^{1}Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory of Physics and Devices in Post-Moore Era, College of Hunan Province, School of Physics and Electronics, Hunan Normal University, Changsha 410081, China

^{2}Tsinghua-Berkeley Shenzhen Institute, Institute of Data and Information, Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China

^{3}Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore

Responsivity is a critical parameter for sensors utilized in industrial miniaturized sensors and biomedical implants, which is typically constrained by the size and the coupling with external reader, hindering their widespread applications in our daily life. Here, we propose a highly-responsive sensing method based on Hamiltonian hopping, achieving the responsivity enhancement by 40 folds in microscale sensor detection compared to the standard method. We implement this sensing method in a nonlinear system with a pair of coupled resonators, one of which has a nonlinear gain. Surprisingly, our method surpasses the sensing performance at an exceptional point (EP)—simultaneous coalescence of both eigenvalues and eigenvectors. The responsivity of our method is notably enhanced thanks to the large frequency response at a Hamiltonian hopping point (HHP) in the strong coupling, far from the EP. Our study also reveals a linear HHP shift under different perturbations and demonstrates the detection capabilities down to sub-picofarad ($<1\text{}\mathrm{pF}$) of the microscale pressure sensors, highlighting their potential applications in biomedical implants.

1. INTRODUCTION

Inductor–capacitor (LC) sensors represent a category of electronic devices that convert target signals into resonant frequency shifts, which are interrogated by an external wireless-coupled reader [1,2], thus eliminating the necessity for integrated chip and battery [3–5]. The simple implementation of LC sensors has led to various miniaturized devices [6–9], especially biomedical implants for measuring intraocular pressures, intracranial pressures, and pulmonary artery pressures, to name a few [10–15]. These LC sensors, however, usually suffered from low responsivity—characterized by the resonant frequency shift in the reader’s reflection spectrum. Subtle capacitive perturbations applied to LC sensors typically cannot lead to perceptible spectral responses due to the weak responsivity inherent in the miniaturized devices and the low $Q$-factor resonance resulting from the high magnetic-field absorption in surrounding environments. These challenges seriously limit their wide applications in both industrial small-scale sensors and biomedical implants.

Exceptional points (EPs) have recently garnered attention for their ability to enhance responsivity in resonant-type sensors across diverse fields, ranging from photonics to electronics [16–18]. At an EP, the system’s eigenvalues and eigenvectors are simultaneously degenerated, and the system’s Hamiltonian shows a nonlinear splitting under a small perturbation, resulting in amplified eigenfrequency response [19–21]. Amplified responsivity has been successfully observed in optical microcavity sensors, biophotonic sensors, and microscale pressure sensors, among others [22–26]. However, recent studies mentioned that, given the amplified responsivity, EP sensors do not perform better than conventional wireless sensors because of the high noise at the EPs [27–29]. Various methods have been proposed to get out of this dilemma, but the responsivity has only been enhanced up to 10-fold [30–33]. An effective method to further amplify the responsivity to meet the real-world applications is still lacking.

Here, we show a highly-responsive sensing method based on Hamiltonian hopping and achieve a 40-fold responsivity enhancement in sensing of a microscale pressure sensor. Hamiltonian hopping happens during the encircling of the EP: the system evolves adiabatically and experiences a state flip in one direction but returns to the initial state through a nonadiabatic transition in the reverse direction, hence Hamiltonian hopping [34–39]. Benefiting from the large frequency response at the Hamiltonian hopping point (HHP), we achieve a notable responsivity enhancement compared to a standard sensing method without HHP. We also reveal a linear HHP shift under the different sensor inputs (Fig. 1) and demonstrate the detection of a microscale pressure sensor that could potentially be used as biomedical implants. Furthermore, we study noise at the HHP via Monte Carlo simulations and experiments and propose strategies to suppress the noise in our sensors.

