Kicked rotator

Phase portraits (p vs. x) of the classical kicked rotor at different kicking strengths. The top row shows, from left to right, K = 0.5, 0.971635, 1.3. The bottom row shows, from left to right, K = 2.1, 5.0, 10.0. The phase portrait at the chaotic boundary is the upper middle plot, with K_C = 0.971635. At and above K_C, regions of uniform, grainy-coloured, qausi-random trajectories appear and eventually consume the entire plot, indicating chaos.

The kicked rotator, also spelled as kicked rotor, is a prototype model for chaos and quantum chaos studies. It describes a particle that is constrained to move on a ring (equivalently: a rotating stick). The particle is kicked periodically by an homogeneous field (equivalently: the gravitation is switched on periodically in short pulses). The model is described by the Hamiltonian

${\mathcal {H}}(p,x,t)={\frac {1}{2}}p^{2}+K\cos(x)\sum _{n=-\infty }^{\infty }\delta (t-n)$

Where $\textstyle \delta$ is the Dirac delta function, $\textstyle x$ is the angular position (for example, on a ring), taken modulo $\textstyle 2\pi$, $\textstyle p$ is the momentum, and $\textstyle K$ is the kicking strength. Its dynamics are described by the standard map

$p_{n+1}=p_{n}+K\sin(x_{n}),\;\;x_{n+1}=x_{n}+p_{n+1}$

With the caveat that $\textstyle p$ is not periodic, as it is in the standard map. See more details and references on the standard map here, or better in the associated Scholarpedia entry.

Main properties (classical)

In the classical analysis, if the kicks are strong enough, $K>K_{c}\approx 0.971635\dots$, the system is chaotic and has a positive Maximal Lyapunov exponent (MLE).

The averaged diffusion of the momentum-squared is a useful parameter in characterizing the delocalization of nearby trajectories. The inductive result of the standard map yields the following equation for momentum

$p(n)=p(0)+K\sum _{i=0}^{n-1}\sin(x(i))$

Kicker Rotor Phase Portrait Animation

The diffusion can then be calculated by squaring the difference in momentum after the ${n}^{th}$ kick and the initial momentum, and then averaging, yielding

$\left\langle {(\Delta p)}^{2}\right\rangle =\left\langle {(p(n)-p(0))}^{2}\right\rangle =K^{2}\sum _{i=0}^{n-1}\left\langle {\sin }^{2}(x(i))\right\rangle +K^{2}\sum _{i\neq j}^{}\left\langle \sin(x(i))\sin(x(j))\right\rangle$

In the chaotic domain, the momenta at different time points may be anywhere from entirely uncorrelated to highly correlated. If they are assumed uncorrelated due to the quasi-random behaviour, the sum involving the cross-terms $\textstyle i\neq j$ is neglected. In this limit, since the first term is a sum of $\textstyle n$ terms all equalling $\textstyle {\frac {1}{2}}$, the momentum diffusion becomes $\textstyle \left\langle {(\Delta p)}^{2}\right\rangle ={\frac {1}{2}}K^{2}n$. However, if the momenta at different time points are assumed highly correlated, the sum involving the cross-terms is not neglected, and so it contributes more terms equalling $\textstyle {\frac {1}{2}}$. Altogether, there are $\textstyle n^{2}$ terms to sum, all of the form $\textstyle \sim \left\langle {\sin }^{2}x\right\rangle ={\frac {1}{2}}$. This gives an upper bound on the momentum diffusion of $\textstyle \left\langle {(\Delta p)}^{2}\right\rangle ={\frac {1}{2}}K^{2}n^{2}$. Therefore, in the chaotic domain, the momentum diffusion is between

${\frac {1}{2}}K^{2}n\leq \left\langle {(\Delta p)}^{2}\right\rangle \leq {\frac {1}{2}}K^{2}n^{2}$

That is, the momentum diffusion in the chaotic domain has somewhere between a linear and a quadratic dependence on the number of kicks. An exact expression for $\textstyle \left\langle {(\Delta p)}^{2}\right\rangle$ can be obtained in principle by calculating the sums explicitly for an ensemble of trajectories.

