# Cold atom experiments in quantum chaos

Mark Raizen and Daniel A. Steck (2011), Scholarpedia, 6(11):10468. | doi:10.4249/scholarpedia.10468 | revision #122155 [link to/cite this article] |

## Contents |

## Introduction

The experimental tests of quantum chaos require a system where controlled nonlinear potentials can be created, so that the classical equations of motion can be chaotic. This system must also be Hamiltonian in order for quantum chaotic effects to be manifested. The combination of these conditions places stringent constraints on the experimental system, and are readily satisfied for ultra-cold atoms in time-dependent optical lattices. First proposed by Graham et al. (1992), and realized experimentally by Moore et al. (1994), ultra-cold atoms in optical lattices have become one of the primary testing grounds for quantum chaos.

## Optical potentials

One of the fundamental tools for studying quantum chaos with cold atoms arises in the interaction of atoms with laser light. The electromagnetic (optical) field perturbs an electron in the atom, inducing an oscillating dipole moment. The energy associated with the dipole–field interaction leads to a potential energy in space for the atom (this is also called the **ac Stark effect**). The structure of the potential reflects the spatiotemporal variation of the laser field. In the useful limit where the laser is tuned close to a single atomic resonance, but still far away from the resonance compared to the atomic line width (decay rate), the potential is proportional to \(I(\mathbf{r})/\Delta\ ,\) where I is the intensity profile of the laser, and \(\Delta\) is the **laser detuning**, or the difference between the laser and atomic-resonance frequencies. There is great freedom in designing laser intensity profiles, and so this method enables the production of a wide range of spatial potentials for atoms. Note that the detuning here can be positive or negative. For a laser tuned *below* ("red of") resonance, \(\Delta<0\ ,\) and atoms experience forces toward bright regions. For a laser *above* ("blue of") resonance, \(\Delta>0\ ,\) and the bright regions *repel* atoms. Another useful case occurs when the laser is tuned far away from *all* atomic resonances. Here, the total potential is the sum over the potentials for all transitions (including the associated Bloch–Siegert shifts).

A particularly important example of an optical potential is the **optical lattice** (Jessen & Deutsch, 1996). The lattice is formed by two identical, counterpropagating laser beams. The interference between the two beams produces a sinusoidally varying intensity along the common axis of the beams, as shown in Fig. Figure 1. An atom in this field experiences a corresponding sinusoidal potential along the optical axis, as illustrated in Fig. Figure 2. A typical application in quantum-chaos experiments is to arrange the atoms within a region much smaller than the beam diameter, so they do not experience a transverse potential. Thus, the atoms are subject to an effective, one-dimensional, sinusoidal potential. Since the atoms are not transversely confined, the experiments are restricted to ms time scales, before the atoms have moved away from the beam centers.

One main consideration in designing optical potentials is the suppression of spontaneous scattering of the laser light, which causes decoherence and dissipative forces, both of which tend to suppress the desired quantum effects. When the light is detuned far from a single, dominant resonance, the rate of spontaneous scattering scales as \(I(\mathbf{r})/\Delta^2\ .\) Thus, it can be suppressed by using a large detuning, and compensating with a large intensity to produce the desired potential.

## Phase space

A central concept in understanding cold-atom experiments in quantum chaos is the classical **phase space** or **phase portrait** for Hamiltonian systems. The phase space acts
as a sort of road map for the classical and quantum dynamics of such systems. More information on this subject is available
in the State space article, but here we will construct an example that covers the concepts necessary to understand the experiments described below.

