  Superconducting nanowire single-photon detectors (SNSPDs) have been widely
used to study the discrete nature of quantum states of light in the form of
photon-counting experiments. We show that SNSPDs can also be used to study
continuous variables of optical quantum states by performing homodyne detection
at a bandwidth of $400~\mathrm{kHz}$. By measuring the interference of a
continuous-wave field of a local oscillator with the field of the vacuum state
using two SNSPDs, we show that the variance of the difference in count rates is
linearly proportional to the photon flux of the local oscillator over almost
five orders of magnitude. The resulting shot-noise clearance of
$(46.0\pm1.1)~\mathrm{dB}$ is the highest reported clearance for a balanced
optical homodyne detector, demonstrating their potential for measuring highly
squeezed states in the continuous-wave regime. In addition, we measured a
$\mathrm{CMRR}=22.4~\mathrm{dB}$. From the joint click counting statistics, we
also measure the phase-dependent quadrature of a weak coherent state to
demonstrate our device's functionality as a homodyne detector.
