How does the existence of chaos in classical dynamics affect the quantum spectrum of the system? The atom in a magnetic field is a prototype of systems where complete classical and quantal calculations are possible. Transition from regular to chaotic trajectories are linked quantally to the progressive breaking of some peculiar symmetries of the system. We propose a method and its experimental realization on lithium atom to clearly see a transition between regularity and chaos on a quantum spectrum by selecting the starting state of the excitation. In addition, experimental and calculated spectra on hydrogen shows a remarkably good agreement.
How does the existence of chaos in classical dynamics affect the quantum spectrum of the system? The atom in a magnetic field is a prototype of systems where complete classical and quantal calculations are possible. Transition from regular to chaotic trajectories are linked quantally to the progressive breaking of some peculiar symmetries of the system. We propose a method and its experimental realization on lithium atom to clearly see a transition between regularity and chaos on a quantum spectrum by selecting the starting state of the excitation. In addition, experimental and calculated spectra on hydrogen shows a remarkably good agreement.
A discussion regarding the energy levels spectrum of quantum systems whose classical analogous has states of chaotic motion is presented. The chaotic dynamics of the classical underlying system has its manifestation in the wave functions (in the form of “scars”) and in the energy levels (in the form of “statistical repulsion” of the energy levels). The above mentioned signatures are named “quantum chaos”. A typical study of quantum chaos requires finding accurate energy eigenvalues of highly excited states, to calculate the nearest neighbors spacing between levels, to perform the “unfolding” of the spectrum in order to separate the fluctuations, and finally to find the probability distribution of the unfolded spectrum. This is exemplified by the hydrogen atom under uniform magnetic field and a quadrupole electric field.
The possibility of experimental observation of chaotic motion of cold atoms in a one-dimensional rigid optical lattice without any modulation of its parameters and additional impact is demonstrated theoretically and numerically. This effect of deterministic chaos arises near the optical resonance due to the nonlinear coupling of electronic and mechanical degrees of freedom in the atom. In a certain range of resonance detunings, at each crossing of the lattice-forming standing wave node by the atom, the atomic dipole moment changes pseudorandomly in a step-like manner, accompanied by an appropriate change in the atom momentum. This leads to the effect of chaotic wandering of atoms, possessing the properties of the classical deterministic chaos, namely, to the exponential sensitivity of atomic trajectories to small variations of the initial conditions and/or the control parameters. With time, part of atoms from the initial cloud changes the direction of their motion to the opposite one, which can be detected in a real experiment. Numerical experiments with lithium atoms with the spontaneous emission taken into account show that by varying the laser radiation intensity (normalised detuning of resonance) in the rigid optical lattice, one can implement the transition from the regular regime of the atomic motion to the chaotic wandering in a real experiment.
The influence of quantum chaos on small atomic systems is reviewed. It now seems clear that chaos in the usually understood sense (e.g. exponential sensitivity to perturbations) is not found in isolated quantum systems. However, there are phenomena which appear only when the corresponding classical system is chaotic. The stability of the classical dynamics, in other words whether it is regular or chaotic, has a profound effect on the character of the corresponding quantum spectrum and wavefunctions.
Laser-cooled atoms offer an excellent platform for testing new ideas of quantum control and measurement. I will discuss experiments where we use light and magnetic fields to drive and monitor non-trivial quantum dynamics of a large spin-angular momentum associated with an atomic hyperfine ground state. We can design Hamiltonians to generate arbitrary spin states and perform a full quantum state reconstruction of the results. We have implemented and verified time optimal controls to generate a broad variety of spin states, including spin-squeezed states useful for metrology. Yields achieved are of the range 0.8-0.9.We present a first experimental demonstration of the quantum kicked top, a popular paradigm for quantum and classical chaos. We make `movies' of the evolving quantum state which provides a direct observation of phase space dynamics of this system. The spin dynamics seen in the experiment includes dynamical tunneling between regular islands, rapid spreading of states throughout the chaotic sea, and surprisingly robust signatures of classical phase space structures. Our data show differences between regular and chaotic dynamics in the sensitivity to perturbations of the quantum kicked top Hamiltonian and in the average electron-nuclear spin entanglement during the first 40 kicks. The difference, while clear, is modest due to the small size of the spin.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
Most classical dynamical systems are chaotic. The trajectories of two identical systems prepared in infinitesimally different initial conditions diverge exponentially with time. Quantum systems, instead, exhibit quasiperiodicity due to their discrete spectrum. Nonetheless, the dynamics of quantum systems whose classical counterparts are chaotic are expected to show some features that resemble chaotic motion. Among the many controversial aspects of the quantum-classical boundary, the emergence of chaos remains among the least experimentally verified. Time-resolved observations of quantum chaotic dynamics are particularly rare, and as yet unachieved in a single particle, where the subtle interplay between chaos and quantum measurement could be explored at its deepest levels. We present here a realistic proposal to construct a chaotic driven top from the nuclear spin of a single donor atom in silicon, in the presence of a nuclear quadrupole interaction. This system is exquisitely measurable and controllable, and possesses extremely long intrinsic quantum coherence times, allowing for the observation of subtle dynamical behavior over extended periods. We show that signatures of chaos are expected to arise for experimentally realizable parameters of the system, allowing the study of the relation between quantum decoherence and classical chaos, and the observation of dynamical tunneling.