Complex Order Parameters in Quantum Optics First Order Phase Transition Analogies

Abstract

Our purpose in this paper is to present the thermodynamic treatment of quantum optical systems which have first order phase transition analogies and whose order parameters are two- dimensional or complex valued such as the electric field variable in optical bistability. In so doing we show that it is advantageous to analyze phase transitions by exactly the same procedure as one uses in equilibrium thermodynamics by means of the appropriate free energies which are related to each other by Legendre transformations. We find there exist two classes of quantum optical systems which differ in the role played by the phase variable in the complex order parameter. In the first class, consisting of systems such as saturable absorbers, the mean field first order transition is completely independent of the phase variable as in the analogy between the laser threshold region and the second order phase transition. In the second class of systems, such as optical bistability, we find the first order phase transition depends on the relative phase between the incident field and the internal field. We show how the necessary metastable states arise without the addition of physical mechanisms to break gauge invariance when the external field is zero. We conclude with an analysis of the effect fluctuations in the phase variable have on the mean field phase transition.