The FLOQUET Operator of the Kicked Harmonic
Oscillator

Having discussed the properties of the FLOQUET operator from a general point of view, I now proceed to the investigation of the specific ${\hat{U}}$ defined by equations (2.8) and (2.14) with the Hamiltonian (2.5) of the kicked harmonic oscillator.

As its classical counterpart $M$ in equations (1.20), ${\hat{U}}$ can be decomposed into a contribution ${\hat{U}}_{\mbox{\scriptsize kick}}$, describing the kick, and the propagator of the free harmonic oscillator dynamics for time $T$, ${\hat{U}}_{\mbox{\scriptsize free}}$:

$${\hat{U}}\; = \; {\hat{U}}_{\mbox{\scriptsize free}} \, {\hat{U}}_{\mbox{\scriptsize kick}},$$ (2.28)

corresponding to the Hamiltonians

The free propagator -- as many more expressions to follow -- is most conveniently expressed in terms of the ladder operators

which can be used to write

$$\H_{\mbox{\scriptsize free}} \; = \; \hbar \left( {\hat{a}}^\dagger{\hat{a}}+\frac{1}{2} \right)$$ (2.27)

and thus

Note that this means that the free propagator in the harmonic oscillator eigenrepresentation solely depends on $T$ and not on $\hbar$ (naturally it does not depend on $V_0$ either): ${\hat{U}}_{\mbox{\scriptsize free}}={\hat{U}}_{\mbox{\scriptsize free}}(T)$; cf. equation (2.44) below.

The computation of the kick propagator requires slightly more effort because of the explicit time-dependence of the kick part of the Hamiltonian. Dividing the SCHRÖDINGER equation in coordinate representation by the corresponding wave function $\psi(x,t):=\left< x \left\vert \psi(t) \right> \right.$, and integrating over time at the $n$-th kick I
obtain

$$i\hbar \int\limits _{nT-\varepsilon}^{nT+\varepsilon} \fr... ...{nT-\varepsilon}^{nT+\varepsilon} \delta(t-nT) \, {\mbox{d}}t.$$ (2.27)

Taking the limit $\varepsilon\to 0$, the first term on the right hand side vanishes, such that

$$\left\vert \psi(nT+0) \right> \; = \; % \exp \left\{ \frac{... ...frac{i}{\hbar}V_0\cos{\hat{x}}} \left\vert \psi(nT-0) \right>$$ (2.28)

and therefore

In the position representation the kick propagator thus depends on the quotient of $V_0$ and $\hbar$ only: ${\hat{U}}_{\mbox{\scriptsize kick}}={\hat{U}}_{\mbox{\scriptsize kick}}(V_0/\hbar)$.

Summarizing, this gives for the full FLOQUET operator

$${\hat{U}} \; = \; e^{ \textstyle -iT \left( {\hat{a}}^\... ...r}{2}} \left( {\hat{a}}^\dagger+{\hat{a}}\right) \right) },$$ (2.28)

and the full quantum map for the kicked harmonic oscillator is obtained as

$$\hspace*{-0.2cm} \fbox{$ \displaystyle \rule[-0.5cm]{0.0cm... ...} \, % e^{ \T \left\vert \psi_n \right>. \hspace*{0.1cm} $}$$ (2.29)

In the last equation a ``mixed'' notation, using both ${\hat{a}},{\hat{a}}^\dagger$ and ${\hat{x}}$, is used, not only because this is the most concise form of the quantum map, but also because it is in this form that the quantum map (2.37) is evaluated in the following subsection.

Note that equation (2.37) cannot be simplified significantly by restricting the discussion to values of $T$ that satisfy a resonance condition (1.23) as in the classical case (cf. equations (1.25-1.29)). This statement also holds with respect to the explicit expressions for the matrix elements of ${\hat{U}}$ that I derive in subsection 2.1.3; there, $T$ may take any value as well. For the comparison of classical and quantum results on stochastic webs, one has to concentrate on resonant values of $T$. The way in which the choice of $T$ controls the existence of quantum mechanical periodic stochastic webs is discussed in chapter 4; for the discussion of ANDERSON localization in chapter 5, nonresonant values of $T$ are in the focus of attention.