We show that single and multislit experiments involving matter waves may be
constructed to assess correlations between the position and momentum of a
single free particle. These correlations give rise to position dependent phases
which develop dynamically and may play an important role in the interference
patterns. For large enough transverse coherence lenght such interference
patterns are noticeably different from those of a classical dispersion free
wave.Comment: 7 pages, 5 figures, revised manuscrip

A.F.R. de Toledo Piza

A.H. Blin

Arndt

B. Hiller

Davisson

G. Glionna

Gahler

M.C. Nemes

Marcos Sampaio

Nairz

Viale

English

arXiv

We show that single and multislit experiments involving matter waves may be constructed to assess dispersively generated correlations between the position and momentum of a single free particle. These correlations give rise to position dependent phases which develop dynamically as a result of dispersion and may play an important role in the interference patterns. To the extent that initial transverse coherence is preserved throughout the proposed diffraction setup, such interference patterns are noticeably different from those of a classical dispersion free wave.http://www.sciencedirect.com/science/article/B6TVG-4R1MF2G-4/1/9160b29f2e5d4fd201ba541caa7bc6d

Glionna, G.

Blin, A. H.

Hiller, B.

Nemes, M. C.

Sampaio, Marcos

Piza, A. F. R. de Toledo

Estudo Geral

Physica A 387 (2008) 1485–1490
www.elsevier.com/locate/physa
Dispersion and uncertainty in multislit matter wave diffraction
G. Glionna, A.H. Blin, B. Hiller, M.C. Nemes∗, Marcos Sampaio, A.F.R. de Toledo Piza
Federal University of Minas Gerais, Physics Department - ICEx, P.O. Box 702, 30 161-970, Belo Horizonte MG, Brazil
Physics Department, CFT, University of Coimbra, 3000, Portugal
Institute of Physics, University of Sa˜o Paulo, P. O. Box 66 318, 5315-970, Sa˜o Paulo, Brazil
Universite´ Paris Sud XI - LPT - Centre d’Orsay, 91 405, France
Received 4 June 2007; received in revised form 21 September 2007
Available online 1 November 2007
Abstract
We show that single and multislit experiments involving matter waves may be constructed to assess dispersively generated
correlations between the position and momentum of a single free particle. These correlations give rise to position dependent phases
which develop dynamically as a result of dispersion and may play an important role in the interference patterns. To the extent
that initial transverse coherence is preserved throughout the proposed diffraction setup, such interference patterns are noticeably
different from those of a classical dispersion free wave.
c© 2007 Published by Elsevier B.V.
PACS: 03.65.Xp; 03.65.Yz; 32.80.-t
Fundamental aspects of quantum mechanics are revealed by diffraction experiments with particles. Much work has
been devoted to the matter considering electron [1], neutron [2] and more recently large molecule [3,4] diffraction from
multislit gratings. In the present contribution we are concerned with a double diffraction problem. The question to be
addressed is how sensitive are the measured interference patterns to the dispersive dynamics of particle propagation
before it reaches the grating and proceeds from there to the screen. We assume that a collimated beam with sufficiently
large transverse coherence length γ is first sent through a narrow slit of width σ0  γ after which it can be
described by a fully coherent wave packet of initial transverse width σ0 as described in detail below [5]. This wave
packet subsequently goes through a second, multi-slit diffraction grating after a flight path sufficiently long to allow
for the development of matter-wave dispersion effects. Also, as done in Ref. [7], we assume that the longitudinal
(beam direction) wave packet localization is sharp enough compared with the flight path so that the time-of-flight
approximation is valid. In other words, the flight paths involved in the problem are much larger than the position
spread in longitudinal direction. Coherence loss mechanisms are neglected, since their effects are well known. In this
idealized scenario a Gedanken experiment is described in terms of the time evolution of an initially Gaussian wave
packet which travels freely from the first slit to the multislit grating.
∗ Corresponding author at: Federal University of Minas Gerais, Physics Department - ICEx, P.O. Box 702, 30 161-970, Belo Horizonte MG,
