Bi-Directional Waveguide via Feedback Reduction

This example reconstructs the N=2 bidirectional-waveguide model using the SLH feedback reduction rule. The resulting Hamiltonian and Lindblad operators are then used to simulate the transmitted and reflected intensities for a coherent input pulse.

using QuantumInputOutputusing SecondQuantizedAlgebrausing Symbolics: Symbolicsusing QuantumOpticsusing QuantumOpticsBase: daggerusing Plotsusing LinearAlgebra

N = 2# symbolic Hilbert spaceha(i) = NLevelSpace(Symbol("a$(i)"), 2)h = tensor([ha(i) for i = 1:N]...)# symbolic operatorsσ(α, i, j) = Transition(h, Symbol("σ_$(α)"), i, j, α)# symbolic parametersγR(i) = Symbolics.variable(Symbol("γ^{($(i))}_R"); T = Real)γL(i) = Symbolics.variable(Symbol("γ^{($(i))}_L"); T = Real)Δ(i) = Symbolics.variable(Symbol("Δ_{$(i)}"); T = Real)ϕ(i, j) = Symbolics.variable(Symbol("ϕ_{$(i)$(j)}"); T = Real)Ein = Symbolics.variable(Symbol("E_{in}"); T = Real)

Each quantum dot is treated as a two-port component: port 1 couples to the right-moving field and port 2 couples to the left-moving field. We concatenate the source, the two dots, and the phase shifter, and then eliminate the internal waveguide links with six feedback reductions.

I2 = Matrix{Int}(I, 2, 2)G_in = SLH(I2, [Ein, 0], 0)G_qd(i) = SLH(I2, [√(γR(i)) * σ(i, 1, 2), √(γL(i)) * σ(i, 1, 2)], -Δ(i) * σ(i, 2, 2))G_phase(i, j) = SLH([exp(1im * ϕ(i, j)) 0; 0 exp(1im * ϕ(i, j))], [0, 0], 0)G_unc = G_in ⊞ G_qd(1) ⊞ G_phase(1, 2) ⊞ G_qd(2)G_t = feedback(G_unc, 1 => 3, 3 => 5, 5 => 7, 8 => 6, 6 => 4, 4 => 2)

The reduced model has two external ports. In the remaining port order, the first Lindblad operator corresponds to the reflected left-moving output and the second one to the transmitted right-moving output.

H = hamiltonian(G_t)L = jump_operator(G_t)L_L = L[1]L_R = L[2]

γ_ = 1.0β = 0.9γRn = fill(γ_ * β / 2, N)γLn = fill(γ_ * β / 2, N)γ_add = fill(γ_ * (1 - β), N)Δn = fill(0.0, N)ϕn = fill(π / 10, max(N - 1, 0))σt = 0.8α0 = √(0.1)t0 = 4σtTend = 3t0u1(t) = 1 / (sqrt(σt) * π^(1 / 4)) * exp(-(t - t0)^2 / (2 * σt^2))Ein_t(t) = α0 * u1(t)p_sym = [    [γR(i) for i = 1:N];    [γL(i) for i = 1:N];    [Δ(i) for i = 1:N];    [ϕ(i, i + 1) for i = 1:(N-1)]]p_num = [γRn; γLn; Δn; ϕn]dict_p = Dict(p_sym .=> p_num)dict_p_t = Dict(Ein => Ein_t)

ba = NLevelBasis(2)b = tensor([ba for _ = 1:N]...)H_QO = to_numeric(H, b; parameter = dict_p, time_parameter = dict_p_t)L_R_QO = to_numeric(L_R, b; parameter = dict_p, time_parameter = dict_p_t)L_L_QO = to_numeric(L_L, b; parameter = dict_p, time_parameter = dict_p_t)σ_qo(α, i, j) = to_numeric(σ(α, i, j), b)J_add = [√(γ_add[i]) * σ_qo(i, 1, 2) for i = 1:N]function input_output(t, ρ)    Ht = H_QO(t)    J = [L_R_QO(t), L_L_QO(t), J_add...]    return Ht, J, dagger.(J)end

T = collect(0.0:0.005:1.0) .* Tendψ0 = tensor([nlevelstate(ba, 1) for _ = 1:N]...)t, ρt = timeevolution.master_dynamic(T, ψ0, input_output)

I_R = zeros(length(t))I_L = zeros(length(t))for (i, ti) in enumerate(t)    LR = L_R_QO(ti)    LL = L_L_QO(ti)    I_R[i] = real(expect(LR' * LR, ρt[i]))    I_L[i] = real(expect(LL' * LL, ρt[i]))end

p = plot(t, I_R; label = "Transmission")plot!(p, t, I_L; label = "Reflection")plot!(p, t, abs2.(Ein_t.(t)); color = :grey, ls = :dash, label = "Input")plot!(    p;    xlabel = "time",    ylabel = "intensity",    legend = :best,    grid = true,    size = (500, 320),)p

Package versions

These results were obtained using the following versions:

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

Julia Version 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-20.1.8 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_DEBUG = Documenter,Literate
  JULIA_NUM_THREADS = 1
Status `~/work/QuantumInputOutput.jl/QuantumInputOutput.jl/docs/Manifest.toml`
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  [5717a53b] QuantumInterface v0.4.4
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  [ae029012] Requires v1.3.1
  [0bca4576] SciMLBase v3.56.1
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⌅ [f7aa4685] SecondQuantizedAlgebra v0.11.0
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