Recent developments in photonic chips have exhibited their great capability for achieving versatile physical models with highly compact integration, small footprint size and enabling novel functionalities. For these on-chip devices, photonic simulations of physical models with the large scale and the on-demand tunability are usually desired to trigger more opportunities for the integrated photonic applications. Present configurations that hold time-independent Hamiltonian features are limited to static physical models, periodically driven physical systems characterized by time-varying Hamiltonians (i.e., Floquet system), hold profoundly distinctive features and thus inspire strong study interests, with exotic photonic phenomena. In this work, we make an important step forward by constructing various time-varying Hamiltonians in a high quality factor microresonator fabricated on a thin-film lithium niobate chip. By applying a bichromatic near-resonant electro-optic modulation with two modulation frequencies oppositely detuned from the resonant frequency, a periodically driven synthetic lattice model with time-varying coupling strength is constructed in the synthetic frequency dimension.

Fig. 1 (a) Time-dependent Hamiltonian model. (b) The evolution of static energy bands over time. (c) The dynamic energy band trajectories formed through the intersection point in (b).

In experiments, we fabricate the racetrack microresonator on TFLN chip with a circumference of 26.5 mm [see Fig. 1(a)]. We apply a bichromatic EO modulation signal on the electrodes, which is composed by two radio-frequency signals with their modulation frequencies oppositely detuned from the resonant frequency. To explore such a time-varying Hamiltonian with the periodicity, we track the evolution trajectory of the time-varying band structure using the dynamic band structure method. At any given time slice, the frequency can excite the corresponding eigenvalues on the band [the intersection points are labeled by the colored circles in Fig. 1(b)], which are two wave vectors. By connecting the wave vector and eigenvalues for all time slices, one obtains the trajectory of the dynamic band structure as illustrated in Fig. 1(c).

Fig. 2 Energy band trajectory of the system under different frequency detunings.

The measured trajectories of the band structure are shown in Fig. 2 with varying. The system exhibits a static lattice model with varying is 0. The dynamic feature manifests once varying isn’t 0, where the measurements give splitting patterns. The trajectory of the dynamic band structure associated to different time-varying Hamiltonians is along the vertical energy axis. It also reflects the stationary wave feature of the band structure due to the time-varying coupling strength. The splitting band number increases with the frequency detuning while the vertical energy window remains unchanged. In addition, the phase in our constructed time-varying Hamiltonian here significantly affects the pattern of band trajectory. At the same time, our chip can provide further opportunity for studying more complicated time-varying Hamiltonians. Our work can be potentially extended to explore two-dimensional time-varying Hamiltonian models by coupling multiple high quality factor microresonators as additional spatial dimension. Therefore, our work plays a cornerstone role in exploring higher-dimensional time-dependent physics in photonic chips.

The research was published in “R. Ye, G. Li, S. Wan, X. Xue, P.-Y. Wang, X. Qiao, L. Wang, H. Li, S. Liu, J. Wang, R. Ma, F. Bo, Y. Zheng, C.-H. Dong, L. Yuan, and X. Chen, "Construction of Various Time-Varying Hamiltonians on Thin-Film Lithium Niobate Chip," Phys. Rev. Lett. 134, 163802 (2025).”

Link: https://doi.org/10.1103/PhysRevLett.134.163802