# Squeezed states H.O problem

1. Nov 3, 2012

### center o bass

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

$$|0 \rangle _a = \sum_{n = 0}^\infty \alpha_n |n\rangle _b$$

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

$$a = cb + sb^\dagger, \quad a^\dagger = sb + cb^\dagger$$
and
$$b=ca - sa^\dagger, \quad a^\dagger = - sa + ca^\dagger$$

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

$$U = e^{-\frac{\xi}{2}(a^2 - (a^\dagger)^2)} = e^{\frac{\xi}{2}(b^2 - (a^\dagger)^b)}.$$

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

$$\alpha_n = \langle n|_b 0\rangle_a = \langle 0|_b \frac{b^n}{\sqrt{n!}}|0\rangle_a$$

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

$$\alpha_{2n} = (-\frac{s}{2c})^n \frac{\sqrt{2n!}}{n!} \alpha_0$$

and

$$\alpha_0^{-2}= \sum_{n=0}^\infty (\frac{s}{2c})^{2n}\frac{(2n)!}{(n!)^2}.$$

Could someone help me in showing these results? :)