Quantum Chaos and Atom Optics: from Experiments to Number Theory Italo Guarneri, Laura Rebuzzini, Michael Sheinman Sandro Wimberger, Roberto Artuso and S.F., Advice and comments: M.V. Berry, Y. Gefen, M. Raizen, W. Phillips, D. Ullmo, P.Schlagheck, E. Narimanov Experiments: M. d’Arcy, G. Summy, M. Oberthaler, R. Godun, Z.Y. Ma Collaborators: K. Burnett, A. Buchleitner, S.A. Gardiner, T. Oliker, M. Sheinman, R. Hihinishvili, A. Iomin

Quantum Chaos Atom Optics Kicked Rotor Classical Diffusion (1979 ) Quantum Deviations from classical behavior Anderson localization (1958,1982) Observation of Anderson localization for laser cooled Cs atoms (Raizen, 1995) Effects of gravity, Oxford 1999 New resonance Fictitious Classical mechanics Far from the classical limit (2002) Quantum nonlinear resonance Short wavelength perturbation

Experiment R.M. Godun, M.B.d’Arcy, M.K. Oberthaler, G.S. Summy and K. Burnett, Phys. Rev. A 62, (2000), Phys. Rev. Lett. 83, 4447 (1999) Related experiments by M. Raizen and coworkers 1. Laser cooling of Cs Atoms 2. Driving Electric fielddipole potential On center of mass 3. Detection of momentum distribution

relative to free fall any structure? =momentum Accelerator mode What is this mode? Why is it stable? What is the decay mechanism and the decay rate? Any other modes of this type? How general?? Experimental results

Kicked Rotor Model Dimensionless units

Classical Motion Standard Map Assume Accelerated, also vicinity accelerated Robust, holds also for vicinity of kick

For typical kick Effectively random Diffusion in For values of Where acceleration, it dominates NonlinearityAccelerator modes robust

Classical Motion Standard Map For typical Effectively random Diffusion in for integer Diffusion Acceleration for examplesome and vicinity accelerated

Quantum Evolution operator rational Quantum resonance irrational pseudorandom Anderson localization like for 1D solids with disorder Anderson localization

Quantum classical quantum Eigenstates of Exponentially localized Anderson localization like for 1D solids with disorder rationalQuantum resonance Simple resonances: Talbot time irrational pseudorandom

Kicked Particle rotor Classical-similar to rotor Quantum :Not quantized periodictransitions fractional part of (quasimomentum ) CONSERVED rational, resonance only for few values of classical quantum Anderson localization irrational

kicked rotor kicked particle typical diffusion in acceleration integer arbitrary typical Localization in rational resonances resonances only for few initial conditions classical quantum

F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, PRL 75, 4598 (1995) momentum (momentum) 2 <

Effect of Gravity on Kicked Atoms Quantum accelerator modes A short wavelength perturbation superimposed on long wavelength behavior

Experiment-kicked atoms in presence of gravity dimensionless units in experiment NOT periodicquasimomentum NOT conserved

NOT periodicquasimomentum NOT conserved gauge transformation to restore periodicity integer introduce fictitious classical limit whereplays the role of

Gauge Transformation same classical equation for Formomentum relative to free fall quasimomentumconserved

Quantum Evolution “momentum” up to terms independent of operators but depending on

“momentum” quantization effective Planck’s constant dequantization Fictitious classical mechanics useful fornear resonance destroys localization dynamics of a kicked system whereplays the role of meaningful “classical limit”

-classical dynamics motion on torus change variables

Accelerator modes motion on torus Solve for stable classical periodic orbits follow wave packets in islands of stability quantum accelerator mode stable -classical periodic orbit period 1 (fixed points): solution requires choice ofand accelerator mode

Color --- Husimi (coarse grained Wigner) -classicsblack

Color-quantum Lines classical

relative to free fall any structure? =momentum Accelerator mode What is this mode? Why is it stable? What is the decay mechanism and the decay rate? Any other modes of this type? How general?? Experimental results

Color-quantum Lines classical

decay rate transient decay mode

Accelerator mode spectroscopy period fixed point Higher accelerator modes: (period, jump in momentum) observed in experiments motion on torus map: as Farey approximants of gravity in some units Acceleration proportional to difference from rational

-classics

color-quantum black- classical experiment

Farey Rule

Boundary of existence of periodic orbits Boundary of stability width of tongue “size” of tongue decreases with Farey hierarchy natural

After 30 kicks

Tunneling out of Phase Space Islands of Maps Resonance Assisted Decay of Phase Space Islands

Numerical data Analytical approximation (ground state) Continuum formula

Effects of Interatomic Interactions

linear focusing defocusing

linear attractive repulsive

position momentum maximum

initial linear focusing defocusing

Probability inside island Number of non-condensed particles Stability

Summary of results 1. Fictitious classical mechanics to describe quantum resonances takes into account quantum symmetries: conservation of quasimomentum and 2. Accelerator mode spectroscopy and the Farey hierarchy 3. Islands stabilized by interactions 4. Steps in resonance assisted tunneling