The Kicked Rotor

The kicked rotor is one of the most frequently studied model systems in dynamical systems theory. It emerges in many physical systems (see for example [Zas85,MRB⁺95]), and despite being very simple it has successfully been used for modelling the onset of chaos, retaining many of the typical and complex features of the underlying physical system.

After suitable scaling, leading to dimensionless variables, the kicked rotor can be defined by the Hamiltonian

$$H_{\rm kr}(\vartheta,I,t) \: = \; % \frac{1}{2}I^2 - V_0\c... ...ac{1}{2}I^2 + V(\vartheta) \sum_{n=-\infty}^\infty \delta(t-n)$$ (5.1)

with the angular displacement $\vartheta$ and the angular momentum $I$ conjugate to $\vartheta$. In the present subsection I choose the kick potential in the conventional way,

$$V(\vartheta) \; = \; -V_0\cos\vartheta,$$ (5.2)

such that $H_{\rm kr}$ models, for example, a mathematical pendulum that is driven by ``impulsively acting gravity'' [CCIF79,LL92]. The first summand $I^2/2$ of $H_{\rm kr}$ describes free rotation; the second specifies the periodic ``gravitational'' kicks the strength of which depends on the kick function. See figure 5.1 for a schematic illustration.

In subsection 5.1.3 other kick functions are considered as well.

Note that while the Hamiltonian (5.1) with the harmonic forcing (5.2) is similar to the Hamiltonian (1.17) of the kicked harmonic oscillator there are two essential differences: the phase space of the rotor is a cylinder -- as opposed to the phase plane of the oscillator -- and there is no harmonic potential term like $x^2/2$ in $H_{\rm kr}$. Notice further that the rotor depends on the single parameter $V_0$ only that controls the amplitude of the kicks. This is in contrast to the oscillator, where a second parameter (for example the period $T$ of the kicks) cannot be eliminated by scaling. As I outline below these seemingly minor differences account for remarkably different dynamics of the two model systems, both in classical and quantum mechanics.

Defining $\vartheta_n$, $I_n$ as the values of $\vartheta$, $I$ immediately before the $n$-th kick,

one obtains from $H_{\rm kr}$ the discrete time dynamics

$$\begin{array}{rclcl} \vartheta_{n+1} & = & \vartheta_n+I_{... ...I_n-V'(\vartheta_n) & = & I_n-V_0\sin\vartheta_n. \end{array}$$ (5.2)

This is the CHIRIKOV-TAYLOR map [Chi79] or standard map.^5.1The behaviour of this discrete dynamical system has been studied extensively in the literature [Sch89,LL92] (the second reference contains a long list of references on the standard map).

The bifurcation scenario of this map for increasing values of $V_0$ is a typical KAM scenario in the sense that, as $V_0$ is increased, more and more invariant tori, guaranteed to exist by the KAM theorem [GH83], are destroyed. Some phase portraits -- periodic with period $2\pi$ both in the $\vartheta$- and $I$-directions -- of this transition to chaos are shown in figure 5.2.

Figure 5.3: Figure 5.2: (continued) (c) $V_0=5.0$. The initial value at $(\vartheta ,I)^t=(3.1415,0.0)^t$ is iterated 50000 times.

For $V_0=0.2$ (figure 5.2a) the phase portrait is dominated by the invariant lines of regular dynamics. For the intermediate parameter value $V_0=1.0$ (figure 5.2b) the regime of weak chaos has been reached where invariant lines, POINCARÉ-BIRKHOFF island chains and chaotic regions coexist. For large enough $V_0$ unbounded motion in the direction of $I$ becomes possible: at a critical value $V_{0,{\rm c}}\approx 0.9716$ only a single global torus -- the ``golden'' KAM torus -- persists, and for $V_0>V_{0,{\rm c}}$ it is destroyed, giving way for global diffusion in phase space (figure 5.2c, for $V_0=5.0$).

Energy diffusion of the rotor in the classical $(\vartheta,I)$-phase space can be described by considering the rotational energy^5.2before the $n$-th kick:

$$E_n \; := \; \frac{1}{2} I_n^2.$$ (5.3)

The standard map (5.4) makes $E_n$ evolve according to

$$E_{n+1} \; = \; E_n - V_0I_n\sin\vartheta_n + \frac{V_0^2}{2}\sin^2\vartheta_n,$$ (5.4)

which by averaging over an ensemble of orbits and employing the random phase approximation (cf. subsection 1.2.4) becomes the diffusion law

such that normally diffusive dynamics is to be expected. This averaging is justified for large enough $V_0$, when unhindered diffusion through phase space is possible. Corrections to formula (5.7b) resulting from accelerator modes and angular correlations are discussed in [LL92], for example.

