Energy Growth within Quantum Stochastic Webs

The analysis described in the previous section can be extended in order to give a qualitative explanation of the energy growth that has been observed in subsection 4.1.2 as a result of the unbounded quantum dynamics in the channels of the quantum stochastic web. In this way, contact is made with the classical counterpart not only with respect to the symmetries of the phase portrait -- thereby taking a static point of view -- but also with respect to the most important dynamical property of the system.

The existence of a complete set of extended states implies that, with respect to almost all initial states $\left\vert \psi(0) \right>$, unbounded growth of the energy expectation value as a function of time is to be expected.

In order to check the explicit time dependence of this energy growth the commutation properties of different translation operators must be considered. Using the BCH formula

$$e^{\textstyle {\hat{A}}} e^{\textstyle {\hat{B}}} \; = \; e... ...{\hat{B}}} \; e^{\textstyle \frac{1}{2}[{\hat{A}},{\hat{B}}]},$$ (4.7)

which holds for operators ${\hat{A}}$ and ${\hat{B}}$ satisfying $[{\hat{A}},[{\hat{A}},{\hat{B}}]\,]=[{\hat{B}},[{\hat{A}},{\hat{B}}]\,]=0$      [Per93], it is easily confirmed that

$$\left[ {\hat{D}}(x_1',p_1'),{\hat{D}}(x_0',p_0') \right] \... ..._1'p_0'-x_0'p_1'}{2\hbar} \, {\hat{D}}(x_0'+x_1',p_0'+p_1') .$$ (4.8)

This equation is now used to identify those translations defined by equations (4.32-4.34), respectively, that are independent of each other in the sense that they form commutative groups of translations, with all elements satisfying

$$\sin \frac{x_1'p_0'-x_0'p_1'}{2\hbar} \; = \; 0 \qquad \mbox{for all} \quad x_0',p_0',x_1',p_1' .$$ (4.9)

For these groups of commuting translation operators the obvious multiplication rule is used:

$${\hat{D}}(x_1',p_1') \, {\hat{D}}(x_0',p_0') \; = \; {\hat{D}}(x_0'+x_1',p_0'+p_1') .$$ (4.10)

$\bullet$

$q=4$:

For this value of $q$ it follows from equations (4.34) and (4.37) that $[{\hat{D}}(x_1',p_1'),{\hat{D}}(x_0',p_0')]=0$ is equivalent to

$$\sin\left( \frac{2\pi^2}{\hbar}(k_1l_0-k_0l_1) \right) \; = \; 0 .$$ (4.11)

There are two ways for this equality to hold for all $k_0,l_0,k_1,l_1$:

-

First there is the case of

$$k_1l_0 \; = \; k_0l_1 .$$ (4.12)

The translation operators of this type can be organized as a family -- indexed by $r\in\mathbb{Q}\cup\{\infty\}$ -- of commutative one-parameter groups of translations. For $r\in\mathbb{Q}$, these groups are given by

$$\left\{ {\hat{D}}(2\pi k,2\pi rk) \; \Big\vert \; k\in\mathbb{Z} \right\} .$$ (4.13)

All translations in such a group shift along the same direction in the $(x,p)$-plane, defined by the rational gradient

$$r \; = \; \frac{l_0}{k_0} \; = \; \frac{l_1}{k_1} .$$ (4.14)

For $r=\infty$, the group is

$$\left\{ {\hat{D}}(0,2\pi k) \; \Big\vert \; k\in\mathbb{Z} \right\}$$ (4.15)

and consists of vertical translations.

-

Alternatively, equation (4.39) is satisfied for all $k_0,l_0,k_1,l_1$ if there is an $s\in\mathbb{N}$ such that

$$\hbar \; = \; \frac{2\pi}{s},$$ (4.16)

i.e. if $2\pi$ is an integer multiple of $\hbar$. Then the set

$$\left\{ {\hat{D}}(2\pi k,2\pi l) \; \Big\vert \; k,l\in\mathbb{Z} \right\}$$ (4.17)

of all translations satisfying equations (4.34) is in fact a commutative two-parameter group.

