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the interaction of a collection of atoms with a radiation field has always been an important topic in quantum optics . the dicke model ( dm ) which describes interaction of @xmath0 identical two level atoms with a single radiation field mode , established the importance of collective effects of atom - field interaction , where the intensity of the spontaneously emitted light is proportional to @xmath1 rather than @xmath0 @xcite . the spatial dimensions of the ensemble of atoms are smaller than the wavelength of the radiation field . as a result , all the atoms experience the same field and this gives rise to the collective and cooperative interaction between light and matter . the dm exhibits a second - order quantum phase transition ( qpt ) from a non - superradiant normal phase to a superradiant phase when the atom - field coupling constant exceeds a certain critical value @xcite . the experimental observation of the qpt predicted in the dm required that the collective atom - photon coupling strength to be of the same order of magnitude as the energy separation between the two atomic levels . in conventional atom - cavity setup this condition was impossible to satisfy until it was observed recently in a trapped bose - einstein condensate ( bec ) in an optical cavity @xcite . in the bec setup , the two spin states of the original dm are the two momentum states of the bec which are controlled by the atomic recoil energy and raman pumping schemes . this approach is similar to a novel scheme proposed by dimer et . an important aspect of these experimental developments is the possibility to explore exotic phases mediated by the cavity field . the superradiance phase transition in a bec is accompanied by self - organization of the atoms into a checker board pattern @xcite . interesting equilibrium and non - equilibrium phases have been predicted in the dm with bec @xcite , including crystallization and frustration @xcite , as well as spin glass phase @xcite . multimode dm has also been explored recently , revealing interesting physics such as abelian and non - abelian gauge potentials @xcite , spin - orbit induced anomalous hall effect @xcite , and prediction of the nambu - goldstone mode @xcite . an interesting extension of the bec dicke model is the optomechanical dicke model which has been proposed for detection of weak forces @xcite . in the present paper , we investigate the non - equilibrium properties of the two - atom dicke model , which can be realized by two species bec in an optical cavity . apart from the usual non - equilibrium normal and inverted phases , the dynamical phase diagram reveals a new kind of non - equilibrium mixed phase . this gives rise to a new quantum phase transition from the mixed phase to the superradiant phase by manipulation of the two distinct atom - photon coupling strengths . in addition , we show that a quantum phase transition from the non - superradiant phase to the superradiant phase is not allowed for certain values of the atom - photon coupling strengths of the two set of atoms . [ fig1 ]    we consider two different ensembles of @xmath2 and @xmath3 atoms coupled simultaneously to the quantized field of an optical cavity mode ( fig.1 ) . the two sets of atoms have transition frequencies @xmath4 and @xmath5 while the frequency of the cavity mode is @xmath6 . the cavity is pumped by an transverse external laser with frequency @xmath7 . the light - matter coupling strengths for the two sets of atoms are @xmath8 and @xmath9 . these coupling strengths @xmath8 and @xmath9 can be written as @xmath10 ( @xmath11 ) , @xmath12 is the single atom - cavity mode coupling while @xmath13 is the transverse pump beam rabi frequency . the detuning @xmath14 is considered to be large so as to avoid spontaneous emission . the effective hamiltonian of the system takes the form of a two - atom dicke model with    @xmath15    where @xmath16 is the effective collective spin of length @xmath17 for the two sets of atom and @xmath18 . we now discuss the non - equilibrium dynamics arising from the above two - atom dicke model.the semi - classical equations of motion for the system are given by    @xmath19    @xmath20    @xmath21    @xmath22    @xmath23    here @xmath24 is the decay rate of the cavity photons . in addition the magnitude of pseudo - angular momentum is conserved ,    @xmath25    @xmath26    the long time steady state solutions from the equations of motion can be studied with @xmath27 and @xmath28 . these fixed point solutions can be stable or unstable . separating @xmath29 , @xmath30 @xmath31 , one obtains the steady state equations as    @xmath32    @xmath33    @xmath34    @xmath35    @xmath36    @xmath37    an analysis of these equations leads us to four types of steady states , namely @xmath38 . the state @xmath39 is the normal phase while @xmath40 is the inverted phase . the states @xmath41 and @xmath42 are called mixed phases . as we shall show later that these mixed phases generate interesting non - equilibrium phase diagrams . the critical coupling strength corresponding to the onset of superradiance starting from the normal , inverted or mixed phase is obtained by putting @xmath43 @xmath31 . this leads us to the following possible critical constants    @xmath44    @xmath45    @xmath46    @xmath47    @xmath48    @xmath49    @xmath50    @xmath51    @xmath52    @xmath53    @xmath54    @xmath55    ) in the @xmath56 plane . the parameters used are @xmath57 @xmath58 @xmath59 @xmath58 @xmath60 . the white region is the non - superradiant normal phase while the superradiant phase is indicated by the contours . the darker region in the contour corresponds to low superradiance while light region corresponds to high superradiance . the left plot is the @xmath61 root while the right plot is the @xmath62 root . ] [ fig2 ]    these set of expressions reveals one interesting point that the critical coupling strength for one set of atoms depends on the coupling strength of the other set of atoms . for a given set of @xmath63 and @xmath64 , eqns . [ n1]-[m2 ] determines the boundary between the nonsuperradiant ( normal / inverted / mixed ) and superradiant phase . a trivial manipulation of eqns . [ s1]-[s6 ] leads us to the following equation for @xmath65 and @xmath66 ,    @xmath67    now there are two possibilities depending on whether @xmath68 and @xmath69 or @xmath70 and @xmath71 . the first condition implies that both the set of atoms are in the non - superradiant phase . the second solution corresponds to the case where the first set of atoms are in the non - superradiant normal phase while the second set of atoms are in the superradiant phase . substituting the second expression for @xmath64 from above in the expression for @xmath72,one obtains @xmath73 this implies that by keeping one coupling strength arbitrarily low , one could enter the superradiant phase by manipulating the second coupling strength alone . this point would be more clear when we discuss the dynamical phase diagrams in the next section . in this section , we explore the fluctuation dynamics above the steady state ( fixed points ) . in particular , we will consider the instability of the normal ( @xmath74 ) , inverted ( @xmath75)and mixed phases ( @xmath76 or @xmath77 ) . to this end , we write @xmath78 , @xmath79 @xmath31 , where @xmath80 , @xmath81 and @xmath82 @xmath31 . substituting these into eqns.