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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 e g L d E E Electric field potential dipole E d E 2 On center of mass x 3. Detection of momentum distribution V cos Gx (t mT ) m 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 (t =2) and extended (t =2π/3) case, (b) Classical calculation (t =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 quasienergy states characterized by a large overlap with the rotor ground state |0> for interaction strength k=2 (a) t =2, (b) t =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 e g L d E E Electric field potential dipole E d E 2 On center of mass x 3. Detection of momentum distribution V cos Gx (t mT ) m pˆ 2 H = K cos x (t mT ) 2M m 1 2 H = t pˆ k cos x (t m) 2 m kicked rotor 0 = x 2 classical K = kt typical K K 2 l diffusion in t = typical t / 2 =rational x p diffusion in acceleration p = integer quantum kicked particle Localization in p acceleration p arbitrary p pn p resonances Localization in p resonances only for few initial conditions 1 (momentum) 2 2 k2 t 2 2 < t momentum t F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, PRL 75, 4598 (1995) Moore, … Raizen PRL 75, 4598 (1995) Observed localization Indeed Quantum For values of K 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, 013411 (2000), Phys. Rev. Lett. 83, 4447 (1999) Related experiments by M. Raizen and coworkers 1. Laser cooling of Cs Atoms 2. Driving e g L d E E Electric field potential x E d E 2 V cos Gx (t mT ) m Mgx 3. Detection of momentum distribution On center of mass dipole Experimental results relative to free fall any structure? t / 2 1 67 s p=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?? Experiment-kicked atoms in presence of gravity p2 H = Mgx (1 cos Gx (t mT ) 2M 2 m G = 4 / = 895nm Gx x dimensionless units T 66.5 s l t /T t H 1 2 H = t pˆ x k cos x (t m) 2 m TG 2 t = M k= in experiment x NOT periodic 2 k MT = g G 0.1 quasimomentum NOT conserved x NOT periodic quasimomentum NOT conserved gauge transformation to restore periodicity t = 2 l l= integer introduce fictitious classical limit where 1 plays the role of Gauge Transformation 1 2 H I = t pˆ x k cos x (t m) 2 m 1 2 H II = t ( pˆ t k cos x (t m) 2 m same classical equation for x i = H I t i xt ( x, t ) = e ( x, t ) i = H II t For H II momentum relative to free fall ( t ) quasimomentum conserved p = n n̂ = i = x mod(2 ) Uˆ = Uˆ kick Uˆ free Quantum Evolution ik cos ˆ U kick = e Uˆ free = e Uˆ free = e Uˆ free = e 1 i t nˆ 2 nˆt ( t / 2 2 t = 2 l up to terms independent of operators but depending on i n2l e 1 ˆ2 ˆ i n n l nˆt ( t / 2 2 i ˆ2 ˆ I I ( l t ( t / 2) | | 2 Uˆ kick = e “momentum” i k | | cos =e i nl = sign( ) Iˆ =| | nˆ = i | | k = k | | ˆI = i | | “momentum” Uˆ = e i k cos | | e i | | k = k | | destroys localization H 1 ˆ2 ˆ H = I I ( l t ( t / 2) 2 dynamics of a kicked system where | | plays the role of | |=effective Planck’s constant quantization p i x Fictitious classical mechanics useful for meaningful “classical limit” dequantization i | | I | | 1 near resonance -classical dynamics H = H k cos (t m) m It 1 = It k sin t 1 change variables t 1 = t It tt l t t / 2 Jt = It tt l t t / 2 Jt 1 = Jt k sin t 1 t motion on torus = mod(2 ) t 1 = t J t J = J mod(2 ) Accelerator modes Jt 1 = Jt k sin t 1 t motion on torus t 1 = t J t J = J mod(2 ) = mod(2 ) Solve for stable classical periodic orbits follow wave packets in islands of stability stable quantum accelerator mode period 1 (fixed points): J0 = 0 solution requires choice of and -classical periodic orbit sin 0 = t / k 0 accelerator mode n = n0 tt / Color --- Husimi (coarse grained Wigner) black -classics Color-quantum Lines classical Experimental results relative to free fall any structure? t / 2 1 67 s p=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?? Color-quantum Lines classical P e e A/ t e decay mode A /| | transient decay rate Accelerator mode spectroscopy map: Jt 1 = Jt k sin t 1 t period motion on torus n = I / | | p fixed point t 1 = t Jt J p = J 0 2 j p = 0 2 n 2 | j | n = n0 t | | p Higher accelerator modes: = difference from rational ( p, j ) = (period, jump in momentum) observed in experiments j/ p Acceleration proportional to as Farey approximants of gravity in some units t = mod(1) 2 ( p, j ) = (5, 2) -classics (10,1) t = 60 color-quantum black- experiment classical Farey Rule 0 1 0 1 1 2 0 1 0 1 1 1 1 1 1 1 1 4 1 3 1 2 2 3 1 3 1 2 2 3 j ( p, j ) p 3 4 1 1 Boundary of existence of periodic orbits j k = 2 p p Boundary of stability width of tongue 1 3/ 2 p km “size” of tongue decreases with p Farey hierarchy natural 1 p After 30 kicks k = 0.3902.. k Summary of results 1. Fictitious classical mechanics to describe quantum resonances takes into account quantum symmetries: conservation of quasimomentum and i n2l i nl e =e 2. Accelerator mode spectroscopy and the Farey hierarchy