12 10 100 1000 10000 20000 24000 28000 32000 36000 N ′ (a) 10 100 1000 10000 20000 24000 28000 32000 36000 N ′ (b) 10 100 1000 10000 20000 24000 28000 32000 36000 N ′ Δ$_{C}$ in units of ω$_{R}$ (c) 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 10 1 20000 24000 28000 32000 36000 n ′ ph (a) 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 10 1 20000 24000 28000 32000 36000 n ′ ph (b) 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 10 1 20000 24000 28000 32000 36000 n ′ ph Δ$_{C}$ in units of ω$_{R}$ (c) FIG. 7: (color online) The depletion of the condensate vs. the cavity detuning of the optomechanical model. All parameters are the same as for Fig. 2. FIG. 6: (color online) The nonclassical part of the photon number vs. the cavity detuning of the optomechanical model. For reference we have also plotted the mean-field solution, | α$_{ss}$ | $^{2}$, with a dashed line. All parameters are the same as for Fig. 2. 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 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 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 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 C$_{k,l}$ ( t ) = 1 2 〈 u$_{k}$ ( t ) u$_{l}$ ( t ) + u$_{l}$ ( t ) u$_{k}$ ( t ) 〉 , (35) C = [ P X X T A$^{]}$ , (36) n ′ ph = 〈 δa † δa 〉 = 〈 δx $^{2}$〉 + 〈 δy $^{2}$〉 − 1 2 = C$_{1}$$_{,}$$_{1}$ + C$_{2}$$_{,}$$_{2}$ − 1 2 . (37)