  Interesting experimental signatures of quantum cavity optomechanics arise
because the quantum back-action induces correlations between incident quantum
shot noise and the cavity field. While the quantum linear theory of
optomechanics (QLT) has provided vital understanding across many experimental
platforms, in certain new set-ups it may be insufficient: analysis in the time
domain may be needed, but QLT obtains only spectra in frequency space; and
nonlinear behavior may be present. Direct solution of the stochastic equations
of motion in time is an alternative, but unfortunately standard methods do not
preserve the important optomechanical correlations. We introduce two-timescale
stochastic Langevin (T2SL) propagation as an efficient and straightforward
method to obtain time traces with the correct correlations. We show that T2SL,
in contrast to standard stochastic simulations, can efficiently simulate
correlation phenomena such as ponderomotive squeezing and reproduces accurately
cavity sideband structures on the scale of the applied quantum noise and even
complicated features entirely submerged below the quantum shot noise
imprecision floor. We investigate nonlinear regimes and find where comparison
is possible, that the method agrees with analytical results obtained with
master equations at low temperatures and in perturbative regimes.
