Coherent-Feedback Squeezing with a Beam-Splitter Loop

This example implements the feedback loop to enhance field squeezing using coherent feedback described in J. Gough and S. Wildfeuer, PRA 80, 042107 (2009), see also Example VI.1 of J. Combes, et al. Advances in Physics: X, 2:3, 784-888 (2017). A degenerate parametric oscillator is concatenated with a beam splitter, and the internal wires are eliminated with the SLH feedback reduction rule.

using QuantumInputOutputusing SecondQuantizedAlgebrausing SymbolicUtilsusing Plots

# symbolic Hilbert spacehc = FockSpace(:c)# symbolic operatora = Destroy(hc, :a)# symbolic parameters@variables κ::Real ϵ::Real η::Real

The open-loop OPO has one port, while the beam splitter has two ports. We first form the unconnected network and then apply the two feedback reductions.

r = √(1 - η^2)G_opo = SLH(1, √(κ) * a, 1im * ϵ * (a'^2 - a^2))G_bs = SLH([-r η; η r], [0, 0], 0)G_unconnected = G_opo ⊞ G_bsG_loop = feedback(G_unconnected, 1 => 2, 2 => 1)

S_loop = scattering(G_loop)L_loop = jump_operator(G_loop)[1]

((-sqrt(1 - (η^2))*η*(κ^(1//2))) / ((1.0 + 0.0im) + sqrt(1 - (η^2))) + η*(κ^(1//2))) * a

H_loop = hamiltonian(G_loop)

(-ϵ)im * a * a + (0.0 - 0.5im)*((sqrt(1 - (η^2))*((1 - (η^2))*(κ^(1//2)) + (η^2)*(κ^(1//2)))*(κ^(1//2))) / ((1.0 - 0.0im) + sqrt(1 - (η^2))) + (-sqrt(1 - (η^2))*((1 - (η^2))*(κ^(1//2)) + (η^2)*(κ^(1//2)))*(κ^(1//2))) / ((1.0 + 0.0im) + sqrt(1 - (η^2)))) * a' * a + ϵim * a' * a'

The feedback loop leaves the OPO Hamiltonian unchanged but rescales the coupling operator by $l = \eta / (1 + \sqrt{1-\eta^2})$. For a numerical illustration, we evaluate the effective damping factor $l^2$ as a function of the beam-splitter transmission coefficient.

η_grid = collect(0.0:0.005:0.999)l2_grid = @. (η_grid / (1 + sqrt(1 - η_grid^2)))^2p = plot(    η_grid,    l2_grid;    lw = 2,    label = "",    xlabel = "η",    ylabel = "effective damping fraction",    grid = true,    size = (520, 320),)hline!(p, [1.0]; color = :grey, ls = :dash, label = "open loop")p

Package versions

These results were obtained using the following versions:

using InteractiveUtilsversioninfo()using PkgPkg.status(    ["QuantumInputOutput", "SecondQuantizedAlgebra", "Plots"],    mode = PKGMODE_MANIFEST,)

Julia Version 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 UTC)
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  LLVM: libLLVM-20.1.8 (ORCJIT, znver3)
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Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_DEBUG = Documenter,Literate
  JULIA_NUM_THREADS = 1
Status `~/work/QuantumInputOutput.jl/QuantumInputOutput.jl/docs/Manifest.toml`
⌅ [861a8166] Combinatorics v1.0.2
  [d38c429a] Contour v0.6.3
⌅ [82cc6244] DataInterpolations v9.5.0
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  [aea7be01] PrecompileTools v1.3.4
  [18f9eda6] QuantumInputOutput v0.5.3 `~/work/QuantumInputOutput.jl/QuantumInputOutput.jl`
  [5717a53b] QuantumInterface v0.4.4
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  [ae029012] Requires v1.3.1
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⌅ [f7aa4685] SecondQuantizedAlgebra v0.11.0
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  [d1185830] SymbolicUtils v4.48.0
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