# Two points function of “2 particles” eigenstate of two harmonic oscillators

Suppose we have two harmonic oscillators in 1+1 dimension at positions $x_1$ and $x_2$ of frequency $\omega_1$ and $\omega_2$ respectively. This can be seen as a field creating particles at locations $x_1$ and $x_2$. The excitations of each oscillator are given by the usual wave functions, in particular we have: $$\psi^i_2=\langle x_i|a^\dagger_i(x_i)a^\dagger_i(x_i)|0\rangle=(\frac{\alpha}{m})^{\frac{1}{4}} \frac{2y^2-1}{\sqrt{2}}e^{-\frac{y^2}{2}},$$ where $i=1,2$ denotes the oscillator number, $\alpha=\frac{m\omega_i}{\hbar}$ and $y=\sqrt{\alpha}x$.

Does anyone have a reference for computing the two points function $\langle x_1,x_2|a^\dagger_1(x_1)a^\dagger_2(x_2)|0\rangle$?

I know the $\omega_i$'s will get mixed but I need the exact expression to show that it has infinite entanglement because of vacuum correlations.

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