Universal tuning of Förster resonance energy transfer in gate-programmable conductor-dielectric-conductor heterostructures

We develop a quantum-electrodynamical theory for universal tuning of spontaneous emission and Förster resonance energy transfer (FRET) in a material-agnostic conductor-dielectric-conductor heterostructure. The platform consists of a dielectric spacer of thickness $W$ bounded by two gate-tunable two-dimensional conductors. The only microscopic input from the surrounding materials is the transverse-magnetic and transverse-electric reflection amplitudes $r_{\text{TM}\mathrm{/}\text{TE}}(q_{\rho },\omega )$ of the two sheets. Starting from the QED photon propagator, we derive the retarded Maxwell dyadic, the vacuum/Hadamard field propagator, and the time-ordered Feynman propagator in the same geometry. The spontaneous-emission rate is controlled by the local vacuum spectral density, while FRET is controlled by the nonlocal retarded/advanced product ${\text{obG}}_{\text{R}}({\text{bx}}_D,{\text{bx}}_A;{\omega }_D)\mathrm{\Im }{\text{balpha}}_A({\omega }_D){\text{obG}}_{\text{A}}({\text{bx}}_A,{\text{bx}}_D;{\omega }_D)$. In the transparent limit $r_{\text{TM}}\to 0$, the near-field FRET rate recovers the bulk $x_{\rho }^{-6}$ law. In the Dirichlet/PEC branch $r_{\text{TM}}\to -1$, the gapless transverse mode is removed and the donor-acceptor coupling acquires a Bessel-$K$ envelope, giving an exponentially screened FRET rate ${\mathrm{\Gamma }}_{D\to A}\propto \exp (-2\pi x_{\rho }\mathrm{/}W)$ at large lateral separation. In the opposite Neumann/PMC-like branch $r_{\text{TM}}\to 1^{-}$, a nearly gapless transverse mode survives and produces a wide quasi-two-dimensional logarithmic propagator, enhancing the nonlocal electromagnetic coupling over a gate-programmable range $x_{\rho ,\ast }\sim W\mathrm{/}(1-r_{\text{TM}})$. For graphene-Er implementations, the same retarded Green tensor also separates dissipative on-shell Er-to-graphene decay, governed by its absorptive part, from dispersive virtual-plasmon-mediated Er-Er coupling, governed by its reactive part; a plasmonic band gap can suppress the former while retaining the latter.