Mean-field Two-sided Cavity

Here we show how to solve the dynamics of the example Two-sided Cavity with Atoms in the Heisenberg picture with a higher-order mean-field approach (cumulant expansion), which is done with the package QuantumCumulants.jl.

using QuantumInputOutputusing SecondQuantizedAlgebrausing QuantumCumulantsusing ModelingToolkitBaseusing OrdinaryDiffEqusing QuantumOpticsBaseusing QuantumOpticsBase: daggerusing Plots

@variables E::Real κ_L::Real κ_R::Real Δ::Real g::Real γ::RealNatoms = 2hc = FockSpace(:cavity)ha(i) = NLevelSpace(Symbol("a_$i"), 2)h = hc ⊗ tensor([ha(i) for i = 1:Natoms]...);a = Destroy(h, :a, 1) # cavityσ(α, i, j) = Transition(h, Symbol("σ_$(α)"), i, j, 1+α) # two-level atom α∑σ(i, j) = sum(σ(α, i, j) for α = 1:Natoms) # collective atomic operator

Empty two-sided cavity

We couple a classical drive into the cavity through the left mirror $(\kappa_L)$. The decay through the right mirror can be added in several ways: with the concatenate rule, including it already in the initial cavity SLH triple or by simply including the decay term to the Lindblad by hand. In this example, we use the first option.

G_d = SLH(1, E, 0) # classical driveH_cavity = -Δ*a'aG_c_L = SLH(1, [√(κ_L)*a], H_cavity)G_cav_L_drive = G_d ▷ G_c_LG_c_R = SLH(1, [√(κ_R)*a], 0)G_cav_L_R_drive = G_cav_L_drive ⊞ G_c_R

Note that one needs to be careful to not double-count the Hamiltonian terms with the concatenation rule.

H1 = hamiltonian(G_cav_L_R_drive)

(0.5E*sqrt(κ_L))im * a + (-0.5E*sqrt(κ_L))im * a' - Δ * a' * a

L1_L = jump_operator(G_cav_L_R_drive)[1]

E + sqrt(κ_L) * a

L1_R = jump_operator(G_cav_L_R_drive)[2]

sqrt(κ_R) * a

The typical cavity drive-term $\sqrt{\kappa_L} E (a^\dagger + a)$ is a combination of Hamiltonian term and Lindblad. We use the function meanfield to obtain the equation for the intra-cavity field, which leads to a closed set of equations in this particular case.

eqs_a = meanfield([a], H1, [L1_L, L1_R])complete!(eqs_a)

\[\begin{align} \frac{d}{dt} \langle a\rangle &= 1 i \Delta \langle a\rangle -1.0 \sqrt{\kappa{L}} E -0.5 \left( \left( \sqrt{\kappa{L}} \right)^{2} + \left( \sqrt{\kappa_{R}} \right)^{2} \right) \langle a\rangle \end{align}\]

We defined the numerical parameters and the initial state of the system, create the ODE problem and solve the dynamics.

# numerical parametersEn = 0.5κ_Rn = 1.0κ_Ln = 1.0Δn = 0.0Δn_ls = [-5.0:0.1:5.0;];lΔ=length(Δn_ls)p_sym = [E, κ_R, κ_L, Δ]p_num = [En, κ_Rn, κ_Ln, Δn]dict_p1 = Dict(p_sym .=> p_num)# initial stateu0 = [0.0im]

# numerical systemT = [0:0.01:1;]*20sys_a = mtkcompile(System(eqs_a; name = :sys_a))dict = merge(Dict(p_sym .=> p_num), initial_values(eqs_a, u0))prob_a = ODEProblem(sys_a, dict, (0.0, T[end]))

sol_a = solve(prob_a, Tsit5(); saveat = T)

n_cavity = abs2.(get_solution(sol_a, a, eqs_a).(T))n_ref = abs2.(√(κ_Ln) .* get_solution(sol_a, a, eqs_a).(T) .+ En)n_trans = abs2.(√(κ_Rn) .* get_solution(sol_a, a, eqs_a).(T))

p1 = plot(T, n_cavity; label = "", xlabel = "t", ylabel = "cavity photons", grid = true)p2 = plot(T, n_ref; label = "reflection")plot!(p2, T, n_trans; label = "transmission", ls = :dash)plot!(p2; xlabel = "t", ylabel = "intensity rate", grid = true, legend = :best)plot(p1, p2; layout = (2, 1), size = (600, 500))

