The kicked rotator, also spelled as kicked rotor, is a paradigmatic model for both Hamiltonian chaos (the study of chaos in Hamiltonian systems) and quantum chaos. It describes a free rotating stick (with moment of inertia ${\displaystyle I}$) in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses. The model is described by the Hamiltonian

${\displaystyle {\mathcal {H))(\theta ,p_{\theta },t)={\frac {p_{\theta }^{2)){2I))+K\cos \theta \sum _{n=-\infty }^{\infty }\delta \left({\frac {t}{T))-n\right)}$,

where ${\displaystyle \theta \in [0,2\pi ]}$ is the angular position of the stick (${\displaystyle \theta =\pi }$ corresponds to the position of the rotator at rest), ${\displaystyle p_{\theta ))$ is the conjugated momentum of ${\displaystyle \theta }$, ${\displaystyle \textstyle K}$ is the kicking strength, ${\displaystyle T}$ is the kicking period and ${\displaystyle \textstyle \delta }$ is the Dirac delta function.

Classical properties

Stroboscopic dynamics

The equations of motion of the kicked rotator write

$${\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t))={\frac {\partial {\mathcal {H))}{\partial p))={\frac {p}{I))\quad {\text{and))\quad {\frac {\mathrm {d} p}{\mathrm {d} t))=-{\frac {\partial {\mathcal {H))}{\partial \theta ))=K\sin \theta \sum _{n=-\infty }^{\infty }\delta \left({\frac {t}{T))-n\right)}$$

${\displaystyle p}$

${\displaystyle KT\sin \theta }$

${\displaystyle \theta }$

$${\displaystyle p_{n+1}=p_{n}+KT\sin \theta _{n}\quad {\text{and))\quad \theta _{n+1}=\theta _{n}+{\frac {T}{I))p_{n+1))$$

${\displaystyle \theta _{n))$

${\displaystyle p_{n))$

${\displaystyle t=nT^{-))$

${\displaystyle n}$

${\textstyle p\rightarrow p/{\frac {I}{T))}$

${\textstyle t\rightarrow t/T}$

${\textstyle K\rightarrow K/{\frac {I}{T^{2))))$

$${\displaystyle p_{n+1}=p_{n}+K\sin \theta _{n}\quad {\text{and))\quad \theta _{n+1}=\theta _{n}+p_{n+1))$$

${\displaystyle p_{n))$

${\displaystyle \theta _{0))$

${\displaystyle p_{0))$

${\textstyle p_{0}+2\pi l}$

${\displaystyle l}$

${\textstyle 2\pi l}$

${\textstyle p\in [-\pi ,\pi ]}$

Transition from integrability to chaos

The kicked rotator is a prototype model to illustrate the transition from integrability to chaos in Hamiltonian systems and in particular the Kolmogorov–Arnold–Moser theorem. In the limit ${\displaystyle K=0}$, the system describes the free motion of the rotator, the momentum is conserved (the system is integrable) and the corresponding trajectories are straight lines in the ${\displaystyle (\theta ,p)}$ plane (phase space), that is tori. For small, but non-vanishing perturbation ${\displaystyle K}$, instabilities and chaos starts to develop. Only quasi-periodic orbits (represented by invariant tori in phase space) remain stables, while other orbits become unstables. For larger ${\displaystyle K}$, invariant tori are eventually destroyed by the perturbation. For the value ${\displaystyle K=K_{c}\approx 0.971635\dots }$, the last invariant tori connecting ${\displaystyle \theta =-\pi }$ and ${\displaystyle \theta =\pi }$ in phase space is destroyed.

Diffusion in momentum direction

For ${\displaystyle K>K_{c))$, chaotic unstable orbits are no longer constraints by invariant tori in the momentum direction and can explore the full phase space. For ${\displaystyle K\gg K_{c))$, the particle after each kicks typically moved over a large distance, which strongly modifies the amplitude and sign of the following kick. At long time enough, the particle as thus been submitted to a series of kicks with quasi-random amplitudes. This quasi-random walk is responsible for a diffusion process in the momentum direction ${\displaystyle \langle (\Delta p_{n})^{2}\rangle =2D_{\text{cl))n}$ (where the average runs over different initial conditions).

