Induced Violation of Time-Reversal Invariance in the Regime of Weakly Overlapping Resonances

We measure the complex scattering amplitudes of a flat microwave cavity (a “chaotic billiard”). Time-reversal ($\mathcal{T}$) invariance is partially broken by a magnetized ferrite placed within the cavity. We extend the random-matrix approach to $\mathcal{T}$ violation in scattering, determine the parameters from some properties of the scattering amplitudes, and successfully predict others. Our work constitutes the most precise test of the random-matrix theoretical approach to $\mathcal{T}$ violation so far available.

Figure 1

The tilted-stadium billiard (schematic). The two antennas 1, 2 connect the resonator to the VNA. The ferrite is fixed, the scatterer can be moved freely.

Figure 2

Transmission spectra for $B=190\mathrm{mT}$ in the range 16–17 GHz. The amplitudes and phases of $S_{12}$ (solid) and $S_{21}$ (dashed) are seen to differ.

Figure 3

Experimental values of $C_{\mathrm{cross}}(0)$ (upper panel) from Eq. (1) and the parameter $\xi$ for $\mathcal{T}$ violation deduced from these (lower panel) with the help of Eq. (3). The error bars indicate the rms variation of $C_{\mathrm{cross}}(0)$ over the 6 realizations.

Figure 4

Dependence of the cross-correlation coefficient $C_{\mathrm{cross}}(0)$ on the parameter $\xi$ as predicted by the random-matrix model for partial violation of $\mathcal{T}$ invariance. Also shown is how $C_{\mathrm{cross}}(0)=0.49(3)$ translates into $\xi =0.29(2)$.

Figure 5

Autocorrelation function for $S_{12}$ in the range of 16–17 GHz and at $B=190\mathrm{mT}$. In the time domain (a) the data (dots) scatter around the theoretical fit (solid) for $T_1=0.37$, $T_2=0.41$, ${\tau }_{\mathrm{abs}}=2.9$ and $\xi =0.25$. Transforming the results back into frequency domain (b) confirms the good agreement between data and theory. We observe that neighboring data points in (b) are correlated, whereas those in (a) are not.

Figure 6

Elastic enhancement factors $\mathcal{W}$ determined from the data (open circles) and from the analytic result for partial $\mathcal{T}$ violation (filled circles) using the fitted parameters. The error bars indicate the variations within the 6 realizations. The dashed horizontal lines indicate the limits of $\mathcal{W}$ for $\mathcal{T}$ invariant systems.