INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 38 (2005) 9803­9819 doi:10.1088/0305-4470/38/45/006
Single-particle matter wave pulses
Adolfo del Campo and Juan Gonzalo Muga
Departamento de Química-Física, Universidad del País Vasco, Apdo. 644, Bilbao, Spain
E-mail: qfbdeeca@lg.ehu.es and jg.muga@ehu.es
Received 3 August 2005, in final form 27 September 2005
Published 26 October 2005
Online at stacks.iop.org/JPhysA/38/9803
Abstract
We study the properties of quantum single-particle wave pulses created by
sharp-edged or apodized shutters with single or periodic openings. In particular,
we examine the following: the visibility of diffraction fringes depending
on evolution time and temperature; the purity of the state depending on the
opening-time window; the accuracy of a simplified description which uses
`source' boundary conditions instead of solving an initial value problem; and
the effects of apodization on the energy width.
PACS numbers: 03.75.-b, 03.65.Yz
1. Introduction
Diffraction in time was first discussed by Moshinsky [1]. The hallmark of this phenomenon
consists of temporal oscillations, deviating from the classical regime, of Schr¨odinger, matter
waves released in one or several pulses from a preparation region in which they are initially
confined. The original setting consisted of a sudden opening of a shutter to release a semi-
infinite beam, and provided a quantum, temporal analogue of spatial Fresnel-diffraction by
a sharp edge [1]; later more complicated shutter windows, initial confinements and time-slit
combinations have been considered [2], in particular two temporal slits and the corresponding
interferences. Experimental observations have been carried out with ultracold neutrons [3]
and with ultracold atoms [4­6]. Recently, it has also been observed for electrons in a double
temporal-slit experiment [7]. The `shutter problem' has indeed important applications, as it is
the modelization of turning on and off a beam of atoms as done, for example, in integrated atom-
optical circuits or a planar atom waveguide [8], and may allow us to translate the principles of
spatial diffractive light optics to the time domain for matter waves [9]. In addition, it provides
a time­energy uncertainty relation [10, 11] which has been verified experimentally for atomic
waves, by realizing a Young interferometer with temporal slits [4­6]. The Moshinsky shutter
has also been discussed with the Wigner function and tomographic probabilities [12], for
relativistic equations [1, 13, 14], with dissipation [15, 16], in relation to Feynman paths [17],
or for time-dependent barriers [18], including the time dependence of states initially confined
0305-4470/05/459803+17$30.00  2005 IOP Publishing Ltd Printed in the UK 9803
9804 A del Campo and J G Muga
in a box when the walls are suddenly removed [19, 20]. With adequate interaction potentials
added to the model, it has been used to study and characterize transient dynamics of tunnelling
matter waves [14, 21­27], and the transient response to abrupt changes of the interaction
potential in semiconductor structures and quantum dots [28, 29].
Within the source boundary approximation, in which the form of the wave is imposed
for all times at a source point or surface, many works have been carried out within the field
of neutron interferometry [30, 31], considering also triangular aperture functions [30, 32],
atom-wave diffraction [33], tunnelling dynamics [14, 25, 34­38], and absorbing media, in
which an ultrafast peak-propagation phenomenon has been recently described [16].
The importance of pulse formation is nowadays enhanced due to the possibilities of
controlling the aperture function of optical shutters in atom optics, and to the development
of atom lasers. For such devices some of the first mechanisms proposed explicitly implied
periodically switching off and on the cavity mirrors, that is, the confining potential of the lasing
mode. As an outcome, a pulsed atom laser is obtained. Much effort has been devoted to design
a continuous atom laser, whose principle has been demonstrated using Raman transitions
[39, 40]. With this output coupling mechanism, an atom initially trapped suffers a transition to
a nontrapped state, receiving a momentum kick during the process due to the photon emission.
These transitions can be mapped to the pulse formation, so that the `continuous' nature of the
laser arises as a consequence of the overlap of such pulses. There is, in summary, a strong
motivation for a thorough understanding of matter wave pulse creation, even at an elementary
single-particle level.
In this paper, we aim to describe the characteristics of one-dimensional, matter wave
pulses within the approximation in which interatomic interactions can be neglected, as it is
usually the case in standard low-pressure atomic beams. This will be useful as a reference
and first step to consider the interacting case later on. While the most idealized pulses have
been studied in several works since the seminal paper of Moshinsky [1], some aspects of
realistic pulse production have not been examined with enough detail yet, such as the effects
of statistical mixing in the preparation beam, or the fact that optical shutters do not have
absolutely sharp edges, and could be tailored for designing the pulse features. The other main
objective of the paper is to compare two ways to model boundary conditions: as a standard
initial value problem where the wave is specified in the preparation region at time t = 0, or
according to the `source approach' in which the wavefunction is given for all times at the
source point.
The organization of the paper is as follows: In section 2 we review the Moshinsky shutter,
whose main feature--diffraction in time--is quantified by means of the fringe visibility in
section 3; section 4 deals with the evolution of finite pulses; in section 5 the exact results
are compared with the usual `source' approximation, and, finally, in section 6 the aperture
function is modified for different types of apodization and the time­energy uncertainty product
is evaluated. We also consider periodical shutter apertures and find analytical solutions for the
wavefunction.
2. Moshinsky shutter
In this section, we review the `Moshinsky shutter' [1, 10, 41] and discuss its physical
interpretation. Consider a plane wave impinging from the left on a totally absorbing infinite
potential barrier located at the origin (shutter) which is suddenly turned off at time zero. For
t = 0 one has the initial condition
(x, t = 0) = eipx/h
(-x),
Single-particle matter wave pulses 9805
4 2 2 4
0.25
0.5
0.75
1
1.25
P
0.75 0.5 0.25 0.25 0.5 0.75
C
0.5
0.25
0.25
0.5
S(a) (b)
Figure 1. (a) Diffraction in time seen in P (), see (3), for an initial cut-off plane wave state.
(b) Cornu spiral. The points associated with the maximum and minimum of the largest fringe are
plotted. They correspond to a maximum and a minimum of the distance to (-1/2, -1/2).
where is the step function, whereas for t > 0 the wavefunction can be written using the
free-particle Green's function,
(x, t) =
0
-
dx G0(x, t; x , 0)(x, 0) =
e
imx2
2th
2
w[-u(p, t)], (1)
where
G0(x, t; x , 0) :=
m
2iht
ei m(x-x )2
2ht , u(p, t) :=
1 + i
2
t
mh
p -
mx
t
, (2)
and
w(z) := e-z2
erfc(-iz) =
1
i -
du
e-u2
u - z
,
is the w or `Faddeyeva' function [42, 43]. The contour - goes from - to  passing below
the pole.
The associated `density'1
|(x, t)|2
exhibits diffraction in time, namely, a characteristic
oscillation in time, and also in the space domain, see figure 1(a), in contrast with the simple
step function for a spatially homogeneous ensemble of classical particles with momentum p
released at time t = 0. Thanks to the relation between the w-function and Fresnel integrals
[43],
C() + iS() =
1 + i
2
1 - ei2
/2
w
1 + i
2
1/2
 ,
 =
t
mh
1/2
(p - mx/t) =
1 - i
1/2
u(p, t),
the solution may also be rewritten as
|(x, t)|2
= P() =
1
2
S() +
1
2
2
+ C() +
1
2
2
, (3)
and therefore be mapped onto the Cornu spiral, which is plotted in figure 1(b). The `density'
can then be read as half the square of the distance from the point (-1/2, -1/2) to any other
point of the spiral; and the origin corresponds to the classical particle with momentum p
released at time t = 0 from the shutter position.
Suppose now that the shutter is closed suddenly at time . The integrated `density'
N+() :=

