Microwave chaos

Berggren, K.-F., and A.F. Sadreev. Chaos in quantum billiards and similarities with pure-tone random models in acoustics, microwave cavities and electric networks. Mathematical modelling in physics, engineering and cognitive sciences. Proc. of the conf. Mathematical Modelling of Wave Phenomena , 7 229, 2002. 

Figure 1. Comparison at identical parameter values of experimental and quantum-mechanical values for the microwave field strength for 10% ionization probability as a function of microwave frequency. The field and frequency are classically scaled, u>o = and = q6, where no is the initially excited state. Ionization includes excitation to states with n above nc. The theoretical points are shown as solid triangles. The dashed curve is drawn through the entire experimental data set. Values of no, nc are 64, 114 (filled circles) 68, 114 (crosses) 76, 114 (filled squares) 80, 120 (open squares) 86, 130 (triangles) 94, 130 (pluses) and 98, 130 (diamonds). Multiple theoretical values at the same uq are for different compensating experimental choices of no and a. The dotted curve is the classical chaos border. The solid line is the quantum 10% threshold according to localization theory for the present experimental conditions.

Considering the sensitivity of classical chaotic systems to external perturbations, and the ubiquitous nature of chaotic dynamics in larger systems, it is important to 1 establish that quantum mechanics allows for control in chaotic systems as well. [ One simple molecular system that displays quantirm chaos is the rotational exci- tation of a diatomic molecule using pulsed microwave radiation [227], Under the conditions adopted below, this system is a molecular analog of the delta-lacked ij rotor, that is, a rotor that is periodically lacked by a delta fiinction potential, which 4 is a paradigm for chaotic dynamics [228, 229], The observed energy absorption of such systems is called quantum chaotic diffusion. 

Since the SSE system by itself is essentially one-dimensional and autonomous, there is no chaos in this system. Even exposing the surface state electrons to weak microwave radiation does not change the situation. As a matter of fact, exposure to weak microwave fields was discussed above in connection with the spectroscopy experiments by Grimes and Brown (1974). In these experiments the microwave fields were defiber-ately chosen to be weak in order not to disturb the SSE system too much. Weak-field irradiation does not produce chaos, but results in regular absorption fines. 

The situation changes drastically, however, if we irradiate the surface state electrons with a sufficiently strong microwave field. Strong fields change the physics of the SSE system profoundly, eventually driving it into chaos. That chaos can indeed occur in the SSE system was first demonstrated by Jensen in 1982. The system he proposed is shown in Fig. 6.1(b). It is an extension of the system shown in Fig. 6.1(a). A microwave field is applied perpendicular to the helium surface S such that the field direction is parallel to the x direction. The resulting classical Hamiltonian of the combined SSE plus microwave field can be written as... 

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. 

The data shown in Fig. 6.9 and Fig. 6.10 confirm our suspicion that for weak microwave fields no chaos mechanisms have to be invoked for an adequate physical understanding of microwave ionization data. The situation, however, is quite different in the case of strong microwave fields. In this case the ionization routes are very comphcated, and the multiphoton pictmre loses its attractiveness. It has to be replaced by a picture based on chaos. Chaos provides a simpler description of the ionization process and consequently a better physical insight. The discussion of the chaotic strong-field regime is the topic of the following section. 

In Section 11.1 we discuss recent advances in quantum chaology, i.e. the semiclassical basis for the analysis of atomic and molecular spectra in the classically chaotic regime. In Section 11.2 we discuss some recent results in type II quantum chaos within the framework of the dynamic Born-Oppenheimer approximation. Recent experimental and theoretical results of the hydrogen atom in strong microwave and magnetic fields are presented in Sections 11.3 and 11.4, respectively. We conclude this chapter with a brief review of the current status of research on chaos in the helium atom. 

Bayfield and Koch (1974) provided the first experimental results on a manifestly quantum, but classically chaotic, system hydrogen Rydberg atoms in a strong microwave field. Both pioneers, Bayfield at Pittsburgh and Koch at Stony Brook, continue to contribute actively to the investigation of time dependent chaos in Rydberg atoms. 

While for a long time microwave ionization experiments addressed the linear polarization (LP) case only, experimental results on elliptic polarization (EP) are now available from Stony Brook (Koch and van Leeuwen (1995), Bellermann et al. (1996)). According to a widely used rule-of-thumb, EP ionization thresholds are expected to be higher than LP ionization thresholds. Bellermann et al. have shown that this is not generally the case. Bellermann et al. also provide experimental evidence for the importance of classical phase-space structures in the EP case. The EP case adds a new dimension to the microwave ionization problem. It provides an additional testing ground for the manifestations of chaos in atomic physics. 

Koch, P.M., Moorman, L. and Sauer, B.E. (1992). Microwave ionization of excited hydrogen atoms experiments versus theories for high scaled fi equencies, in Irregular Atomic Systems and Quantum Chaos, ed. J.-C. Gay (Gordon and Breach, Philadelphia). 

Kudrolli, A., Sridhar, S., Pandey, A. and Ramaswamy, R. (1994). Signatures of chaos in quantum billiards Microwave experiments, Phys. Rev. E49, R11-R14. 

Stockmann, H.-J. and Stein, J. (1990). Quantmn chaos in billiards studied by microwave absorption, Phys. Rev. Lett. 64, 2215-2218. 

The second class of atomic systems studied in the search for manifestations of chaos consists of time-dependent Hamiltonian systems such as one-electron atoms in an oscillating field. The hydrogen atom in a microwave or laser field is the standard physical example and has been a focus of attention since the ionization of highly excited hydrogen atoms by intense microwave fields was first observed by Bayfield and Koch in 1974 [10]. 

Noerochim, L., Wang, J. Z., Wexler, D., Chao, Z., and Liu, H. K. (2013). Rapid synthesis of free-standing MoOs/Graphene films by the microwave hydrothermal method as cathode for bendable lithium batteries,/ Power Sources, 228, pp. 198-205. 

Tsai SH, Chao CW, Lee CL, Shih HC. Bias-enhanced nucleation and growth of the aligned carbon nanotubes with open ends under microwave plasma synthesis. Appl Phys Lett 1999 74 3462. ... 

Chen J, Li Z, Chao D, Zhang W, Wang C (2008) Synthesis of size-tunable metal nanopeirticles based on polyacrylonitrile nanofibers enabled by electrospinning and microwave irradiation. Mater Lett 62(4-5) 692- 94. doi 10.1016/j.matlet.2007.06.047... 

Cheng Z L, Chao Z S and Wan H L (2002), Research of A type zeolite as well as zeolite membrane by microwave heating . Chin J Inorg Chem, 18,528-532. 

---