GEOMETRIC OPTICS WITH ATOMIC BEAMS
---
SCATTERED BY A DETUNED STANDING LASER WAVE

S.V. Prants, V.O. Vitkovsky, L.E. Konkov

Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
690041 Vladivostok, Russia, URL: dynalab.poi.dvo.ru

∗Corresponding author e-mail<EMAIL_ADDRESS>
###### Abstract

We report on theoretical and numerical study of propagation of atomic beams
crossing a detuned standing-wave laser beam in the geometric optics limit. The
interplay between external and internal atomic degrees of freedom is used to
manipulate the atomic motion along the optical axis by light. By adjusting the
atom-laser detuning, we demonstrate how to focus, split and scatter atomic
beams in a real experiment. The novel effect of chaotic scattering of atoms at
a regular near-resonant standing wave is found numerically and explained
qualitatively. Some applications of the effects found are discussed.

Keywords: atomic scattering, standing wave, optical nanolithography

##  1 Introduction

Manipulation of atoms by light becomes possible due to the dipole forces which
are well described by the semiclassical model with quantum description of
internal atomic transitions induced by a near resonant laser field and
classical description of their center-of-mass motion [1]. For the first time,
the ideas to trap and channel cold atoms with the help of standing laser waves
(SLW) have been proposed by V. Letokhov and his co-workers [2, 3, 4]. The
ability of a SLW to deflect, channel and split atomic beams [5, 6] has been
used for a variety of applications including atom microscopy, interferometry,
isotope separation and optical lithography [6, 7, 8, 9]. Lasers can be used to
manipulate atomic trajectories to create atomic analogues of such familiar
optical phenomena as focusing of light, beam splitting and light scattering.
It is remarkable that now we are able to reverse the roles of light and matter
from their familiar roles. The semiclassical description, used in this paper,
is similar to the geometric optics limit in conventional optics. Atomic
trajectories play the role of light rays with the SLW being a light mask.

In the present paper we intend to demonstrate theoretically and numerically
that adjusting in an experiment only one parameter, the detuning between the
frequencies of a working atomic transition and the SLW, one can explore a
variety of the regimes of the atom-laser interaction to focus, split and
scatter atomic beams. Near the atom-field resonance, where the interaction
between the internal and external atomic degrees of freedom is intense, there
is a possibility to create conditions for chaotic scattering of atoms [10, 11,
12] without any additional efforts like a SLW modulation. It becomes possible
due to the peculiarities of the dipole force in a near-resonant optical
lattice [13, 14, 15, 16].

##  2 Focusing, splitting, bunching and scattering of atomic beams

###  2.1 Equations of motion

A beam of two-level atoms in the $z$ direction crosses a SLW laser field with
optical axis in the $x$ direction. The laser beam has the Gaussian profile
$\exp[-(z-z_{0})^{2}/r^{2}]$ with $r$ being the $e^{-2}$ radius at the laser
beam waist. The characteristic length of the atom-field interaction is
supposed to be $\pm 1.5r$ because the light intensity at $z=\pm 1.5r$ is two
orders of magnitude smaller than the peak value. The longitudinal velocity of
atoms, $v_{z}$, is much larger than their transversal velocity $v_{x}$ and is
supposed to be constant. Thus, the spatial laser profile may be replaced by
the temporal one. The Hamiltonian of a two-level atom in the one-dimensional
SLW can be written in the frame rotating with the angular laser frequency
$\omega_{f}$ as follows:

$\hat{H}=\frac{P^{2}}{2m_{a}}+\frac{\hbar}{2}(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\\\
\hbar\Omega_{0}\exp[-(t-\frac{3}{2}\sigma_{t})^{2}/\sigma^{2}_{t}]\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}X_{a}}-\frac{i\hbar\Gamma}{2}\hat{\sigma}_{+}\hat{\sigma}_{-},$
(1)

where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $X_{a}$ and $P$ are the classical atomic position and
momentum, $\Gamma$, $\omega_{a}$, and $\Omega_{0}$ are the decay rate, the
atomic transition and maximal Rabi frequencies, respectively. The simple wave
function for the electronic degree of freedom is
${\left|\Psi(t)\right>}=a(t){\left|2\right>}+b(t){\left|1\right>}$, where
$a\equiv A+i\alpha$ and $b\equiv B+i\beta$ are the complex-valued probability
amplitudes to find the atom in the excited, ${\left|2\right>}$, and ground,
${\left|1\right>}$, states, respectively.

