This is the same non-Hermitian 2×2 coupled-oscillator Hamiltonian used to describe plasmon–exciton strong coupling, but represented two ways that are both native to a 2D canvas instead of a rotating 3D nanoparticle:
H = [ Epl − i·γpl/2 g ]
[ g Eex − i·γex/2 ]
E± = (Epl+Eex)/2 − i(γpl+γex)/4
± sqrt[ g² + ( (Epl−Eex)/2 − i(γpl−γex)/4 )² ]
Left panel — anticrossing diagram: Re(E±) plotted directly against detuning Δ, swept analytically across the whole slider range, tracing the classic avoided-crossing "X". The current Δ is marked with a moving dot on each branch.
Right panel — time-domain quantum beats: instead of diagonalizing H once, this side numerically integrates the equivalent temporal coupled-mode equations (Schrödinger-like evolution with ħ=1 in the frame rotating at the mean energy) via 4th-order Runge–Kutta:
i·d(a_pl)/dt = (Δ/2 − i·γpl/2)·a_pl + g·a_ex
i·d(a_ex)/dt = (−Δ/2 − i·γex/2)·a_ex + g·a_pl
Starting from a plasmon photoexcitation (a_pl=1, a_ex=0), the populations |a_pl|² and |a_ex|² genuinely oscillate back and forth — a live demonstration of vacuum Rabi flopping — while decaying at a rate set by the linewidths. In the strong-coupling regime the energy completes at least one full round trip between the plasmon and the exciton before it decays away; in the weak-coupling regime it decays before ever crossing back. The two masses-on-a-spring animation encodes Re(a_pl), Re(a_ex) as vertical position and |a|² as circle size — a direct 2D read-out of the same complex amplitudes driving the strip-chart below it.
- Coupling g — widens the anticrossing gap and speeds up the beat.
- Detuning Δ — sweeping it traces the "X"; also detunes the two masses' natural pull.
- γpl, γex — how fast each population decays; push either high enough and the beat disappears before completing even one cycle (weak coupling).