Now we scan the laser-cavity detuning $\Delta$ to plot the transmission and reflection spectrum.

dict_p_Δ(Δn) = merge(Dict(p_sym .=> [En, κ_Rn, κ_Ln, Δn]), initial_values(eqs_a, u0))n_ref_Δ = zeros(lΔ)n_trans_Δ = zeros(lΔ)for it = 1:lΔ    Δn_ = Δn_ls[it]    prob_ = ODEProblem(sys_a, dict_p_Δ(Δn_), (0.0, T[end]))    sol_ = solve(prob_, Tsit5(); saveat = T)    n_ref_Δ[it] = abs2(√(κ_Ln) * get_solution(sol_, a, eqs_a)(T[end]) + En)    n_trans_Δ[it] = abs2(√(κ_Ln) * get_solution(sol_, a, eqs_a)(T[end]))end

p = plot(Δn_ls, n_ref_Δ; label = "reflection")plot!(p, Δn_ls, n_trans_Δ; label = "transmission", ls = :dash)plot!(p; xlabel = "Δ", grid = true, legend = :best, size = (600, 350))p

Two-sided cavity with atoms

In the following, we include $N=2$ two-level atoms in the cavity and simulate the transmission and reflection of a coherent Gaussian pulse with a mean photon number of $|\alpha|^2 = 1/10$. We assume that the atoms are on resonance with the cavity, i.e. $\Delta = \Delta_c = \Delta_a$.

Define the ModelingToolkit independent variable and register the classical drive as a function of it.

@independent_variables t@register_symbolic Et(tt)G_d_t = SLH(1, Et(t), 0)H_ac = -Δ*(a'a + ∑σ(2, 2)) + g*(a'∑σ(1, 2) + a*∑σ(2, 1))G_ac = SLH(1, √κ_L*a, H_ac)G_ac_drive = (G_d_t ▷ G_ac) ⊞ SLH(1, √κ_R*a, 0)

H2 = G_ac_drive.hamiltonian

(0.5Main.var"##142".Et(t)*sqrt(κ_L))im * a + (-0.5Main.var"##142".Et(t)*sqrt(κ_L))im * a' - Δ * σ_1₂₂ - Δ * σ_2₂₂ + g * a * σ_1₂₁ + g * a * σ_2₂₁ - Δ * a' * a + g * a' * σ_1₁₂ + g * a' * σ_2₁₂

L2_L = G_ac_drive.jump_operator[1]

Main.var"##142".Et(t) + sqrt(κ_L) * a

L2_R = G_ac_drive.jump_operator[2]

sqrt(κ_R) * a

We derive the equations of motion for system with a second-order mean-field approximation.

J_add = [√(γ)*σ(α, 1, 2) for α = 1:Natoms]eqs2 = meanfield([a'a, σ(1, 2, 2)], H2, [L2_L, L2_R, J_add...]; order = 2)

\[\begin{align} \frac{d}{dt} \langle a^\dagger a\rangle &= -1 i g \langle a^\dagger {\sigma1}^{{12}}\rangle + \langle a^\dagger a\rangle \left( -1.0 \left( \sqrt{\kappa{L}} \right)^{2} -1.0 \left( \sqrt{\kappa{R}} \right)^{2} \right) -1 i g \langle a^\dagger {\sigma2}^{{12}}\rangle + 1 i \langle a {\sigma2}^{{21}}\rangle g -1.0 \sqrt{\kappa{L}} \langle a\rangle \mathrm{Et}\left( t \right) -1.0 \sqrt{\kappa{L}} \langle a^\dagger\rangle \mathrm{Et}\left( t \right) + 1 i g \langle a {\sigma1}^{{21}}\rangle \\ \frac{d}{dt} \langle {\sigma1}^{{22}}\rangle &= 1 i g \langle a^\dagger {\sigma1}^{{12}}\rangle -1.0 \left( \sqrt{\gamma} \right)^{2} \langle {\sigma1}^{{22}}\rangle -1 i g \langle a {\sigma1}^{{21}}\rangle \end{align}\]

eqs2_c = complete(eqs2)length(eqs2_c)

18

Again, we defined the numerical parameters and the initial state of the system, create the ODE problem and solve the dynamics.

