Universal intensity statistics of multifractal resonance states

We conjecture that in chaotic quantum systems with escape, the intensity statistics for resonance states universally follows an exponential distribution. This requires a scaling by the multifractal mean intensity, which depends on the system and the decay rate of the resonance state. We numerically support the conjecture by studying the phase-space Husimi function and the position representation of resonance states of the chaotic standard map, the baker map, and a random matrix model, each with partial escape.

Figure 1

(a) Distribution of quantum decay rates $\gamma$ for the chaotic standard map with partial escape, defined in Appendix pp1, with $h=1/16000$. Classical decay rates ${\gamma }_{\text{nat}}\approx 0.22,{\gamma }_{\text{typ}}\approx 0.48,{\gamma }_{\text{inv}}\approx 0.88$ are indicated. (b) Husimi functions $H_{\psi }$ for three exemplary resonance states $\psi$ with decay rates ${\gamma }_{\text{nat}}$, ${\gamma }_{\text{typ}}$, and ${\gamma }_{\text{inv}}-0.1$. (c) Averaged Husimi function ${\langle H_{}\rangle }_{\gamma }$ of $n_{\text{avg}}=200$ resonance states close to each $\gamma$. The same color map with maximum $I_{\text{c}}$ is used for each pair of individual and average Husimi function, where $I_{\text{c}}={max}_{\mathrm{\Gamma }}{\langle H_{}\rangle }_{\gamma }$. (d) Scaled Husimi functions ${\tilde{H}}_{\psi }$, see Eq. (4), visualized with $I_{\text{c}}=8$.

Figure 2

Distribution of scaled Husimi functions ${\tilde{H}}_{\psi }$ at three phase-space points $\mathit{x}=(q,p)\in \left\{\right(0.1,0.2),(0.5,0.5),(0.8,0.7\left)\right\}$ using $n_{\text{avg}}=200$ and $1/h=16000$. Resonance states with decay rates $\gamma \in [{\gamma }_{\text{nat}},{\gamma }_{\text{inv}}-0.1]$ are used. The exponential distribution with mean one, Eq. (7), is shown as a black line. The inset shows the comparison on a semilogarithmic scale. The system is the chaotic standard map with partial escape, see Appendix pp1.

Figure 3

Distribution of scaled Husimi functions ${\tilde{H}}_{\psi }$ close to three decay rates ${\gamma }_{\text{nat}}\approx 0.22$, ${\gamma }_{\text{typ}}\approx 0.48$, and ${\gamma }_{\text{inv}}-0.1\approx 0.78$ for $n_{\text{s}}=300$ resonance states each, evaluated on a $50\times 50$ phase-space grid using $n_{\text{avg}}=200$ and $1/h=16000$. The exponential distribution with mean one, Eq. (7), is shown as a black line. The inset shows the comparison on a semilogarithmic scale. The system is the chaotic standard map with partial escape, see Appendix pp1.

Figure 4

Distribution of scaled position intensities ${\tilde{I}}_q\left(\psi \right)$ for all 16000 positions $q$, using the same parameters as in Fig. 3. The dashed line in the inset shows additionally the expected distribution when scaling with an average for finite $n_{\text{avg}}=200$, Eq. (C4).

Figure 5

Distribution of scaled Husimi functions ${\tilde{H}}_{\psi }$ for $h=1/250$, $n_{\text{avg}}=n_{\text{s}}=24$, and considering a $15\times 15$ phase-space grid, otherwise as in Fig. 3. The dashed line in the inset shows additionally the expected distribution when scaling with an average for finite $n_{\text{avg}}=24$, Eq. (C4).

Figure 6

Same as Fig. 1 for the triadic baker map with partial escape, defined in Appendix pp2, using $h=1/16002$. The classical decay rates are ${\gamma }_{\text{nat}}\approx 0.31$, ${\gamma }_{\text{typ}}\approx 0.54$, and ${\gamma }_{\text{inv}}\approx 0.85$.

