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it is widely accepted that quantum entanglement is an essential ingredient for the implementation of quantum information processing devices @xcite . creating highly entangled multi - particle states is therefore one of the most challenging goals of modern experimental quantum mechanics . quantum entanglement of two or more systems leads to correlations between observables of the systems that can not be explained on the basis of local realistic theories . quantum entanglement lies at the heart of the difference between the quantum and the classical world . due to its vast application , quantum entanglement has been studied in different systems such as optomechanical systems@xcite , bose einstein condensates ( bec ) trapped in double quantum well @xciteand in optical lattices @xcite . entanglement has also been observed experimentally at nist group @xcite , where entanglement state of four - ions was created successfully by using the scheme proposed by sorensen and molmer @xcite .    in order to build a quantum information network using atomic systems , quantum state exchange between light and matter is an essential ingredient . one prefers matter over photons as quantum memory elements since experimentally it is difficult to localize and store photons . stimulated raman adiabatic passage is considered to be a very convenient technique of storing light and has been used in single atom cavity quantum electrodynamics for transfer of qubits between atoms and photons and for building quantum logic gates @xcite . mapping quantum states to collective atomic spin systems @xcite and quantized vibrational states of trapped atoms @xcite by means of stimulated raman adiabatic passage technology and electromagnetically induced transparency @xcite has been discussed earlier . new possibilities for cavity opto - mechanics may emerge by combining the tools of cavity quantum electrodynamics(qed ) with those of ultracold gases @xcite . placing an ensemble of atoms inside a high - finesse cavity enhances the atom - light interaction because the atoms collectively couple to the same light mode . the motional degrees of freedom of ultracold atomic gases represent a new source of long - lived coherence affecting light - atom interaction . nonlinear optics arising from this long - lived coherent motion of ultracold atoms trapped within a high - finesse fabry - perot cavity was reported recently @xcite . strong optical nonlinearities were observed even at low mean photon number of @xmath0 . this nonlinearity also gives rise to bistability in the transmitted probe light through the cavity . experimental implementation of a combination of cold atoms and cavity qed ( quantum electrodynamics ) has made significant progress @xcite . theoretically there have been some interesting work on the correlated atom - field dynamics in a cavity . it has been shown that the strong coupling of the condensed atoms to the cavity mode changes the resonance frequency of the cavity @xcite . finite cavity response times lead to damping of the coupled atom - field excitations @xcite . the driving field in the cavity can significantly enhance the localization and the cooling properties of the system@xcite . it has been shown that in a cavity the atomic back action on the field introduces atom - field entanglement which modifies the associated quantum phase transition @xcite . the light field and the atoms become strongly entangled if the latter are in a superfluid state , in which case the photon statistics typically exhibits complicated multimodal structures @xcite . a coherent control over the superfluid properties of the bec can also be achieved with the cavity and pump @xcite . recently , a new approach was proposed which is based on all optical measurements that conserve the number of atoms . it was shown that atomic quantum statistics can be mapped on transmission spectra of high - q cavities , where atoms create a quantum refractive index . this was shown to be useful for studying phase transitions between mott insulator and superfluid states since various phases show qualitatively distinct spectra @xcite . motivated by such interesting developments in the field of cavity electrodynamics of bose einstein condensates , we propose in this study a new scheme of entangling two bose - einstein condensates ( bec ) in a double cavity set - up by transferring the entanglement of quantum - correlated light fields produced from a nondegenerate parametric amplifier ( nopa ) to the collective density excitations of the two becs . in this section , we show that the quantum state of the input light field can be transferred to a bec trapped in a cavity . we consider an elongated cigar shaped bose - einstein condensate(bec ) of @xmath1 two - level @xmath2 atoms in the @xmath3 state with mass @xmath4 and frequency @xmath5 of the @xmath6 transition of the @xmath7 line of @xmath2 , strongly interacting with a quantized single standing wave cavity mode of frequency @xmath8 as shown in fig . 