合成频率维中的非线性状态和动力学
Nonlinear states and dynamics in a synthetic frequency dimension
摘要：
最近的进展，在研究合成维度揭示了一种可能性，采用频率空间作为一个额外的自由度，允许调查和利用高维现象在先验低维系统。然而，非线性效应对合成频率维度的影响仅在显著限制下进行了研究。在本文中，我们开发了一个广义平均场模型的光场包络内的驱动耗散谐振腔的二次和三次非线性，其频率耦合通过电光谐振时间调制。首阶方程采用带余弦势的驱动Gross-Pitaevskii方程的形式。我们用数值方法研究了这种具有合成频率维度的微波谐振器在参量频率转换发生时的非线性动力学。在反常色散情况下，我们发现电光模耦合的存在限制并稳定了混沌调制不稳定区。这导致出现一种新型的稳定的相干结构，出现在合成空间中恢复平移对称性，在一个区域的参数，传统上只有混沌调制不稳定状态存在。这种结构出现在合成带的中心，因此被称为带孤子。最后，我们将我们的结果的情况下，多个调制频率与可控的相对相位创建合成晶格与非平凡的几何形状。我们发现，一个不对称的合成带导致腔体内的电磁场的混沌和相干态的共存，即可以被解释为嵌合体状的状态的动力学。最近开发的$χ^{(2)}$微谐振器可以打开的方式来实验探索我们的研究结果。
英文摘要：
Recent advances in the study of synthetic dimensions revealed a possibility to employ the frequency space as an additional degree of freedom which allows for investigating and exploiting higher-dimensional phenomena in a priori low-dimensional systems. However, the influence of nonlinear effects on the synthetic frequency dimensions was studied only under significant restrictions. In the present paper, we develop a generalized mean-field model for the optical field envelope inside a single driven-dissipative resonator with quadratic and cubic nonlinearities, whose frequencies are coupled via an electro-optical resonant temporal modulation. The leading order equation takes the form of driven Gross-Pitaevskii equation with a cosine potential. We numerically investigate the nonlinear dynamics in such microring resonator with a synthetic frequency dimension in the regime where parametric frequency conversion occurs. In the case of anomalous dispersion, we find that the presence of electro-optical mode coupling confines and stabilizes the chaotic modulation instability region. This leads to the appearance of a novel type of stable coherent structures which emerge in the synthetic space with restored translational symmetry, in a region of parameters where conventionally only chaotic modulation instability states exist. This structure appears in the center of the synthetic band and, therefore, is referred to as Band Soliton. Finally, we extend our results to the case of multiple modulation frequencies with controllable relative phases creating synthetic lattices with nontrivial geometry. We show that an asymmetric synthetic band leads to the coexistence of chaotic and coherent states of the electromagnetic field inside the cavity i.e. dynamics that can be interpreted as chimera-like states. Recently developed $χ^{(2)}$ microresonators can open the way to experimentally explore our findings.
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