NEUTRON WAVEPACKETS AND LONGITUDINAL COHERENCE

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NEUTRON WAVEPACKETS AND LONGITUDINAL COHERENCE

A. Klein, G. Opat

To cite this version:

A. Klein, G. Opat. NEUTRON WAVEPACKETS AND LONGITUDINAL COHERENCE. Journal

de Physique Colloques, 1984, 45 (C3), pp.C3-235-C3-238. �10.1051/jphyscol:1984339�. �jpa-00224053�

S O U R ~ A L DE PHYSIQUE

Colloque C3, supplCment au n03, Tome 45, mars 1984 page C3-235

NEUTRON WAVEPACKETS AND LONGITUDINAL COHERENCE A.G. Klein and G.I. Opat*

School of Physics, Universfty of ^M elbourne, ParkviLZe 3052, A u s t r a l i a and Physics Department, MIL", Cambridge, M A 02139, U.S.A.

*school of Physics, University of Melbourne, ParkviLZe 3052, A u s t r a l i a

Resume

-

La longueur des paquets d'ondes n ' e s t pas identique

a

l a longueur de coherence. Est-ce-que l a longueur des paquets d'ondes e s t accessible aux mesures experimentales ? Une p o s s i b i l i t e d'observation par i n t e r f e r o - metrie

a

resolution temporelle e s t proposee.

Abstract

-

The concept of wavepackets i s widely used in visualizing the behavior of p a r t i c l e s in optical experiments. The length of wavepackets, however, i s shown not t o be synonymous with coherence length as measured by interferometry. I s the length of wavepackets accessible t o experfmental measurement? A possible approach via time-resolved interferometry i s pro- posed.

Wavepackets a r e a convenient conceptual bridge between wave-like behaviour on the one hand and localizable p a r t i c l e - l i k e behaviour on t h e other. A homogene- ous beam of identical p a r t i c l e s may be represented as an ensemble of wavepackets each of which i s a coherent superposition of waves with a f i n i t e range of wave- lengths and frequencies. In a non-dispersive medium such wavepackets r e t a i n t h e i r shape and may be characterized by a length Ax related t o the range of wavevectors, A k , by t h e uncertainty r e l a t i o n : A x Ak)1/2. For t h e special case of Gaussian wavepackets, i . e . sinusoidal waves with a Gaussian envelope, the equality holds and Ax = Axmi, = 1 / 2 A k ; t h i s i s referred t o as a minimum wavepacket.

Recent experiments in electron /1,2/ and neutron /3/ interferometry have raised the question: I s the shape, in p a r t i c u l a r the length, of wavepackets amen- able t o experimental observation? In a non-dispersive medium, subject t o the assumption t h a t the p a r t i c l e s which c o n s t i t u t e the beam can a l l be represented as Gaussian (minimum) wavepackets, a l l being identical except f o r a r b i t r a r y phases, the answer i s yes. As will be shown below, the output of a two-beam interferometer as a function of path difference, L , i s related t o the mutual coherence function r ( L ) , defined as the b i l i n e a r ensemble average:

I t i s then e a s i l y shown t h a t in the above special case averaging over the ensemble does not change the width of the mutual coherence function, i . e . the coherence length. Thus t h e coherence length, LC i s ( u p t o a small numerical f a c t o r which depends on definitions) equal to the length of the wavepacket, i .e. LC = Axmi,.

However, t h i s i s by no means t r u e in general, even in a non-dispersive medium.

In general the beam may well consist of a mixture of wavepackets, each with a d i f f e r e n t mean value of k and maybe even d i f f e r e n t lengths, Ax. In t h a t case

the ensemble averaging involved in the definition of the coherence function removes any immediate relation between i t and the lengths of the individual wavepackets.

As we shall see, LC will then depend purely on the overall range of wavevectors contained in the beam.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984339

C3-236 J O U R N A L DE PHYSIQUE

The r e l a t i o n between t h e coherence l e n c t h and t h e l e n g t h o f t h e wavepackets i s even more tenuous i n t h e case o f de B r o g l i e waves, by v i r t u e o f t h e well-known f a c t t h a t t h e Schrodinger equation gives r i s e t o d i s p e r s i v e propagation, i . e . t h a t t h e wavepackets r e p r e s e n t i n g m a t e r i a l p a r t i c l e s w i l l spread upon propagation.

