Optimal Impedance Matching and Quantum Limits of Electromagnetic Axion and Hidden-Photon Dark Matter Searches

For the first time, we determine the properties of the optimal single-moded, linear, passive search for electromagnetic coupling to axion and hidden-photon dark matter, subject to the Standard Quantum Limit on phase-insensitive amplification. We establish the parameters that must be considered to determine the optimal search: the impedance match to dark matter; receiver frequency-response and tuning; irreducible noise sources; and prior information on the dark-matter signal. Using complex-power flow equations, we identify two categories of coupling to the dark-matter signal: radiative and reactive. We motivate a focus on single-moded reactive couplings, as receivers using solely radiative couplings are limited in sensitivity by mismatch with the dark-matter source impedance. We define integrated sensitivity as a figure of merit in comparing searches over a wide frequency range and show that the Bode-Fano criterion sets a limit on integrated sensitivity in a reactively coupled receiver. We examine single-pole resonators, a broadly used form of reactive coupling, and show that when thermal noise dominates amplifier noise and noise matching is optimized, substantial sensitivity is available away from the resonator bandwidth. The Bode-Fano constraint establishes the single-pole resonator as near-ideal for single-moded dark-matter detection. Additionally, the optimized resonator is superior to the optimized reactive broadband receiver at all frequencies at which a resonator may practically be made. We optimize time allocation in a tunable resonator search using priors and derive quantum limits on resonant search sensitivity. At low frequencies, the application of our optimization may enhance scan rates by a few orders of magnitude. While our results broadly inform laboratory searches for light fields, they are the basis for DMRadio, a DOE-funded program in axion and hidden-photon detection.