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Figure 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. ${\omega}_{1,2}$, resonant frequencies of the reader and sensor; $\kappa $, coupling rate; $\gamma $, loss rate; ${g}_{\mathrm{sat}}$, saturable gain; $\mathrm{\Delta}\omega $, 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.

We begin by describing an EP system using coupled mode theory. The EP system consists of two resonators with resonant frequencies ${\omega}_{1,2}$, coupling rate $\kappa $, loss rate $\gamma $, and gain rate $g$, as shown in Fig. 1(a). The system’s Hamiltonian can be expressed as $$H=\left(\begin{array}{cc}{\omega}_{1}+ig& \kappa \\ \kappa & {\omega}_{2}-i\gamma \end{array}\right).$$

Considering two resonant states with a time-harmonic signal ${e}^{-i\omega t}$, the above equation is equivalent to the circuit theory [26]. The eigenfrequencies can thus be calculated from the characteristic equation: $$[i({\omega}_{1}-\omega )+g][i({\omega}_{2}-\omega )-\gamma ]+{\kappa}^{2}=0,$$where $\omega $ is the eigenfrequency of Eq. (1) that has two branches denoted as ${\omega}_{\pm}$. The complex-valued eigenfrequencies are given by $${\omega}_{\pm}=\frac{1}{2}[({\omega}_{1}+{\omega}_{2})+i(-g+\gamma )]\pm \sqrt{\frac{1}{4}{[({\omega}_{1}-{\omega}_{2})+i(-g-\gamma )]}^{2}+{\kappa}^{2}}.$$

We now additionally introduce a nonlinearity—saturable gain ${g}_{\mathrm{sat}}$—into the Hamiltonian, a more realistic approach in practice gain media. Previous studies have also shown that such a nonlinearity could lead to bistability [40–44], resulting in the occurrence of a pair of HHPs [Fig. 1(b)]. In particular, the induced nonlinearity forces the system to oscillate at a steady state with purely real eigenfrequencies and saturate the gain. In contrast, other states with complex-valued eigenfrequencies are prevented from reaching steady-state resonance [40–44].

Taking eigenfrequencies to be real, we separate the real and imaginary parts of Eq. (3), which leads to $${g}_{\mathrm{sat}}=\gamma \frac{\omega -{\omega}_{1}}{\omega -{\omega}_{2}}.$$

The characteristic equation now becomes $$(\omega -{\omega}_{1}){(\omega -{\omega}_{2})}^{2}+{\gamma}^{2}(\omega -{\omega}_{1})-{\kappa}^{2}(\omega -{\omega}_{2})=0.$$

We numerically solve Eq. (5) and obtain a self-intersecting solution surface [i.e., Riemann surface, Fig. 1(b)] in parameter space of ${\omega}_{1}$ and $\kappa $ by assuming that other parameters are fixed (${\omega}_{2}=1$ and $\gamma =0.2$). The Riemann surface consists of three possible solutions shown by different colors. The states with purely real eigenfrequencies (blue and orange) are steady states supported by the system, while the other state (yellow) is always unstable. An EP is located at ${\omega}_{1}={\omega}_{2}$ and $\gamma =\kappa $, where the real parts and imaginary parts of eigenfrequencies simultaneously coalesce.

A pair of HHPs occurs during the encircling of the EP in both counterclockwise (CCW) and clockwise (CW) directions [Fig. 1(b)]. To give a clear description of Hamiltonian hopping, we plot the cross section of Riemann surface under $\kappa =0.25$, as shown in Fig. 2(b). The system dynamics during the encircling of EP in both directions are shown by the black arrows. Initially, the system is at the upper branch [the white dot in Fig. 1(b) or the blue curve in Fig. 2(b)] and evolves adiabatically in the CCW direction. Although other states (the red and yellow curves) have zero imaginary eigenfrequencies, the system will not switch the state because the current one is already steady, preventing other states from reaching steady. When the system reaches the right HHP, the system experiences a nonadiabatic transition and spontaneously switches to the other steady-state resonance. The system dynamics in the CW direction are similar, but the nonadiabatic transition happens at the right HHP, leading to unique bistability and a pair of HHPs.

