The Quantum Skeleton

The classical stochastic webs discussed in chapter 1 are objects within classical phase space, which is spanned by both the position and momentum variables. This is an inherently classical concept, as in quantum mechanics either the position or the momentum representation can be used, but these representations are mutually exclusive. Therefore it is not a priori clear how the classical and quantum results are to be compared.

A solution for this problem of classical-quantum comparison is to consider quantum phase space distribution functions. In appendix A I describe in some detail how such distributions can be defined and used to compare a quantum state with a classical phase portrait. It turns out that many different phase space distributions can be defined, but the most important is the HUSIMI distribution $F^{\rm H}(x,p,t;\zeta)$. In many respects, this quasiprobability distribution function is as close as possible to the (LIOUVILLE) probability distributions in phase space obtained for classical systems. In the same way as $\left< x \left\vert \psi(t) \right> \right.$ in the position representation contains exactly the same information as $\left< p \left\vert \psi(t) \right> \right.$ in the momentum representation, $F^{\rm H}(x,p,t;\zeta)$ gives an equivalent description of the quantum state $\left\vert \psi(t) \right>$. $\zeta\in\mathbb{R}$ is a numerical parameter that can be chosen as desired; here I use $\zeta=1$ alone, in which case the HUSIMI distribution is also known as the coherent state representation. See appendix A for more information on the theory of quantum phase distribution functions.

I now begin the discussion of the numerical methods with the example of $\hbar =1.0$, $V_0=1.0$ and the resonance given by $T=\pi /2$ ($q=4$). This value of $T$ classically leads to rectangular classical stochastic webs as shown in figures 1.7, 1.8, 1.10b and 1.12. The skeletons of the classical stochastic webs for $T=\pi /2$ are given by the square grid (1.45). In the figures of the present section for $T=\pi /2$, this grid is displayed via thin lines, in addition to the contour lines of $F^{\rm H}(x,p,nT-0;1)$.

In figure 4.1,

the result of a numerical simulation is shown for which the GOLDBERG finite differences algorithm of subsection 3.1.2 has been used. In the interval $[-10\pi,10\pi]$, $\left< x \left\vert \psi_n \right> \right.$ is sampled at $j_{\mbox{\scriptsize max}}=16384$ equidistant points, and boundary conditions according to equation (3.24a) have been assumed. Figure 4.2

shows the same as the preceding figure, the difference being that here the dynamics has been obtained using the propagator of the quantum map in the eigenrepresentation of the harmonic oscillator, with $m_{\mbox{\scriptsize max}}=6000$ being the size of the basis used -- see section 3.3. For both figures, the ground state $\left \vert 0 \right >$ of the harmonic oscillator has been chosen as the initial state. In the figures, contour lines of the HUSIMI distributions are drawn at 2.5%, 10%, 20%, ..., 90%, 99% of the respective maximum values of $F^{\rm H}(x,p,nT-0;1)$ for each state.

Using the parameters stated above, the harmonic oscillator representation method runs much faster than the finite differences method: on a Pentium 4 with 2 GHz, the algorithms require approximately 10 sec / 150 sec per iteration of the quantum map, respectively. After a closer inspection of the numerical data it also turns out that the GOLDBERG algorithm produces a faster growing numerical error, manifesting itself in an increasing deviation of the norm of the numerically modelled quantum state from its nominal value of 1. In principle, this error can be controlled by adaptively using smaller time steps $T/s$, but only at considerable numerical cost. The second method also produces a numerical error, but at a distinctly smaller rate: after $n=2000$ kicks, for example, the respective errors of the two runs shown in the figures are of the order of $10^{-3}$ and $10^{-5}$, respectively.

This is reflected in the figures: the initial state $\left \vert 0 \right >$ -- the HUSIMI distribution corresponding to the ground state $\left \vert 0 \right >$ of the harmonic oscillator is a two-dimensional Gaussian in phase space, centered around $(0,0)^t$; see figures 4.1 ($n=0$), 4.2 ($n=0$) and A.1c -- evolves into a web-like structure, as the more exact figure 4.2 nicely shows. The spreading of the wave packet is visible in figure 4.1 as well, but there it takes place much more slowly and in a much less pronounced fashion; the periodic pattern clearly displayed in figure 4.2 evolves much more slowly in figure 4.1, again indicating a numerical error of the GOLDBERG result. This error could be reduced by increasing $j_{\mbox{\scriptsize max}}$ and $s$, and by spending more numerical effort on solving equation (3.21) in order to control cancellations and other numerical errors; but in any case the method would be slowed down further.

