Fig. 1
Flow of eigenvalues and transitions in neutral-loss and gain-loss coupled electrical oscillators. a Schematic of an inductor-capacitor (LC) oscillator (gray) inductively coupled to an resistive-LC oscillator (blue). In the weak coupling limit, \(M = \sqrt {L/L_x} \ll 1\), this system maps onto a dissipative dimer (Methods, Circuit to dimer mapping). b Flow of eigenvalues \(\Re \lambda _k\) as a function of loss shows that the top two (red and blue) levels attract each other and reach a minimum gap ∝M3 near loss strength \(\gamma /\omega _0\sim 2M^2\) before diverging again, thus indicating an avoided level crossing (ALC). c Flow of eigenmode decay rates −\(\Im\)λk shows that the slowly decaying modes (blue) emerge at \(\gamma /\omega _0\sim 2M^2\) signaling a passive parity-time \(\left( {{\cal P}{\cal T}} \right)\) transition at the location of the ALC. The insets show expanded view of the transition region. d Schematic circuit with gain and loss17. Flows of \(\Re \lambda _k\), e, and \(\Im\)λk, f, as a function of the gain-loss strength show that the \({\cal P}{\cal T}\)-breaking transition occurs only an exceptional point (EP), γ = (ωM−ω0) ∝ ω0M2. The comparison of a and d shows that the transition accompanied by an ALC, instead of an EP, occurs only in the dissipative case
