The quantum kicked rotator First approach to “Quantum Chaos”: take a system that is classically...
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![Page 1: The quantum kicked rotator First approach to “Quantum Chaos”: take a system that is classically chaotic and quantize it.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697c0141a28abf838ccd3a2/html5/thumbnails/1.jpg)
The quantum kicked rotator
First approach to “Quantum Chaos”: take a system that is classically chaoticand quantize it.
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Classical kicked rotator
One parameter map; can incorporate all others into choice of units
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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
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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..).
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Diffusion in the kicked rotator
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Diffusion in the kicked rotator
Assume that angles are random variables;i.e. uncorrelated
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Diffusion in the kicked rotator
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Central limit theorem
Characteristic function for the distribution
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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!
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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.
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The Floquet map
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The Floquet map
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The Floquet map
F is clearly unitary, as it must be, withthe Floquet phases as the diagonalelements.
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The Floquet map
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Floquet map for the kicked rotator
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Rational : a quantum resonance
Continuous spectrum
Quadratic growth; has no classical counterpart
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Irrational : a transient diffusion
•Only for short time scales can diffusive behavior be seen•Spectrum of Floquet operator is now discrete.
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…and localization!
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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.
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Rational : a quantum resonance
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Rational : a quantum resonance
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Irrational : a transient diffusion
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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.
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Tight-binding model of crystal lattice
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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
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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.
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TB Hamiltonian from a quantum map
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TB Hamiltonian from a quantum map
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TB Hamiltonian from a quantum map
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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.
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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..
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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