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. We observe that the modulation brings additional control to the system, enabling one to readily create and manipulate bright and dark dissipative solitons inside the cavity. 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 ${\chi }^{(2)}$ microresonators can open the way to experimentally explore our findings.

中文翻译：

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合成频率维度中的非线性状态和动力学

综合维研究的最新进展表明，有可能将频率空间用作附加自由度，从而可以研究和利用先验低维系统中的高维现象。但是，仅在明显的限制条件下研究了非线性效应对合成频率尺寸的影响。在本文中，我们为具有二次非线性和三次非线性的单个驱动耗散谐振器中的光场包络线建立了一个广义的平均场模型，该谐振器的频率通过电光谐振时间调制耦合。前导方程采用带余弦势的从动Gross-Pitaevskii方程的形式。我们在参数频率转换发生的情况下，以合成频率尺寸数值研究了这种微环谐振器中的非线性动力学。我们观察到，调制为系统带来了额外的控制，使人们可以轻松地在腔体内创建和操纵明暗耗散孤子。在异常色散的情况下，我们发现电光模式耦合的存在限制并稳定了混沌调制不稳定性区域。这导致出现新型类型的稳定相干结构，该结构在合成空间中以恢复的平移对称性出现，在通常仅存在混沌调制不稳定性状态的参数区域中。该结构出现在合成带的中心，因此，被称为带孤子。最后，我们将结果扩展到具有可控相对相位的多个调制频率的情况，从而创建具有非平凡几何形状的合成晶格。我们表明，不对称的合成带导致腔内电磁场的混沌状态和相干状态并存，即可以解释为类似嵌合状态的动力学。最近开发${\chi }^{（2）}$ 微谐振器可以为实验性探索我们的发现开辟道路。