0 dx|(x, )|2
grows continuously with  and may reach arbitrarily large values,
obviously greater than 1. Moreover, |(x, t)|2
is dimensionless, so N+ cannot be taken as a
1 Actually |(x, t)|2 is a relative density with respect to the asymptotic, long-time, stationary value.
9806 A del Campo and J G Muga
probability in any case. This poses the question of physically interpreting the mathematical
results. Clearly, we face analogous problems to interpret a plane wave or stationary scattering
states, which are not in Hilbert space. The solution is then found similarly, by assuming that
(x, t), except for a normalization factor, is just a component of a normalizable state in Hilbert
space, to be determined by the experimental preparation set-up. A consistent interpretation
for several--noninteracting--particles requires the appropriate quantum symmetrization [44],
which will be treated elsewhere. In the present paper, we shall limit ourselves to the simplest
case and consider only one-particle states.
For a shutter with a non-zero reflectivity, the cut-off plane wave of opposite momentum
has to be taken into account, so the initial state takes the form [1]
(R)
p (x, t = 0) := (eip/hx
+ R e-ip/hx
) (-x). (4)
The cases R = {1, 0, -1} turn out to be the most relevant. We will refer to R = -1 as sine
initial conditions, which arise when the shutter is totally reflective; and to R = 1 as cosine
initial conditions. This last case may seem hard to implement but, surprisingly, it is related to
the usual source boundary condition,
s
p(x = 0, t) = e-it
(t), s
p(x > 0, t 0) = 0, (5)
where  = p2
/(2mh), and is more tractable mathematically than the sine case. As we shall
see, s
p(x, t) = (1)
p (x, t).
3. Visibility of main diffraction fringe
Let us examine first the time evolution from a cut-off plane wave initial condition, i.e., R = 0,
as in equations (1) and (3). The motion of points of constant `density' is obtained by imposing
a constant value of , according to (3). In particular, for the maximum and nearby minimum,
see figure 1(b),
xmax(t) = pt/m -
ht
m
max, xmin(t) = pt/m -
ht
m
min,
where max = 1.217 and min = 1.872 are universal for R = 0. Since the probability is
exclusively  dependent, see (3), Pmax = 1.370, and Pmin = 0.778, independent of time, mass
or momentum, and therefore the fringe visibility, defined as
V(T ) =
P1st max - P1st min
P1st max + P1st min
, (6)
is concluded to be universal (equal to 0.276 for R = 0), and of course time independent.
Another important fringe feature is the width of the main peak. It can be computed from
the intersection with the classical probability density (for a cut-off beam of particles released
at time 0 with momentum p) [1, 10],
x 0.85
ht
m
1/2
. (7)
Next, we consider an initial state given by a statistical mixture,
(t = 0) := dp f (p) (R)
p (t = 0) (R)
p (t = 0) , (8)
and focus our attention on the effect of temperature and evolution time on the diffraction
pattern. Let us assume, by integrating out the y, z-degrees of freedom of a Maxwell­Boltzmann
momentum distribution [45], that the Gaussian momentum probability density
fMB(p) =
1

