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