1 . the standing wave that forms in the cavity results in an one - dimensional optical lattice potential . the cavity field is also coupled to external fields incident from one side mirror . it is well known that high - q optical cavities can significantly isolate the system from its environment , thus strongly reducing decoherence and ensuring that the light field remains quantum - mechanical for the duration of the experiment . we also assume that the induced resonance frequency shift of the cavity is much smaller than the longitudinal mode spacing , so that we restrict the model to a single longitudinal mode . in order to create an elongated bec , the frequency of the harmonic trap along the transverse direction should be much larger than one in the axial ( along the direction of the optical lattice ) direction . the system is also coherently driven by a laser field with frequency @xmath9 through the cavity mirror . here the two mirrors @xmath10 and @xmath11 are fixed . this system is modelled by a jaynes - cummings type hamiltonian ( @xmath12 ) in the rotating and dipole approximation . is incident from the left mirror and from the same mirror the reflected light from the bec is taken out as @xmath13 . faraday oscillator ( @xmath14 ) enables a unidirectional coupling . ] @xmath15 + i \hbar \sqrt{2 \gamma}(c_{in}c^{\dagger}-c_{in } ^ { * } c ) , \ ] ]    where @xmath16 and @xmath17 are the pauli matrices and @xmath18 and @xmath19 are the large atom - pump and cavity - pump detuning , respectively . . @xmath20 is the space dependent atom field coupling . @xmath21 and @xmath22 are the annihilation and creation operators of the intracavity field . the input laser field populates the intracavity mode which couples to the atoms through the dipole interaction . the field in turn is modified by the back - action of the atoms . the system is open as the cavity field is damped by the photon leakage through the coupling mirror . we are considering a system with large detuning and hence spontaneous emission is ignored and hence we can adiabatically eliminate the excited state using heisenberg equation of motion @xmath23 $ ] , which yields the single particle hamiltonian    @xmath24\ ] ]    the parameter @xmath25 is the optical lattice barrier height per photon and represents the atomic backaction on the field . also @xmath26 is the external classical potential . here we will always take @xmath27 . in this case the condensate is attracted to the nodes of the light field and hence the lowest bound state is localized at these positions which leads to a reduced coupling of the condensate to the cavity compared to that for @xmath28 . along @xmath29 , the cavity field forms an optical lattice potential of period @xmath30 and depth ( @xmath31 ) . we now write the hamiltonian in a second quantized form including the two body interaction term . @xmath32    where @xmath33 is the field operator for the atoms . here @xmath34 is the two body @xmath35-wave scattering length . the corresponding opto - mechanical - bose - hubbard ( ombh ) hamiltonian can be derived by writing @xmath36 , where @xmath37 is the wannier function and @xmath38 is the corresponding annihilation operator for the bosonic atom at the @xmath39 site . retaining only the lowest band with nearest neighbor interaction , we have    @xmath40 j_{o } \sum_{i }   b_{i}^{\dagger } b_{i}+\frac{u}{2 } \sum_{i } b_{i}^{\dagger}b_{i}^{\dagger } b_{i } b_{i}\ ] ]    where    @xmath41    here , we have neglected the tunneling terms @xmath42 and @xmath43 , as they can be made small by varying the optical lattice depth . under this approximation , the matter - wave dynamics is not essential for light scattering . in experiments , such a situation can be realized because the time scale of light measurements can be much faster than the time scale of atomic tunneling . the dynamics of the system can be described by the following quantum langevin equations    @xmath44    @xmath45j_{o}b_{i}-i\frac{u}{\hbar}b^{\dagger}_{i } b_{i } b_{i}-\gamma b_{i}+\xi.