As an example, consider the **amplitude-modulated pendulum**, illustrated in
Fig. Figure 3. This is a particle in a sinusoidal potential, where the amplitude of the potential is itself modulated
sinusoidally in time, according to the Hamiltonian
\[\tag{1}
H = \frac{p^2}{2}+2\alpha\cos^2(\pi t)\cos(x),
\]

where \(x\) and \(p\) are scaled position and momentum coordinates, respectively, for the particle,
and \(\alpha\) is a scaled potential amplitude.
Experimentally, this system can be realized in atom optics by simply modulating the intensity of an optical lattice
The temporal modulation here produces a mix of stable chaotic behavior in this system, with more initial conditions giving rise
to chaotic trajectories as \(\alpha\) increases.
The phase portrait for this system is shown in
Fig. Figure 4. To understand this diagram, consider its construction.
In Fig. Figure 5, we start by plotting an initial condition, corresponding to the
\((x,p)\) coordinates at a particular time (say, \(t=0\)). This point uniquely determines the state of the system
and the potential. Noting that the system is time-periodic with period \(1\ ,\) we then evolve the system forward over
one time period, again plotting the state of the system. Continuing with this process, we build up a history of the trajectory,
with its location plotted stroboscopically (at times \(t=0, 1, 2, 3, \ldots\)). In this case the points on the trajectory
are confined to an ellipse, as is characteristic of a stable trajectory.
If we repeat this process with a *different* initial condition, as in
Fig. Figure 6, we see the second trajectory behaves differently: it spreads erratically through a nonzero area in the phase
plane, as is characteristic of a chaotic trajectory.
The full phase space
Fig. Figure 4 is simply the result of repeating this process for *many* initial conditions, as illustrated in
Fig. Figure 7. This gives a convenient picture at a glance of the qualitative behavior of *all* initial conditions of the system,
showing for example the regions of stable behavior, or **islands of stability**, interspersed with chaotic regions.

## Kicked rotor

One of the most important models in the study of classical and quantum chaos is the **kicked rotor**, described by the Hamiltonian
\[\tag{2}
H = \frac{p^2}{2}+K\cos(x)\sum_{n=-\infty}^\infty\delta(t-n),
\]

Again, \(x\) is a scaled position and \(p\) is a scaled momentum, while the **stochasticity parameter** \(K\)
controls the amplitude of the kicking.
This is similar to the amplitude-modulated pendulum above, but instead of a smooth temporal modulation of the potential, the potential is
turned on in a periodic sequence of arbitrarily short and intense pulses. This system is especially amenable to theoretical study since it reduces
to a relatively simple discrete mapping in the classical and quantum cases; for more details, see the article on the
Chirikov standard map.
Experimentally, this can again be realized by modulating the intensity of an optical lattice, this time exposing the atoms to a train of short pulses of laser light.
In practice, the delta-function pulses are idealizations that describe the experiment well so long as atoms do not move significantly compared to the
optical-lattice period during each optical pulse.

For large values of the stochasticity parameter \(K\ ,\) the (classical) phase space of the kicked rotor exhibits widespread chaos
(Fig. Figure 8). In this regime, the chaotic behavior appears as *diffusive* behavior in the momentum of the particle:
each kick of the potential is quasi-random in amplitude and direction, leading to a random walk in momentum space.
On average, then, a particle subjected to the kicking potential gains kinetic energy at a constant rate.
Quantum mechanically, this system behaves differently, however. At late times the rate of energy increase is dramatically reduced
due to quantum interference effects (Casati et al., 1979; Shepelyansky, 1983), so that the atomic momentum distribution effectively
freezes in spite of the kicking potential.
This **dynamical localization** effect is formally equivalent to **Anderson localization**, where a particle in a one-dimensional disordered
crystal becomes localized due to the disorder and does not conduct
(Anderson, 1958). The exception to this generic quantum phenomenon occurs at specific values of the kicking period, known as **quantum resonances**, which are described in the following section.
In the one-dimensional optical lattice, only states separated by \(2\hbar k\ ,\) where \(k\) is the optical wave number,
are connected by the optical potential. (Heuristically, \(\hbar k\) is the photon momentum, and so momentum is transferred to the atom
in multiples of \(2\hbar k\) when the atom scatters a photon into the backward direction.)
These discrete momentum states play the role of the lattice sites in the disordered crystal in the Anderson model (Fishman et al., 1982; Grempel et al., 1984).