If $\hbar$ is given by equation (4.44), then the commuting translation operators in any case form the two-parameter group (4.45), regardless of equation (4.42) being satisfied in addition or not.

$\bullet$

$q=3, 6$:

For these two values of $q$, commuting translation operators are obtained from equations (4.33) and (4.37) if and only if

$$\sin\left( \frac{4\pi^2}{\sqrt{3}\hbar}(k_1l_0-k_0l_1) \right) \; = \; 0 ,$$ (4.18)

which can be satisfied in two ways:

-

Either $k_0,l_0,k_1,l_1$ satisfy equation (4.40), making this case very similar to the corresponding case for $q=4$. Again, the family of Abelian one-parameter groups obtained in this way is indexed by the parameter $r\in\mathbb{Q}\cup\{\infty\}$. For $r\in\mathbb{Q}$, one has the groups

$$\left\{ {\hat{D}}\left(2\pi k,\frac{2\pi}{\sqrt{3}}(2r+1)k\right) \; \bigg\vert \; k\in\mathbb{Z} \right\},$$ (4.19)

and for $r=\infty$, the group of vertical translations is

$$\left\{ {\hat{D}}\left(0,\frac{4\pi}{\sqrt{3}} \, k \right) \; \bigg\vert \; k\in\mathbb{Z} \right\} .$$ (4.20)

-

Equation (4.46) is also satisfied for all $k_0,l_0,k_1,l_1$ if there is an $s\in\mathbb{N}$ such that

$$\hbar \; = \; \frac{4\pi}{\sqrt{3}s} .$$ (4.21)

This being granted, with equations (4.33) the commutative two-parameter group

$$\left\{ {\hat{D}}\left(2\pi k,\frac{2\pi}{\sqrt{3}}(k+2l)\right) \; \bigg\vert \; k,l\in\mathbb{Z} \right\}$$ (4.22)

is obtained, which is similar to the group (4.45) for $q=4$.

$\bullet$

$q=1, 2$:

While the above shows that the cases of $q=3,$ 4 and 6 are essentially equivalent with respect to commutation of translation operators, the situation is substantially different for $q=1$ or $q=2$, because here $p_0'$ and $p_1'$ can take on any real value. I do not discuss these cases any further.

Equations (4.44) and (4.49) represent a new kind of quantum resonance with respect to $\hbar$. It has no classical counterpart and is thus entirely different from the resonance condition (1.23/4.22) that concerns the parameter $T$ and plays quite the same role both classically and quantum mechanically, as discussed earlier. The consequences of the quantum resonances (4.44, 4.49) have been studied numerically in subsection 4.1.2.

In this way, for $q\in\tilde{{\mathcal Q}}$ the existence of the commutative groups (4.41, 4.45, 4.47, 4.50) of translation operators, each commuting with the $q$-th power of the FLOQUET operator, has been established. This is the setting of BLOCH's theorem [Mad78]. The associated energy growth can now be estimated qualitatively as follows.

The first step is to express the energy expectation value in terms of the HUSIMI distribution function $F^{\rm H}(x,p,t;1)$ corresponding to the state $\left\vert \psi(t) \right>$. With equation (2.31) and the overcompleteness relation (A.52) of the coherent states I obtain for all $t$:

Integrals of this kind can be approximately evaluated by considering the phase space region ${\mathcal A}(t)$ significantly occupied by the HUSIMI distribution at time $t$. From the normalization property (A.53) of $F^{\rm H}(x,p,t;1)$,

$$% \int\!\! \dop x\!\! \int\!\! \dop p\, \int\limits _{-\in... ...fty}^\infty\!\! {\mbox{d}}p\, F^{\rm H}(x,p,t;1) \; = \; 1 ,$$ (4.22)

which is easily confirmed using the overcompleteness relation (A.52),

$$\int\!\!\!\int\limits _{{\!\!\!\!\mathcal A}(t)} \! {\mbox{d}}x\, {\mbox{d}}p \, F^{\rm H}(x,p,t;1) \; \approx \; 1$$ (4.23)

