Microwave-driven surface state electrons

What happens if one sprinkles electrons onto the surface of liquid helium Surprisingly the electrons are not absorbed into the bulk of the fluid, but form a quasi-two-dimensional sheet of electrons concentrated at some distance above the helium sm-face. In general, electrons hovering above the surface of a dielectric are called surface state electrons. An excellent review of surface state electrons is that by Cole (1974). 

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Because of the apparent chaos in Fig. 6.5, simple analytical solutions of the driven SSE system probably do not exist, neither for the classical nor for the quantum mechanical problem. Therefore, if we want to investigate the quantum dynamics of the SSE system, powerful numerical schemes have to be devised to solve the time dependent Schrddinger equation of the microwave-driven SSE system. While the integration of classical trajectories is nearly trivial (a simple fourth order Runge-Kutta scheme, e.g., is sufficient), the quantum mechanical treatment of microwave-driven surface state electrons is far from trivial. In the chaotic regime many SSE bound states are strongly coupled, and the existence of the continuum and associated ionization channels poses additional problems. Numerical and approximate analytical solutions of the quantum SSE problem are proposed in the following section. 

Fig. 6.5. Poincare section of a microwave-driven surface state electron in the chaotic regime.

The electron is restricted to move in the half-space x > 0. There is a totally reflecting wall at x = 0. Since the Hamiltonian (8.1.1) of the kicked hydrogen atom and the Hamiltonian of microwave-driven surface state electrons are so similar, we can use many of the results that were derived in Chapter 6. The most important result is the transformation to action and angle variables I and 6, respectively, defined in (6.1.18). The... 

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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