Brazil.
E-mail addresses: gglionna@fisica.ufmg.br (G. Glionna), alex@teor.fis.uc.pt (A.H. Blin), brigitte@teor.fis.uc.pt (B. Hiller),
carolina@fisica.ufmg.br (M.C. Nemes), sampaio@th.u-psud.fr (M. Sampaio), piza@fma.if.usp.br (A.F.R. de Toledo Piza).
0378-4371/$ - see front matter c© 2007 Published by Elsevier B.V.
doi:10.1016/j.physa.2007.10.069
1486 G. Glionna et al. / Physica A 387 (2008) 1485–1490
Fig. 1. Sketch of a double diffraction arrangement for a two-slit grating. Parameters: σ 0 = 0.5 × 10−5 m, d = 10−7 m, b = 1.8 × 10−8 m,
L = 0.1 m, l = 1.25 m, and T = L/v, τ = l/v, where v is the velocity of the molecules.
This model sheds light on specific matter wave dispersion effects, notably the dynamic evolution of the
position–momentum correlations. For a real initial Gaussian wave packet, these correlations appear in the form of
a position dependent phase as soon as the particle leaves the first slit. The importance of this position dependent phase
depends on the ratio between the time of flight and an intrinsic time τ = mσ 20 /h¯, m being the particle mass and σ0
is the initial width of the wave packet. Therefore if the particle travels long enough before it reaches the multislit
grating, it will arrive at each slit with a different phase. As will be shown in what follows, this may radically alter the
interference patterns observed on the screen.
In order to review the essential dispersive dynamic effects we discuss the time evolution in the transverse direction,
from the first slit at t = 0 to the grating, of a Gaussian wave packet given by
ϕ(x, 0) =
(
1
σ0
√
pi
)1/2
exp
(
−x2
2σ 20
)
, (1)
where x is the transverse direction (see Fig. 1) and the position in beam direction (y in Fig. 1) enters indirectly via the
time coordinate. σ0 is the width of the first slit. Its time evolution according to Schroedinger’s equation will yield for
the wave packet just before the grating [6]
ϕ(x, T ) =
(
1
B(T )
√
pi
)1/2
exp
[
− x
2
2B2(T )
(
1− i h¯T
mσ 20
)]
≡ ϕ˜(x, T ) exp(i S(x, T )),
where
ϕ˜(x, T ) =
(
1
B(T )
√
pi
)1/2
exp
[
− x
2
2B2(T )
]
,
B2(T ) = σ 20
(
1+ h¯
2 T 2
m2σ 40
)
(2)
and
S = x
2
2B2(T )
h¯T
mσ 20
≡ x
2
2B2(T )
T
τ0
. (3)
Notice that the position dependent phase S contains the time scale τ0 = mσ
2
0
h¯ . The ratio T/τ0 will determine
the importance of this x-dependent phase to the interference pattern. In the experimental setups using fullerene
molecules [3] T/τ0 ≈ 104 which is also the condition for Frauhoffer diffraction (see Ref. [7]). The time scale τ0 is
fundamentally determined by Heisenberg’s uncertainty relation, given the initial position dispersion1x(0) = σ0/
√
2.
G. Glionna et al. / Physica A 387 (2008) 1485–1490 1487
In fact, the corresponding momentum dispersion is 1p = h¯/(σ0
√
2). Because the momentum is a constant of motion
this momentum spread will be preserved in time. Both 1x and 1p constitute intrinsic properties of the initial wave
packet, in terms of which the time scale τ0 is expressed as [8]
τ0 = 1x(0)
(1p)/m
. (4)
The numerator in the above relation represents the spatial dimensions of the initial wave packet, whilst the denominator
stands for the scale of velocity differences enforced by the uncertainty principle. Therefore the time scale τ0
corresponds essentially to the time during which a distance of the order of the wave packet extension is traversed
with a speed corresponding to the dispersion in velocity. It can therefore be viewed as a characteristic time for the
“aging” of the initial state, which consists in components with larger velocities (relatively to the group velocity of the
wave packet) concentrating at the frontal region of the packet. This can be seen explicitly by deriving the velocity
field associated with the phase S in Eq. (2), which reads
v(x, T ) = h¯
m
∂S
∂x
= T x
τ 20 + T 2
. (5)
This expression shows that for T > 0 the velocity field with the initial value v(x, 0) = 0 varies linearly with respect
to the distance from centre of the wave packet (x = 0).
Next we relate quantitatively this “ageing” effect to position–momentum correlations. This is readily achieved
using the generalized uncertainty relation devised by Schroedinger [9], which is expressed in this case in terms of the
determinant of the covariance matrix Σ
detΣ ≡ det
 1x2 12 〈xˆ pˆ + pˆxˆ〉1
2
〈xˆ pˆ + pˆxˆ〉 1p2
 ≥ h¯2
4
.
For the minimumuncertainty wave packet of Eq. (2) we obtain, at all times,
det