In the semiclassical approximation, atom with quantized internal dynamics is
treated as a point-like particle to be described by the Hamilton–Schrödinger
equations of motion written for the real and imaginary parts of the
probability amplitudes

$\begin{gathered}\dot{x}=\omega_{r}p,\,\dot{p}=-2\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}](AB+\alpha\beta)\sin
x,\\\ \dot{A}=\frac{1}{2}(\omega_{r}p^{2}-\Delta)\alpha-\frac{1}{2}\gamma
A-\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\beta\cos x,\\\
\dot{\alpha}=-\frac{1}{2}(\omega_{r}p^{2}-\Delta)A-\frac{1}{2}\gamma\alpha+\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]B\cos
x,\\\
\dot{B}=\frac{1}{2}(\omega_{r}p^{2}+\Delta)\beta-\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\alpha\cos
x,\\\
\dot{\beta}=-\frac{1}{2}(\omega_{r}p^{2}+\Delta)B+\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]A\cos
x,\end{gathered}$ (2)

where $x\equiv k_{f}X_{a}$ and $p\equiv P/\hbar k_{f}$ are scaled atomic
center-of-mass position and transversal momentum, respectively and dot denotes
differentiation with respect to the dimensionless time
$\tau\equiv\Omega_{0}t$. The recoil frequency, $\omega_{r}\equiv\hbar
k_{f}^{2}/m_{a}\Omega_{0}\ll 1$, the atom-laser detuning,
$\Delta\equiv(\omega_{f}-\omega_{a})/\Omega_{0}$, the decay rate
$\gamma=\Gamma/\Omega_{0}$, and the characteristic interaction time,
$\sigma_{\tau}\equiv r\Omega_{0}/v_{z}$, are the control parameters.

Let us introduce instead of the complex-valued probability amplitudes $a$ and
$b$ the following real-valued variables:

$u\equiv 2\operatorname{Re}\left(ab^{*}\right),\quad
v\equiv-2\operatorname{Im}\left(ab^{*}\right),\quad
z\equiv\left|a\right|^{2}-\left|b\right|^{2},$ (3)

where $u$ and $v$ are synchronized and quadrature components of the atomic
electric dipole moment, respectively, and $z$ is the atomic population
inversion. In the absence of any losses ($\gamma=0$), Eqs. (2) can be cast in
the form

$\begin{gathered}\dot{x}=\omega_{r}p,\quad\dot{p}=-u\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\sin
x,\quad\dot{u}=\Delta v,\\\ \dot{v}=-\Delta
u+2\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]z\cos
x,\quad\dot{z}=-2\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]v\cos
x.\end{gathered}$ (4)

The system (4) has two integrals of motion, namely, the total energy

$H\equiv\frac{\omega_{r}}{2}p^{2}-u\cos x-\frac{\Delta}{2}z,$ (5)

and the length of the Bloch vector, $u^{2}+v^{2}+z^{2}=1$, whose conservation
follows immediately from Eqs. (3).

