Physical measurement of the squeezing parameter The squeezing parameter is defined as $\zeta = r e^{i\psi}$ where r is the squeezing strength and $\psi $ is the squeezing angle.
Is there a way to analytically calculate the value of r. For example in the case of SPDC can I calculate r for a given input power of laser light?
Thanks
 A: A two-mode squeezing operator takes the form
$$S=\exp\left(\zeta^* ab-\zeta a^{\dagger}b^\dagger\right).$$ This looks like a unitary interaction with Hamiltonian
$$-iHt=\zeta^* ab-\zeta a^{\dagger}b^\dagger.$$ Indeed, it comes from a nonlinear interaction with a strong pump field, annihilated by some other operator $c$, corresponding to the laser light. The interaction Hamiltonian for SPDC looks like
$$H=h c^\dagger ab-h^* c a^\dagger b^\dagger,$$ where $h$ is related to the strength of the $\chi(2)$ nonlinearity in the material. When the pump field is a strong coherent state with strength $\alpha$, we can approximately replace $c$ and $c^\dagger$ with complex numbers, such that
$$H=h\alpha^* ab-h^*\alpha a^\dagger b^\dagger.$$ Comparing with the above, we see that $$\zeta=-ith^*\alpha,$$ so the squeezing parameter's strength is related to the interaction time, nonlinearity strength, and strength of the laser, while its phase is related to the nonlinearity and the phase of the laser. Similar results hold for other nonlinear processes and things like single-mode squeezing operators.