For example, a Gaussian wavepacket w i t h i n i t i a l w i d t h Ax(0) w i l l , a f t e r propagating f o r a t i m e t, have t h e width:

W i l l t h e coherence length, as measured by t h e w i d t h o f t h e c o n t r a s t f u n c t i o n i n a two-beam i n t e r f e r o m e t e r , a l s o increase? The answer, both t h e o r e t i c a l l y /4/

and e x p e r i m e n t a l l y /1,2,3/, i s no; t h e coherence l e n g t h stays constant (and equal t o Ax(0)) i n s p i t e o f t h e i n d e f i n i t e spreading o f the wavepackets. To see t h i s , consider t h e wavepacket

$(x,t) = p ( k ) e i (kx -wkt ) dk

e n t e r i n g an i n t e r f e r o m e t e r where i t i s s p l i t i n t o two equal havles, one o f which i s s h i f t e d by a r e l a t i v e path d i f f e r e n c e L. The s t a t e emerging from t h e i n t e r f e r - ometer i s given by:

and t h e observed o u t p u t o f t h e i n t e r f e r o m e t e r may be shown t o be t h e average:

where r ( L ) i s t h e mutual coherence f u n c t i o n d e f i n e d i n (1) and r ( 0 ) = I(O) i s j u s t t h e average beam i n t e n s i t y . Now, i n s e r t i n g eq. ( 3 ) i n t o eq. (1) gives:

r(L)

. <Idk/

d k ' a ( k ) a * ( k ) e i k ( ~ + L)- iwt e- i k ' x + i w ' t

>

= j l a ( k ) 1 2 e i k L d k

.

⁽⁶⁾

Thus t h e i n t e r f e r o m e t e r output, eq. ( 5 ) , measures o n l y t h e F o u r i e r t r a n s f o r m o f t h e k-space i n t e n s i t y spectrum o f t h e beam. Note t h a t t h e averaging has removed any X-dependence and any dependence on t h e l e n g t h o f t h e wavepackets. As noted above, t h i s i s t h e case even f o r non-dispersive s i t u a t i o n s (e.g. l i g h t i n vacuo).

The i n t e r f e r o m e t e r o u t p u t , i n c l u d i n g t h e coherence length, depends o n l y upon t h e d i s t r i b u t i o n i n k-space o f t h e beam i n t e n s i t y and o n l y i n t h e s p e c i a l case o f iden- t i c a l minimum wavepackets i s LC = Ax.

What happens t o our mental p i c t u r e i n which we g e t i n t e r f e r e n c e c o n t r a s t when t h e wavepackets overlap? For t h e p a r t i c u l a r case o f a Gaussian wavepacket we g a i n i n s i g h t by s t u d y i n g t h e e v o l u t i o n o f the packet. I n Fig. 1 we see t h a t n o t o n l y i s t h e envelope o f t h e i n i t i a l l y s i n u s o i d a l packet lengthening b u t a l s o t h a t t h e s p a t i a l and temporal frequencies change along t h e evolved packet. The s h o r t e r wave- lengths, r e p r e s e n t i n g f a s t e r motion, precede t h e longer wavelengths, representing slower motion, as expected. Consider now t h e s u p e r p o s i t i o n o f t h e evolved packet w i t h a r e p l i c a of i t s e l f , s h i f t e d by a l e n g t h L, as shown i n Fig. 2. I t i s ob- vious t h a t t h e p a t h d i f f e r e n c e L i s t o o g r e a t t o g i v e a s t a t i o n a r y phase r e l a t i o n - s h i p and we g e t "beats" which move through t h e d e t e c t o r and average t o zero. Thus, even though t h e evolved wavepackets overlap, we do n o t g e t a s t a t i o n a r y i n t e r f e r - ence p a t t e r n unless L < LC = Ax(0).

The remaining question i s : I s t h e r e some o t h e r way t o observe t h e l e n g t h o f wave- packets, g i v e n t h a t simple i n t e r f e r o m e t r y w i l l n o t s u f f i c e ? The answer l i e s i n t h e d i r e c t i o n of time-dependent measurements. The coherence p r o p e r t i e s o f a beam

Fig. 1

-

E v o l u t i o n o f Gaussian wavepacket w i t h dispersion.

