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.

We begin by describing an EP system using coupled mode theory. The EP system consists of two resonators with resonant frequencies ω1,2, coupling rate κ, loss rate γ, and gain rate g, as shown in Fig. 1(a). The system’s Hamiltonian can be expressed as H=(ω1+igκκω2−iγ).(1)

Considering two resonant states with a time-harmonic signal e−iωt, the above equation is equivalent to the circuit theory [26]. The eigenfrequencies can thus be calculated from the characteristic equation: [i(ω1−ω)+g][i(ω2−ω)−γ]+κ2=0,(2)where ω is the eigenfrequency of Eq. (1) that has two branches denoted as ω±. The complex-valued eigenfrequencies are given by ω±=12[(ω1+ω2)+i(−g+γ)]±14[(ω1−ω2)+i(−g−γ)]2+κ2.(3)

We now additionally introduce a nonlinearity—saturable gain gsat—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 gsat=γω−ω1ω−ω2.(4)

The characteristic equation now becomes (ω−ω1)(ω−ω2)2+γ2(ω−ω1)−κ2(ω−ω2)=0.(5)

We numerically solve Eq. (5) and obtain a self-intersecting solution surface [i.e., Riemann surface, Fig. 1(b)] in parameter space of ω1 and κ by assuming that other parameters are fixed (ω2=1 and γ=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 ω1=ω2 and γ=κ, 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 κ=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.

We now consider ω2 as the sensor. The sensing mechanism is illustrated in Fig. 1(c). When the sensor is perturbed by Δω, the HHP shifts linearly with Δω. The linear relation between HHP and Δω 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 (κ>γ) is the eigenfrequency difference of the states between the upper branch and the lower branch, which can be approximated as ω+−ω−=2κ2−γ2.(6)

In comparison, the standard method uses a single inductive coil to read out the LC sensor, and the responsivity is [26] ω+=ω2−κ2ω1−ω2.(7)

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.

We implement the above sensing method using the radio-frequency circuit shown in Fig. 3. Two printed spiral inductors (L1=L2=5.5μH) are fabricated on a printed circuit board and tuned using programmable capacitors Cn(n=1,2) to resonant frequencies ωn=1/LnCn. 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=ω1ω2Ms/(2ωL1L2), where Ms 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 R2 from 75 to 1000Ω with a step of 3.9Ω (AD5254, Analog Device). The reader (i.e., gain resonator) is self-oscillating with a steady-state frequency readout by an oscilloscope (MDO3012, Tektronix).

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 R2 and C2. 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 R2<REP, the resonant frequencies during CCW and CW encircling are almost the same except for the noise induced by programmable components. In the region of R2>REP, 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 R2<REP=1.141 are displayed in Fig. 4(c), which show a pair of HHPs marked by the dots.

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 C2 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 C2 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)].

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 ω1 and consider Gaussian distribution of the perturbation with a probability density function: P(ω2)=12πσexp(−12ω22σ2),(8)where σ is the standard deviation representing the noise intensity. The numerical results of eigenfrequencies under κ/γ=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.

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 κ/γ=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 C2 in both directions, as shown in Fig. 7. The noise of HHPs, i.e., uncertainty, is measured by sweeping the programmable capacitor C2 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.

We now demonstrate the detection of a microscale pressure sensor (Protron Mikrotechnik GmbH), which can be potentially used as a biomedical implant [0.6mm×1.2mm, 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 1×106Ω.

The sensing performance in the method without HHP is first evaluated. Under a weak capacitive perturbation (<3pF), 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 (<1pF). 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 C2 is swept slowly and repeatedly to monitor the HHP shift. The resolution of C2 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.

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