Quantum signatures of classical multifractal measures

A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension $D_0$ of the classical invariant set of open systems. Quantum systems of interest are often partially open (e.g., cavities in which trajectories are partially reflected or absorbed). In the corresponding classical systems $D_0$ is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of Rényi dimensions $D_q$. In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical simulations in area-preserving maps show for a wide range of configurations and system sizes $M$ that (i) the Weyl law is governed by a dimension different from $D_0=2$, and (ii) the observed dimension oscillates as a function of $M$ and other relevant parameters. We propose a classical model that considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a nontrivial dimension $D_{\mathrm{asymptotic}}<D_0$ in the semiclassical limit $M\to \infty$.

Figure 1

Classical dynamics of area-preserving maps $(x_n,y_n)\mapsto (x_{n+1},y_{n+1})$ with absorbing regions. The pairs of plots (a) and (b) show the action of one application of the baker map $(x_{n+1},y_{n+1})=(3x-\lfloor 3x_n\rfloor ,y_n/3+\lfloor 3y_n\rfloor /3)$ and the cat map $(x_{n+1},y_{n+1})=(2x_n+y_{n}mod1,3x_n+2y_{n}mod1)$, respectively. The areas surrounded by dotted lines in the left plots indicate the absorbing regions $\left(R<1\right)$ of each map. The dark regions in the right plots show the first images of the absorbing region.

Figure 2

Multifractal properties of the maps defined in the legend of Fig. 1. Top row: finite-time approximation of a typical phase-space distribution of the measure ${\mu }_i$ defined in Eq. (3). Bottom row: Rényi spectrum $D_q$ for $R=0.3$ (see Appendix pp1 for the analytical calculations on the baker map).

Figure 3

Distribution of eigenvalues $\left|\nu \right|$ [see Eq. (5)] in the quantized baker (a) and cat (b) maps. Two matrix sizes were used: $M=3^7$ (gray line) and $M=3^9$ (black line) [19]. The vertical dashed line marks the cutoff value used for the fractal Weyl law calculations in Figs. 4 and 4.

Figure 4

Scaling behavior of the quantum states for the baker [(a)–(c)] and cat [(d)–(f)] maps. Top row: Fraction of eigenvalues above the cutoff value ${\left|\nu \right|}_{\mathrm{cutoff}}$ as a function of the system size $M$ at different reflectivities. The dotted lines mark the expected scaling for $R=0$ [$d_0^{(R=0)}=ln\left(2\right)/ln\left(3\right)$ for the baker map and $d_0^{(R=0)}\approx 0.42$ for the cat map]. Middle row: Local scaling exponent $d_{\mathrm{loc}}\left(M\right):=[lnN\left(M\right)-lnN(M/3)]/ln3$, where $N\left(M\right)$ denotes $N({\nu }_i:|{\nu }_i|\ge |\nu {|}_{\mathrm{cutoff}})$, at $M=3^9$ for different values of $R$ as a function of ${\left|\nu \right|}_{\mathrm{cutoff}}$ (colors and symbols as in top row). We show results up to a cutoff where $N\left(M\right)<0.005M$, after which point the lack of resonances leads to strong oscillations of $d_{\mathrm{loc}}$. The vertical dashed lines indicate the cutoff values used in the top row. The small ticks on the upper axis mark the positions of the medians of the eigenvalue distributions $\left({\approx \left|\nu \right|}_{\mathrm{typ}}\right)$. Bottom row: $d_{\mathrm{loc}}\left(M\right)$ as a function of the cutoff for $M=\{3^7,3^8,3^9\}$ (lighter shades correspond to smaller $M$).

Figure 5

Scaling behavior of the classical expected values [Eq. (9)] for the baker map. The black dotted line denotes $d_0^{(R=0)}$. (a) Rescaled expected value of accepted boxes at an arbitrary sample size of $S\left(M\right)=S_{0}M$ with $S_0=0.3$. $\mathrm{E}\left[N_b\right]$ grows at fixed $\varepsilon$ monotonically with $R$. The shaded areas cover the range between $\mathrm{E}\left[N_b\right]\pm \sigma \left[N_b\right]$, where $\sigma$ is the standard deviation. (b) Scaling exponent $d_{\mathrm{loc}}(M=3^9)$ as a function of $1/S_0$. The lower the value of $R$, the lower the curve intersects the dashed vertical line (the $S_0$ choice used in a). (c) $d_{\mathrm{loc}}(M=3^n)$ with $n\in \{7,8,9\}$. Lighter shades correspond to smaller $M$.

Figure 6

Comparison of the quantum data to the classical model. The symbols show the number of long-lived states. The solid lines are the classical values of $aE\left[N_b\right]$ obtained by choosing $S_0$ and $a$ for each $R$ so that the rightmost two points of the quantum data are matched as close as possible.

Figure 7

Asymptotic scaling of the classical prediction. (a) Extrapolation of $E\left[N_b\right]$ of the curves in Fig. 6 without the shift (i.e., $a=1$). $R$ increases from the black (bottom) to the brown (top) curve. The dashed lines follow the asymptotic scaling $M^{d_{\mathrm{asymptotic}}-1}$ with $d_{\mathrm{asymptotic}}=\langle \frac{lnN\left(M\right)-lnN\left(M^{\prime }\right)}{lnM-ln{M}^{\prime }}\rangle$, where the average is computed over all possible combinations $\{(M,M^{\prime }):M^{\prime }<M\wedge (M,M^{\prime })\in {\{3^{10},3^{11},\cdots ,3^{40}\}}^2\}$. In (b) we show these dimensions together with their standard deviations. In gray are analytically obtained $d_q$ curves; see Eq. (A10). Note, that for fixed $R$ and $q\in {\mathbb{R}}^{+},d_0\ge d_q\ge d_{\infty }$ [1].