  The dynamical behavior of entanglement between the atomic ensemble and the
spontaneous emission is investigated by applying the femtosecond Gaussian
pulses in double- lambda quantum system. In fact by solving the density matrix
equations of motion in such a system and by using the von Neumann entropy, the
degree of entanglement (DEM) could be measured in semi classical regime and in
the multi-photon resonance condition. We interested in investigating the
maximum value for DEM and controlled the DEM via relative phase. At first we
consider three laser fields as CW optical laser fields and one femtosecond
Gaussian field. In this case DEM has interesting behavior. When the system is
applied by the other laser fields and Gaussian pulse is still there DEM can be
controlled via phase but after finishing the time duration of the femtosecond
Gaussian pulse DEM will be phase independent. Then we consider all the laser
fields as femtosecond Gaussian pulses which interact with atomic system and we
achieve to this point that the steady state entanglement can be occurred by
using such ultra-short laser pulses; in this situation DEM will be phase
independent. The system would be disentangled in special parameters of Rabi
frequencies and all the atoms are coherently populated in a dark state in
dressed state picture. Moreover when the system is entangled, there is a
population distribution of the atoms in dressed states.
