Transformation to rotating frame

I want to apply a transformation to the rotating frame of a two level system such that a state in the transformed frame is $ |hat{phi} rangle = U |phi rangle$, where U is the generator of rotations $ U = e^{iomega J_{z}t}$ with its Hermitian conjugate $U^dagger $.

Given the Hamiltonian in the stationary frame $H = omega_0 J_z + epsilon (J_{+}e^{-iomega t} +J_{-} e^{iomega t})$, I want to derive the expression for the Hamiltonian in the rotating frame, which should be $ hat{H} = (omega_0 -omega)J_z + epsilon(J_{+} +J_{-})$.

So far, I proceeded as follows:

Demanding that both$|phirangle$ and $|hat{phi} rangle$ satisfy the time-dependent Schroedinger equation, we may write (let $hbar =1$) :
$i frac{d}{dt}|hat{phi} rangle = hat{H}|hat{phi} rangle$,

where the LHS evaluates to $i frac{d}{dt}|hat{phi} rangle = i frac{dU}{dt}|phirangle + U i frac{d}{dt}|phirangle = – omega J_{z} e^{iomega J_z t}|phirangle + UH|phirangle$.

Since U is unitary, we can also write $|phirangle = U^dagger|hat{phi}rangle $ and substitute this into the expression to above to find
$i frac{d}{dt}|hat{phi} rangle = (- omega J_{z} + UHU^{dagger})|hat{phi} rangle $.

This would imply that $hat{H} = (- omega J_{z} + UHU^{dagger}) $

I would highly appreciate any help how to get to the desired result.