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