A PT-symmetric system couples a "gain" site to a "loss" site of equal strength γ, and is invariant under combined parity (site swap) + time reversal even though its Hamiltonian is not Hermitian:
H(γ) = [ iγ κ ]
[ κ -iγ ]
Eigenvalues: E± = ± sqrt(κ² − γ²)
Evolution: dψ/dt = -i H ψ (integrated live below, RK4)
- γ < κ — PT-unbroken phase. The discriminant κ²−γ² is positive, so both eigenvalues are purely real despite H being non-Hermitian. The total probability |ψ₁|²+|ψ₂|² still oscillates rather than staying fixed — dynamics are non-unitary even with a real spectrum.
- γ = κ — exceptional point (EP). Both eigenvalues collide at E=0 and their eigenvectors become parallel — the Hamiltonian stops being diagonalizable, so evolution picks up a secular t·e^{-iEt} term (Jordan-block behaviour). That makes the wavefunction amplitude grow linearly in time — but since probability is amplitude squared, the total norm |ψ|² you see plotted grows quadratically in time right at the EP (verified numerically here; the source 3D sim's theory text calls the norm itself "linear-in-time", which is the more common shorthand but conflates it with the amplitude — this map plots norm, so we state the correct power here).
- γ > κ — PT-broken phase. The eigenvalues become a complex-conjugate pair E± = ±i·sqrt(γ²−κ²). One mode grows exponentially (gain dominates) while its partner decays — the norm readout climbs without bound.
The top-left panel plots both eigenvalue branches against γ on one shared axis: solid lines are the real part, dashed lines the imaginary part — they trade off exactly at the exceptional point (gold marker). The top-right panel scrolls the live site populations and total norm through time. The bottom panel shows the two coupled sites as points orbiting a shared center, radius set by their instantaneous population — drag it to tilt the view, the same way you'd orbit a camera around a 3D plot.
Real-world relevance: PT-symmetric Hamiltonians describe coupled optical waveguides with gain/loss, microwave and acoustic resonator pairs, and open quantum systems — exceptional-point sensors built on this effect exploit the divergent sensitivity right at γ=κ.