Kicked rotor

The stability of scarred states to external noise and other environmental disturbances was the next natural issue that was raised and partially addressed earlier (L. Sirko, et.al., 1993 R. Scharf, et.al., 1994). The main conclusion was that scarred states are quite robust to reasonable levels of noise. This question took on added relevance with the coming of age of mesoscopic systems where, be it spontaneous emission in atom optics or leads or scattering and other forms of dissipation in heterostructures, the open nature of the system must be accounted for. These new experiments also provided non-ideal realizations of simple theoretical paradigms such as stadium billiards and the kicked rotor, with additional issues that had to be accounted for in the theory. 

It turns out that this analysis applies only to systems with a bounded phase space. It is possible that topological restrictions on the accessible phase space - and not only the form of the particular Hamiltonian -play a crucial role in determining when the weak form of the QCT actually applies. For example, this might explain why the open-system quantum delta-kicked rotor is a counter-example to naive expectations regarding the QCT (S. Habib et.al., 2002). 

Keywords Dynamical chaos, kicked rotor, relativistic systems... 

In the classical case, the evolution of the kicked rotor dynamics is described by the well-known standard map (Chirikov, 1979). This map greatly facilitates the qualitative treatment of the system. A map describing the evolution of the wave function can be obtained in the quantum case, too (Casati et.al., 1979). In spite of the fact, that the first work with detailed treatment of the quantum kicked rotor appeared 23 years ago (Casati et.al., 1979), this system is still studied extensively (Casati et.al., 1987 Izrailev, 1990). 

In this work we treat classical and quantum relativistic kicked rotor problem. Using the relativistic standard map, we calculate the average energy of the classical rotor for various values of the relativistic factor. The relativistic quantum rotor is treated using the same approach as in the pioneering work (Casati et.al., 1979). 

As is well known (Chirikov, 1979 Izrailev, 1990), the phase-space evolution of the norelativistic classical kicked rotor is described by nonrelativistic standard map. The analysis of this map shows that the motion of the nonrelativistic kicked rotor is accompanied by unlimited diffusion in the energy and momentum. However, this diffusion is suppressed in the quantum case (Casati et.al., 1979 Izrailev, 1990). Such a suppression of diffusive growth of the energy can be observed when one considers the (classical) relativistic extention of the classical standard map (Nomura et.al., 1992) which was obtained recently by considering the motion of the relativistic electron in the field of an electrostatic wave packet. The relativistic generalization of the standard map is obtained recently (Nomura et.al., 1992)... 

Quantum dynamics of the relativistic kicked rotor is described by the following Dirac equation ... 

The main feature of the quantum kicked rotor is the quantum localization phenomenon, which implies suppression of the diffusive growth of energy of the quantum kicked rotor compared to the energy of the classical rotor (Izrailev, 1990). The time dependence of the energy can be calculated as... 

Thus we have treated relativistic kicked rotor problem both in classical and quantum contexts. It is found that in the classical case the diffusion is strongly suppressed in highly relativistic and ultrarelativis-tic regimes. However, the energy growth can be observed in resonances... 

Many realistic systems and their models have been considered to study dynamical chaos phenomenon. Such systems as, kicked rotor and various billiard geometries allow one to treat chaotic behavior of deterministic systems successfully. 

To date there are only a few studies of the quantum dynamics associated with classical anomalous diffusion. We consider here a recent study by Brumer and co-workers, who showed that quantum effects can further accelerate classical anomalous diffusion [97]. This is highly counterintuitive, since people tend to believe that in aU cases quantum effects suppress classical chaotic transport. The system they studied is a modified kicked rotor system, whose Hamiltonian is given by... 

Figure 51. Classical phase-space structures of a modified kicked rotor system. Note that the regular islands are transporting islands. [From J. B. Gong, H. J. Worner, and P. Brumer, Phys. Rev. E. 68, 026209 (2003).]...

This chapter is organized as follows. In Section II, we show how quantum chaos systems can be controlled under the optimal fields obtained by OCT. The examples are a random matrix system and a quantum kicked rotor. (The former is considered as a strong-chaos-limit case, and the latter is considered as mixed regular-chaotic cases.) In Section III, a coarse-grained Rabi state is introduced to analyze the controlled dynamics in quantum chaos systems. We numerically obtain a smooth transition between time-dependent states, which justifies the use of such a picture. In Section IV, we derive an analytic expression for the optimal field under the assumption of the CG Rabi state, and we numerically show that the field can really steer an initial state to a target state in random matrix systems. Finally, we summarize the chapter and discuss further aspects of controlling quantum chaos. 

In the following subsections, we numerically demonstrate to control multilevel-multilevel transition problems in quantum chaos systems One is a random matrix system, and the other is a quantum kicked rotor. 

The kicked rotor (or the standard map) is one of famous models in chaotic dynamical systems, and it has been studied in various situations [17]. One feature of its chaotic dynamics is the deterministic diffusion along the momentum direction. It is also well known that if we quantize this system, this diffusion is... 

Here we employ the quantum kicked rotor as a simple model of quanmm chaos systems. The Hamiltonian of a kicked rotor is written as... 

The kicked rotor is often described only at discrete time immediately after/ before the periodic kicks. In our control problem, however, we must represent dynamics driven by e t) between those kicks. Then, we can apply the Zhu-Botina-Rabitz scheme as usual. According to Eq. (5), the optimal external field is given by... 

In Figs. 3 and 4, we show numerical results for the quantum kicked rotor as in Section II. A. The system parameters are chosen to pick up a regular dynamics (Fig. 3) and a chaotic dynamics (Fig. 4), and the others are T = 400 and a = 1. The optimal field after 100 iterations for the regular case (Fig. 3a) is much simpler than that for the chaotic case (Fig. 4a). (See also Figs. 3b and 4b.) This is because more states are involved in the latter chaotic process. 

Figure 3. Optimal control in a regular kicked rotor with = 1 and % — 0.3436 by the Zhu-Botina-Rabitz scheme with T — 400 and a — 1. (a) the optimal field after 100 iterations b) its power spectrum (c) the optimal evolution of the squared overlap with the target (([)(r) (py) as well as its magnified values near the target time in the inset (d) the convergence behavior of the overlap Jq (solid curve) and the functional J (dashed curve) versus the number of iteration steps.

Figure 5. Time evolution of the Husimi distribution for quantum kicked rotors with H = 0.3436 under an optimal field after 100 iterations. The Zhu-Botina-Rabitz scheme was used with the penalty factor a = 1 and the target time T = 400. From left to right, quantum states immediately after the kick at t = 0, 1, 2, 10, 100, 200, 300, 398, 399, and 400 are depicted, (a) The parameters are = 1 (regular case), (0 ,/) ) = (l- j l- ) (S/jf/) = (1-0, 1-0) (b) K = 1...

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