QUANTUM CHAOS :QUANTUM CHAOS :

Glows at Sunset

QUANTUM CHAOS

1x 2x 3x 4x5x

)sin( 11 nnn xkppnnn pxx 1

Kicked Cold Atoms

Cs

Bloch TheoryBloch Theory• 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 :

2

is the detuning from exact resonance

2

1

2

3

?

Bloch

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

cc

GR

AV

ITY

Experiments at Oxford: the Kicked Accelerator

895 nm

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

mom

entu

m

Hamiltonians for kicked Hamiltonians for kicked atomsatoms

Bloch TheoryBloch Theory• 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 :

2

is the detuning from exact resonance

Theory of QAMTheory of QAMFishman, Guarneri, Rebuzzini 2002Fishman, Guarneri, Rebuzzini 2002

QAMs as Resonances : classical, nonlinear

example

Accelerator ModesAccelerator Modes• Each stable

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

)(||

2pmonaccelerati

p : period of the orbit

m/p : winding number

Phase Diagram of Quantum Accelerator Modes K

Mode LockingMode LockingA 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

Frequency Locking:

)cos()sin( tBAa

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

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)

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]

Phase Diagram of Quantum Accelerator Modes : TonguesK

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

22GgM

Fibonacci sequence of QAMsFibonacci sequence of QAMs

Decay of population inside a QAM with time : totally unitary dynamics

QAMs as resonances II: quantum metastable states