A Purely Dissipative Third-Order Exceptional Point in a Plasmonic Nanocavity with Two Quantum Dots: Exact Conditions and the Absence of an Enhanced Observable Response

Third-order exceptional points in cavity systems have so far been obtained by introducing optical or mechanical gain, which restricts them to platforms where gain can be engineered. We show that two semiconductor quantum dots in a lossy plasmonic nanocavity host such a point with loss alone, and we solve the coalescence problem for the 3 × 3 effective Hamiltonian in closed form. The conditions leave no free parameter: the cavity sits at the mean transition frequency of the dots, the direct dipole-dipole coupling vanishes, the loss contrast obeys κ − γ = 3√3g, and the two dots are detuned from each other by exactly one coupling constant, ω₁ − ω₂ = g. The tuning knob is the detuning between the emitters rather than a gain rate. We confirm that the effective Hamiltonian becomes a single Jordan block of size three and that the coalescence also appears in the spectrum of the full Liouvillian, where three eigenvalues agree to one part in 10⁵ of the linewidth, and the geometric multiplicity remains one, so quantum jumps do not remove it. The eigenvalue splitting follows (εg²)^(1/3) for a cavity perturbation ε, and measured against the linewidth it exceeds the second-order response of the same device by a factor of 4 to 12 over four decades. The steady-state cavity population, however, responds linearly in ε with a fitted exponent of 1.000 and an advantage over the second-order point of only 1.09. The reason is structural: the characteristic polynomial depends on ε linearly at fixed frequency, so the resolvent stays analytic even though the pole positions do not. The cube-root law therefore describes where the poles sit and not what a spectroscopic measurement returns. We also find that a residual dipole-dipole coupling of 10⁻³ g restores linear eigenvalue response, which sets a minimum interdot separation of 12 to 36 nm and conflicts with the tight confinement that produces large coupling in plasmonic gaps.