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Kicked Rotor Phase Portrait

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 The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...

I

) in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses. The model is described by the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

:

\mathcal(\theta,p,t)= \frac + K \cos \theta \sum_^\infty \delta \left(\frac-n\right)

, where

\theta \in conjugated momentum of \theta, \textstyle K is the kicking strength, T is the kicking period and \textstyle \delta is the Dirac delta">Dirac delta function

In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

.

Classical properties

Stroboscopic dynamics

The equations of motion of the kicked rotator write

\frac = \frac = \frac \quad \text \quad \frac = -\frac =  K \sin \theta \sum_^\infty \delta \left(\frac-n\right)

Theses equations show that between two consecutive kicks, the rotator simply moves freely: the momentum

p

is conserved and the angular position growths linearly in time. On the other hand, during each kick the momentum abruptly jumps by a quantity

K T \sin \theta

, where

\theta

is the angular position near the kick. The kicked rotator dynamics can thus be described by the discrete map

p_=p_n+ KT \sin \theta_n \quad \text \quad \theta_ = \theta_n + \frac p_

where

\theta_n

and

p_n

are the canonical coordinates at time

t=nT^-

, just before the

n

-th kick. It is usually more convenient to introduce dimensionless momentum

p \rightarrow p/\frac

, time

t \rightarrow t/T

and kicking strength

K \rightarrow K/\frac

to reduce the dynamics to the single parameter map

p_=p_n+ K \sin \theta_n \quad \text \quad \theta_ = \theta_n + p_

known as

Chirikov standard map The standard map (also known as the Chirikov–Taylor map or as the Chirikov standard map) is an area-preserving chaotic map from a square with side 2\pi onto itself. It is constructed by a Poincaré's surface of section of the kicked rot ...

, with the caveat that

p_n

is not periodic as in the standard map. However, one can directly see that two rotators with same initial angular position

\theta_0

but shifted dimensionless momentum

p_0

and

p_0+ 2\pi l

(with

l

an arbitrary integer) will have the same exact stroboscopic dynamics, but with dimensionless momentum shifted at any time by

2\pi l

(this is why stroboscopic phase portraits of the kicked rotator are usually displayed in a single momentum cell

p \in \pi ,\pi /math>).

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 The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory ...

. In the limit

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

(\theta,p)

plane (phase space), that is tori. For small, but non-vanishing perturbation

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

K

, invariant tori are eventually destroyed by the perturbation. For the value

K=K_c\approx0.971635\dots

, the last invariant tori connecting

\theta=-\pi

and

\theta= \pi

in phase space is destroyed.

Diffusion in momentum direction

For

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

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 Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

process in the momentum direction

\langle (\Delta p_n)^2 \rangle = 2 D_\text n

(where the average runs over different initial conditions). More precisely, after

n

kicks, the momentum

p_n

of a particle with initial momentum

p_0

writes

p_n = p_0 + K\sum_^\sin \theta_i

(obtained by iterating

n

times the standard map). Assuming that kicks are randoms and uncorrelated in time, the spreading of the momentum distribution writes

\left \langle ^ \right \rangle = \left \langle ^ \right \rangle = K^2\sum_^\left \langle ^ \theta_i  \right \rangle + K^2 \sum_^\left \langle \sin \theta_i \sin \theta_j  \right \rangle \approx  K^2\sum_^\left \langle ^ \theta_i \right \rangle = \frac K n

The classical diffusion coefficient in momentum direction is then given in first approximation by

D_\text = \frac

. Corrections coming from neglected correlation terms can actually be taken into account, leading to the improved expression

D_\text = \frac -2J_2(K)+2J_2^2(K)

where

J_2

is the

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

of first kind.

