  Optical heterodyne detection-based spectrometers are attractive due to their
relatively simple construction and ultra-high resolution. Here we demonstrate a
proof-of-principle single-mode optical-fiber-based heterodyne spectrometer
which has picometer resolution and quantum-limited sensitivity around 1550 nm.
Moreover, we report a generalized quantum limit of detecting broadband
multi-spectral-temporal-mode light using heterodyne detection, which provides a
sensitivity limit on a heterodyne detection-based optical spectrometer. We then
compare this sensitivity limit to several spectrometer types and dim light
sources of interest, such as, spontaneous parametric downconversion, Raman
scattering, and spontaneous four-wave mixing. We calculate the heterodyne
spectrometer is significantly less sensitive than a single-photon detector and
unable to detect these dim light sources, except for the brightest and
narrowest-bandwidth examples.
