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Numerical Simulations of an Atomic Beam Focused by an Optical
Potential
T. Kidan, Joan Adler and A. Ron
Physics Department, Technion-Israel Institute of Technology, Haifa 32000,
Israel.
e-mail Address: kidan@tx.technion.ac.il
May 6, 1998
Abstract:
We present a model to describe the behavior of an atomic beam in a focusing
experiment and discuss its implementation. Such modeling is relevant to the
development of atom lithography. For simplicity, the light-atom interaction may
be treated as a conserving potential (neglecting spontaneous emission) so that
the Schrodinger equation is sufficient for the description of the problem. We
start from a two dimensional time dependent Schrodinger equation that is
reduced to a one dimensional equation by a paraxial approximation, and then
solved iteratively via Pade approximation. The solution is implemented in an
interactive routine with Matlab(5.x) to compute and visualize the atomic wave
packet development as it propagates through the laser light optical potential.
We present examples of our implementation, starting with cases which have
known analytic solutions such as propagation of the atomic wave packet
through free space and through a parabolic potential. We also present a case
which does not have a simple analytic solution, namely, the propagation of the
atomic wave packet through a focusing optical potential, which has a Gaussian
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2
profile in the propagation direction and a parabolic profile in the perpendicular
direction.
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3
I. Introduction
Optical manipulations of atoms have been widely studied in recent years [1,2].
One possible application of this growing field is atom lithography, which has
many advantages over other lithography techniques. For example, electron
beam lithography is limited in speed and resolution by space charge effects,
which are absent in atom lithography because the atom is neutral. X-ray and
electron beam lithography induce damage resulting in limitation to the
lithography resolution, while this is probably absent for low energy atoms. As a
consequence of the small de Broglie wavelength, atom lithography has the
potential for high resolution [3].
The general configuration of the atom lithography process is illustrated
in Figure 1. The atomic beam is initially mechanically collimated, and then
further collimated by passing through a region in which there is a laser light
radiation field which collimates the atomic beam transversely, reducing the
transverse velocity spread and hence the divergence of the beam. The highly
collimated beam of atoms then passes through a standing wave of the radiation
field, which is formed by a retro-reflected near resonant laser beam grazing
across the surface of the substrate on to which the atoms are to be deposited.
While passing through the standing wave the atoms experience a dipole force,
at the node, which acts as a lens, causing them to focus and to form patterns as
they deposit onto the surface.
We will describe the experiment in detail in section II, where the atomic
beam will be represented by a wave packet, interacting with the conserving
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Atom
source
Vacuum chamber
Slit
Laser light for optical collimating
Standing
wave laser light
Mirror
Substrate
Sample holder
Figure 1: A typical apparatus for atom lithography utilizing laser
focusing of atoms in a standing wave
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light potential (neglecting the spontaneous emission) [1,4]. The form of
Schrodinger equation used by these computer simulations will be obtained in
section III from a two dimensional time dependent equation into an effectively
one dimensional equation, by utilizing the paraxial approximation [5,6]. In
section IV we will present the numerical solution. This solution is found by
reducing the parabolic equation to a system of differential equations (via Pade
approximants), whose solutions are standard finite differences equations [7].
The computer simulations and visualizations of the various cases will be
described in section V, starting with the examination of the numerical algorithm
for cases with known analytic solutions and proceeding with the case of the
focusing experiment. In section VI we summarize and discuss applications of
our routines.
II. Description of the focusing experiment
The beam of atoms in a typical focusing experiment are predominantly
traveling in the z–direction (downwards in Figure 1), which traverses a
focusing light potential, aligned both in the x (left to right in Figure 1) and z
directions. Close enough [8] to the anti-node of the standing wave light field
the potential has an approximately parabolic shape along the x–direction and
acquires the Gaussian profile of the laser beam light field along the propagation
direction, which is the z–direction. The focusing light potential at the anti-node
region of the large period [9] optical standing wave is produced by the
reflection of the red detuned laser beam from a glass surface. Assuming both
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that the kinetic energy of the incoming atoms in the z–direction is much larger
than the interaction energy and that the focusing region is uniform in the ydirection, the focusing is then described by an effectively one dimensional
model of the quantum mechanical motion along the x–direction. Utilizing a
paraxial approximation, the spatial evolution of the atomic wave function
 ( x, z ) along the z–direction is then described by the Schrodinger equation:
 2 2


iv 0  ( x, z )  
 V ( x, z )   ( x, z )
2
z
 2m  x

(1)
where v0 is the center of mass velocity of the incoming atoms in the zdirection, m is the mass of the atoms and V ( x, z ) is the interaction operator
between the atoms and the focusing optical potential. (The derivation of this
equation is presented in detail in section III). The presence of this potential is
but a small perturbation (see section III) to the propagation of the wave packet
along the z–direction. Equation (1) gives an effectively one-dimensional model
for this focusing experiment.
III. Construction of the effectively one dimensional model
The two dimensional time dependent Schrodinger equation is:
H ( x, z , t )  i

