11institutetext:  Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
43 Baltiiskaya st., 690041 Vladivostok, Russia, URL: dynalab.poi.dvo.ru

Quantum chaos; semiclassical methods Mechanical effects of light on atoms,
molecules, and ions Nonclassical interferometry, subwavelength lithography

# Chaotic scattering of atoms at a standing laser wave

S. V. Prants

###### Abstract

Atoms, propagating across a detuned standing laser wave, can be scattered in a
chaotic way even in the absence of spontaneous emission and any modulation of
the laser field. Spontaneous emission masks the effect in some degree, but the
Monte Carlo simulation shows that it can be observed in real experiments by
the absorption imaging method or depositing atoms on a substrate. The effect
of chaotic scattering is explained by a specific behavior of the dipole
moments of atoms crossing the field nodes and is shown to depend strongly on
the value of the atom-laser detuning.

###### pacs:

05.45.Mt

###### pacs:

37.10.Vz

###### pacs:

42.50.St

##  1 Introduction

The deflection of an atomic beam at a laser standing wave (SW) is explained by
the dipole forces which are well described by the classical atom-field
interaction model [1, 2]. The ability of a SW to diffract, focus or splitting
an atomic beam [3] has been used for a variety of applications including atom
microscopy, interferometry, isotope separation, and optical lithography [4, 5,
6]. On the other hand, cold atoms are ideal candidates to test fundamental
principles of quantum physics including the phenomenon of dynamical chaos at
the microscopic level that is known as a kind of random-like motion in a
deterministic environment [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Dynamical
chaos is characterized by exponential sensitivity of trajectories in the phase
space to small variations in initial conditions and/or control parameters.

It has been proposed in Ref. [7] to study quantum chaos and the corresponding
effect of dynamical localization placing cold atoms in a far-detuned SW with a
periodic kick-like modulation of positions of the SW nodes. A number of
experiments [9] have been carried out in accordance with this proposal. At
large detunings, the atoms are not excited being quantum analogues of
classical kick rotors. Since those experiments on atom optics realization of
the $\delta$-kicked quantum rotor, cold atoms provide new grounds for
experiments and theory on quantum chaos.

It has been shown in Ref. [10] that even a single-pulse far-detuned SW can
induce chaos in atomic motion. For sufficiently large detuning, the excited
state amplitude can be adiabatically eliminated [7], leading to a Hamiltonian
with an external degree of freedom only. The corresponding equations of motion
for an externally modulated nonlinear pendulum constitute the well-known model
with one and half degree of freedom that can be chaotic under some conditions.
The other possibility is to induce chaos in spin degree of freedom of atoms
periodically kicked by applying short magnetic field pulses [11]. That is the
one and half degree of freedom model of a kicked top.

In difference from those and other papers on the related topic, we consider
the physical situation with comparatively small detunings and should take into
account a coupling between external and internal atomic dynamics, leading to a
model with three degrees of freedom. It will be shown in the present paper
that in this case one needs no modulation or any other perturbation of the SW
to induce chaotic internal and external dynamics of atoms crossing the SW
laser field.

Near the atom-field resonance, when the interaction between the internal and
external atomic degrees of freedom is intense, there is a possibility to
create conditions for chaotic behavior without any kicking and modulation [15,
16, 17]. If so, it is open the way to test the novel regime of atomic motion
caused by the peculiarities of the dipole force in the strong coupling regime.
In the semiclassical approximation, atom with quantized internal dynamics is
treated as a point-like particle with the Hamilton–Schrödinger equations of
motion constituting a nonlinear dynamical system [15, 16, 17, 18]. In a
certain range of the atom-field detunings, a set of atomic trajectories
becomes exponentially sensitive to small variations in initial quantum
internal and classical external states or/and in the control parameters.
Hamiltonian evolution is a smooth process that is well described in a
semiclassical approximation by the Hamilton-Schrödinger equations. The problem
becomes much more complicated because of spontaneous emission of atoms with a
specific shot quantum noise acting in a dynamical system which is chaotic in
the absence of noise. A number of nonlinear Hamiltonian and dissipative
effects have been found numerically and analytically near the resonance
including chaotic Rabi oscillations, chaotic atomic transport, dynamical
fractals, and Lévy flights [15, 16, 17, 18, 19, 20, 21, 22].

