The quantum kicked rotator First approach to “Quantum Chaos”: take a system that is classically chaotic and quantize it.

Classical kicked rotator One parameter map; can incorporate all others into choice of units

Diffusion in the kicked rotator K = 5.0; strongly chaotic regime. Take ensemble of 100,000 initial points with zero angular momentum, and pseudo-randomly distributed angles. Iterate map and take ensemble average at each time step

Diffusion in the kicked rotator System can get “trapped” for very long times in regions of cantori. These are the fractal remnants of invarient tori. K = 1.0; i.e. last torus has been destroyed (K=0.97..).

Diffusion in the kicked rotator

Assume that angles are random variables; i.e. uncorrelated

Diffusion in the kicked rotator

Central limit theorem Characteristic function for the distribution

Central limit theorem Characteristic function of a joint probability distribution is the product of individual distributions (if uncorrelated) And Fourier transform back gives a Gaussian distribution – independent of the nature of the X random variable!

Quantum kicked rotator How do the physical properties of the system change when we quantize? Two parameters in this Schrodinger equation; Planck’s constant is the additional parameter.

The Floquet map

F is clearly unitary, as it must be, with the Floquet phases as the diagonal elements.

The Floquet map

Floquet map for the kicked rotator

Rational  quantum resonance Continuous spectrum Quadratic growth; has no classical counterpart

Irrational  transient diffusion Only for short time scales can diffusive behavior be seen Spectrum of Floquet operator is now discrete.

…and localization!

Quantum chaos in ultra-cold atoms All this can be seen in experiment; interaction of ultra-cold atoms (micro Kelvin) with light field; dynamical localization of atoms is seen for certain field modulations.

Rational  quantum resonance

Irrational  transient diffusion

System does not “feel” discrete nature of spectrum Rapidly oscillating phase cancels out, only zero phase term survives Since F is a banded matrix then the U’s will also all be banded, and hence for l, k, k’ larger than some value there is no contribution to sum.

Tight-binding model of crystal lattice

Disorder in the on-site potentials One dimensional lattice of 300 sites; Ordered system: zero on-site potential. Disordered system: pseudo-random on-site potentials in range [-0.5,0.5] with t=1. Peaks in the spectrum of the ordered system are van Hove singularities; peaks in the spectrum of the disordered system are very different in origin

Localisation of electrons by disorder On-site orderOn-site disorder Probability of finding system at a given site (y-axis) plotted versus energy index (x-axis); magnitude of probability indicated by size of dots.

TB Hamiltonian from a quantum map

If  is irrational then x distributed uniformly on [0,1] Thus the analogy between Anderson localization in condensed matter and the angular momentum (or energy) localization is quantum chaotic systems is established.

Next weeks lecture Proof that on-site disorder leads to localisation Husimi functions and (p,q) phase space Examples of quantum chaos: Quantum chaos in interaction of ultra-cold atoms with light field. Square lattice in a magnetic field. Some of these topics..

Resources used “Quantum chaos: an introduction”, Hans-Jurgen Stockman, Cambridge University Press, (many typos!) “The transition to chaos”: L. E. Reichl, Springer-Verlag (in library) On-line: A good scholarpedia article about the quantum kicked oscillator; Other links which look nice (Google will bring up many more)