Fig. 2

-

Superposition o f evolved wavepackets showing t h e appearance o f "beats".

o f r a d i a t i o n may be represented w i t h g r e a t e r g e n e r a l i t y by t h e complete space-.

t i m e coherence f u n c t i o n r ( x l ,tl; x2,t2) d e f i n e d as

r(xl,tl; x2,t2) = ( Y * ( x ~ , ~ ~ ) Y ( x ~ . ~ ~ ) > (7)

o r , f o r s t a t i o n a r y , t i m e - i n v a r i a n t systems w i t h x 2 = x l + L and t 2 = t l + T . r(L,T) = < Y * ( x , ~ ) Y ( x + L, t +T)>

C3-238 JOURNAL DE PHYSIQUE

I f t h e wave equation obeyed by ~ ( x , t ) i s denoted by

a a 2

( f o r example, f o r l i g h t i n vacuo: W(x,t) =

- ^-

^-

c2ak2 ax2

5 2

a 2 .

⁾

o r f o r n o n - r e l a t i v i s t i c p a r t i c l e s o f mass m: W(x,t) = ih&

+x-

ax2 then i t i s e a s i l y shown t h a t , provided W i s a l i n e a r operator,

I n o t h e r words t h e coherence f u n c t i o n s a t i s f i e s t h e same wave equation as t h e amplitude f u n c t i o n . I n p a r t i c u l a r , i f Y develops d i s p e r s i v e l y i n t i m e then

r

w i l l develop d i s p e r s i v e l y also. I t i s n o t p o s s i b l e t o observe t h i s behavior i n an experiment u s i n g a s i n g l e detector, measuring a t a s i n g l e p o i n t i n time.

The cases considered e a r l i e r correspond t o T = 0 and t h e q u a n t i t y r ( ~ ) d e f i n e d e a r l i e r i s , i n f a c t , e q u i v a l e n t t o r(L,O). I n t h e case o f electromagnetic wave i n f r e e space r(L,T) i s a c t u a l l y a f u n c t i o n o f o n l y one v a r i a b l e , (L

-

cT)

,

and space and time displacement i n t e r v a l s p l a y a s i m i l a r r o l e . Thus, t h e complete coherence f u n c t i o n i s e x p e r i m e n t a l l y accessible, as we have shown elsewhere /5/.

I n t h e case o f de B r o g l i e waves, where no d i s p e r s i o n l e s s propagation i s possible, t h e time-dependent aspects o f t h e coherence f u n c t i o n have n o t y e t been i n v e s t i g a t e d . A p o s s i b l e c l u e t o how t h i s m i g h t be done i s contained i n Fig. 2. I f t h e time- o r i g i n o f t h e wavepackets c o u l d be d e f i n e d by means o f an u l t r a - f a s t chopper, then t h e "beats" i n t h e s u p e r p o s i t i o n o f t h e two halves o f t h e wavepackets may be observable w i t h t h e a i d o f a f a s t , time-resolved d e t e c t i o n system. This would be analogous t o performing a "double s l i t experiment" i n time and studying t h e temp- o r a l , r a t h e r than s p a t i a l coherence p r o p e r t i e s o f t h e beam. This experimental poss- i b i l i t y i s c u r r e n t l y under i n v e s t i g a t i o n .

We wish t o thank Profs. S. A. Werner, A. Z e i l i n g e r , and M. A. Horne f o r s t i m u l a t - i n g and h e l p f u l discussions. AGK wishes t o thank P r o f . C. G. S h u l l f o r h i s h o s p i t - a l i t y a t M.I.T. where t h i s paper was prepared and Ms. Rose f o r t y p i n g it.

References

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2. Daberkow, I., Gauch, H., and Hasselbach, F.: J o i n t Meeting on E l e c t r o n Micro- scopy, Antwerp, 1983, A b s t r a c t No. 104.

3. Kaiser, H., Werner, S. A. and George, E. A.: Phys. Rev. L e t t s .

50,

560, 1983.

4. K l e i n , A. G., Opat, G. I. and Hamilton, W. A.: Phys. Rev. L e t t s .

50,

563, 1983.

5. Hamilton,

W.

A., K l e i n , A. G., and Opat, G. I.: Phys. Rev. A. ( ~ a p i d ~omm.), ( i n press).