Induced Time-Reversal Symmetry Breaking Observed in Microwave Billiards

Using reciprocity, we investigate the breaking of time-reversal ($T$) symmetry due to a ferrite embedded in a flat microwave billiard. Transmission spectra of isolated single resonances are not sensitive to $T$ violation, whereas those of pairs of nearly degenerate resonances do depend on the direction of time. For their theoretical description a scattering matrix model from nuclear physics is used. The $T$-violating matrix elements of the effective Hamiltonian for the microwave billiard with the embedded ferrite are determined experimentally as functions of the magnetization of the ferrite.

Figure 1

Scheme of the experimental setup (not to scale). The antennas connected to the vector network analyzer (VNA) were located at the positions $a$ and $b$. The inner circle is the copper disk introduced into the resonator to transform the circular into an annular billiard.

Figure 2

A singlet in the annular billiard. For an external field of 119.3 mT we show $S_{ab}$ (open circles) and $S_{ba}$ (solid circles). Both amplitudes and phases coincide and reciprocity holds. The statistical errors of the data are smaller than the symbols.

Figure 3

The doublet at 2.67 GHz in the circular billiard with an external magnetic field of 36.0 mT. The upper part shows the absolute values of $S_{aa}$ (solid) and $S_{bb}$ (dashed), the lower one those of $S_{ab}$ (solid) and $S_{ba}$ (dashed). The uncertainties of $S_{ab}$ and $S_{ba}$ are about $5\times {10}^{-4}$. Reciprocity is clearly violated.

Figure 4

The $T$ violating matrix element $H_{12}^a$ for (a) the second doublet at 2.67 GHz (upper panel) and (b) the third doublet at 2.9 GHz (lower panel). The error bars include the systematic, statistical, and numerical errors. The line shape of the ferromagnetic resonance for the second doublet (upper panel) has been convoluted with a Gaussian distribution of magnetization with width 15.5 Oe, see Eq. (10).