  We investigate the properties of the condensates of cold atoms at zero
temperature in the tunable binary Bose-Einstein condensate system with a Rabi
transition between atomic hyperfine states. We use this system to examine the
effect of quantum fluctuations in a tunable quantum gas on phonon propagation.
We show that the system can be represented by a coupled two-field model of a
gapless phonon and a gapped mode, which are analogous to the Goldstone and
Higgs particles in particle physics. We then further trace out the gapped modes
to give an effective purely phononic theory using closed-time-path formalism.
In particular, we are interested in the sound cone fluctuations due to the
variation of the speed-of-sound acoustic metric, induced by quantum
fluctuations of the gapped modes. These fluctuations can be interpreted as
inducing a stochastic space-time, and thus are regarded as analogue phenomena
of light cone fluctuations presumably arising from quantum gravity effects. The
effects of fluctuations can be displayed in the variation in the travel time of
sound waves. We suggest the relevant experiments to discuss the possibility of
experimental observations.
