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the help of the field operators R . We assemble them into the following row vector: u = ( δx, δy, δX, δY ) $^{T}$. With the quadratures being hermitian, one can define a real correlation matrix by 

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which is in the following block form 

with P representing the correlations of the photonic de- gree of freedom, A representing atomic fluctuations, and X describing the cross correlations. For example, on top of the mean field contribution | α | $^{2}$, there is a nonclassical part of the photon number which is given by 

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In an empty resonator close to zero temperature, where 〈 ξ † ( t ) ξ ( t ′ ) 〉 = 0, the nonclassical part of the photon num- ber (37) vanishes, and the resonator field is in a pure 

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coherent state. The fluctuation of the quadratures are distributed equally and the Heisenberg uncertainty prin- ciple is fulfilled in a sharp sense ( 〈 δx $^{2}$〉 = 〈 δy $^{2}$〉 = 1 / 2). However, due to atom-photon interaction, the photon fluctuations couple to those of the atomic motion. By iterative substitution of Eqs. (32) one can simply check that n ′ ph = 〈 δa † δa 〉 is no longer zero in the presence of a Bose-Einstein condensate. The photon field is no longer in a purely coherent state, and the correlation matrix P is not simply half of the unit matrix. Indeed, since δa couples now to δa † (through δ ˜ c ) and the coupling term is proportional with α 2 $_{ss}$, which is complex, the correlation matrix P is not isotropic. The angle of the major axis coincides with twice the phase of α$_{ss}$ , taking the values from − π (when the system is far from the resonance) to zero (at resonance). By performing a rotation of the correlation matrix by this angle to bring it into a diag- onal form, one of the eigenvalues will remain 1 / 2, as in the pure coherent-state case, while the other one will al- ways become bigger than 1 / 2. Since the trace of a matrix is invariant under rotations, the nonclassical part of the photon number n ′ ph = ( λ$_{>}$ − 1 / 2) / 2, with λ$_{>}$ being the bigger eigenvalue of P . Figure 6 shows the nonclassical contribution to the photon number. The classical average | α$_{ss}$ | of Fig. 2 is also plotted here with a dashed line for 