Main properties (quantum)

In the quantum analysis, the Hamiltonian must first be rewritten in operator form, using the substitution $\textstyle p=i\hbar {\frac {\partial }{\partial x}}$ to give (in dimensionless form)

$H(x,t)=-{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}+K\cos(x)\sum _{n=-\infty }^{\infty }\delta (t-n)$

The wavefunction can then be solved for using Schrödinger's equation

$i\hbar {\frac {\partial }{\partial t}}\psi (t)=H\psi (t)$

where $\textstyle \hbar$ is here scaled according to the period between kicks, $\textstyle T$, and the wave-vector of the driving potential, $\textstyle k_{0}$, as

$\hbar \rightarrow \hbar {{k}_{0}}^{2}T/m$

The wavefunction at the $\textstyle n^{th}$ kick can be expanded in terms of the momentum eigenstates, $\textstyle |l\rangle$, as

$\psi (t_{n})=\sum _{l=-\infty }^{\infty }a_{l}(t_{n})|l\rangle$

It can be shown that the coefficients are given recursively by

$a_{m}({t}_{n+1})=\exp[-im^{2}\hbar /2]\sum _{l=-\infty }^{\infty }{(-i)}^{l-m}{J}_{l-m}(K/\hbar )a_{l}(t_{n})$

Where $\textstyle {J}_{\alpha }$ is a Bessel function of order $\textstyle \alpha$.

Given some set of initial conditions, it is relatively straightforward to numerically solve the recursive equation above for all time, and substitute the calculated coefficients back into the momentum eigenstate decomposition to find the total wavefunction. Squaring this gives the time evolution of the probability distribution, thus providing a complete quantum mechanical description.

Another way to calculate the time evolution is to iteratively apply the unitary operator

${\hat {U}}=\exp \left[-i{\frac {1}{2\hbar }}{\hat {p}}^{2}\right]\exp \left[-i{\frac {1}{\hbar }}K\cos {\hat {x}}\right]$

It has been discovered that the classical diffusion is suppressed, and later it has been understood that this is a manifestation of a quantum dynamical localization effect that parallels Anderson localization. There is a general argument that leads to the following estimate for the breaktime of the diffusive behavior

$t^{*}\ \approx \ D_{cl}/\hbar ^{2}$

Where $D_{cl}$ is the classical diffusion coefficient. The associated localization scale in momentum is therefore $\textstyle {\sqrt {D_{cl}t^{*}}}$.

The effect of noise and dissipation

If noise is added to the system, the dynamical localization is destroyed, and diffusion is induced. This is somewhat similar to hopping conductance. The proper analysis requires to figure out how the dynamical correlations that are responsible for the localization effect are diminished.

Recall that the diffusion coefficient is $D_{cl}\approx K^{2}/2$, because the change $(p(t)-p(0))$ in the momentum is the sum of quasi-random kicks $K\sin(x(n))$. An exact expression for $D_{cl}$ is obtained by calculating the "area" of the correlation function $C(n)=\langle \sin(x(n))\sin(x(0))\rangle$, namely the sum $D=K^{2}\sum C(n)$. Note that $C(0)=1/2$. The same calculation recipe holds also in the quantum mechanical case, and also if noise is added.

In the quantum case, without the noise, the area under $C(n)$ is zero (due to long negative tails), while with the noise a practical approximation is $C(n)\mapsto C(n)e^{-t/t_{c}}$ where the coherence time $t_{c}$ is inversely proportional to the intensity of the noise. Consequently the noise induced diffusion coefficient is

$D\approx D_{cl}t^{*}/t_{c}\quad [{\text{assuming }}t_{c}\gg t^{*}]$

Also the problem of quantum kicked rotator with dissipation (due to coupling to a thermal bath) has been considered. There is an issue here how to introduce an interaction that respects the angle periodicity of the position $x$ coordinate, and is still spatially homogeneous. In the first works a quantum-optic type interaction has been assumed that involves a momentum dependent coupling. Later a way to formulate a purely position dependent coupling, as in the Calderia-Leggett model, has been figured out, which can be regarded as the earlier version of the DLD model.

Experiments

Experimental realizations of the quantum kicked rotator have been achieved by the Raizen group, and by the Auckland group, and have encouraged a renewed interest in the theoretical analysis. In this kind of experiment, a sample of cold atoms provided by a Magneto-optical trap interacts with a pulsed standing wave of light. The light being detuned with respect to the atomic transitions, atoms undergo a space-periodic conservative force. Hence, the angular dependence is replaced by a dependence on position in the experimental approach. Sub-milliKelvin cooling is necessary to obtain quantum effects: because of the Heisenberg uncertainty principle, the de Broglie wavelength, i.e. the atomic wavelength, can become comparable to the light wavelength. For further information, see. Thanks to this technique, several phenomena have been investigated, including the noticeable:

• quantum Ratchets;
• the Anderson transition in 3D.

See also

• Circle map