Unique Steady-State Squeezing

In this example we show the unique squeezing observed in a driven Dicke model described by $N$ two-level systems coupled to a quantized harmonic oscillator [K. Gietka. et. al., Phys. Rev. Lett. 131, 223604 (2023)]. First we present the full dynamics with a second order cumulant expansion. The Hamiltonian describing the system is

\[\begin{align} H = \omega a^\dagger a + \frac{\Omega}{2} \sum_j \sigma^j_z + \frac{g}{2} \sum_j (a^\dagger + a) \sigma^j_x + \eta ( a \, e^{i \omega_{d}\, t} + a^\dagger e^{-i \omega_{d} \,t}), \end{align}\]

for $N = 1$ it describes the driven quantum Rabi model. Additionally the system features a decay channel, losses of the harmonic oscillator with rate $\kappa$.

We start by loading the packages.

using QuantumCumulants
using OrdinaryDiffEq, ModelingToolkit
using Plots

We define the Hilbert space and the symbolic parameters of the system.

hf = FockSpace(:harmonic) # Define hilbert space
ha = NLevelSpace(Symbol(:spin), 2)
h = hf ⊗ ha

@cnumbers ω Ω ωd η κ g γ N ξ # Parameter
@syms t::Real # time

On the Hilbert space we create the destroy operator $a$ of the harmonic oscillator and the (indexed) transition operator $\sigma_i^{xy}$ for the $i$-th two-level system.

@qnumbers a::Destroy(h)
σ(x, y, i) = IndexedOperator(Transition(h, :σ, x, y), i)

With the symbolic parameters, operators and indices we define the Hamiltonian and Liouvillian of the system. Note, however, that in the strong coupling regime $g \sim g_c\equiv \sqrt{\omega \Omega}$ the driving term and jump operators have to be redefined. For a strongly interacting system, the ground state is very different from the ground state of a non-interacting system. Therefore, using jump operators of a non-interacting system would lead to extraction of energy from the ground state of a strongly interacting system. The correct operators are the ones that diagonalise the Hamiltonian with adiabatiacally eliminated spins.

b = a*cosh(ξ) + a'*sinh(ξ) # Operators diagonalizing the Hamiltonian,i.e. approximate new eigenmodes of the system

i = Index(h, :i, N, ha) # Indices
j = Index(h, :j, N, ha)

Hf = ω*a'*a + η*(b'*exp(-1im*ωd*t) + b*exp(1im*ωd*t))
Ha = Ω*Σ(σ(2, 2, i)-σ(1, 1, i), i)/2
Hi = g*Σ((σ(1, 2, i)+σ(2, 1, i))*(a + a'), i)/2
H = Hf + Ha + Hi # Hamiltonian

J = [b, σ(1, 2, i)] # Jump operators & and rates
rates = [κ, γ]

ps = [ω, Ω, ωd, g, η, κ, γ, N, ξ] # symbolic and numeric parameter list

9-element Vector{SymbolicUtils.BasicSymbolic{Complex{Real}}}:
 ω
 Ω
 ωd
 g
 η
 κ
 γ
 N
 ξ

First we derive the mean-field equations in second order for $\langle a \rangle$, $\langle a^\dagger a \rangle$ and $\langle \sigma^{22}_j \rangle$, then we complete the system to obtain a closed set of equations.

eqs = meanfield([a, a'a, σ(2, 2, j)], H, J; rates = rates, order = 2)

