Altermagnets represent a novel category of magnetic materials characterized by zero net magnetization yet featuring spin-split band structures, and they demonstrate distinctive orbital-spin locking phenomena. Commencing from the minimal multiorbital tight-binding Hamiltonian of $d$-wave altermagnets, we conduct an analysis of the general formulas for the third-order injection and shift currents. These currents are solely determined by the quantum metric and quantum connection, and involve no Berry curvature contribution. In the ideal scenario where the $\delta$-bond hopping $V_{\delta }$ approaches zero $(V_{\delta }=0)$, we derive closed-form analytical solutions for the third-order photoconductivities. For the general situation with a finite value of $V_{\delta }$, we present a perturbative analytical solution within the limit of $V_{\delta }\ll V_{\pi }$, and this solution is verified through numerical calculations. Our research establishes a comprehensive theoretical description of the third-order optospintronic responses in $d$-wave altermagnets based on a microscopic model. Moreover, it offers a viable approach for the experimental observation of pure quantum geometric effects.

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• Quantum metric
• Third order nonlinear optical processes
• Altermagnets