### Observation of dynamical localization

In order to observe dynamical localization experimentally with ultra-cold atoms, the first step is trapping and cooling. This is conveniently accomplished using laser cooling techniques for alkali atoms. First experiments were performed with laser cooled sodium, and subsequent experiments used cesium and rubidium. The initial condition for the experiment consist of a cloud of ultra-cold atoms in a magneto-optical trap. The trapping beams are then turned off, and the atoms are exposed to a time-dependent optical lattice on a time scale short compared with free-fall due to gravity. To observe dynamical localization in the kicked rotor, atoms were exposed to a periodically pulsed standing wave of light where the adjustable parameters were the period of the pulses (the "kicks"), the number of pulses, and the light intensity as realized by Moore et al. (1995). The effects of the interaction is measured by a time-of-flight expansion, where the momentum distribution induced by the kicks is determined by the spatial expansion of the cloud. The motion of the atoms is frozen by optical molasses created by six laser beams near-resonance, and the resulting fluorescence is imaged on a charged coupled device (CCD) camera. The time evolution is measured by a series of independent measurements, where the number of pulses is stepped up consecutively. An experimental plot is shown in Fig. Figure 9, where time is measured in terms of pulse number, N. It shows that the distributions evolve from an initial Gaussian at \(N=0\) to an exponentially localized distribution after approximately \(N=8\) kicks. The distributions are measured out to \(N=50\ ,\) and no further change in the momentum is observed. This freezing of the distribution into exponential lineshapes is the hallmark of dynamical localization. The growth of the mean kinetic energy of the atoms as a function of the number of kicks is displayed in Fig. Figure 10. It shows an initial diffusive growth until the quantum break time of \(N^*=8.4\) kicks, after which dynamical localization is observed. The classical and quantum calculations both agree with the data over the diffusive regime. The measured distributions stop growing, as predicted by the quantum analysis and the theory of dynamical localization of quantum chaos developed for the Chirikov standard map (Shepelyansky, 1987).

One effect studied in detail is a "boundary" to the momentum diffusion imposed by the finite duration of the laser pulses. By contrast to the ideal case of delta-function kicks mentioned above, atoms with large momentum can move over a substantial portion of the optical lattice during a finite-duration pulse. This process reduces the effective strength of each kick, cutting off the chaotic region shown in Fig. Figure 8 at large momenta (Klappauf et al., 1999). An important consideration in the design of experiments on dynamical localization is to keep the pulses short enough that any localization can be attributed to quantum effects rather than this classical boundary.

### Quantum resonance

In the kicked rotor, atoms undergo free evolution between kicks. During this period, the atoms accumulate a dynamical quantum phase. An initial plane wave at \(p=0\) couples to a ladder of states separated by multiples of the two-photon momenta. For particular pulse periods, the quantum phase for each state in the ladder is a multiple of 2π, a condition known as "quantum resonance" (Izrailev & Shepelyanskii, 1980; Fishman et al., 2002). More generally, a quantum resonance is predicted when the accumulated phase between kicks is a rational multiple of 2π. The experiments with ultra-cold atoms observed such quantum resonances by scanning the pulse period (Moore et al., 1995; Oskay et al., 2000). The original prediction for a plane-wave initial condition was that energy grows quadratically with time, never following classical diffusion. However, when the initial condition is a Gaussian with a characteristic width in momentum, the quantum resonance is manifested differently. Instead of exponential lineshapes, the distributions settle after a few kicks into a Gaussian lineshape. These results are in agreement with a theoretical analysis.

