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By W.-H Steeb

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4 3N . 603 In the region above the transition at θ = 1 the mean occupation number n grows proportionally with the pumping rate N , so in this region the cavity acts as a maser. There is a single minimum of the effective potential described by the branch x0(θ), defined by the region 0 < φ < π in Eq. 36). 21) 10 6 8 N = 10, 20, . . 15 0 2 4 γξ 0 1 2 3 4 5 6 θ Figure 14: The correlation length in the thermal and maser phases as a function of θ for various values of√ N . The dotted curves are the limiting value for N = ∞.

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. ” G. Polya We shall from now on limit the discussion to the case of initially completely excited atoms, a = 1, b = 0, which simplifies the following discussion considerably.

If we disregard, for the moment, the decay of the cavity field due to interactions with the environment, the evolution is governed by the JC Hamiltonian in Eq. 1). It is natural to assume that the atom and the radiation field of the cavity initially are completely uncorrelated so that the initial density matrix factories in a cavity part and a product of k atoms as ρ = ρ C ⊗ ρ A1 ⊗ ρ A2 ⊗ · · · ⊗ ρ Ak . 1 0 0 10 20 gτ Figure 4: The upper figure shows the revival probabilities P(+) and P(+, +) for a coherent state |z with a mean number |z|2 = 49 of photons as a function of the atomic passage time gτ .

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