribution function W(~):

(39)

T j

9

However, in practice, the incident beam is the same as the wave packet [7,10] and some effects of the deviation of its wavelength just appear at interactions with devices. Equation (38) does not take into account these effects of the interactions between the wave packet and the device and it is based on the following concept. Each incident particle is considered a plane wave with its wavelength ~0. The incident beam is an assembly of this plane wave, i.e. there is a distribution of the wavelength in the incident beam. So the observed intensity is the integration of the wavelength over this distribution using the intensity of the plane wave. In the experiment by the Vienna group it is made use of pulsed neutrons. These incident neutrons are actually equivalent to wave packets but not to the plane wave. So, we contend that such a wave packet is not the same as the assembly of the plane wave. If we want to investigate the phenomenon of the wave packet reduction relative to the wave packet rather than the assembly of the plane wave, the method represented by eq. (38) is not suitable.

DAMPING EFFECTS

ON T H E C O H E R E N C E

TERM ETC.

521

This is the reason why we adopt the wave function of the wave packet instead of either adopting that of the plane wave or using the formulation of eq. (38). We compare our results with the intensity of eq. (39). When F(k) is a linear function of k, the intensity without path compensation eq. (11) coincides with eq. (39). In the case of the quadratic function, however, the coherence term eq. (16) is different from eq. (39). The difference between the assembly of the plane wave and the wave packet will become significant when F(k) is the quadratic function and the wave number deviation ~k is not very small. From these results, we contend that our method is a more precise treatment of the wave packet, and this enables us to investigate the results of the experiment using wave packets.

5. -

Conclusion.

We analyse the damping effect of the device on the coherence terms in the interference-type experiment of the wave packet instead of the plane wave. Special damping effects appear in case of the wave packet, though they do not exist in case of a plane wave. When F(k) is described by a linear function of k, the form of the damping factor is eq. (12):

ex"[ This damping factor consists of the coefficient of the first-order k in F(k), ~, and it is derived from the optical-path difference between T~ and TII. If this opticalpath difference is compensated for, the damping term eq. (12) disappears, that is to say, the interference fringes are restored. In other words, the visibility changes from eq. (12) into 1. The way to compensate for the optical path will be possibly developed when the interferometer for the ultracold neutron is actualized [16], and it is easily realized by inserting an optical fiber into one path for the photon experiment. When F(k) is the quadratic function of k, the damping term becomes of more complicated form. In this case even if the optical-path difference is compensated for, the damping factor is retained in the interference term, while the same factor as eq. (12) disappears. So, there are two types of damping factors in the coherence term, one is eq. (12) and this term is extinguishable by compensating for the optical path and the interference fringes are restored. The other damping term appears only in the case where F(k) is the quadratic function of k, it is not extinguishable and it indicates the occurrence of the wave packet reduction. The way we judge the occurrence of the wave packet reduction from the result of the experiment is the estimation of visibility. When interference fringes disappear, in other words, the visibility has a small value, it is almost impossible

522

Y. MORIKAWAand Y. OTAKE

to recognize the interference fringes, so we consider that the wave packet reduction occurs. But we point out that this is not the case when the object system is the wave packet. Because the wave packet has the fluctuation in itself as the wave number deviation Sk, this fluctuation makes the visibility damped down as eq. (12). However, this damping phenomenon is not unrestorable as we show in this paper. Such restorable phenomenon is not related to wave packet reduction. When F(k) is the quadratic function, the damping factor is retained as in eq. (18) even after compensating for the path. We contend that this phenomenon indicates wave packet reduction, which is derived from ~k. So, we must investigate very carefully the damping phenomena relative to the wave packet reduction in the case of the wave packet, and we should distinguish these damping factors when we analyse the experiment with wave packet. We also investigate the case where the device is considered to have the fluctuations, such as the characteristics of the macroscopic system, because such fluctuations give rise to the wave packet reduction when the object system is described by the wave function of a plane wave. Equation (26) shows that the interference term damps exponentially even by compensating for the optical path. So, if ~V is large, the interference term disappears as is the case of the plane wave. We find in eq. (29) that when the device has the fluctuation, the damping factor includes/~', while it is always extinguished by compensating for the optical path in the case where such a fluctuation is not considered. We contend that even if the optical path is compensated for, the damping factor or the interference term is retained under the following conditions. F(k) is the quadratic function and/or the fluctuation of the device is not so small that we can neglect it. Such damping effects were measured by the Missouri-Columbia group using a neutron beam which is not precisely monochromatic. They measured the longitudinal coherence length of the neutron. In their experiment the incident beam is not the plane wave as they mentioned, then we should take into account k-dependence of a phase hx, otherwise it is impossible to yield a phase shift which derives the real spatial phase shift. Such spatial phase shifts are equivalent to the optical-path difference. As we have shown before, the constant phase cannot cause the spatial phase shift. Of course, when we treat the experiment of the plane wave, the wave number is always constant and the effect of the scattering in the device only yields the constant phase shift in the wave function of the object particle as in usual neutron interference experiments with very small Sk. We adopt the wave function of the wave packet to obtain its intensity but we do not use eq. (38). As we explained, eq. (38) does not precisely describe the intensity of the wave packet. When F(k) is the quadratic function of k, there exists a difference between the intensity of our method and eq. (38). If the intensity is measured in the experiment with rather large ~k, this

DAMPING EFFECTS ON THE COHERENCE TERM ETC.