 ( x, z , t )
t
(2)
where the hamiltonian is
H  H0  V  
2 2
  V ( x, z )
2m
6
(3)
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and thus the evolution of the wave function in time is:
( x, z, t )  e
i
 Ht

 ( x, z,0) .
(4)
The initial wave packet describing the atom in the focusing experiment is
chosen to be a Gaussian in both z and x directions, propagating as a plane wave
with momentum k z along the z-direction,
0
 ( x, z,0)   1 ( x,0) 2 ( z,0) 

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4
2 x2
e
x2
1
4 x2
2 z2
4
e
ikz 0 z

e
z2
4 z2
(5)
where the widths of these Gaussians are chosen to be  z   x and thus
 ( x, z , t )  e
i
 E ( k z0 ) t

 ( x, z,0)
(6)
Substituting (6) into (2) yields the the stationary Schrodinger equation in two
dimensions
 2  x, z  


2m
E k  V  x, z    x, z   0
2
z0
(7)
For the selected conditions of the typical experiment, described above
(equations (5), (6)), it is convenient to express the wave function  ( x, z) as a
product of ( x, z ) and a carrier wave moving in the positive z-direction:
 ( x, z)  ( x, z)e
 ikz0 z
(8)
where
kz 
0
2mEk

z
0
(9)
Substitution of the wave function (8) into the two dimensional stationary
Schrodinger equation (7) gives:
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2

2
 2  ( x, z )  2ik z
 ( x, z )  2  ( x , z ) 
z
z
x
0

2mE k
2
z0
2mV
2
 ( x, z ) 
 ( x , z )  k z  ( x, z )
2

(10)
0
The last equation can be simplified to become equation (1) (which we will
solve numerically in the next section), in a few steps: First, utilization of a
paraxial approximation will lead to the negligence of the second order
derivative (the first term on the left-hand side of equation (10)). This is made
possible by the fact that the function ( x, z ) changes slowly (see equations (5)
to (6)), in the z-direction compared to the x-direction. Second, the use of
relation (9) leads to the cancellation of the first term on the right-hand side of
equation (10) with the third term from the right hand side. Note that the slow
change of ( x, z ) comes about because the motion of the center of the wavepacket along the z-direction, for which most of the momentum is concentrated
in only one spatial mode k z , is on the trajectory z  v0t . In this case where
0
most of the momentum of the wave packet is concentrated in one spatial mode
k z there is practically no reflection of the wave packet.
0
IV. The algorithm for the solution of the effectively one dimensional model
Equation (1) is a parabolic partial differential equation that can be solved by
different techniques. One convenient method is to use the finite-differences
approximation via the ordinary differential equations. Consider equation (1)
rewritten as
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 2


 ( x, z )  a 2  bV ( x, z )  ( x, z )
z
 x

where a 
(11)
i
i
and b  
.
2mv 0
v0
The function ( x, z ) satisfies the initial condition (x,0) , which is
a Gaussian form of the wave packet. The second derivative with respect to x at
(x,z),
on
the
right
hand side of
equation (11),
is replaced
by
1
( x  dx, z )  2( x, z )  ( x  dx, z ) so the equation becomes in its
(dx) 2
matrix form:
:
b

  
 1 
 2  (dx ) 2V1
1

 1 


a




b

2


  2 
1
 2  (dx ) V2 1
 2 
a

 . 
  



d
a




2 



 


dz 
(dx )