(4)-(6 ) , one obtains the linearized equations    @xmath83    @xmath84    @xmath85     ( eqns.[o3 ] ) in the @xmath56 plane . the parameters used are @xmath57 @xmath58 @xmath59 @xmath58 @xmath60 . the left plot is the @xmath61 root while the right plot is the @xmath62 root . the white region is the non - superradiant mixed phase @xmath86 while the superradiant phase is indicated by the contours . ] [ fig3 ]    we now write @xmath87 , @xmath88 and @xmath89 and equate coefficients with the same time dependence to obtain algebraic equations for @xmath90 , @xmath91 , @xmath92 , @xmath93 , @xmath94 and @xmath95 . the corresponding self consistency equations yields a quadratic equation for @xmath96 , whose roots characterize the possible instabilities for @xmath97 . these instabilities describe the boundaries in the emerging dynamical phase diagrams . in particular , the various boundaries between exponentially growing and decaying fluctuations are given as :    normal phase :    @xmath98    inverted phase :    @xmath99    mixed phase 1 :    @xmath100    mixed phase 2 :    @xmath101    note that the `` inverted phase '' is the inversion of the `` normal phase '' around @xmath102 boundary while `` mixed phase @xmath103 '' is the mirror inversion of `` mixed phase @xmath86 '' . the contour plot of the stable roots @xmath104 of eqns.[o1 ] as a function of @xmath8 and @xmath9 for the normal phase is shown in fig.2 . fig.2(a ) shows the boundary separating the non - superradiant phase and the superradiant phase for the @xmath61 root . this boundary is the curve that joins @xmath105 ( with @xmath106 ) and @xmath107 ( with @xmath108 ) . below this boundary is the non - superradiant normal phase while above this curve is the superradiant phase . as we move along the @xmath109-axis @xmath110 , we reach the superradiant phase at @xmath111 . this analysis agrees with our steady state analysis of the previous section . thus along the @xmath112 or the @xmath109 axis , the system behaves as if only one set of atoms are present . in any other direction , both set of atoms contribute to the dynamics . note that white region is the non - superradiant normal phase while the superradiant phase is indicated by the contours . the darker region in the contour corresponds to low superradiance while light region corresponds to high superradiance . fig2b shows the plot of @xmath62 root . the combination of @xmath61 and @xmath62 determine the complete boundary between the non - superradiant phase and superradiant phase . the phase diagrams of the inverted phase ( not shown ) is the mirror inversion of the normal phase . in a similar manner , one can determine the dynamical phase diagrams for the mixed phases . in fig.3(a ) and 3(b ) , we demonstrate this for the mixed phase @xmath86 . a new kind of dynamical phase diagram emerges for the mixed phase . the phase diagram now splits into two distinct superradiant regimes separated by the non - superradiant phase . the two critical points @xmath105 @xmath113 and @xmath107 @xmath110 along the @xmath112 and @xmath109 axis are still the same . in fig.3a , @xmath61 root is shown and on moving along the @xmath112 axis ( increasing @xmath8 ) , we encounter the usual superradiant phase with increasing energy . on the other hand , moving along the @xmath109 axis ( increasing @xmath9 ) , we get a superradiant phase of constant low energy . there are regions in the phase diagram , where even when @xmath114 and @xmath115 , the system stays in the non  superradiant phase . interestingly for @xmath116 , the superradiant phase can never be reached . infact the energy landscape in the @xmath56 plane gives an impression of anti - crossing of energy levels . fig.3(b ) shows the plot of @xmath62 root whose behavior is opposite to that of the @xmath61 root . a superradiant phase with decreasing energy is encountered along the @xmath109 axis while a constant high energy phase is encountered along the @xmath112 axis . note that if we choose @xmath117 , then the energy plots of fig.2 and fig.3 becomes asymmetric ( figure not shown ) . the current predictions can be tested in an experiment similar to that of ref.@xcite but with two species condensate . in addition one has to look into the long duration of these experiments beyond the @xmath118 @xmath119 time scale as noted in ref.@xcite in summary , we have investigated the non - equilibrium quantum phases of a two - atom dicke model , which is realized in a collection two set of cold atoms coupling simultaneously to a single quantized cavity mode . within the framework of the non - equilibrium two - atom dicke model , we reveal a rich and new set of phase diagrams . we have shown the existence of a new kind of quantum phase transition from the non - 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in this paper , we investigate the non - equilibrium quantum phases of the two - atom dicke model , which can be realized in a two species bose - einstein condensate interacting with a single light mode in an optical cavity . apart from the usual non - equilibrium normal and inverted phases , a non - equilibrium mixed phase is possible which is a combination of normal and inverted phase . a new kind of quantum phase transition is predicted from non - superradiant mixed phase to the superradiant phase which can be achieved by tuning the two different atom - photon couplings . 
 we also show that a quantum phase transition from the non - superradiant mixed phase to the superradiant phase is forbidden for certain values of the two atom - photon coupling strengths . 
 * keywords : * non - equilibrium dicke model , quantum phase transition .