This behaviour of the GOLDBERG algorithm, suggesting that a considerably greater numerical effort has to be spent for obtaining sufficiently accurate results, has also been observed in several other calculations performed for other parameter combinations; it seems to be typical for this algorithm when applied to the quantum map of the kicked harmonic oscillator. Consequentially, the GOLDBERG algorithm is not used further on in this study. All the rest of the numerical iterations of the quantum map shown in chapters 4 and 5 and in appendix C have been carried out using the superior algorithm, namely the method using the propagator in the eigenrepresentation of the harmonic oscillator.

If the computation of the quantum analogues of stochastic webs had been the only focus of this work, then using the GOLDBERG algorithm with periodic boundary conditions (3.24b) might have been an alternative approach, potentially faster in obtaining the complete web than the other methods, because periodicity in $x$-direction is already built into the algorithm. But this advantage is compensated by the disadvantage that solving equation (3.21) becomes numerically more difficult for periodic boundary conditions and easily leads to cancellations, overflows etc. In principle, these difficulties can be mastered -- for example by solving equation (3.21) using iterative methods like SOR (successive overrelaxation) [SB00] -- but again only at considerable numerical cost for each iteration of the quantum map.

In addition to the better numerical performance, the harmonic oscillator eigenrepresentation method also has the advantage that the same numerical algorithm can be used for all three relevant cases, i.e. for periodic and aperiodic stochastic webs and for the case of nonresonance. Of these, only the periodic webs allow the application of an algorithm relying on periodic boundary conditions, whereas the method used here can cope with all three cases.

Although this algorithm runs comparably fast, while producing quite accurate results, it should be noted that the computer time needed for these simulations can be quite long. As long as the achieved numerical accuracy allows it, in the following often the long-time dynamics for up to $10^5$ kicks is studied. Typically, on a fast workstation, it takes up to and beyond ten days to complete such a simulation run. Series of simulations of this scope, where the parameters $T$, $\hbar$, $V_0$ each take on several different values, can be performed only if a larger number of fast workstations is available.

In figure 4.2, the dynamics can be iterated up to roughly $10^4$ kicks, before the cut-off error reduces the numerical norm of $\left\vert \psi_n \right>$ too much. For $n=10^4$, the figure indicates that the HUSIMI distribution already exhibits a nearly periodic pattern within the square grid shown. Furthermore, the figure also suggests a 4-fold rotational symmetry, just as for the classical stochastic webs for $T=\pi /2$.^4.1 $F^{\rm H}(x,p,nT-0;1)$ tends to be concentrated in the meshes of the classical web, rather than in the channels, where the -- classical and quantum mechanical -- phase space density gets transported away along the classical skeleton (1.45) rapidly.

Figure 4.2 leads to the conjecture that, as in the classical case, for $n\to\infty$ the web-like structure uniformly extends over the complete phase space and thus establishes (the meshes of) the quantum stochastic web, only the central portion of which is shown in the figure. The classical and the quantum stochastic webs for $T=\pi /2$ seem to be characterized by the same symmetries.

In figures 4.1 and 4.2, an initial state $\left\vert \psi_0 \right>$ corresponding to a HUSIMI distribution centered in a mesh of the stochastic web has been used, leading to a web-like structure which obviously is concentrated in the meshes of the web.^4.2This makes it natural to ask for an initial state that is located somewhere in the channels of the web.

To this end, WEYL's unitary displacement operator, or translation operator,

needs to be considered; the parameters $x',p'\in\mathbb{R}$ are essentially the real and imaginary parts of $\alpha\in\mathbb{C}$:^4.3

$$\alpha \; = \; \alpha(x',p') \; =: \; \frac{1}{\sqrt{2\hbar}} \left( x'+ip' \right).$$ (4.0)

$\hat{D}(x',p')$ acts by translating a state in quantum phase space by $x'$ along the $x$-axis and by $p'$ along the $p$-axis. This is discussed in some more detail in subsection A.3.1 of the appendix.

Using the translation operator, initial states like

$$\left\vert \psi_0 \right> \; = \; {\hat{D}}(0,p_0)\left\vert m \right>$$ (4.1)

can be constructed. While $\left\vert m \right>$ gives a HUSIMI distribution that is an annular structure centered around the origin -- see figures A.2 and A.3 of the appendix -- ${\hat{D}}(0,p_0)\left\vert m \right>$ accordingly gives an annular HUSIMI distribution centered around $(0,p_0)^t$, where $p_0\in\mathbb{R}$ can be chosen as desired. In the present context, either I choose $p_0=0$ in order to obtain an initial state centered in the central mesh of the classical stochastic web, or I choose $p_0\neq 0$ in such a way that the initial state is centered around a point where two channels of the classical web intersect. Note that all states $\hat{D}(x,p)\left\vert 0 \right>$, i.e. the ground state of the harmonic oscillator shifted to $(x,p)^t$ in the phase plane, are called coherent states-- cf. subsection A.3.1 of the appendix.

