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 ${\hat{U}}$, as given by equations (2.28, 2.32a, 2.35a), with respect to the displacement operator $\hat{D}(\cdot)$ (equations (4.1)) need to be considered.

For the first contribution to ${\hat{U}}$, the kick propagator ${\hat{U}}_{\mbox{\scriptsize kick}}=e^{\textstyle -\frac{i}{\hbar}V_0\cos{\hat{x}}}$,

$$\hat{D}(x',p') \, e^{\textstyle -\frac{i}{\hbar}V_0\cos{\h... ...yle -\frac{i}{\hbar}V_0\cos({\hat{x}}-x') } \, \hat{D}(x',p')$$ (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 ${\hat{U}}_{\mbox{\scriptsize free}}={\hat{U}}_{\mbox{\scriptsize free}}(T)$ which describes the free harmonic oscillator dynamics. At this stage of the present discussion, $T$ is not yet restricted to one of the resonance values specified in equation (1.23), but can take on any value. Since ${\hat{U}}_{\mbox{\scriptsize free}}$ brings about clockwise rotation in phase space through the angle $T$, corresponding to the free part $M_{\mbox{\scriptsize free}}(T)$ of the classical web map (1.20c), the operators $\hat{D}(x',p')$ and ${\hat{U}}_{\mbox{\scriptsize free}}$ can be interchanged by introducing a new set $\tilde{x}',\tilde{p}'$ of parameters which is obtained from $x',p'$ by anti-clockwise rotation through $T$:

with the real parameters

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

$$\tilde{\alpha}(\tilde{x}',\tilde{p}') \; = \; \frac{1}{\sqrt{2\hbar}} \left( \tilde{x}'+i\tilde{p}' \right) ,$$ (4.5)

the new translation operator

and the new $\tilde{\alpha}$ in terms of the old $\alpha$ is

$$\tilde{\alpha} \; = \; \frac{1}{\sqrt{2\hbar}} \left( x'+... ...ght) e^{\textstyle iT} \; = \; \alpha \, e^{\textstyle iT} .$$ (4.5)

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

$$\hat{D}(x',p') \, {\hat{U}} = \hat{D}(x',p') \, {\hat{U}}_{\mbox{\scriptsize free}} \, {\hat{U}}_{\mbox{\scriptsize kick}}$$

$$= {\hat{U}}_{\mbox{\scriptsize free}} \, \hat{D}(\tilde{x}',\tilde{p}') \, e^{\textstyle -\frac{i}{\hbar}V_0\cos{\hat{x}}}$$

$$= {\hat{U}}_{\mbox{\scriptsize free}} \, e^{\textstyle -\frac{i}{\hbar}V_0\cos({\hat{x}}-\tilde{x}') } \, \hat{D}(\tilde{x}',\tilde{p}') .$$ (4.6)

for a single iteration of ${\hat{U}}$. For $q\in\mathbb{N}$ iterations of ${\hat{U}}$, this implies

$$\hat{D}(x_0',p_0') \, {\hat{U}}^q = {\hat{U}}_{\mbox{\scriptsize free}} \, e^{\textstyle -\frac{i}{\hbar}V_0\cos({\hat{x}}-x_1') } \, \hat{D}(x_1',p_1') \, {\hat{U}}^{q-1}$$

$$= \cdots$$

$$= \prod_{j=1}^q \left\{ \, {\hat{U}}_{\mbox{\scriptsize free}} \, e... ...tyle -\frac{i}{\hbar}V_0\cos({\hat{x}}-x_j') } \, \right\} \hat{D}(x_q',p_q') ,$$ (4.7)

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

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

$$\hat{D}(\alpha_0) \, \left\{ \, {\hat{U}}_{\mbox{\scriptsi... ...right\} \hat{D} \!\left(\alpha_0 \, e^{\textstyle iqT}\right)$$

(4.7)

with

$$\alpha_0 \; =\; \frac{1}{\sqrt{2\hbar}} \, (x_0'+ip_0').$$ (4.8)

This expression can now be used to derive conditions that must be satisfied for a $\hat{D}(\alpha_0)$ commuting with some power $q$ of the FLOQUET operator ${\hat{U}}$.

Concluding from equation (4.19), the operators $\hat{D}(\alpha_0)$ and ${\hat{U}}^q$ commute if and only if there are integers $k_1,\dots,k_q,l\in\mathbb{Z}$ 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 ${\hat{U}}_{\mbox{\scriptsize free}}$ of the FLOQUET operator and thus leads to rotational symmetries. For general values of $l$, equation (4.21a) is identical with the general classical resonance condition (1.22). For $l=1$ it restricts the values of $T$ to

$$T \; = \; \frac{2\pi}{q} .$$ (4.8)

This means that if the classical resonance condition (1.23) is satisfied then $\hat{D}(\alpha_0)$ and ${\hat{U}}^q$ 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 $q$ was just the number of iterations of ${\hat{U}}$ being considered. Equations (4.21a) and (4.22) are noteworthy because they link this number of iterations with the parameter $q$ 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 $q$ successive applications of ${\hat{U}}_{\mbox{\scriptsize free}}$ in equation (4.19) have accounted for exactly one full rotation in phase space. For general values of $l$, $T$ belongs to the more general category of resonances as defined by equation (1.22), corresponding to $l$ rotations in phase space after $q$ applications of ${\hat{U}}_{\mbox{\scriptsize free}}$. Below I show that it suffices to consider the simplest case $l=1$ 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 $j=q$ 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 $q$ iterations of ${\hat{U}}$.