\ ] ]    here @xmath46 is the cavity decay rate , @xmath47 is the condensate decay rate and @xmath48 is the classical thermal noise operator of the bec and has the following correlation function at temperature @xmath49 . @xmath50=@xmath51=@xmath52,@xmath53 and @xmath54 . the noise operators for the input field obey the following correlation functions : @xmath55 , @xmath56.the thermal noise input for the bec is provided by the thermal cloud of atoms . we now write each canonical operator of the system as a sum of its steady state mean value and a small fluctuation with zero mean value i.e @xmath57 and @xmath58 and linearize to obtain the following heisenberg - langevin equations for the fluctuation operators . @xmath59    @xmath60b -i u_{eff } b^{\dagger}-i g_{c}(c^{\dagger}c_{s}+c_{s}^{\dagger}c)-\gamma b+ \xi,\ ] ]    here , @xmath61 , @xmath62 , @xmath63 , @xmath64 is the detuning with respect to the renormalized resonance . @xmath1 is the total number of atoms in @xmath65 sites . as before , we assume negligible tunneling ( @xmath66 ) and hence we drop the site index @xmath67 from the atomic operators . we now make the transformations @xmath68 , @xmath69 and @xmath70 . neglecting the fast rotating terms , we get the following equations    @xmath71    @xmath72    if we assume that decay rate @xmath46 of the cavity field is very large so that we can adiabatically eliminate the dynamics of the cavity mode . consequently we have the steady state value of @xmath73 .    @xmath74    on substituting [ steady_1 ] in [ fluc_2 ] and assuming @xmath75 for simplicity we get    @xmath76    here @xmath77 . if @xmath78 , the statistics of the input light field can be transferred to the collective density excitations of the bec . here we now utilize the results obtained in the previous section to show that how we can transfer entanglement of a pair of quantum correlated light fields into a pair of becs in two independent cavity which are physically separated . the scheme is shown in figure 2 . the quantum correlated light fields are generated by a nopa . the output fields are frequency degenerate but polarization nondegenerate . the coupling between the two intracavity field modes @xmath79 and @xmath80 can be represented by @xmath81 . here @xmath82 is the coupling strength that is proportional to the nonlinear susceptibility of the intracavity medium and the intensity of the coherent pump field . we take @xmath83 as the damping rate of the cavity modes . the equations of motion for the mode operators can be written as    @xmath84    @xmath85    where @xmath86 and @xmath87 are the vacuum input fields of the two cavity modes of nopa . the output fields from the nopa follows from the boundary conditions    @xmath88    @xmath89        the quadrature amplitudes @xmath90 , @xmath91 , @xmath92 in fourier space are found as :    @xmath93\ ] ]    @xmath94\ ] ]    the highly correlated light fields from the nopa are incident on the two space separated cavities with two identical becs . we can now write the bec mode operators of the two condensates as    @xmath95    @xmath96    here we assume that the coupling between the nopa and the cavities is unidirectional . if system parameters are chosen such that @xmath97 and @xmath98 can be regarded as quantum white noise operators , then the variances of the positions ( @xmath99 ) and momenta ( @xmath100 ) of the two becs are given by    @xmath101 } \right\}+\frac { \gamma ( 1 + 2 n_{t})}{b}\ ] ]     as a function of @xmath102 for two values of the detuning @xmath103 ( thin line ) and @xmath104 ( thick line ) . the other parameters are @xmath105 , @xmath106 , @xmath107 , @xmath108 and @xmath109 . quantum entanglement is said to exist when @xmath110 . ]    here , @xmath111 , @xmath112 , @xmath113 and @xmath114 . the two identical becs are entangled if @xmath110 . for certain set of parameters , the variances @xmath115 @xmath116 @xmath117 @xmath118 @xmath119 i.e an epr state in the position and momenta of the two becs . as evident from eqn . [ variance_1 ] , the thermal contribution destroys the entanglement . a plot of epr variance @xmath120 as a function of @xmath102 for two values of the detuning @xmath103 ( thin line ) and @xmath104 ( thick line ) is shown in fig . quantum entanglement is said exist when @xmath110 . clearly we see that significant entanglement is achieved when @xmath121 . further we observe that for a large detuning @xmath122 , the entanglement is more but the range of @xmath123 where this entanglement can be achieved is reduced . on the other hand for a smaller detuning , the entanglement is reduced but the range over which entanglement is achieved is enhanced . an epr state can thus be reached for the collective density excitations of the two becs by keeping @xmath124 and @xmath125 .    