  


b

 N  2 
 
1  2  (dx ) 2VN  2
1
N 2


a





b


2
1
 2  (dx ) VN 1   N 1 

 N 1 
a


(12)
where i and Vi are the wave packet and the potential vectors respectively.
The solution of this ordinary scalar differential equation, can be written in a
more concise form as
d
 A
dz
(13)
where the matrix A is in general dependent on z and  (z) satisfies the initial
condition  (0) , is simply:
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( z  dz)  exp(dzA)( z)
(14)
For the restriction on dz see [7]. The exponential function of the matrix A
appearing on the right-hand side can be approximated by a [1,1] Pade’
approximant to give:
1
1 A
2  O[ A 3 ]    
exp( A) 
1
1 A
2
(15)
(Pade approximants are ratios of polynomials, for example, the [1,1] Pade
approximant has polynomials of order A in both numerator and denominator.)
This approximation is second order accurate in z, having an error term of the
1
third order in A with a coefficient of 12 . Although the [1,1] Pade
approximant produces unwanted finite oscillations near points of discontinuity
it is adequate for this simulation since the wave function is continuous and
smooth over the grid and zero at the edges. The evolution of the wave packet
through the two dimensional potential region can now be calculated iteratively
from (14) and (15). The output is a matrix with the values of the propagating
wave packet and its distortion caused by the potential.
V. Computer simulations and visualization of the wave packets
In order to demonstrate this solution, an interactive visualization using
Matlab(5.x) has been prepared. Free propagation of the atomic wave packet and
propagation through a parabolic potential in the x-direction for all z were
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calculated, visualized and compared to the analytical known solutions of the
evolution for a Gaussian wave packet. Both the simulation and the visualization
were done using the convenient tools of Matlab. The program m-files are
available on the world wide web [10].
The input parameters are the velocity of the atoms along the z - direction
which is of the order of 1m/sec, the laser intensity, the width of the atomic
wave packet, the length of the z grid, the number of lattice points and the
location of the laser beam (the optical potential), again along the z – direction.
All these parameters can take a wide range of values depending on the details
of the experimental implementation. Some specific parameters are given in the
example m-files.
The optical potential for the focusing experiment has the form of
( zz
  2
1   2
2
V ( x, z )   2 x  e
2
  L
0)
2
L
2

