A system of low dimensional condensed ultracold atomic gases inside a field of a laser-driven optical cavity exhibits dispersive optical bistability. During such a process the system also shows quantum fluctuations. Condensate fluctuations are highly manifested particularly in low dimensional systems. In this paper we have investigated the theory and manifestation of fluctuations in quantum optical systems in low dimensional Bose-Einstein condensates. We have described the system using the mean-field approximation. In this study we have verified that low dimensional quantum gases exhibit not only highly fascinating properties but also we have indicated that the use of mean field theory to describe quantum gases in low dimensions is highly restricted since the possibility of generating low dimensional bosonic condensates is dominated by the existence of highly sensitive and intrinsic quantum fluctuations.
The index of refraction of an optical medium at a temperature above absolute zero undergoes statistical fluctuations, which in turn introduce fluctuations in the phase of an optical signal propagating in the medium. The magnitude and spectral density of these phase fluctuations are calculated, and it is shown that they can be larger than the phase uncertainty due to quantum noise. Barring the onset of nonlinearities, these fluctuations and not quantum noise set the limit to the measurement of optical phase.>