QUANTUM CHAOS : QUANTUM CHAOS Glows at Sunset Colloquium at Technion 18 nov 2004

Kicked Cold Atoms Cs

Bloch Theory is the detuning from exact resonance The Hamiltonian commutes with translations by : the spatial period of the kicks The Quasi-momentum is conserved Any wave function may be decomposed in Bloch waves of the form each of these evolves independently of the others. The corresponding dynamics is formally that of a Rotor with angle coordinate Evolution of the Rotor : is the detuning from exact resonance

Bloch 1 2 ? 3

Localization & Resonances The quantum KR: Casati, Chirikov, Ford, Izrailev 1978 Localization & Resonances Localization : Fishman, Grempel, Prange 1982 Resonances : Izrailev, Shepelyansky 1979 Experimental realizations with cold atoms: Moore, Robinson, Bharucha, Sundaram, Raizen 1995

c 895 nm Experiments at Oxford: the Kicked Accelerator GRAVITY

Quantum Accelerator Modes Atomic momentum The atoms are far from the classical limit, and the modes are absent in the classical limit !!! Pulse period

Hamiltonians for kicked atoms

Bloch Theory is the detuning from exact resonance The Hamiltonian in the falling frame commutes with translations by : the spatial period of the kicks The Quasi-momentum is conserved Evolution of the Rotor : is the detuning from exact resonance

Theory of QAM Fishman, Guarneri, Rebuzzini 2002

QAMs as Resonances : classical, nonlinear example

Accelerator Modes Each stable periodic orbit of the map gives rise to an accelerator mode. m/p : winding number p : period of the orbit

Phase Diagram of Quantum Accelerator Modes K

Mode Locking Such regions are termed Arnol’d tongues. A periodically driven nonlinear oscillator with dissipation may eventually adjust to a periodic motion, whose period is rationally related to the period of the driving. The rational “locking ratio” is then stable against small changes of the system’s parameters and so is constant inside regions of the system’s phase diagram. C. Huyghens Such regions are termed Arnol’d tongues. V.I. Arnol’d

Frequency Locking: Circle Map : A popular example: a Periodically forced damped pendulum . Dissipation leads to shrinking of phase area. Motion in 2d phase space eventually collapses onto a 1d line (a circle) wherein the one-period dynamics is given by a Circle Map :

Paradigm: the Sine Circle Map For k<1 any rational winding number is observed in some region of the phase diagram. In that parameter region, all orbits are attracted by a periodic orbit with that very winding number . Such regions are termed Arnol’d Tongues

Tongues of increasing order are exponentially narrow From Jensen, Bak, Bohr PR-A 30 (1984) Tongues of increasing order are exponentially narrow Chaos here Critical line No overlaps here

Arithmetics : Farey Tales J.Farey On a Curious Property of Vulgar Fractions, Phil.Mag. 47 (1816) A Farey Interval is an interval [r,r’] with rational endpoints r=h/k and r’=h’/k’ (both fractions irreducible) such that all rationals h”/k” lying between r and r’ have k” larger than both k and k’ e.g, [1/4 , 1/3] Theorem. The following statements are equivalent : [r,r’] is a Farey interval The fraction with the smallest divisor, to be found inbetween h/k and h’/k’, is the fraction (h+h’)/(k+k’). This is called the Farey Mediant of h/k and h’/k’ .

Phase Diagram of Quantum Accelerator Modes : Tongues K

Farey approximation: getting better and better rational approximants, at the least cost in terms of divisors. 1/1 0/1 1/2 0/1 1/1 0/1 1/1 1/3 1/2 Continuing this construction a sequence of nested red intervals is generated . These are Farey intervals and their endpoints are a sequence of rationals, which converges to

The observed modes are the sequence of Farey rational approximants to the number

Fibonacci sequence of QAMs

QAMs as resonances II: quantum metastable states Decay of population inside a QAM with time : totally unitary dynamics