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