Chaos threshold

On the basis of the phase-space portraits presented in Fig. 7.5, we conclude that the ionization thresholds displayed in Fig. 7.3 do indeed correlate with the onset of chaos. The equation ionization thresholds = chaos thresholds is therefore justified. 

On summarizing the material presented in this section we obtained two major results, (a) With the help of numerical simulations we proved the equivalence of ionization thresholds and chaos thresholds in the regime uuq < 1 and (b) we formulated an analytically solvable model which predicts threshold fields in close agreement with experimental results. This model, however, should be considered only as a first step for a more complete understanding of the behaviour of the critical fields. For instance, closer inspection of Fig. 7.6 shows that the critical fields predicted by the model have the wrong slope. In the limit u> 0), e.g., the model... 

The quantity R in Eq.(35) should provide a fair estimate of the ratio of classical/quantum rate constants, if the following basic conditions are fuMlled the initial energy s should be above the chaos threshold energy e s > s, and the final energy should be positive, s + Ae >0. Of course, the whole discussion is vahd only if O, the ratio of the driving frequency to the frequency of the vdW moiety, is noticeably larger than one. 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

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.

One of the most basic features of the helium spectrum is its organization into an infinite sequence of ionization thresholds. This feature is not the result of intricate computations. It is already apparent on the level of the independent particle model of the helium atom (see Section 10.1). All predictions on the quantum manifestations of chaos have to... 

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. 

The helium atom is a classically chaotic system. In Chapter 10 we saw that chaos chooses the way of overlapping resonance series to make its presence felt. The existence of overlapping series in the helium atom was confirmed experimentaUy by Domke et al. (1995). Using synchrotron radiation, helium resonances up to the N = 9 threshold were investigated. At the same time good agreement with theoretical complex rota-... 

In 1989, Chao [2] reported that LiLSX (lithium ion-exchanged low silica X zeolite, having a Si/Al ratio close to I.O) showed an unexpected high capacitiy and selectivity for nitrogen over oxygen. He found a Li exchange threshold value in LiNaLSX at about 2/3. Below... 

Situations favourable to the emergence of chaos are those in which a dense manifold of bound Rydberg states is subject to perturbations of a strength comparable to the level spacing. This can occur mainly close to the first ionisation threshold and in the first autoionising range. 

---