2mkBT
e
- (p-pc)2
2mkB T
, (9)
Single-particle matter wave pulses 9807
-6 -4 -2 0 2 4
logT(K)
0
0.1
0.2
0.3
0.4
Visibility
Sin
Exp(ikx)
Cos
(a)
0 5 10 15 20

0
0.1
0.2
0.3
0.4
Visibility
200s
150s
100s
50s
(b)
Figure 2. (a) Visibility as a function of temperature and the reflectivity of the shutter for a beam
of argon with the distribution (9). pc/m = 10 cm s-1, t = 20 s. (b) Visibility as a function of
the  parameter for an argon atom with a velocity pc/m = 10 cm s-1. Each curve is computed by
varying the temperature for a given opening time. R = -1.
holds for the momentum in the x-direction. The introduction of a non-zero mean average
momentum pc implies that the atomic ensemble, say in a magneto-optical trap, is launched
along the x-axis with a velocity pc/m. This is realized, for example, in the `moving
molasses technique' [50] within an atomic fountain clock. We shall generally assume that the
contribution of negative p in (9) is negligible.
Figure 2(a) shows the effect of the temperature on the visibility of the main diffraction
fringe for statistical mixtures weighted by the distribution (9), with R = {0, 1}. It is clear
that the diffraction peaks are more visible for sine conditions than for the isolated cut-off plane
wave and cosine conditions, the difference being more noticeable at small temperatures. As
expected, there is a suppression of the diffraction pattern when the temperature increases. A
possible criterion for this suppression is a small value of the ratio, , between the width of
the fringe x and the separation xcl of the classical arrival points corresponding to fast and
slow components of the statistical mixture, namely,
 :=
x
xcl
=
0.85
p
mh
t
.
For the Gaussian distribution, equation (9), one can choose p as the full width at half
maximum (FWHM), that is, 2

2ln2mkBT , so that
 = 0.36
h
KBT t
.
According to figure 2(b), the visibility grows with  for small values and saturates for large
, i.e., for small times and/or temperatures.
Moreover, if instead of (9), the momentum distribution of an effusive atomic beam is used
[46, 45],
fbeam(p) =
p3
2(mkBT )2
e
- p2
2mkB T
(p), (10)
then p =

9mkBT /8, and the upshot is a complete suppression of the diffraction-in-time
phenomena for all times and temperatures. This fact points out the relevant role played by the
momentum distribution, that is, by the experimental preparation set-up being considered.
Summing it up, the fringe visibility, at variance with the pure state case, becomes time
dependent for mixed states. The characteristic oscillations of diffraction in time tend to be
9808 A del Campo and J G Muga
washed out with the observation time after opening the shutter, and also by increasing the
temperature, or for the momentum distribution of an effusive beam.
4. Evolution of a finite pulse
In this section, we focus on the formation and evolution of a matter wave, single-particle pulse.
Suppose that the shutter is opened at the instant t = 0 and closed at the instant . We will refer
to the duration  as the `opening time'. It is, in other words, the width of the time slit. The
notation (R)
p0, (t) will denote the state formed in that manner, at time t , from the initial
state of (4) with main momentum p0.2
For an initially pure state the time evolution of the pulse can be written in terms of the
inner products with the eigenstates of the free Hamiltonian which vanish at the origin,
x|E =
2mh
p
sin (px/h), E = p2
/(2m). (11)
They obey orthonormality and completeness relations,
E |E = (E - E ),

0
dE|E E| = ^1.
As shown by Moshinsky [10], the overlap between these eigenstates and the state that results
from a state formed after an opening time  with R = 0 is
E|(0)
p0, (t = ) =
mph
2
w[-u0(p0, )]
p2 - p2
0
-
w[-u0(p, )]
2p(p - p0)
-
w[-u0(-p, )]
2p(p + p0)
,
where
u0(p, ) :=
1 + i
2

mh
p, (12)
to be compared with (2). (u0(p, ) is a particular case of u(p, ) in which x = 0.) The
overlaps for the cases R = 1 are obtained easily from (12).
Writing the final time of observation as t =  + t, the evolved pulse can then be
calculated as
(R)
p0, (x, t) =

0
dE e-iE t/h
x|E E (R)
p0, (t = ) . (13)
This expression is not well-behaved numerically, but such a problem can be swiftly solved by
modifying the contour of integration in the complex p-plane, since several terms can be written
as w-functions (see appendices A and B for details), and for the rest we gain an integrand with
Gaussian decay. Explicitly, for R = -1,
(-1)
p0, (x, t) =
e
imx2
2th
2
{[w[-u(p0, t)] + w[-u(-p0, t)]}
-
e-i
p2
0
2mh - w[u0(p0, )]
2
e
imx2
2 th {w[-u(p0, t)] + w[-u(-p0, t)]}
-
i
2 +
dp e-i p2
2mh +ipx/h w[u0(-p, )]
p + p0
+
w[u0(p, )]
p - p0
, (14)
2 Another way to form a pulse is to remove at time t = 0 the potential walls that confine a state which is initially
stationary. For an infinite-wall square box the result is the subtraction of two semi-infinite R = -1 pulses, i.e., at
least for free motion, a simple combination of w-functions. For applications of this idea in free-motion cases or
scattering configurations, see [19, 20, 26, 44, 47].
Single-particle matter wave pulses 9809
where the contour passes above the poles. This result is so far exact. The integral term, Y,
can be rewritten by integrating in the u variable,
Y = -
i e
imx2
2 th
2 u
du e-u2 w 
t
u + v
u - u(p0, t)
+
w - 
t
u - v
u - u(-p0, t)
, (15)
see appendix A, with
v =
1 + i
2

