Pseudoclassical theory for fidelity of nearly resonant quantum rotors

Using a semiclassical ansatz we analytically predict for the fidelity of $\delta$-kicked rotors the occurrence of revivals and the disappearance of intermediate revival peaks arising from the breaking of a symmetry in the initial conditions. A numerical verification of the predicted effects is given and experimental ramifications are discussed.

Figure 1

(Color online) Fidelity as predicted by Eq. (12) because of the singularities of the analytical formula the curve is folded with normalized Gaussians with a standard deviation of $t\approx 6$ kicks (solid black line) and numerical data (gray/green curve) for $k_1=0.8\pi$, $k_2=0.6\pi$, and detuning $ϵ=0.01$ from $\tau -ϵ=2\pi$. In the inset, the nonsmoothed result [Eq. (12)] is shown.

Figure 2

(Color online) Same as in Fig. 1 for an ensemble of 5000 equidistantly chosen rotors (solid gray/green lines) with a width of (a) $\Delta \beta =0.05$ (or $\Delta \overline{\beta }\approx 0.31$) around the resonant value covering half the width of the resonance island in the phase space induced by Eq. (5) and (b) $\Delta \beta =1$ covering the full phase space, compared with the smoothed (see caption of Fig. 1) version of Eq. (11) (solid black lines). In (a) the intermediate revival peaks observed in Fig. 1 disappear as predicted by Eq. (11). The dashed line in (a) reproduces the smoothed analytic formula from Fig. 1. For $\beta$ distributed over the full Brillouin zone in (b), the revivals are barely visible since the average includes many nonresonant rotors performing rotational motion in phase space, which is not decribed by our theory valid just for the librational island motion.