By Cvitanovic P., Artuso R., Dahlqvist P.
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If the atomic density matrix has off-diagonal elements, the above formalism breaks down. The reduced cavity density matrix will then also develop off-diagonal elements, even if initially it is diagonal. We shall not go further into this question here (see for example Refs. –). 17) describes the pumping of a lossless cavity with a beam of atoms. After k atoms have passed through the cavity, its state has become M k p. In order to see whether this process may reach statistical equilibrium for k → ∞ we write Eq.
15 here. Notice that for θ > θ1 the logarithm of the correlation length grows linearly with N for large N . 10. 6).
The eigenvalues are λn = n(1 − q (x0)), n = 0, 1, . , and grow linearly with n. This may be observed in Figure 9. The correlation length becomes ξ= 1 1 = 1 − q (x0 ) 1 − φ cot φ 59 for 0 < φ < π . 24) As in the thermal phase, the correlation length is independent of N (for large N ). 5 Mean Field Calculation We shall now use a mean field method to get an expression for the correlation length in both the thermal and maser phases and in the critical region. 25) or with ∆x = 1/N 1d x = q(x + ∆x) + nb ∆x − x γ dt .
Classical and quantum chaos by Cvitanovic P., Artuso R., Dahlqvist P.