Noise limits for dc SQUID readout of high-$Q$ resonators below 300 MHz

We present the limits on noise for the readout of cryogenic high-$Q$ resonators using dc Superconducting Quantum Interference Devices (SQUIDs) below 300 MHz. This analysis uses realized first-stage SQUIDs (previously published), whose performance is well described by Tesche-Clarke (TC) theory, coupled directly to the resonators. We also present data from a prototype second-stage dc SQUID array designed to couple to this first-stage SQUID as a follow-on amplifier with high system bandwidth. This analysis is the first full consideration of dc SQUID noise performance referred to a high-$Q$ resonator over this frequency range, and is presented relative to the standard quantum limit. We include imprecision, backaction, and backaction-imprecision noise correlations from TC theory, the noise contributed by the second-stage SQUIDs, wiring, and preamplifiers, and optimizations for both on-resonance measurements and off-resonance scan sensitivity. This architecture has modern relevance due to the increased interest in axion searches and the requirements of the DMRadio-m$^3$ axion search, which will use dc SQUIDs in this frequency range.