mh
mx
t
. (16)
A way of dealing with this kind of integral to excellent accuracy was discussed by Brouard
and Muga in [21]. It consists in extracting the singularities of the integrand, which we shall
denote as g(u), in the following way:
g(u) =
i
Ai
u - ui
+ h(u). (17)
Ai is the residue of g(u) at the pole ui and h(u) is an entire function which admits a power
series expansion convergent in the whole u-plane,

-
h(u) e-u2
du =

 h(u = 0) +

n=1
(2n - 1)!!
2n(2n)!
h(2n)
(u = 0) .
Retaining just the first term of the previous expansion leads to
u
du g(u)
i
Ai -iw(-ui) +


ui
+

g(u = 0). (18)
In particular,
Y -
e
imx2
2 th
2
{w[u0(p0, )]w[-u(p0, t)] + w[-u0(p0, )]w[-u(-p0, t)]}
+
i e
imx2
2 th
2


w(v) - w[u0(p0, )]
u(p0, t)
+
w(-v) - w[-u0(p0, )]
u(-p0, t)
, (19)
which we shall use later in section 5 to compare the initial value problem and source approaches.
4.1. Purity of the chopped state: coherence creation from incoherent mixtures
by using small opening times
It is possible to `create coherence' from statistical mixtures, simply by means of small opening
times. This was seen experimentally with cold atoms by Dalibard and coworkers [6]. To
quantify this effect we use the purity, defined by
Pt := Tr 2
N, (t),
where N, (t) is the normalized density operator at time t for an opening time . Our starting
point is the unnormalized state
 (t) := dp f (p) (0)
p, (t) (0)
p, (t) . (20)
Thanks to the invariance under cyclic permutations of the trace and unitarity of the time
evolution, the purity is invariant for t , so it can be calculated in particular at t = ,
P =
1
N2
dp dp f (p)f (p ) (0)
p, (0)
p ,
2
, (21)
9810 A del Campo and J G Muga
0 5 10 15 20
(s)
0
0.2
0.4
0.6
0.8
1
Purity
5K
10K
20K
Figure 3. Variation of purity versus the opening time for different temperatures. An effusive beam
of argon atoms is considered, see (10). R = 0.
where N is given by
N = Tr  = dp f (p) (0)
p, (0)
p, . (22)
Considering the case of an effusive atomic beam, f (p) = fbeam(p), equation (10), figure 3
shows that the purity Pt decreases as a function of the opening time and temperature. For
short opening times a highly coherent state is created in spite of the mixed preparation state.
This result, in the limit   0, may be understood as follows. As long as a (pure state)
wavefunction vanishes in x > 0 at time t = 0, the wavefunction for t > t > 0 can be written
as [47]
(x, t) =
t
0
dt K+(t, x; t , x = 0)(x = 0, t ), (23)
where
K+(t, x; t , x = 0) =
h
mi
d
dx
G0(x, t; 0, t ) =
m
ih(t - t )3
1/2
x eimx2
/2h(t-t )
.
By performing a Taylor series expansion in t around t = 0, one finds that
(x, t) = tK+(t, x; 0, 0) + O(t2
) (24)
for (x = 0, t = 0) = 1. Since (0)
p, (x = 0, t = 0) = 1 for all p, then to first order all
the momenta that form the preparation mixture give the same contribution, in other words,
|p, (t) in (20) becomes independent of p, and thus the resulting state tends to be independent
of the initial distribution f (p), and totally coherent (pure) in the limit of small opening times.
For the case of the Gaussian distribution in (9), the same qualitative behaviour (monotonic
decay from 1 in the same time scale) is observed.
5. Comparison between `source' and `initial value problem' approaches
Golub and G¨ahler [30] tackled the diffraction-in-time problem in two dimensions using the
Green's function formalism, working within the Fraunhoffer limit, and imposed source
boundary conditions. This approach was later thoroughly implemented for different
combinations of time and space rectangular slits by Brukner and Zeilinger [33]. Other
works have used the simple source boundary conditions of (5) in one dimension, see [38] and
references therein. In [47] the relations between source boundary conditions and the initial
value problem (IVP) were examined and equivalence conditions established.
Single-particle matter wave pulses 9811
In order to compare pulses formed by source or IVP approaches let us introduce the
window function [0,](t) = (t) ( - t) and impose the source boundary condition
s
p0, (x = 0, t) = e-i0t
[0,](t), (25)
(the wavefunction is also assumed to vanish at x > 0 for all t 0) with Fourier transform
s
p0, (x = 0, ) =
i

2
1
 - 0
[1 - ei(-0)
].
At a later time one finds, for x > 0, that
s
p0, (x, t) =
i
2

