  We study a frequency-dependent noise-to-current ratio for asymmetric,
symmetric, and noncommutative current-noise in a ferromagnetic insulating
junction, and extract quantum-mechanical properties of magnon transport at low
temperatures. We demonstrate that the noncommutative noise, vanished in the
dc-limit (i.e., a classical regime), increases monotonically as a function of
frequency, and show that the noncommutative noise associated directly with
quantum fluctuations of magnon currents breaks through the classical upper
limit determined by the symmetric noise and realizes asymmetric quantum shot
noise. Finally, we show that our theoretical predictions are within
experimental reach with current device and measurement scheme while exploiting
low temperatures. Our work provides a platform toward experimental access to
quantum fluctuations of magnon currents.
