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Squeezed states H.O problem

  1. Nov 3, 2012 #1
    I'm working on a problem with a Harmonic oscillator which suddenly goes from the frequency ω_a to ω_b and I'm trying to find the expansion coefficients in

    [tex] |0 \rangle _a = \sum_{n = 0}^\infty \alpha_n |n\rangle _b [/tex]

    where |0>_a is the ground state right before the change in frequency. The raising and lowering operators of the oscillators with different frequency are related by

    [tex] a = cb + sb^\dagger, \quad a^\dagger = sb + cb^\dagger[/tex]
    and
    [tex]b=ca - sa^\dagger, \quad a^\dagger = - sa + ca^\dagger [/tex]

    and they can also be expressed as related by the unitary transformation

    [tex]U = e^{-\frac{\xi}{2}(a^2 - (a^\dagger)^2)} = e^{\frac{\xi}{2}(b^2 - (a^\dagger)^b)}.[/tex]

    where [itex]s = \sinh \xi, \quad c = \cosh \xi[/itex]. I tried to find the coefficients by observing that

    [tex]\alpha_n = \langle n|_b 0\rangle_a = \langle 0|_b \frac{b^n}{\sqrt{n!}}|0\rangle_a[/tex]

    but I failed to proceed from here. The correct answer is supposed to be that

    [tex]\alpha_{2n} = (-\frac{s}{2c})^n \frac{\sqrt{2n!}}{n!} \alpha_0[/tex]

    and

    [tex]\alpha_0^{-2}= \sum_{n=0}^\infty (\frac{s}{2c})^{2n}\frac{(2n)!}{(n!)^2}.[/tex]

    Could someone help me in showing these results? :)
     
  2. jcsd
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