For the following figures, depending on the value of $\hbar$, $m=m(\hbar)$ in the initial states $\left\vert m(\hbar) \right>$ and ${\hat{D}}(0,p_0)\left\vert m(\hbar) \right>$ is chosen in such a way that the energy of $\left\vert m(\hbar) \right>$ is as close to 1/2 as possible:

$$E_0 \; = \; \hbar\left(m(\hbar)+\frac{1}{2}\right) \; \approx \; \frac{1}{2}.$$ (4.2)

Any other sufficiently large value of $E_0$ could have been chosen as well in (4.4). The point about this expression is that one wants to compare the dynamics -- and in particular, in the next subsection, the behaviour of the energy as a function of time -- corresponding to initial states with similar energies, although the values of $\hbar$ may be different.

This formalism is applied for figure 4.3,

where for $\hbar =0.01$ the initial condition $\left\vert \psi_0 \right>=\hat{D}\big(0,\pi\big)\left\vert 50 \right>$ is used; for $T=\pi /2$, the point $(0,\pi)^t$ marks the intersection of the two stochastic channels right above the origin. In the figure, HUSIMI contour lines are drawn at 10%, 20%, ..., 90%, 99% of the respective maximum values of $F^{\rm H}(x,p,nT;1)$ for each state. Unless stated otherwise, this contour line scheme is applied in all HUSIMI contour plots throughout this study.

Figure 4.3 convincingly shows how the central portions of the stochastic web get filled by the phase space density evolving with time. Iterating for more than the $10^4$ kicks shown in the figure, one would also see that not only the central portion of the channels of the web gets visited by the dynamics, but that later on the phase space density to a larger degree flows into the outer parts of the web, too. (Because of the declining numerical accuracy at large $n$ for fixed $m_{\mbox{\scriptsize max}}$, such figures are not shown here.) Note how closely the quantum web sticks to the square grid marking the skeleton of the classical web. In this sense, the part of phase space covered in figure 4.3 at $n=10^4$ -- plus its periodic continuation along the grid lines -- marks the skeleton of the quantum stochastic web.

It is also interesting to see how in this particular case at least for some time the quantum dynamics quite closely mimics the classical evolution of an ensemble of phase space points: the figure for $n=100$ shows how the quantum distribution is stretched along the skeleton line described by $p(x)=\pi+x$, while being compressed in the direction of $p(x)=\pi-x$. This is just the classical behaviour near the separatrices -- near the stable and unstable manifolds -- of the fixed point $(0,\pi)^t$ (with respect to the mapping $M_4^4$; cf. equation (1.29)). Similarly, the figures for $n=300$, $n=1000$ and $n=3000$ demonstrate the dynamics in the neighbourhood of other classical separatrices: the quantum distribution roughly follows the unstable manifold of a fixed point until it comes close enough to the next fixed point, where it again changes direction, as determined by the respective separatrix. Sections C.1 and C.2 of the appendix contain several additional examples where this behaviour can clearly be recognized; see for example figures C.15, C.17, C.35 and C.36.

Similar HUSIMI contour plots, but for the resonance case $T=\pi /3$ ($q=6$), are displayed in figures 4.4-4.6.

This value of $T$ classically produces hexagonal stochastic webs, one example of which is given in figure 1.3a. The classical skeleton is given by the straight level lines (1.48) of the Hamiltonian (1.47a), the contours of which are plotted in figure 1.10a. In the figures displaying hexagonal quantum stochastic webs, the classical skeleton is plotted using thin lines. According to the figures, the quantum skeleton again develops around these grid lines.

The 6-fold (respectively 2-fold: cf. the footnote on page ) rotational symmetry of the quantum state shown in figure 4.4 becomes clearly visible, once the quantum map is iterated often enough: $n{ {\protect\begin{array}{c} >\protect\\ [-0.3cm]\sim \protect\end{array}} }1000$. The picture for $n=1000$ also gives a good indication already of what the quantum web would look like for $n\to\infty$.

In figures 4.5 and 4.6, in addition to the usual level lines also the 2.5%-contours are plotted. In this way, for a suitably chosen initial state, the central portion of the channels of the web becomes densely filled by level lines after sufficiently large time, thereby exhibiting the stochastic region of the quantum stochastic web more clearly.