to demonstrate that the dynamics investigated here are within experimental reach , we discuss the experimental parameters from @xcite : a bec of typically @xmath126 @xmath127 atoms is coupled to the light field of an optical ultra high - finesse fabry - perot cavity . the atom - field coupling @xmath128 @xcite ( @xmath129 @xcite ) is greater than the decay rate of the intracavity field @xmath130 @xcite ( @xmath131 @xcite ) . typically atom - pump detuning is @xmath132 . the rate @xmath133 at which atoms are coupled out of the bec is about @xmath134 @xcite . the kinetic energy and potential energy contribution @xmath135 is about @xmath136 @xcite(@xmath137 @xcite).the energy of the cavity mode decreases due to the photon loss through the cavity mirrors , which leads to a reduced atom - field coupling . photon loss can be minimized by using high - q cavities . our proposed scheme relies crucially on the fact that coherent dynamics dominate over the losses . it is important that the characteristic time - scales of coherent dynamics are significantly faster than those associated with losses . in conclusion , we have proposed a novel scheme to transfer the quantum entanglement between two light fields emerging from a nondegenerate parametric amplifier to the collective density excitations of two bose einstein condensates which are physically separated in two independent optical cavities . an epr state of the collective density excitations can be created by a judicious choice of the parameters . this study could be of use in studying entanglement in macroscopic quantum objects and for high precision metrology . 99 r. f. werner quantum information processing(springer tracts in modern physics ) * 173 * , ( 2001 ) . s. mancini et al . , phys . lett . * 88 * , 12 ( 2002 ) ; m. pinard et al . , europhys . lett . * 72 * 747 ( 2005 ) ; j. zhang et al . a * 68 * 013808 ( 2003 ) . j. chen et al . a * 360 * 429 2007 . l. you phys . letts * 90 * 030402 2003 . c. a. sackett et al . , naturelondon * 404 * , 256 ( 2000 ) . a. srensen and k. mlmer , phys . . lett . * 82 * , 1971 ( 1999 ) ; k. mlmer and a. srensen , phys . lett . * 82 * , 1835 ( 1999 ) ; j. mod . opt . * 47 * , 2515 ( 2000 ) ; a. srensen and k. mlmer , phys . a * 62 * , 022311 ( 2000 ) . a. s .parkins et al . , phys . rev . letts . * 71 * 3095 ( 1993 ) ; t. pellizzari et al . rev . letts . * 75 * 3788 ( 1995 ) ; j. i. cirac et al . letts . * 78 * 3221 ( 1997 ) . a. e. kozhekin et al . , phys . a * 62 * , 033809 ( 2000 ) . parkin and h. j. kimble , j. opt . b : quantum semiclassical opt . , * 1 * , 496 ( 1999 ) . m. d. lukin et al . letts . , textbf84 , 4232 ( 2000 ) . a. bhattacherjee , phys . a , * 80 * , 043607 ( 2009 ) . f. brennecke et al . , science * 322 * , 235 ( 2008 ) . k. w. murch et al . , nature physics * 4 * , 561 ( 2008 ) . b. nagorny , tn . elssser and a. hemmerich , phys . letts . , * 91 * , 153003 ( 2003 ) . j. a. sauer , k.m . fortier , m.s.chang , c.d . hamley and m.s . chapman , phys . a , * 69 * , 051804(r ) ( 2004 ) . anton ttl , stephan ritter , michael khl and tilman esslinger , phys . letts . , * 95 * , 090404 ( 2005 ) . p. horak , s. m. barnett and h. ritsch , phys . a , * 61 * , 033609 ( 2000 ) . p. horak , h. ritsch , phys . a * 63 * , 023603 ( 2001 ) . a. griessner , d. jaksch and p. zoller , j. phys . b,*37 * , 1419 ( 2004 ) . c. maschler and h. ritsch , opt . comm . * 243 * , 145 ( 2004 ) . c. maschler and h. ritsch , phys . * 95 * , 260401 ( 2005 ) . w. chen , d. meiser and p. meystre , phys . a , * 75 * , 023812 ( 2007 ) . a. bhattacherjee , optics communications , * 281 * , 3004 ( 2008 ) . i. b. mekhov , c. maschler and h. ritsch , nature physics , * 3 * , 319 ( 2007 ) .
we propose a scheme to transfer the quantum state of light fields to the collective density excitations of a bose einstein condensate ( bec ) in a cavity . 
 this scheme allows to entangle two becs in a double cavity setup by transferring the quantum entanglement of two light fields produced from a nondegenerate parametric amplifier ( nopa ) to the collective density excitations of the two becs . 
 an epr state of the collective density excitations can be created by a judicious choice of the system parameters .