Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 441
dummy variable introduced for convenience coincides with the classical angle variable ˆ . In
general, r is not an integer, so that the various exp(ir ) in Eq. (94) are not periodic functions of .
However, as all r values have the same fractional part, if follows that f( ) must satisfy “modi ed”
periodic boundary conditions of the form
f( +2 ) = exp(−2i n0)f( ) : (96)
The reason for this surprising boundary condition is clear: r =n−n0 is the quantum analog of Î −Î1.
In general, the resonant action Î 1 is not an integer or half-integer multiple of ˝—exactly as n0 is
not an integer. The semiclassical quantization, Eq. (79), which expresses the ˆ periodicity of the
eigenstate, applies for the Î variable. When expressed in terms of the variable Î − Î 1, it contains the
additional phase shift present in Eq. (96).
Few words of caution are in order: the equivalence of the semiclassical quantization with the pure
quantum approach holds in the semiclassical limit only, when the quantum problem can be mapped
on a pendulum problem. In the general case, it is not possible to de ne a quantum angle variable
[94]. Hence, the quantum treatment presented here is no more general or more powerful than the
semiclassical treatment. They both rely on the same approximations and have the same limitations:
perturbative regime (no overlap of resonances) and semiclassical approximation.
Finally, Eq. (95) can be written in its standard form, known as the “Mathieu equation” [95]:
d2y +(a− 2q cos 2v)y =0: (97)
dv2 The correspondence with Eq. (95) is established via
=2v; (98)
a = 8[E − H0(Î 1)+!(Î 1 − ˝=4)]
˝2H ′′
0 (Î 1)
4 V1
q =
˝2H ′′
0 (Î 1)
; (99)
: (100)
The boundary condition, Eq. (96), is xed by the so-called “characteristic exponent” in the Mathieu
equation,
= −2n0 (mod 2) : (101)
The Mathieu equation has solutions (for a given characteristic exponent) for a discrete set of values
of a only. That implies quantization of the quasi-energy levels, according to Eq. (99). The
quantized values a ( ; q) depend on q and , and are labeled 12 by a non-negative integer . They
are well known—especially asymptotic expansions are available both in the small and in the large
q regime—and can be found in standard handbooks [95]. For example, Fig. 9(a) shows the rst
12 In the standard text books as [95], the various solutions of the Mathieu equation are divided in odd and even solutions,
and furthermore in - and 2 -periodic functions. In our case, only the “a2p” and “b2p” (in the language of [95]) are to
be considered.