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Prove relation for squeezed state - Quantum Information

  1. Apr 14, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove the following relation for ##\zeta:=r e^{i \theta}##:
    [tex]
    S(\zeta)^\dagger a S(\zeta)= a \cosh r - a^\dagger e^{i \theta} \sinh r
    [/tex]

    with ##S(\zeta)=e^{1/2[\zeta^\ast a^2-\zeta(a^\dagger)^2]}## and ##a## being the annihilation operator with eigenvalue ##\alpha##.

    2. Relevant equations
    See above

    3. The attempt at a solution
    I used the following identity: ##e^{At}Be^{-At}=B+\frac{t}{1!}[A,B+\frac{t^2}{2!}\big[A,[A,B]\big]+\mathcal{O}## to write

    [tex]
    \begin{align*}
    S(\zeta)^\dagger a S(\zeta)&=\underbrace{\left(e^{1/2[\zeta^\ast a^2-\zeta(a^\dagger)^2]}\right)^\dagger}_{e^A} a \underbrace{\left(e^{1/2[\zeta^\ast a^2-\zeta(a^\dagger)^2]}\right)}_{e^{-A}} \\
    &= e^A \; a \; e^{-A}\\
    &=a+[A,a]+\frac{1}{2} \big[A,[A,a]\big]
    \end{align*}
    [/tex]

    Computing the commutator ##[A,a]## yields:
    [tex]
    [A,a]=\frac{1}{1}re^{i\theta}[a^\dagger a^\dagger a-a a^\dagger a^\dagger]
    [/tex]

    I am not sure what to do with this expression, though. Is there any way to simplify it?

    If not, what other way is there to prove the identity?

    Thanks in advance!
     
  2. jcsd
  3. Apr 14, 2016 #2

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    Consider substituting ##a^\dagger a## with the help of the commutation ##[a,a^\dagger]=1##. Unfortunately, the above commutator expansion will have no constant or vanishing term so that you have to compute "all" terms (if you are clever enough, you should be ablo to find the pattern, though),
    You will also need to know the Taylor expansion of hyperbolic functions.
     
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