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QUANTUM CHAOS :

QUANTUM CHAOS :. QUANTUM CHAOS. Last Glows at Sunset. Quantum Accelerator Modes. : Italo Guarneri Center for Nonlinear and Complex Systems Universita’ dell’Insubria a COMO - Italia. M.B. d’Arcy Oxford G.Summy Oxford. Shmuel Fishman Haifa L.Rebuzzini Como

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QUANTUM CHAOS :

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  1. QUANTUM CHAOS : QUANTUMCHAOS Last Glows at Sunset

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

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

  4. Bloch 1 2 3

  5. Bloch Theory • The one-period evolution operator commutes with translations by : the spatial period of the kicks • TheQuasi-momentumis 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 Classical Kicked Rotor: Unbounded Diffusion in Momentum at 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 MK Oberthaler RM Godun MB d’Arcy GS Summy K Burnett PRL 83 (99) 4447 895 nm Experiments at Oxford: the Kicked Accelerator GRAVITY

  8. Quantum Accelerator Modes Atomic momentum The atoms arefarfrom the classicallimit, and the modes areabsentin theclassical limit !!! Pulse period

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

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

  11. 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

  12. QAMs as Resonances : classical, nonlinear

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

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

  15. 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. C. Huyghens Such regions are termed Arnol’d tongues. V.I. Arnol’d

  16. 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

  17. 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 1/1 0/1 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

  18. 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

  19. Fibonacci sequence of QAMs

  20. 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 iscalledthe FareyMediant of h/k and h’/k’ .

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