More precisely, after ${\displaystyle n}$ kicks, the momentum ${\displaystyle p_{n))$ of a particle with initial momentum ${\displaystyle p_{0))$ writes ${\textstyle p_{n}=p_{0}+K\sum _{i=0}^{n-1}\sin \theta _{i))$^[2] (obtained by iterating ${\displaystyle n}$ times the standard map). Assuming that kicks are randoms and uncorrelated in time, the spreading of the momentum distribution writes

$${\displaystyle \left\langle {(\Delta p)}^{2}\right\rangle =\left\langle {(p_{n}-p_{0})}^{2}\right\rangle =K^{2}\sum _{i=0}^{n-1}\left\langle {\sin }^{2}\theta _{i}\right\rangle +K^{2}\sum _{i\neq j}^{}\left\langle \sin \theta _{i}\sin \theta _{j}\right\rangle \approx K^{2}\sum _{i=0}^{n-1}\left\langle {\sin }^{2}\theta _{i}\right\rangle ={\frac {1}{2))Kn}$$

${\textstyle D_{\text{cl))={\frac {K^{2)){4))}$

$${\displaystyle D_{\text{cl))={\frac {K^{2)){4))[1-2J_{2}(K)+2J_{2}^{2}(K)]}$$

${\textstyle J_{2))$

The quantum kicked rotator

Stroboscopic dynamics

The dynamics of the quantum kicked rotator (with wave function ${\displaystyle |\psi (t)\rangle }$) is governed by the time dependent Schrödinger's equation

${\displaystyle i\hbar {\frac {\partial }{\partial t))|\psi (t)\rangle =\left[{\frac ((\hat {p))^{2)){2I))+K\cos {\hat {\theta ))\sum _{n=-\infty }^{\infty }\delta \left({\frac {t}{T))-n\right)\right]|\psi (t)\rangle }$

with ${\displaystyle [{\hat {\theta )),{\hat {p))]=i\hbar }$ (or equivalently ${\textstyle \langle \theta |{\hat {p))|\psi \rangle =i\hbar {\frac {\partial \psi }{\partial \theta ))}$).

As for classical dynamics, a stroboscopic point of view can be adopted by introducing the time propagator over a kicking period ${\displaystyle {\hat {U))}$ (that is the Floquet operator) so that ${\displaystyle |\psi (t+T)\rangle ={\hat {U))|\psi (t)\rangle }$. After a careful integration of the time-dependent Schrödinger's equation, one finds that ${\displaystyle {\hat {U))}$ can be written as the product of two operators

$${\displaystyle {\hat {U))=\exp \left[-i{\frac ((\hat {p))^{2}T}{2I\hbar ))\right]\exp \left[-i{\frac {KT}{\hbar ))\cos {\hat {\theta ))\right]}$$

${\displaystyle T}$

${\displaystyle {\hat {U))}$

Because of the periodic boundary conditions at ${\displaystyle \theta =\pm \pi }$, any wave function ${\displaystyle |\psi \rangle }$ can be expanded in a discrete momentum basis ${\textstyle |l\rangle }$ (with ${\displaystyle p=l\hbar }$, ${\displaystyle l}$ integer) see Bloch theorem), so that

${\displaystyle \langle \theta |\psi \rangle =\sum _{l=-\infty }^{\infty }\langle l|\psi \rangle \mathrm {e} ^{il\theta }\Leftrightarrow \langle l|\psi \rangle =\int _{-\pi }^{\pi }{\frac {\mathrm {d} x}{2\pi ))\langle \theta |\psi \rangle \mathrm {e} ^{-il\theta ))$

Using this relation with the above expression of ${\displaystyle {\hat {U))}$, we find the recursion relation^[4]

$${\displaystyle \langle l|\psi (t+T)\rangle =\exp \left(-i{\frac {l^{2}\hbar T}{2))\right)\sum _{m=-\infty }^{\infty }(-i)^{m-l}J_{m-l}\left({\frac {KT}{\hbar ))\right)\langle m|\psi (t)\rangle }$$

${\displaystyle \textstyle {J}_{n))$

Demonstration

Indeed, we have

$${\displaystyle \langle l|\psi (t+T)\rangle =\langle l|{\hat {U))|\psi (t)\rangle =\exp \left(-i{\frac {l^{2}\hbar T}{2I))\right)\langle l|\exp \left(-i{\frac {KT\cos {\hat {\theta ))}{\hbar ))\right)|\psi (t)\rangle }$$

$${\displaystyle \langle l|\psi (t+T)\rangle =\exp \left(-i{\frac {l^{2}\hbar T}{2I))\right)\int _{-\pi }^{\pi }{\frac {\mathrm {d} \theta }{2\pi ))\mathrm {e} ^{-il\theta }\exp \left(-i{\frac {KT\cos {\hat {\theta ))}{\hbar ))\right)\langle \theta |\psi (t)\rangle }$$