In analogy with the quantum map of the kicked harmonic oscillator introduced in section 2.1, the quantum dynamics of the kicked rotor is given by

$$\left\vert \psi_{n+1} \right> \; = \; {\hat{U}}_{\rm kr,fr... ...\rm kr,kick} \left\vert \psi_n \right>, \quad n\in\mathbb{Z},$$ (5.4)

with the quantum state $\left\vert \psi_n \right>$ just before the $n$-th kick,

$$\left\vert \psi_n \right> \; := \; \lim_{t\nearrow n} \left\vert \psi(t) \right>,$$ (5.5)

and the time evolution operators

for the kick and the unperturbed dynamics, respectively, where $\hat{\vartheta}$ and $\hat{I}$ are the angle and angular momentum operators, and $V(\hat{\vartheta})$ is the potential energy operator. Unlike the classical system that contains the single parameter $V_0$ only, the quantum system depends on both the parameters $V_0$ and $\hbar$. The dependence on $\hbar$ is essential for the proof of quantum localization in subsection 5.1.3 below.

The quantum map (5.8) can be iterated comparatively easily since the unperturbed dynamics of the rotor is free rotation and the propagator ${\hat{U}}_{\rm kr,free}$ becomes a mere multiplication operator in the angular momentum representation: for $\left\vert \psi_n \right>$ expanded according to

$$\left< \vartheta \left\vert \psi_n \right> \right. \; = \; ... ...qrt{2\pi}} \sum_{m=-\infty}^\infty \Psi_{nm} e^{im\vartheta},$$ (5.5)

using the FOURIER coefficients $\Psi_{nm} = \rule[-0.1cm]{0.0cm}{0.1cm}_{\rm r}\!\left< m \left\vert \psi_n \right> \right.$ with the eigenstates of angular momentum

$$\left< \vartheta \left\vert m \right> \right._{\rm r} \; = ... ...ac{1}{\sqrt{2\pi}} \; e^{im\vartheta}, \quad m\in\mathbb{Z},$$ (5.6)

with respect to the eigenvalues $\hbar m$, one obtains for the free rotation part of the dynamics:

Switching between the angle and angular momentum representations can be accomplished with little numerical effort by fast FOURIER transformation (cf. the footnote on page ). This makes the kicked rotor a numerically much more accessible model than the kicked harmonic oscillator, where the quantum map typically has to be iterated by multiplying with huge matrices, as described in chapter 3. In [CCIF79] another numerical method for iterating the quantum map of the cosine-kicked rotor is described, which is even more efficient than the FFT-based method, but has the disadvantage of being less general because it is tailored to the special kick potential (5.2).

Results of numerical experiments for both the classical and quantum rotors are shown in figure 5.4, where classical normal diffusion in the case $V_0=4.0$ is contrasted with quantum mechanically suppressed diffusion.

The classical diffusion coefficient is found numerically as $D_{\rm kr}(4.0)\approx 3.07$, a somewhat smaller value than given by the large $V_0$ approximation (5.7b), due to the small value of $V_0$ (cf. the discussion, at the end of subsection 1.2.4, of a similar situation with respect to the kicked harmonic oscillator). On the other hand, the quantum energy expectation value as a function of discretized time $n$,

$$\left< E \right>_n \; := \; \Big< \psi_n \Big\vert \frac{1}... ...\sum_{m=-\infty}^\infty \left\vert\Psi_{nm}\right\vert^2 m^2,$$ (5.6)

exhibits a notably distinct behaviour. Up to a quantum break time $n^*$, $\left< E \right>_n$ follows the classical curve (5.7a), but is suppressed for larger times; $\left< E \right>_n$ even appears to be bounded for all $n$. (In the present example one has $n^*\approx 5$ for $\hbar =0.1$ and $n^*\approx 10$ for $\hbar=0.2$; but note that these values of $n^*$ are subject not only to the value of $\hbar$, but also depend on $V_0$ and the initial state $\left\vert \psi_0 \right>$.)

It is this quantum mechanically suppressed energy diffusion, or quantum localization, of the kicked rotor that is to be explained in the following two subsections.

Footnotes

... map.^5.1

The sign of the force term in the standard map (5.4) or in the potential (5.2) is a matter of convention. Changing this sign is equivalent to shifting $\vartheta$ by $\pi$.

... energy^5.2

Due to the boundedness of the rotor's phase space in the direction of $\vartheta$, energy diffusion occurs only in the (angular) momentum coordinate $I$ here, as opposed to the case of the kicked harmonic oscillator, where both the position and momentum variables $x$, $p$ are unbounded and subject to diffusion -- cf. equations (1.74, 1.77).