B2(T )
2
h¯T
2τ0
h¯T
2τ0
h¯2
2σ 20
 = h¯24 . (6)
The Gaussian wave packet therefore saturates Schroedinger’s uncertainty relation at all times. We can thus describe
this result by saying that the time dependence of the Heisenberg uncertainty relation (obtained by dropping the off-
diagonal elements of the covariance matrix) which is due, in this case, to the dispersive increase of the wave packet
width in time, just reflects the x − p correlation process. The relevant quantity in this connection is the correlation
matrix element
〈xˆ pˆ + pˆxˆ〉
2
= −h¯ T
2τ0
. (7)
Recently an interesting experiment [10] has been performed in order to study the Heisenberg uncertainty relation
for fullerene molecules using the fullerene C70 and measuring the momentum spread after the passage through a
narrow slit with a variable width (down to 70 nm). The results are interpreted in the light of Heisenberg’s uncertainty
relation. The results are summarized by the empirical relation
1p = Ch
1x
, (8)
where C = 0.89 and h = 2pi h¯. Having analytical expressions for each of the elements of the covariance matrix Σ and
using the empirical value1x1p = 0.89h we can evaluate 〈xp+ px〉. Using the values of the experiment of Ref. [10]
we get 〈xp + px〉 ≈ 11.14h¯. This value is a measure of the x − p correlation and thus of the dispersive increase of
the wave-packet width.
1488 G. Glionna et al. / Physica A 387 (2008) 1485–1490
Fig. 2. Diffraction patterns for the arrangement of Fig. 1, for different values of the width of the first slit σ0. As the intensity at x = 0 diminishes
we have σ0 = 6.0 µm, σ0 = 0.02 µm, σ0 = 0.0175 µm, σ0 = 0.013 µm, respectively. Recall that the distance between two slits is ≈0.1 µm.
Two slit grating. We next consider the double diffraction experiment for a two-slit grating following the first slit (see
Fig. 1). In this case the intensity at the screen is given by [7]
I (x) = |Ψ+(x, T, τ )+Ψ−(x, T, τ )|2 (9)
where
Ψ±(x, T, τ ) =
∫ +∞
−∞
dxi
∫ +∞
−∞
dwK (x, T + τ ;w, T )G(w ± d/2)K (w, T ; xi , 0)ϕ(xi ) (10)
and
K (z, t;w, t0) =
√
m
2pi i h¯(t − t0) exp
[
i
m(z − w)2
2h¯(t − t0)
]
, (11)
G(w) = exp
[
− w
2
2b2
]
, and (12)
ϕ(xi ) = 1√
σ0
√
pi
exp
[
− x
2
i
2σ 20
]
. (13)
The kernel K (z, t, w, t0) is the free propagator for the particle, the functions G(w ± d/2) describe the double slit
apertures which are taken to be Gaussian of width b separated by a distance d; the width of the first slit is σ0, m is
the mass of the particle, T (τ ) is the time of flight from the first slit (double slit) to the double slit (screen). Parameter
values are taken from Ref. [10].
Let us now allow for wave-packet “ageing”, i.e. for significant transverse spreading with the accompanying x − p
correlation effects before reaching the two-slit grating. This can be achieved in a variety of ways, but we choose for
simplicity to make σ0 smaller while keeping the width just before the slits fixed at 1 µm. The result is shown in Fig. 2
for different values of σ0, other parameters remaining unchanged. Note the qualitative changes of the interference
pattern as the x−p correlations grow, implying increasing phase difference between the contributions of the impinging
wave packet at the two slits with decreasing σ0.
Multislit grating. An alternate way to bring about the effects of quantum mechanical dispersion with fixed σ0 is to
consider diffraction by an increasing number of equally spaced diffraction slits instead of just two. In order to explore
G. Glionna et al. / Physica A 387 (2008) 1485–1490 1489
Fig. 3. Intensity pattern for two slits and parameters of Fig. 1.
this strategy we evaluate
I (x) =
∣∣∣∣∣N−1∑
n=0
Ψn(x, T, τ )
∣∣∣∣∣
2
(14)
where
Ψn(x, T, τ ) =
∫ +∞
−∞
dw
∫ +∞
−∞
dxiK (x, T + τ ;w, T )G(w − Xn)K (w, T ; xi , 0)ϕ(xi ) (15)
with Xn = X0 − (N − 1) d2 + nd for N slits centered around X0 = 0. As discussed before, the wave function at the
grating is given by
Ψ(x, t) ∝ exp
[
− x
2
2B2(t)
(
1− i h¯t
mσ 20
)]
(16)
where B(t) = σ0
√
1+ (h¯2 t2)/(m2σ 40 ). Note again the second term in the exponent giving rise to the quantum
dispersive phase which will be different at each slit position. An estimate of the number of slits above which the
effects of the phase in Eq. (16) become effective is
