# Quantum noise of a Bose-Einstein condensate in an optical cavity, correlations and entanglement 

$^{1}$Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary and 

$^{2}$ICFO-Institut de Ci` encies Fot` oniques, 08860 Castelldefels (Barcelona), Spain 

G. Szirmai 1 , $^{2}$, D. Nagy$^{1}$, and P. Domokos1 

A Bose-Einstein condensate of ultracold atoms inside the field of a laser-driven optical cavity exhibits dispersive optical bistability. We describe this system by using mean-field approximation and by analyzing the correlation functions of the linearized quantum fluctuations around the mean- field solution. The entanglement and the statistics of the atom-field quadratures are given in the stationary state. It is shown that the mean-field solution, i.e. the Bose-Einstein condensate is robust against entanglement generation for most part of the phase diagram. 

Nonlinear open systems often produce bistabilities or dynamical phase transitions. A nice example is the be- havior of a Bose-Einstein condensate (BEC) in a pumped high-finesse optical cavity, where the nonlinearity is pro- duced by the dispersive atom-light interaction [1–8]. The weak cavity pumping causes a classical electromagnetic field to build up between the mirrors. The atoms cou- pled dispersively to the radiation field detune the cavity according to the overlap of their spatial distribution with the mode function of the electric field. Consequently, the cavity resonance frequency can be shifted away from or towards the frequency of the pumping laser; thereby a big variation in intensity can be induced merely by the spatial redistribution of the atoms. In turn, the inten- sity change translates into the variation of the depth of the optical dipole potential, and so it acts back upon the atomic distribution itself. In a tiny region of the pa- rameter settings close to the cavity resonance two stable or metastable configurations can exist, giving rise to a dynamical phase transition. 

Atom-light interaction itself is a major issue in the studies of ultracold quantum gases, being the most uni- versal tool in accessing the properties of the system either by slowing the cloud of atoms, trapping them in classi- cal potentials, putting obstacles in their path, or finally detecting and imaging them. Moreover, recent proposals have raised the possibility of quantum state preparation of the atomic ensemble with the help of measuring the output photon signal of a pumped optical resonator [9– 13]. The cornerstone of such a quantum state preparation with the help of a quantum nondemolition measurement is also the mutual backaction between the atomic and photonic degrees of freedom. The study of correlations between atomic motion and light generated by atom-light interaction in an optical cavity is therefore of fundamen- tal importance. 

An important research area on the manifestation of light-matter interaction is optomechanics, where the ra- diation pressure force of a single mode Fabry-P´ erot res- 

PACS numbers: 03.75.Gg,42.50.Wk,67.85.Hj 

## I. INTRODUCTION 

onator is used to manipulate the center of mass motion of a mechanical oscillator. For a short review on optome- chanical systems and on their experimental realizations see Ref. [14], and also references therein. The reason of popularity of optomechanical studies, besides exper- imental realizability, is that the theoretical description can be performed relatively simply, involving only the few modes of the cavity field and one mode for the mo- tion of the mirror [15–17]. Such a paradigmatic system is an ideal playground to test correlations between light and mesoscopic objects, to understand the underlying physics and to speculate on possible applications in quantum in- formation processing. 

In recent experiments done with ultracold bosons in optical resonators the above concepts unify nicely [1–8]. The cavity radiation field couples to a single collective motional excitation of the Bose condensed atomic sam- ple. Starting the experiment with a pure Bose-Einstein condensate, other motional excitations can be safely dis- regarded and so a situation analogous to optomechan- ics can be realized without a movable mirror, but rather with the collective motion of an ensemble of atoms. The difference between the experimental tools (manipulation and detection methods) of traditional optomechanical systems and those with ultracold gases nicely comple- ment each other, while the theoretical description is very similar. 

The aim of the present paper is to discuss correlations generated in a hybrid system of ultracold bosons and the radiation field, especially close to the region of parame- ter settings where the system shows bistability and where the optomechanical simplification can be harvested. Pho- tons leaking out of the resonator make the cavity field noisy, which infiltrates the dynamics of atomic motion [5, 18, 19]. In turn, quantum fluctuations of the atom field have a back-action on the photon statistics. Cor- related fluctuations of the light and matter wave fields appear then, which are strongly enhanced close to the critical regime of bistability. The study of correlations is further motivated by the need for justifying the ba- sic assumption of the generally used mean-field theories 

