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quantum entanglement is one of the most mysterious quantum phenomena and has become an indispensable resource for quantum communication and quantum computation . in the past few years
, there has been much effort devoted to the study of continuous - variable ( cv ) quantum communication @xcite , of which the basis is the multi - partite entanglement .
multi - partite cv entanglement is essential and necessary for measurement - based quantum computation and controlled quantum teleportation .
it was predicted theretically and demonstrated experimentally that multi - color cv entanglement can be produced by nondegenerate optical parametric oscillator ( opo ) , optical superlattice and second - order nonlinearity media @xcite .
alternatively , cv entanglement can also be produced in coherent atomic systems , whose entangled beams have narrower linewidths and longer coherence times ; in particular , they can be used for free - space quantum teleportation @xcite , such as , from a satellite to a ground station . such entanglement using four - wave mixing ( fwm )
has been predicted by ref .
all those results above endow us the feasibility and necessity to realize our scheme . based on those processes , we extend the scheme to generate six - partite cv entanglement by intracavity four - wave - mixing cascaded with double such generations in a four - level atomic system .
and we theoretically demonstrate such possibility of multy - partite entanglement using third - order nonlinearity according to the inseparability criterion for multipartite cv entanglement proposed by van loock and furusawa .
this paper is arranged as follows .
the physical model and its hamiltonian are discussed in sec .
iii gives the equations of motion of the system and sec .
iv provides a linearized fluctuation analysis to calculate the measurable output fluctuation spectra .
these output spectra are further discussed in sec .
v and they demonstrate the violation of the optimized vlf criteria and some entanglement characteristics . at last ,
we draw some brief conclusions in sec . vi .
we consider an ensemble of independent four - level atoms inside a resonant optical cavity . the proposed experimental setup for this scheme is shown in fig . 1(a ) .
a vapor cell containing a four - level atomic system is placed inside a fabry perot cavity , using lasers whose frequency are @xmath0 and @xmath1 as pumps , where the virtual energy level diagram of three cascaded fwm effects are shown in fig .
1(b ) . those generated six beams are transmitted out through the coupling mirror 2 with partial tramittion coefficient , then being splitted by the prism .
spatially separated beams can be measured using method as ref .
@xcite to determine the entanglement relationship .    as shown in fig .
1 , the lower levels , @xmath2 , @xmath3 and @xmath4 , of the atoms are coupled to the excited level @xmath5 by two driving lasers of frequencies @xmath0 , @xmath1 .
there are three fwm processes which are indicated in fig .
1(b ) , that @xmath6 and @xmath7 compose the first fwm process , where @xmath6 implies @xmath0 and @xmath1 stimulated from @xmath2 and @xmath3 to virtual levels , @xmath7 implies @xmath8 and @xmath9 falling from virtual levels to @xmath3 and @xmath2 , respectively .
similar are to @xmath10 , @xmath11 , @xmath12 and @xmath13 .
and we suppose all the virtual levels are detuned with @xmath5 .    in present scheme ,
we only concentrate on the entanglement characteristics among optical fields instead of these between atoms and optical fields .
so the interaction hamiltonian of the three cascaded fwm processes could be written as @xmath14 + \textup{h.c . } \ ; , \ ] ] where @xmath15 ( i=1,2,3 ) are the dimensionless nonlinear coupling coefficients and are commonly taken to be a real number , @xmath16 and @xmath17 are corresponding to the light field generation and annihilation operators repectively .    considering the driving fields in the cavity ,
the hamiltonian for the pump beams is given by @xmath18 + \textup{h.c . } \ ; , \ ] ] where @xmath19 ( i=1,2 ) is the classical pump and coupling laser amplitude .
the loss of the @xmath20th mode @xcite in the cavity can be written as @xmath21 \ ; , \ ] ] where @xmath22 stand for the damping rates for the corresponding cavity modes which are related to the amplitude transmission coefficients .
we now consider the full physical system , where the four - level atomic system is contained inside a pumped resonant fabry - perot cavity .
the master equation for the six cavity modes is @xmath23+\sum_{i=1}^{6}l_{i}\hat{\rho } \ ; .\ ] ]    the equations of motion for the six cavity modes can be obtained by solving the fokker - planck equation in the p - representation @xcite as @xmath24 where @xmath25 can be expressed as @xmath26^{\mathbf{t } } $ ] and @xmath27 be expressed as @xmath28^{\mathbf{t } } $ ] , where elements in the vector @xmath29 are real noise terms and have @xmath30 , @xmath31 .
and @xmath32 is the main part of system s evolution , as the form of @xmath33 $ ] , where @xmath34    @xmath35 contains the coefficients of the noise terms and is not calculated here .