 where L is the laser's wavelength,  L stands

for the width of the laser's beam profile and z 0 is the location of the laser beam
along the z-direction.
The output is the square of the amplitude of the wave function, indicated
by P
2
on the graph. Each simulation is visualized with two subplots, one is
three dimensional and the other two dimensional. The color bar at the side of
the lower plot indicates the P
2
value. The dual visualization technique
reinforces visual perception.
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On a PC with a Pentium II 233MHz processor and 64Mb SDRAM both
the simulation and the visualization for 100  100 lattice took approximately 3
seconds for the free propagation simulation up to 40 seconds for the
propagation through the parabolic - Gaussian potential. (On a 200Mhz with
32Mb RAM these times were 10 and 200 seconds respectively)
We now present a series of figures illustrating the use of the program to
visualize both free and constrained propagation of Rb atoms through a region
one wavelength wide with the antinode in the middle (where the laser's
wavelength is L = 0.760 m ). The laser's Gaussian profile (along the zdirection) is  L =500 m wide. In Figure 2 the main menu of the program is
pictured. In Figure 3 the free propagation of the atomic wave packet is shown.
Here V(x,z)=0 and it is easily seen that the Gaussian wave packet spreads as it
propagates, as expected. Figure 4 illustrates the propagation of the atomic wave
packet through the parabolic (along the x – direction) potential. The Gaussian
wave packet’s width oscillates as it propagates, as expected. In Figure 5 the
propagation of the atomic wave packet through the parabolic (along the x –
direction) potential is again shown but with the atomic beam off center. Again
the Gaussian wave packet oscillates as it propagates. Figure 6 illustrates the
propagation of the atomic wave packet through a potential that is parabolic
along the x – direction, and Gaussian along the z – direction. The Gaussian
wave packet is focused as it propagates through the potential and then spreads
as it leaves the region of the potential. In Figure 7 we give a closer look at the
propagation of the atomic wave packet through the same potential as in Figure
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6, parabolic along the x – direction and Gaussian along the z – direction. Again
the Gaussian wave packet is focused as it propagates through the potential and
then spreads as it leaves the region of the potential.
For those readers who are interested in the details of the routines, we
conclude with a brief outline of the different files that we have placed on the
web [10]. The README file tells the user to enter matlab and then write main
at the matlab prompt. This displays the menu pictured in Figure 2. The intro.m,
intro1.m etc files provide a short introduction to the physics of the problem;
the help1.m file gives advice on running the interactive routine and the about.m
gives contact information for Tal Kidan. The program contains five sets of
demonstrations for the three different potentials, which are accessed thru the
file demon.m, and the outputs of these are presented in figures 3-7 above.
These have parameters already encoded for typical cases in the corresponding
files evop.m (free), evop1.m (parabolic), evop1off.m (off-center parabolic),
evop2.m (parabolic-Gaussian), evop3.m (parabolic-Gaussian), ready to
reproduce the demonstration figures. The user can set her/his own values for
each of the three preprogrammed potential cases (free, parabolic, parabolicGaussian) thru the interactive simulation option with sliders that move within
the allowable parameter ranges. For all of these no programming skills are
needed, just download the files into a single directory and enter matlab! The
programs work both under windows and under UNIX (tested on AIX, HP/UX
and LINUX). For other potentials it is necessary to edit a file of the evop.m
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type, this can be done by simply replacing evop3.m if you do not want to bother
with editing the interface.
VI. Applications and summary
An effectively one dimensional model was introduced in order to simulate the
propagation of an atomic wave packet through a two dimensional potential. A
simple algorithm was used to calculate the evolution of the wave packet and the
visualization tools of Matlab allowed convenient visualization. This routine
could be used to teach students about wave packet propagation. It has already
found research application in a laboratory of Atom Optics where it is used to
estimate the spot size of the beam of Rb atoms deposited on the surface of the
substrate. The user interface which allows simultaneously changing parameters,
such as the intensity of the laser light potential or the velocity of the atoms, has
been very helpful to this experimental Atom Optics group.
VII. Acknowledgements
The motivation for this study came from Dr. N. Davidson, head of the
experimental Atom Optics group at the Physics Department, Weitzman Institute
of Science - Israel. The numerical solution and visualization were carried out as
a project by T. Kidan in a computational physics class at the Physics
Department, Technion - Israel Institute of Technology.
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VIII. References
[1] “Atom Optics”, C. S. Adams, M. Sigel and J. Mlynek, Phys. Rep. 240, 143
(1994).
[2] Reference [1] covers the subject in depth including discussions on the
validity of all the approximations needed to model various experiments.
There are extensive references to previos calculations and experiments.
[3] S. Nowak, T. Pfau and J. Mlynek, Appl. Phys. B 63, 203 (1996).
[4] “Atomic motion in Laser light”, C. Cohen-Tannoudji, Fundamental systems
in QO, Elsevier Science Publications B. V. (1991), p. 4-166.
[5] “Computation of mode properties in optical fiber waveguides by a
propagating beam method”, M.D. Feit and J. A. Fleck, Applied Optics, 19,
7, 1154 (1989).
[6] “Atomic Fresnel images and possible applications in atom lithography”, U.
Janicke, J. Phys. II France, 4, 1975 (1994).
[7] “Numerical Solution of PDE: Finite Difference Method”, G.D. Smith,
Oxford Press, 3rd ed. (1993), p. 111-126.
[8] Ref [1] - Section III B1.
[9] T. Sleator, T. Pfau, V. Balykin and J. Mlynek, Appl. Phys. B 54, 375,
(1992).
[10] http://comphy.technion.ac.il/~comphy/talkdn.html
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Figure captions
Figure 1: A typical apparatus for atom lithography utilizing laser focusing of
atoms in a standing wave.
Figure 2: The program’s main menu.
Figure 3: Free propagation of the atomic wave packet. It is easily seen that the
Gaussian wave packet is spreading as it propagates, as expected.
Figure 4: Propagation of the atomic wave packet through the parabolic (along
the x – direction) potential. The Gaussian wave packet’s width is oscillating as
it propagates, as expected.
Figure 5: Propagation of the atomic wave packet through the parabolic (along
the x – direction) potential is again shown but with the atomic beam off center.
Again the Gaussian wave packet is oscillating as it propagates.
Figure 6: Propagation of the atomic wave packet through a potential that is
parabolic, along the x – direction, and Gaussian along the z – direction. The
Gaussian wave packet is focused as it propagates through the potential and then
spreads as it leaves the region of the potential.
Figure 7: A closer look at the propagation of the atomic wave packet through
the same potential as in Figure 6, parabolic along the x – direction and
Gaussian along the z – direction.. Again the Gaussian wave packet is focused as
it propagates through the potential and then spreads as it leaves the region of
the potential.
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