The main aim of the paper is to demonstrate theoretically and numerically that
the new type of atomic diffraction at a rigid SW without any modulation,
chaotic atomic scattering, can be observed in a real experiment. The scheme of
such an experiment is shown in Fig. 1 with a beam of atoms crossing a SW laser
field. One either measures a spatial atomic distribution after the interaction
by the absorption imaging technique or measures an atomic distribution on a
silicon substrate in the far-field zone. The results are expected to be
different depending on the value of the atom-field detuning. The distribution
is expected to be comparatively narrow at those values of the detuning at
which atomic scattering is regular (r.s. distribution in Fig. 1) or wide at
the detuning values providing chaotic scattering of atoms due to their chaotic
walking along the SW (c.s. distribution in Fig. 1).

Figure 1: Schematic representation of the proposed experiment on regular
(r.s.) and chaotic (c.s.) atomic scattering at a Gaussian standing laser wave.

##  2 Results

###  2.1 Main equations and the regimes of atomic motion

A beam of two-level atoms in the $z$ direction crosses a SW laser field with
optical axis in the $x$ direction (Fig. 1). The laser field amplitude 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 $3r$ because the light intensity at $z=z_{0}\pm
1.5r$ 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 an atom in the 1D SW field
can be written in the frame rotating with the laser frequency $\omega_{f}$ as
follows:

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

where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $X$ 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, and
$\sigma_{t}\equiv r/v_{z}$, i.e., $3\sigma_{t}$ is the transit time. The
wavefunction for the electronic degree of freedom is
${|\Psi(t)\closeket}=a(t){|2\closeket}+b(t){|1\closeket}$, where $a\equiv
A+i\alpha$ and $b\equiv B+i\beta$ are the probability amplitudes to find the
atom in the excited, ${|2\closeket}$, and ground, ${|1\closeket}$, states,
respectively.

Figure 2: Finite-time Lyapunov exponent $\lambda$ vs atom-field detuning
$\Delta$ (in units of the Rabi frequency $\Omega_{0}$) and initial atomic
transversal momentum $p_{0}$ (in units of the photon momentum $\hbar k_{f}$)
at the normalized recoil frequency $\omega_{r}=10^{-3}$ and $\gamma=0$.

In the semiclassical approximation, atom with quantized internal dynamics is
treated as a point-like particle (the transversal atomic momentum $p$ is
supposed to be, in average, much larger than the photon one, $\hbar k_{f}$)
with the 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 $x\equiv k_{f}X$ and $p\equiv P/\hbar k_{f}$ are 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.

Figure 3: Evolution ($\tau$ is in units of $\Omega_{0}^{-1}$) of the atomic
dipole-moment component $u=2(AB+\alpha\beta)$ in (a) the chaotic
($\Delta=0.2$) and (b) regular ($\Delta=1$) regimes of atomic motion.

Without the decay, $\gamma=0$, Eqs. (2) constitute the nonlinear Hamiltonian
dynamical system with three degrees of freedom describing an atom moving in
the six-dimensional phase space. The simple way to know how complicated this
motion may be is to compute the quantitative measure of chaos, maximal
Lyapunov exponent characterizing the mean rate of the exponential divergence
of initially close trajectories. Because of a transient character of chaos we
compute the finite-time Lyapunov exponent $\lambda$, i.e. the value of the
exponent at the moment when atoms leave the interaction region. The result of
computation with Eqs. (2) at the given value of the recoil frequency,
$\omega_{r}=10^{-3}$, and zero decay rate in dependence on the detuning
$\Delta$ and the initial atomic transversal momentum $p_{0}$ is shown in Fig.
2. Color codes the magnitude of the finite-time Lyapunov exponent. In white
regions the values of $\lambda$ are almost zero, and the internal and
translational motion is regular in the corresponding ranges of $\Delta$ and
$p_{0}$. In shadowed regions positive values of $\lambda$ imply unstable
motion.

Figure 4: Trajectories in the real space for 50 atoms without spontaneous
emission. Hamiltonian (a) chaotic ($\Delta=0.2$) and (b) regular scattering
($\Delta=1$). The atomic position $x$ is in units of the optical wavelength.

Figure 5: The same as Fig. 4 but in the momentum space.