-
d
eipx/h-it
 - 0
[1 - ei(-0)
],
where Im p 0. This can be written in the complex p-plane by introducing the contour of
integration + which goes from - to  passing above the poles,
s
p0, (x, t) =
i
2 +
dp 2p
eipx/h-i p2t
2mh
p2 - p2
0
[1 - ei(-0)
]
=
i
2 +
dp
1
p + p0
+
1
p - p0
eipx/h-i p2t
2mh [1 - ei(-0)
].
The integral can be carried out exactly using appendix A,
s
p0, (x, t) =
e
imx2
2th
2
{w[-u(p0, t)] + w[-u(-p0, t)]}
-
e-i
p2
0
2mh + imx2
2 th
2
{w[-u(p0, t)] + w[-u(-p0, t)]},
so the time evolution is straightforward and given by a difference of w-functions (in fact,
expressing the window function as a difference of step functions [0,] = (t) - (t - ),
the solution can be foreseen to be given by the difference of one source open at t = 0 and
a second one at a later time t = , with the associated phase taken into account). The nice
thing about this approach is that it allows a clear and intuitive interpretation of the different
contributions to the total wavefunction, and it is easy to implement complicated combinations
of time and space windows. The flip side is that deviations from the exact result of the initial
value problem arise, specially for the back part of the wavefunction cut by the chopper, except
in the case R = 1. Although the calculation is somewhat tedious--see appendix C--it can be
proved that for cosine initial conditions, the wavefunction that results from a shutter opened
between 0 and , (1)
p0, (x, t), is exactly equivalent to using source boundary conditions in that
interval, i.e., to (26). The difference with the solution for the reflecting shutter (R = -1) can
be written as
(-1)
p0, (x, t) - (1)
p0, (x, t)
e
imx2
2 th
2
{w[u0(p0, )] - w[-u0(p0, )]} w[-u(-p0, t)]
+
ie
imx2
2 th
2


w(v) - w[u0(p0, )]
u(p0, t)
+
w(-v) - w[-u0(p0, )]
u(-p0, t)
, (26)
where we have used equations (14), (19) and (26). The overlap probability between pulses
generated for different values of R and fixed  and p0,
O(R,R )
:=
| (R)
|(R )
|2
(R )|(R ) (R)|(R)
, (27)
9812 A del Campo and J G Muga
0 1 2 3 4 5
2S/h
0.92
0.94
0.96
0.98
1
Overlap
Figure 4. Overlap probability between the wavefunction of one argon atom associated with
sine and cosine initial conditions (the continuous line), cut-off plane wave and sine conditions
(circles), and cut-off plane wave and cosine conditions (the dashed line), as a function of the phase
S/h := p2
0/(2mh), see (27).
is independent of the evolution time t, and can be calculated exactly at t = . It turns out to be
a function of the adimensional phase S/h := p2
0 (2mh), see figure 4. The overlaps tend to 1
as   0, . Remarkably, there is a lower bound which is 93.56% between sine and cosine
source conditions. We may conclude that, in general, the result of using source boundary
conditions is not at all a bad approximation to the solution of the initial value problem with
reflecting or absorbing conditions, except when very accurate results are required.
6. Apodization of matter wave pulses
Apodization is a technique long used in light optics to avoid the diffraction effect in several
devices. To put it simply, it consists in suppressing high-frequency components by smoothing
the aperture function (or `window' for short) of the pulse [48]. The price to pay is a slight
broadening of the energy distribution. An important step in this direction was given in
[30, 32], where the case of a mechanical shutter with a triangular aperture function was treated.
In this section, apodization is applied to matter wave pulses within the source approach. In
particular, the case of a sine-type aperture function can be analytically solved in terms of
w-functions. Let us consider in (25), instead of the sudden, rectangular shape, a sine aperture
function,
[0,](t) = sin( t) (t) ( - t) (28)
with  =  -1
. (Note that the `sine' aperture function has nothing to do with sine initial
conditions.) Then, defining  = 0  and the corresponding momenta p =

2mh,
with the branch cut taken along the negative imaginary axis of , it is found that
s
p0, (x = 0, ) =
1

8
1 - ei(--)
 - -
-
1 - ei(-+)
 - +
,
and the wavefunction is (we shall distinguish apodized pulses from the rectangular aperture
pulses used so far by means of a tilde)
s
p0, (x, t) =
-1
2 =


-
d e-it+ipx/h 1 - ei(-)
 - 
=
-1
2 =

+
dp eipx/h
(e-it
- e-i t-i
)
1
p + p
+
1
p - p
Single-particle matter wave pulses 9813
-1 0 1 2
t/
00.51
Aperturefunction
Figure 5. Different aperture function studied. The smoothness at the edges increases in this order:
rectangular (the continuous line), sine (the dotted line), Hanning (the dashed line) and Blackman
(the long dashed line).
=
i
2 =
 e
imx2
2th {w[-u(p, t)] + w[-u(-p, t)]}
- e-i
p2

2mh + imx2
2 th {w[-u(p, t)] + w[-u(-p, t)]}
= i
=
s
p, (x, t).
As it is clear from the last line, the apodization with the sine aperture function is equivalent
to the introduction of two different momenta components whose values are related to the
length of the pulse. Pictorially, a sine window pulse is equivalent to the coherent difference
of two simultaneous pulses produced with sudden rectangular windows from speeded up and
decelerated sources.
Similarly, other window functions, as those shown in figure 5 can be worked out. For
instance, the Hanning aperture function,
[0,](t) = sin2
( t) (t) ( - t) = 1
2
(1 - cos 2 t) (t) ( - t), (29)
after introducing p =

