Extended Data Fig. 4: Calculated eigenstates of the \({{\boldsymbol{a}}}_{{\bf{1}}},{{\boldsymbol{a}}}_{{\bf{2}}},{{\boldsymbol{a}}}_{{\bf{3}}}^{\dagger }\) loop in the singly conjugated trimer studied in Fig. 4.
From: Non-Hermitian chiral phononics through optomechanically induced squeezing
a, Phase diagram for the imaginary part of the eigenfrequencies, showing the stability-to-instability boundary in ξ–Φ space, where \(\xi =J/(2\sqrt{2}\eta )\) and γi = 0. Such boundary is associated with a second-order exceptional contour. b, Cuts of the eigenfrequency Riemann surfaces along Φ = 0, shown as a red dashed trajectory in the phase diagram, as a function of the ratio \(\xi =J/(2\sqrt{2}\eta )\). The squared weights of the J = 0 eigenstates in the full eigenvectors are shown in the colour scale. The weights are calculated from the symplectic projections (Σz product) on the gainy/lossy combinations ag, al and the passive mode a−. A second-order exceptional point (denoted EP2), found for \(J=2\sqrt{2}\eta \), is highlighted. As \(J < 2\sqrt{2}\eta \), \({{\mathscr{P}}}_{{\rm{gl}}}{\mathscr{T}}\) symmetry is spontaneously broken, inducing eigenstate localization. The antisymmetric 1–2 mode a− is detached from this mechanism and remains uncoupled. Real and imaginary parts are rescaled by η. c, As in b, but along the cut Φ = π/2 (corresponding to the blue dashed line in a, which shows the third-order exceptional point (EP3, at \(J=\sqrt{2}\eta \)). The \({{\mathscr{P}}}_{{\rm{gl}}}{\mathscr{T}}\)-symmetry broken states are now hybrid combinations of ag/a− and al/a− modes. Such combinations break \({{\mathscr{P}}}_{12}{\mathscr{T}}\) symmetry as well, as explained in the text.
