Universal tuning of F\"{o}rster resonance energy transfer in gate-programmable conductor-dielectric-conductor heterostructures

Abstract

We develop a quantum-electrodynamical theory for universal tuning of spontaneous emission and F\"{o}rster 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 $r_{\TM/\TE}(q_\rho,\omega)$ 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 $\obG_{\text R}(\bx_D,\bx_A;\omega_D)\Im\,\balpha_A(\omega_D)\obG_{\text A}(\bx_A,\bx_D;\omega_D)$. In the transparent limit $r_{\TM}\to 0$, the near-field FRET rate recovers the bulk $x_\rho^{-6}$ law. In the Dirichlet/PEC branch $r_{\TM}\to -1$, the gapless transverse mode is removed and the donor-acceptor coupling acquires a Bessel-$K$ envelope, giving an exponentially screened FRET rate $\Gamma_{D\to A}\propto\exp(-2\pi x_\rho/W)$ at large lateral separation. In the opposite Neumann/PMC-like branch $r_{\TM}\to 1^{-}$, 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 $x_{\rho,*}\sim W/(1-r_{\TM})$. 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.