Emulation of quantum mechanical billiards by electrical resonance circuits

We propose that a two-dimensional electric network may be used for fundamental studies of wave function properties, transport, and related statistics. Using Kirchhoff’s current law and the $j\omega$ method we find that the network is analogous to a discretized Schrödinger equation for quantum billiards and dots. Thus complex electric potentials play the role of quantum mechanical wave functions. Ways of realizing the electric network are discussed briefly. The role of symmetries is outlined, and a direct way of selecting states with a given symmetry is presented.

Figure 1

Internal points in a grid.

Figure 2

Modeling of a “two-lead” cavity in the shape of an open quantum dot.

Figure 3

Grid with connected voltage.

Figure 4

(Color online) A typical plot of the current $\mid I\mid$ as a function of $\omega$ for a small net. Each minimum is an eigenfrequency.

Figure 5

Grid with alternative boundary conditions.

Figure 6

(Color online) Current $\mid I\mid$ through the load resistance, with marked dips for a selected $\omega$ region.

Figure 7

(Color online) The first eight modes, showing ${\mid V\mid }^2$, to be compared with ${\mid \psi \mid }^2$ in QM. The number of grid points $(i,j)$ is $100\times 100$, and the values $L=10\mathrm{mH}$, $C=1\mathrm{mF}$, and $R=0.05\mathrm{m}\Omega$ were used in the simulations.

Figure 8

(Color online) Poynting vector fields, corresponding to modes 67, 75, and 175. Grid points indices $(i,j)$ are indicated on the horizontal and vertical axes (cf. Figs. 1, 2).

Figure 9

A desymmetrized two-lead cavity, isolating the odd wave functions.

Figure 10

Symmetries in the Bunimovich billiard.

Figure 11

A stadium shaped grid giving the odd/odd symmetry class. The four different positions of the symbol $E$ with respect to the voltage sources define the phases, in our case equal or opposite.

Figure 12

(Color online) Statistics for the normalized angular frequency spacings ${\omega }^2$, corresponding to the spacings between eigenenergies in quantum mechanics. The curve is the Wigner-Dyson distribution given by Eq. (19); 1199 spacings for the half billiard in Fig. 9 were used for generating the histogram.

Figure 13

(Color online) (a) shows ${\mid V_{i,j}\mid }^2$ for a regular resonant eigenmode and (b) for a chaotic resonant mode. The curve in (c) and (d) is the Porter-Thomas distribution in Eq. (20) for (a) and (b), respectively. (e) and (f) show the phase of the “wave function” $V_{i,j}$ for (a) and (b). The behavior of the statistics in (c), i.e., the way the tail suddenly drops to zero, is a general characteristic for regular modes [10, 20]. The number of grid points is indicated on the horizontal and vertical axes.

Figure 14

(Color online) (a) shows ${\mid V_{i,j}\mid }^2$ for a regular state with $\omega =26.65\mathrm{rad}∕\mathrm{s}$, and (b)–(e) the distributions for the real and imaginary parts of the potential field $V_{i,j}$, before and after a rotation by the angle $\alpha$, defined by Eq. (24). The curves are Gaussian distributions, Eq. (23), with parameters given in each subfigure. The number of grid points is indicated on the horizontal and vertical axes.

Figure 15

(Color online) (a) shows ${\mid V_{i,j}\mid }^2$ for an irregular state with $\omega =112.41\mathrm{rad}∕\mathrm{s}$, and (b)–(e) the corresponding distributions for the real and imaginary parts of the potential field $V_{i,j}$, before and after a rotation by the angle $\alpha$, given by Eq. (24). The curves are Gaussian distributions, Eq. (23), with parameters given in each subfigure. The number of grid points are indicated on the horizontal and vertical axes.

Figure 16

(Color online) Statistics for a typical high-frequency mode. The curve in (b) is given by Eq. (26), and the curves in (c) and (d) by Eq. (25). The fitting parameter is $\tau =2.0$.