2mh(0  2 ) with  = , leads to
s
p0, (x, t) =
1
2

s
p0, (x, t) -
1
2 =
s
p , (x, t)

 , (30)
where the effect of the apodization is to subtract from the pulse with the source momentum
two other matter wave trains associated with p, all of them formed with rectangular aperture
functions.
A related configuration which can be solved analytically in terms of w-functions is the
source apodized periodically to create successive pulses. The case of two rectangular slits
has already been been discussed in [31, 33]. Here we extend (29) to all positive times, which
allows us to consider an arbitrary number of apodized slits. The result is
s
p0, (x, t) =
e
imx2
2th
4
{w[-u(p0, t)] + w[-u(-p0, t)]}
-
1
8 =
e
imx2
2th {w[-u(p, t)] + w[-u(-p, t)]}.
9814 A del Campo and J G Muga
60 70 80 90 100
x(m)
0
0.01
0.02
0.03
0.04
|(x,t)|
2
Figure 6. Front part of the probability density for the periodic Hanning function at t = 1.2 ms,
with an aperture time  = 10 s. The circle, diamond and square mark, respectively, the classical
position of the slow (p-), original (p0) and accelerated (p+) components. p0/m = 10 cm s-1.
10 12 14 16 18 20
x(m)
0
0.1
0.2
0.3
0.4
0.5
|(x,t)|
2
(a)
16 17 18 19 20 21
x(m)
0
0.01
0.02
0.03
0.04
|(x,t)|
2
(b)
Figure 7. (a) Probability density of a pulse of ultracold argon atoms (p0/m = 10 cm s-1), with
 = 10 s registered at t = 200 s for different window functions: rectangular (the continuous
line), sine (the dotted line), Hanning (the dashed line) and Blackman (the dot-dashed line).
(b) Detail of the sidelobes of the previous figure.
Note again the intervention of three different momenta. A consequence is that the w-function
for the fastest component will eventually separate spatially as a smoother--rather than pulsed--
forerunner, see figure 6.
Also suitable for suppression of the sidelobes is the so-called Blackman aperture function,
[0,](t) = 0.42 - 0.5 cos(2 t) + 0.08 cos(4 t). (31)
Using the p defined above and the momenta
p = 2mh(0  4 ), (32)
with ,  = , one can write the time evolved wavefunction in terms of 20 w-functions,
s
p0, (x, t) = 0.42s
p0, (x, t) - 0.25
=
s
p , (x, t) + 0.04
 =
s
p , (x, t).
So, once again, the smoothed shutter is related to a combination of different pulses from
rectangular time slits.
The probability density for a single pulse and the details of the secondary diffraction peaks
for four different aperture functions are illustrated in figure 7, where all states are normalized
Single-particle matter wave pulses 9815
0 5 10 15 20
(s)
2
4
6
8
2E/h
Figure 8. Uncertainty product, E, for a pulse of ultracold argon atoms (p0/m = 10 cm s-1):
sine aperture function and E = FWHM (the solid line); sine aperture function and E = E
(the long-dashed line); rectangular aperture function with E = FWHM (the dashed line).
to one. As expected, the smoother the time window function, the higher the suppression of
the sidelobes. Note that the Blackman apodized pulse is slightly more advanced that the other
ones due to the faster components induced by this particular apodization.
6.1. Time­energy uncertainty relation
A time­energy uncertainty relation was discussed for the shutter problem by Moshinsky, who
calculated the energy distribution of a particle after closing the shutter [10]. Using cosine
conditions with energy E and taking the overlap of the wavefunction with the free-particle
eigenstates which vanish at the origin, he calculated the energy distribution when the shutter
was open at time ,
(E, E , ) = N

E
sin2
[(E - E )/2h]
(E - E )2
, (33)
with N being the normalization constant. Then it was concluded that, for some measure of the
energy width [10] (see [11] for a comprehensive review of time­energy uncertainty relation),
E h. (34)
In fact, if one introduces the root of the variance as the measure of energy spreading, E = E,
such an uncertainty relation cannot be established since E diverges. This is in consonance
with the non-existence of the average displacement, x , for sharply cut waves [49]. One
can also calculate the energy distribution for the case of a sine aperture function and source
boundary conditions, equation (28), analytically. It turns out to be slightly broadened, but the
variance is now well defined as a consequence of the suppression of high-frequency sidelobes.
Alternatively, as done in [6], the FWHM can be taken as the measure of energy spread.
One observes that in both cases, sudden and smooth shutter, both measures of the spread
decrease with the opening time. The uncertainty product is represented versus the opening
time in figure 8. Note the two regimes, first a linear growth and then saturation, separated by
the characteristic time required for a classical particle to travel a de Broglie wavelength,
=
hm
p2
0
. (35)
9816 A del Campo and J G Muga
7. Concluding remarks
Starting with a brief review of the Moshinsky shutter problem, we have examined several one-
dimensional, matter wave pulse characteristics such as the visibility of the diffraction-in-time
fringes versus evolution time and temperature, or the creation of coherence by small aperture
times. Moreover, a longstanding missed comparison among several initial conditions used
in previous works, which differed on the reflected components in the preparation region, has
been carried out. We have studied several apodizations of the aperture function, and obtained
analytical expressions for the time evolution of the corresponding pulses, also in the periodic
case. The effects of apodization to suppress secondary diffraction peaks and in the energy
width have been discussed.
The present work is the first step towards a more ambitious objective, namely, tailoring the
pulses for particular needs, including the case in which interatomic interaction is important.
Applications for laser atoms, in particular, will require in general considering non-linear
effects. Physically, optical shutters formed by effective potentials due to detuned lasers offer
a unique opportunity, not available with mechanical shutters, to control the formation and
subsequent behaviour of the pulse.
Acknowledgments
We thank Gastón García-Calderón for valuable comments. This work has been supported
by Ministerio de Educación y Ciencia (BFM2003-01003), and UPV-EHU (00039.310-
15968/2004). AC acknowledges a fellowship from the Basque Government (BF104.479).
Appendix A. Integrals
Consider the following integral along a contour which goes from - to  passing above the
pole at p0,
I =
+
dp
e-iap2
+ibp
p - p0
, a > 0. (A.1)
The saddle point is at ps = b/2a. By completing the square, one is led to introduce the
variable
u = u(p) =
1 + i

