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) 409–547 495<br />
Section 3.4. A detailed stability analysis can be found <strong>in</strong> [30,46,154] and we summarize here the<br />
ma<strong>in</strong> results only.<br />
S<strong>in</strong>ce chang<strong>in</strong>g the sign of F <strong>in</strong> Eq. (214) is equivalent to chang<strong>in</strong>g the sign of x from positive<br />
to negative, we only consider the equilibrium position at xeq ¿ 0 (compare Eqs. (185)–(188)). For<br />
non-vanish<strong>in</strong>g magnetic eld, its position is given by<br />
!(! − !c)xeq − 1<br />
x2 eq<br />
− F =0: (215)<br />
Rede n<strong>in</strong>g the dimensionless parameter q (see Eq. (186)) via<br />
1<br />
q =<br />
!(! − !c)x3 eq<br />
; (216)<br />
we obta<strong>in</strong> for the micro<strong>wave</strong> amplitude<br />
F =[!(! − !c)] 2=3 (1 − q)=q 1=3<br />
and<br />
(217)<br />
Eeq =[!(! − !c)] 1=3 (1 − 4q)=2q 2=3<br />
for the equilibrium energy.<br />
(218)<br />
Harmonic expansion of Eq. (214) around the equilibrium po<strong>in</strong>t (xeq;yeq =0;zeq = 0) allows for<br />
a l<strong>in</strong>ear stability analysis <strong>in</strong> its vic<strong>in</strong>ity. Alike the pure CP case, the z motion decouples from the<br />
motion <strong>in</strong> the (x; y) plane. For the latter, we recover the generic harmonic Hamiltonian discussed <strong>in</strong><br />
Section 3.4, Eq. (191), provided we substitute<br />
˜! = ! − !c=2 : (219)<br />
When expanded at second order around the equilibrium po<strong>in</strong>t, Hamiltonian (214) takes the standard<br />
form of a rotat<strong>in</strong>g anisotropic oscillator, Eq. (191), with ˜! replac<strong>in</strong>g !, and with the stability<br />
parameters:<br />
a = 1<br />
˜! 2<br />
2 ! c<br />
4<br />
b = 1<br />
˜! 2<br />
2 ! c<br />
4<br />
2<br />
−<br />
x3 <br />
eq<br />
1<br />
+<br />
x3 <br />
eq<br />
;<br />
: (220)<br />
The regions of stability of the equilibrium po<strong>in</strong>t (xeq;yeq;zeq) are thus obta<strong>in</strong>ed from the doma<strong>in</strong>s<br />
of stability of the 2D rotat<strong>in</strong>g anisotropic oscillator, given by Eqs. (192) and (193). They are<br />
visualized <strong>in</strong> terms of the physical parameters F and !c, (us<strong>in</strong>g the standard scaled electric eld<br />
F0 = Fn 4 0 = F!−4=3 ) <strong>in</strong> Fig. 39, with the black region correspond<strong>in</strong>g to Eq. (192), and the grey<br />
region to Eq. (193). Observe that the presence of the magnetic eld tends to enlarge the region<br />
of stability <strong>in</strong> parameter space; for !c = 0 (pure CP case, no magnetic eld) the stability region<br />
is quite t<strong>in</strong>y, <strong>in</strong> comparison to large values of |!c|: 24 On the other hand, the stability diagram<br />
24 As long as we are <strong>in</strong>terested <strong>in</strong> long-lived <strong>wave</strong> <strong>packets</strong>, the region of small F0 is of <strong>in</strong>terest only. At higher F0<br />
and for !c ¡!, the strong driv<strong>in</strong>g eld will ionize the atom rather fast—see Section 7.1. This makes the gray region<br />
F0 ¿ 0:1 of little practical <strong>in</strong>terest.