  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.
