A Picture Book of Quantum Stochastic Webs and Localization

Our interpretation of the experimental material rests essentially on the classical concepts.

NIELS BOHR

In this appendix I present a number of typical examples of quantum stochastic webs and localized quantum dynamics that are generated by the quantum map (2.37). See section 3.3 for a more detailed description of how the quantum states $\left\vert \psi_n \right>$ shown in these pictures are generated. In chapters 4 and 5 some important aspects of the quantum dynamics leading to these $\left\vert \psi_n \right>$ are described and explained.

Having studied the dynamics for several different initial states $\left\vert \psi_0 \right>$ it has turned out that in the resonance cases (1.33) essentially there are just two important types of $\left\vert \psi_0 \right>$: those located in quantum phase space in one of the meshes of the classical stochastic web, and those centered in a stochastic region of the classical web, i.e. in a channel of the web. All other initial states are combinations of these two types, and regardless of the actual position of the ``initial mesh'' or ``initial channel'' chosen for $\left\vert \psi_0 \right>$, the dynamics yield comparable results. Therefore, in the following sections two types of initial states are considered: states centered around the origin $(0,0)^t$ of phase space, i.e. in a mesh, and states centered around one of those intersections of stochastic channels that are closest to the origin, at $(0,p_0)^t$ with suitable $p_0$.

For better comparison, depending on the value of $\hbar$, harmonic oscillator eigenstates

$$\left\vert \psi_0 \right> \; = \; \left\vert m(\hbar) \right>$$ (C.1)

are chosen as initial states in such a way that the corresponding energies all take on the same value, approximately:

$$E_0 \; = \; \hbar\left(m(\hbar)+\frac{1}{2}\right) \; \approx \; \frac{1}{2}$$ (C.2)

(cf. equation (4.4) and the remark following that equation). Similarly, for initial states centered around $(0,p_0)^t$ rather than $(0,0)^t$

$$\left\vert \psi_0 \right> \; = \; {\hat{D}}(0,p_0)\left\vert m(\hbar) \right>,$$ (C.3)

with the translation operator ${\hat{D}}(\cdot,\cdot)$ defined in equation (4.1b) and $m(\hbar)$ according to expression (C.2), is used.

In chapter 3 I have discussed the way in which the size $m_{\mbox{\scriptsize max}}$ of the basis $\{\left\vert m \right> \vert \; 0\leq m\leq m_{\mbox{\scriptsize max}}\}$ used for expanding the quantum states affects the accuracy of the algorithm. Only the phase space region

$$\left\{ (x,p)^t \; \bigg\vert \ \sqrt{x^2+p^2} \; { {\prot... ...iptsize max}}+1) \rule[-0.1cm]{0.0cm}{0.42cm} } \, \right\}$$ (C.4)

can be expected to be well described by this basis. Table C.1 gives $r_{\mbox{\scriptsize max}}(\hbar,m_{\mbox{\scriptsize max}})$ for several values of $\hbar$ and $m_{\mbox{\scriptsize max}}$.

Table: The radius $r_{\mbox{\scriptsize max}}(\hbar,m_{\mbox{\scriptsize max}})$ of the region of phase space that can be described using the basis $\{\left\vert m \right> \vert \; 0\leq m\leq m_{\mbox{\scriptsize max}}\}$, for $m_{\mbox{\scriptsize max}}=3000$ and $m_{\mbox{\scriptsize max}}=6000$.

	$\hbar$		$r_{\mbox{\scriptsize max}}(\hbar,3000)$		$r_{\mbox{\scriptsize max}}(\hbar,6000)$
	1.0		77.5		109.5
	0.1		24.5		34.6
	0.01		7.7		11.0
	0.001		2.5		3.5

The necessity for using as large a value of $m_{\mbox{\scriptsize max}}$ as possible in order to be able do describe a large portion of phase space is obvious. For all but a few calculations described in this study $m_{\mbox{\scriptsize max}}=6000$ has been used. Much larger values of $m_{\mbox{\scriptsize max}}$ are not practical with currently available workstations, as both the execution time per kick and the computer memory needed for storing the FLOQUET matrix elements grow quadratically with $m_{\mbox{\scriptsize max}}$. Note that $m_{\mbox{\scriptsize max}}$ being finite implies that the numerical algorithm cannot yield exactly periodic phase space structures, but at best can approximate them.

For the series of figures shown below, the parameters $\hbar$ and $V_0$ are varied more or less systematically in order to yield states which are as prototypical as possible. The states $\left\vert \psi_n \right>$ obtained in this way are then converted into their corresponding HUSIMI distributions (cf. appendix A). The lines in the following contour plots of HUSIMI distributions are drawn at 10%, 20%, ..., 90%, 99% of the respective maximum values of $F^{\rm H}(x,p,nT-0;1)$ for each state $\left\vert \psi_n \right>$.