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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A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 435<br />
started <strong>in</strong> the resonance island will be locked on the phase of the driv<strong>in</strong>g eld. This is the very<br />
orig<strong>in</strong> of the phase lock<strong>in</strong>g phenomenon discussed <strong>in</strong> Section 1.4 above. A crucial po<strong>in</strong>t here<strong>in</strong> is<br />
that the resonance island covers a signi cant part of phase space with nite volume: it is the whole<br />
structure, not few trajectories, which is phase locked. This is why, further down, we will be able to<br />
build <strong>quantum</strong> <strong>wave</strong> <strong>packets</strong> on this structure, which will be phase locked to the classical orbit and<br />
will not spread. The classical version of a non-spread<strong>in</strong>g <strong>wave</strong> packet thus consists of a family of<br />
trajectories, trapped with<strong>in</strong> the resonance island, such that this family is <strong>in</strong>variant under the evolution<br />
generated by the pendulum Hamiltonian. The simplest possibility is to sample all trajectories with<strong>in</strong><br />
the “energy” range [H0(Î 1) − !Î 1 −| V1(Î 1)|;H0(Î 1) − !Î 1 + | V1(Î 1)|] of the pendulum. In real<br />
space, this will appear as a localized probability density follow<strong>in</strong>g the classical stable periodic orbit,<br />
reproduc<strong>in</strong>g its shape exactly after each period of the drive.<br />
So far, to derive the characteristics of the resonance island, we have consistently used rst-order<br />
perturbation theory, which is valid for small . At higher values of , higher-order terms come <strong>in</strong>to<br />
play and modify the shape and the precise location of the resonance island. However, it is crucial to<br />
note that the island itself considered as a structure is robust, and will survive up to rather high values<br />
of (as a consequence of the KAM theorem [3]). S<strong>in</strong>ce the size of the resonance island grows with<br />
| |, Eq. (71), the island may occupy a signi cant area <strong>in</strong> phase space and eventually <strong>in</strong>teract with<br />
islands associated with other resonances, for su ciently large | |. The mechanism of this “resonance<br />
overlap” is rather well understood [88,89]: <strong>in</strong> general, the motion close to the separatrix (where the<br />
period of the classical motion of the pendulum tends to <strong>in</strong> nity) is most sensitive to higher-order<br />
corrections. The general scenario is thus the non-<strong>in</strong>tegrable perturbation of the separatrix and the<br />
emergence of a “stochastic” layer of chaotic motion <strong>in</strong> phase space, as | | is <strong>in</strong>creased. At still larger<br />
values of | |, chaos may <strong>in</strong>vade large parts of phase space, and the resonance island may shr<strong>in</strong>k and<br />
nally disappear. While consider<strong>in</strong>g realistic examples later on, we shall enter the non-perturbative<br />
regime. Let us, however, consider rst the <strong>quantum</strong> perturbative picture.<br />
3.1.2. Quantum dynamics<br />
As shown <strong>in</strong> the previous section, the dynamics of a one-dimensional system exposed to a weak,<br />
resonant, periodic driv<strong>in</strong>g is essentially regular and analogous to the one of a pendulum, Eq. (67)<br />
(<strong>in</strong> the rotat<strong>in</strong>g frame, Eqs. (60)–(62)). In the present section, we will show that the same physical<br />
picture can be employed <strong>in</strong> <strong>quantum</strong> mechanics, to construct non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>. They will<br />
follow the stable classical trajectory locked on the external drive, and exactly reproduce their <strong>in</strong>itial<br />
shape after each period.<br />
Our start<strong>in</strong>g po<strong>in</strong>t is the time-dependent Schrod<strong>in</strong>ger equation associated with Hamiltonian (54): 8<br />
d| (t)〉<br />
i˝<br />
dt =(H0 + V cos !t)| (t)〉 : (72)<br />
S<strong>in</strong>ce Hamiltonian (54) is periodic <strong>in</strong> time, the Floquet theorem 9 guarantees that the general solution<br />
of Eq. (72) is given by a l<strong>in</strong>ear comb<strong>in</strong>ation of elementary, time-periodic states—the so-called<br />
8<br />
For simplicity, we use the same notation for classical and <strong>quantum</strong> quantities, the dist<strong>in</strong>ction between them will<br />
become clear from the context.<br />
9<br />
The Floquet theorem [90] <strong>in</strong> the time doma<strong>in</strong> is strictly equivalent to the Bloch theorem for potentials periodic <strong>in</strong><br />
space [91].