N 2d2
2σ 20
τ
t
≥ 1. (17)
With the parameter values of Ref. [10] and σ0 = 0.5× 10−5 m we obtain N ∼ 50. The intensity I (x) is depicted as a
function of transverse position x in Figs. 3–5 for different values of the number of slits N (only half of the symmetrical
interference pattern is shown). The parameters used correspond to the experimental setup for C60 molecules of
Ref. [3]. In particular we take σ0 = 0.5× 10−5 m in the initial wave packet Eq. (13). The grating is characterized by
the half width of the slits b and by the slit spacing d , the times T and τ in Eq. (2) are calculated from the velocity v
of the molecules and the distance from the first slit to the grating and from there to the detector position respectively.
Each figure shows the intensity for the most probable velocity v = 200 m/s (full line) and an incoherent sum over
velocities (dotted line), which takes into account the experimental spread in the initial velocities of about 60% as
parameterized in Ref. [3]. The dash-dotted lines exhibit the corresponding classical Fraunhofer interference pattern
based on the de Broglie wavelength of the C60 molecule with v = 220 m/s.
The classical Fraunhofer expression for the intensity is very similar to the one with a definite velocity in the first
case, N = 2. For the second case, N = 30, deviations from the classical Fraunhofer pattern can be seen. Note also
that with such a large spread in the velocities this difference will not be experimentally accessible. For larger values
of N as shown in Fig. 5 (N = 100), the difference between a matter wave and a classical wave becomes qualitative.
These are purely quantum effects; in fact they are quantum effects coming from the first stage of the experiment
and the position dependent phase. Of course these effects could be observed for a smaller number of slits provided the
x–p correlations in the first part of the experiment be large enough.
1490 G. Glionna et al. / Physica A 387 (2008) 1485–1490
Fig. 4. Intensity pattern for thirty slits and parameters of Fig. 1.
Fig. 5. Intensity pattern for a hundred slits and parameters of Fig. 1.
In summary, for multislit diffraction of matter waves we have shown that in an experiment with sufficiently
monoenergetic trains of wave packets it is possible in principle to distinguish typical classical wave patterns from
typical quantum matter wave patterns as a function of the number of slits. Decoherence and a spread in beam
momentum will tend to obscure the effect we propose here. A possible way to circumvent this problem is to use
paraxial light. When the conditions for this approximation to the Helmholtz equation are met there is a one-to-one
correspondence between matter and light waves [11]. Besides, as extensively studied in Ref. [7], decoherence will not
preclude the observation of interference fringes in the present experimental setup.
Acknowledgements
The work of MCN was partially supported by CNPq, FAPESP, PRAXIS/BCC/4301/94, PRAXIS/FIS/12247/98
and POCTI/1999/FIS/3530 and MS was supported by CAPES-MEC.
References
[1] C. Davisson, L. Germer, Nature 119 (1927) 558.
[2] R. Gahler, C.G. Shull, W. Treimer, W. Mampe, A. Zeilinger, Rev. Modern Phys. 60 (1988) 1067.
[3] M. Arndt, O. Nairz, J. Vos-Andrae, C. Kaller, G. van der Zouw, A. Zeilinger, Nature (London) 401 (1999) 680.
[4] O. Nairz, M. Arndt, A. Zeilinger, Amer. J. Phys. 71 (2003) 319;
O. Nairz, M. Arndt, A. Zeilinger, J. Modern Opt. 47 (2000) 2811.
[5] Full coherence of the wave packet, in the sense of Eq. (1) below also requires that γ > l, l being the length of the slit of width σ0  l.
[6] See for instance, Elementary Quantum Mechanics by David S. Saxon, 1968.
[7] A. Viale, M. Vicari, N. Zanghi, Phys. Rev. A 68 (2003) 063610.
[8] Since we assume a long, narrow slit, σ0  l, the characteristic time corresponding to τ0 but associated to l is much larger that τ0. Accordingly,
we ignore wave-packet spreading in this direction.
[9] E. Schroedinger, Proceedings of the Prussian Academy of Sciences XIX, 1930, pp. 296–303.
[10] O. Nairz, M. Arndt, A. Zeilinger, Phys. Rev. A 65 (2002) 032109.
[11] I.G. Paz, Master thesis. Departamento de Fı´sica, Universidade Federal de Minas Gerais, 2006.