The quantum kicked rotator

Stroboscopic dynamics

The dynamics of the quantum kicked rotator (with wave function

,  \psi(t) \rangle

) is governed by the time dependent Schrödinger's equation :

\psi(t) \rangle

with

i\hbar

(or equivalently

\langle \theta ,  \hat ,  \psi \rangle= i\hbar \frac

). As for classical dynamics, a stroboscopic point of view can be adopted by introducing the time propagator over a kicking period

\hat

(that is the Floquet operator) so that

, \psi(t+T) \rangle =  \hat , \psi(t) \rangle

. After a careful integration of the time-dependent Schrödinger's equation, one finds that

\hat

can be written as the product of two operators

\hat=\exp\left i\frac\right \exp\left i\frac \cos\hat\right /math>We recover the classical interpretation: the dynamics of  the quantum kicked rotor between two kicks is the succession of a free propagation during a time T followed by a short kick. This simple expression of the Floquet operator \hat (a product of two operators, one diagonal in momentum basis, the other one diagonal in angular position basis) allows to easily numerically solve the evolution of a given wave function using

split-step method In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies ...

. Because of the periodic boundary conditions at

\theta=\pm \pi

, any wave function

,  \psi \rangle

can be expanded in a discrete momentum basis

, l \rangle

(with

p=l \hbar

,

l

integer) see

Bloch theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...

), so that :

\langle \theta ,  \psi \rangle =\sum_^  \langle l ,  \psi \rangle \mathrm^ \Leftrightarrow \langle l ,  \psi \rangle = \int_^ \frac  \langle \theta ,  \psi \rangle \mathrm^

Using this relation with the above expression of

\hat

, we find the recursion relation

\langle l,  \psi(t+T) \rangle 
= \exp\left(-i\frac\right) \sum_^\infty (-i)^ J_ \left(\frac \right)  \langle m,  \psi(t) \rangle

where

\textstyle _n

is a

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

of first kind.

Dynamical localization

It has been discoveredG. Casati, B.V. Chirikov, F.M. Izrailev and J. Ford, in ''Stochastic Behaviour in classical and Quantum Hamiltonian Systems'', Vol. 93 of Lecture Notes in Physics, edited by G. Casati and J. Ford (Springer, N.Y. 1979), p. 334 that the classical diffusion is suppressed in the quantum kicked rotator. It was later 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_/\hbar^2

Where

D_

is the classical diffusion coefficient. The associated localization scale in momentum is therefore

\textstyle \sqrt

.

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

, n \rangle

, where Anderson localization takes place (in one dimension)

\hat = \sum_ \varepsilon_n , n \rangle \langle n,  + \sum_ t_ ,  n \rangle \langle m ,

where the

\varepsilon_n

are random on-site energies, and the

t_

are the hoping amplitudes between sites

n

and

m

. In the quantum kicked rotator it can be shown, that the plane wave

,  p \rangle

with quantized momentum

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

\omega

)

\mathrm^ \quad \text \quad \varepsilon_n = \tan(\omega/2 - n^2/4)

Dynamical localization in the quantum kicked rotator then actually takes place in the momentum basis.

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_\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_

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^

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_t^* / t_c \quad textt_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.

Experimental realization with cold atoms

The first experimental realizations of the quantum kicked rotator have been achieved by

Mark G. Raizen Mark George Raizen is an American physicist who conducts experiments on quantum optics and atom optics. Early life and education Raizen was born in New York City. Raizen's uncle, Dr. Robert F. Goldberger, was provost of Columbia University and ...

group in 1995, later followed 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 A magneto-optical trap (MOT) is an apparatus which uses laser cooling and a spatially-varying magnetic field to create a trap which can produce samples of cold, trapped, neutral atoms. Temperatures achieved in a MOT can be as low as several microk ...

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 In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...

, the de Broglie wavelength, i.e. the atomic wavelength, can become comparable to the light wavelength. For further information, see.M. Raizen in ''New directions in quantum chaos'', Proceedings of the International School of Physics ''Enrico Fermi'', Course CXLIII, Edited by G. Casati, I. Guarneri and U. Smilansky (IOS Press, Amsterdam 2000). Thanks to this technique, several phenomena have been investigated, including the noticeable: * quantum Ratchets; * the Anderson transition in 3D.

See also

*

Circle map In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamica ...

References

External links

Scholarpedia entry
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