Two-dimensional isotropic harmonic oscillator

H = a_{x}^{†} a_{x} + a_{y}^{†} a_{y} + 1.

The level with total occupation

N = n_{x} + n_{y}

N + 1

and degeneracy

N + 1

in units where

ℏ = 1

Oscillator bilinear commutator

For bosonic modes satisfying

[a_{i}, a_{j}^{†}] = δ_{ij}

, the number-preserving bilinears

T_{ij} = a_{i}^{†} a_{j}

obey

[T_{ij}, T_{k l}] = δ_{jk} T_{i l} - δ_{i l} T_{kj} .

Contracting two-mode oscillator bilinears with the Pauli matrices,

T^{a} = \frac{1}{2} σ_{ij}^{a} a_{i}^{†} a_{j},

gives

[T^{a}, T^{b}] = i ε_{ab c} T^{c}

. The subspace with total occupation

N

is the spin-

N /2

N + 1

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