Non-dispersive wave packets in periodically driven quantum systems

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504 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 40. A non-dispersive s = 1 wave packet of the gravitational bouncer, Eq. (249). The quantum number of the resonant state is chosen as n0 = 1000, to match typical experimental dimensions [158]. The left column shows the time evolution of the wave packet for !t =0; =2; ;3 =2 (from top to bottom). The right column shows the corresponding phase space (Husimi, see Eq. (249)) representation (z-axis horizontal as in the left column, momentum p on the vertical axis). The parameters are ! 0:1487, =0:025. The periodic, nondispersive dynamics of the wave packet bouncing o the mirror in the gravitational eld is apparent. while the full, time-dependent Hamiltonian reads (3 I)2=3 I H = + 2 2=3 ∞ 2 (3I)2=3 sin(!t) − sin(!t) (3 ) 1=3 4=3 Thus, the resonant action (226) is given by Î s = 2 s 3 3! 3 with the associated strength of the e ective coupling n=1 cos(n ) n 2 : (253) (254) Vs = (3I)2=3 : (255) s2 4=3 Using the framework of Sections 3.1.1, 3.1.2 (for s = 1), and 5.1 (for s¿1), the reader may easily compute the various properties of non-dispersive wave packets in this system. 29 An example for s=1 is presented in Fig. 40, for the resonant principal quantum number n0=1000, (i.e., Î 1=1000:75) where both, the (time-periodic) probability densities in con guration and phase space are shown. Note that such high n0 values (or even higher) correspond to typical experimental falling heights (around 0:1 mm for n0 = 1000) in experiments on cold atoms [158]. Therefore, the creation of an atomic wave packet in such an experiment would allow to store the atom in a quasi-classical 29 There is, however, a tricky point: the Maslov index in this system is 3, with a contribution 1 coming from the outer turning point, and 2 from z = 0, since the oscillating plane acts as a hard wall. Hence, the relation between the principal quantum number and the action is I1 = n +3=4, see Eq. (38).
|ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0.0 0.3 0.2 0.1 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 505 0.0 0 100 200 0 100 200 z z |Tunneling splitting| 10 0 10 -5 10 -10 10 -4 10 -9 10 -14 0.0 0.1 0.2 0.3 λ Fig. 41. Floquet eigenstates anchored to the ! =2 resonance in the gravitational bouncer (left column), in con guration space, at !t =0. Each eigenstate exhibits two wave packets shifted by a phase along the classical trajectory, to abide the Floquet periodicity imposed by Eq. (74). Symmetric and antisymmetric linear combinations of these states isolate either one of the wave packets, which now evolves precisely like a classical particle (right column), periodically bouncing o the wall at z = 0. The parameters are ! 0:2974, (corresponding to n0 = 1000, for the 2:1 resonance), =0:025. Fig. 42. Tunneling splitting between the energies (modulo !=2) of two Floquet states of the gravitational bouncer, anchored to the s=2 resonance island, as a function of the driving amplitude . Driving frequency ! 1:0825 (top) and ! 0:8034 (bottom), corresponding to resonant states n0 = 20 and n0 = 50, respectively. The dashed line reproduces the prediction of Mathieu theory [42], Eq. (256). Observe that the latter ts the exact numerical data only for small values of .At larger ; small avoiding crossings between one member of the doublet and eigenstates originating from other manifolds dominate over the pure tunneling contribution and the Mathieu prediction is not accurate. “bouncing mode” over arbitrarily long times, and might nd some application in the eld of atom optics [159]. For s = 2, we expect, following the general discussion in Section 5.1, two quasi-energy levels, separated by !=2, according to the semiclassical result, Eq. (239), which are both associated with the s=2 resonance. As a matter of fact, such states are born out from an exact numerical diagonalization of the Floquet Hamiltonian derived from Eq. (249). An exemplary situation is shown in Fig. 41, for n0=1000 (i.e., Î 2=1000:75, in Eq. (254)). The tunneling coupling between the individual wave packets shown in the right column of Fig. 41 is given by the tunneling splitting between the energies of both associated Floquet states (left column of Fig. 41) modulo !=2. From the Mathieu approach— Eqs. (247) and (248)—this tunneling coupling is directly related to the variations of the Mathieu eigenvalues when the characteristic exponent is changed. In the limit where the resonance island is big enough, q1 in Eq. (247), asymptotic expressions [95] allow for the following estimate [42]: = 8√ 2 3=4 √ ! exp √ 16 − ! 3 = 8[3(n0 1=6 3=4 +3=4)] exp(−6(n0 +3=4) 4=3 √ = ) ; (256) which we can test with our numerically exact quantum treatment. The result is shown in Fig. 42, for two di erent values of n0.
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Non-dispersive wave packets in periodically driven quantum systems

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