is obtained, making it possible to write

$$\left\vert {\mathcal A}(t) \right\vert \; \approx \; \frac{1}{\left< F^{\rm H} \right>_t} ,$$ (4.24)

with $\left< F^{\rm H} \right>_t$ being the average value of $F^{\rm H}(x,p,t;1)$ in ${\mathcal A}(t)$. For sufficiently large $t$, and on the assumption that $F^{\rm H}$ covers ${\mathcal A}(t)$ more or less uniformly,^4.4 $\left< E \right>_t$ can now be approximated as

where in the intermediate step it has also been assumed that on the average ${\mathcal A}(t)$ grows isotropically in the phase plane -- a behaviour that is typical for the dynamics with $q\in\tilde{{\mathcal Q}}$: cf. figures 4.2, 4.4 and 4.6 (this behaviour is in contrast to the nonisotropic growth of ${\mathcal A}(t)$ for $q=1, 2$, examples of which are shown in figures C.38-C.40 in appendix C).

From here on the discussion has to distinguish between the cases of the one-parameter groups (4.41, 4.47) and the two-parameter groups (4.45, 4.50) of commuting translation operators. In the first case, there exists a complete set of common eigenstates $\big\vert \tilde{\phi}_m(\tilde{k},t \big>$ of ${\hat{U}}^q$ and, for example, ${\hat{D}}(2\pi,2\pi r)$ or ${\hat{D}}\left(2\pi,2\pi(2r+1)/\sqrt{3}\right)$; the indices $m\in\mathbb{N}$, $\tilde{k}\in\mathbb{R}$ label the eigenstates of ${\hat{U}}^q$ and ${\hat{D}}(\cdot,\cdot)$, respectively. These states share the most important properties of the quasienergy states (2.20). In particular, they can be written as

analogous to the $\left\vert u_E(t) \right>$ of equation (2.23). The phase in the exponential in the full states (4.56a) is essentially determined by $F_m(\tilde{k})$ which parallels the quasienergy $E$ in the states (2.20). Due to the completeness of $\big\{ \big\vert \tilde{\phi}_m(\tilde{k},0) \big> \, \big\vert \, m\in\mathbb{N}, \, \tilde{k}\in\mathbb{R}\big\}$, any initial state $\left\vert \psi(0) \right>$ can be expanded as

$$\left\vert \psi(0) \right> \; = \; \sum_m \int\!\! {\mbox{... ...m(\tilde{k}) \; \big\vert \tilde{\phi}_m(\tilde{k},0) \big> ,$$ (4.23)

with suitable coefficients $A_m(\tilde{k})\in\mathbb{C}$, and by equation (4.56a) evolves according to

$$\left\vert \psi(t) \right> \; = \; \sum_m \int\!\! {\mbox{... ..._m(\tilde{k})t} \, \big\vert \tilde{u}_m(\tilde{k},t) \big> .$$ (4.24)

The HUSIMI distribution at stroboscopic times $nT$ is then obtained as^4.5

$$F^{\rm H}(x,p,nT;1) = \! \frac{1}{2\pi\hbar} \; \Big\vert \left< \alpha(x,p) \left\vert \psi(nT) \right> \right. \Big\vert^2$$

$$= \! \frac{1}{2\pi\hbar} \; \left\vert \, \sum_m \int\!\! {\mbox{d}}\t... ..., \big< \alpha(x,p) \big\vert \tilde{u}_m(\tilde{k},0) \big> \, \right\vert^2 .$$ (4.25)

