Experiment F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, Phys. Rev. Lett. 25, 4598 (1995) 1. Laser cooling of Na Atoms 2. Driving Electric fielddipole potential On center of mass 3. Detection of momentum distribution

R. Blumel, S. Fishman and U. Smilansky, J. Chem. Phys., 84, 2604 (1986).

Figure 2: Average energy of the rotor as a function of time for k=2 and Δ(t)=Δ (N=7) (t). (a) Quantum mechanical calculations for the localized  2) and extended  2π/3) case, (b) Classical calculation  2). R. Blumel, S. Fishman and U. Smilansky, J. Chem. Phys., 84, 2604 (1986).

Figure 3 R. Blumel, S. Fishman and U. Smilansky, J. Chem. Phys., 84, 2604 (1986).

Figure 4: Some quasi- energy states characterized by a large overlap with the rotor ground state |0> for interaction strength k=2 (a)  =2, (b)  =2π/3. R. Blumel, S. Fishman and U. Smilansky, J. Chem. Phys., 84, 2604 (1986).

Experiment F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, Phys. Rev. Lett. 25, 4598 (1995) 1. Laser cooling of Na Atoms 2. Driving Electric fielddipole potential On center of mass 3. Detection of momentum distribution

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 <

Moore, … Raizen PRL 75, 4598 (1995) Observed localization Indeed Quantum

For values of where there are accelerator Modes – No exponential localization Remember motion bounded in momentum Klappauf…. Raizen PRL 81, 4044 (1998)

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

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

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

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