CritPt / Challenge_19_main / Consider two cascaded optical parametric amplifiers (OPAs) pumped by lasers at…

# Problem setup:
Consider two cascaded optical parametric amplifiers (OPAs) pumped by lasers at frequency 2Ω. The first OPA is pumped with a phase of ϕ₁, while the second OPA is pumped with a phase of ϕ₂. The signal and idler beams generated by the first OPA undergo on-chip loss before entering the second OPA, with transmission coefficients μ_(s)=μ_(i)=μ. After passing through the second OPA, there are additional detection efficiencies η_(s)=η_(i)=η. Suppose the gain parameters of the two OPAs are r₁ and r₂, respectively. Define the mean squared power of the photocurrent's sideband ⟨|I_(θ)(ν)|²⟩. Here I_(θ)(ν)=a_(Ω+ν)e^(−iθ)+a_(Ω−ν)^(†)e^(iθ) and a_(ω) is the annihilation operator of the photon mode of optical frequency ω.

# Main problem:
Derive ⟨|I_(θ)(ν)|²⟩, then find out the maximum squeezed and anti-squeezed ⟨|I_(θ)(ν)|²⟩ if ϕ₂−ϕ₁=π.

# Problem setup:
Consider two cascaded optical parametric amplifiers (OPAs) pumped by lasers at frequency $2\Omega$. The first OPA is pumped with a phase of $\phi_1$, while the second OPA is pumped with a phase of $\phi_2$. The signal and idler beams generated by the first OPA undergo on-chip loss before entering the second OPA, with transmission coefficients $\mu_s=\mu_i=\mu$. After passing through the second OPA, there are additional detection efficiencies $\eta_s=\eta_i=\eta$. Suppose the gain parameters of the two OPAs are $r_1$ and $r_2$, respectively. Define the mean squared power of the photocurrent's sideband $\left\langle {{{\left| {I_{\theta}\left( \nu  \right)} \right|}^2}} \right\rangle$. Here $I_{\theta}(\nu ) = {a_{\Omega  + \nu }}{e^{ - i\theta }} + a_{\Omega  - \nu }^\dagger {e^{i\theta }}$ and $a_\omega$ is the annihilation operator of the photon mode of optical frequency $\omega$.

# Main problem:
Derive $\left\langle {{{\left| {I_{\theta}\left( \nu  \right)} \right|}^2}} \right\rangle$, then find out the maximum squeezed and anti-squeezed $\left\langle {{{\left| {I_{\theta}\left( \nu  \right)} \right|}^2}} \right\rangle$ if $\phi_2 - \phi_1 = \pi$.