1. QUANTUM CHAOS : Last Glows at Sunset QUANTUM CHAOS

2. : Italo Guarneri Center for Nonlinear and Complex Systems Universita’ dell’Insubria a COMO - Italia Quantum Accelerator Modes Shmuel Fishman Haifa L.Rebuzzini Como M.Sheinman Haifa S Wimberger Heidelberg A Buchleitner Freiburg M.B. d’Arcy Oxford G.Summy Oxford Talk given at the 98 th Statistical Mechanics Conference, Rutgers University NJ, Dec 2007

3. Kicked Cold Atoms Moore, Robinson, Bharucha, Sundaram, Raizen 1995….. Cs Boris V. Chirikov

4. 1 2 3 Bloch

5. Bloch Theory The one-period evolution operator 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 Unitary Evolution of the Rotor in - representation:

6. 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 The Classical Kicked Rotor: Unbounded Diffusion in Momentum at

7. c GRAVITY Experiments at Oxford: the Kicked Accelerator 895 nm MK Oberthaler RM Godun MB d’Arcy GS Summy K Burnett PRL 83 (99) 4447

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

10. Hamiltonians for kicked atoms S Fishman I Guarneri L Rebuzzini Phys Rev Lett 89 (2002) 0841011 J Stat Phys 110 (2003) 911

12. Bloch Theory The 1-period evolution in the falling frame commutes with translations in space by the spatial period of the kicks Quasi-momentum is conserved Evolution of the Rotor : is the detuning from exact resonance

13. Pseudoclassical Limit The small- asymptotics is the same as a quasi-classical asymptotics using as the Planck’s constant. In this “ epsilon - classical limit” the map over one period is Is the gravity acceleration with time and space given in units of the time- and space kicking periods

14. QAMs as Resonances : classical, nonlinear

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

16. Phase Diagram of Quantum Accelerator Modes I I Guarneri L Rebuzzini S Fishman Nonlinearity 19 (2006) 1141 K

18. Mode Locking 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. Such regions are termed Arnol’d tongues. C. Huyghens V.I. Arnol’d

19. 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

20. Farey approximation : getting better and better rational approximants, at the least cost in terms of divisors. 1/1 0/1 1/1 1/2 1/3 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 1/2 0/1 1/1

21. The observed modes are the sequence of Farey rational approximants to the number A Buchleitner MB d’Arcy S Fishman S Gardiner I Guarneri ZY Ma L Rebuzzini GS Summy Phys Rev Lett 96 (2006) 164101

22. Fibonacci sequence of QAMs Fibonacci sequence of QAMs

23. Decay of QAM due to tunneling through invariant tori QAM associated with non-unimodular eigenvalues of the unitary evolution operator. Relation to Wannier-Stark resonances M.Sheinman, S Fishman, I Guarneri, L Rebuzzini Phys Rev A 73 (2006) 052110 Effects of Interactions (Gross-Pitaevski )

24. Arithmetics : Farey Tales J.Farey On a Curious Property of Vulgar Fractions, Phil.Mag. 47 (1816) 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’. 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]