Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
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|ψ (z)| 2<br />
|ψ (z)| 2<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
0.3<br />
0.2<br />
0.1<br />
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 505<br />
0.0<br />
0 100 200 0 100 200<br />
z z<br />
|Tunnel<strong>in</strong>g splitt<strong>in</strong>g|<br />
10 0<br />
10 -5<br />
10 -10<br />
10 -4<br />
10 -9<br />
10 -14<br />
0.0 0.1 0.2 0.3<br />
λ<br />
Fig. 41. Floquet eigenstates anchored to the ! =2 resonance <strong>in</strong> the gravitational bouncer (left column), <strong>in</strong> con guration<br />
space, at !t =0. Each eigenstate exhibits two <strong>wave</strong> <strong>packets</strong> shifted by a phase along the classical trajectory, to abide the<br />
Floquet periodicity imposed by Eq. (74). Symmetric and antisymmetric l<strong>in</strong>ear comb<strong>in</strong>ations of these states isolate either<br />
one of the <strong>wave</strong> <strong>packets</strong>, which now evolves precisely like a classical particle (right column), <strong>periodically</strong> bounc<strong>in</strong>g o<br />
the wall at z = 0. The parameters are ! 0:2974, (correspond<strong>in</strong>g to n0 = 1000, for the 2:1 resonance), =0:025.<br />
Fig. 42. Tunnel<strong>in</strong>g splitt<strong>in</strong>g between the energies (modulo !=2) of two Floquet states of the gravitational bouncer, anchored<br />
to the s=2 resonance island, as a function of the driv<strong>in</strong>g amplitude . Driv<strong>in</strong>g frequency ! 1:0825 (top) and ! 0:8034<br />
(bottom), correspond<strong>in</strong>g to resonant states n0 = 20 and n0 = 50, respectively. The dashed l<strong>in</strong>e reproduces the prediction<br />
of Mathieu theory [42], Eq. (256). Observe that the latter ts the exact numerical data only for small values of .At<br />
larger ; small avoid<strong>in</strong>g cross<strong>in</strong>gs between one member of the doublet and eigenstates orig<strong>in</strong>at<strong>in</strong>g from other manifolds<br />
dom<strong>in</strong>ate over the pure tunnel<strong>in</strong>g contribution and the Mathieu prediction is not accurate.<br />
“bounc<strong>in</strong>g mode” over arbitrarily long times, and might nd some application <strong>in</strong> the eld of atom<br />
optics [159].<br />
For s = 2, we expect, follow<strong>in</strong>g the general discussion <strong>in</strong> Section 5.1, two quasi-energy levels,<br />
separated by !=2, accord<strong>in</strong>g to the semiclassical result, Eq. (239), which are both associated with the<br />
s=2 resonance. As a matter of fact, such states are born out from an exact numerical diagonalization<br />
of the Floquet Hamiltonian derived from Eq. (249). An exemplary situation is shown <strong>in</strong> Fig. 41, for<br />
n0=1000 (i.e., Î 2=1000:75, <strong>in</strong> Eq. (254)). The tunnel<strong>in</strong>g coupl<strong>in</strong>g between the <strong>in</strong>dividual <strong>wave</strong> <strong>packets</strong><br />
shown <strong>in</strong> the right column of Fig. 41 is given by the tunnel<strong>in</strong>g splitt<strong>in</strong>g between the energies<br />
of both associated Floquet states (left column of Fig. 41) modulo !=2. From the Mathieu approach—<br />
Eqs. (247) and (248)—this tunnel<strong>in</strong>g coupl<strong>in</strong>g is directly related to the variations of the Mathieu<br />
eigenvalues when the characteristic exponent is changed. In the limit where the resonance island is<br />
big enough, q1 <strong>in</strong> Eq. (247), asymptotic expressions [95] allow for the follow<strong>in</strong>g estimate [42]:<br />
= 8√ 2 3=4<br />
√ ! exp<br />
√<br />
16<br />
−<br />
! 3<br />
<br />
= 8[3(n0<br />
1=6 3=4<br />
+3=4)]<br />
exp(−6(n0 +3=4) 4=3 √ = ) ; (256)<br />
which we can test with our numerically exact <strong>quantum</strong> treatment. The result is shown <strong>in</strong> Fig. 42,<br />
for two di erent values of n0.