The scheme of the proposed experiment in Fig. 1 resembles the scattering
problem with particles entering an interaction region along completely regular
trajectories and leaving it along asymptotically regular trajectories [23, 24,
25, 26, 27, 28]. However, in difference from the standard chaotic scattering
with a nonattractive fractal invariant set existing over an infinite time,
this process may be interpreted as a transient chaos or a finite-time chaotic
scattering.

There are three possible chaotic regimes of the center-of-mass motion along
the SW optical axis [17, 18]. In dependence on the initial conditions and the
values of the control parameters, atoms may oscillate chaotically in wells of
the optical potential or move ballistically over its hills with chaotic
variations of their velocity. Chaotic motion becomes possible in a narrow
range of the detuning values, $0<|\Delta|<1$. At $\Delta=0$, the synchronized
electric-dipole component, $u=2(AB+\alpha\beta)$ becomes a constant. That
implies the additional integral of motion in the Hamiltonian version of
Eqs.(2) and the regular motion with $\lambda=0$. Far from the resonance, at
$|\Delta|>1$, the motion is again regular both in the trapping and flight
modes. That speculation is confirmed by the Lyapunov map in Fig. 2.

Moreover, 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 SW 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 SW without any modulation of its parameters
in difference from the case with periodically kicked and far detuned optical
lattices [7, 10, 9, 11]. The trivial time dependence in the Hamiltonian (2)
cannot produce chaotic motion, it simply accounts for a modulation of the
interaction of atoms with a Gaussian laser beam. Even if the atoms would cross
an absolutely homogeneous (in the $z$-direction) laser beam there would be
under appropriate conditions chaotic atomic center-of-mass motion in the
transversal $x$-direction.

Chaotic walking occurs due to the specific behavior of the Bloch-vector
component, $u$, of a moving atom whose shallow oscillations between the SW
nodes are interrupted by sudden jumps with different amplitudes while atom
crosses each node [18]. We illustrate in Fig. 3 the behavior of the $u$
component with different values of the detuning $\Delta=0.2$ and $\Delta=1$ at
which the atomic motion in accordance with the $\lambda$-map in Fig. 2 is
chaotic and regular, respectively. The time of the atomic interaction with the
SW field is estimated to be $3\sigma_{\tau}=1200$. So, the jumps of the $u$
variable (if any) disappear after that time in Fig. 3. It follows from the
second equation in the set (2) that jumps in the variable
$u=2(AB+\alpha\beta)$ result in jumps of the atomic momentum while crossing a
node of the SW. If the value of the atomic energy is close to a 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 deterministic SW.

It is easy to estimate the range of initial momenta at which atoms are
expected to change their direction of motion or move ballistically. At small
detunings, $|\Delta|\ll 1$, the total energy consists of the kinetic one,
$K=\omega_{r}p^{2}/2$, and the potential one, $U=u\cos x$. 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
$p_{0}>\sqrt{2/\omega_{r}}>44$. If the initial conditions are chosen to give
$0\leq K(\tau=0)+U(\tau=0)\leq 1$, the atoms with $0\lesssim p_{0}\lesssim 44$
are expected to perform a chaotic walking at positive $\Delta$.

###  2.2 Hamiltonian chaotic scattering

Figure 6: Scattering of 50 spontaneously emitting atoms at the SW with the
decay rate $\gamma=0.05$ and the same other conditions as in Fig. 4. (a)
Chaotic ($\Delta=0.2$) and (b) regular ($\Delta=1$) regimes.

Figure 7: The distributions of $10^{4}$ lithium atoms (a) without and (b) with
spontaneous emission at $\tau=1000$ under the conditions of chaotic scattering
at $\Delta=0.2$ (bold curves) and regular scattering at $\Delta=1$ (dashed
curves).

Let us consider a spatially uniform and previously focused beam of atoms
crossing a Gaussiam laser beam. The position and momentum distributions of
atoms are measured after interaction with the SW field. We predict that those
distributions would be much broader at those values of $\Delta$ at which one
expects chaotic walking to occur. Firstly, we perform simulation with a
negligible probability of spontaneous emission. To be concrete let us take
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.
The normalized recoil frequency is $\omega_{r}=4\pi\nu_{\rm
rec}/\Omega_{0}=10^{-3}$ and $\sigma_{\tau}=400$.

