Quantum chaotic scattering in microwave resonators

In a frequency range where a microwave resonator simulates a chaotic quantum billiard, we have measured moduli and phases of reflection and transmission amplitudes in the regimes of both isolated and of weakly overlapping resonances and for resonators with and without time-reversal invariance. Statistical measures for $S$-matrix fluctuations were determined from the data and compared with extant and/or newly derived theoretical results obtained from the random-matrix approach to quantum chaotic scattering. The latter contained a small number of fit parameters. The large data sets taken made it possible to test the theoretical expressions with unprecedented accuracy. The theory is confirmed by both a goodness-of-fit-test and the agreement of predicted values for those statistical measures that were not used for the fits, with the data.

Figure 1

The tilted stadium billiard (schematic). The two antennas 1, 2 connect the resonator to the VNA. Optionally a ferrite is inserted at a fixed location to violate $\mathcal{T}$ invariance and/or a movable scatterer is used to gather independent data sets (see main text). Taken from Ref. [13].

Figure 2

Reflection spectra (upper panels) and transmission spectra (lower panels) of the $\mathcal{T}$-invariant billiard taken at three frequency ranges (panel columns). While the left and center panels show data at $\Gamma /d\approx 0.02$ and 0.29, respectively, for the two-dimensional regime (where the billiard mimics a quantum billiard), the data at $\Gamma /d\approx 0.6$ in the right panels are obtained in a frequency range where the cavity supports three-dimensional field distributions.

Figure 3

Spectra of the tilted stadium billiard with a TRIV ferrite magnetized through an external field of $B=190\text{mT}$ (left panels) and of $B=0\text{mT}$ (right panels). The upper two panels show the squared modulus of the elements of the scattering matrix $S_{12}$ (full line) and $S_{21}$ (dashed line) taken in the range 15.5–16.0 GHz, where $\Gamma /d\approx 0.50$. In this range the resonator supports, due to its height of 5 mm, only two-dimensional modes. Reciprocity is violated for nonvanishing magnetic field since $S_{12}\ne S_{21}$. To clarify this, we show in the lower panels the difference of the squared modulus of the elements of the scattering matrix $S_{12}$ and $S_{21}$.

Figure 4

Three autocorrelation functions for the $\mathcal{T}$-invariant billiard determined from the measured values for $S_{12}$ in the frequency intervals 3–4 GHz ($\Gamma /d\approx 0.02$, circles), 9–10 GHz ($\Gamma /d\approx 0.29$, triangles) and 14–15 GHz ($\Gamma /d\approx 0.6$, squares). The frequency difference $\epsilon$ is plotted in units of the local mean level spacing $d$ as obtained from the Weyl formula [43]. All curves are normalized to unity at $\epsilon =0$.

Figure 5

Cross-correlation coefficient $C_{\text{cross}}\left(0\right)$ as a measure of TRIV. In each frequency interval of 1 GHz length $C_{\text{cross}}\left(0\right)$ was evaluated as an average over six realizations. The points denote the mean values and the error bars denote the standard deviations. The zero on the ordinate is suppressed. Based on Ref. [13].

Figure 6

Fourier coefficients of the autocorrelation function of $S_{12}$ (in semilogarithmic scale). Data points are from the billiard without ferrite in the frequency ranges 3–4 GHz (upper panel) and 9–10 GHz (lower panel). For clarity only every fifth data point is shown. The solid lines are best fits to the data. In the data shown in the lower panel the decay is dominated by noise for times larger than about 800 ns. These data are not taken into account in the fitting procedure.

Figure 7

Elastic enhancement factors $\mathcal{W}$ for the $\mathcal{T}$-invariant billiard. The open circles are obtained with method (i), the filled circles with method (ii) described in the text. The error bars indicate uncertainties due to the finite range of data [34, 35]. Above 10 GHz the analogy to a quantum billiard breaks down and Eq. (26) needed for the analytic evaluation of $\mathcal{W}$ is no longer applicable. The dashed horizontal lines indicate the limits of $\mathcal{W}$ for $\mathcal{T}$-invariant systems: Upper line for $\Gamma ⪡d$, lower line for $\Gamma ⪢d$.

Figure 8

Elastic enhancement factors $\mathcal{W}$ for the billiard with partial TRIV. The open circles are obtained with method (i), the filled circles with method (ii) as described in the text. The error bars show the root-mean-square values for the six realizations. The dashed horizontal lines mark the limits of $\mathcal{W}$ for $\mathcal{T}$-invariant systems as in Fig. 7. Taken from Ref. [13].

Figure 9

Dependence of the cross-correlation coefficient $C_{\text{cross}}\left(0\right)$ on the parameter $\xi$ as predicted by a random-matrix model for partial violation of $\mathcal{T}$ invariance. The analytic result (line) is compared with an RMT simulation (dots) for the same set of transmission coefficients. Also shown is how an experimental value of $C_{\text{cross}}\left(0\right)=0.49\left(3\right)$, c.f. Fig. 5, translates into $\xi =0.29\left(2\right)$. Based on Ref. [13].

