Firstly, we introduce the physical mechanism of a \({J}_{x}\) chain. The schematic and equivalent circuit of \({J}_{x}\) chain composed of N coil resonators are shown in Fig. 1a. The frequency of each coil resonator is \({\omega }_{0}\). We define the function \({\kappa }_{n}={\kappa }_{0}\sqrt{n(N-n)}\) for the coupling strength between the coil resonator n and n + 1, where \({\kappa }_{0}\) is the characteristic coupling strength and N is the number of coil resonators [35]. In Fig. 1b, the blue triangles indicate the magnitudes of the coupling strengths between adjacent resonators. In this case, the characteristic coupling strength is set to \({\kappa }_{0}=1\) kHz. These coupling strengths follow a parabolic spatial profile. The parameter \(\gamma\), representing the rate of loss in the system, is fixed at 2 kHz throughout the analysis for actual WPT system. The system can be described by the tight-binding Hamiltonian,

where \({c}_{n}\)(\({c}_{n}^{\dagger}\)) denotes the annihilation (creation) operator for the nth coil resonator. Then the eigenvalues and eigenvectors of the system can be derived. Under this specific coupling distribution, the calculated eigenvalues exhibit equal spacing, as indicated by the brown dots. A useful representation of the entire eigenvectors is also shown in Fig. 1c. Under distinct eigenmodes, energy is predominantly localized at different sites. This property can be effectively exploited to achieve multi-load WPT across varying frequency regimes.

To better understand the equally spaced eigenvalues in the system, we investigate a chain consisting of eight coil resonators. The eigenvalue spectrum of the Hermitian system (\(\gamma\) =0 kHz) evolves with the parameter \({\kappa }_{0}\) is shown in Fig. 2a, where a diabolic points (DP) with multiple degeneracy at \({\kappa }_{0}\) = 0 kHz is determined. Considering the non-Hermitian system (\(\gamma\) \(\ne\) 0), we calculate the real eigenfrequencies dependent on parameter \({\kappa }_{0}\) as shown in Fig. 2b and c for \(\gamma\) =1 kHz and 2 kHz, respectively. The corresponding comparison of imaginary eigenfrequencies is shown in Fig. 2d. From Fig. 2b to d, we can find the point of coalescence of the eigenfrequencies, which corresponds to the exceptional points (EPs) of the non-Hermitian system. Take the actual case with \(\gamma\) =2 kHz for example, for \({\kappa }_{0}\) < 0.9 kHz, the system exhibits eight eigenvalues, including a degenerate pair with nonzero imaginary components. As \({\kappa }_{0}\) increases, the real parts of the eigenvalues split and the imaginary parts vanish. The system eventually reaches a regime with eight purely real eigenvalues. Based on the calculation results, a larger \({\kappa }_{0}\) is required to ensure stable and efficient energy transfer.

A one-dimensional chain comprising eight coil resonators is fabricated in Fig. 3a. Each resonator consists of an acrylic-skeleton Litz-wound coil with a diameter of 10 cm, paired with a 22 nF lumped capacitor. This configuration yields an inductance of 8.9 \(\mu H\)and a unified resonant frequency of 358 kHz across all resonators. We measure the reflection spectrum \({S}_{11}\) with a near-field probe coil connected with vector network analyzer (VNA), which serves as a source and a detector simultaneously. By positioning the probe coil at the center of each coil resonator, local density of states (LDOS) can be extracted via \(1-{\left|{S}_{11}\right|}^{2}\), and density of states (DOS) spectrum is derived by summing and averaging the LDOS of all coils [42]. The inset depicts the spatial configuration of the probe. To fulfill the parabolic coupling distribution required for the \({J}_{x}\) chain, precise control of the coupling strength between adjacent coils is essential. The coupling strength between coils follows an exponential relationship with their separation distance. By measuring the reflection spectrum at varying distances between two coil resonators, the function between coupling strength \(\kappa\) and distance d is determined as \(\kappa =79{e}^{-d/2.29}+6.08\) [43], as shown in Fig. 3b. The red dots and dashed line denote the experimental measurements and theoretical fitted results, respectively. The inset illustrates the reflection spectrum of the system when d = 4 cm, where the coupling strength corresponds to half the difference between the central values of the two resonance peaks.

