Motivated by the recent advancements in experimental techniques in the cold atomic systems, we study the dependence of the conductivity on momentum in a system of strongly interacting bosons in an optical lattice. By employing the quantum rotor approach to the Bose-Hubbard model we calculate the current-current correlation function and subsequently obtain the analytic formula for the momentum-dependent longitudinal conductivity. We analyze the behavior of the conductivity for the square and cubic lattices in both, the superfluid and Mott insulator phases. As a consequence of the particle-hole symmetry, the conductivity for a uniformly filled lattice in the superfluid phase exhibits a linear dependence for a surprisingly wide range of momenta around $\mathbf{k}=0$. This allows us to predict the value of the group velocity of the particle excitations. We also consider the impact of finite temperature and discover that it leads to an additional conductivity channel, which is aligned along the direction of the probe field and located within the energy gap.

中文翻译：

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光学晶格中强相互作用玻色子的动量分辨电导率

受冷原子系统实验技术最新进展的推动，我们研究了光学晶格中强相互作用玻色子系统中电导率对动量的依赖性。通过对 Bose-Hubbard 模型采用量子转子方法，我们计算了电流-电流相关函数，随后获得了动量相关纵向电导率的解析公式。我们分析了超流体和莫特绝缘体相中方格和立方晶格的电导率行为。由于粒子 - 孔对称性，超流体相中均匀填充晶格的电导率表现出对周围动量范围惊人的线性依赖性$\mathbf{克}=0$. 这使我们能够预测粒子激发的群速度值。我们还考虑了有限温度的影响，并发现它会导致额外的电导通道，该通道沿着探测场的方向排列并位于能隙内。