Multifractality of open quantum systems

We study the eigenstates of open maps whose classical dynamics is pseudointegrable and for which the corresponding closed quantum system has multifractal properties. Adapting the existing general framework developed for open chaotic quantum maps, we specify the relationship between the eigenstates and the classical structures, and we quantify their multifractality at different scales. Based on this study, we conjecture that quantum states in such systems are distributed according to a hierarchy of classical structures, but these states are multifractal instead of ergodic at each level of the hierarchy. This is visible for sufficiently long-lived resonance states at scales smaller than the classical structures. Our results can guide experimentalists in order to observe multifractal behavior in open systems.

Figure 1

Regions $R_{-}^m$ (top row, $1\le m<t_E$) and $R_{+}^m$ (bottom row, $0\le m<t_E$) for the classical map opened in $x$ with $\ell =2^{-6}$ (left column), $2^{-2}$ (right column), and $\gamma =1/3$. The color scale indicates the number of iterates $m$. White regions include all those regions $R_{\pm }^m$ corresponding to $m\ge t_E$ ($t_E=2$ for $\ell =2^{-6}$ and $t_E=28$ for $\ell =2^{-2}$), i.e., the sets $K_{-}^{t_E}$ (top row) and $K_{+}^{t_E}$ (bottom row) whose intersection gives the trapped set $K_0^{t_E}$. As $t_E\to \infty$, the sets $K_{\pm }^{t_E}$ tend in fact to (the same) horizontal connected sets of periodic points that stay in the system forever, either under backward (top) or forward (bottom) evolution.

Figure 2

Spectral norms $|{\lambda }_j|$ (top row) of the dynamical map with $\gamma =1/3$, opened in $x$ with $\ell =2^{-6}$ (a) and $2^{-2}$ (c), for various system sizes $N=2^7,\cdots ,2^{12}$ (light to dark circles). The scaling of the number of resonances with norm above 0.5 is shown in the bottom row for each case $\ell =2^{-6}$ (b) and $2^{-2}$ (d). Circles correspond to the data, while the line is the linear fit (with the solid segment indicating the fitting range), whose slope corresponds to $d_0/2$.

Figure 3

Numerically calculated weights $\langle {\mathrm{\Psi }}_j^{-}\left|{\mathrm{\Pi }}_{-}^m\right|{\mathrm{\Psi }}_j^{-}\rangle$ of left eigenstates of the open dynamical map on regions $R_{-}^m$ for $m=1,2$, and 3 (blue, red, green; dark to light, respectively). Dashed black lines correspond to the theoretical prediction (12) for each of these corresponding regions (from top to bottom in each panel). The map (with $N=2^{12}$ and $\gamma =1/3$) is opened in $x$ with $\ell =2^{-6}$ [(a) and (b)] and $\ell =2^{-2}$ (c). For $\ell =2^{-6}$, panel (a) corresponds to the weights calculated on the original regions (i.e., in the same way as for the large opening), while panel (b) corresponds to the weights on the broadened regions.

Figure 4

Husimi representation of four eigenstates of the dynamical map opened in $x$ for $\ell =2^{-6}$ (a)–(d) and $\ell =2^{-2}$ (e)–(h). Superimposed on each panel are the classical regions $R_{-}^1$, $R_{-}^2$, and $R_{-}^3$ (blue, red, green; dark to light, respectively). The eigenvalue norm $\left|\lambda \right|$ associated with each eigenstate is (a) 0.022 23, (b) 0.718 80, (c) 0.973 35, (d) 0.999 98, (e) 0.020 07, (f) 0.720 74, (g) 0.0973 31, and (h) 0.999 89.

Figure 5

Multifractal analysis for the dynamical map opened in $x$ with $\ell =2^{-6}$ (a)–(g) and $\ell =2^{-2}$ (h)–(n). The top three rows in each case ($|{\lambda }_j|$ decreases from top to bottom): box-size $n$ behavior of ${\tilde{D}}_2\left(n\right)$ for the left eigenstates of the open map $|{\mathrm{\Psi }}_j\rangle$ (thin colored lines) and those of the closed map $|{\mathrm{\Psi }}_k^{\left(0\right)}\rangle$ (thick black line); for $|{\mathrm{\Psi }}_k^{\left(0\right)}\rangle$ the average is over the $N$ available states, whereas for $|{\mathrm{\Psi }}_j\rangle$ we average over all those states (out of the $N$) falling within a small norm range indicated by the color and shade of the curve in panel (g) displaying the spectral norms $|{\lambda }_j|$. To the right of each ${\tilde{D}}_2\left(n\right)$ plot, we display $|\langle {\mathrm{\Psi }}_j|p\rangle {|}^2$ for an exemplifying state in the corresponding $|{\lambda }_j|$ window. Results correspond to system size $N=2^{12}$ and $\gamma =1/3$.

Figure 6

Average spectral norms $\langle \left|{\lambda }_j\right|\rangle$ of the random map for $\gamma =1/3$, opened in $x$ with $\ell =2^{-6}$ (a), $2^{-2}$ (b), and for various system sizes $N=2^7$ (lightest $\mathrm{gray}),\cdots ,2^{12}$ (darkest gray). For each $N$ we diagonalize $M=2^{17}/N$ random-phase realizations. The index $j\in \{1,\cdots ,N\}$ is defined by the ordering $|{\lambda }_j|\le |{\lambda }_{j+1}|$, and the average $\langle \left|{\lambda }_j\right|\rangle$ is taken over the $M$ realizations at fixed $j$. The arrow at the top of each plot marks the (rescaled) theoretical prediction $\ell$ for the size of the short-lived sector.

