Pattern formation is ubiquitous in nature at all scales, from morphogenesis and cloud formation to galaxy filamentation. How patterns emerge in a homogeneous system is a fundamental question across interdisciplinary research including hydrodynamics¹, condensed matter physics², nonlinear optics³, cosmology⁴ and bio-chemistry^5,6. Paradigmatic examples, such as Rayleigh–Bénard convection rolls and Faraday waves^7,8, have been studied extensively and found numerous applications^9,10,11. How such knowledge applies to quantum systems and whether the patterns in a quantum system can be controlled remain intriguing questions. Here we show that the density patterns with two- (D₂), four- (D₄) and six-fold (D₆) symmetries can emerge in Bose–Einstein condensates on demand when the atomic interactions are modulated at multiple frequencies. The D₆ pattern, in particular, arises from a resonant wave-mixing process that establishes phase coherence of the excitations that respect the symmetry. Our experiments explore a novel class of non-equilibrium phenomena in quantum gases, as well as a new route to prepare quantum states with desired correlations.

从形态发生和云形成到星系丝状化，模式形成在自然界中无所不在。模式如何在同质系统中出现是跨学科研究的基本问题，包括流体力学¹，凝聚态物理学²，非线性光学³，宇宙学⁴和生物化学^5,6。范式示例，例如瑞利-贝纳德对流辊和法拉第波^7,8，已经得到了广泛的研究，并发现了许多应用^9,10,11。这些知识如何应用于量子系统以及量子系统中的模式是否可以控制仍然是有趣的问题。在这里我们表明，当原子相互作用在多个频率上进行调制时，按需在Bose-Einstein凝聚体中出现对称性为（D ₂），（D ₄）（D ₄）和（D ₆）为六倍的密度模式。所述d ₆图案，特别地，产生从规定，尊重对称激励的相位相干性的谐振波混频过程。我们的实验探索了量子气体中的一类新的非平衡现象，以及一种制备具有所需相关性的量子态的新途径。