The defining characteristic^1,2 of Cooper pairs with finite centre-of-mass momentum is a spatially modulating superconducting energy gap Δ(r), where r is a position. Recently, this concept has been generalized to the pair-density-wave (PDW) state predicted to exist in copper oxides (cuprates)^3,4. Although the signature of a cuprate PDW has been detected in Cooper-pair tunnelling⁵, the distinctive signature in single-electron tunnelling of a periodic Δ(r) modulation has not been observed. Here, using a spectroscopic technique based on scanning tunnelling microscopy, we find strong Δ(r) modulations in the canonical cuprate Bi₂Sr₂CaCu₂O_8+δ that have eight-unit-cell periodicity or wavevectors Q ≈ (2π/a₀)(1/8, 0) and Q ≈ (2π/a₀)(0, 1/8) (where a₀ is the distance between neighbouring Cu atoms). Simultaneous imaging of the local density of states N(r, E) (where E is the energy) reveals electronic modulations with wavevectors Q and 2Q, as anticipated when the PDW coexists with superconductivity. Finally, by visualizing the topological defects in these N(r, E) density waves at 2Q, we find them to be concentrated in areas where the PDW spatial phase changes by π, as predicted by the theory of half-vortices in a PDW state^6,7. Overall, this is a compelling demonstration, from multiple single-electron signatures, of a PDW state coexisting with superconductivity in Bi₂Sr₂CaCu₂O_8+δ.

具有有限质心动量的库珀对的定义特征^{1,2是空间调制超导能隙}Δ ( r )，其中r是位置。最近，这一概念已推广到预测存在于氧化铜（铜酸盐）中的对密度波 (PDW) 状态^3,4。虽然铜酸盐 PDW 的特征已在库珀对隧道⁵中检测到，但尚未观察到周期性Δ ( r ) 调制的单电子隧穿中的独特特征。在这里，使用基于扫描隧道显微镜的光谱技术，我们发现强Δ ( r) 在标准铜酸盐 Bi ₂ Sr ₂ CaCu ₂ O _{8+ δ}中的调制，具有八晶胞周期或波矢量Q  ≈ (2π/ a ₀ )(1/8, 0) 和Q  ≈ (2π/ a ₀ ) (0, 1/8)（其中_{0是相邻 Cu 原子之间}的距离）。局部状态密度N ( r ,  E ) （其中E是能量）的同时成像揭示了具有波矢量Q和 2 Q的电子调制，正如预期的那样，当 PDW 与超导共存时。最后，通过可视化这些N ( r ,  E ) 密度波在 2 Q处的拓扑缺陷，我们发现它们集中在 PDW 空间相位变化 π 的区域，正如 PDW 中的半涡理论所预测的那样状态^6,7。总体而言，这是一个令人信服的演示，从多个单电子特征来看，PDW 状态与 Bi ₂ Sr ₂ CaCu ₂ O _{8+ δ}中的超导性共存。