Decay Rates and Survival Probabilities in Open Quantum Systems

We provide the first statistical analysis of the decay rates of strongly driven 3D atomic Rydberg states. The distribution of the rates exhibits universal features due to Anderson localization, while universality of the time dependent decay requires particular initial conditions.

Figure 1

Distribution of the ionization rates $\Gamma$ of microwave-driven (a) 1D and (b) 3D $m_0=0$ Rydberg states of atomic hydrogen, for three different values of the localization parameter $\mathcal{L}=0.25$ (diamonds), $0.5$ (stars), $1.0$ (circles), and $2.0$ [pyramids; only in (a), because of the high cost of 3D calculations at the corresponding high field amplitudes]. The distributions were generated, at fixed $\mathcal{L}$, by sampling the Floquet spectra over the frequency (and corresponding amplitude, $F$) ranges $\omega /2\pi =13.16\dots 16.45\mathrm{G}\mathrm{H}\mathrm{z}$ (1D) and $35.5\dots 36.1\mathrm{G}\mathrm{H}\mathrm{z}$ (3D), within a Floquet zone of width $\omega$ centered around $n_0=70$ (3D) and $100$ (1D), respectively.

Figure 2

(a) Expansion coefficients $w_j$ of the 3D atomic initial state $|n_0=70,{\ell }_0=0,m_0=0⟩$ in the Floquet eigenstates $|{ϵ}_j⟩$, for $\mathcal{L}=1.0$ and $\omega /2\pi =35.6\mathrm{G}\mathrm{H}\mathrm{z}$, vs the associated decay rates ${\Gamma }_j$. (Qualitatively similar results are obtained for the one-dimensional model atom, as well as for microwave driven alkali Rydberg states [17].) (b) Same as (a), but for initial states prepared at the sites $900$ (plusses), $999$ (crosses), $1000$ (diamonds clustered along the diagonal) of a 1D Anderson model of sample length $N=1000$ (uniformly distributed on-site potentials $V_{n_i}$ with $|V_{n_i}|\le 1.5$, constant coupling $V_{n_i,n_i\pm 1}=1$, and an absorbing boundary at one end, introduced by adding an imaginary part $-i\gamma =-i\times 0.31$ to the on-site potential at $n_i=1000$; results for 10 realizations of the potential are gathered together). Only in the special case of the initial state placed right at the edge $n_{\mathrm{i}}=1000$ of the sample is there a one-to-one relation between decay rates and expansion coefficients.

Figure 3

Survival probability $P_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}}(t)$ obtained from distributions as in Figs. 1, 2 via Eq. (1). (a) 1D hydrogen, initial state $|n_0=100⟩$, driving frequencies and localization parameters $\omega /2\pi =16.45\mathrm{G}\mathrm{H}\mathrm{z}$, $\mathcal{L}=1.0$ (full line), $13.16\mathrm{G}\mathrm{H}\mathrm{z}$ and $2.0$ (dashed); (b) 3D hydrogen, $n_0=70$, $m_0=0$, ${\ell }_0=0$ (solid line), $15$ (dashed), $45$ (dotted), $\omega /2\pi =35.6\mathrm{G}\mathrm{H}\mathrm{z}$, $\mathcal{L}=1.0$. In this latter case, different $w_j$ distributions are realized by changing the angular momentum ${\ell }_0$ of the initial atomic state, always leading to near-algebraic decay (like in 1D), though with distinct (and nonuniversal) decay exponents. Only fictitious expansion coefficients $w_j\equiv {\Gamma }_j/⟨\Gamma ⟩$ induce asymptotically “universal” decay $P_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}}(t)\sim t^{-\gamma }$ (dash-dotted line), $\gamma \simeq 2-\alpha \simeq 1.1$ (indicated by the full line), with ${\Gamma }_j$, $⟨\Gamma ⟩$, and $\alpha \simeq 0.9$ extracted from the $\Gamma <{10}^{-10}\mathrm{a}\mathrm{.}\mathrm{u}\mathrm{.}$ part of the $\mathcal{L}=1.0$ data in Fig. 1b.