$\mathcal{P}\mathcal{T}$ Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard

We demonstrate the presence of parity-time ($\mathcal{P}\mathcal{T}$) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its $\mathcal{P}\mathcal{T}$ symmetry. Spontaneous symmetry breaking occurs at the EP.

Figure 1

Two typical spectra $|S_{11}(f){|}^2$ for $B=38\mathrm{mT}$ and the parameter settings $(s,\delta )=(1.66,41.79)\mathrm{mm}$ (solid line) and $(s,\delta )=(1.99,41.76)\mathrm{mm}$ (dashed line), respectively, in the vicinity of the EP, which is located at $(s_{\mathrm{EP}},{\delta }_{\mathrm{EP}})=(1.72\pm 0.01,41.78\pm 0.01)\mathrm{mm}$. Inset: Top view (to scale) of the microwave billiard. The setup is described in the text.

Figure 2

Differences of the complex eigenvalues $E_{1,2}=f_{1,2}-i{\Gamma }_{1,2}/2$ in an area of the parameter plane ($s,\delta$) around the EP located at $(s_{\mathrm{EP}},{\delta }_{\mathrm{EP}})=(1.68\pm 0.01,41.19\pm 0.01)\mathrm{mm}$ for $B=0$ (left panel) and at $(s_{\mathrm{EP}},{\delta }_{\mathrm{EP}})=(1.72\pm 0.01,41.78\pm 0.01)\mathrm{mm}$ for $B=38\mathrm{mT}$ (right panel). The darker the color, the smaller the respective difference.

Figure 3

Radicand $\mathcal{D}$ defined in Eq. (5) for the parameter values ($s,\delta$) along the curve of darkest color in Fig. 2. Shown are $|\mathrm{Re}\overrightarrow{h}{|}^2$ (crosses), $|\mathrm{Im}\overrightarrow{h}{|}^2$ (circles), and $\mathrm{Re}\overrightarrow{h}·\mathrm{Im}\overrightarrow{h}$ (triangles), in the left panel for $B=0$, in the right one for $B=38\mathrm{mT}$. To obtain a dimensionless $\overrightarrow{h}$, it was divided by $|h_1|$.

Figure 4

The parameter $\tau$ in units of maximum $\mathcal{T}$ violation $\pi /4$ along the curve of darkest color in Fig. 2 for $B=38\mathrm{mT}$.