PT-Symmetric Quantum Mechanics: Exceptional Points

Non-Hermitian Hamiltonian

Gain/loss strength γ 0.40

Coupling strength κ 1.00

PT-unbroken (real spectrum)

Eigenvalues E± = ±√(κ²−γ²)

Re(E+)

0.00

Im(E+)

0.00

Re(E−)

0.00

Im(E−)

0.00

Live dynamics

Total norm |ψ|²

1.00

Sim time t

0.0

How it works

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. Norm growth switches from exponential to linear-in-time right at this threshold, the experimental fingerprint used to detect an EP.
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 3D plot traces both eigenvalue branches against γ: real parts rise from the horizontal (Re) plane and, past the exceptional point, bend into the vertical (Im) plane. The two small orbiting spheres show the live population on the gain site (green) and loss site (red) as the coupled ODE is integrated frame by frame.

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.

Branch E+ Branch E− Gain site population Loss site population

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