## Tunneling

One of the best-known, manifestly quantum effects is barrier tunneling, where a quantum particle passes through a potential barrier even though it is forbidden to do so classically.
An analogous quantum effect, **dynamical tunneling**, occurs in classically chaotic systems (Davis & Heller, 1981).
To understand this effect, consider the phase space in Fig. Figure 11, where two islands of stability are situated symmetrically about \(p=0\) in a chaotic sea.
This is the phase space for the modulated pendulum, but for a larger value of \(\alpha\) than in Fig. Figure 4.
Recall from
Fig. Figure 6
that a classical particle in an island of stability is confined within it (it is in fact confined to a particular ellipse-like curve within the island).
But owing to the symmetry of the two islands in Fig. Figure 11, a quantum particle can tunnel periodically between them.
Dynamical tunneling was observed with cold atoms in this phase space, as shown in Fig. Figure 12, where the momentum distribution of an atomic ensemble oscillates between the momenta of the two islands
(Steck et al., 2001; Steck et al., 2002). Dynamical tunneling was also observed in a closely related, modulated-optical-lattice system (Hensinger et al., 2001), and in the spin degree of freedom of cold atoms (Chaudhury et al., 2009).
Crucial to the success of these experiments was a controlled localization of the atoms in phase space near the uncertainty limit, and, in the optical-lattice experiments,
coherence of the atomic wave packets across multiple sites of the modulated lattices.

The presence of the chaotic region can also facilitate tunneling between islands of stability, increasing the tunneling rate by orders of magnitude.
Quantum mechanically, ordinary dynamical tunneling can be understood in terms of only two states associated with the islands, but in **chaos-assisted tunneling**,
a third state in the chaotic region interacts with the two island states to effect the tunneling enhancement
(Bohigas et al., 1993; Tomsovic & Ullmo, 1994).
Signatures of chaos-assisted tunneling—including multiple tunneling frequencies and strong parameter dependence of the tunneling rate—were also observed with the cold-atom realization of the amplitude-modulated pendulum
(Steck et al., 2001; Steck et al., 2002), as confirmed by subsequent theoretical analysis
(Luter & Reichl, 2002; Averbukh et al., 2002).

## Other experiments

Since the first experiments in 1994, cold atoms have been used to test many aspects of quantum chaos. One of the first directions was to study the effects of noise and decoherence on dynamical localization (Ott et al., 1984; Cohen, 1991). A transition to classical behavior was observed (Amman et al., 1998; Klappauf et al., 1998a). The quantum signatures of classical anomalous diffusion in the kicked rotor were studied, leading to regimes of sub-diffusion and super-diffusion (Klappauf et al., 1998b; Zhong, 2001). In most of the optical-lattice-based experiments described in this article, the lattice was oriented horizontally to avoid effects of gravity on the dynamics. However, "quantum accelerator modes" of the kicked rotor in the presence of gravity were studied, and a connection was shown with blazed matter wave gratings (Oberthaler et al, 1999). The use of Bose-Einstein condensates, instead of laser-cooled atoms, provided much better initial conditions for a number of recent experiments. For example, this was implemented in an experiment that observed high-order quantum resonances (Ryu et al, 2006) and in an experimental realization of approximate time-reversal in the kicked rotor (Ullah et al, 2011). In recent work, the Anderson metal-insulator transition was observed with atomic matter waves (Chabé et al., 2008). The quantum signatures of chaos were also observed in a cold-atom realization of the kicked top (Chaudhury et al., 2009).

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**Internal references**

- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

- Boris Chirikov and Dima Shepelyansky (2008) Chirikov standard map. Scholarpedia, 3(3):3550.

- Shmuel Fishman (2010) Anderson localization and quantum chaos maps Scholarpedia, 5(8):9816

## Recommended reading

- Ott, E., 2002.
*Chaos in Dynamical Systems*, 2nd edition. Cambridge: University Press.

- Raizen, M. G., 1999. Quantum Chaos with cold atoms, in Bederson, B. & Walther, H.,
*Advances in Atomic, Molecular, and Optical Physics*, vol. 41, pp. 43-81.

- Reichl, L. E., 2004.
*The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations*, 2nd edition. New York: Springer-Verlag.

- Tabor, M., 1989. "Chaos and Integrability in Nonlinear Dynamics: An Introduction", New York: Wiley.