\[\begin{align} \frac{d}{dt} \langle a\rangle =& -0.5 i \left( \underset{i}{\overset{N}{\sum}} g \langle {\sigma}_{i}^{{12}}\rangle + \underset{i}{\overset{N}{\sum}} g \langle {\sigma}_{i}^{{21}}\rangle \right) -1 i \omega \langle a\rangle -1 i \eta \sinh\left( \xi \right) e^{1 i t {\omega}d} -1 i \eta e^{-1 i t {\omega}d} {cosh(\xi)^{*}} -0.5 \kappa \cosh\left( \xi \right) \langle a\rangle {cosh(\xi)^{*}} + 0.5 \kappa \sinh\left( \xi \right) \langle a\rangle {sinh(\xi)^{*}} \\ \frac{d}{dt} \langle a^\dagger a\rangle =& 0.5 i \left( \underset{i}{\overset{N}{\sum}} g \langle a {\sigma}_{i}^{{12}}\rangle + \underset{i}{\overset{N}{\sum}} g \langle a {\sigma}_{i}^{{21}}\rangle \right) -0.5 i \left( \underset{i}{\overset{N}{\sum}} g \langle a^\dagger {\sigma}_{i}^{{12}}\rangle + \underset{i}{\overset{N}{\sum}} g \langle a^\dagger {\sigma}_{i}^{{21}}\rangle \right) + \kappa \sinh\left( \xi \right) {sinh(\xi)^{*}} + \kappa \sinh\left( \xi \right) {sinh(\xi)^{*}} \langle a^\dagger a\rangle -1 i \eta \sinh\left( \xi \right) \langle a^\dagger\rangle e^{1 i t {\omega}d} + 1 i \eta \cosh\left( \xi \right) \langle a\rangle e^{1 i t {\omega}d} -1 i \eta \langle a^\dagger\rangle e^{-1 i t {\omega}d} {cosh(\xi)^{*}} + 1 i \eta \langle a\rangle e^{-1 i t {\omega}d} {sinh(\xi)^{*}} -1.0 \kappa \cosh\left( \xi \right) {cosh(\xi)^{*}} \langle a^\dagger a\rangle \\ \frac{d}{dt} \langle {\sigma}_{j}^{{22}}\rangle =& -1.0 \gamma \langle {\sigma}_{j}^{{22}}\rangle -0.5 i g \left( \langle a^\dagger {\sigma}_{j}^{{21}}\rangle + \langle a {\sigma}_{j}^{{21}}\rangle \right) + 0.5 i g \left( \langle a^\dagger {\sigma}_{j}^{{12}}\rangle + \langle a {\sigma}_{j}^{{12}}\rangle \right) \end{align}\]

eqs_c = complete(eqs)
length(eqs_c)

13

All two-level systems behave identically, due to this permutation symmetry of the system we can scale-up the equations.

eqs_sc = scale(eqs_c)
scale(eqs) # Example scaling on the first three equations

\[\begin{align} \frac{d}{dt} \langle a\rangle =& -0.5 i \left( N g \langle {\sigma}_{1}^{{21}}\rangle + N g \langle {\sigma}_{1}^{{12}}\rangle \right) -1 i \omega \langle a\rangle -1 i \eta \sinh\left( \xi \right) e^{1 i t {\omega}d} -1 i \eta e^{-1 i t {\omega}d} {cosh(\xi)^{*}} -0.5 \kappa \cosh\left( \xi \right) \langle a\rangle {cosh(\xi)^{*}} + 0.5 \kappa \sinh\left( \xi \right) \langle a\rangle {sinh(\xi)^{*}} \\ \frac{d}{dt} \langle a^\dagger a\rangle =& -0.5 i \left( N g \langle a^\dagger {\sigma}_{1}^{{21}}\rangle + N g \langle a^\dagger {\sigma}_{1}^{{12}}\rangle \right) + 0.5 i \left( N g \langle a {\sigma}_{1}^{{21}}\rangle + N g \langle a {\sigma}_{1}^{{12}}\rangle \right) + \kappa \sinh\left( \xi \right) {sinh(\xi)^{*}} + \kappa \sinh\left( \xi \right) {sinh(\xi)^{*}} \langle a^\dagger a\rangle -1 i \eta \sinh\left( \xi \right) \langle a^\dagger\rangle e^{1 i t {\omega}d} + 1 i \eta \cosh\left( \xi \right) \langle a\rangle e^{1 i t {\omega}d} -1 i \eta \langle a^\dagger\rangle e^{-1 i t {\omega}d} {cosh(\xi)^{*}} + 1 i \eta \langle a\rangle e^{-1 i t {\omega}d} {sinh(\xi)^{*}} -1.0 \kappa \cosh\left( \xi \right) {cosh(\xi)^{*}} \langle a^\dagger a\rangle \\ \frac{d}{dt} \langle {\sigma}_{1}^{{22}}\rangle =& -0.5 i g \left( \langle a^\dagger {\sigma}_{1}^{{21}}\rangle + \langle a {\sigma}_{1}^{{21}}\rangle \right) + 0.5 i g \left( \langle a^\dagger {\sigma}_{1}^{{12}}\rangle + \langle a {\sigma}_{1}^{{12}}\rangle \right) -1.0 \gamma \langle {\sigma}_{1}^{{22}}\rangle \end{align}\]

To calculate the dynamics of the system we create a system of ordinary differential equations with its initial state and numerical parameters.

