Locking equation with colored noise: Continued fraction solution versus decoupling theory

The locking equation in the presence of colored noise is studied. This system models, for example, the mean beat frequency 〈φ̇〉 of a ring-laser gyroscope in which weak noise with (dimensionless) noise correlation times τ≊${10}^{\mathrm{-}2}$–${10}^2$ is used experimentally to overcome the locking. The non-Markovian, colored-noise dynamics is solved by use of a matrix-continued-fraction technique. The thusly calculated stationary probability and the mean beat frequency are compared with the decoupling theory introduced recently by Haaumlnggi and co-workers [Phys. Rev. A 32, 695 (1985); 33, 4459 (1986)], as well as with the conventional small-noise-correlation-time approximation. The decoupling approximation, resulting in a Fokker-Planck equation with an effective diffusion which must be evaluated self-consistently, yields satisfactory agreement over the whole regime of physically relevant correlation times τ. The small-correlation-time approximation however, breaks down for moderate-to-large τ. The mean beat frequency 〈φ̇〉 decreases at constant noise intensity with increasing noise color; i.e. increasing the noise correlation time τ increases the tendency to lock.