Density and correlation functions of vortex and saddle points in open billiard systems

We present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random-wave model (RWM). In particular, due to an improvement of the experimental setup, we determine experimentally the spatial correlation of saddle points of the current field. An asymptotic expression for the vortex-saddle and saddle-saddle correlation functions based on the RWM is derived, with experiment and theory agreeing well. We also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the RWM and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying Dirichlet conditions.

Figure 1

(Color online) Morphology of typical complex wave function $\psi$ in the open quantum billiard of our experiments: (a) modulus (intensity) ${\mid \psi \mid }^2$, (b) current flow $\mathrm{Im}{\psi }^{*}\nabla \psi$, and (c) blowup of (b), demonstrating the critical points we study. Counterclockwise vortices are marked by triangles (red), clockwise ones by squares (green). The saddle points are marked by crosses (orange). Our points of measurement are the crossing points in the background grid.

Figure 2

(Color online) The normalized mean field intensity $B(Y;a)$, plotted against $Y$ for various choices of Robin parameter $a$: $a=0$—i.e., Dirichlet conditions (solid, black line); $a=\pi ∕4$ (dotted, blue line); $a=\pi ∕2$—i.e., Neumann conditions (dashed, black line); $a=-\pi ∕4$ (dashed-dotted, red line). The horizontal black line at $B(Y;a)=1$ corresponds to the asymptotic limit for $Y\to \infty$.

Figure 3

(Color online) Analytic two-point correlation functions $g_{\mathrm{vv}}\left(R\right)$ (blue line, top) and $g_Q\left(R\right)$ (red line, bottom), plotted against $R$. The black line is the asymptotic value of 1 for $g_{\mathrm{vv}}\left(R\right)$ as $R\to \infty$. The analytic functions compare well with their asymptotic approximations (dashed line) for $R\gtrsim 4$.

Figure 4

(Color online) Density oscillations of critical points as a function of distance $Y$ from a straight wall satisfying mixed Robin conditions: (a) ${\rho }_{\mathrm{v}}(Y;a)$ and (b) ${\rho }_{\mathrm{s}}(Y;a)$. Line styles and colors represent the same choices of $a$ as in Fig. 2; the black line is at 1 (the asymptotic limit for $Y\to \infty$); the two densities are clearly out of phase for $Y\gtrsim 4$. (c) Peak in the densities for $a=\pi ∕10$: ${\rho }_{\mathrm{v}}$ (left peak, blue line) and ${\rho }_{\mathrm{s}}$ (right peak, red line). The dashed lines correspond to the small-$a$ forms, as in Eqs. (42, 43).

Figure 5

(Color online) Experimental vortex pair correlation $g_{\mathrm{vv}}\left(R\right)$ (top histogram) and charge correlation function $g_Q\left(R\right)$ (bottom histogram) in (a) the low-frequency regime $(5\mathrm{GHz}<\nu <9\mathrm{GHz})$ and (b) the higher-frequency regime $(15\mathrm{GHz}<\nu <18.6\mathrm{GHz})$. The solid lines correspond to the analytic prediction of the RWM discussed in Sec. 4B, using the exact formula for $g_{\mathrm{vv}}\left(R\right)$ rather than the asymptotic form.

Figure 6

(Color online) Pair correlations involving saddle points: (a) saddle-saddle correlation function $g_{\mathrm{ss}}\left(R\right)$ and (b) vortex-saddle function $g_{\mathrm{vs}}\left(R\right)$. Experimental results (histogram) are plotted against the asymptotic forms [solid, blue line; see Eqs. (34, 35)].

Figure 7

(Color online) Critical-point density fluctuations as a function of scaled distance $Y$ from a straight wall satisfying Dirichlet boundary conditions: (a) vortex density and (b) saddle density. Experimental results are plotted against the analytic forms for Dirichlet boundary conditions [see Eqs. (38, 39)].