The classical kicked rotor

At first glance, the kicked molecule experiment sketched in Fig. 5.1 does not appear to be a system worthy of much attention. The set-up is simple, there are no comphcated boundary conditions, and noise effects are neglected. But its simpHcity notwithstanding, it turns out that the classical as well as the quantum dynamics of the system sketched in Fig. 5.1 are very complicated, and cannot in either case be solved analytically in the presence of a strong driving field. 

In order to extract the essence of the dynamics of the kicked molecule, we consider a simple model constructed by replacing the molecule with a two-dimensional dipole (see Fig. 5.2). Moreover, we replace the sequence of finite-width pulses provided by the pulse generator (see Fig. 5.1) by a train of zero-width -function kicks. Thus, the dipole is perturbed by a 

A time diagram of the electric field bursts (5.1.1) is shown in Fig. 5.3. A planar rotor perturbed by a sequence of delta kicks according to (5.1.1) is known as the kicked rotor. In order to familiarize ourselves with the physics of kicked systems, we focus in this section on the classical mechanics of the kicked rotor. If we denote by L the angular momentum of the rotor, the Hamiltonian function of the kicked rotor is given by 

Measuring the angular momentum in units of I/T, such that Ln = Iln/T, and introducing the control parameter K = uSqT /I, we obtain the following set of dimensionless mapping equations  

Iq = 270,1 + 2g7T, for four different values of the control parameter, 

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The quantum Hamiltonian of the classical kicked rotor, defined by the classical Hamiltonian function (5.1.2), is easily obtained by canonical quantization. On replacing the classical angular momentum L by the quantum angular momentum operator L according to... 

In analogy to the classical kicked rotor we define the mean kinetic energy of the rotor given by... 

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). 

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)... 

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... 

The plan of Chapter 5 is the following. In order to get a feeUng for the dynamics of the kicked molecule, we approximate it by a one-dimensional schematic model by restricting its motion to rotation in the x, z) plane and ignoring motion of the centre of mass. In this approximation the kicked molecule becomes the kicked rotor, probably the most widely studied model in quantum chaology. This model was introduced by Casati et al. in 1979. The classical mechanics of the kicked rotor is discussed in Section 5.1. Section 5.2 presents Chirikov s overlap criterion, which can be applied generally to estimate analytically the critical control parameter necessary for the onset of chaos. We use it here to estimate the onset of chaos in the kicked rotor model. The quantum mechanics of the kicked rotor is discussed in Section 5.3. In Section 5.4 we show that the results obtained for the quantum kicked rotor model are of immediate... 

Summarizing, we predict on the basis of the one-dimensional classical kicked rotor that, for AT < ATc w 1, the molecule in Fig. 5.1 cannot absorb any substantial quantities of energy from the driving field, no matter how long the molecule is exposed to the field. Strong absorption of energy with a linear growth as a function of exposure time is expected to occur for K>Kc. 

In the concluding remark of Section 5.2 we asked the question whether the transition from confined chaos to global chaos K = Kc can be seen in an experiment with diatomic molecules. The technical feasibility of such an experiment is discussed in Section 5.4. Here we ask the more modest question whether, and if so, how, the transition to global chaos manifests itself within the framework of the quantum kicked rotor. Since the transition to global chaos is primarily a classical phenomenon, we expect that we have the best chance of seeing any manifestation of this transition in the quantum kicked rotor the more classical we prepare its initial state and control parameters. Thus, we choose a small value... 

Fig. 5.9. Exponential localization of the wave function of the quantum kicked rotor in the classically chaotic regime.

The Hamiltonian (5.3.2) of the quantum kicked rotor is time reversal invariant. The question is, what happens to the localization length of the kicked rotor if a time reversal violating interaction is switched on The difficulty here is to add time reversal violating terms to (5.3.2) in such a way that the classical properties of the Hamiltonian are not affected. This is important, since otherwise a change in the localization length is not surprising since it can always, at least partially, be blamed on the... 

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). 

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... 

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).]...

It is worth mentioning that although the original problem (8.1.1) depended on the two field parameters P and w, the scaled equations of motion (8.1.8) as well as the scaled Hamiltonian (8.1.10) depend only on the single control parameter This is a major difference compared with the problem of microwave-driven surface state electrons, but is reminiscent of the classical mechanics of the kicked rotor discussed in Chapter... 

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