$${\displaystyle \langle l|\psi (t+T)\rangle =\exp \left(-i{\frac {l^{2}\hbar T}{2I))\right)\int _{-\pi }^{\pi }{\frac {\mathrm {d} \theta }{2\pi ))\mathrm {e} ^{-il\theta }\sum _{n=-\infty }^{\infty }(-i)^{n}J_{n}\left({\frac {KT}{\hbar ))\right)\mathrm {e} ^{-in\theta }\sum _{m=-\infty }^{\infty }\mathrm {e} ^{im\theta }\langle m|\psi (t)\rangle }$$

$${\displaystyle \langle l|\psi (t+T)\rangle =\exp \left(-i{\frac {l^{2}\hbar T}{2I))\right)\sum _{n,m=-\infty }^{\infty }(-i)^{n}J_{n}\left({\frac {KT}{\hbar ))\right)\left[\int _{-\pi }^{\pi }{\frac {\mathrm {d} \theta }{2\pi ))\mathrm {e} ^{i(m-l-n)\theta }\right]\langle m|\psi (t)\rangle }$$

$${\displaystyle \langle l|\psi (t+T)\rangle =\exp \left(-i{\frac {l^{2}\hbar T}{2I))\right)\sum _{n,m=-\infty }^{\infty }(-i)^{n}J_{n}\left({\frac {KT}{\hbar ))\right){\frac {\sin([m-l-n])\pi }{(m-l-n)\pi ))\langle m|\psi (t)\rangle }$$

So that we recover the result, keeping only non vanishing terms ${\displaystyle m-l-n=0}$ in the double sum.

Dynamical localization

It has been discovered^[1] that the classical diffusion is suppressed in the quantum kicked rotator. It was later understood^[5][6][7][8] that this is a manifestation of a quantum dynamical localization effect that parallels Anderson localization. There is a general argument^[9][10] that leads to the following estimate for the breaktime of the diffusive behavior

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

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

Link with Anderson tight-binding model

The quantum kicked rotor can actually formally be related to the Anderson tight-binding model a celebrated Hamiltonian that describes electrons in a disordered lattice with lattice site state ${\displaystyle |n\rangle }$, where Anderson localization takes place (in one dimension)

$${\displaystyle {\hat {H))=\sum _{n}\varepsilon _{n}|n\rangle \langle n|+\sum _{n\neq m}t_{n-m}|n\rangle \langle m|}$$

${\displaystyle \varepsilon _{n))$

${\displaystyle t_{n-m))$

${\displaystyle n}$

${\displaystyle m}$

In the quantum kicked rotator it can be shown,^[11] that the plane wave ${\displaystyle |p\rangle }$ with quantized momentum ${\displaystyle p=n\hbar }$ play the role of the lattice sites states. The full mapping to the Anderson tight-binding model goes as follow (for a given eigenstates of the Floquet operator, with quasi-energy ${\displaystyle \omega }$)

$${\displaystyle t_{n}=-\int _{-\pi }^{\pi }{\frac {\mathrm {d} x}{2\pi ))\tan[K\cos(x)/2]\mathrm {e} ^{-ixn}\quad {\text{and))\quad \varepsilon _{n}=\tan(\omega /2-n^{2}/4)}$$

The effect of noise and dissipation

If noise is added to the system, the dynamical localization is destroyed, and diffusion is induced.^[12][13][14] 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 ${\displaystyle D_{cl}\approx K^{2}/2}$, because the change ${\displaystyle (p(t)-p(0))}$ in the momentum is the sum of quasi-random kicks ${\displaystyle K\sin(x(n))}$. An exact expression for ${\displaystyle D_{cl))$ is obtained by calculating the "area" of the correlation function ${\displaystyle C(n)=\langle \sin(x(n))\sin(x(0))\rangle }$, namely the sum ${\displaystyle D=K^{2}\sum C(n)}$. Note that ${\displaystyle C(0)=1/2}$. 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 ${\displaystyle C(n)}$ is zero (due to long negative tails), while with the noise a practical approximation is ${\displaystyle C(n)\mapsto C(n)e^{-t/t_{c))}$ where the coherence time ${\displaystyle t_{c))$ is inversely proportional to the intensity of the noise. Consequently, the noise induced diffusion coefficient is

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

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 ${\displaystyle x}$ coordinate, and is still spatially homogeneous. In the first works ^[15][16] a quantum-optic type interaction has been assumed that involves a momentum dependent coupling. Later^[17] 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.

Experimental realization with cold atoms

The first experimental realizations of the quantum kicked rotator have been achieved by Mark G. Raizen group^[18][19] in 1995, later followed by the Auckland group,^[20] 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.^[21] Thanks to this technique, several phenomena have been investigated, including the noticeable:

• quantum Ratchets;^[22]
• the Anderson transition in 3D.^[23]