Fig. 1. Mechanism of highly-responsive wireless sensing. (a) System configuration. The system consists of a pair of LC resonators, where one resonator has a lossy component acting as the sensor and the other one with a saturable gain serves as the reader. ω1,2, resonant frequencies of the reader and sensor; κ, coupling rate; γ, loss rate; gsat, saturable gain; Δω, sensor’s perturbation. (b) Eigenfrequencies of the proposed system. The EP marks the point where the two branches split (black dot). A pair of HHPs is located at the edges of the bistability (the positions of arrows) induced by the nonlinearity. (c) Sensing mechanism. This sensing method has a high responsivity because of the Hamiltonian hopping. The sensor can also be continuously measured by monitoring the shifts of the HHP.

Fig. 2. Numerical calculations. (a) Side view of the eigenfrequency solution surface. The red and blue areas represent the two resonances in the strong coupling regime. Two white dots represent a pair of HHPs on the edges of bistability regime. (b) Cross section of Riemann surface at κ=0.25. The figure shows a pair of HHPs. (c) Linear relation between sensor’s capacitance perturbation and HHP shift.

Fig. 3. Experimental setup. (a) Photo of a wirelessly coupled circuit. Two spiral inductors are fabricated on a printed circuit board. The resonant frequencies can be tuned by programmable resistor R2 and capacitor C2. (b) Circuit diagram of wirelessly coupled resonators. The sensor is connected to the loss side in parallel. The gain is provided by a negative impedance converter (NIC). The resonant frequency is measured by an oscilloscope.

Fig. 4. Experimental characterization. (a), (b) Riemann surface and bistable mode switching in (a) CCW and (b) CW encircling of the EP. The black circled dot shows the EP. To encircle the EP, we first fix R2 and scan C2 and then fix C2 and scan R2 in a square-loop trajectory as shown by the arrows. (c) Hamiltonian hopping when sweeping C2 in CW and CCW directions. Frequency response in the standard method is also shown by the gray curve. (d) Enhancement factor of the responsivity at the HHP over the frequency shift by the standard method. The shaded area shows the noise of the HHP.

Fig. 5. Relation of the sensor’s perturbation and the HHP. (a) Frequency response under different perturbation (step 3.8 pF) when sweeping C2 in CCW and CW directions. (b) Linear dependence of the HHP on the sensor’s perturbation.

Fig. 6. Monte Carlo simulation of noise. (a), (b) Real part of eigenfrequencies under a Gaussian distribution of ω2 (standard deviation σ=0.005) and coupling rate of (a) κ/γ=1.25 or (b) κ/γ=1.5. The HHP is distributed with uncertainty reflecting the noise. The noise of the HHP can be reduced in a stronger coupling regime.

Fig. 7. Experimental study of noise. (a), (b) Experiment results in the strong coupling regime by (a) quick (200 ms) and (b) slow (500 ms) sweeping of programmable capacitor C2 in both directions. The shadow indicates the standard deviation of five trials. The noise of HHPs can be reduced under slow tuning of C2.

Fig. 8. Measurement of a microscale pressure sensor. (a) Characterization of the pressure sensor. The capacitance of the sensor is linear to the air pressure. The inset shows the microscopic image of the pressure sensor. (b) Measurement of the sensor by the standard method. The results show low responsivity and high noise. (c) Measurement of the sensor at an HHP. The C2 is fixed, and the pressure keeps increasing until the resonant frequency of system hops. The results show a 40-fold enhanced responsivity and a detection limit down to sub-picofarad. (d) Continuous measurement of the sensor by monitoring the shift of the HHP. The pressure on the sensor is fixed to some discretized values, and C2 is swept repeatedly to obtain the positions of HHP.

Fig. 9. Measurement of the microscale sensor with five trials. The results show that the measurement uncertainty (i.e., HHP noise) is around 0.15 pF, indicating the sensing limit down to sub-picofarad (<1  pF).