2

ap -
b
2

a
. (A.2)
Deforming the contour to go along the steepest descent path from the saddle, which in the
u-plane is at the origin,
I = ei b2
4a
u
du
e-u2
u - u(p0)
, (A.3)
If C0 denotes a counterclockwise circle around the pole u(p0), then u = (-, )  C0 if
the pole has been crossed (Im(u) > 0) and the real line of u otherwise. We can immediately
recognize this integral as a Faddeyeva function since the following equalities hold thanks to
Cauchy's theorem,
1
i +
du
e-u2
u - u(p0)
=
1
i -
du
e-u2
u - u(p0)
- 2e-[u(p0)]2
= w[u(p0)] - 2e-[u(p0)]2
= -w[-u(p0)],
Single-particle matter wave pulses 9817
where + and - go from - to  passing above and below the pole, respectively. One
concludes that
I = -i ei b2
4a w[-u(p0)]. (A.4)
Appendix B. sine initial conditions
After opening the shutter and closing it at time , the wavefunction for a reflecting shutter,
R = -1, is given by
(-1)
p, (x, t) = (0)
p, (x, t) - (0)
-p, (x, t),
which is a difference of w-functions.
Since, using equations (12) and (13),
(0)
p, (x, t) =
1
2

-
dp p e-i p 2 t
2mh sin(p x/h)
×
w[-u0(p, )]
p 2 - p2
-
w[-u0(p , )]
2p (p - p)
-
w[-u0(-p , )]
2p (p + p)
,
the wavefunction can be written as
(-1)
p, (x, t) =
i
8

-
dp e-i p 2 t
2mh (eip x/h
- e-ip x/h
)
×
w[u0(p , )] - w[-u0(p , )] - w[-u0(p, )] + w[u0(p, )]
p + p
+
w[-u0(p , )] - w[u0(p , )] - w[-u0(p, )] + w[u0(p, )]
p - p
.
Using the symmetry under p  -p the previous expression is simplified to
x|(-1)
p, (t) =
i
4

-
dp e-i p 2 t
2mh +ip x/h
×
w[u0(p , )] - w[-u0(p , )] - w[-u0(p, )] + w[u0(p, )]
p + p
+
w[-u0(p , )] - w[u0(p , )] - w[-u0(p, )] + w[u0(p, )]
p - p
,
where the relation
w(z) + w(-z) = 2e-z2
(B.1)
can be applied leading to
(-1)
p, (x, t) =
-i
2

-
dp e-i p 2 t
2mh +ip x/h
× e-i p2
2mh - w[u0(p, )] - e-i p 2
2mh
1
p + p
+
1
p - p
+ Y, (B.2)
with
Y = -
i
2

-
dp e-i p 2 t
2mh +ip x/h w[u0(p , )]
p - p
+
w[-u0(p , )]
p + p
.
The resulting integrals, but for the term Y, are of the form solved in appendix A. The final
result is (14).
9818 A del Campo and J G Muga
Appendix C. cosine initial conditions
The solution to the Moshinsky shutter with cosine initial conditions, R = 1,
(1)
p (x, t) =
e
imx2
2th
2
{w[-u(p, t)] + w[-u(-p, t)]}, (C.1)
is immediately computed from (1). The same result is obtained using the source boundary
conditions of (5) and following the steps leading to (25).
In fact the same agreement can be extended to finite opening times . The procedure to
obtain (1)
p0, (x, t) is very much the same as for the sine case. Using
(1)
p, (x, t) = (0)
p, (x, t) + (0)
-p, (x, t),
the expressions for the (0)
p, (t) in (B.1), the identity in (B.1), and t =  + t, one finds that
(1)
p, (x, t) =
i
4

