Quantum Potential and Chaos of the Kicked Rotor

Bruce G. Klappauf, D. Steck, Mark G. Raizen

Yindong Zheng

The de Broglie-Bohm (BB) approach to quantum mechanics gives trajectories similar to classical trajectories except that they are also determined by a quantum potential. The quantum potential is a "fictitious potential" in the sense that it is part of the quantum kinetic energy. We use quantum trajectories to treat quantum chaos in a manner similar to classical chaos. For the kicked rotor, which is a bounded system, we use the Benettin et al. method to calculate both classical and quantum Lyapunov exponents as a function of control parameter K and find chaos in both cases. Within the chaotic sea we find in both cases nonchaotic stability regions for K equal to multiples of π. For even multiples of π the stability regions are associated with classical accelerator mode islands and for odd multiples of π they are associated with new oscillator modes. We examine the structure of these regions. Momentum diffusion of the quantum kicked rotor is studied with both BB and standard quantum mechanics (SQM). A general analytical expression is given for the momentum diffusion at quantum resonance of both BB and SQM. We obtain agreement between the two approaches in numerical experiments. For the case of nonresonance the quantum potential is not zero and must be included as part of the quantum kinetic energy for agreement. The numerical data for momentum diffusion of classical kicked rotor is well fit by a power law DNβ in the number of kicks N. In the anomalous momentum diffusion regions due to accelerator modes the exponent β(K) is slightly less than quadratic, except for a slight dip, in agreement with an upper bound (K2/2)N2. The corresponding coefficient D(K) in these regions has three distinct sections, most likely due to accelerator modes with period greater than one. We also show that the local Lyapunov exponent of the classical kicked rotor has a plateau for a duration that depends on the initial separation and then decreases asymptotically as O(t-1lnt), where t is the time. This behavior is consistent with an upper bound that is determined analytically.

Anjali Nambudiripad, J. Bharathi Kannan, M. S. Santhanam

Despite the periodic kicks, a linear kicked rotor (LKR) is an integrable and exactly solvable model in which the kinetic energy term is linear in momentum. It was recently shown that spatially interacting LKRs are also integrable, and results in dynamical localization in the corresponding quantum regime. Similar localized phases exist in other nonintegrable models such as the coupled relativistic kicked rotors. This work, using a two-body LKR, demonstrates two main results; first, it is shown that chaos can be induced in the integrable linear kicked rotor through interactions between the momenta of rotors. An analytical estimate of its Lyapunov exponent is obtained. Second, the quantum dynamics of this chaotic model, upon variation of kicking and interaction strengths, is shown to exhibit a variety of phases: classically induced localization, dynamical localization, subdiffusive, and diffusive phases. We point out the signatures of these phases from the perspective of entanglement production in this system. By defining an effective Hilbert space dimension, the entanglement growth rate can be understood using appropriate random matrix averages.

Windell H. Oskay, Valery Milner, D. Steck +1 more

T. Boness

The Heisenberg spin-chain and the Quantum Kicked Rotor (QKR) are key models in condensed matter and quantum chaos respectively. The former, consisting of a line of qubits that interact with their neighbours through an exchange interaction, has recently received much attention in the context of quantum information and communication. In particular, it has been shown that a Heisenberg chain can be used as a channel for the transfer quantum states and the sharing of entanglement. The QKR, on the other hand, is a paradigm of quantum chaos. It is one of a valuable collection of quantum systems with a well de�ned classical limit that shows both regular and chaotic dynamics. This thesis investigates the dynamics of a Heisenberg chain subjected to periodic ±-kicks from a parabolic magnetic �eld. It will be shown that, when perturbed in this way, spin-chains can provide realizations of QKR systems. Using this connection, well-known features of the dynamics of kicked rotors can be used to manipulate the evolution of excitations and entanglement in the spin-chain. Conversely, results from the spin-chain analysis can inform our knowledge of the kicked rotor systems. The focus of the �rst part of this thesis is the dynamics of singly excited chains. In this case, the kicked Heisenberg chain is shown to be dynamically equivalent to a QKR. Similar correspondences can be found. Two examples, spin-chain realizations of the Double-Kicked Rotor and a double-well kicked rotor, are explored. Later sections are concerned with the dynamics of multiple excitations. These excitations interact - scattering or forming bound states of inverted spins. The kicked chain now corresponds to a system of coupled kicked rotors. It will be shown that by tuning the parameters it is possible to control the time-evolution of correlations in the chain.

A. Ullah, S.K. Ruddell, Jean Anne C. Incorvia +1 more