The recent emerging field of synthetic dimension in photonics offers a variety of opportunities for manipulating different internal degrees of freedom of photons such as the spectrum of light. While nonlinear optical effects can be incorporated into these photonic systems with synthetic dimensions, these nonlinear effects typically result in long-range interactions along the frequency axis. Thus it has been difficult to use of the synthetic dimension concept to study a large class of Hamiltonians that involves local interactions. Here we show that a Hamiltonian that is locally interacting along the synthetic dimension can be achieved in a dynamically-modulated ring resonator incorporating $\chi^{(3)}$ nonlinearity, provided that the group velocity dispersion of the waveguide forming the ring is specifically designed. As a demonstration we numerically implement a Bose-Hubbard model, and explore photon blockade effect, in the synthetic frequency space. Our work opens new possiblities for studying fundamental many-body physics in the synthetic space in photonics, with potential applications in optical quantum communication and quantum computation.

中文翻译：

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在光子的合成频率维度中创建局部相互作用的哈密顿量

最近出现的光子学合成维度领域为操纵光子的不同内部自由度（例如光谱）提供了各种机会。虽然非线性光学效应可以结合到这些具有合成尺寸的光子系统中，但这些非线性效应通常会导致沿频率轴的长距离相互作用。因此，很难使用合成维度概念来研究一大类涉及局部相互作用的哈密顿量。在这里，我们表明可以在动态调制的环形谐振器中实现沿合成维度局部相互作用的哈密顿量，该谐振器包含 $\chi^{(3)}$ 非线性，前提是形成环的波导的群速度色散为专门设计的。作为演示，我们在合成频率空间中数值实现了 Bose-Hubbard 模型，并探索光子阻塞效应。我们的工作为研究光子学合成空间中的基础多体物理学开辟了新的可能性，并在光量子通信和量子计算中具有潜在应用。