Equations (4) constitute a nonlinear Hamiltonian autonomous system with two
and half degrees of freedom which, owing to the two integrals of motion, move
on a three-dimensional hypersurface with a given energy value $H$. In general,
motion in a three-dimensional phase space in characterized by a positive
Lyapunov exponent $\lambda$, a negative exponent equal in magnitude to the
positive one, and zero exponent [17]. The maximal Lyapunov exponent
characterizes the mean rate of the exponential divergence of initially close
trajectories and serves as a quantitative measure of dynamical chaos in the
system. The values of the maximal Lyapunov exponent in dependence on the
detuning, the recoil frequency and the initial atomic momentum have been
computed in Refs. [14, 15].

There are different regimes of the center-of-mass motion along the SLW optical
axis [12, 15]. In dependence on the initial conditions and the values of the
control parameters, atoms may oscillate in a regular or a chaotic way in wells
of the optical potential or move ballistically over its hills with regular or
chaotic variations of their velocity. Chaotic motion with a positive value of
the maximal Lyapunov exponent becomes possible in a narrow range of the
detuning values, $0<|\Delta|<1$ [15]. At $\Delta=0$, the synchronized
electric-dipole component, $u$, becomes a constant. That implies the
additional integral of motion in the Hamiltonian version (4) of Eqs. (2) and
the regular motion with zero maximal Lyapunov exponent. Far off the resonance,
at $|\Delta|>1$, the motion is regular both in the trapping and ballistic
modes.

It is remarkable that there is a specific type of motion, chaotic walking in a
deterministic optical potential, when atoms can change the direction of motion
alternating between flying through the SLW and being trapped in its potential
wells. We would like to stress that the local instability produces chaotic
center-of-mass motion in a rigid SLW without any modulation of its parameters.
Chaotic walking occurs due to the specific behavior of the Bloch-vector
component of a moving atom $u$ whose shallow oscillations between the SLW
nodes are interrupted by sudden jumps with different amplitudes while atom
crosses each node of the SLW [15]. It looks like a random like shots happened
in a fully deterministic environment. It follows from the second equation in
the set (4) that those jumps result in jumps of the atomic momentum while
crossing a node of the SLW. If the value of the atomic energy is close to the
separatrix one, the atom after the corresponding jump-like change in $p$ can
either overcome the potential barrier and leave a potential well or it will be
trapped by the well, or it will move as before. The jump-like behavior of $u$
is the ultimate reason of chaotic atomic walking along a rigid SW.

The total atomic energy (5) consists of the kinetic one,
$K=\omega_{r}p^{2}/2$, and the potential one, $U=-u\cos x-z\Delta/2$. The
optical potential changes its depth in course of time. Averaging over fast
oscillations of the internal atomic variables, we get the averaged potential
$\bar{U}=-\bar{u}\cos x-\bar{z}\Delta/2$ that can be used to explain why atoms
move in such or another way.

At small detunings $|\Delta|\ll 1$, the potential is approximately $U\simeq
u\cos x$. If $K(\tau=0)>|U_{\rm max}|=1$, then the atom will move
ballistically. This occurs if the initial atomic momentum, $p_{0}$, satisfies
to the condition $p_{0}>\sqrt{2/\omega_{r}}$. If the initial conditions are
chosen to give $0\leq K(\tau=0)+U(\tau=0)\leq 1$, the corresponding atoms with
$0\lesssim p_{0}\lesssim\sqrt{2/\omega_{r}}$ are expected to move chaotically
at the appropriate values of $\Delta$.

###  2.2 Focusing and splitting

In this section we demonstrate how to focus and split atomic beams crossing a
Gaussian laser beam by varying only one of the control parameters, the atom-
field detuning $\Delta$. Firstly, we perform simulation with a negligible
probability of spontaneous emission and solve the Hamiltonian equations of
motion (4) at comparatively large value of the detuning, $\Delta=1$. To be
concrete we take as an example calcium atoms with the working intercombination
transition $4^{1}S_{0}-4^{3}P_{1}$ at $\lambda_{a}=657.5$ nm, the recoil
frequency $\nu_{\rm rec}\simeq 10$ KHz, and the lifetime of the excited state
$T_{\rm sp}=0.4$ ms. Taking the maximal Rabi frequency to be
$\Omega_{0}/2\pi=2\cdot 10^{7}$ Hz, the radius of the laser beam $r=0.3$ cm,
and the mean longitudinal velocity $v_{z}=10^{3}$ m/s, the interaction time is
estimated to be $0.9$ ms, longer than the atomic lifetime. The normalized
recoil frequency is $\omega_{r}=4\pi\nu_{\rm rec}/\Omega_{0}=10^{-3}$ and the
normalized characteristic time is $\sigma_{\tau}=400$.