523

difference will appear because eq. (38) shows the intensity of the beam which consists of the assembly of a plane wave not by the actual wave packet. ***

We greatly thank Prof. M. Namiki. He originally suggested us to investigate the influence on wave packet reduction in the interference-type experiment when the incident beam is actually described by the wave packet, and gave us fruitful discussions. We are also grateful to Prof. I. Ohba. We deeply thank Dr. S. Pascazio for useful discussions. We also thank Mr. M. N. Morris for checking the English of our manuscript.

REFERENCES [1] For instance, S. MACHIDAand M. NAMIKI: Prog. Theor. Phys., 63, 1457, 1833 (1980); H. ARAKI:Prog. Theor. Phys., 64, 719 (1980); M. NAMIgI:Found. Phys., 18, 29 (1988); Y. MORIKAWA,M. NAYIIKIand Y. OTAKE:Prog. Theor. Phys., 78, 951 (1987). [2] M. NAMIKI, Y. OTAKE and H. SOSHI: Prog. Theor. Phys., 77, 508 (1987). [3] R. FUKUI)A: Phys. Rev. A, 35, 8 (1987); 36, 3023 (1987). [4] G. BADREK, H. RAUCH and J. SUMMHAMMER:Phys. Rev. Left., 51, 1015 (1983). [5] C. DEWDNEY, PH. GUERET, A. KYPRIANIDIS and J. P. VIGIER: Phys. Left. A, 102, 291 (1984). [6] Originally, M. Namiki indicated to analyze the effect of wave packet on the wave packet reduction to us. [7] H. KAISER, S. A. WERNER and E. A. GEORGE:Phys. Rev. Lett., 50, 560 (1983). [8] M. HEINRICH, H. RAUCH and H. WOLWITSCH:Physica, B, 156, 157, 588 (1989). [9] H. RAUCH, E. SEIDL, D. TUPPINGER, D. PETRASCHECK and R. SCHERM: Z. Phys. B, 69, 313 (1987). [10] R. J. GLAUBER:in Lectures in Theoretical Physics, edited by W. E. BRITTINand L. G. DUNHAM (Interscience Publishers, Inc., New York, N.Y., 1959), p. 135. [11] This procedure is based on the suggestion by M. Namiki. [12] L. I. SCHIFE:Quantum Mechanics, 2nd. ed. (McGraw-Hill, New York, N.Y., 1955), p. 54. [13] A. G. KLEIN, G. I. OPAT and W. A. HAMILTON:Phys. Rev. Left., 50, 563 (1983). [14] G. COMSA:Phys. Rev. Left., 51, 1105 (1983); H. KAISER, S. A. WERNER and E. A. GEORGE: Phys. Rev. Left., 51, 1106 (1983). [15] For instance, C. O. ALLEY:Proceedings of the International Symposium on Foundation of Quantum Mechanics-In the Light of New Technology, ISQM, Tokyo '83, edited by S. KAMEFUCHIet al. (Phys. Soc. Japan, Tokyo, 1984). [16] T. EBISAWA: KEK Report (1989); 89-13P.33 (in Japanese); Y. YAMADA, T. EBISAWA, N. ACHIWA, T. AKIYOSHIand S. OKAMOTO:Annu. Rep. Res. React. Inst. Kyoto Univ., 11, 8 (1978). [17] M. GRUBER, K. EDER, A. ZEILINGER, R. G•HLER and W. MAMPE:Phys. Left. A, 140, 363 (1989). [18] Suggested by M. Namiki.

524

Y. MORIKAWA and Y. OTAKE

[19] Y. MURAYAMA: private communication; Y. MURAYAMA and M. NAMIKI: Proceedinas of the III International Meeting on Epistemology, edited by E. I. BITSAKISand C. A. NIKOLAIDES (Kluwer Academic Publishers, Dordrecht, 1989). [20] S. Pascazio suggested to present our results in terms of the visibility. [21] H. RAUCH: Lecture Note of Bangalore Meeting (1984); H. RAUCH: in Open Questions in Quantum Physics, edited by G. TAROZZIand A. VAN DER MERWE (D. Reidel Publishing Company, Dordrecht, 1985), p. 345. [22] H. RAUCH: in Neutron Interferometry, edited by U. BONSE and H. RAUCH (Clarendon Press, Oxford, 1979), p. 161.