@named sys = System(eqs_sc) # symbolic ordinary differential equation system
u0 = zeros(ComplexF64, length(eqs_sc)); # initial state

ω_ = 1.0 # Parameters
Ω_ = 2e3ω_
gc_ = sqrt(Ω_*ω_/N) # renormalization of coupling to keep the system intensive
g_ = 0.9gc_
η_ = 4ω_
κ_ = ω_
γ_ = ω_
ωd_ = sqrt(1-g_^2/gc_^2)*ω_
ξ_ = 1/4*log(1-N*g_^2/(ω_*Ω_))

\[ \begin{equation} 0.25 \log\left( 1 - 0.00040500000000000003 \left( \sqrt{\frac{2000.0}{N}} \right)^{2} N \right) \end{equation} \]

We solve the dynamics for four different numbers of two-level systems $N = [1, 2, 10, 100]$.

sol_ls = []
N_ls = [1, 2, 10, 100]
for N_ in N_ls
    p0 = [ω_, Ω_, ωd_, g_, η_, κ_, γ_, N_, ξ_]
    dict = merge(Dict(unknowns(sys) .=> u0), Dict(ps .=> p0))
    prob = ODEProblem(sys, dict, (0.0, 4π/ωd_))
    sol = solve(prob, Tsit5(); saveat = π/30ωd_, reltol = 1e-10, abstol = 1e-10)
    push!(sol_ls, sol)
end

c_ls=[:black, :red, :blue, :cyan] # plot results
p1 = plot(xlabel = "ω t", ylabel = "Δ² O")
p2 = plot(xlabel = "ω t", ylabel = "⟨σz⟩")
for i = 1:length(N_ls)
    sol = sol_ls[i]
    t_ = sol.t

    sqx = sol[a'*a'] + sol[a*a] + 2*sol[a'*a] .+ 1 - (sol[a'] + sol[a]) .^ 2
    sqy = sol[a'*a'] + sol[a*a] - 2*sol[a'*a] .- 1 - (sol[a'] - sol[a]) .^ 2
    plot!(p1, t_, real.(sqx), label = "N = $(N_ls[i])", color = c_ls[i])
    plot!(p1, t_, -real.(sqy), ls = :dash, label = nothing, color = c_ls[i])

    s22 = sol[σ(2, 2, 1)]
    plot!(p2, t_, real.(2s22 .- 1), color = c_ls[i], label = nothing)
end
plot(
    p1,
    p2,
    layout = (1, 2),
    size = (700, 250),
    bottom_margin = 5*Plots.mm,
    left_margin = 5*Plots.mm,
)

Effective model

For a sufficiently low excitation we can adiabatically eliminate the dynamics of the two-level system(s). This leads to an effective Hamiltonian

\[\begin{align} H_\mathrm{a} = \omega a^\dagger a - \frac{g^2}{4 \Omega}(a + a^\dagger)^2 + \eta ( a \, e^{i \omega_{d} \, t} + a^\dagger e^{-i \omega_{d} \, t}). \end{align}\]

We calculate now the dynamics for this effective model and compare it with the full system. Note that this Hamiltonian is quadratic, which means that a second order description is exact.

@cnumbers gΩ # g^2/4Ω
H_a = Hf - N*gΩ*(a + a')^2 # effective Hamiltonian, N is added for the sake of intensitivity

eqs_a = meanfield([a, a'a, a*a], H_a, [b]; rates = [κ], order = 2)

