Time dependence of the displacement operator

Question

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?

Answer 1

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.

This means that we cannot use the usual rewriting of the equation of motion, but eq. 81 in the lecture notes gives the correct transformation. The new generator of time evolution is the operator

$ 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.

The clever part here is that the drive drops out of the equation completely so that the dynamics can be solved for just the "quantum part" and then at the end one transforms back to the original frame to get the full state of the system.