Figure 7

Husimi representation of three left eigenstates of a particular realization of the open random map with $\gamma =1/3$, $N=2^{12}$, and opened in $x$ with $\ell =2^{-6}$ (a)–(c), $2^{-2}$ (d)–(f). The eigenvalue norm $\left|\lambda \right|$ associated with each eigenstate is (a) 0.740, (b) 0.900, (c) 0.995, (d) 0.661, (e) 0.750, and (f) 0.949.

Figure 8

Multifractal analysis for the random map opened in $x$ with $\ell =2^{-6}$ (a)–(g) and $\ell =2^{-2}$ (h)–(n). The top three rows in each case ($\langle \left|{\lambda }_j\right|\rangle$ decreases from top to bottom): box-size $n$ behavior of ${\tilde{D}}_2\left(n\right)$ for the left eigenstates of the open map $|{\mathrm{\Psi }}_j\rangle$ (thin colored lines) and those of the closed map $|{\mathrm{\Psi }}_k^{\left(0\right)}\rangle$ (thick black line); for $|{\mathrm{\Psi }}_k^{\left(0\right)}\rangle$ the average is over the $M\times N$ available states, whereas for $|{\mathrm{\Psi }}_j\rangle$ we average over all those states (out of the $M\times N$) falling within a small norm range indicated by the color and shade of the curve in panel (g) displaying the spectral norms $\langle \left|{\lambda }_j\right|\rangle$. To the right of each ${\tilde{D}}_2\left(n\right)$ plot, we display $|\langle {\mathrm{\Psi }}_j|p\rangle {|}^2$ for an exemplifying state in the corresponding $\langle \left|{\lambda }_j\right|\rangle$ window. Results correspond to $M=32$ random realizations with $N=2^{12}$ and $\gamma =1/3$.

Figure 9

Regions $R_{-}^m$ ($m=1,2,3$ as blue, blue-green, and green, respectively) for the classical map with $\gamma =1/3$, opened in $p$ with $\ell =2^{-6}$ (a) and $2^{-2}$ (b). As for the structures corresponding to forward evolution, we simply have $\mathrm{\Omega }=R_{+}^0=R_{-}^3$, $R_{+}^1=R_{-}^2$, $R_{+}^2=R_{-}^1$, and $K_{+}=K_{-}=K_0$.

Figure 10

Average spectral norms $\langle \left|{\lambda }_j\right|\rangle$ of the random map for $\gamma =1/3$, opened in $p$ with $\ell =2^{-6}$ (a), $2^{-2}$ (b), and for various system sizes $N=2^7$ (lightest $\mathrm{gray}),\cdots ,2^{12}$ (darkest gray). For each $N$ we diagonalize $M=2^{17}/N$ random-phase realizations. The index $j\in \{1,\cdots ,N\}$ is defined by the ordering $|{\lambda }_j|\le |{\lambda }_{j+1}|$, and the average $\langle \left|{\lambda }_j\right|\rangle$ is taken over the $M$ realizations at fixed $j$. The red arrow at the top of each plot marks the theoretical prediction $\ell b$ for the (rescaled) size of the short-lived sector.

Figure 11

Husimi representation of four eigenstates of a given realization of the random map opened in $p$ for $\ell =2^{-6}$ (a)–(d) and $\ell =2^{-2}$ (e)–(h). Superimposed on each panel are the classical regions $R_{-}^1$, $R_{-}^2$, and $R_{-}^3$ (blue, red, green; dark to light, respectively). The eigenvalue norm $\left|\lambda \right|$ associated with each eigenstate is (a) 0.190 09, (b) 0.957 90, (c) 0.995 27, (d) 0.999 52, (e) 0.017 85, (f) 0.958 03, (g) 0.995 22, and (h) 0.999 58.

Figure 12

Multifractal analysis for the random map opened in $p$ with $\ell =2^{-6}$ (a)–(g) and $\ell =2^{-2}$ (h)–(n). Top three rows in each case ($\langle \left|{\lambda }_j\right|\rangle$ decreases from top to bottom): box-size $n$ behavior of ${\tilde{D}}_2\left(n\right)$ for the left eigenstates of the open map $|{\mathrm{\Psi }}_j\rangle$ (thin colored lines) and those of the closed map $|{\mathrm{\Psi }}_k^{\left(0\right)}\rangle$ (thick black line); for $|{\mathrm{\Psi }}_k^{\left(0\right)}\rangle$ the average is over the $M\times N$ available states, whereas for $|{\mathrm{\Psi }}_j\rangle$ we average over all those states (out of the $M\times N$) falling within a small norm range indicated by the color and shade of the curve in panel (g) displaying the spectral norms $\langle \left|{\lambda }_j\right|\rangle$. To the right of each ${\tilde{D}}_2\left(n\right)$ plot, we display $|\langle {\mathrm{\Psi }}_j|p\rangle {|}^2$ for an exemplifying state in the corresponding $\langle \left|{\lambda }_j\right|\rangle$ window. Results correspond to $M=32$ random realizations with $N=2^{12}$ and $\gamma =1/3$.