For all $n\leq 10^4$ shown, the dynamics in figure 4.5 is obviously still in a transient stage. While for growing $n$ the HUSIMI distributions spread more and more in phase space, different phase space cells are visited one after the other in a way that mimics the corresponding classical dynamics in a periodic web: classically, for $q=6$ the iteration of the web map (1.24) yields sequences of points that, qualitatively speaking, rapidly encircle the origin of phase space, completing approximately one rotation after every six iterations; superimposed to this is the much slower component of the dynamics in radial direction. This quantum-classical mimicry leads to the phase portraits in figure 4.5 being asymmetric with respect to 6-fold rotations around the origin of phase space. For values of $n$ exceeding $10^4$, the phase space distributions can be expected to become more symmetric -- in a way that is similar to the development of the web displayed in the following figure.

Finally, figure 4.6 is remarkable in that it shows the way in which a larger part of the quantum web becomes explored in the course of the dynamics. For $n<10^4$, the phase portraits look similar to those of figure 4.5 and the phase space regions with essentially nonzero $F^{\rm H}(x,p,nT-0;1)$ are still approximately confined to only two meshes of the web at any given time. For larger times, though, $F^{\rm H}(x,p,nT-0;1)$ increasingly becomes distributed over more than just two meshes, and it is natural to assume that, for times larger than $n=10^5$, this process continues, comprising even more -- and finally all -- meshes of the web.

When the quantum map is iterated as often as $n=10^5$, the question of accuracy of the computed states needs to be addressed again. After so many iterations, the numerical $\left\vert \psi_n \right>$ should not be expected to be exactly the state ${\hat{U}}^n\left\vert \psi_0 \right>$. But in the spirit of the Shadowing Lemma of dynamical systems theory [GH83], one may hope that the sequence $\big\{\left\vert \psi_n \right>\}$ the computer finds is nonetheless an approximation to some true quantum dynamics of the system, presumably with respect to a somewhat different initial state $\vert \tilde{\psi}_0 \rangle$. Relying on this heuristic argument, one can more or less safely iterate as long as desired, provided the norms of the computed states do not deviate too much from 1.

Some more contour plots of quantum stochastic webs generated by the quantum kicked harmonic oscillator for $T=\pi /2$ ($q=4$), $T=2\pi /3$ ($q=3$) and $T=2\pi$ ($q=1$) can be found in sections C.1, C.2 and C.3 of the appendix.

The portion of phase space significantly covered by the HUSIMI distribution can be visualized approximately by plotting the 1%-contour lines of $F^{\rm H}(x,p,nT-0;1)$. For $T=\pi /2$, $\hbar =0.1$, $V_0=1.5$ and $p_0=\pi$, this is done in figure 4.7.

In this way, the borders of (the central) channels of the web are outlined more clearly, and the spreading of the state along these channels with increasing $n$ becomes more evident. The following subsection addresses the question how this rather vague notion of spreading can be cast into a more exact, measurable form.

Footnotes

....^4.1

Similar to the patterns in the classical figures, the rotational symmetry exhibited in figure 4.2 is only 2-fold if the shape and the orientation of the distribution contours in the phase space meshes is also taken into account. The skeleton (1.45) describes the 4-fold symmetry pattern, disregarding the shape of the contour lines.

... web.^4.2

Note that in contour plots of HUSIMI distributions showing quantum stochastic webs, typically not all meshes of the web show up with significantly nonzero $F^{\rm H}(x,p,nT-0;1)$. In figures 4.1 and 4.2, for example, the meshes at $((2k+1)\pi,(2l+1)\pi)^t$ with $k,l\in\mathbb{Z}$ appear to be essentially unoccupied by the quantum state even after $10^4$ iterations of the quantum map. A closer inspection of the numerical data reveals that, although generally speaking $F^{\rm H}(x,p,nT-0;1)$ takes on nonzero values in all meshes, the meshes at $((2k+1)\pi,(2l+1)\pi)^t$ are characterized by much smaller values of $F^{\rm H}(x,p,nT-0;1)$ than elsewhere. This is a consequence of the choice of the initial state $\left\vert \psi_0 \right>=\left\vert 0 \right>$. A different initial state, for example $\left\vert \psi_0 \right>=\hat{D}(\pi,\pi)\left\vert 0 \right>$ (with $\hat{D}$ defined in equations (4.1) below), would lead to nonzero $F^{\rm H}(x,p,nT-0;1)$ in the meshes at $((2k+1)\pi,(2l+1)\pi)^t$.

...:^4.3

By equation (4.2), in place of $F^{\rm H}(x,p,t;1)$ one may equivalently write $F^{\rm H}(\alpha,t;1)$; this is used below in subsection 4.2.2, for example.