Noise resilient exceptional-point voltmeters enabled by oscillation quenching phenomena

Exceptional point degeneracies (EPD) of linear non-Hermitian systems have been recently utilized for hypersensitive sensing. This proposal exploits the sublinear response that the degenerate frequencies experience once the system is externally perturbed. The enhanced sensitivity, however, might be offset by excess (fundamental and/or technical) noise. Here, we developed a self-oscillating nonlinear platform that supports transitions between two distinct oscillation quenching mechanisms – one having a spatially symmetric steady-state, and the other with an asymmetric steady-state – and displays nonlinear EPDs (NLEPDs) that can be employed for noise-resilient sensing. The experimental setup incorporates a nonlinear electronic dimer with voltage-sensitive coupling and demonstrates two-orders signal-to-noise enhancement of voltage variation measurements near NLEPDs. Our results resolve a long-standing debate on the efficacy of EPD-sensing in active systems above self-oscillating threshold.

We start the derivation of Eq. ( 1 Next, we incorporate in our circuit, an energy dissipation (enhancement) associated with a nonlinear damping (antidamping) channel.We consider nonlinear components that are characterized by the I-V curve ( ) where ( ) represents loss (gain) and .The corresponding Kirchhoff's equation for the individual RLC resonator collapse to the following expression where ( ) ( ( ) ) is the relaxation rate of the individual (linear) LRC oscillator, and .Equation (S2) can be written in terms of and its conjugate as By invoking a rotating wave approximation and eliminating the fast-oscillating terms, the above equation can be further simplified as where .
We proceed by developing a TCMT associated to the two coupled RLC tanks.
From Kirchhoff's laws we obtain the equations for the voltage in resonator where .
( ) / is the relaxation rate of each resonator (below we will assume the high-Q limit, i.e., ), is the (voltage controlled) coupling capacitance, with being the coupling strength coefficient.Using the complexmode representation for each resonator , we rewrite the circuit equations Eq. ( S5) as where we have invoked, as in the case of single RLC resonator, the rotating wave approximation, together with the weak coupling limit and high-Q approximations.We further simplify Eq. (S6) by defining the rescaled complex field .

Supplementary Note 2: Coupling of the Circuit to Transmission Lines
The effect of a weak coupling of the RLC resonators to a transmission line (TL) via a coupling capacitance , , can be also modeled using CMT.At the node connecting the TL with the coupling capacitance, the voltage and current flowing toward the th ( ) RLC resonator can be written as a superposition of forward and backward propagating voltage waves, ( ) and where is the TL's characteristic impedance.In turn, the voltages can be represented by complex wave amplitudes

Supplementary Note 3: Equations of Motion in Polar Form
It is convenient for our analysis, to rewrite Eq. ( 1) of the main text in polar representation.To this end, we express the complex amplitudes as ( ) where the magnitudes and the phases of the fields in resonator are real numbers.Substitution of these expressions back to Eq.
(1) leads to the following set of coupled differential equations ̇ .Finally, in this polar representation the emitted power spectrum Eq. (S10) takes the form ( ) ./.
) by expressing the energy stored in the LC resonator as | | where the complex modal amplitude of a single LC resonator ( ) is the node voltage and √ the resonant frequency of the resonator in the absence of dissipation (amplification).Using this representation, we rewrite the circuit equation of the LC resonator, i.e., , as a set of two (uncoupled) first order differential equations i.e., and its complex conjugate.The latter relation emphasizes the time dependence of the modal amplitude and .
invoked the rotating wave approximation, and we introduced the TL-RLC coupling coefficient .Using the transformations √ and , and combining Eqs.(S7) and (S10) we arrive to Eq. (1) of the main text which describes the whole system of coupled RLC resonators and TLs.Let us finally point out that when the input wave