We theoretically investigate Bloch oscillations in a one-dimensional Bose–Hubbard chain, with single-particle losses from the odd lattice sites described by a Lindblad equation. For a single particle the time evolution of the state is completely determined by a non-Hermitian effective Hamiltonian. We analyse the spectral properties of this Hamiltonian for an infinite lattice and link features of the spectrum to observable dynamical effects, such as frequency doubling in breathing modes. We further consider the case of many particles in the mean-field limit leading to complex nonlinear Schrödinger dynamics. Analytic expressions are derived for the generalised nonlinear stationary states and the nonlinear Bloch bands. The interplay of nonlinearity and particle losses leads to peculiar features in the nonlinear Bloch bands, such as the vanishing of solutions and the formation of additional exceptional points. The stability of the stationary states is determined via the Bogoliubov–d...

中文翻译：

---

具有单粒子损失的Bose-Hubbard链中的Bloch振荡

我们从理论上研究了一维Bose-Hubbard链中的Bloch振荡，其中Lindblad方程描述了奇数晶格位点的单粒子损失。对于单个粒子，状态的时间演化完全由非埃尔米特有效哈密顿量确定。我们分析了此哈密顿量对于无限晶格的光谱特性，并将光谱特征链接到可观察到的动态效果，例如呼吸模式下的倍频。我们进一步考虑在平均场范围内有许多粒子的情况，这会导致复杂的非线性Schrödinger动力学。推导了广义非线性平稳态和非线性布洛赫带的解析表达式。非线性和粒子损失的相互作用导致非线性Bloch谱带具有独特的特征，例如解决方案的消失和其他特殊点的形成。稳态的稳定性是通过Bogoliubov-d来确定的。