For sufficiently large $n$, the exponential in equation (4.59) becomes a rapidly oscillating function of $\tilde{k}$, such that the integral can be evaluated using a stationary phase argument [JJ72]. Assuming without loss of generality that, for each $m$, $F_m(\tilde{k})$ has a single critical point at $\tilde{k}_{0,m}$, I have:

$$F^{\rm H}(x,p,nT;1) \approx \frac{1}{nT} \, \Bigg\vert \, \sum_m A_m(\tilde{k}_{0,m}) \; \big< \alpha(x,p) \big\vert \tilde{u}_m(\tilde{k}_{0,m},0) \big>$$

$$\hspace*{3.5cm} \times \; \frac{e^{\textstyle -\frac{i}{\hbar}F_m... ...u_m } } {\sqrt{F_m''(\tilde{k}_{0,m})}} \, \Bigg\vert^2 , % , \quad n\to\infty$$ (4.26)

where $\nu_m\!=\pm 1$ for $F_m''(\tilde{k}_{0,m})\!\! {\protect\begin{array}{c} <\protect\\ [-0.3cm]> \protect\end{array}} \!0$. Initial states $\left\vert \psi(0) \right>$ with $A_m(\tilde{k}_{0,m})=0$ can be excluded on generic grounds, but even in such cases comparable results may be found [Olv74]. Similarly, the method can be generalized to cover situations with $F_m''(\tilde{k}_{0,m})=0$ as well. The important point about equation (4.60) is that it indicates that the HUSIMI distribution at a given phase space point $(x,p)^t$ asymptotically decays like

$$F^{\rm H}(x,p,nT;1) \; \sim \; \frac{1}{n}$$ (4.27)

with time $nT$. This in turn, together with equation (4.55b), gives

$$\left< E \right>_n \; \sim \; n ,$$ (4.28)

that has already been found numerically -- cf. equation (4.6). In this way, asymptotically unbounded energy growth has been proven, and this energy growth follows a linear, diffusive pattern indeed. This is the generic result, which holds for most values of $\hbar$ and is demonstrated for example in figures 4.8a and 4.9a in subsection 4.1.2. The other, exceptional cases are discussed in the following.

For the two-parameter groups of commuting translation operators in the cases of the $\hbar$-resonances (4.44) and (4.49), the above reasoning can be repeated in a similar, but not identical, fashion. The differences in some details account for a result that is remarkably different from equation (4.62).

With the two-parameter groups (4.45, 4.50), there is a complete set of common eigenstates $\big\vert \tilde{\phi}_m(\tilde{k},\tilde{l},t \big>$ of ${\hat{U}}^q$ and, for example, ${\hat{D}}(2\pi,0)$ and ${\hat{D}}(0,2\pi)$ (or  ${\hat{D}}\left(0 ,4\pi/\sqrt{3}\right)$ and ${\hat{D}}\left(2\pi,2\pi/\sqrt{3}\right)$); corresponding to the two group parameters there are now two indices $\tilde{k},\tilde{l}\in\mathbb{R}$ in addition to $m$. Parallel to equations (4.56) I now have

The initial state

$$\left\vert \psi(0) \right> \; = \; \sum_m \int\!\! {\mbox{... ...}) \; \big\vert \tilde{\phi}_m(\tilde{k},\tilde{l},0) \big> ,$$ (4.28)

with coefficients $A_m(\tilde{k},\tilde{l})\in\mathbb{C}$, becomes

$$\left\vert \psi(t) \right> \; = \; \sum_m \int\!\! {\mbox{... ...e{l})t} \, \big\vert \tilde{u}_m(\tilde{k},\tilde{l},t) \big>$$ (4.29)

after time $t$ and gives the HUSIMI distribution

$$F^{\rm H}(x,p,nT;1) = \frac{1}{2\pi\hbar} \; \Bigg\vert \, \sum_m \int\!\! {\mbox{d}}\t... ...ilde{k},\tilde{l}) \; e^{\textstyle -\frac{i}{\hbar}F_m(\tilde{k},\tilde{l})nT}$$

$$\hspace*{4.4cm} \times \; \big< \alpha(x,p) \big\vert \tilde{u}_m(\tilde{k},\tilde{l},0) \big> \, \Bigg\vert^2 .$$ (4.30)