-
dp e-i p 2t
2mh - e-i p2
2mh e-i p 2 t
2mh (eip x/h
- e-ip x/h
)
1
p + p
+
1
p - p
.
Due to the symmetry under p  -p the term with -e-ip x/h
is equal to that with eip x/h
and
the resulting integral can be carried out by completing the square and deforming the contour
of integration in the complex p-plane as described in appendix A, so new Faddeyeva functions
can be identified. This leads to the same expression found in (26).
References
[1] Moshinsky M 1952 Phys. Rev. 88 625
[2] Kleber M 1994 Phys. Rep. 236 331
[3] Hils T, Felber J, G¨ahler R, Gl¨aser W, Golub R, Habicht K and Wille P 1998 Phys. Rev. A 58 4784
[4] Steane A, Szriftgiser P, Desbiolles P and Dalibard J 1995 Phys. Rev. Lett. 74 4972
[5] Arndt M, Szriftgiser P, Dalibard J and Steane A M 1996 Phys. Rev. A 53 3369
[6] Szriftgiser P, Guéry-Odelin D, Arndt M and Dalibar J 2003 Phys. Rev. Lett. 77 4
[7] Lindner F et al 2005 Preprint quant-ph/0503165
[8] Schneble D, Hasuo M, Anker T, Pfau T and Mlynek J 2003 J. Opt. Soc. Am. B 20 648
[9] Bernet S, Abfalterer R, Keller C, Schmiedmayer J and Zelinger A 1998 J. Opt. Soc. Am. B 15 2817
[10] Moshinsky M 1976 Am. J. Phys. 44 1037
[11] Busch P 2002 Time in Quantum Mechanics ed J G Muga, R Sala and I Egusquiza (Berlin: Springer) chapter 3
[12] Mánko V, Moshinsky M and Sharma A 1999 Solid State Commun. 94 979
[13] García-Calderón G, Rubio A and Villavicencio J 1999 Phys. Rev. A 59 1758
[14] Delgado F, Muga J G, Ruschhaupt A, García-Calderón G and Villavicencio J 2003 Phys. Rev. A 68 032101
[15] Moshinsky M and Schuch D 2001 J. Phys. A: Math. Gen. 34 4217
[16] Delgado F, Muga J G and Ruschhaupt A 2004 Phys. Rev. A 69 022106
[17] García-Calderón G, Villavicencio J and Yamada N 2003 Phys. Rev. A 67 052106
[18] Scheitler G and Kleber M 1988 Z. Phys. D 9 267
[19] Gerasimov A S and Kazarnovskii M V 1976 Sov. Phys.--JETP 44 892
[20] Godoy S 2002 Phys. Rev. A 65 042111
[21] Brouard M and Muga J G 1996 Phys. Rev. A 54 3055
[22] García-Calderón G and Rubio A 1997 Phys. Rev. A 55 3361
[23] García-Calderón G and Villavicencio J 2001 Phys. Rev. A 64 012107
[24] García-Calderón G, Villavicencio J, Delgado F and Muga J G 2002 Phys. Rev. A 66 042119
[25] Stevens K W H 1980 Eur. J. Phys. 1 98
Stevens K W H 1983 J. Phys. C: Solid State Phys. 16 3649
[26] Teranishi N, Kriman A M and Ferry D K 1987 Superlatt. Microstruct. 3 509
[27] Jauho A P and Jonson M 1989 Superlatt. Microstruct. 6 303
[28] Delgado F, Cruz H and Muga J G 2002 J. Phys. A: Math. Gen. 35 10377
[29] Delgado F, Muga J G, Austing D G and García-Calderón G 2005 J. Appl. Phys. 97 013705
[30] G¨ahler R and Golub R 1984 Z. Phys. B 56 5
Single-particle matter wave pulses 9819
[31] Felber J, G¨ahler R and Golub R 1988 Physica B 151 135
[32] Felber J, M¨uller G, G¨ahler R and Golub R 1990 Physica B 162 191
[33] Brukner C and Zeilinger A 1997 Phys. Rev. A 56 3804
[34] Ranfangi A, Mugnai D, Fabeni P and Pazzi P 1990 Phys. Scr. 42 508
[35] Ranfangi A, Mugnai D and Agresti A 1991 Phys. Lett. A 158 161
[36] Moretti P 1992 Phys. Scr. 45 18
[37] B¨uttiker M and Thomas H 1998 Superlatt. Microstruct. 23 781
[38] Muga J G and B¨uttiker M 2000 Phys. Rev. A 62 023808
[39] Moy G M, Hope J J and Davage C M 1997 Phys. Rev. A 55 3631
[40] Hagley E W, Deng L, Kozuma M, Wen J, Helmerson K, Rolston S L and Phillips W D 1999 Science 283 1706
[41] Moshinsky M and Sadurní E 2005 Symmetry Integrability Geom.: Methods Appl. 1 003
[42] Faddeyeva V N and Terentev N M 1961 Mathematical Tables: Tables of the Values of the Function w(z) for
Complex Argument (New York: Pergamon)
[43] Abramowitz A and Stegun I A 1965 Handbook of Mathematical Functions (New York: Dover)
[44] Stevens K W H 1984 J. Phys. C: Solid State Phys. 17 5735
[45] Metcalf H J and van der Straten P 1999 Laser Cooling and Trapping (New York: Springer)
[46] Ramsey N F 1956 Molecular Beams (London: Oxford)
[47] Baute A D, Egusquiza I L and Muga J G 2001 J. Phys. A: Math. Gen. 34 4289
[48] Fowles G R 1968 Introduction to Modern Optics (New York: Dover)
[49] Marchewka A and Schuss Z 2005 Preprint quant-ph/0504105
[50] Clairon A, Laurent P, Nadir A, Drewsen M, Grison D, Lounis B and Salomon C 1992 EFTF: Proc. 6th European
Frequency and Time Forum