Dark matter is an enduring mystery in our quest to understand the fundamental constituents of our universe. Low mass bosons, such the axion or hidden photon, are compelling dark matter candidates. We leverage their potential interactions with electromagnetic fields, whereby the dark matter (of unknown mass) on rare occasion converts into a single photon, to devise a method of detecting these candidates. Current dark matter searches operating at microwave frequencies use a resonant cavity to coherently accumulate the field sourced by the dark matter and a near standard quantum limited (SQL) linear amplifier to read out the cavity signal. To further increase sensitivity to the dark matter signal and enable future searches, sub-SQL detection techniques are required. In this thesis, I report the development of a novel microwave photon counting technique and a new exclusion limit on hidden photon dark matter. We operate a superconducting qubit to make repeated quantum non-demolition measurements of cavity photons and apply a hidden Markov model analysis to reduce the noise to 15.7 dB below the quantum limit, with overall detector performance limited by a residual background of real photons. With the present device, we perform a hidden photon search and constrain the kinetic mixing angle to less than 1.68 X 10^{-15} in a band around 6.011 GHz (24.86 micro-eV) with an integration time of 8.33 s. This demonstrated noise reduction technique enables future dark matter searches to be sped up by a factor of 1300. By coupling a qubit to an arbitrary quantum sensor, more general sub-SQL metrology is possible with the techniques presented in this work.




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