Figure 7

Distribution of scaled Husimi functions ${\tilde{H}}_{\psi }$, for the triadic baker map with partial escape, defined in Appendix pp2, using $h=1/16002$ and considering a $60\times 60$ phase-space grid. Otherwise parameters as in Fig. 3. The considered decay rates are ${\gamma }_{\text{nat}}\approx 0.31$, ${\gamma }_{\text{typ}}\approx 0.54$, and ${\gamma }_{\text{inv}}-0.1\approx 0.75$.

Figure 8

(a) Distribution of quantum decay rates $\gamma$ for a random matrix with partial escape using $N=1000$. The inset shows the considered reflectivity function $r\left(q\right)$. The classical decay rates ${\gamma }_{\text{nat}}\approx 0.64,{\gamma }_{\text{typ}}\approx 0.98,{\gamma }_{\text{inv}}\approx 1.5$ are indicated. (b) Intensities in position space $I_q\left(\psi \right)N$ for three exemplary resonance states $\psi$ with decay rates ${\gamma }_{\text{nat}}$, ${\gamma }_{\text{typ}}$, and ${\gamma }_{\text{inv}}$ and classical density ${\rho }_{\gamma }\left(q\right)$, Eq. (D10) (thick line). (c) Scaled intensities ${\tilde{I}}_q\left(\psi \right)$ for the same decay rates, compared to the uniform density (thick line).

Figure 9

Distribution of scaled intensities ${\tilde{I}}_q\left(\psi \right)$ close to the three decay rates ${\gamma }_{\text{nat}}\approx 0.64$, ${\gamma }_{\text{typ}}\approx 0.98$, and ${\gamma }_{\text{inv}}\approx 1.5$ for $n_{\text{s}}=300$ resonance states each, evaluated for all position states $q$ for $N=16000$ using ${\rho }_{\gamma }\left(q\right)$, Eq. (D10), for scaling. The exponential distribution with mean one, Eq. (7), is shown as a black line. The inset shows the comparison on a semilogarithmic scale. The system is the random matrix model with partial escape.

Figure 10

Single $H_{\psi }$, averaged ${\langle H_{}\rangle }_{\gamma }$, and scaled ${\tilde{H}}_{\psi }$ Husimi functions (left to right) for resonance states $\psi$ with decay rate closest to ${\gamma }_{\text{typ}}$ for the chaotic standard map with different $r(q,p)$, using $h=1/16000$ and $n_{\text{avg}}=200$. The typical decay rates are (a) ${\gamma }_{\text{typ}}\approx 0.90$, (b) ${\gamma }_{\text{typ}}\approx 0.11$, (c) ${\gamma }_{\text{typ}}\approx 0.24$, (d) ${\gamma }_{\text{typ}}\approx 0.97$, and (e) ${\gamma }_{\text{typ}}\approx 0.98$. Color maps as in Fig. 1.

Figure 11

Distribution of scaled Husimi functions ${\tilde{H}}_{\psi }$ with decay rates close to ${\gamma }_{\text{typ}}$ for the chaotic standard map with different $r(q,p)$, as specified in Fig. 10. Other parameters as in Fig. 3.

Figure 12

Same as Fig. 1 for the asymmetric baker map with partial escape using $h=1/16002$. The classical decay rates are ${\gamma }_{\text{nat}}\approx 0.31$, ${\gamma }_{\text{typ}}\approx 0.54$, and ${\gamma }_{\text{inv}}\approx 0.85$.

Figure 13

Distribution of scaled Husimi functions ${\tilde{H}}_{\psi }$ for the asymmetric baker map with partial escape using $h=1/16002$ and considering a $60\times 60$ phase-space grid. Other parameters as in Fig. 3. The considered decay rates are ${\gamma }_{\text{nat}}\approx 0.31$, ${\gamma }_{\text{typ}}\approx 0.54$, and ${\gamma }_{\text{inv}}-0.1\approx 0.75$.