Influence of random fluctuations on delayed bifurcations: The case of additive white noise

The influence of additive white noise on the delay of a bifurcation point in the presence of a swept control parameter has been studied by analog experiment and digital simulation. Measurements of the time taken for the second moment 〈$x^2$(t)〉 to reach a given threshold are in excellent agreement with the calculations of Zeghlache, Mandel, and Van den Broeck [Phys. Rev. A 40, 286 (1989)]. It is demonstrated, however, that the mean first-passage time (MFPT) for $x^2$(t) to attain the same threshold, corresponding to the quantity that is usually determined in laser experiments, can be markedly different. It may be either larger or smaller, depending on the conditions under which the measurements are made. A calculation of the MFPT is presented and shown to be in excellent agreement with the experimental measurements and to reduce, in the relevant limit, to the theoretical results previously published by Torrent and San Miguel [Phys. Rev. A 38, 245 (1988)].