Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 415
At still longer times, a pure quantum phenomenon appears, due to the discrete nature of the energy
spectrum. Since the (n−n0) 2 factors in Eq. (15) are all integers, the second-order contributions to the
phase are all integer multiples of the phase of the n − n0 = 1 term. If the latter is an integer multiple
of 2 , all the second-order contributions will rephase, inducing a revival of the wave packet in its
original shape. A re ned estimation of the revival time actually shows that this analysis overestimates
the revival time by a factor two. 2 The correct result is [8–11]
2 ˝
Trevival =
(d2En=dn 2 : (18)
)(n0)
Based on the very elementary considerations above, we can so far draw the following conclusions:
• An initially localized wave packet will follow the classical equations of motion for a nite time
t ∼ Trecurrence;
• its localization properties cannot be stationary as time evolves;
• in general, the initial quasi-classical motion is followed by collapse and revival, with the corresponding
time-scales Trecurrence ¡Tcollapse ¡Trevival.
In the sequel of this report, we will show how under very general conditions it is indeed possible
to create wave packets as single eigenstates of quantum systems, i.e., as localized ground states
in an appropriately de ned reference frame. The most suitable framework is to consider quantum
evolution in classical phase space, that provides a picture which is independent of the choice of
the basis and allows for an immediate comparison with the classical Hamiltonian ow. In addition,
such a picture motivates a semiclassical interpretation, which we will expand upon in Section 2. The
appropriate technical tool for a phase space description are quasiprobability distributions [12] as the
Wigner representation of the state | (t)〉,
W (˜r; ˜p)=
1
(2 ˝) f
∗
˜r + ˜x
˜r −
2
˜x
˜x · ˜p
exp i d
2 ˝
f ˜x; (19)
where f is the number of degrees of freedom.
The Wigner density W (˜r; ˜p) is real, but not necessarily positive [12,13]. Its time evolution follows
from the Schrodinger equation [12]:
9W (˜r; ˜p; t)
˝
˝ = −2H(˜r; ˜p; t) sin W (˜r; ˜p; t) ; (20)
9t
2
where
→
∇˜p ∇˜r − ← →
∇˜r ∇˜p
= ←
and the arrows indicate in which direction the derivatives act. Eq. (20) can serve to motivate
the semiclassical approach. Indeed, the sin function can be expanded in a Taylor series, i.e., a
power expansion in ˝. At lowest non-vanishing order, only terms linear in contribute and
2 It must also be noted that, at simple rational multiples (such as 1
3
, 1
2
(21)
2
, ) of the revival time, one observes “fractional
3
revivals” [8–11], where only part of the various amplitudes which contribute to Eq. (13) rephase. This generates a wave
function split into several individual wave packets, localized at di erent positions along the classical orbit.