We numerically solve the equations of motion (2) with $\gamma=0$ at two values
of $\Delta$ corresponding to chaotic and regular regimes of the center-of-mass
motion. In accordance with the Lyapunov map in Fig. 2, behavior of the Bloch
component $u$ in Fig. 3, and the simple estimates given above, we expect the
chaotic scattering of atoms at $\Delta=0.2$ and their regular motion at
$\Delta=1$. Trajectories in the real and momentum spaces for 50 atoms with the
same initial momentum, $p_{0}=10$, and initial positions in the range
$-\pi/10\leq x\leq\pi/10$ are shown in Figs. 4 and 5, respectively, at the
fixed value of the recoil frequency $\omega_{r}=10^{-3}$. The upper panels in
Figs. 4 and 5 illustrate the broad distributions of the atoms in the $x$ and
$p$ spaces in the regime of chaotic scattering that contrasts strictly with
those obtained in the case of the regular scattering at $\Delta=1$ (Figs. 4b
and 5b). In order to create narrow atomic beams, one may use a pair of light
masks. The first SW with a red large detuning ($\Delta<0$) splits the initial
atomic beam into a number of narrow beams with the widths much smaller than
the optical wavelength which then cross the second SW. The method for creating
narrow wave packets in the nonadiabatic regime of scattering has been proposed
in Ref.[29].

###  2.3 Dissipative chaotic scattering and simulation of a real experiment

We have illustrated in Figs. 4 and 5 the Hamiltonian chaotic scattering that
may occur in the absence of any losses. To simulate trajectories of
spontaneously emitting atoms we use the standard stochastic wave-function
technique (see, for example, [30, 31, 32]) 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 one $\tau=\tau_{1}$ 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
performs the jump 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 $\gamma=0.05$. Simulated trajectories in the real
space for 50 spontaneously emitting atoms under the same conditions as in Fig.
4 are shown in Fig. 6. Even though deterministic Hamiltonian chaos is masked
by random events of spontaneous emission, nevertheless the spatial and
momentum (not shown) distributions are much broader at those values of
$\Delta$ at which the Hamiltonian center-of-mass motion is chaotic. Namely the
chaotic Hamiltonian walking is eventually responsible for divergency of atomic
beams in the real and momentum spaces.

To simulate a real experiment we consider a beam of $10^{4}$ lithium atoms
with the initial Gaussian distribution (the rms $\sigma_{x}=\sigma_{p}=2$ and
the average values, $x_{0}=0$, and momentum, $p_{0}=10$) and compute their
distribution at a fixed moment of time. In Fig. 7a we compare the atomic
position distributions at $\tau=1000$ for the chaotic scattering at
$\Delta=0.2$ (bold curve) and the regular scattering at $\Delta=1$ (dashed
curve) when neglecting spontaneous emission. The difference is evident. In the
regime of the chaotic scattering at $\Delta=0.2$ atoms are distributed more or
less homogeneously over a large distance of 8 wavelengths along the $x$-axis
whereas they form a few peaks in a much more narrow interval under the
conditions of the regular scattering at $\Delta=1$. Figure 7b demonstrates the
distributions of spontaneously emitting atoms at the normalized decay rate
$\gamma=0.05$ under the same conditions as in Fig. 7a. The regularly scattered
atoms at $\Delta=1$ (dashed curve) form the contrast atomic relief with the
bifurcated peaks around the first few SW nodes at $x=\pm 1/4$, $x=\pm 3/4$ and
$x=5/4$. The distribution of chaotically scattered atoms at $\Delta=0.2$ (bold
curve) has the peaks without any bifurcation at $x=\pm 1/4$ and $x=3/4$ with a
smaller number of atoms in each one. Moreover, this distribution is less
contrast as compared to the previous one. 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 SW 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.

We predict that experiments on the scattering of atomic beams at a SW laser
field can directly image chaotic walking of atoms along the SW. 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. 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 [9].

##  3 Conclusion

We have simulated the new type of atomic diffraction at a SW without any
modulation of its parameters and shown that it 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 effect could be
used in optical nanolithography to fabricate complex atomic structures on
substrates.

##  4 Acknowledgments

The work was supported by the Integration grant from the Far-Eastern and
Siberian branches of the Russian Academy of Sciences (12-II-0-02-001), and by
the Program “Fundamental Problems of Nonlinear Dynamics in Mathematics and
Physics”. I thank L.E. Konkov and M.Yu. Uleysky for the help in preparing some
figures.

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