  A collective description of density matrix is presented for identical
multi-level atoms, which are either excited initially, driven coherently or
pumped incoherently. The density matrix is defined as expectation value of
projection or transition operators in a basis of atom's product states. The
identical matrix elements are identified with several integers, which specify
uniquely the involved operators. To remove the redundancy, these identical
elements are treated as single quantity and the equation for this quantity is
dervied by mapping the transition or projection operators to a single vector
specified with these integers. As a result, the number of computed elements
increases polynomially rather than exponentially with the number of atoms. As
an example, we carry out exact simulation of hundreds of two-level atoms and
demonstrate the different conditions for observing superradiance and
superfluorescence.
