Hamiltonian represents energy. I can understand this when considering about harmonic oscillator, whose Hamiltonian is expressed as: $$ \hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{m\omega^2}{2}\hat{q}^2$$ This equation can be interpreted as energy very clearly, because $p$ is momentum and $q$ is position, so these terms represent kinetic energy and quadratic potential.
However, when considering about the single-mode squeezing of light using spontaneous parametric down conversion in Optical Parametric Amplification(OPA), Hamiltonian is given as follows: $$ \hat{H} = i(\hat{a}^2 - \hat{a}^{\dagger 2})$$ Here, $\hat{a}$ is an annihilation operator, which means an annihilation of a photon. Solving the Heisenberg motion of equation, I can confirm that the state is indeed squeezed under this Hamiltonian. However, I cannot find the physical interpretation of this squeezing Hamiltonian. I can guess that $\hat{a}^{\dagger 2}$ means the creation of two photons due to the interaction of signal light and pump light in the OPA. Similarly, $\hat{a}^2$ is annihilation of two photons. But I cannot understand why $\hat{a}^{\dagger 2}$ has negative sign and imaginary unit $i$. In my understanding the Hamiltonian become $\hat{H}=\hat{a}^{\dagger 2} - \hat{a}^2$, because the energy (variation?) is sum of increase on the creation of two photon and decrease on the annihilation of two photon.
I'd like to know the physical interpretation of $ \hat{H} = i(\hat{a}^2 - \hat{a}^{\dagger 2})$ .