The quantum kicked rotor

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 the vicinity of a kick, say, kick number n, the kinetic energy operator L / 2I) can be neglected. In this case the Schrodinger equation reads 

If we denote by [ the state of the rotor immediately before kick number n, and by 0 ) the state of the rotor immediately after kick number n, then 0 ) can be calculated by a simple quadrature. It is given by  

Between kicks the external force is zero and the state of the rotor evolves freely  

All the information about the long-time behaviour of the rotor states is contained in the properties of the one-cycle propagator U. Introducing the dimensionless angular momentum I according to 

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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). 

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... 

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

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. 

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... 

The matrix multiplication scheme is sufficient for some basic numerical experiments with the quantum kicked rotor. A more eflficient scheme... 

Another unexpected feature of the quantum kicked rotor is the existence of resonances that lead to quadratic energy gain. Quantum resonances in the kicked rotor were first analysed by Izrailev and Shepelyan-skii (1979). They showed that whenever the quantum control parameter T is a rational multiple of tt, the quantum energy grows quadrati-... 

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.

Fig. 5.10. Average energy of the quantum kicked rotor as a function of K = fcr for T = 1/10. The rotor was prepared at t = 0 in the rotational state lo = 10).

In this section we investigate the physics of impulsively perturbed diatomic molecules. These provide a real physical system whose characteristics are expected to be close to the ones established for the kicked rotor. We establish that the dynamical effects exhibited by the quantum kicked rotor can be demonstrated experimentally with diatomic molecules within the possibilities of present day technology. 

In summary, Csl molecules appear to be the most promising candidates for an experimental verification of the dynamical predictions of the quantum kicked rotor. The most significant result would be a demonstration of Anderson localization with the help of diatomic molecules. Our results indicate that this demonstration is within technical reach. 

Recently, interesting symmetry effects have been discovered in connection with the quantum kicked rotor (see, e.g., Dittrich and Smilansky (1991a,b), Bliimel and Smilansky (1992), Thaha et al. (1993), Thaha and Bliimel (1994)). The issue concerns the influence of symmetries and their destruction on the localization length of the quantum kicked rotor. Can these symmetry effects be observed in atomic and molecular physics We think that this issue is important and propose the search for the influence of symmetry on the localization length of dynamically localizing systems as an interesting and important topic for future research. Therefore, the purpose of this section is to sketch briefly the essence of these recent discoveries as an incentive for their application in atomic and molecular physics. 

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... 

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