@xmath35 can be obtained by factoring the diffusion matrix @xmath36 according to @xmath37 .
we express d as @xmath38 where @xmath39 is a 6@xmath406 zero matrix and the nonzero block is given by @xmath41
now we conduct the stability analysis of the system in order to analyze the output spectral correlations . by neglecting the noise terms in eq .
( 5 ) , we obtain a set of classical equations for the mean values :    and from these we can obtain steady - state solutions .
@xmath42    in order to simplify the calculation , the same damping rate and the same classical pumping laser amplitudes in the cavity are assumed ( i.e. , @xmath43 , @xmath44 , @xmath45 and @xmath46 ) .
similarly , the same nonlinear coupling coefficient are assumed ( i.e. , @xmath47 ) .
threshold can be obtained if @xmath48 .
the values of the two threshold are expressed as following respecively , @xmath49    when @xmath50 , there is only one set of nonzero solutions for all six modes . if @xmath51 , there will be two sets of nonzero solutions for all six modes .
when there exists a threshold the analytical expressions of stationary solutions for different values of the pump amplitude are    \(i ) @xmath52 @xmath53    \(ii ) @xmath54 @xmath55    \(iii ) @xmath56 @xmath57    if @xmath58 , there is no threshold in this system . on this condition
the signal and the idler modes will not be excited no matter how large the pump amplitude is . thus ,
when a threshold does not exist the analytical expressions of the stationary solutions are @xmath59    in the following , we decompose the system variables @xmath60 into their steady - state values @xmath61 and small fluctuations around the steady - state values @xmath62 as @xmath63
. then equation ( 5 ) can be linearized as @xmath64 where @xmath65^{t}$ ] ; m , the drift matrix , is given by    @xmath66    where @xmath67 and @xmath68    if the requirement that the real part of the eigenvalues stay positive is satisfied , the fluctuation equations will describe an ornstein - uhlenbeck process @xcite for which the intracavity spectral correlation matrix is @xmath69    all the correlations required to study the measurable extracavity spectra are contained in this intracavity spectral matrix .
the output fields can be obtained along with the standard input - output relations @xcite . in particular , the output fields spectral variances and covariances have the general form @xmath70 similar expressions can be derived for the @xmath71 quadratures .
in order to investigate multipartite entanglement , and particularly show that the system under consideration demonstrates true six - partite entanglement , we define quadrature operators for each mode as @xmath72 such that @xmath73=2i$ ] .
the conditions proposed by van loock and furusawa @xcite , which are a generalization of the conditions for bipartite entanglement , are sufficient to demonstrate multipartite entanglement .    using the quadrature definitions ,
the six - partite inequalities are @xmath74 where @xmath75 denotes the variance and @xmath76 are arbitrary real parameters used to optimize the violation of these inequalities .
it is important to note that in the uncorrelated limit these optimized vlf criteria approach 4 .
hence , without optimization , some entanglement which is presented may be missed .    from now on , we numerically calculate the values of vlf inequalities , and these are the quantities that can be measured in experiments .
note that in our symmetrical situation , inequalities i2-p1 and s2-p2 are equal ; the same are to p1-s1 and i1-p2 . to simplify , we write the subscript s1-i1 as a , p1-s1 and i1-p2 as b , i2-p1 and s2-p2 as c separately .    when below the threshold or without threshold , the powers of the pump are much higher than others and the steady - state values of the other output beams are all equal to zero .
therefore , the quantum characteristics of the pump and other beams in such situation are not considered .     with + @xmath77 , and @xmath78 . + here @xmath79 and @xmath54 . ]     with + @xmath77 , and @xmath78 . + here @xmath79 and @xmath54 . ]    in fig 2 , we plot the minimum of the inequalities as a function of frequency normalized to @xmath80 with @xmath81 , and @xmath82 . in this case , @xmath83 so that the two supposed thresholds coincide to be one threshold .
it is obvious that the minimal values of inequalities are all less than 4 in a wide range of analysis frequencies , which is sufficient to demonstrate that all the six beams are cv entangled with each other . in general , the smaller the value of the inequality , the stronger the degree of entanglement and quantum correlation is . for large frequencies
the inequalities approaches 4 , which is the result of our optimization . in fig .