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, $k_0:=k_q$ is used, which is consistent with equation (4.23a). Equation (4.24a) can be rewritten in the form

which holds for $q\geq 3$ and $j=1,2,\dots,j_{\mbox{\scriptsize max}}$ with

$$j_{\mbox{\scriptsize max}} = \left\{ \begin{array}{ccl} \... ...playstyle \frac{q-1}{2} & & \mbox{odd} . \end{array} \right.$$ (4.6)

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

$$k_{\textstyle \frac{q}{2}} \; = \; k_0 \, (-1)^l .$$ (4.7)

It can be concluded from (4.25a) that, since the $k_j$ are integers, in order to obtain solutions of equation (4.24a) $\cos2\pi jl/q$ must take on rational values for all $1\leq j\leq j_{\mbox{\scriptsize max}}$.

Note that -- with any $z\in\mathbb{R}$, and in fact with any $j_{\mbox{\scriptsize max}}\in\mathbb{N}$ -- the following two assertions are equivalent:

This is a useful observation, since it allows to discuss the rationality of $\cos2\pi jl/q$ for all $j$ by considering just the case of $j=l=1$. The equivalence of (4.28a) and (4.28b) can be shown in the following way. Clearly, (4.28b) is implied by (4.28a) with $j=1$. Conversely, assuming that $\cos z$ is rational, the same is true for $\cos 2z=2\cos^2\!z-1$. Rationality of $\cos jz$ for all $j\geq 3$ then follows by induction using

$$\cos jz \; = \; 2\, \cos (j-1)z \, \cos z - \cos(j-2)z .$$ (4.7)

Finally, for general values of $q\in\mathbb{N}$ one has the result that

$$\cos\frac{2\pi}{q}\in\mathbb{Q} \quad \Leftrightarrow \quad q \in{\mathcal Q}.$$ (4.8)

The ``$\Leftarrow$'' part of (4.30) is trivial; for a proof of the ``$\Rightarrow$'' part see [CR62].

For simplicity, from here on I consider the case of $l=1$ only. Higher order symmetries that are associated with $l\neq 1$ are thus excluded from the following considerations. On the other hand, the choice $l=1$ 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 $T$ satisfies the classical resonance condition (1.23) with $q\in {\mathcal Q}$, then integers $k_j$ can in fact be found which satisfy equation (4.25a), and therefore equation (4.24a) as well, provided the $k_0,p_0'$ are chosen suitably. This being granted, the result

$$\left[ \hat{D}(x_0',p_0'), \, {\hat{U}}^q \right] \; = \; 0$$ (4.9)

is established. This means that there exists a complete set of common eigenstates of $\hat{D}(x_0',p_0')$ and ${\hat{U}}^q$, and consequently these eigenstates of ${\hat{U}}^q$ are invariant with respect to the translations defined by $x_0',p_0'$. Furthermore, the eigenstates of ${\hat{U}}$ are also periodic in phase space and invariant with respect to rotations through $T=2\pi/q$. All eigenstates are extended in phase space. Below, this scenario is analyzed in some more detail for each of the elements $q$ of ${\mathcal Q}$.

In this way the classical resonance condition with $q\in {\mathcal Q}$ 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 $V_0$ and $\hbar$. As in the classical case, the overall structure of the stochastic web is entirely and solely determined by the parameter $T$.

In order to check that the $k_j$ 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 $k_j$ 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 $q\in {\mathcal Q}$.

Evaluating equation (4.24a) it turns out -- not surprisingly -- that the cases of $q=1, 2$, belonging to ${\mathcal Q}$ but not to $\tilde{{\mathcal Q}}$, are in a characteristic way different from the cases of $q=3,4,6$ belonging to $\tilde{{\mathcal Q}}$. In the following equations, both $k_0$ and $l_0$ are integers that can be chosen as desired.

• $q=1, 2$:

  The symmetric quantum phase space structures obtained for $q=1$ and $q=2$ are identical. The restrictions discussed above imply that the parameters $x_0',p_0'$ describing the translational symmetries have to be chosen according to
  
  The phase portrait is periodic in $x$-direction with period $2\pi$; translations by $p_0'$ in $p$-direction are admissible with any $p_0'$. This is exactly the symmetry pattern visible in the contour plot 1.9 of the classical time averaged Hamiltonians ${\mathcal H}_1$, ${\mathcal H}_2$. And while it is not confirmed -- due to the limited number of iterations -- by the quantum phase portraits C.38-C.40 for $q=1$ (see section C.3 of the appendix), these figures at least do not stand in contradiction to the symmetries described here.
  
  

• $q=3, 6$:

  For these values of $q$, 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 $q=3$ and $q=6$.

• $q=4$:

  In this case, the translations
  
  are obtained, reproducing the classical square grid of figure 1.10b and of the quantum stochastic webs for $q=4$ 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 $q\in {\mathcal Q}$ and for all values of $V_0$ and $\hbar$.

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 ${\hat{U}}_{\mbox{\scriptsize kick}}$ and ${\hat{U}}_{\mbox{\scriptsize free}}$. None of these operators alone gives rise to the symmetries of the webs. The same is true for $M_{\mbox{\scriptsize kick}}$ and $M_{\mbox{\scriptsize free}}$ 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 $H_{\mbox{\scriptsize kick}}$ and $\H_{\mbox{\scriptsize free}}$ to the Hamiltonian. In particular, the symmetries of the quantum webs do not reflect any symmetries of the elements of the basis $\big\{ \! \left\vert m \right> \! \big\}$ used for expanding the quantum states in the web.