A quantum dot (QD) exciton at energy Ex coupled to a single cavity mode at energy Ec with coupling strength g is described by a 2×2 coupled-oscillator (Jaynes–Cummings, low-excitation) Hamiltonian. Adding radiative loss — cavity linewidth κ and dot linewidth γ — as imaginary parts on the diagonal turns it into the standard non-Hermitian coupled-mode Hamiltonian used to model real micropillar/microcavity QD systems:
H = [ E_x − iγ/2 g ]
[ g E_c − iκ/2 ]
E± = (E_x+E_c)/2 − i(γ+κ)/4 ± sqrt{ g² + [(E_x−E_c) − i(γ−κ)/2]² / 4 }
Γ± = −2·Im(E±) (branch decay rate)
Splitting = |Re(E+) − Re(E−)| Strong coupling ⇔ 2g > |κ−γ|/2
This sim diagonalizes H exactly, every frame, for whatever detuning Δ = Ex−Ec and coupling g you set (Ec is the zero of the energy axis). Two regimes fall straight out of the same matrix, with no separate code path:
- Weak coupling (2g ≪ κ) — the eigenvalues stay near Ex and Ec, but Im(E±) mixes: the dot inherits some of the cavity's fast decay. On resonance this reduces to the familiar Purcell formula FP ≈ 4g²/(κγ), and the emission spectrum shows one Purcell-enhanced peak.
- Strong coupling (2g > κ) — Re(E±) split apart even at Δ = 0: the vacuum Rabi splitting 2g, the signature normal-mode anticrossing of cavity QED (Reithmaier et al. 2004; Yoshie et al. 2004, InAs QD–micropillar/photonic-crystal systems). The spectrum shows two resolved peaks that pull apart as g grows and merge back together as |Δ| grows.
The top plot sweeps Δ across a fixed window and re-diagonalizes H at every sample to draw the real anticrossing curve Re(E±)(Δ) — the computed eigenvalues, not a drawn hyperbola. The bottom plot sums two Lorentzians centred on the current Re(E±), with linewidths Γ± and weights equal to each branch's exciton (dot) Hopfield fraction — the actual eigenvector mixing, so a cavity-like branch contributes less to what the dot radiates.