Odd-Time Reversal $\mathcal{P}\mathcal{T}$ Symmetry Induced by an Anti-$\mathcal{P}\mathcal{T}$-Symmetric Medium

We introduce an optical system (a coupler) obeying parity-time ($\mathcal{P}\mathcal{T}$) symmetry with odd-time reversal, ${\mathcal{T}}^2=-1$. It is implemented with two birefringent waveguides embedded in an anti-$\mathcal{P}\mathcal{T}$-symmetric medium. The system possesses properties that are untypical for most physical systems with the conventional even-time reversal. Having a symmetry-protected degeneracy of the linear modes, the coupler allows for the realization of a coherent switch operating with a superposition of binary states that are distinguished by their polarizations. When a Kerr nonlinearity is taken into account, each linear state, being doubly degenerated, bifurcates into several distinct nonlinear modes, some of which are dynamically stable. The nonlinear modes are characterized by amplitude and by polarization and come in $\mathcal{P}\mathcal{T}$-conjugate pairs.

Figure 1

Coupled transparent waveguides embedded in an anti-$\mathcal{P}\mathcal{T}$-symmetric medium (see the text for notations).

Figure 2

The upper panel shows schematically the coherent switch. Shadowed domains correspond to anti-$\mathcal{P}\mathcal{T}$-symmetric media. The central empty part illustrates the uncoupled waveguides with gain and losses. The polarization vectors at the output indicate (schematically) $\pi /2$-phase rotation of the nonswitched signal (at $\mathrm{\Gamma }=0$, red), and switching of the superposition rotated by angle $-\alpha$ (at $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{sw}}$, blue). The lower panel shows power distributions in the first $P_1$ (red line) and second $P_2$ (blue line) arms at ${\mathrm{\Gamma }}_{\mathrm{sw}}$, obtained for $\delta ={\delta }_0=2$ and $\kappa =1$.

Figure 3

(a) Families of solutions for $\delta =2$, $\kappa =1$, $\alpha =\pi /6$, $\vartheta =\phi =0$ visualized as dependencies $P$ vs $b$. We show two families bifurcating from each eigenvalue ${\tilde{b}}_{+}$ and ${\tilde{b}}_{-}$. Stable and unstable modes are represented by solid and dotted segments, respectively. Vertical dotted lines indicate values $b=0$ and $b=-6$ analyzed in (b) and (c) and Fig. 4. (b) Polarization vectors ${\mathcal{E}}_{1,2}$ in each waveguide for two stable nonlinear modes bifurcating from ${\tilde{b}}_{+}$, at $b=0$. Arrows corresponding to the same mode have the same color as the respective family (and the same arrow head). (c) Polarization vectors ${\mathcal{E}}_{1,2}$ for two stable modes bifurcating from and ${\tilde{b}}_{-}$, at $b=-6$. The lengths of arrows ${\mathcal{E}}_{1,2}$ are equal to powers $P_{1,2}$ in each arm.

Figure 4

(a) Branches of nonlinear modes for fixed $b=-6$, $\delta =2$, and changing $\kappa$. Stable and unstable modes are represented by solid and dotted segments, respectively. The vertical dotted line indicates the $\mathcal{P}\mathcal{T}$-symmetry-breaking threshold $\kappa =2$. Panels (${\mathcal{E}}_1$) and (${\mathcal{E}}_2$) show polarization vectors corresponding to two merging branches (red and green curves) from (a).