The topological protection of wave transport, originally observed in the context of the quantum Hall effect in two-dimensional electron gases¹, has been shown to apply broadly to a range of physical platforms, including photonics^2,3,4,5, ultracold atoms in optical lattices^6,7,8 and others^9,10,11,12. That said, the behaviour of such systems can be very different from the electronic case, particularly when interparticle interactions or nonlinearity play a major role^{13,14,15,16,17,18,19,20,21,22}. A Thouless pump²³ is a one-dimensional model that captures the topological quantization of transport in the quantum Hall effect using the notion of dimensional reduction: an adiabatically, time-varying potential mathematically maps onto a momentum coordinate in a conceptual second dimension^{24,25,26,27,28,29,30,31,32,33,34}. Importantly, quantization assumes uniformly filled electron bands below a Fermi energy, or an equivalent occupation for non-equilibrium bosonic systems. Here we theoretically propose and experimentally demonstrate quantized nonlinear Thouless pumping of photons with a band that is decidedly not uniformly occupied. In our system, nonlinearity acts to quantize transport via soliton formation and spontaneous symmetry-breaking bifurcations. Quantization follows from the fact that the instantaneous soliton solutions centred upon a given unit cell are identical after each pump cycle, up to translation invariance; this is an entirely different mechanism from traditional Thouless pumping. This result shows that nonlinearity and interparticle interactions can induce quantized transport and topological behaviour without a linear counterpart.

波传输的拓扑保护，最初是在二维电子气¹中的量子霍尔效应的背景下观察到的，已被证明广泛应用于一系列物理平台，包括光子学^2,3,4,5，超冷原子在光学晶格^6,7,8和其他^9,10,11,12中。也就是说，此类系统的行为可能与电子情况有很大不同，特别是当粒子间相互作用或非线性发挥主要作用时^{13,14,15,16,17,18,19,20,21,22}。一个 Thouless 泵²³是一个一维模型，它使用降维概念捕获量子霍尔效应中传输的拓扑量化：绝热时变势能在数学上映射到概念第二维中的动量坐标^{24,25,26,27 ,28,29,30,31,32,33,34}. 重要的是，量子化假设均匀填充的电子能带低于费米能量，或非平衡玻色子系统的等效占据。在这里，我们从理论上提出并通过实验证明了光子的量化非线性 Thouless 泵浦，其带明显不被均匀占据。在我们的系统中，非线性通过孤子形成和自发的对称性破坏分岔来量化传输。量化源于这样一个事实，即以给定晶胞为中心的瞬时孤子解在每个泵循环之后是相同的，直到平移不变性；这是与传统的 Thouless 泵送完全不同的机制。该结果表明，非线性和粒子间相互作用可以在没有线性对应物的情况下引起量化传输和拓扑行为。