# numerical parameterκ_Rn2 = κ_Ln2 = 2π*1gn2 = 2π*0.4Δn2 = 0.0γn = 2π*0.1p_sym2 = [κ_R, κ_L, Δ, g, γ]p_num2 = [κ_Rn2, κ_Ln2, Δn2, gn2, γn]σp = 10/κ_Ln2 # pulse widthTp = 4σp # pulse peakTend = 3Tpα0 = √(0.1) # √ of total photon numberΩ0 = α0*2*√(κ_Ln2)/(π^(1/4)*√(σp))Ω1(t) = Ω0/2*exp(-(t-Tp)^2 / (2*σp^2))Et(t) = Ω1(t)/√(κ_Ln2)T2 = [0:0.001:1;]*TendΔT = T2[2] - T2[1]n_pulse = round(sum(abs2.(Et.(T2)))*ΔT, digits = 7)@show n_pulsedict_p2 = Dict(p_sym2 .=> p_num2)

n_pulse = 0.1

# initial statebc1 = FockBasis(4)ba = NLevelBasis(2)b = bc1 ⊗ tensor([ba for i = 1:Natoms]...)ψ0 = LazyKet(b, (fockstate(bc1, 0), [nlevelstate(ba, 1) for i = 1:Natoms]...))u0_2 = initial_values(eqs2_c, initial_values(eqs2_c, ψ0)) # state -> vector -> u0 dict

sys2 = mtkcompile(System(eqs2_c; name = :sys2)) # initial statedict2 = merge(Dict(p_sym2 .=> p_num2), u0_2)prob2 = ODEProblem(sys2, dict2, (0.0, T2[end]))

sol2 = solve(prob2, Tsit5(); saveat = T2)

n_ref2 = abs2.(√(κ_Ln2) .* get_solution(sol2, a, eqs2_c).(T2) .+ Et.(T2))n_trans2 = abs2.(√(κ_Rn2) .* get_solution(sol2, a, eqs2_c).(T2))p = plot(    T2,    n_trans2;    label = "transmission = $(round(sum(n_trans2) * ΔT / n_pulse * 100))%",)plot!(    p,    T2,    n_ref2;    ls = :dash,    label = "reflection = $(round(sum(n_ref2) * ΔT / n_pulse * 100))%",)plot!(p; xlabel = "t", legend = :best, grid = true, size = (600, 350))p

We can see that all results agree with the full quantum dynamics of the example Two-sided Cavity with Atoms, which means the second order cumulant expansion is good approximation here.

Package versions

These results were obtained using the following versions:

using InteractiveUtilsversioninfo()using PkgPkg.status([    "QuantumInputOutput",    "SecondQuantizedAlgebra",    "QuantumCumulants",    "ModelingToolkitBase",    "OrdinaryDiffEq",    "QuantumOpticsBase",    "Plots",])

Julia Version 1.13.2
Commit 45b6d15e34b (2026-10-14 20:33 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × Intel(R) Xeon(R) Platinum 8370C CPU @ 2.80GHz
  WORD_SIZE: 64
  LLVM: libLLVM-20.1.8 (ORCJIT, icelake-server)
  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/Project.toml`
  [7771a370] ModelingToolkitBase v1.77.5
  [1dea7af3] OrdinaryDiffEq v7.8.1
  [91a5bcdd] Plots v1.41.7
⌃ [35bcea6d] QuantumCumulants v0.7.1
  [18f9eda6] QuantumInputOutput v0.5.3 `~/work/QuantumInputOutput.jl/QuantumInputOutput.jl`
  [4f57444f] QuantumOpticsBase v0.5.16
⌅ [f7aa4685] SecondQuantizedAlgebra v0.11.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

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