\[\begin{align} \frac{d}{dt} \langle a\rangle =& -1 i \omega \langle a\rangle + 2 i N g\Omega \left( \langle a^\dagger\rangle + \langle a\rangle \right) -1 i \eta \sinh\left( \xi \right) e^{1 i t {\omega}d} -1 i \eta e^{-1 i t {\omega}d} {cosh(\xi)^{*}} -0.5 \kappa \cosh\left( \xi \right) \langle a\rangle {cosh(\xi)^{*}} + 0.5 \kappa \sinh\left( \xi \right) \langle a\rangle {sinh(\xi)^{*}} \\ \frac{d}{dt} \langle a^\dagger a\rangle =& \kappa \sinh\left( \xi \right) {sinh(\xi)^{*}} + 2 i N g\Omega \langle a^\dagger a^\dagger\rangle -2 i N g\Omega \langle a a\rangle + \kappa \sinh\left( \xi \right) {sinh(\xi)^{*}} \langle a^\dagger a\rangle -1 i \eta \sinh\left( \xi \right) \langle a^\dagger\rangle e^{1 i t {\omega}d} + 1 i \eta \cosh\left( \xi \right) \langle a\rangle e^{1 i t {\omega}d} -1 i \eta \langle a^\dagger\rangle e^{-1 i t {\omega}d} {cosh(\xi)^{*}} + 1 i \eta \langle a\rangle e^{-1 i t {\omega}d} {sinh(\xi)^{*}} -1.0 \kappa \cosh\left( \xi \right) {cosh(\xi)^{*}} \langle a^\dagger a\rangle \\ \frac{d}{dt} \langle a a\rangle =& 2 i N g\Omega -2 i \omega \langle a a\rangle + 4 i N g\Omega \left( \langle a^\dagger a\rangle + \langle a a\rangle \right) -1.0 \kappa \sinh\left( \xi \right) {cosh(\xi)^{*}} + \kappa \sinh\left( \xi \right) {sinh(\xi)^{*}} \langle a a\rangle -1.0 \kappa \cosh\left( \xi \right) {cosh(\xi)^{*}} \langle a a\rangle -2 i \eta \sinh\left( \xi \right) \langle a\rangle e^{1 i t {\omega}d} -2 i \eta \langle a\rangle e^{-1 i t {\omega}d} {cosh(\xi)^{*}} \end{align}\]

@named sys_a = System(eqs_a) # symbolic ordinary differential equation system

u0_a = zeros(ComplexF64, length(eqs_a)) # initial state

gΩ_ = g_^2/(4Ω_) # Additional parameter
N_ = 69 # the final result does not depend on N

ps_a = [ω, ωd, η, κ, N, g, Ω, ξ, gΩ] # symbolic and numeric parameter list
p0_a = [ω_, ωd_, η_, κ_, N_, g_, Ω_, ξ_, gΩ_]

prob_a = ODEProblem(sys_a, u0_a, (0.0, 4π/ωd_), ps_a .=> p0_a) # define and solve numeric ordinary differential equation problem
sol_a = solve(prob_a, Tsit5(); saveat = π/30ωd_, reltol = 1e-8, abstol = 1e-8)

sol = sol_ls[4] # plot results
t_ = sol.t
sqx = sol[a'*a'] + sol[a*a] + 2*sol[a'*a] .+ 1 - (sol[a'] + sol[a]) .^ 2
sqy = sol[a'*a'] + sol[a*a] - 2*sol[a'*a] .- 1 - (sol[a'] - sol[a]) .^ 2

t_a = sol_a.t
sqx_a = sol_a[a'*a'] + sol_a[a*a] + 2*sol_a[a'*a] .+ 1 - (sol_a[a'] + sol_a[a]) .^ 2
sqy_a = sol_a[a'*a'] + sol_a[a*a] - 2*sol_a[a'*a] .- 1 - (sol_a[a'] - sol_a[a]) .^ 2

p = plot(xlabel = "ω t", ylabel = "Δ² O")
plot!(p, t_, real.(sqx), label = "X - Full model")
plot!(p, t_, -real.(sqy), label = "P - Full model", ls = :dash)
plot!(p, t_a, real.(sqx_a), label = "X - Effective model")
plot!(p, t_a, -real.(sqy_a), label = "P - Effective model", ls = :dash)
plot(p, size = (500, 200))

Package versions

These results were obtained using the following versions:

using InteractiveUtils
versioninfo()

using Pkg
Pkg.status(["QuantumCumulants", "OrdinaryDiffEq"], mode = PKGMODE_MANIFEST)

Julia Version 1.11.7
Commit f2b3dbda30a (2025-09-08 12:10 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-16.0.6 (ORCJIT, znver3)
Threads: 4 default, 0 interactive, 2 GC (on 4 virtual cores)
Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_DEBUG = Documenter
  JULIA_NUM_THREADS = auto
Status `~/work/QuantumCumulants.jl/QuantumCumulants.jl/docs/Manifest.toml`
⌃ [1dea7af3] OrdinaryDiffEq v6.101.0
  [35bcea6d] QuantumCumulants v0.4.2 `~/work/QuantumCumulants.jl/QuantumCumulants.jl`
Info Packages marked with ⌃ have new versions available and may be upgradable.

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