Figure 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 $\kappa =0.25$. The figure shows a pair of HHPs. (c) Linear relation between sensor’s capacitance perturbation and HHP shift.

We now consider ${\omega}_{2}$ as the sensor. The sensing mechanism is illustrated in Fig. 1(c). When the sensor is perturbed by $\mathrm{\Delta}\omega $, the HHP shifts linearly with $\mathrm{\Delta}\omega $. The linear relation between HHP and $\mathrm{\Delta}\omega $ is also numerically illustrated in Fig. 2(c). Therefore, due to the Hamitonian hopping, we can record the sensor’s perturbation via HHP shifts in a high-responsivity manner. The responsivity at the HHP in the strong coupling ($\kappa >\gamma $) is the eigenfrequency difference of the states between the upper branch and the lower branch, which can be approximated as $${\omega}_{+}-{\omega}_{-}=2\sqrt{{\kappa}^{2}-{\gamma}^{2}}.$$

In comparison, the standard method uses a single inductive coil to read out the LC sensor, and the responsivity is [26] $${\omega}_{+}={\omega}_{2}-\frac{{\kappa}^{2}}{{\omega}_{1}-{\omega}_{2}}.$$

When the perturbation is extremely weak, our HHP-based sensing method can generate a much larger response than the standard method (see experimental results below). Higher coupling strength or lower loss can further increase the responsivity of HHP sensing.

3. SYSTEM CHARACTERIZATION

We implement the above sensing method using the radio-frequency circuit shown in Fig. 3. Two printed spiral inductors (${L}_{1}={L}_{2}=5.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu H}$) are fabricated on a printed circuit board and tuned using programmable capacitors ${C}_{n}\text{\hspace{0.17em}\hspace{0.17em}}(n=1,2)$ to resonant frequencies ${\omega}_{n}=1/\sqrt{{L}_{n}{C}_{n}}$. The programmable capacitor can be tuned from 12.5 to 194 pF with a step of 0.355 pF (NCD2400M, IXYS Corporation). The separation distance between the inductors determines the internal coupling coefficient $k={\omega}_{1}{\omega}_{2}{M}_{s}/(2\omega \sqrt{{L}_{1}{L}_{2}})$, where ${M}_{s}$ is the mutual inductance between the reader and sensor inductors. A gain element is achieved by a negative impedance converter (NIC), in which the nonlinearity is introduced by a saturable amplifier (ADA4817, Analog Devices). Meanwhile, the loss component is controlled by a programmable resistor ${R}_{2}$ from 75 to $1000\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\Omega}$ with a step of $3.9\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\Omega}$ (AD5254, Analog Device). The reader (i.e., gain resonator) is self-oscillating with a steady-state frequency readout by an oscilloscope (MDO3012, Tektronix).

Figure 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 ${R}_{2}$ and capacitor ${C}_{2}$. (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.

We observe the pair of HHPs by tuning the programmable resistor and capacitor on the loss side and measuring the resonant frequencies in the parameter space of ${R}_{2}$ and ${C}_{2}$. The resonant frequencies are shown by two separated Riemann surfaces in CCW encircling [Fig. 4(a)] and CW encircling [Fig. 4(b)] of the EP. The white dot indicates the EP, where the continuous frequency shift has maximum abruptness. In the region of ${R}_{2}<{R}_{\mathrm{EP}}$, the resonant frequencies during CCW and CW encircling are almost the same except for the noise induced by programmable components. In the region of ${R}_{2}>{R}_{\mathrm{EP}}$, the mismatched area of the two surfaces indicates a unique bistability and thus a series of pairs of HHPs. The resonant frequencies under a specific case of ${R}_{2}<{R}_{\mathrm{EP}}=1.141$ are displayed in Fig. 4(c), which show a pair of HHPs marked by the dots.

