Exact finite-N Tavis–Cummings model in the symmetric subspace

CollectiveTransition keeps the atomic Hilbert space polynomial in atom number while retaining exact finite-N dynamics. For two-level atoms the symmetric sector has dimension N + 1, instead of 2^N for the full product space.

using SecondQuantizedAlgebra
using QuantumOpticsBase

N = 20
nphotons = 12

hc = FockSpace(:cavity)
ha = CollectiveNLevelSpace(:atoms, (:g, :e))
h = hc ⊗ ha

@qnumbers a::Destroy(h, 1)
Sge = CollectiveTransition(h, :S, :g, :e, 2)
Seg = Sge'
See = CollectiveTransition(h, :S, :e, :e, 2)
Sgg = CollectiveTransition(h, :S, :g, :g, 2)

@variables ωc ωa g
H = ωc * a' * a + ωa * See + g * (a' * Sge + a * Seg)

\[\mathtt{{\omega}a} {S}^{{22}} + g a{S}^{{21}} + \mathtt{{\omega}c} a^{\dagger}a + g a^{\dagger}{S}^{{12}}\]

The symbolic algebra applies the collective SU(2) commutator.

iszero(simplify(commutator(Sge, Seg) - Sgg + See))

true

Fixing total atomic occupation to N constructs the symmetric sector.

bc = FockBasis(nphotons)
b1 = NLevelBasis(2)
ba = ManyBodyBasis(b1, bosonstates(b1, N))
b = bc ⊗ ba

[Fock(cutoff=12) ⊗ ManyBody(onebodybasis=NLevel(N=2), states:21)]

Substitute physical parameters and materialize the exact sparse Hamiltonian.

Hnum = to_numeric(H, b; parameter = Dict(ωc => 1.0, ωa => 1.0, g => 0.1))

@assert length(ba) == N + 1
@assert size(Hnum.data) == (length(b), length(b))

Diagonal collective populations sum to N, not one.

number_identity = to_numeric(Sgg + See, b)
@assert number_identity ≈ N * one(b)

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