Trajectories for 50 calcium atoms to be prepared in the ground states
($u_{0}=v_{0}=0$, $z_{0}=-1$) with the same initial momentum, $p_{0}=10$, and
initial positions in the range $-\pi/10\leq x\leq\pi/10$ are shown in Fig. 1.
In units of the optical wavelength, $X=x/2\pi$, this range is $-0.05\leq X\leq
0.05$. The focusing occurs at those moments of time when the average
transverse momentum in the atomic beam is approximately equal to zero. If one
turns off the laser at one of these moments, it becomes possible to reduce the
beam width practically in ten times. The reason of focusing is simple. It is
well known [4] that at positive blue detunings atoms are attracted to the
nodes of the SLW where the minima of the optical potential are situated at
$\Delta>0$. The first node the atoms reach at $p_{0}>0$ is situated at
$X=1/4$. The initial kinetic atomic energy, $K_{0}=0.05$, is not enough to
overcome the potential barrier whose depth can be estimated to be $\simeq
0.35$ because the simulation gives $\bar{u}\simeq 0$ and $\bar{z}\simeq-0.7$.
So, all the atoms in the beam oscillate in the first potential well in the
$x$-direction around the first node. The initial width of the beam, $\delta
X_{0}=0.1$, is gradually reduced because in course of time the atoms with
initial negative positions catch up with the ones with positive $X_{0}$ near
the first turning point where the average beam momentum is close to zero. The
time interval of the atomic interaction with the SLW field is estimated to be
$3\sigma_{\tau}=1200$. So, the atoms leave the potential well after that time
and move freely (see Fig. 1).

In order to take into account spontaneous emission we use the standard
stochastic wave-function technique [18, 19, 20] for solving Eqs. (2). The
integration time is divided into a large number of small time intervals
$\delta\tau$. At the end of the first interval, $\tau=\tau_{1}$, the
probability of spontaneous emission,
$s_{1}=\gamma\delta\tau|a_{\tau_{1}}|^{2}/(|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2})$,
is computed and compared with a random number, $\varepsilon$, from the
interval $[0,1]$. If $s_{1}<\varepsilon_{1}$, then one prolongs the
integration but renormalizes the state vector in the end of the first interval
at $\tau=\tau_{1}^{+}$:
$a_{\tau_{1}^{+}}=a_{\tau_{1}}/\sqrt{|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2}}$
and
$b_{\tau_{1}^{+}}=b_{\tau_{1}}/\sqrt{|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2}}$.
If $s_{1}\geq\varepsilon_{1}$, then the atom emits a spontaneous photon and
jumps to the ground state at $\tau=\tau_{1}$ with
$A_{\tau_{1}}=\alpha_{\tau_{1}}=\beta_{\tau_{1}}=0$, $B_{\tau_{1}}=1$. Its
momentum in the $x$ direction changes for a random number from the interval
$[0,1]$ due to the photon recoil effect, and the next time step commences.

