# Which unitary transformation should I use to change the frame reference properly?

I'm dealing with time-periodic Hamiltonian $$H(t)=H(t+T)$$ , where

$$i\hbar \partial_t\psi(r,t)=H(r,t)\psi(r,t).$$

The periodicity lies on the potential (i.e. $$V(t)=V(t+T)$$ inside the $$H(r,t)$$).

The analogous to the Bloch's theorem to treat this kind of system is the Floquet's theorem. In the problem that I'm considering, it is convinent to write out the wave function in the rotating frame by the unitary transformation $$\psi = U_{f}F$$, where it obeys the following equation

$$H_{eff}F_{n} = E_{n} F_n,$$ which satisfies the relation: $$H_{eff} =U_{f}^{\dagger}H(t)U_{f} - i\hbar U_{f}^{\dagger} \partial_{t} U_{f}.$$

However, I need to compute $$U_{f}$$ to describe the system in the rotating frame. How could I perform this calculation properly ? I've read three diferents manners that I can do this. I'll list the three differents ways I have read.

1. $$U_{f} = \exp \left(\frac{-i}{\hbar}\int_{to}^{t} V(t')dt'\right)$$
2. $$U_{f} = \exp \left(\frac{-i\omega t}{2\hbar}\sigma_{z} \right)$$
3. $$U_{f} = \exp \left(\frac{-i\omega t}{2\hbar}(I - \sigma_{z}) \right)$$

Its not clear the origin of each one, so I've trying to obtain some of the expressions listed above studying Magnus-Floquet expansion, however, I'm still in doubt If I'm the right way and I don't know the difference between these three differents expressions.

• What system are you talking about? The equations you show remind me of the the transitions in an atom in a circularly polarised beam... but $V(t)$ does not look right. Have you got a reference where you are basing this from? – SuperCiocia Sep 25 '19 at 5:53
• SuperCiocia, thanks for note this. In fact, I wrote something wrong there. You're right !. I edited the question above. Indeed, I'm dealing with circular polarized light, but I insert this beam in the Hamiltonian via time-periodic potential vector A(t). After that, I'll treat the problem with the Floquet Theorem since A(t) in that case, makes H(t) to be time-periodic. I have a reference, however, they does not show what kind of transformation $U_{f}$ they used to perform the transformation $H(t) \rightarrow H_{eff}$. Therefore, I'm not able to reproduce their results correctly – Leicam Sep 25 '19 at 16:52
• Still, you need to give us more information about the system you are talking about. Like a reference or something. Having $\sigma_z$ in your expressions $2$ and $3$ means you've chosen a specific Hamiltonian already, while in $1$ you are still keeping it general. – SuperCiocia Sep 25 '19 at 18:24