Microwave-Driven Atoms: From Anderson Localization to Einstein’s Photoeffect

We study the counterpart of Anderson localization in driven one-electron Rydberg atoms. By changing the initial Rydberg state at fixed microwave frequency and interaction time, we numerically monitor the crossover from Anderson localization to the photoeffect in the atomic ionization signal.

Figure 1

Scaled ionization threshold field $F_0^{10\%}=F^{10\%}{n}_0^4$ of atomic hydrogen [red dashed line (□)] and lithium [blue solid line (○)], at fixed laboratory microwave frequency $\omega =17.5\mathrm{GHz}$ and interaction time $t=500\mathrm{ns}$. The scaled frequency ${\omega }_0=\omega n_0^3=1.9–39.1$ is tuned by changing the initial state’s principal quantum number from $n_0=90$ to $n_0=245$, at fixed values of the angular momentum quantum numbers ${\ell }_0=1$ and $m_0=0$. We observe three distinct regimes. (I), $1.9\le {\omega }_0\le 13.1$: the monotonous increase of $F_0^{10\%}$ with ${\omega }_0$ is a characteristic signature of Anderson localization in strongly driven quantum systems [20]. Regime (II), $13.1\le {\omega }_0<31.5$: $F_0^{10\%}$ still increases with ${\omega }_0$, on average, but is garnished by large modulations due to the passage of the atomic initial state across subsequent $N_{\rightsquigarrow }$-photon ionization thresholds indicated by vertical arrows. Anderson localization and finite $N_{\rightsquigarrow }$-photon ionization coexist. Regime (III), ${\omega }_0\ge 31.5$: The photon energy exceeds the ionization potential of the initial state, the Anderson scenario is inapplicable, and single photon absorption mediates the ionization process.

Figure 2

Ratio of the atomic localization length $\xi$ as estimated by Eq. (2) to the number of absorbed photons $N_{\rightsquigarrow }$, versus scaled frequency ${\omega }_0=\omega n_0^3$. Data are extracted from the 10%-ionization thresholds of Fig. 1, for atomic hydrogen [red dashed line (□)] and lithium [blue solid line (○)]. On average, $\xi /N_{\rightsquigarrow }$ is constant in regime (I)—a hallmark of exponential localization of the electronic wave function on the energy axis. This is still true in regime (II), where, however, the discreteness of the lattice (on the energy axis) strongly affects the transport behavior: Large-scale modulations of the signal emerge due to direct $N_{\rightsquigarrow }$ photon transitions to the continuum. Only in regime (III) does $\xi$ drop to zero, thus invalidating the Anderson picture.

Figure 3

Shannon width, Eq. (4), at the 10%-ionization threshold, as a function of the scaled frequency ${\omega }_0=\omega n_0^3$, for the same parameters as in Fig. 1. Remarkably, $\mathcal{W}$ takes appreciable values for initial states up to right below the single photon ionization threshold at the lower edge of regime (III). In regime (I), atomic hydrogen exhibits considerably larger values than lithium, which we attribute to the angular momentum degeneracy of the hydrogenic initial state.