# Eigenvectors of "squeezed" amplitude operator

1. Mar 19, 2016

### carllacan

1. The problem statement, all variables and given/known data
Prove that the states $$|z, \alpha \rangle = \hat S(z)\hat D(\alpha) | 0 \rangle$$ $$|\alpha, z \rangle = \hat D(\alpha) \hat S(z)| 0 \rangle$$
are eigenvectors of the squeezed amplitude operator $$\hat b = \hat S(z) \hat a \hat S ^\dagger (z) = \mu \hat a + \nu \hat a ^\dagger$$, with μ, ν and z being complex numbers and where $$\hat D(\alpha) = e^{\alpha \hat a ^\dagger - \alpha ^* \hat a }$$ is the displacement operator and $$\hat S(z) = e ^{ \frac{z^*}{2} \hat a ^2 - \frac{z}{2} \hat a ^{\dagger 2}}$$ is the compression operator.
2. Relevant equations

3. The attempt at a solution
For the first one I've tried $$\hat b \hat S(z)\hat D(\alpha) | 0 \rangle = \hat S(z) \hat a \hat S ^\dagger (z) \hat S(z)\hat D(\alpha) | 0 \rangle = \hat S(z) \hat a | \alpha \rangle$$, but I can't get farther than that. I've tried and reordering the exponentials, writing them as taylor series, and writing the coherent state α as a sum of number states n. but I haven't got nowhere.

I have the feeling that the solution is much easier than what I am doing. Could anyone point me in the right direction?

2. Mar 19, 2016

### blue_leaf77

Just a few more (if not one) steps needed. Remember that the coherent state is an eigenstate of $\hat{a}$.

3. Mar 20, 2016

### carllacan

Yes, I realized that I already had the solution just after I went to bed. I'm retarded, haha. And the other state is easy to prove once you have this one. Using D(α)S(z) = S(z)D(μα+να*) it's easy to see that it's an eigenvector with eigenvalue μα+να*.