Dispersion and uncertainty in multislit matter wave diffraction

Crossref

We show that single and multislit experiments involving matter waves may be constructed to assess dispersively generated correlations between the position and momentum of a single free particle. These correlations give rise to position dependent phases which develop dynamically as a result of dispersion and may play an important role in the interference patterns. To the extent that initial transverse coherence is preserved throughout the proposed diffraction setup, such interference patterns are noticeably different from those of a classical dispersion free wave. (c) 2007 Published by Elsevier B.V

GLIONNA, G.

BLIN, A. H.

HILLER, B.

Nemes, Maria Carolina

SAMPAIO, Marcos

Piza, Antonio Fernando Ribeiro de Toledo

Biblioteca Digital da Produção Intelectual da Universidade de São Paulo (BDPI/USP)

Universidade de São Paulo

Physica A Statistical Mechanics and its Applications

7 pages, 5 figures, revised manuscriptWe show that single and multislit experiments involving matter waves may be constructed to assess correlations between the position and momentum of a single free particle. These correlations give rise to position dependent phases which develop dynamically and may play an important role in the interference patterns. For large enough transverse coherence lenght such interference patterns are noticeably different from those of a classical dispersion free wave

Hal-Diderot

RCAAP - Repositório Científico de Acesso Aberto de Portugal

https://estudogeral.sib.uc.pt/jspui/bitstream/10316/4338/1/filef0f6125a076e4c07a7720ca55120bdb0.pdf

Dispersion and uncertainty in multislit matter wave diffraction

Abstract

Similar works

Full text

Available Versions

Estudo Geral

Crossref

Biblioteca Digital da Produção Intelectual da Universidade de São Paulo (BDPI/USP)

Universidade de São Paulo

Estudo Geral

Hal-Diderot

RCAAP - Repositório Científico de Acesso Aberto de Portugal