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## Translational Invariance of the Quantum Skeleton

One of the defining properties of classical periodic stochastic webs is their translational invariance. Therefore it is natural to look for a similar property in the quantum realm. To this end, the commutation properties of the FLOQUET operator , as given by equations (2.28, 2.32a, 2.35a), with respect to the displacement operator (equations (4.1)) need to be considered.

For the first contribution to , the kick propagator ,

 (4.6)

is obtained, which is a direct consequence of the definition of the translation operator and can be seen by inspection of equations (A.39, A.40a), for example.

Next I consider the propagator which describes the free harmonic oscillator dynamics. At this stage of the present discussion, is not yet restricted to one of the resonance values specified in equation (1.23), but can take on any value. Since brings about clockwise rotation in phase space through the angle , corresponding to the free part of the classical web map (1.20c), the operators and can be interchanged by introducing a new set of parameters which is obtained from by anti-clockwise rotation through :

with the real parameters

-- cf. equation (1.20c). Using the definition

 (4.5)

the new translation operator

and the new in terms of the old is
 (4.5)

Combination of equations (4.10) and (4.11b) with the splitting (2.28) then gives

 (4.6)

for a single iteration of . For iterations of , this implies
 (4.7)

where -- with equations (4.12) -- the real parameters

have been introduced. Expressing in terms of where this seems appropriate, and using equation (4.15), equation (4.17) can also be formulated as

 (4.7)

with
 (4.8)

This expression can now be used to derive conditions that must be satisfied for a commuting with some power of the FLOQUET operator .

Concluding from equation (4.19), the operators and commute if and only if there are integers such that

These conditions need to be analyzed further in order to understand the consequences for the quantum phase portrait, but some consequences can be read off from equations (4.21) directly.

Equation (4.21a) imposes a restriction on the rotational part of the FLOQUET operator and thus leads to rotational symmetries. For general values of , equation (4.21a) is identical with the general classical resonance condition (1.22). For it restricts the values of to

 (4.8)

This means that if the classical resonance condition (1.23) is satisfied then and do commute, provided certain additional conditions, resulting from equation (4.21b), are satisfied as well; these additional conditions are discussed below. In the present chapter, up to this point was just the number of iterations of  being considered. Equations (4.21a) and (4.22) are noteworthy because they link this number of iterations with the parameter of the classical resonance condition.

In the context of equation (4.19), a lucid interpretation of the resonance condition (1.23/4.22) can be given: it refers to symmetries that come about after the successive applications of in equation (4.19) have accounted for exactly one full rotation in phase space. For general values of , belongs to the more general category of resonances as defined by equation (1.22), corresponding to rotations in phase space after applications of . Below I show that it suffices to consider the simplest case in order to discuss and explain the symmetries of the quantum stochastic webs that have been observed in the previous two sections.

Combination of equation (4.21a) with the definition (4.18) for shows that the resonance condition implies

which is another way of expressing that a full rotation in phase space is obtained after not more than iterations of .

Equation (4.21b) determines the allowed translations and therefore concerns the translational symmetries of the phase portrait. Substituting equation (4.21b) into the definitions (4.18),

is obtained; here, is used, which is consistent with equation (4.23a). Equation (4.24a) can be rewritten in the form

which holds for and with

 (4.6)

For even , an additional nontrivial relation follows from equation (4.24a):
 (4.7)

It can be concluded from (4.25a) that, since the are integers, in order to obtain solutions of equation (4.24a) must take on rational values for all .

Note that -- with any , and in fact with any -- the following two assertions are equivalent:

This is a useful observation, since it allows to discuss the rationality of for all by considering just the case of . The equivalence of (4.28a) and (4.28b) can be shown in the following way. Clearly, (4.28b) is implied by (4.28a) with . Conversely, assuming that is rational, the same is true for . Rationality of for all then follows by induction using

 (4.7)

Finally, for general values of one has the result that

 (4.8)

The '' part of (4.30) is trivial; for a proof of the '' part see [CR62].

For simplicity, from here on I consider the case of only. Higher order symmetries that are associated with are thus excluded from the following considerations. On the other hand, the choice is sufficient to discuss and explain the symmetries of the quantum stochastic webs that have been observed in the previous two subsections.

Combining the above arguments (4.25-4.30), it is now shown that if satisfies the classical resonance condition (1.23) with , then integers can in fact be found which satisfy equation (4.25a), and therefore equation (4.24a) as well, provided the are chosen suitably. This being granted, the result

 (4.9)

is established. This means that there exists a complete set of common eigenstates of and , and consequently these eigenstates of are invariant with respect to the translations defined by . Furthermore, the eigenstates of are also periodic in phase space and invariant with respect to rotations through . All eigenstates are extended in phase space. Below, this scenario is analyzed in some more detail for each of the elements of .

In this way the classical resonance condition with is proven to be a necessary condition for the existence of periodic quantum stochastic webs; in this sense it plays the same role both in classical and quantum mechanics. This observation is supported by the fact that the results described above are obtained irrespective of the values of and . As in the classical case, the overall structure of the stochastic web is entirely and solely determined by the parameter .

In order to check that the determined in accordance with equation (4.25a) also satisfy (4.24a) it remains to check that equation (4.25b) is obeyed, too. In particular, it must be confirmed that the determined by equation (4.24a) are indeed integers, as required by equation (4.21b). This cannot be discussed in general terms, but needs to be checked for each individual .

Evaluating equation (4.24a) it turns out -- not surprisingly -- that the cases of , belonging to but not to , are in a characteristic way different from the cases of belonging to . In the following equations, both and are integers that can be chosen as desired.

• :

The symmetric quantum phase space structures obtained for and are identical. The restrictions discussed above imply that the parameters describing the translational symmetries have to be chosen according to

The phase portrait is periodic in -direction with period ; translations by in -direction are admissible with any . This is exactly the symmetry pattern visible in the contour plot 1.9 of the classical time averaged Hamiltonians , . And while it is not confirmed -- due to the limited number of iterations -- by the quantum phase portraits C.38-C.40 for (see section C.3 of the appendix), these figures at least do not stand in contradiction to the symmetries described here.

• :

For these values of , translational invariance is obtained with respect to the translations given by

in agreement with both the skeleton of the classical stochastic webs displayed in figure 1.10a and the quantum phase portraits 4.4-4.6 and C.18-C.37 (in the appendix) for and .

• :

In this case, the translations

are obtained, reproducing the classical square grid of figure 1.10b and of the quantum stochastic webs for in figures 4.2, 4.3, 4.7 and C.1-C.17.

In this way, the symmetries of the quantum stochastic webs that have been obtained in subsection 4.1.1 using numerical means are explained analytically by exploiting the translation invariance of the FLOQUET operator. This explanation of the infinitely extended eigenstates of the system works for all and for all values of and .

Note that this analytical explanation of the quantum skeletons nicely parallels the analytical explanation of the skeletons of classical stochastic webs (see subsection 1.2.2) in the following sense: the rotational and translational symmetries of the quantum webs arise from the combination of and . None of these operators alone gives rise to the symmetries of the webs. The same is true for and with respect to the classical webs -- cf. the footnote on page . In summary, both the classical and quantum stochastic webs are the result of the combination of both indispensable contributions and to the Hamiltonian. In particular, the symmetries of the quantum webs do not reflect any symmetries of the elements of the basis used for expanding the quantum states in the web.

Next: Energy Growth within Quantum Up: An Analytical Explanation of Previous: An Analytical Explanation of   Contents
Martin Engel 2004-01-01