Fig. 1. Imaginary parts of the spectrum Eq. (1) for nonzero perturbations δ/J={0.01,0.1}. The schematic PT dimer with gain (red) and loss (blue) sites is shown. When γ/J≪1, Iλ± grow linearly with the gain-loss strength, but are nonzero. The divergence of their derivative at the threshold γ=J is smoothed out as δ>0 increases. In this case, the system is always in the PT-broken phase, no matter how small Iλ±(γ)≠0 are.

Fig. 2. Decay rates Γ± of the two eigenmodes of Eq. (5) show the emergence of a slowly decaying mode for γ∼J, not only at δ=0, the prototypical exceptional point case, but also for a wide range of δ/J≠0. Inset: the PT transition threshold is determined by dΓ/dγ changing its sign from positive to negative. These results are even in the offset δ and thus remain the same for δ<0.

Fig. 3. Net transmission T(γ) shows an upturn with increasing loss strength γ signaling the passive PT transition [29]. It shows minimal change from its δ=0 value [29] when δ/J is increased all the way to unity, i.e., the system is removed far from the exceptional point. These results are even in δ and thus remain unchanged for δ<0. Inset: the lattice-model results (line) are consistent with the BPM results (stars) obtained with sample parameters in Ref. [29]; see Section 5 for details.

Fig. 4. PT transition in three coupled waveguides with lossy center waveguide, Eq. (6). The decay rates Γk show the emergence of a slow mode near γPT∼22J for a wide range of center-site potential. These results are even in δ and thus remain valid for δ<0.

Fig. 5. PT transition in three coupled waveguides with first lossy waveguide, Eq. (7). When δ=0, the two modes with equal decay rates undergo a sign change for dΓk/dγ near γ/J∼1.4. When onsite potential δ>0 is introduced, the two degenerate modes split, and the eigenmode decay-rate diagram hints at the existence of an exceptional point at δ/J=0.75.

Fig. 6. BPM results for the intensity I(x,z) of a symmetric initial pulse in a waveguide trimer defined by Hamiltonian Eq. (6) with (a), (b) δ=0 and (c), (d) δ/J=1. (a) When γ/J=2.75<γPT, the pulse decays quickly. (b) At a much larger loss, γ/J=10, the pulse propagates longer, giving rise to increased transmission. (c) For γ/J=2.75, the pulse decays quickly. (d) At γ/J=10, the pulse propagates longer. In both cases, δ=0 and δ/J=1, the emergence of a slowly decaying mode is clear; in the latter case, the Hamiltonian is far removed from an exceptional point.