Cold atoms passing through a thin laser beam:
a Fourier optics approach
Shuyu Zhou (周蜀渝)
1,2,
*, Jun Qian (钱 军)
1
, Shanchao Zhang (张善超)
2
,
and Yuzhu Wang (王育竹)
1,
**
1
Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China
2
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong, 999077, China
*Corresponding author: syz@siom.ac.cn; **corresponding author: yzwang@mail.shcnc.ac.cn
Received January 29, 2016; accepted May 5, 2016; posted online June 13, 2016
A Fourier optics approach can be a concise and powerful tool to solve problems in atom optics. In this report, we
adopt it to investigate the kinetic behavior of cold atoms passing through a far red-detuned Gaussian beam. We
demonstrate that the aberration has significant influence on the evolution of the atomic cloud, which is rooted in
the deviation of the Gaussian profile from the quadratic form. In particular, we observe an intriguing effect
analogous to Fresnel’s double prism with cold atoms. The experimental results are in good agreement with
the numerical simulation.
OCIS codes: 020.1335, 020.3320, 070.7345.
doi: 10.3788/COL201614.070202.
Atoms have wave-particle duality
[1,2]
. However, the wave-
length of the therm al de Broglie wave of hot atoms is so
short that their kinetic problems can be dealt with using
the approaches developed for classical particles
[3,4]
. With
the rapid development of atom cooling techniques, atomic
samples with the temperature of several micro Kelvins were
achieved by using laser cooling
[5–9]
. They can be further
cooled via evaporative cooling
[10,11]
and prepared as ultra-
cold atoms or Bose–Einstein condensates (BECs) with
temperatures of nano-Kelvins
[12,13]
. Ultracold atoms have
longer thermal de Broglie wavelengths and exhibit more
significant wave properties, such as interference and
diffraction
[14–17]
, so, the kinetic problems of ultracold
atoms should be addressed in a wave picture. The path-
integral approach, which can give strict solutions to kinetic
problems where the wave properties need to be considered,
has been developed
[18–20]
. However, it is technically compli-
cated and not intuitive.
In this Letter, we propose a simple approach to handling
this kind of problem. Some results in geometric optics
should be reproduced in Fourier optics when the interfer-
ence and diffraction can be neglected
[21,22]
. Naturally, the
Fourier optic approach can be used to solve problems
in atom optics, such as the transport of atomic clouds
in optical potentials. Here we derive the effective focal
length of a quadratic and a Gaussian beam. In the former
case, the potential is equivalent to an ideal atomic lens.
Then we discuss the case in which the Gaussian potential
cannot be approximated as a quadratic potential. When a
spherical atomic cloud passes through a far-detuned Gaus-
sian laser beam, a novel effect analog of a light beam
passing through a Fresnel’s double prism is observed.
Classically, for a light wave propagating in a medium
with refractive index nð
rÞ, the wave vector is
kð
rÞ¼
2πnð
rÞ
λ
0
k; (1)
where λ
0
is the wavelength of light in a vacuum and
k
0
is
the unit vector pointin g in the direction of the maximum
light-phase gradient. However, for an atom moving in a
potential well, one can update the wave vector with
kð
rÞ¼
2M ½E − V ð
rÞ
�
2
r
k
0
; (2)
where E is the total atomic energy and V ð
rÞ is the poten-
tial. Given that the wave vector of a free particle is
2ME∕ℏ
2
p
k
0
in the de Broglie relation, the effective
refractive index of the potential can be expressed as
nð
rÞ¼
1 − V ð
rÞ∕E
p
: (3)
This equation also can be obtained by comparing the basic
function in geometric optics (eikonal equation) with the
Jacobi–Hamilton equation
[21,23]
. When jV j ≪ E,
nð
rÞ ≈ 1 −
V ð
rÞ
2E
: (4)
For simplicity, we begin with a cylindrical atomic lens.
Here, we assume that the atomic lens is constituted by
a one-dimensional harmonic potential in the x direction,
which is independent of its position in the z direction.
The potential is given by
V ðx; zÞ¼
κ
2
x
2
; (5)
where κ is a constant. Then, we have
COL 14 (7), 070202(2016) CHINESE OPTICS LETTERS July 10, 2016
1671-7694/2016/070202(5) 070202-1 © 2016 Chinese Optics Letters
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