# Time dependence of the displacement operator

I am following the derivation of the master equation (and application of this) in these lecture notes. Unfortunately I do not follow the step of eliminating the driving terms of the harmonic oscillator (p. 17, eq 164).

Assume that we have a quantum system described in the density operator formalism in the Schrödinger picture with system Hamiltonian (i.e. harmonic oscillator and coherent drive):

$H = H_0 + H_D = \hbar \omega (\hat{a}^\dagger \hat{a} + 1/2) + \hbar f_0 (\hat{a} e^{i\omega_D t} + \hat{a}^\dagger e^{-i\omega_D t} )$

Define the displacement operator as

$D(\alpha) = \exp(\alpha\hat{a}^\dagger - \alpha^* \hat{a})$

where $\hat{a}$ is the annihilation operator of a quantum harmonic oscillator and $\alpha$ is a complex number.

If I want to do a unitary transformation $U=D(\alpha)$ into a rotating frame, then I believe I should make this transformation:

$A' = U^\dagger A U$

and from this I should be able to derive a new master equation for my system etc.

But in the lecture notes it seems that they write the new Hamiltonian in the rotated frame as (eq. 164):

$\tilde{H} = U^\dagger H U + i\hbar \frac{\partial U^\dagger}{\partial t} U$

where $U=D(\alpha)$. Why include the second term if $D(\alpha)$ doesn't depend on time (we are in the Schrodinger picture per. eq. 134)?

If somehow I have misunderstood and the displacement operator is considered to depend on time through something like $\alpha = \alpha_0 e^{i\omega_D t}$ how would one handle this i.e. how does one show that $\tilde{H}$ has the above form?

To summarize: How to transform to another frame when there is time dependence such that $i\hbar \frac{\partial U}{\partial t} = HU(t)$ doesn't hold? (Because of time dependence of H)

and

Why is it assumed that $D(\alpha)$ has this time dependence in the Schrödinger picture?

The answer to the problem is that the operator used to change to the new frame, $U(t) = D(\alpha)$ is dependent on time through the time dependence of $\alpha(t)$ and for the purposes of the derivation this time dependence is kept quite general.
$U^\dagger H U + i\hbar \frac{dU^\dagger}{dt} U$
which is what is defined as $\tilde{H}$ in the notes.
The derivation is then continued without settling on a specific time dependence of $\alpha$. At the end this yields a master equation with extra terms due to the displacement operator and the drive field. If however $\alpha$ obeys a classical equation of motion for a harmonic oscillator then all these terms cancel.