As in equation (4.60), this integral is evaluated using a stationary phase argument. For each $m$, it is assumed without loss of generality that $F_m(\tilde{k},\tilde{l})$ has a single critical point at $(\tilde{k}_{0,m},\tilde{l}_{0,m})$ and that the Hessian at this point is either negative or positive definite, such that $\nu_m=\pm 1$ for negative/positive definite $(\mbox{Hess}\, F_m)(\tilde{k}_{0,m},\tilde{l}_{0,m})$ can be defined. Then I obtain for large enough $n$:

$$F^{\rm H}(x,p,nT;1) \approx \frac{2\pi}{n^2T} \, \Bigg\vert \, \sum_m A_m(\tilde{k}_{0,m},\ti... ...big< \alpha(x,p) \big\vert \tilde{u}_m(\tilde{k}_{0,m},\tilde{l}_{0,m},0) \big>$$

$$\hspace*{2.7cm} \times \; \frac{e^{\textstyle -\frac{i}{\hbar} F_... ...rt{\det(\mbox{Hess}\, F_m)(\tilde{k}_{0,m},\tilde{l}_{0,m})}} \; \Bigg\vert^2 .$$ (4.31)

In this way, as a result of the two-dimensional integration, the HUSIMI distribution scales like

$$F^{\rm H}(x,p,nT;1) \; \sim \; \frac{1}{n^2} ,$$ (4.32)

as opposed to the previous result (4.61). This, with equation (4.55b), implies

$$\left< E \right>_n \; \sim \; n^2,$$ (4.33)

thereby explaining the asymptotically quadratic, ballistic energy growth in the resonance cases (4.44) and (4.49) that is demonstrated in figures 4.8a and 4.9a and in equation (4.8).

The above explanation for the energy growth relies on the possibility to expand the states as in equations (4.57) and (4.64), and on the applicability of the stationary phase approximation. In particular the expansions (4.57, 4.64) are somewhat questionable; their -- at least approximate -- validity depends on details of the spectral properties of the respective operators. Nevertheless, the arguments based on these assumptions yield suggestive results, in accordance with the numerical findings in section 4.1.2. These points are discussed in some more detail in [GB93,BR95].

This chapter provides a description of the typical quantum dynamics of the kicked harmonic oscillator in the cases of resonance with $q\in {\mathcal Q}$. It is natural to ask in which way the quantum dynamics in the complementary cases of nonresonance -- where there are no phase space structures characterized by combined translational and rotational symmetries -- differs from the scenario of the present chapter. It might be conjectured that in the absence of resonance, without the condition (1.23/4.22) enforcing the existence of infinitely extended quantum states, the typical quantum dynamics is characterized by some kind of localization phenomenon. That conjecture can be considered to be motivated by the multitude of localization results that have been obtained for the quantum kicked rotor, for which the absence of a classical resonance condition like (1.23) is an essential feature. This question is addressed in the following chapter 5, where the localization approach to the dynamics is taken.

Footnotes

... uniformly,^4.4

As the contour plots in subsection 4.1.1 and in appendix C show, this assumption is somewhat ill-justified, but at least within the family $F^f$ of phase space distributions discussed in appendix A, $F^{\rm H}$ is the smoothest (cf. equation (A.92)) and thus comes closest to satisfying this assumption. What is more, for the estimate (4.55a) it is also used that the HUSIMI distribution -- by equations (A.86) -- is non-negative. Table A.1 shows that this advantageous feature is a unique property of $F^{\rm H}$ (and thus of $F^{\rm AN}$) within the family $F^f$

These observations, together with the fact that the conclusions obtained on the basis of the assumptions leading to the approximation (4.55a) obviously agree with the numerical results of section 4.1.2 (this is discussed on pages ff below), further explain the importance of the HUSIMI distribution for the analysis of the quantum dynamics of classically chaotic systems.

... as^4.5

Due to the approximative nature of the derivations in this subsection it is not necessary here to consider, more exactly, the stroboscopic times $nT-0$ just before the $n$-th kick, rather than $nT$.