It is well known that when light passes through a spatial interface between two regions with different refractive indices, reflection and refraction can occur. In recent years, the temporal counterpart of such spatial interfaces—namely, temporal interfaces (TIs)—has attracted growing research interest. When the refractive index of a homogeneous medium undergoes an abrupt change in time, light can exhibit temporal reflection and refraction. To date, most studies on TIs have been realized through fast modulation of system parameters, and the associated dynamics—such as temporal reflection and refraction as well as frequency shifts—have been experimentally observed. However, band operations over the entire energy-band range, including band folding and unfolding, as well as momentum conservation at two-dimensional TIs, have not yet been rigorously demonstrated in experiments. Moreover, the realization of TIs typically requires system parameters to be varied at rates much faster than the intrinsic dynamics of the waves, posing significant challenges for implementation of TIs at optical frequencies. Therefore, constructing a TI that does not rely on ultrafast modulation is of great importance.

Fig. 1：Schematic illustration of constructing a TI in the synthetic frequency dimension and verifying momentum conservation by the spectral tomography method.

To verify momentum conservation at a TI over the entire energy-band range, the research team developed an spectra tomography method. As shown in Fig. 1(c), by comparing the wave function after the TI with the sum of the two group velocities of band v=v_a+v_b at energy E_s, one can obtain the projection relations of all eigenstates throughout the entire band. This enables a direct verification of momentum conservation at the TI.

The research team experimentally verified this concept using a synthetic frequency dimension platform, with the experimental setup illustrated in Fig. 1(e). a tunable single-frequency continuous-wave laser field with central frequency w_0 first passes through an electro-optic modulator (EOM1), which is equivalent to the wave function undergoing a time evolution t_1 in frequency space under the Hamiltonian H_1. The modulated light field then enters a ring with a second electro-optic modulator (EOM2). EOM2 further modulates the light field, corresponding to an additional time evolution of the wave function under the Hamiltonian H_2. In this way, a TI is constructed in the synthetic frequency space.

Fig. 2: Observation of momentum conservation at a TI with one-dimensional band shifting in synthetic frequency space.

The team first verified momentum conservation at a TI constructed via band shifting, as shown in Fig. 2. The two lattices before and after the time interface have the same lattice period, but the coupling between neighboring sites differs by a phase ϕ, which is equivalent to a translation of the band in momentum space. When ϕ=0, the two lattices are identical and no TIs exists. Over the entire band, the condition v=v_a+v_b=0 is satisfied, corresponding to C=0. In contrast, when ϕ=0.5pi, a relative shift exists between the two bands E_1(k) and E_2(k). As a result, the sum of the two group velocities at different energies becomes nonzero v≠0, demonstrating the presence of a TI. The measured center of mass over the entire band satisfies C∝v, thereby confirming momentum conservation over the full band for a TI with band shifting.

Furthermore, by implementing band unfolding and folding, the team realized multiple types of time interfaces and verified momentum conservation at TIs in one-dimensional systems.

The research team further extended TIs to higher dimensions. By introducing nearest-neighbor and long-range couplings in the electro-optic modulators, a one-dimensional frequency lattice was folded into two dimensions, enabling the realization of a two-dimensional TI. Momentum conservation at a two-dimensional TI was thereby demonstrated.

This work represents the first experimental realization of TIs based on one-dimensional band translation, folding, and unfolding, as well as two-dimensional TIs, and provides a direct experimental verification of momentum conservation at TIs. Unlike previous experimental studies, the construction of TIs in this optical system does not rely on ultrafast modulation. This is because the time evolution takes place in the frequency domain: in the absence of modulation, time is effectively “frozen”, and temporal evolution only proceeds when the light field is modulated. This work establishes a platform for realizing TIs at optical frequencies and lays the foundation for future studies of topology, non-Hermitian, and non-Abelian physics based on TIs.

This work is published in “Yanyan He, Zhaohui Dong, Guangzhen Li, Penghong Yu, Xiaoxiong Wu, Xianfeng Chen, and Luqi Yuan. Observing momentum conservation at temporal interfaces in synthetic frequency Dimension. Sci. Adv. 11, eadz5445 (2025)”.

Link: https://www.science.org/doi/full/10.1126/sciadv.adz5445