Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics

The exotic physics associated with exceptional points (EPs) is always under the scrutiny of theoretical and experimental science. Recently, considerable effort has been invested in the combination of nonlinearity and non-Hermiticity. The concept of nonlinear EPs (NEPs) has been introduced, which can avoid the loss of completeness of the eigenbasis in dynamics while retaining the key features of linear EPs. Here, we present the first direct experimental demonstration of a NEP based on two non-Hermition coupled circuit resonators combined with a nonlinear saturable gain. At the NEP, the response of the eigenfrequency to perturbations demonstrates a third-order root law and the eigenbasis of the Hamiltonian governing the system dynamics is still complete. Our results bring this counterintuitive aspect of the NEP to light and possibly open new avenues for applications.

Figure 1

(a) Schematic of a linear model. It consists of two resonators with resonance frequency ${\omega }_{1,2}$ and linear gain $g_1$ and loss $l_2$. The coupling between them is non-Hermitian. (b) shows the variation of the two eigenstates $|{\varphi }_{+}⟩$ and $|{\varphi }_{-}⟩$ versus $ϵ$. (c) Schematic of our nonlinear model. ${\psi }_A$ of the three steady states (d) and the real part of corresponding frequency (e) versus $ϵ$. (f) shows ${\psi }_A$ of the two eigenstates of the instantaneous Hamiltonian $H_s$, which is anchored by the stable eigenmode. The EP and NEP are located at $ϵ=0$. The parameters used are: ${\omega }_1={\omega }_A=0$, ${\kappa }_0={\kappa }_{\nu }=g_1=l_2=1$, ${\omega }_2=2$, $l_B=1.14$, and ${\omega }_B=1.97$. The specific form of the saturable gain $g_A(|{\psi }_A|)$ in (c) is irrelevant for the results in (d)–(f) as long as Eq. (5) is satisfied.

Figure 2

(a) The circuit used in the experiment, showing the inductors ($L$), capacitors ($C$), resistors ($R$), diodes ($D$), and an amplifier ($A$). The black dashed rectangular marks the negative resistor $-R_A(|V_A|)$. (b) A photo of the experimental setup. The oscilloscope records the waveforms of $V_A$ and $V_B$, which then gives us the corresponding frequencies, voltages, and relative phase. The dc power supplies power for the amplifier, and the arbitrary waveform generator is used to generate the required external driving signal. We add a homemade variable inductor to fine-tune $L_A$ to control the resonance frequency of resonator $A$ [42]. Details of the circuit elements on the printed circuit board (PCB) can be found in Supplemental Material Sec. 5 [61]. (c) Removing the external driving signal of 1 V and 70 kHz, the red and blue lines show the temporal dynamics of $V_A$ and $V_B$, respectively. At around 0.4 ms, $V_A$ and $V_B$ reach a stable state. The bold red and blue lines highlight the envelope of $V_A$ and $V_B$, which remain nearly constant after reaching the stable state. (d) Starting at 0.66 ms, zoom in on the dynamics of the stable state. Parameters in the experiments are $L_A=245.4\mathrm{\mu }\mathrm{H}$, $C_0=18.5\mathrm{nF}$, $C_c=3.9\mathrm{nF}$, $R_c=760\mathrm{\Omega }$, $R_B=1314\mathrm{\Omega }$, and $L_B=197.7\mathrm{\mu }\mathrm{H}$.

Figure 3

The voltage amplitude ratio $|V_A/V_B|$ (a), the relative phase ${\theta }_A-{\theta }_B$ with ${\theta }_{A,B}=\arg (V_{A,B})$ (b) and the resonance frequency (c) versus the perturbation $ϵ$ by varying $L_A$. Here, the perturbation $ϵ$ is defined as $ϵ\equiv ({\omega }_A-{\omega }_{A0})/2\pi$, where ${\omega }_{A0}$ represents the resonance frequency of resonator $A$ when the ${\mathrm{NEP}}_3$ is reached. (d) The double-log plot of the frequency shift $\mathrm{\Delta }\omega =(\omega -{\omega }_{{\mathrm{NEP}}_3})/2\pi$ versus the perturbation $|ϵ|$, where ${\omega }_{{\mathrm{NEP}}_3}$ represents the eigenfrequency at the ${\mathrm{NEP}}_3$. The measured values are averaged over 16 independent measurements. The experimental errors (standard deviation) are smaller than the marker size. For demonstration purposes, we exaggerate the error bars in (a)–(d) by factors of 20, 10, 10, and 10, respectively. $L_0=253.5\mathrm{\mu }\mathrm{H}$ and the other parameters are the same as the Fig. 2.

Figure 4

The dynamics of the stable modes (a) and the corresponding Fourier spectra (b) under an artificial Gaussian noise. The circuit model used is plotted in Fig. S11 [61]. The real part of eigenvalues (c) and $V_A/V_B$ of the eigenstates (d) of $H_s$, the PF (e), and the SNR (f) versus the external perturbation $ϵ$. In (c) and (d), the solid red lines and the dashed gray lines represent the stable and the accompanied eigenmode of $H_s$. For demonstration purposes, we exaggerate the error bars by a factor of 10, 10, and 15 in (c)–(e), respectively. The cyan area in (f) is for eye guiding. The definition of $ϵ$ and the parameters used are the same as those in Fig. 3.