2 , we can also see that the value of @xmath84 is comparatively smaller than @xmath85 , which indicates that the degree of entanglement of second - class output is bigger .
it is similar to those situations in the four wave mixing process .    in fig .
3 , we plot the minimum of the inequalities as a function of frequency normalized to @xmath80 with @xmath86 and the other parameters remain unchanged . again , the minima of the inequalities are less than 4 when above threshold , and it verifies that the all field modes are entangled .     with + the parameters unchanged as in fig .
we use the + stationary solution with @xmath87 . ]     with + the parameters unchanged as in fig .
we use the + stationary solution with @xmath87 . ]
however , we are unable to decide the real entanglement when @xmath56 because of the two possible stationary solutions . because the solution of ordinary differential equations ( ode ) usually depends on the initial value of variables . while the variables of ode here are set initially as gaussian states and randomly generated at each specific situation .
it should be expected that all those variables will fall into one of the stationary solutions as the time approaches infinity , with a certain distribution of these two stationary solutions .
therefore , we calculate the minimum of the inequalities beyond the larger threshold @xmath88 using each stationary solution separately .     with + the parameters unchanged as in fig .
we use the + stationary solution with @xmath87 . ]     with + the parameters unchanged as in fig .
we use the + stationary solution with @xmath87 . ]    in the following , we plot the minimum of the inequalities as a function of frequency normalized to @xmath80 using each stationary solution at different @xmath89 , with other parameters unchanged .
we find an interesting result . as shown in fig . 4 and fig .
5 , when @xmath89 is a little bigger than @xmath88 , the stationary solution @xmath90 shows a relatively competitive result , just as results in fig . 2 and fig .
3 . the stationary solution @xmath87 does not work effective and the minimum value of @xmath91 is almost near 4 .
it is because the output beams other than the pump is so lightly occupied if @xmath87 .
in other words , @xmath61 ( i = s1,i1,s2,i2 ) is close to zero .    on the other hand , in fig .
6 and fig . 7 , the situation is reverse . for solution @xmath90
, it seems that , as the intensity of the pump power increases , the gain of other beams begins to saturate .
usually , when the intensity of the pump beams is higher than those of others , their quantum characteristics tend to vanish and the degree of quantum entanglement will be small
. however the solution @xmath87 is greater at this time , which seems to be more effective in a wide range of @xmath89 .     with + the parameters unchanged as in fig .
we use the + stationary solution with @xmath87 . ]     with + the parameters unchanged as in fig .
we use the + stationary solution with @xmath87 . ]    in fig . 8 and fig .
9 , we numerically search the minimum of @xmath92 in all frequencies , as a function of the pump power parameter @xmath89 , also with other parameters unchanged . in these figures
, we can see clearly that the stationary solution @xmath90 is valid when @xmath89 is small just like the usual cases , see ref .
@xcite . and
solution @xmath87 is effective even @xmath89 is tens of times larger than @xmath88 .
this directly confirms previous idea .
we hope this can be considered as an effect of optical bistability .
although we can not confirm the proportion of photons resulting in the stationary solution @xmath87 , it really enlarges the range of @xmath89 to be effective .
we propose a scheme to produce bright six - partite cv entanglement in an optical cavity operating above threshold in a four - level atomic system , which has been theoretically demonstrated using both vlf criterion of multipartite cv entanglement and the input - output relations .
this result is experimentally feasible and can have a larger range of pump power to produce an effective multi - partite entanglement ; it is much well - developed , compared with a basic system using the third - order nonlinearity media .
moreover , the bright six - partite entangled beams could be separated spatially , with narrow linewidths after generation , which have significant use in quantum network and free - space teleportation .
this work is supported by the national natural science foundation of china ( grants no .
601102053 ) , and shanghai jiao tong university prp ( grants no . t030prp19035 ) .
we theoretically show that bright six - partite continuous - variable entanglement can be generated using cascaded four - wave mixing effects of third - order nonlinearity atomic systems above threshold . 
 the six - partite continuous - variable entanglement among the six cavity fields with different frequencies is analyzed by applying optimized inseparability criteria proposed by van loock and furusawa .