Introduction to Quantum Chaos Classical chaos: Not a theory in the fundamental sense, a unifying collection of concepts, in fact a dominant theme of classical mechanics that was missed for about 200 years! Non-linear dynamics, nearly always impossible to obtain exact solutions analytically, computer simulations are important (and have driven the field). Quantum chaos: Is there any? Short answer is no. It’s a misleading name! Classical Hamiltonian chaos rests on the notion of trajectory, and the fine structure of phase space; concepts that have no place in quantum mechanics. What is the nature of a quantum system for which the corresponding classical system is chaotic?

Classical chaos Example: double pendulum. Only one more degree of freedom than the standard periodic pendulum. Unpredictable trajectory and sensitivity to initial conditions. Complex non-predictable behavior from simple systems: deterministic chaos. Why are some systems chaotic? What is the origin of chaotic behavior? Why the sensitivity to initial conditions? Are chaotic systems in some (statistical) way as predictable and controllable as integrable systems.

Classical chaos Origins of chaos theory can be found in the end of the 19th centuary with Poincare and others who tried to answer the basic question: is the solar system stable? 3-body problem admits no closed form solution. Perturbation theory in general is expected to diverge (though is very successful when it can be controlled). In fact, signatures of chaos can be found in the solar system; e.g. asteroid distribution.

Chaos in other contexts Chaotic behavior is found in many contexts which can be modeled by a non-linear system of some kind. Since most of nature is non-linear, chaos is everywhere. (However, question: How to distinguish chaos from noise?)

Quantum chaos Quantum scarring. Localization. Universal forms of the distribution function of eigenvalue spacing.

Outline of the course Classical chaos: Lecture 1: Introduction, logistic map. Lecture 2-3: Renormalization in the logistic map, kicked rotator and the standard map, introduction to Hamiltonian chaos and the KAM theorem. Quantum chaos: Rough order of topics; division into lectures not decided! The kicked quantum rotator. Quantum cat map. Universal features of quantum chaos. Gutzwiller trace formulae and quantum scarring. Format of lectures: Generally most of the pictures on slides, and most of the formulas on the blackboard.

Books, other course resources Many books can be found in library, both quantum and classical chaos: dig in! Very good online book: http://chaosbook.org/ by one of the pioneers of the field. Popular science books abound, e.g. Ian Stewart “Does god play dice”, and James Gleik “Chaos” (ignores all the Russian contributions to non-linear dynamics and chaos!). Myself: tend to be found in the office most weekday afternoons during semester.

Completing the course Two ways to do this: Attend lectures and “problems classes”. Attend lectures and do mini-project at the end of the course (I prefer this one).

Logistic map Many features of non-Hamiltonian chaos can be seen in this simple map (and other similar one dimensional maps). Why? Universality. Period doubling Intermittency Crisis Ergodicity Strange attractors Periodic/Aperiodic mix Topological Cantor set

Time series of the logistic map Superstable points; convergence to attractor very rapid Critical slowing down near a bifurcation; convergence very slow

Time series of the logistic map Superstable points; convergence to attractor very rapid Critical slowing down near a bifurcation; convergence very slow.

Time series of the logistic map Bifurcation to a two cycle attractor.

Time series of the logistic map A series of period doublings of the attractor occur for increasing values of r, the “biotic potential” as it is sometimes known for the logistic map.

Time series of the logistic map At a critical value of r the dynamics become aperiodic. However, note that initially close trajectories remain close. Dynamics not ergodic.

Time series of the logistic map At r = 4 one finds aperiodic motion, in which initially close trajectories exponentially diverge.

Liapunov exponents

Liapunov exponents

Self similarity, intermittency, “crisis”

The period doubling regime Derivative identical for all points in cycle. All points become unstable simultaneously; period doubling topology can only be that shown above.

Period doubling in experiment First experimental evidence was via convection currents in mercury, so called Bernard experiment, many others followed e.g. nonlinear driven RCL-oscillator, etc. Period doubling constants (very) close to those of the logistic map!

Other one dimensional maps Other one dimensional maps show a rather similar structure; LHS is the sine map and RHS one that I made up randomly. In fact the structure of the periodic cycles with r is universal.

Universality 2nd order phase transitions: details, i.e. interaction form, does not matter near the phase transition. Depend only on (e.g.) symmetry of order parameter, dimensionality, range of interaction Can be calculate from simple models that share these feature with (complicated) reality.

Renormalisation Change coordinates such that superstable point is at origin

Renormalisation

Renormalisation

Renormalisation

Summary Logistic map shows many features of chaotic driven dissipative systems: Fixed point analysis of 1d map. Period doubling regimes with both qualitative and quantitative universality. In the chaotic regime commingling of unstable periodic orbits and chaotic orbits. Strange attractors: fractal structure (topological Cantor set), ergodic, but nearby trajectories exponentially diverge. Next week: Renormalization for the logistic map, and start of the kicked rotator (shares many features with Hamiltonian chaos). Lorenz attractor