Collective N-Level Systems

There are two complementary ways to describe many identical N-level systems. Choose between them based on the approximation and Hilbert space you need.

Approach	Symbolic operator	Numeric representation	Best suited to
Indexed sites	$\sum_k \sigma_k^{ij}$ with `Index` and `Σ`	Product-space or cumulant workflow	site-dependent parameters, correlations, and large-$N$ mean-field/cumulant models
Collective symmetric sector	`CollectiveTransition` $S^{ij}$	`QuantumOpticsBase.ManyBodyBasis`	exact permutation-symmetric dynamics at finite $N$

Collective transition algebra

CollectiveTransition represents

\[S^{ij}=\sum_{k=1}^{N}|i\rangle_k\langle j|,\]

without expanding the site sum. Its closed Lie algebra is

\[[S^{ij},S^{kl}]=\delta_{jk}S^{il}-\delta_{li}S^{kj}.\]

julia> using SecondQuantizedAlgebra

julia> h = CollectiveNLevelSpace(:atoms, (:g, :e));

julia> S(i, j) = CollectiveTransition(h, :S, i, j);

julia> iszero(simplify(commutator(S(:g, :e), S(:e, :g)) - S(:g, :g) + S(:e, :e)))
true

Unlike a single-site Transition, a product such as $S^{ge}S^{eg}$ does not reduce to a projector: it contains contributions from distinct atoms. There is likewise no algebra-layer completeness rule. For a fixed-$N$ symmetric numeric basis,

\[\sum_i S^{ii}=N I,\]

not $I$.

Exact symmetric-subspace conversion

Use a bosonic ManyBodyBasis whose one-body basis is an NLevelBasis. Fixing the total occupation to $N$ selects the permutation-symmetric sector, whose dimension is $\binom{N+n-1}{n-1}$ for $n$ internal levels.

julia> using QuantumOpticsBase

julia> using SecondQuantizedAlgebra

julia> h = CollectiveNLevelSpace(:atoms, (:g, :e));

julia> S(i, j) = CollectiveTransition(h, :S, i, j);

julia> N = 10; b1 = NLevelBasis(2);

julia> b = ManyBodyBasis(b1, bosonstates(b1, N));

julia> size(to_numeric(S(:g, :e), b).data)
(11, 11)

julia> sum(to_numeric(S(x, x), b) for x in (:g, :e)) ≈ N * one(b)
true

For two levels this is the usual collective-spin representation with spin $N/2$: $S^{12}=S_+$, $S^{21}=S_-$, and $(S^{11}-S^{22})/2=S_z$ in the QuantumOpticsBase basis convention.

Indexed alternative

Keep the site label when atoms are distinguishable or when the calculation needs site-resolved moments:

N = 100
h = NLevelSpace(:atom, (:g, :e))
i = Index(h, :i, N, h)
σge = IndexedOperator(Transition(h, :σ, :g, :e), i)
collective_sum = Σ(σge, i)

This form and CollectiveTransition intentionally do not mix on one subspace: they encode different physical roles and live on different Hilbert-space types.