We analyze the laser process with three different initial states using the entangled state representation, obtain the evolution law of the mean photon number, the entropy, the specific entropy, the second degree of coherence and the Wigner functions, and find out the common characteristics of these three processes.
We investigate systematically the evolution of the number state in a laser process by deriving the analytic expression of the density operator and putting it into a normal ordered form. The eigenvalue of the density operator is related to Jacobi polynomials. Then we derive the expression for the mean photon number, the second degree of coherence, the entropy, Wigner function and the photoncount distribution. The nonclassicality is discussed by virtue of the negativity of Wigner function. It is found that the Wigner function is always negative for t < t0, which is independent on the parameter m. On the other hand, the condition for the second degree of coherence larger than 1 is dependent on the parameter m.
Though laser physics began at 1960s, its entropy evolution has not been touched until very recently the Kraus-form solution of laser's master equation is reported (Ann. Phys. 334 (2013)). We study the new physics based on the discovery in this paper. We analyze time evolution of physical observables and quantum optical properties in the laser process with arbitrary initial states, such as the photon number, the second degree of coherence, etc. The evolution of entropy of these states is also studied. Our results well conform with the known behaviour of laser, which confirms that the master equation describes laser suitably, and the Kraus-form operator solution is correct, elegant and useful.
Based on the density operator’s operator-sum representation recently obtained by Fan and Hu for a laser process (Opt. Commun., 2008, 281: 5571; Opt. Commun., 2009, 282: 932; Phys. Lett. B, 2008, 22: 2435), we derive the evolution law of Wigner operator, the law is concisely expressed in the normally ordered form $\Delta (\alpha ,\alpha ^* ,t) = \tfrac{T} {\pi }:\exp [ - 2T(a^\dag e^{ - (\kappa - g)t} - \alpha ^* )(ae^{ - (\kappa - g)t} - \alpha )] $ :, where g and κ are the cavity gain and the loss, respectively, and T ≡ (κ − g)(κ + g − 2ge−2(κ−g)t )−1:, When $t = 0,\Delta (\alpha ,\alpha ^* ,t) \to \tfrac{1} {\pi }:\exp [ - 2(a^\dag - \alpha ^* )(a - \alpha )] $ ;, which is the initial Wigner operator. Using this formalism the evolution law of Wigner functions in laser process can be directly obtained.
Based on the Kraus operator of laser theory which we derived in [2] and by virtue of the technique of integration within ordered product of operators we exactly calculate the evolving result of a chaotic field, governed by the master equation describing the laser process, we find that the initial density operator evolves into a new chaotic field, with where g and κ represent the cavity gain and loss of the laser, respectively. When g = 0, the system reduces to an amplitude dissipative channel, and f' becomes −ln [(e−f − 1) e2κt + 1]. The law of photon number decay in this process is evaluated, and the corresponding Wigner function's evolution is also demonstrated.
The quantum reduced entropy evolution of a V-type three-level atom interacting with the two- mode odd-even entangled coherent field is investigated.The effects of the detuning,mean photon numbers of field,the atom's initial state and the degree of the entanglement of the two-mode odd-even entangled coherent fields on the entropy are discussed.The results show that the oscillation of the entropy presents collapses and revivals with the increasing of the detuning or the mean photon number.The mean-value of field entropy decreases with the increasing of the detuning.But it increases with the increasing of the mean photon number.The mean-values of field entropy are higher when the atom is initially in excited state or superposition state than that of in ground state.The mean-value and the amplitude of field entropy decrease with the increasing of the degree of the entanglement of the two-mode odd-even entangled coherent fields.