# Interpreting Hamiltonian of single-mode squeezing

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 $$\hat{H} = i(\hat{a}^2 - \hat{a}^{\dagger 2})$$ .

## 1 Answer

What you have here is only the interaction term of the Hamiltonian. The full Hamiltonian also contains the usually terms of the free Hamiltonian.

Moreover, the interactions would only have this form when parametric down-conversion is considered under the semiclassical approximation where the pump is treated as a classical field, which becomes part of the squeezing parameter (which is not shown in your expression).

So, the interpretation is that the the down-converted photons are always produced or annihilated in pairs. This pairwise behavior produces the squeezing effect on phase space.

• Thank you for this information. Textbooks I have first introduces squeezing operator and shows its effect on the phase space. They suddenly introduces the Hamiltonian above. I do not have any textbook that constructs the complete Hamiltonian from clear physical point of view, like that of harmonic oscillator. Would you recommend me some book or learning resource, which describes full construction of Hamiltonian of squeezing, based on physical point of view? – ytaguchi Nov 2 '20 at 11:27
• I'm very curious about the origin of imaginary unit $i$ here. – ytaguchi Nov 2 '20 at 11:28
• There are many textbooks on nonlinear optics. One that springs to mind is the one by Robert W. Boyd. However, you can just search "squeezed state" and download some review articles from arXiv. There should be lots of them. – flippiefanus Nov 2 '20 at 12:50