I am reading this Paper recently. The author says that: for this Hamiltonian: $$H(t) = \frac{p^2}{2m} + \frac{m\omega^2}{2}x^2 + \alpha p_x \sigma_y$$
If we make a unitary transformation $\mathcal{A_{\alpha}} = e^{-imx\alpha \sigma_y/\hbar}$, The Hamiltonian will be transformed to $$ H_0 = \frac{p^2}{2m} + \frac{m\omega^2}{2}x^2 $$ And after that, we can solve the Schrodinger Equation and the evolution of the states can be calculated.
I cannot figure out how to do the transform. Is it just $\mathcal{A_{\alpha}}H(t)\mathcal{A_{\alpha}}^{\dagger}$? (I failed while trying to calculate this). Does anybody knows how to do the transformation?