quantum mechanics - Degenerate parametric amplifier: quadratures

The degenerate parametric amplifier is described by the Hamiltonian:

$H=\hbar \omega a^\dagger a-i\hbar \chi /2 (e^{(2i\omega t)}a^2-e^{(-2i\omega t)}(a^\dagger)^2)$

Where $a$ and $a^\dagger$ as just the operators of creation and anhiquilation and $\chi$ is just a real constant.

If we define the quadratures as:

$X_1=a+a^\dagger \ \ \ \ \ \ \ \ \ ; \ \ \ \ \ \ \ \ \ X_2=a-a^\dagger$

How can we calculate the quadratic fluctuations (uncertainties) of these quadratures? In particular, I read that they satisfy the equations:

$(\Delta X_i)^2(t)=e^{(2\chi t)}(\Delta X_i)^2(0)$

I tried applying the Dirac picture with, as we can easily separate:

$H=\underbrace{\hbar \omega a^\dagger a}_{H_0}+\underbrace{-i\hbar \chi /2 (e^{(2i\omega t)}a^2-e^{(-2i\omega t)}(a^\dagger)^2)}_{H_1}$

Where $H_0$ is just the hamiltonian for the harmonic oscillator (with known solution) and $H_1$ is just a perturbation. This allows to find the equations of motion for $a$ and $a^\dagger$, but I'm not sure how to get the form of $(\Delta X_i)^2 (t)$ shown above.

PD: I haven't studied time-dependent perturbation theory so I'm not sure if it's necessary to solve this problem.