Noise-dependent uncertainty relations for the harmonic oscillator

The Gaussian noise model for a harmonic oscillator subjected to random linear fluctuations (e.g., thermal noise) is used to derive generalized Heisenberg and entropic uncertainty relations for the position-momentum and number-phase observables. The lower bounds are partitioned into pure quantum and pure noise terms, and the minimum-uncertainty states are determined. Formally similar results of Abe and Suzuki [Phys. Rev. A 41, 4608 (1990)] for the position-momentum case, derived using the thermofield formalism, are shown to correspond to amplification noise rather than thermal noise.