Download Quantum chaos: an introduction

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
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
The Floquet map
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 a: quantum resonance
Quadratic growth; has no classical counterpart
Continuous spectrum
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
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.
System does not “feel” discrete nature of spectrum
Rapidly oscillating phase
cancels out, only zero phase
term survives
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
Probability of finding system at a given site (y-axis) plotted versus energy index (x-axis);
magnitude of probability indicated by size of dots.
On-site order
On-site disorder
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 the
angular momentum (or energy) localization is quantum chaotic systems is established.
Next weeks lecture
Some of these topics..
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.
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.pdf
http://lesniewski.us/papers/papers_2/QuantumMaps.pdf
http://steck.us/dissertation/das_diss_04_ch4.pdf