  Stochastic master equations are often used to describe conditional spin
squeezing of atomic ensemble, but are limited so far to the systems with few
atoms due to the exponentially increased Hilbert space. In this article, we
present an exact numerical solution of these equations for systems with
identical atoms by mapping identical density matrix elements to a single
quantity characterized by collective quantum numbers, and apply it to the
system with hundred atoms in a bad cavity subject to a homodyne detection. We
demonstrate that the spin squeezing can be vividly illustrated by the
Gaussian-like distribution of the collective density matrix elements, and we
examine the influence of the probe field strength and polarization, the
detection efficiency, the spontaneous emission rate and the number of atoms.
Our exact approach can play an important role in gauging the approximate
approaches applied for systems with more atoms, such as Gaussian-state
formalism and stochastic mean-field approach, and it permits also exploration
of entanglement effects beyond these approaches.
