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Eigenvectors of "squeezed" amplitude operator

  1. Mar 19, 2016 #1
    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?

    Thank you for your time.
     
  2. jcsd
  3. Mar 19, 2016 #2

    blue_leaf77

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    Just a few more (if not one) steps needed. Remember that the coherent state is an eigenstate of ##\hat{a}##.
     
  4. Mar 20, 2016 #3
    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*.

    Thanks for your help.
     
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