Figure 4b displays the experimentally measured DOS spectrum, where the peaks correspond to the theoretically calculated eigenvalues in Fig. 4a. Here, \({\kappa }_{0}\) is set as 9.55 kHz. The red dashed line denotes the zero-energy level, which is also the resonant frequency of individual coil resonator. With an even number of coil resonators, these energy levels are symmetric about the zero-energy level. Peaks above and below the dashed line represent positive and negative energy levels, respectively. The positive energy levels, which correspond to high frequency modes, exhibit larger spectra weights (peak area) and reduced intensity due to their enhanced intrinsic losses under the experimental configurations. Furthermore, we study the transmittance performance of the non-Hermitian \({J}_{x}\) chain. Figure 5 shows the comparison of calculated transmission between the chains when the loads are placed at different positions (i.e., site 5–8). The dashed lines indicate the operating frequencies at which each system achieves maximum output power.

Due to the variation in energy localization across different frequencies, the load is typically connected to the site with the highest LDOS to maximize output power. Systems 1–4 correspond to configurations where the load is connected to sites 5, 6, 7, and 8, respectively. The transmission efficiency at various frequencies is theoretically evaluated for each configuration. As all eigenvalues are purely real, high transmission efficiency is maintained across the eight eigenfrequencies. Based on the energy distribution profiles at different frequencies, the optimal operating frequencies are selected and indicated by dashed lines. The corresponding LDOS and experimental results are presented in Fig. 6.

At last, we experimentally characterize the state distribution of eigenmodes. Owing to the symmetric state distribution between positive and negative energy levels, we select four low-frequency eigenmodes with reduced intrinsic losses at 270.5 kHz, 299.3 kHz, 321.8 kHz, and 343.5 kHz, corresponding to Fig. 6a–d, respectively. The blue bars and orange points represent theoretical calculations and experimental results, respectively. Centrosymmetric state distributions are observed across all eigenmodes, with distinct spatial localizations emerging in different modes. Notably, the localization positions exhibit a progressive transition from the central site to the edge site as the eigenfrequency increases.

Eight LED indicators are employed to visualize the state distribution. Each coil resonator is attached with a LED lamp through non-resonant coil. Since the frequency of the non-resonant coil is significantly different from the coil resonator, there is essentially no effect on the original state distribution of the system. The chain can be excited by a power source (AG Series Amplifier, T&C Power Conversion) with adjustable frequency. When the coil resonator’s magnetic field reaches critical threshold, the LED at that specific location lights up while the others stay off. At an excitation frequency of 270.5 kHz, the fourth and fifth coil resonators exhibit maximum energy density, resulting in illumination of their spatially correlated LEDs. Under excitation at three additional frequencies, the LED illumination patterns exhibit close agreement with the theoretically predicted mode distributions.

In summary, we have experimentally demonstrated a non-Hermitian wireless power transfer system based on engineered iso-spectral modulation. By precisely tailoring inter-resonator couplings to follow the parabolic profile, we achieved an equidistant eigenvalue spectrum in a 1D coil-resonator chain—a signature feature of \({J}_{x}\) photonic lattices translated into the WPT domain. Crucially, frequency-selective excitation at distinct eigenfrequencies enables programmable spatial energy localization at predetermined lattice sites, as directly visualized through site-specific LED illumination patterns. This paradigm establishes a fundamental framework for simultaneous multi-load charging via spectral multiplexing, overcoming the inherent single-target limitation of topological dimer chains. Beyond planar implementations, our approach exhibits inherent scalability to higher dimensions (e.g., 2D/3D resonator arrays) [44], opening avenues for realizing exotic non-Hermitian lattice configurations with tailored state distributions. These advances position iso-spectral modulation as a transformative strategy for developing adaptive WPT systems capable of complex, spatially encoded energy delivery.

All data are available upon request through the authors.

Not applicable.

This work is supported by the National Key R&D Program of China (Nos. 2023YFA1407600 and 2021YFA1400602), the National Natural Science Foundation of China (Nos. 12374294 and 52477014), the Interdisciplinary key project of Tongji University (No. 2023–1-ZD-02), and the Chenguang Program of Shanghai (No. 21CGA22).

Authors and Affiliations

MOE Key Laboratory of Advanced Micro-Structured Materials, School of Physics Science and Engineering, Tongji University, Shanghai, 200092, China
Luyao Wan, Han Zhang, Xian Wu, Yang Xu, Yaping Yang, Hong Chen & Zhiwei Guo
College of Electronic and Information Engineering, Tongji University, Shanghai, 201804, China
Yunhui Li

Contributions

Z.G. and H.C. conceived the idea and supervised the project. L.W. and H.Z. carried out the analytical calculations with the help of X.W., Y.L., and Y.Y. L.W. and Y.X. prepared the sample and conducted experimental measurements. L.W., H.Z., and Z.G. wrote the manuscript. All authors contributed to discussions of the results and the manuscript.

Corresponding author

Correspondence to Zhiwei Guo.

Competing interests

The author declares no competing interests.

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