Figure 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 ${R}_{2}$ and scan ${C}_{2}$ and then fix ${C}_{2}$ and scan ${R}_{2}$ in a square-loop trajectory as shown by the arrows. (c) Hamiltonian hopping when sweeping ${C}_{2}$ 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.

We compare the sensing at an HHP with the standard method as illustrated in Fig. 4(d). We evaluate the responsivity by considering a small capacitance perturbation and obtain the system’s resonant frequency shift by either HHP or the standard method. Considering a weak perturbation of ${C}_{2}$ within 10 pF, the responsivity at an HHP is enhanced up to 70-fold compared to the standard method. It should be noted that this enhancement cannot approach infinity because an infinitesimal perturbation cannot be produced in practice and, more importantly, the noise of HHP limits the responsivity enhancement (see noise analysis below). We also experimentally verify the linear relation between capacitive perturbation and HHP shift. The pair of HHPs can be determined by continuously sweeping ${C}_{2}$ forward and backward and recording the position where the Hamiltonian hopping happens [Fig. 5(a)]. When the sensor is perturbed by a step of 3.8 pF, both HHPs shift to the right with a linear dependence with perturbation [Fig. 5(b)].

Figure 5.Relation of the sensor’s perturbation and the HHP. (a) Frequency response under different perturbation (step 3.8 pF) when sweeping ${C}_{2}$ in CCW and CW directions. (b) Linear dependence of the HHP on the sensor’s perturbation.

We should consider the noise of the HHP because the noise determines its sensing limit. We first study the noise using the Monte Carlo simulation method. We numerically calculate Eq. (5) under different ${\omega}_{1}$ and consider Gaussian distribution of the perturbation with a probability density function: $$P({\omega}_{2})=\frac{1}{\sqrt{2\pi}\sigma}\text{\hspace{0.17em}}\mathrm{exp}(-\frac{1}{2}\frac{{\omega}_{2}^{2}}{{\sigma}^{2}}),$$where $\sigma $ is the standard deviation representing the noise intensity. The numerical results of eigenfrequencies under $\kappa /\gamma =1.25$ [Fig. 6(a)] show a similar lineshape with Fig. 2(b), but the eigenfrequencies are distributed with a probability density indicated by the color brightness in the figure. The role of noise changes a purely real eigenfrequency into a complex value with an imaginary part, which turns the steady-state resonance into unsteady within a specific spectrum and results in a distributed HHP.

Figure 6.Monte Carlo simulation of noise. (a), (b) Real part of eigenfrequencies under a Gaussian distribution of ${\omega}_{2}$ (standard deviation $\sigma =0.005$) and coupling rate of (a) $\kappa /\gamma =1.25$ or (b) $\kappa /\gamma =1.5$. The HHP is distributed with uncertainty reflecting the noise. The noise of the HHP can be reduced in a stronger coupling regime.

We propose two solutions to reduce noise at an HHP via strong coupling or time evolution. First, the HHP can be reduced in a stronger coupling region. We consider another case with $\kappa /\gamma =1.5$ and repeat the above numerical calculation. The results in Fig. 6(b) illustrate the reduced distribution of HHPs at strong coupling. The underlying mechanism is that the EP is extremely sensitive to noise due to the maximal nonorthogonality of the eigenstates at the EP [20]. Operating the system away from the EP could decrease the nonorthogonality and increase the noise robustness.

We also find that noise can be reduced by time evolution, i.e., by slowly tuning the system’s parameters. We experimentally demonstrate the noise suppression at HHPs under slow tuning of ${C}_{2}$ in both directions, as shown in Fig. 7. The noise of HHPs, i.e., uncertainty, is measured by sweeping the programmable capacitor ${C}_{2}$ forward and backward in the strong coupling regime and repeating five times. The minimal resolution of the programmable capacitor is 0.376 pF. For quick tuning [200 ms, Fig. 7(a)], the noise of HHPs is around 1 pF in both directions. Meanwhile, for slow encircling [500 ms, Fig. 7(b)], the noise of HHPs can be ignored, indicating that the noise of HHPs is less than 0.376 pF under slow detuning in both directions. The mechanism behind the beneficial slow parameter variation can also be understood by the system’s adiabatic transitions. The adiabatic evolution is satisfied by slow parameter variations, stabilizing the resonances along the Riemann surface and thus less HHP noise.

