Getting started

We use units with $\hbar = 1$. Consider a cavity mode on resonance with a two-level atom, described by the Jaynes–Cummings Hamiltonian

\[H = \omega a^\dagger a + \omega \sigma^{22} + g \left( a^\dagger \sigma^{12} + a \sigma^{21} \right).\]

The cavity lives on a FockSpace and the atom on an NLevelSpace; their tensor product ⊗ is the space of the full system. Each space in the tensor product represents one subsystem. In the constructors below, the final argument selects that subsystem: 1 for the cavity and 2 for the atom. Thus the annihilation operator $a$ acts on the cavity, while the transitions $\sigma^{ij} = |i\rangle\langle j|$ act on the atom. The couplings are ordinary Symbolics variables.

using SecondQuantizedAlgebrah = FockSpace(:cavity) ⊗ NLevelSpace(:atom, 2)@qnumbers a::Destroy(h, 1)σ(i, j) = Transition(h, :σ, i, j, 2)@variables ω gH = ω * a' * a + ω * σ(2, 2) + g * (a' * σ(1, 2) + a * σ(2, 1))

\[\omega {\sigma}^{{22}} + g a{\sigma}^{{21}} + \omega a^{\dagger}a + g a^{\dagger}{\sigma}^{{12}}\]

Products apply the local algebraic identities as they are formed: ladder operators come out normal ordered, and transitions on the same atom compose.

a * a', σ(1, 2) * σ(2, 1), σ(2, 1) * σ(2, 1)

(1 + a' * a, σ₁₁, 0)

Here $\sigma^{11} = |1\rangle\langle 1|$ projects onto the atomic ground state, and $\sigma^{22} = |2\rangle\langle 2|$ onto the excited state. Since these are the only two states, their projectors satisfy $\sigma^{11} + \sigma^{22} = 1$. The algebra keeps $\sigma^{11}$ as a single operator; call expand_completeness to replace it using this identity.

expand_completeness(σ(1, 1))

\[1 -{\sigma}^{{22}}\]

Because every product is already in canonical form, a commutator reduces to a subtraction, and the Heisenberg equation $\dot{a} = i[H, a]$ is one line.

1im * commutator(H, a)

\[ - \omega ~ \mathit{i} a - g ~ \mathit{i} {\sigma}^{{12}}\]

Taking expectation values with average turns the operator equation for the atomic coherence into an equation for c-numbers, the starting point of any mean-field or cumulant treatment.

average(1im * commutator(H, σ(1, 2)))

\[ \begin{equation} \langle a{\sigma}^{{22}} \rangle ~ g ~ \mathtt{im} - \langle a{\sigma}^{{11}} \rangle ~ g ~ \mathtt{im} - \langle {\sigma}^{{12}} \rangle ~ \mathtt{im} ~ \omega \end{equation} \]

To compute with the model, to_numeric builds its matrix through QuantumOptics on a truncated basis, here with cavity Fock states from zero to ten photons.

using QuantumOpticsBaseusing LinearAlgebra: eigvals, Hermitianbackend = QuantumOpticsBackend()parameter = Dict(ω => 1.0, g => 0.1)Hn = to_numeric(H, h, (10, 2); backend, parameter)eigvals(Hermitian(Matrix(dense(Hn).data)))[1:5]

5-element Vector{Float64}:
 0.0
 0.8999999999999988
 1.0999999999999983
 1.858578643762691
 2.14142135623731

Above the ground state the spectrum splits into doublets at $\omega \pm g$ and $2\omega \pm \sqrt{2} g$, the vacuum Rabi splitting.