Consider the following spinor (supposed to be a 2x1 matrix but formatting is not working correctly) $$\psi \equiv \times_{i}\left[\begin{array}{l}\chi_{i+}\\ \chi_{i-}\end{array}\right]$$ $$\hat{\mathcal{H}}=\frac{\hbar\omega_0}{2}(|r\rangle\langle r|-|g\rangle\langle g|)+\hbar\Omega e^{i\omega't}|g\rangle\langle r|+\hbar\Omega e^{-i\omega' t}|r\rangle\langle g| $$
The goal is to apply a time-dependent transformation (i.e. unitary time-evolution operator) such that the Hamiltonian under this transformation is given by $$\frac{\mathcal{H}}{\hbar}=c\mathbb{I}+h\sigma^x+h_{||}\sigma^x$$ where $c,h,h_{||}$ are all constants, and $\sigma^x$ is the Pauli matrix for $x$-oriented spin.
My current approach is to define $\hat{U}\in\mathcal{L}(\mathscr{H})$ such that $\hat{U}(t,t_0=0)=e^{\frac{-it\mathcal{H}}{\hbar}}$ and define another operator $$\hat{\tilde{U}}(t)\equiv \hat{A}^\dagger \hat{U}(t) \hat{A}(t) $$ $$\hat{\tilde{U}}(t)=e^{\frac{-it\tilde{\mathcal{H}}}{\hbar}} $$
Basically, I think that we need to identify an operator $\hat{A}:\hat{A}^\dagger \mathcal{H}\hat{A}=\tilde{\mathcal{H}}$
If we consider the time-dependent Schrodinger equation, we have $$i\hbar\partial_t|\psi\rangle = \mathcal{H}|\psi\rangle $$ We can make the ansatz $|\psi\rangle = \hat{A}|\tilde{\psi}\rangle$. This means that we have $$i\hbar\partial_t |\tilde{\psi}\rangle = \tilde{\mathcal{H}}|\tilde{\psi}\rangle $$ Here is where the confusion comes. There are two approaches I can think of: the first is to say $$\tilde{\mathcal{H}}=\langle \tilde{\psi}|i\hbar\partial_t \hat{A}|\tilde{\psi}\rangle $$ But writing the matrix of $\hat{A}$ as a linear combination of two arbitrary phase shifts $e^{i\alpha t},e^{i\beta t}$ such that the matrix is diagonal, this yields only two terms, while I should be getting three. Alternatively, we could try separation of variables and integrating, but I am not sure how I would integrate the Hamiltonian.
It is also worth noting that the original Hamiltonian of the system was stated to be (but not entirely relevant to this sub-problem) $$\frac{\mathcal{H}}{\hbar}=h\sum_i \sigma_i^x+J\sum_i \sigma_i^z\sigma_{i+1}^z$$ where $i$ indexes which spin the Pauli matrix acts on and $J$ is a parameter that determines ferromagnetic or antiferromagnetic properties. Any advice is appreciated.
