Non-dispersive wave packets in periodically driven quantum systems

536 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
of classical phase space, manifesting in an abundance of avoided crossings of various sizes in the
level dynamics. Still, as exempli ed in Figs. 9, 30 and 59, the wave-packet states may be followed
rather easily under changes of F (parametrized by t, during the switching of the pulse) in the level
dynamics, in agreement with their “solitonic” character (see Section 7.3). Nontheless, the very same
gures illustrate clearly that the targeted wave-packet state undergoes many avoided crossings as
the microwave amplitude is swept. To remain in a single eigenstate, the avoided crossings should
be passed either adiabatically or diabatically, with a branching ratio at an individual crossing being
described by the well known Landau–Zener scenario [7,227]. Consequently, if we want to populate
an individual wave-packet eigenstate from a eld-free atomic state | 0〉, we need some knowledge of
the energy level dynamics. Then it is possible to identify those eld-free states which are connected
to the wave packet via adiabatic and=or diabatic transitions in the network of energy levels, and
subsequently to design F(t) such as to transfer population from | 0〉 to |E〉 most e ciently. A
precise experimental preparation of | 0〉 is the prerequisite of any such approach.
When the driving eld is increased from zero, the major modi cation in the classical phase space
is the emergence of the resonance island (see Figs. 22, 25 and 26). Quantum mechanically, the
states with initial principal quantum number close to n0 = ! −1=3 will enter progressively inside the
resonance island. For a one-dimensional system, the Mathieu equation, discussed in Section 3.1.4,
fully describes the evolution of the energy levels in this regime. As shown for example in Fig. 9,
the non-dispersive wave packet with the best localization, i.e., N = 0, is—in this simple situation—
adiabatically connected to the eld-free state with principal quantum number closest to n0, i.e. the
eigenstate = 0 of the Mathieu equation. When the Mathieu parameter q, Eq. (100), is of the order
of unity, the state of interest is trapped in the resonance island, which happens at eld amplitudes
given by Eq. (150) for the one-dimensional atom, and for linear polarization of the microwave eld.
A similar scaling is expected for other polarizations, too. In the interval F 6 Ftrapping, the eld has to
be increased slowly enough such as to avoid losses from the ground state to the excited states of the
Mathieu equation, at an energy separation of the order of n −4
0 . The most favorable situation is then
the case of “optimal” resonance (see Section 3.1.2), when n0 is an integer, the situation in Fig. 9.
The wave-packet state is always separated from the other states by an energy gap comparable to its
value at F = 0, i.e., of the order of 3=(2n4 0 ). The situation is less favorable if n0 is not an integer,
because the energy gap between the wave packet of interest and the other states is smaller when
F → 0. The worst case is met when n0 is half-integer: the free states n0 +1=2 and n0 − 1=2 are
quasi-degenerate, and selective excitation of a single wave packet is thus more di cult.
The appropriate time scale for switching on the eld is given by the inverse of the energy splitting,
i.e., for “optimal” resonance
trapping ∼ n 4 0 = n0 × 2 =! (278)
or n0 driving eld periods.
Once trapped in the resonance island, the coupling to states localized outside the island will be
residual—mediated by quantum mechanical tunneling, see Section 7.1—and the size of the avoided
crossings between the trapped and the untrapped states is exponentially small. After adiabatic switching
into the resonance island on a time-scale of n0 Kepler orbits, we now have to switch diabatically
from Ftrapping to some nal F value, in order to avoid adiabatic losses from the wave packet into
other states while passing through the avoided crossings.