The field of "quantum chaos'' was born in 1917 when Albert Einstein tried to unravel which mechanical systems can be subjected to the Bohr-Sommerfeld-Epstein quantization rules. He concluded that in the absence of invariant tori in phase space these quantization rules cannot be used and that, moreover, this absence applies to most systems. "Chaos'' is associated with rapid divergence of arbitrarily close points in phase space. Strictly there can be no such thing as quantum chaos as an infinitely fine level of detail is needed to describe the trajectories of a classical chaotic system. In reality a system is bound by Heisenberg's uncertainty principle restricting the amount of detail of position and momentum needed for classical chaos. Classical chaos can be described as emergence of complexity on infinitely fine scales in classical phase space. In contrast in quantum mechanics structure is smoothed away in an area below the size of hbar.

During the years since the birth of "quantum chaos'', significant amounts of theory have been created to give a better description of chaotic physical systems in a quantum dynamical context. The key question is, what happens to classical chaos in the quantum world? One approach is to seek generic features of quantum dynamics for a system whose classical description exhibits chaotic dynamics. One example of such features is dynamical localization, a quantum suppression of classical diffusion, which was discovered by Fishman et al in numerical studies of the periodically kicked quantum rotor. Conductance fluctuations in ballistic microstructures associated with complex electron trajectories constitute another example of the occurrence of quantum chaos. Finally, molecular excitation experiments can show interesting quantum features (e.g. Anderson localization, an effect related to dynamical localization) if the scaled Planck's constant is kept finite but exhibit chaotic dynamics in the classical limit (hbar =0). To gain a different perspective on the quantum nature of classical chaos some experiments look at manifestations of classical chaos in wave propagation. In these experiments the time independent wave equation, the Helmholtz equation, is mathematically equivalent to the time independent Schroedinger equation for a billiard system. In billiard-shaped cavities eigenfrequencies and eigenfunctions can be measured by microwave absorption. In 1991 quantum scars, which are concentrations of probability along periodic orbits, were experimentally observed by Sridhar and coworkers. Experiments to study the quantum dynamics of classically chaotic systems have been carried out on Rydberg atoms, measuring microwave ionization of highly excited hydrogen atoms. One result of these experiments is the recognition of different regimes determined by how well classical and quantum mechanics agree with each other. These regimes are characterized by the scaled microwave frequency.

It was first proposed by Graham, Schlautmann and Zoller to use atom manipulation experiments to test predictions of quantum chaos. Cold atoms provide new grounds for experiments in quantum chaos which have some advantages compared to Rydberg atom experiments. Firstly, the potentials which are used are extremely well approximated as one dimensional potentials. In contrast, the potentials involved in the Rydberg atom ionization experiments are much harder to approximate by one-dimensional potentials. At present the three dimensional quantum simulations needed for these highly excited atoms are not feasible without severe approximations. Secondly, in atom optics there is considerable control over the potentials. In the Rydberg atom case the Coulomb potential dictates the dynamics and the system is complicated due to electron-electron interactions (the chaotic trajectory of the outer electron in a Rydberg atom can closely approach the shell of inner electrons) . In atom optics one can tailor the potentials to match the theoretical description and indeed achieve simple nonlinear potentials such as the nonlinear pendulum which we consider in this study. We can also achieve a considerable variety of modulation dynamics. Finally, atom optical systems are far less dissipative and noisy than Rydberg atom experiments due to laser cooling and the ability to operate far off resonance. Hence atom optical systems can be well approximated by Hamiltonian dynamics.

Moore et al. and Ammann et al. showed that cold atoms can be used to simulate the quantum delta kicked rotor (Q-DKR). Raizen's group also reported the first experimental observation of dynamic localization in cold atoms. This is the quantum suppression of chaotic diffusion. This can be roughly understood by considering that classical chaotic paths can interfere destructively. Decoherence will tend to suppress this quantum behavior and leads to classical-like dynamics.