We simulate lithium atoms with the relevant transition $2S_{1/2}-2P_{3/2}$,
the corresponding wavelength $\lambda_{a}=670.7$ nm, recoil frequency
$\nu_{\rm rec}=63$ KHz, and the decay time $T_{\rm sp}=2.73\cdot 10^{-8}$ s.
With the maximal Rabi frequency $\Omega_{0}/2\pi\simeq 126$ MHz and the radius
of the laser beam $r=0.05$ cm one gets $\omega_{r}=10^{-3}$,
$\sigma_{\tau}=400$, and $\gamma=0.05$. Trajectories for 50 spontaneously
emitting atoms under the same conditions as in Fig. 1a are shown in Fig. 1b.
As expected, spontaneous emission destroys in part the effect of focusing.
However, the atoms move more or less coherently because spontaneous emission
events are comparatively rare at $\Delta=1$.

Figure 1: (a) Focusing the atomic beam with a long lifetime of the excited
state. (b) The effect of spontaneous emission on the focusing. The detuning is
$\Delta=1$ in both the cases. The atomic position $X$ is in units of the
optical wavelength.

Figure 2: Splitting the atomic beam (a) without and (b) with spontaneous
emission. The detuning is $\Delta=-1$.

The other effect, we would like to demonstrate with atomic beams crossing the
SLW, is a splitting of the beam. To do this one needs to choose such the value
of the detuning in order that some atoms in the beam would be trapped in the
first well of the optical potential but another ones could overcome the
barrier and leave that well. It is possible to split atomic beams as at
positive and negative values of the detuning. As an example, we demonstrate in
Fig. 2 the effect of splitting at $\Delta=-1$ for atoms without and with
spontaneous emission. It is seen that spontaneous emission changes slightly
the effect because a few atoms may leave the potential wells due to random
recoils.

##  3 Bunching and chaotic scattering of atoms

The ability of blue and red detuned lasers to attract atoms to the nodes and
antinodes of the SLW, respectively, can be used to create periodic structures
composed of atoms deposited on substrates in the process of optical
nanolithography [7, 8, 9]. To simulate a real experiment we consider a beam
with $N_{0}=10^{5}$ calcium atoms with the initial Gaussian distribution (with
the rms $\sigma_{x}=\sigma_{p}=2$ and the average values $x_{0}=0$ and
$p_{0}=10$) and compute their distribution against the SLW at a fixed moment
of time $\tau=1000$. The bunching of atoms at the SLW nodes at $\Delta=1$
(blue detuning) is shown in Fig. 3a where the atomic density, $n=N(X)/N_{0}$,
is plotted along the optical axis $X$ at $\tau=1000$. The same effect, but
with the atoms bunching around the SLW antinodes (red detuning,
$\Delta=-0.2$), is shown in Fig. 3b. In both the cases we get a periodic
atomic relief with the period $\lambda_{f}/2$ the width of which is restricted
by the time the atoms interact with the Gaussian laser beam.

Figure 3: The effect of bunching of $10^{5}$ calcium atoms around (a) the SLW
nodes (blue detuning, $\Delta=1$) and (b) the antinodes (red detuning,
$\Delta=-0.2$). The plot of atomic density $n=N(X)/N_{0}$ at the fixed moment
of time $\tau=1000$.

The problem we consider resembles the scattering process with particles
entering an interaction region along completely regular trajectories and
leaving it along asymptotically regular trajectories. It is known from many
studies in celestial mechanics, fluid dynamics and other disciplines that
under certain conditions the motion inside the interaction region may have
features that are typical for dynamical chaos, (homoclinic and heteroclinic
tangles, fractals, strange invariant sets, positive finite-time Lyapunov
exponents, etc.) although the particle’s trajectories are not chaotic in a
rigorous sense because chaos is strictly defined as an irregular motion over
infinite time. It has been found [21, 22, 23, 24] that transient Hamiltonian
chaos in the interaction region occurs due to existence of, at least, one
nonattractive chaotic invariant set consisting of an infinite number of
localized unstable periodic orbits and aperiodic orbits. This set possesses
stable and unstable manifolds extending in the phase space into the regions of
regular motion. The particles with the initial positions close to the stable
manifold follow the chaotic-set trajectories for a comparatively long time,
then deviate from them, and leave the interaction region along the unstable
manifold. It is the common mechanism of chaotic scattering that in our problem
causes the chaotic walking of atoms along the SLW.

