Adi R. Bulsara

Phase transitions induced by white noise in bistable optical systems

Abstract We consider a gas of two-level atoms in a Fabry-Perot cavity subjected to a coherent driving field, in the sub-threshold region where no hysterisis loop is normally present. It is shown that external white noise can induce a phase transition in this region. A comparison is made to the microscopic theory [8] and it is shown that for the noise variance σ2 = 0.005, exactly the same results are obtained using our stochastic differential equation approach.

First passage time analysis of metastable states in laser phase transitions

Abstract We consider a laser with a saturable absorber, exhibiting a first order phase transition whth a metastable state below threshold. The stability of this metastable state relative to the true equilibrium solution (corresponding to zero electric field) is analysed through a calculation of the first passage time for the decay of the metastable state.

Higher time derivatives of the generalized entropy

Using the generalized N‐body expression for a Liapunov functional developed by Prigogine, George, and Henin, a condition is obtained whereby the successive time derivatives of this functional alternate in sign for weakly coupled systems. This condition is applied and seen to hold true for generalized Bose and Fermi systems. An N‐body entropy is defined in terms of this functional, which contains diagonal as well as off‐diagonal elements of the density matrix. This ’’generalized entropy’’ is seen to be concave close to equilibrium, similar to the results of Harris. It does not appear that this property holds for the entropy far from equilibrium.

Entropy and stability in a stochastic laser model

The asymptotic stability of a microscopic stochastic laser model in the Heisenberg picture is discussed using Liapounovs direct method. By reducing the laser Langevin equations to a classical van der Pol equation for the photon number in the rotating wave approximation, it may be shown that the above-threshold steady state characterized by the cubic non-linearity in the van der Pol equation, is stable. This leads to the definition of a monotonic increasing entropy function far from thermal equilibrium.

A Description of Laser Stability near Threshold by Liapounov’s Second Method

The theory of the generalized entropy developed by Prigogine, George and Henin [1] in Brussels and Austin suggests a form for an N-body Liapounov function which has already been discussed for closed systems [2–4]. Starting from the Liouville-von Neumann equation for the density operator ρ, Prigogine introduces a causal or physical particle representation wherein he defines a “physical” density matrix ρ(P) which is generated from ρ by a non-unitary transformation in strongly coupled system., this representation, the retarded and advanced components of ρ(P) obey separate evolution equations. For weakly coupled systems, ρ(P) reduces to the untransformed ρ which is a solution to the Pauli equation. This theory has been applied by Hubert [5] to a dense hard sphere gas obeying the Enskog equation. Further, Bulsara and Schieve have demonstrated [2] that infinitesimally close to thermal equilibrium, the generalized form of the entropy proposed by Prigogine et al. has the same form as the Liapounov function referred to above, for weakly coupled systems, a fact which we shall exploit in section 5 of this paper.

A Lyapounov function approach to the theory of stability in a laser using the Scully-Lamb theory

Abstract The stability of a quantum model of a c.w. laser is demonstrated using Lyapounovs second method.