Figure 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 ${C}_{2}$ in both directions. The shadow indicates the standard deviation of five trials. The noise of HHPs can be reduced under slow tuning of ${C}_{2}$.

We now demonstrate the detection of a microscale pressure sensor (Protron Mikrotechnik GmbH), which can be potentially used as a biomedical implant [$0.6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}\times 1.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, Fig. 8(a)]. This capacitor-type sensor is connected to the proposed circuit as a variable capacitor that can be tuned by the air pressure using a syringe. We first characterize the pressure sensor by applying increased air pressure from 1 bar to 3 bar. The capacitance of the sensor increases linearly from 0 pF to 3 pF [Fig. 8(a)]. The loss induced by the pressure sensor can be ignored in the parallel sensor circuit since the resistance of the sensor is more than $1\times {10}^{6}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\Omega}$.

Figure 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 ${C}_{2}$ 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 ${C}_{2}$ is swept repeatedly to obtain the positions of HHP.

The sensing performance in the method without HHP is first evaluated. Under a weak capacitive perturbation ($<3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{pF}$), a weak responsivity and high noise indicated by the shaded area can be observed in Fig. 8(b). We then measure the sensor at an HHP, and the frequency response to the sensor capacitance change is shown in Fig. 8(c). A clear frequency hopping can be observed, indicating a sensing limit down to sub-picofarad ($<1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{pF}$). Although the system has some inevitable noise in strong coupling, the frequency response is much higher than the noise as long as the sensor parameter passes through the HHP. The responsivity at the HHP can be thus evaluated as 0.8 MHz when the capacitance is perturbed by 0.3 pF. However, the responsivity by the standard method is less than 20 kHz [Fig. 8(b)]. The responsivity of the microscale sensor is enhanced at the HHP by 40-fold compared to the standard method without HHP.

We also demonstrate the measurement of the microscale sensor’s multiple states by monitoring the HHP shift. During the measurement, the programmable capacitor ${C}_{2}$ is swept slowly and repeatedly to monitor the HHP shift. The resolution of ${C}_{2}$ is 0.376 pF, which does not induce additional noise on the HHP under slow encircling of the EP [Fig. 7(b)]. When the sensor capacitance is changed linearly, the HHP shifts linearly. Here, only forward detuning is applied, since the HHP in backward tuning follows the same linearity as the forward detuning. As shown by Fig. 8(d), the change of sensor capacitance within 3 pF can be continuously read out indicated by the HHP shifts. We repeat the above measurements five times (Fig. 9). The uncertainty of the HHP in the measurement of the microscale sensor is around 0.15 pF, indicating the sensing limit of the current setup. The noise can be further reduced by using digital components with higher tuning precision and less electronic noise.

Figure 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 ($<1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{pF}$).

In conclusion, we have proposed a highly-responsive wireless sensing method through repeated Hamiltonian hopping in an EP system. We have demonstrated the wireless measurement of a microscale pressure sensor with a sensing limit down to sub-picofarad. Compared to the sensing using a standard method, our method showed enhanced responsivity by 40-fold. We have also demonstrated the measurement of the sensor’s multiple states by monitoring the HHP shifts. Moreover, we have studied the noise at the HHP and proposed two potential solutions to further overcome the noise. Our method could potentially be adopted in photonics and electronics [45–49] and be applied for sensitive biomolecule detection and physiological state monitoring [50–55].

Acknowledgment

Acknowledgment. We thank Prof. John Ho and Prof. Cheng-Wei Qiu for guidance and discussions.