Emission photon statistics in collectively interacting dipole atom arrays in the low-intensity limit

We investigate the photon statistics of light emitted from a system of collectively interacting dipoles in the low-intensity regime, incorporating double-excitation states to capture beyond-single-excitation effects. By analyzing the eigenstates of the double-excitation manifold, we establish their connection to the accessible single-excitation eigenmodes and investigate the role of decay rates in shaping the zero-time-delay photon correlation function $g^{(2)}(τ= 0)$ under different detection schemes. The photon emission statistics can be arbitrarily controlled by interfering two beams of light that selectively address orthogonal eigenmodes. This can act as a tunable nonlinearity that enables both enhancement or suppression of two-photon emission.

💡 Research Summary

In this work the authors investigate the photon‑photon correlations of light emitted by an ensemble of dipole‑coupled two‑level atoms driven in the low‑intensity regime. The system consists of N atoms arranged in a sub‑wavelength lattice (1‑D or 2‑D) with inter‑atomic spacing d < λ and all dipoles polarized along a common axis. The collective interaction is described by the vacuum dyadic Green’s function g(rij), whose real part accounts for coherent dipole‑dipole exchange (energy shifts) and whose imaginary part gives rise to collective decay (super‑ and sub‑radiance). By constructing the complex symmetric matrix Gij = g(rij) and diagonalising it, the authors obtain the single‑excitation eigenmodes α with complex eigenvalues Δα − iγ α/2, where γα is the decay rate of mode α.

Beyond the usual single‑excitation treatment, the paper explicitly includes the double‑excitation manifold |eeμ⟩ (μ = (m1,m2) with m1 < m2). A second‑order interaction matrix \tilde Gμν is built from the same Green’s function, linking double‑excitation states that share a common excited atom. Diagonalising \tilde G yields double‑excitation eigenmodes β with eigenvalues Δ(2)β − iγ(2)β/2; γ(2)β is the rate at which the first photon is emitted, after which the system collapses into a superposition of single‑excitation modes.

The dynamics are treated with a full density‑matrix master equation (Hamiltonian = laser + coherent dipole‑dipole terms, Lindblad dissipator = collective spontaneous emission). The system is driven by a weak, continuous‑wave laser (Ω0 ≪ Γ0) until a steady state is reached. To compute the second‑order correlation function g^(2)(τ), the authors project the steady‑state density matrix onto the subspace after a photon has been emitted (using a lowering operator σ− that can be chosen to select a specific emission direction k or a specific eigenmode α). The projected state ρ′ is then propagated for a delay τ and a second emission event is evaluated. In the zero‑delay limit τ = 0 the expression simplifies to g^(2)(0) = Tr