14 IV. DISCUSSION densate in the cavity cooling regime. Significant contri- butions beyond that of the mean field were found to the photon and particle numbers. The strong depletion of the condensate shows some analogy with the excess noise in lasers [48, 49], already discussed in our previous work [18]. It is interesting however, that the huge nonclassical contribution in the autocorrelation of the photonic and atomic operators do not manifest in the entanglement of these variables. The lack of entanglement can be at- tributed to the big difference of the occupation of these modes. While the atom number was assumed to be in the order of 10 $^{5}$, the photon number ranged in the order of unity. In this paper we have investigated the one-dimensional dynamics of a Bose-Einstein condensate inside a driven optical cavity. As the dispersive atom-photon interac- tion couples the atomic motion to the dynamics of the photonic field in a nonlinear way, strong correlations can appear. The strength of this coupling is inversely pro- portional to the detuning of the pump frequency from the atomic transition, therefore it can be tuned in ex- perimental implementations. We recited and analyzed the mapping of the original system to a two mode ef- fective model in which only the two highest populated one-particle states are kept from a plane-wave expan- sion of the atomic motion [3–5]. By solving the coupled Gross-Pitaevskii equations it was possible to reproduce the bistable behavior caused by the nonlinear coupling [1, 4, 25] and to provide a phase diagram of the system partitioning the whole parameter space into regions with full dynamical stability of the mean-field solution, the cooling region; to a region with a dynamically instabil- ity attributed to cavity heating; and to a region where both a stable and an unstable solutions can exist. We have compared the mean-field solution and the fluctu- ation spectrum of the optomechanical model to that of the model not restricted to the first two highly occu- pied modes. In the cavity heating region, the unstable polariton mode can have a positive imaginary part in the order of a kilohertz giving an evaporation rate of the Bose-Einstein condensate in milliseconds. Such timescale is in the experimental reach. The experimental progress in combining cavity QED systems with ultracold atoms promises an interesting playground to test the manifestation of light-matter in- teractions in the mesoscopic scale. In such systems both the radiation and the atomic part are dynamical enti- ties. The better understanding of their interplay can have impact not just on our knowledge of nonequilib- rium systems, but also on implementations of quantum information processing devices or quantum simulators of other systems. The authors are grateful for Andr´ as Vukics, Claudiu Genes and Helmut Ritsch for useful discussions. We ac- knowledge funding from the National Scientific Fund of Hungary (NF68736, T077629), and from the National Office for Research and Technology (ERC HU 09 OP- TOMECH). G. Sz. acknowledges Spanish MEC projects TOQATA (FIS2008-00784), QOIT (Consolider Ingenio 2010), ERC Advanced Grant QUAGATUA and EU STREP NAMEQUAM. The dispersive atom-photon interaction not just causes cavity cooling or cavity heating but also alters the statis- tics of the constituent subsystems. In the framework of the optomechanical model, second order correlations were also investigated between the radiation field of the cavity and the motional mode of the Bose-Einstein con- V. ACKNOWLEDGEMENTS [10] I. B. Mekhov and H. Ritsch, Phys. Rev. Lett. 102 , 020403 (2009). [1] F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. K¨ ohl, and T. Esslinger, Nature 450 , 268 (2007). [2] Y. Colombe, T. Steinmetz, G. Dubois, F. 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