Figure 4: The distributions of $10^{5}$ calcium atoms at $\tau=1000$ in (a)
the real and (b) momentum space under the conditions of chaotic scattering at
$\Delta=0.2$.

Figure 5: Comparison of the distributions of $10^{5}$ calcium atoms at
$\tau=1000$ over the phase plane in the regimes of (a) chaotic ($\Delta=0.2$)
and (b) regular scattering ($\Delta=-0.2$).

In Fig. 4a we show the atomic position distribution at $\tau=1000$ in the
regime of the chaotic scattering at $\Delta=0.2$ with $10^{5}$ calcium atoms.
This plot should be compared with Fig. 3a where the atomic position
distribution is shown for regularly scattered atoms at $\Delta=1$. First of
all, the distribution of chaotically scattered atoms has a prominent pedestal
and is much broader. Moreover, it has no such a periodic structure as shown in
Fig. 3a. Only the peaks around the first two SLW nodes are prominent in Fig.
4a. The atomic position distribution in the momentum space in Fig. 4b is much
broader than the one for regularly scattered atoms at $\Delta=1$ (not shown).
Thus, we predict that under the conditions of chaotic scattering there should
appear less contrast and more broadened atomic reliefs as compared to the case
of regular scattering because a large number of atoms are expected to be
deposited between the nodes as a result of chaotic walking along the SLW axis.
The effect is expected to be more prominent under the coherent evolution but
it seems to be observable with spontaneously emitting atoms as well. The
difference between chaotic and regular scattering of atoms at a rigid SLW is
especially prominent on the corresponding phase space portraits shown in Fig.
5 where positions and momenta of $10^{5}$ calcium atoms are plotted at the
fixed time moment.

##  4 Conclusion

We have simulated some geometric optics effects with atomic beams crossing a
SLW in the limit of long relaxation time and with spontaneous emission taken
into account. Trajectories of spontaneously emitting atoms have been simulated
with the help of the standard stochastic wave-function technique [18, 19, 20].
It has been shown that by adjusting the detuning it is possible to focus,
split and scatter atoms. The effects have been explained by a coupling between
external and internal atomic degrees of freedom. The depth of the optical
potential depends on the sign and value of the detuning. Varying $\Delta$, one
can create conditions for focusing, splitting and bunching the atoms. It is
remarkable that near the atom-field resonance we have found the new type of
atomic diffraction at a SLW without any modulation of its parameters that can
be observed in real experiments. That would be the prove of existence of the
novel type of atomic motion, chaotic walking in a deterministic environment.
The effects found could be used in optical nanolithography to fabricate
complex atomic structures on substrates.

We predict that experiments on the scattering of atomic beams at a SLW can
directly image chaotic walking of atoms along the SLW. In a real experiment
the final spatial distribution can be recorded via fluorescence or absorption
imaging on a CCD, commonly used methods in atom optics experiments yielding
information on the number of atoms and the cloud’s spatial size. The other
possibility is a nanofabrication where the atoms after the interaction with
the SW are deposited on a silicon substrate in a high vacuum chamber. In this
case the spatial distribution can be analyzed with an atomic force microscope.
As to the momentum distribution, it can be measured, for example, by a time-
of-flight technique [25]. The modern tools of atom optics enable to create
narrow initial atomic distributions in position and momentum, reduce coupling
to the environment and technical noise, create one-dimensional optical
potentials, and to measure spatial and momentum distributions with high
sensitivity and accuracy.

## Acknowledgments

This work was supported by the Russian Foundation for Basic Research (projects
nos. 09-02-00358 and 09-02-01258), by the Integration grant from the Far-
Eastern and Siberian branches of the Russian Academy of Sciences, and by the
Program “Fundamental Problems of Nonlinear Dynamics”.

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