QUANTUM CHAOS :QUANTUM CHAOS :

Last Glows at Sunset

QUANTUM CHAOS

: Italo Guarneri

Center for Nonlinear and Complex Systems

Universita’ dell’Insubria a COMO - Italia

Quantum Accelerator Modes

Shmuel Fishman Haifa L.Rebuzzini ComoM.Sheinman HaifaS Wimberger HeidelbergA Buchleitner Freiburg

M.B. d’Arcy OxfordG.Summy Oxford

Talk given at the 98th Statistical Mechanics Conference, Rutgers University NJ, Dec 2007

1x 2x 3x 4x5x

)sin( 11 nnn xkppnnn pxx 1

Kicked Cold Atoms Moore, Robinson, Bharucha,

Sundaram, Raizen 1995…..

Cs

Boris V. Chirikov

2

1

2

3

Bloch

Bloch TheoryBloch 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:

2

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 1k

cc

GR

AV

ITY

Experiments at Oxford: the Kicked Accelerator

895 nm

MK Oberthaler RM Godun MB d’ArcyGS Summy K BurnettPRL 83 (99) 4447

Quantum Accelerator ModesQuantum Accelerator Modes

The atoms are far from the classical limit, and the modes are absent in the classical limit !!!

Pulse period

Ato

mic

mo

me

ntu

m

Hamiltonians for kicked atomsHamiltonians for kicked atoms

S Fishman I Guarneri L Rebuzzini Phys Rev Lett 89 (2002) 0841011S Fishman I Guarneri L Rebuzzini Phys Rev Lett 89 (2002) 0841011J Stat Phys 110 (2003) 911J Stat Phys 110 (2003) 911

Bloch TheoryBloch 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

Pseudoclassical LimitPseudoclassical 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

kK

QAMs as Resonances : classical, nonlinear

Accelerator ModesAccelerator Modes• Each stable

periodic orbit of the map gives rise to an accelerator mode.

)(||

2

p

monaccelerati

p : period of the orbit

m/p : winding number

Phase Diagram of Quantum Accelerator Modes

I I Guarneri L Rebuzzini S Fishman Nonlinearity 19 (2006) 1141K

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

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

Farey approximation: getting better and better rational approximants, at the least

cost in terms of divisors.

1/10/1

0/1 1/11/2

1/3Continuing 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

1/20/1 1/1

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

32

22

G

gM

A Buchleitner MB d’Arcy S Fishman S Gardiner I Guarneri ZY Ma L Rebuzzini GS SummyPhys Rev Lett 96 (2006) 164101

Fibonacci sequence of QAMsFibonacci sequence of QAMs

Arithmetics : Farey Tales

J.Farey On a Curious Property of Vulgar Fractions, Phil.Mag. 47 (1816)

nk

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]