Femtosecond Laser Direct Writing Of Bloch Oscillations in Optical Floquet Lattices

On April 28, 2024, Prof. Shu Xuewen's team at Huazhong University of Science and Technology (HUST), in collaboration with Prof. Sun Xiankai's team at the Chinese University of Hong Kong (CUHK), published the latest research progress Visual observation of photonic Floquet -Bloch oscillations.
Bloch oscillations are a classical coherent quantum transport phenomenon manifested as periodic oscillations of quantum particles in a periodic potential field under the action of an applied constant force. As a fundamental physical effect, Bloch oscillations have been discovered and investigated in a variety of systems, such as semiconductor superlattices, ultracold atoms, coupled waveguide arrays, and synthetic dimensional photonic lattices, etc. The related research results not only promote the development of fundamental physics research, but also provide new ideas and methods for flexible manipulation of the evolution of the wave function. However, the studies on the Bloch oscillation phenomenon have mainly focused on static systems, and the Bloch oscillation phenomenon in periodically driven (Floquet) systems needs to be further explored in depth.

Fig. 1. (a) Schematic of the femtosecond laser direct-write waveguide array. (b) Photograph of the cross section of the experimentally prepared sample. (c) Top view photo of experimentally prepared sample.
To address this problem, this study investigates the Bloch oscillation phenomenon in the Froché system using a one-dimensional bent waveguide array prepared by femtosecond laser direct writing, proposes a general theory of the Bloch oscillation phenomenon in the optical Froché lattice, and experimentally visualizes and observes the optical Froché-Bloch oscillation phenomenon. As shown in Fig. 1, the bending trajectory of the waveguide in the array is a composite trajectory consisting of circular bending superimposed on periodic bending. Under the evening-axis approximation, the fluctuation equation describing the evolution of light transport in this waveguide array is mathematically equivalent to the Schrödinger equation describing the time-dependent evolution of electrons in a periodic potential field under the action of an applied electric field, and the direction of light transport in the waveguide array is equivalent to the time term in the Schrödinger equation. The curvature of the waveguide bending trajectory is regarded as the equivalent electric field force acting on the transmitted light wave, where the circular bending trajectory generates the equivalent constant electric field force leading to Bloch oscillations, and the periodic bending trajectory generates the equivalent periodic electric field force introducing the Frohnke modulation. Thus, the waveguide array enables the observation of the Bloch oscillation phenomenon in an optical Floquet lattice.

Figure 2. (a)-(d) Experimental observations at single waveguide excitation. (e)-(h) Experimental observations at broad beam excitation.
In this study, the evolution of continuous transmission of light waves in waveguide arrays was visualized and observed using waveguide fluorescence microscopy. Fig. 2 shows the breathing and oscillation modes of the Bloch oscillation phenomenon in the optical Froquet lattice for single waveguide incidence and wide beam incidence, respectively. When the Floquet modulation period is unequal to any integer multiple of the Bloch oscillation period, the Floquet dispersion is constant and equal to zero, and optical Floquet-Bloch oscillations with a period that is the least common multiple of the Floquet modulation period and the Bloch oscillation period occur. In the rest of the cases, the Floquet dispersion is no longer equal to 0, and the transmission of light is usually characterized by diffuse diffraction. In addition, the researchers theoretically and experimentally investigated the effect of the Flokay modulation parameters on the optical Flokay-Bloch oscillations, revealing the unique evolutionary nature of the phenomenon, including the fractal spectral properties associated with the Flokay modulation period and the fractional-order Flokay tunneling properties associated with the Flokay modulation amplitude.

Fig. 3. (a) Fractal spectral properties of optical Froquet-Bloch oscillations. (b) Fractional order Floquet-Bloch tunneling properties of optical Floquet-Bloch oscillations.
The visual observation of optical Floquet-Bloch oscillations reveals a novel mechanism of wave function evolution, which is of great significance in both fundamental research and practical applications. In terms of fundamental research, the theoretical model and experimental platform support further exploration of the novel phenomena arising from the interplay of the design and modulation of Frohike-Bloch oscillations with binary lattices, non-Ermian lattices, and optical nonlinearities; and in terms of practical applications, the optical Frohike-Bloch oscillations are essentially a coherent transport phenomenon, and thus can be In practical applications, optical Floquet-Bloch oscillations are essentially a coherent transport phenomenon, and thus can be extended to a variety of research platforms such as synthetic frequency-domain lattices, cold atoms, space-time crystals, and quantum walks, and are expected to be used for realizing frequency conversion, precision measurements, and the manipulation of a variety of wave transmission.