By Bo-Sture Skagerstam; Norges Teknisk Naturvitenskapelige Universitet
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40) √ Here θk = φk /| sin φk | a − b is the value of θ for which the k’th branch comes into existence. Hence in the interval θK < θ < θK+1 there are exactly 2K + 1 branches, x0 , x1 , x2 , . . , x2K−1 √ , x2K , forming the K + 1 minima and K maxima of V (x). For 0 < θ < θ0 = 1/ a − b there are no extrema. This classification allows us to discuss the different parameter regimes that arise in the limit of N → ∞. Each regime is separated from the others by singularities and are thus equivalent to the phases that arise in the thermodynamic limit of statistical mechanics.
136]. . 5, and 1 as a function of θ = gτ N , where N = R/γ. 8 Analytic Preliminaries “It is futile to employ many principles when it is possible to employ fewer. ” W. Ockham In order to tackle the task of determining the phase structure in the micromaser we need to develop some mathematical tools. The dynamics can be formulated in two different ways which are equivalent in the large flux limit. Both are related to Jacobi matrices describing the stochastic process. Many characteristic features of the correlation length are related to scaling properties for N → ∞, and require a detailed analysis of the continuum limit.
This equation has an infinity of solutions, φ = φk , k = 0, 1, . 38) for k = 1, 2, . , and each of these branches is double-valued, with a sub-branch corresponding to a minimum (D ′ > 0) and another corresponding to a maximum (D ′ < 0). Since there are always k + 1 minima and k maxima, we denote the minima x2k (θ) and the maxima x2k+1 (θ). Thus the minima have even indices and the maxima have odd indices. They are given as a function of θ through Eq. 36) when φ runs through certain intervals. Thus, for the minima of V (x), we have φk < φ < (k + 1)π, θk < θ < ∞, a − b > x2k (θ) > 0, k = 0, 1, .
Topics in modern quantum optics by Bo-Sture Skagerstam; Norges Teknisk Naturvitenskapelige Universitet