Transient localization in the kicked Rydberg atom

We investigate the long-time limit of quantum localization of the kicked Rydberg atom. The kicked Rydberg atom is shown to possess in addition to the quantum localization time ${\tau }_L$ a second crossover time $t_D$ where quantum dynamics diverges from classical dynamics towards increased instability. The quantum localization is shown to vanish as either the strength of the kicks at fixed principal quantum number or the quantum number at fixed kick strength increases. The survival probability as a function of frequency in the transient localization regime ${\tau }_L<t<t_D$ is characterized by highly irregular, fractal-like fluctuations.

Figure 1

Classical $P_{\mathrm{sur}}^{\mathrm{cl}}$ (dashed line) and quantum $P_{\mathrm{sur}}^{\mathrm{qm}}$ (solid lines) survival probability for the positively kicked Rydberg atom as a function of number $K$ of kicks for $\mid F_0^{\mathrm{av}}\mid =0.005$ $(\Delta p_0\approx 0.021)$. The quantum data are shown for 11 frequencies ${\nu }_0$ uniformly distributed between 1.45 and 1.45008, $n_i=50$. (a) Long-time behavior, the dash-dotted line indicates a fit to a power law with exponent $\alpha =-1.5$, Eq. (6), and (b) short-time behavior.

Figure 2

Mean principal quantum number $⟨n⟩$, Eq. (7), for the same parameters as Fig. 1, ${⟨n⟩}_{\mathrm{cl}}$ (dashed line) and ${⟨n⟩}_{\mathrm{qm}}$ (solid lines). (a) Long-time behavior (of $\log ⟨n⟩$), the dash-dotted line indicates a fit to a power law, and (b) short-time behavior of $⟨n⟩$. The arrow in (b) indicates an approximate localization time defined in Eq. (16).

Figure 3

(Color online) Time-dependent spectral distribution $P(n,t)$, both classically (a) and quantum mechanically (b), for the same parameters as Fig. 2 (data shown for ${\nu }_0=1.45$).

Figure 4

$⟨n⟩$ averaged over the interval $1.45⩽{\nu }_0⩽1.47$. In (a), $n_i=50$, and in (b), $\mid F_0^{\mathrm{av}}\mid =0.02$. Solid line, quantum result, and dashed line, classical result. The arrows indicate the lifetimes of the ground state estimated by direct coupling to the continuum. The average is calculated on the logarithmic scale—i.e., shown is ${10}^{⟨{\log }_{10}n⟩}$.

Figure 5

$P_{\mathrm{sur}}$ averaged over the interval $1.45⩽{\nu }_0⩽1.47$. In (a), $n_i=50$, and in (b), $\mid F_0^{\mathrm{av}}\mid =0.02$. Solid line, quantum result; dashed line, classical result; and dash-dotted line, fit of classical result to a power law. The arrows indicate the lifetimes of the ground state estimated by direct coupling to the continuum. The average is calculated on the logarithmic scale.

Figure 6

As Fig. 1b, however for (a) $\mid F_0^{\mathrm{av}}\mid =0.02$ and $n_i=50$, (b) $\mid F_0^{\mathrm{av}}\mid =0.05$ and $n_i=50$, and (c) $\mid F_0^{\mathrm{av}}\mid =0.02$ and $n_i=200$.

Figure 7

As Fig. 2b, however for (a) $\mid F_0^{\mathrm{av}}\mid =0.02$ and $n_i=50$, (b) $\mid F_0^{\mathrm{av}}\mid =0.05$ and $n_i=50$, and (c) $\mid F_0^{\mathrm{av}}\mid =0.02$ and $n_i=200$. The arrows indicate an approximate localization time defined in Eq. (16).

Figure 8

Survival probability $P_{\mathrm{sur}}$ versus frequency $\nu$ for some numbers $K$ of kicks. $n_i=50$ and ${\mid F_0\mid }^{\mathrm{av}}=0.005$. The dashed lines give the classical results.

Figure 9

Average distance $⟨{\delta }_{{\nu }_0}\left(K\right)⟩$ between extrema in $P_{\mathrm{sur}}$ as a function of number $K$ of kicks. $n_i=50$ and $\mid F_0^{\mathrm{av}}\mid =0.005$. We analyze a set containing ${10}^6$ points on $1.45⩽{\nu }_0⩽1.45008$.

Figure 10

Fractal analysis of ${\log }_{10}\left(P_{\mathrm{sur}}\right)$ for $n_i=50$ and $\mid F_0^{\mathrm{av}}\mid =0.005$. In (a) we show the data analyzed for $2\times {10}^6$ points in the interval $1.45⩽{\nu }_0⩽1.47$ for different numbers $K$ of kicks, and in (b) we combine the wide set of (a) with data for a dense set containing ${10}^6$ points in the interval $1.45⩽{\nu }_0⩽1.45008$ for $K=6.8\times {10}^7$.