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[18, 20–25], namely, that the atom-photon cross corre- lations are negligible and mean values of the light and atomic operators can be decoupled. 

The paper is organized as follows. The backbone of the paper is Sec. II, where the model and the theoret- ical description of the system in mean-field level is pre- sented in great detail. Many of the elements of other theoretical models, e.g., the cavity cooling of BEC ex- citations [21], spatial self-organization of a BEC in the cavity [22, 26], and the transient collective atomic recoil lasing [27, 28] will be recapitulated here to give a full ac- count of the mean field dynamics of a BEC in a cavity. The aim, partially, is to reach the optomechanical model and discuss the assumptions and approximations to ar- rive there. Special attention is paid to the effect of nonlin- earities: (i) the nonlinearity caused by atom-light inter- action, responsible for the creation of a periodic optical potential and also for an effective atom-atom collective interaction, and (ii) the nonlinearity caused by atomic s-wave collisions. In Sec. III we present first mean-field results and compare them to experimental observations, wherever applicable. Furthermore, the auto- and cross- correlations of the quantum fluctuations of the fields will be calculated in the stationary state formed by the bal- ance of cavity loss and vacuum noise driving. Finally, a summary is given in Sec. IV. 

## II. THEORETICAL DESCRIPTION OF THE SYSTEM 

The system consists of a single mode, high-finesse op- tical Fabry-P´ erot resonator with a waist much smaller than the cavity length and a sample of dilute, ultracold bosonic atoms prepared to be Bose-Einstein condensed. The condensate is supposed to be cigar shaped along the cavity axis, with a strong transverse confinement. The radiation field inside the cavity is pumped through one of its mirrors by a laser with frequency ω$_{P}$ and wavenum- ber k = ω$_{P}$ /c , with c being the speed of light. The laser frequency is far detuned from the atomic transi- tions, so the populations of the electronic excited states are negligible. In this limit the atomic internal degrees of freedom are frozen; and the atom-light interaction is purely dispersive. On the other hand, the cavity fre- quency ω$_{C}$ lies close to the pump frequency ω$_{P}$ : the de- tuning Δ$_{C}$ = ω$_{P}$ − ω$_{C}$ is comparable to κ , this latter being half of the inverse lifetime of a photon inside the cavity. 

In the frame co-rotating with the pumping laser field the Hamiltonian of the system can be approximated as 

<img src='content_image/391589.jpg'>

## A. Hamiltonian 

H$_{A}$ is the Hamilton operator of the ground state atoms inside the cavity, given by 

<img src='content_image/391573.jpg'>

with m being the mass of the atoms, V$_{ext}$ ( x ) is the exter- nal confining potential along the cavity axis and g being the s-wave scattering constant in 1-dimension. The term H$_{C}$ of the Hamiltonian describes the radiation field of the empty, single-mode cavity, 

The dispersive interaction between the cavity radiation field and the atoms in this low excitation limit is given by the AC-Stark shift, or light shift: 

<img src='content_image/391575.jpg'>

with U$_{0}$ being the single atom light-shift, U$_{0}$ = g 2 $_{CA}$/ Δ$_{A}$; the unique longitudinal mode function of the single mode cavity is cos( k x ) with wavenumber k = ω$_{P}$ /c = 2 π/λ . The part describing the coupling of the cavity field to that of the pump laser is given by 

with η being the strength of the driving field; and the asterisk stands for complex conjugation. The last part of the Hamiltonian, H$_{vac}$ describes the interaction of the cavity field with the broadband reservoir of external ra- diation field modes via the partially transmissive mir- rors. We will give account for this interaction within the Markov approximation, by means of introducing a loss rate 2 κ and a Gaussian white noise operator ξ ( t ) in the Heisenberg equation of motion for the field operators [29]. 

The equation of motion of the photonic annihilation operator is given by 

<img src='content_image/391587.jpg'>

with U ( x ) = U$_{0}$ cos $^{2}$( k x ) the local single atom light-shift, which is a periodic function. Its period is noticeably L = λ/ 2, since it contains the mode function squared. The operator ξ ( t ) describes Gaussian white noise with zero mean and with the only non-vanishing correlation 

It is nicely exhibited in Eq. (2a), that the dispersive inter- action between the atoms and the radiation field causes a shift in the resonator frequency proportional to the 

<img src='content_image/391574.jpg'>

<img src='content_image/391576.jpg'>

## B. Equations of motion 

<img src='content_image/391588.jpg'>

