Time integration scheme, continuum

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 measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

Figure 3.14 Left panel Helium level scheme and the relevant couplings. The dressing laser (>-5 = 51 064 nm) couples the excited state 4s 5o to the ionization continuum. The induced continuum structure is probed (at>. = 5294 nm) through ionization of the 2s So state. Right panel Variations of the ionization cross section probed with a weak probe pulse, the frequency of which is tuned across the two-photon resonance between the 2s So and 4s 5q states. The axis of the probe laser beam (diameter 0.5 mm) coincides with the axis of the dressing laser beam (diameter 3.5 mm). There is no time delay between the pulses. The laser intensities are Ip = 4 MW/cm and Ip, = 75 MW/cm. The experimental profile (dots) is in good agreement with the result from numerical studies (dotted line), which include averaging over fluctuating laser intensities and integrating over the spatial profile of the probe laser. Taken from Ref. [69].

The complete continuum approach was employed in the Kirkwood model on an ab initio level with the basis set of the floating Gauss functions in 1976 [17]. Around that time, a similar formalism for taking the solvent into account was included in the CNDO/2 method [18]. However, such calculational schemes did not gain wide acceptance by reason of excessive expenditure of computer time, difficulties in evaluating some integrals and overall drawbacks inherent in the macroscopic approximation. Eventually some simplified techniques were developed, each of which takes usually one of the components in Eq. (3.8) into account. Next the simplest of these will be considered. 

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