Spin pumping in superconductor-based systems has been studied extensively, particularly in bilayer structures. In this work, we investigate spin pumping in a trilayer system consisting of a ferromagnetic insulator (FMI), a superconductor (SC), and a normal metal (NM). We derive both the AC and DC spin currents induced in the NM layer by the magnetization dynamics of the FMI under circularly polarized microwave irradiation. When the spin dynamics is treated classically, the AC spin current emerges, whereas treating it quantum mechanically as quasiparticle excitations leads to the DC spin current. Although evaluating these currents is computationally demanding, we overcome this challenge by employing the Quantics Tensor Cross Interpolation method. Numerical calculations reveal the dependence of spin current on temperature, microwave frequency, and superconducting layer thickness. Notably, both AC and DC components exhibit a coherence peak in their temperature dependence. Furthermore, we identify a transition in the thickness dependence of the spin current, where its behavior changes beyond a characteristic frequency.

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• Spin pumping
• Spintronics
• Superconductivity
• Heterostructures
• Tensor network methods