We analyze the entanglement condition of a new kind of non-Gaussian quantum
state, which is prepared by photon number subtraction from a two mode
Gaussian state. Fock space criterion and Shchukin-Vogel criterion are
applied and the results are compared. Two kinds of the original Gaussian
states are utilized. Their photon subtracted states have different
The evolution of the optical NOON state is investigated. The environments of
the evolution are the thermal noise and amplitude damping due to fiber loss.
The characteristic function and the density matrix of the time-dependent
state are deduced strictly. We calculate the relative entropy of
entanglement for the truncated time-dependent state for N=1 case. The
phase measurement performance of the damped states is also studied.
The analytical formulas of laser threshold of the quasi-four-energy-level system for Tm3+,Tb3+:LiYF4 are deduced from rate-equation of transition. The energy transfer between Tm3 and Tb3 is discussed. The optimal concentration of Tm3+ and Tb3+ and the optimal length of the crystal rod for the minimum threshold pump intensity are calculated. It shows that when Tm3+-ion-doped concentration is comparatively high the threshold of the~1.5?m laser corresponding to 3H4 ?3F4 transition of Tm3+ increases because of the cross-relaxation. When co-doped with Tb3+ ion, on one hand, the threshold tends to decreases by contribution of Tm-Tb energy transfer which effectively reducing the population on the terminal laser level, on the other hand, it results in an increase in threshold due to Tm-Tb energy transfer which also reduces the lifetime of the initial laser level. In this paper the theoretical analysis for optimal Tb3+ concentration is fumished. Calculation shows that for LiYF4 crystal with the Tm3+ concentration being 1atm% the optimal concentration of Tb3+ is 0.2atm%. In addition, the dependence of optimal length of the crystal rod on the concentration of Tm3+ and Tb3+ and distributed rod loss were analyzed.