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

3. Kicked Cold Atoms Cs

4. 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

5. Bloch 1 2 ? 3

6. 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

7. c
895 nm
Experiments at Oxford: the Kicked Accelerator
GRAVITY

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

10. Hamiltonians for kicked atoms

11. 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

12. Theory of QAM Fishman, Guarneri, Rebuzzini 2002

13. QAMs as Resonances : classical, nonlinear
example

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

15. Phase Diagram of Quantum Accelerator ModesK

16. 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

17. 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 :

18. 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

19. 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

24. 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’ .

25. Phase Diagram of Quantum Accelerator Modes : TonguesK

27. 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

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

29. Fibonacci sequence of QAMs

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