The quantum kicked rotator

First approach to “Quantum Chaos”: take a system that is classically chaoticand 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 angularmomentum, 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. Theseare the fractal remnants of invarient tori.•K = 1.0; i.e. last torus has been destroyed (K=0.97..).

Diffusion in the kicked rotator

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 probabilitydistribution is the product of individual distributions(if uncorrelated)

And Fourier transform back givesa Gaussian distribution – independent of thenature 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 additionalparameter.

The Floquet map

The Floquet map

The Floquet map

F is clearly unitary, as it must be, withthe Floquet phases as the diagonalelements.

The Floquet map

Floquet map for the kicked rotator

Rational a: quantum resonance

Continuous spectrum

Quadratic growth; has no classical counterpart

Irrational a: 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 a: quantum resonance

Rational a: quantum resonance

Irrational a: transient diffusion

Irrational a: transient diffusion

System does not “feel” discrete nature of spectrum

Rapidly oscillating phasecancels out, only zero phaseterm survives

Since F is a banded matrix then the U’s will also all be banded, and hencefor 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-sitepotentials in range [-0.5,0.5] with t=1.•Peaks in the spectrum of the orderedsystem are van Hove singularities; peaks in the spectrum of the disorderedsystem are very different in origin

Localisation of electrons by disorder

On-site order On-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

TB Hamiltonian from a quantum map

TB Hamiltonian from a quantum map

TB Hamiltonian from a quantum map

If b is irrational then x distributed uniformly on [0,1]

Thus the analogy between Anderson localization in condensed matter and theangular momentum (or energy) localization is quantum chaotic systems is established.

Next weeks lecture

Proof that on-site disorder leads to localisationHusimi 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, 1999. (many typos!)

“The transition to chaos”: L. E. Reichl, Springer-Verlag (in library)

On-line: A good scholarpedia article about the quantum kicked oscillator; http://www.scholarpedia.org/article/Chirikov_standard_map

Other links which look nice (Google will bring up many more).

http://george.ph.utexas.edu/~dsteck/lass/notes.pdfhttp://lesniewski.us/papers/papers_2/QuantumMaps.pdfhttp://steck.us/dissertation/das_diss_04_ch4.pdf