Figure 10

Comparison of the analytic result for the autocorrelation function $C_{11}$ versus $\epsilon$ for transmission coefficients $T_1=0.407$, $T_2=0.346$, ${\tau }_{\text{abs}}=2.41$, and TRIV parameter $\xi =0.293$ (solid line) with RMT simulations (dots). We show only the result for $C_{11}$ as that for $C_{12}$ is barely distinguishable. The curve is normalized such that it equals unity for $\epsilon =0$.

Figure 11

Comparison of the analytic results for the Fourier transform of the autocorrelation function versus time $t$ for transmission coefficients $T_1=0.407$, $T_2=0.346$, ${\tau }_{\text{abs}}=2.41$ and TRIV parameter $\xi =0.293$ to RMT simulations. We show the results for ${\tilde{C}}_{ab}$ for $a=b=1$ (dashed line and filled points) and $a=1,b=2$ (solid line and crosses, respectively).

Figure 12

Values of the TRIV parameter $\xi$ for the billiard with the ferrite magnetized with $B=190\text{mT}$. The error bars indicate the variability of the results within the six realizations.

Figure 13

$\Gamma /d$ (top panel), ${\tau }_{\text{abs}}$ (middle panel) and the transmission coefficients $T_1$ (filled circles) and $T_2$ (open circles) (bottom panel) versus frequency for the billiard with $\mathcal{T}$ invariance. The errors are typically of the size of the symbols.

Figure 14

The same as in Fig. 13 but for the billiard with the ferrite, obtained as averages over six realizations. The scatter of the values for different realizations about the mean value is typically of the order of the symbol size.

Figure 15

Distribution of $S_{11}$-matrix elements according to modulus (upper panels) and phase (lower panels). The histograms give the probability distribution functions in the frequency ranges (from left to right) 9–10 GHz, 17–18 GHz, and 23–24 GHz. The data were measured with the billiard used for TRIV but without ferrite and for a total of eight realizations, i.e., each graph is constructed from $80000$ data points. All of these were used in the histograms, and correlations between $S$-matrix elements at neighboring energies were thus neglected. Nevertheless, the analytical result Eq. (31) (solid lines) agrees with the data.

Figure 16

Distribution of $S_{12}$-matrix elements according to modulus (upper panels) and phase (lower panels) for data points as described in the caption of Fig. 15. An RMT simulation (solid lines) shows acceptable agreement with the data.

Figure 17

Distribution $P$ of the rescaled Fourier coefficients $\tilde{S}/\sqrt{⟨x_k⟩}$ for the elastic case, $\left\{a,b\right\}=\left\{1,1\right\}$. Upper panels: the data (histograms) for the distribution agree well with the solid lines given by Eq. (29). Lower panels: the phases are uniformly distributed in the interval $\left\{-\pi ,+\pi \right\}$. Data source is as in Fig. 15 with Fourier coefficients taken from the first 200 ns, i.e., 1240 data points contribute to each histogram.

Figure 18

Same as in Fig. 17 but for the inelastic case $\left\{a,b\right\}=\left\{1,2\right\}$. Data source is as in Fig. 16 with Fourier coefficients taken from the first 200 ns.

Figure 19

Distribution of the Fourier coefficients $x_k={\left|{\tilde{S}}_{12}\right|}^2$ in the interval 16–17 GHz under TRIV with $B=190\text{mT}$. Panel (a) on the left-hand side displays on a logarithmic scale (base 10) the $x_k$ as obtained from the data. Panel (b) shows analogously the rescaled quantities $z_k=x_k/⟨x_k⟩$. Panel (c) on the right-hand side shows the distribution of $M=1200$ rescaled coefficients on a logarithmic scale for six realizations. The dashed line is the exponential expected for a Gaussian-distributed $\tilde{S}$.

Figure 20

The probability distribution function (PDF, upper panel) and the cumulative distribution function (CDF, lower panel) of the distance measure $I/\overline{I}$ for $M=800$. The solid lines in both panels correspond to the analytic model Eq. (41), the results of a Monte Carlo simulation are shown in the upper (lower) panel as a histogram (circles). For a threshold of $K=0.9$ on the CDF a limit (horizontal dashed line) of $R_K^2=1.064$ is imposed on $I/\overline{I}$ (vertical dashed line) which may not be exceeded if the GOF test is to accept the model.

Figure 21

Top: comparison of autocorrelation functions in the frequency domain for a single realization in the range 24–25 GHz, i.e., $\Gamma /d\approx 1.14$, and for $B=190\text{mT}$. The four panels display the results for $C_{11},C_{12},C_{21},C_{22}$. The discrepancy between data (dots) and VWZ (solid) or complete TRIV (dashed) is more pronounced than for the model for partial TRIV (dash-dotted) with $\xi =0.202$. Bottom: $C_{12}$ in the time domain (same key, but for clarity without the result for complete TRIV).