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condensates of resonantly pumped exciton - polaritons in semiconductor microcavities constitute a novel and exciting system for the study of fundamental physical properties of superfluids out of equilibrium  @xcite , and for future device applications  @xcite . being quantum superpositions of light and matter they are privileged candidates for the realization of the next generation of optical devices  @xcite , for example for quantum information technologies .    from the point of view of the fundamental physical properties , particularly interesting is the case of a coherently pumped polariton superfluid in presence of defects . here , in contrast to the corresponding equilibrium case , a weak residual drag force is always present even at extremely high polariton densities  @xcite . nevertheless , properties paradigmatical of an equilibrium superfluids , such as frictionless flow of polariton bullets  @xcite , quantized vortices and metastable persistent flow  @xcite , and the appearance and disappearance of cerenkov - like waves  @xcite have been recently observed in coherently driven exciton - polaritons .    in view of the potential device applications , especially important is the unique versatility of the polaritonic system , which , combined with its high non - linear properties , have been already demonstrated to produce parametric scattering  @xcite and bistability  @xcite . here , the implementation of logic operations and gates comes in a natural way : by manipulating the non - linear properties of the system using several lasers which frequencies , angles of incidence , and intensities can be freely varied externally .    in this paper we investigate a new realization of a two - component polariton system coherently driven by two lasers with independently tunable frequencies , angles of incidence and intensities . firstly , we study the stability of the two polariton components when the two laser intensities are varied . differently from the case of a single laser pump , where the system can only be bistable , we disclose a rich phase diagram , where either one , two or three stable states can coexist at given pumping conditions . then we suggest possible easily tunable multistable hysteresis cycles when the two pumping lasers are varied up and down in intensities . alternatively , a multicomponent polariton fluid can be realized by considering the polarization degrees of freedom . multistability of different polariton spin states has been recently proposed theoretically  @xcite and confirmed experimentally  @xcite by the observation of three stable spin states for a given excitation condition . for the case of two polarized components , multistability in space have also been theoretically proposed  @xcite and experimentally observed  @xcite . for the system with two pumping lasers presented here , the same kind of spatial multistability is expected but with much more complex features . additionally , superimposed to the spatial multistability , interference fringes will appear due to the difference in frequency and momentum of the two pumping lasers . the analogy with the two - component polarized case , suggests that , aside from the interest in investigating multistability , two - component polariton condensates obtained with independent lasers can also be used to realize switches  @xcite and memories . the paper is organized as follows : in section ii , we present the model used to describe the steady state behavior of polaritons excited by two continuous - wave lasers with different frequencies , wave - vectors and intensities . the results obtained within a linear response framework are shown in section iii . in this section we study the number of the possible solutions , their nature and possible kerr or parametric instabilities associated to them . as a consequence of the presence of multiple stable solutions , different cycles of hysteresis can be produced by varying , along different paths , the intensities of the two pumping lasers . finally , section iv contains the conclusions drawn from our analysis . the dynamics of resonantly - driven microcavity polaritons  @xcite can be described via a gross - pitaevskii equation for coupled cavity ( @xmath0 ) and exciton ( @xmath1 ) fields generalized to include decay and resonant pumping ( @xmath2 ) : @xmath3    \begin{pmatrix } \psi_x \\ \psi_c \end{pmatrix}\ ; . \label{eq : model}\ ] ] the repulsive ( @xmath4 ) exciton - exciton interaction induces a non - linear dynamics of the eigenmodes of the single polariton hamiltonian ( lower and upper polariton , @xmath5 ) : @xmath6 here , we assume the cavity dispersion to be quadratic , @xmath7 , with @xmath8 ( @xmath9 is the bare electron mass ) , we will neglect the exciton dispersion and consider the case of zero detuning at normal incidence , @xmath10 . the rabi frequency @xmath11  @xmath12mev@xmath13 $ ] and the excitonic and photonic decay rates , @xmath14  @xmath12mev@xmath13 $ ] are chosen in the range of experimental values . because of the continuous decay , a stationary state requires a continuous injection of photons . here , we consider two continuous - wave laser fields , @xmath15 with independently tunable frequencies @xmath16 and momenta @xmath17 , which can be experimentally changed by changing the laser angle of incidence with respect to the growth direction . we study the mean - field solutions of eq . @xmath18 and their stability with respect to small fluctuations within a linear response analysis . substituting the expression   into   we obtain 4 contributions , two of which oscillate at the main frequencies @xmath19 and @xmath20 and the additional two at the replica ( or satellite state ) frequencies @xmath21 and @xmath22 , where @xmath23 . similarly to what is done in the opo regime  @xcite where replica states in addition to the pump signal and idler states are neglected , here , we consider only the terms oscillating at the main frequencies @xmath19 and @xmath20 . later , see eq . , we analyse the dynamical stability of the two - pump - frequency solution against the weak population of satellite states @xmath24 via parametric scattering processes . through the paper , we will consider only dynamically stable two - pump - frequency solutions . in this approximation , we obtain the following mean - field equations for @xmath25 : @xmath26\psi_{1_x}^{ss}+      \frac{\omega_r}{2}\psi_{1_c}^{ss}=0 \\    [ \omega_c({\bf k}_1)-\omega_1-i\kappa_c]\psi_{1_c}^{ss}+    \frac{\omega_r}{2}\psi_{1_x}^{ss}+f_1 = 0 \\    [ \omega_x-\omega_2-i\kappa_x+g_{21}]\psi_{2_x}^{ss}+    \frac{\omega_r}{2}\psi_{2_c}^{ss}=0 \\    [ \omega_c({\bf k}_2)-\omega_2-i\kappa_c]\psi_{2_c}^{ss}+    \frac{\omega_r}{2}\psi_{2_x}^{ss } + f_2 = 0 \ ; , \end{cases } \label{eq : stead}\ ] ] where @xmath27 with @xmath28 . note that the repulsive interaction term between excitons in different states is two times larger the interaction term between excitons in the same mode , resulting in a non - uniform blue - shift . the mean - field system of equations   can have up to 9 solutions , i.e. 6 solutions more than in the case of one pumping laser , but , as discussed below , only a maximum of 3 solutions are stable .    the dynamical stability of the two - pump - frequency mean - field solution can be established by adding small fluctuations , @xmath29 + \\ e^{-i\omega_2 t }    \left[e^{i{{\mathbf k}}_2\cdot{{\mathbf r } } } \psi^{ss}_{2_{x , c } } +      \theta_{2_{x , c}}({{\mathbf r , t}})\right ] \ ; , \label{eq : fluct}\end{gathered}\ ] ] where the fluctuation fields can be divided into particle - like and hole - like excitations @xmath30 $ ] . expanding eq .   up to linear terms in @xmath31 , we obtain 4 terms oscillating at frequencies @xmath32 and @xmath33 , which we neglect , and 4 terms oscillating at @xmath24 . in other words , we are checking the stability of our solution , where only the two states with frequencies @xmath16 are occupied , against the weak population of the satellite states @xmath24 which can be populated by parametric scattering processes . the fact that we consider only linear terms in @xmath34 and @xmath35 implies that we can obtain only the threshold conditions for such parametric processes , as well as the nature of the instability , whether of kerr - type or parametric - type  see later . the equations for @xmath34 and @xmath35 can be written as an eigenvalue equation rearranging the excitations into an 8-component vector @xmath36 : @xmath37    \mathbb{u}_{{{\mathbf k } } } = 0\ ; . \label{eq : eigen}\ ] ] here matrices @xmath38 with @xmath39 are given by @xmath40 and @xmath41 are given by    @xmath42    with @xmath43 being the total excitonic density . at given values of the pumping strength @xmath44 and @xmath45 , the solutions of the mean - field equations   are stable if all the eight eigenvalues ( @xmath46 , @xmath47 ) of eq . have negative imaginary part for every value of the momentum @xmath48 . for some choices of the system parameters , we find that the number of stable solutions can be larger than one . in the case of one pumping laser , the typical @xmath49-like shape dependence of the polariton field intensity on the pump strength , also referred to as optical bistability , can be explained in terms of the non - linear blue - shift induced by the polariton - polariton interaction  @xcite . when the laser frequency is well above the bare lower polariton dispersion , @xmath50 , and the pump intensity increases from low values , the polariton population remains small because it is hard for the laser to inject polartions with a different energy . however , increasing the pump power , the blue - shift pulls the polariton energy towards resonance with the pump causing the population to grow superlinearly and eventually to abruptly jump to a high value when the pump intensity reaches a critical value @xmath51 . in the opposite situation , when the laser intensity is decreased from high values , the polariton energy is blue - detuned close to the pumping laser frequency and , therefore , the cavity is efficiently filled by the laser even at low pumping intensities . in this case the polariton population jumps down back at low polariton densities for a value @xmath52 of the pump strength lower than @xmath51 . the two jumps at different values of the pump intensity cause therefore a hysteresis cycle . as explained below , in the case of two - component fluids , the situation becomes even richer . [ mev@xmath53 . white , green , yellow and black regions correspond to respectively zero , one , two or three stable solutions . in panels i and ii @xmath54  @xmath55m@xmath56 and @xmath57  @xmath55m@xmath56 while in panels iii and iv @xmath58  @xmath55m@xmath56 and @xmath57  @xmath55m@xmath56 . in the left panels ( i and iii ) @xmath59  @xmath12mev@xmath13 $ ] , while in the right panels ( ii and iv ) @xmath60  @xmath12mev@xmath13 $ ] . the horizontal black lines lies at the three fixed values of @xmath61 corresponding to the three panels of fig . [ fig : cuts ] , while the blue diagonal line is the path used to plot fig . [ fig : hyst ] . 3d panels : plots of @xmath62 [ mev ] as a function of @xmath63 with parameters equal to panel i. stable solutions with higher populations are shown in green , stable solution with the second higher population in yellow and third stable solution with lower population in black . all the solutions are shown in the left panel . since the upper green branches hide a yellow upper branch , the right panel shows only the yellow and black solutions.,title="fig : " ]   [ mev@xmath53 . white , green , yellow and black regions correspond to respectively zero , one , two or three stable solutions . in panels i and ii @xmath54  @xmath55m@xmath56 and @xmath57  @xmath55m@xmath56 while in panels iii and iv @xmath58  @xmath55m@xmath56 and @xmath57  @xmath55m@xmath56 . in the left panels ( i and iii ) @xmath59  @xmath12mev@xmath13 $ ] , while in the right panels ( ii and iv ) @xmath60  @xmath12mev@xmath13 $ ] . the horizontal black lines lies at the three fixed values of @xmath61 corresponding to the three panels of fig . [ fig : cuts ] , while the blue diagonal line is the path used to plot fig . [ fig : hyst ] . 3d panels : plots of @xmath62 [ mev ] as a function of @xmath63 with parameters equal to panel i. stable solutions with higher populations are shown in green , stable solution with the second higher population in yellow and third stable solution with lower population in black . all the solutions are shown in the left panel . since the upper green branches hide a yellow upper branch , the right panel shows only the yellow and black solutions.,title="fig : " ]    we fix both laser frequencies to be blue detuned with respect to the bare polariton dispersion : @xmath64  @xmath12mev@xmath13 $ ] , with @xmath54  @xmath55m@xmath56 and @xmath57  @xmath55m@xmath56 . we plot in panel i of fig . [ fig:2d3d ] the phase diagram showing the regions with a different number of stable solutions ( either one , two or three ) as a function of the two rescaled pumping intensities @xmath65 [ mev@xmath53 . in order to understand better the structure of this phase diagram , we show in fig . [ fig : cuts ] the total exciton density @xmath66 when the pump intensities @xmath61 is kept constant at different values and @xmath67 is varied . when the constant pump @xmath68 has a small value ( fig . [ fig : cuts ] top left panel ) , the dependence of the population on the varying pump intensity @xmath67 is similar to the one - fluid case showing bistability with an @xmath49-like shape . for higher values of the constant pump @xmath61 ( fig .  [ fig : cuts ] lower left panel ) , the number of possible solutions increases but just two are found to be stable . finally when @xmath61 is further increased the set of possible solutions further goes up but only a maximum of @xmath69 are found to be stable . @xmath12mev@xmath13 $ ] ( red dotted curves unstable solutions , black lines stable solutions ) for fixed pump intensities as a function of @xmath67 , for @xmath70 @xmath12mev@xmath71}$ ] ( top left ) , @xmath72 @xmath73 $ ] ( bottom left ) , and @xmath74 @xmath12mev@xmath75 $ ] ( right ) . points @xmath76 ( @xmath77 ) , @xmath78 correspond to the cases when the lower ( upper ) branch of the stability curve becomes unstable ( see fig .  [ fig : imaginary1 ] and  [ fig : imaginary2 ] ) . point @xmath79 does not have a counterpart in the one - fluid case and correspond to the cases when the second high branch of the stability curve becomes unstable ( see fig . [ fig : imaginary2 ] ) . ] the coexistence of three solutions , corresponding to the black regions of fig . [ fig:2d3d ] , can be understood as follows : when the two pump intensities increase from low values , the polariton population is small because it s energy is far below the laser frequencies , resulting in one stable solution . in the opposite situation , when the intensity of one of the lasers decreases from high values , the polariton population is high and its dispersion is significantly blue detuned with respect to the bare one . such blue - shift can be sustained by any of the two lasers , thus giving two additional stable solutions for the same values of the pump intensities . therefore a maximum of three stable solutions can be expected . this is also evident while considering the partial densities for particular values of the pump strength at which three stable solutions are present ( black region of fig .  [ fig:2d3d ] ) , e.g. @xmath80 @xmath12mev@xmath75 $ ] and @xmath81 @xmath12mev@xmath75 $ ] . here , the solution with lower total polariton density corresponds to partial densities @xmath82  @xmath12mev@xmath13 $ ] and @xmath83  @xmath12mev@xmath13 $ ] . the other two solutions correspond to a high value of just one of the two partial populations : @xmath84  @xmath12mev@xmath13 $ ] and @xmath85  @xmath12mev@xmath13 $ ] in one case and @xmath86  @xmath12mev@xmath13 $ ] and @xmath87  @xmath12mev@xmath13 $ ] in the other . note that this situation is similar to the case of two - component condensates obtained with two spins . however , while in the spin - dependent case the two lasers pump the two spin populations with different intensities but at the same angle and energy , here the two pumps are independent also in angle and in energy . this analogy is also visible in eq .  . however , the difference between our system of equations and the spin - dependent case is that here the interaction between different components is twice the interaction between particles in the same component .    in panel ii of fig . [ fig:2d3d ] we plot the phase diagram for the same parameters as in panel i but with the two pumping lasers @xmath88 @xmath12mev@xmath13 $ ] blue detuned with respect to the bare lp branch . we see that the effect of the increased detuning is simply to stretch the phase diagram . since the two pumps are further apart from the lp branch , it is more difficult to inject polaritons into the cavity , and thus the need for higher pump intensities . in panel iv of fig . [ fig:2d3d ] we show that a similar phase diagram can be obtained by changing @xmath48 vector of pump 1 from @xmath89 to @xmath90  @xmath55m@xmath56 . we observe that the multistability is quite robust with respect to the choice of the parameters and , therefore , it should be within an experimental reach . an interesting configuration is plotted in panel iii of the same figure . here a region with no stable solutions appears in the central part of the plot ( white region ) . the instability of this region can be understood by noting that with a pump at @xmath91 and just slightly blue detuned from the lp and a pump at @xmath92 , close to the inflection point of the lp , it is easy to satisfy phase matching conditions for parametric scattering processes . for this set of parameters the system is in a configuration unstable towards the population of satellite states by scattering processes . for the other three sets of parameters , shown in the remaining three panels of fig . [ fig:2d3d ] , it is also possible to find regions of the phase diagram where no solutions are stable . these are the regions where the proposed solution , given by eq . [ eq : meanf ] , where only the two frequency states , @xmath19 and @xmath20 , are populated , is not a stable solution because satellite states start also to be populated  our analysis giving the threshold for this to happen .    . the three panels correspond to points @xmath76 with @xmath78 of fig . [ fig : cuts ] , where the lower part of the stability curves become unstable . in blue ( red ) the parts corresponding to the scattering of two particles with @xmath93 ( @xmath94 ) . ] to further discuss the stability of the system with respect to small perturbations , we plot the dispersion of the imaginary part of the excitation eigenfrequency @xmath95 for several points of the stability curves shown in fig .  [ fig : cuts ] . we start with the cases where the lower branch of the stability curve became unstable at points @xmath76 . for very small values of @xmath61 the imaginary part of the dispersion ( top panel of fig . [ fig : imaginary1 ] ) shows two peaks for given values of @xmath48 . one peak lies at higher value @xmath96 @xmath55m@xmath56 and one peak at lower value @xmath97 @xmath55m@xmath56 with @xmath98 . this two - peaks structure is a precursor of a parametric instability due to the scattering between two particles in the component of the condensate with momentum @xmath99 . this situation corresponds exactly to the case of one component fluids . when the pump intensity @xmath61 is slightly increased ( middle panel ) we observe 6 other peaks appearing in the imaginary part of the dispersion . two of these new peaks are such that @xmath98 ( blue lines ) while the other four can be combined to identify two different scattering processes with @xmath100 ( red curves ) . this more complicated structure of the imaginary parts of the eigenvalues is consistent with the fact that with two components a richer mechanisms of scattering might occur . when the intensity of @xmath61 is further increased ( lower panel ) , still 4 different scattering may occur but , in this case , it is the scattering between two particles with @xmath101 that induce the instability of the system .    . the three panels correspond to points @xmath77 with @xmath78 and @xmath79 of fig . [ fig : cuts ] , where the higher part of the stability curve becomes unstable . ]    for the transition from stable to unstable regions of the higher branch of the stability curve we plot the dispersion of the imaginary part of the excitation eigenfrequency @xmath95 for points @xmath77 in fig . [ fig : imaginary2 ] . in analogy with the case of fluids with one component , for low intensities of pump 2 , the imaginary part of the dispersion shows a peak at the wavevector of pump 1 , a clear precursor of a kerr instability . when the intensity of pump 2 is increased new peaks appear in the imaginary part . in the case of point @xmath102 two new peaks are precursors of a parametric instability for the state with @xmath103 @xmath104 $ ] even if the mechanism responsible for the instability of the solution is still of kerr type . it is only when the pump intensity @xmath61 is further increased ( point @xmath105 ) that the two peaks at @xmath106 and @xmath107 @xmath104 $ ] became more important and the mechanism of instability is of the parametric type . finally in the lower right panel ( corresponding to point c ) of fig . [ fig : imaginary2 ] a single peak at @xmath93 is the precursor of a kerr - type instability that ends the region with three stable solutions in the right panel of fig . [ fig : cuts ] . ( color online ) hysteresis cycles of @xmath108 [ dimensionless ] as a function of @xmath67 for different values of @xmath61  stable solutions are in black , while the hysteresis cycle performed by the system is in brown . panel a : @xmath109 as in the blue line of fig . [ fig:2d3d ] . panel b : @xmath110 @xmath12mev@xmath75 $ ] as in the horizontal black line of fig . [ fig:2d3d ] . ] multistability also manifests itself in a hysteresis loop for the populations and emission intensities obtained with a cycle of first increasing and later decreasing the pumping intensities . here , the presence of three stable solutions gives more complicated loops than the ones obtained for a bistable system in a single - component polariton fluid . in order to study this aspect , we calculate the exciton emission intensity at a given frequency @xmath111 normalized to the total exciton emission intensity , @xmath112 , along a closed path of varying pumping intensities . the panels a ) and b ) of fig .  [ fig : hyst ] respectively show the hysteresis cycles of @xmath113 when the two pump intensities change along either the blue or the higher horizontal black line of fig . [ fig:2d3d ] . in panel a ) , one starts from a low value of @xmath67 taking @xmath114  @xmath12mev@xmath75 $ ] so that the population of state @xmath115 is much higher than the population of state @xmath116 , i.e. @xmath117 . increasing @xmath67 the two populations smoothly evolve until @xmath118 @xmath12mev@xmath75 $ ] and @xmath119 @xmath12mev@xmath75 $ ] . at this point @xmath61 is too weak to sustain high population densities in state 2 and , therefore , the system jumps to a new stable configuration , in which the populations of both states are low , i.e. @xmath120 . a further increase of @xmath67 produces a smooth evolution of the two populations until @xmath121 @xmath12mev@xmath75 $ ] when the system jumps to a third configuration with a population in state 1 much higher than in state 2 , i.e. @xmath122 . when we revert the variation of the pumping intensities along the same path , the jumps to states corresponding to intermediate and low values of @xmath113 are shifted to the left of the ones just described for increasing @xmath67 . the multistable hysteresis loop shown in fig . [ fig : hyst]a ) is related to the fact that the two pumping lasers are at different pumping angles , @xmath123 , and pumping frequencies , @xmath111 . therefore the jumps from low to high population for each component appear at different values of the pumping intensities , producing the multistable behavior of @xmath113 . a similar situation occurs when the system evolves along a path on which one of the pumping intensities remains constant , while the other varies ( black horizontal line at @xmath74 of panel 1 in fig  [ fig:2d3d ] ) , as shown in fig . [ fig : hyst]b ) . starting the with @xmath124 , @xmath113 increases smoothly from zero following the lower branch up to @xmath125 @xmath12mev@xmath75 $ ] . at this point @xmath113 jumps from values of the order of @xmath126 to @xmath127 , corresponding to a population of state 1 being smaller but non - negligible compared to state 2 . as @xmath67 further increases up to @xmath128 @xmath12mev@xmath75 $ ] , @xmath113 again jumps abruptly to values of the order of @xmath129 . in the reverse process , @xmath67 is decreased completing the loop . also in this case , the jumps from high to low values of @xmath113 are shifted to the left because of the different angles and energies at which the lasers are pumping the two component of the fluid . it is worth noting that the length and the height of the different plateaus of the hysteresis loops can be efficiently tuned by carefully choosing the path followed by the intensities , angles and frequencies of the pumping lasers . to summarize , we have studied the stability of a two - component exciton - polariton fluid under resonant excitation of two pumping lasers with independently tunable frequencies , angles of incidence and intensities . we have studied the effect of the detuning between the laser pump and the bare lp branch , and discussed the different kinds of instability that might occur for different values of the parameters . we have shown that , even though the kind of instabilities are the same as in the one component case ( kerr or parametric ) , here the interplay between different instabilities in the two components of the fluid , can produce a much richer picture . moreover , we have shown that since each component of the fluid jumps between stable states of its population at a different values of the pumping intensity , the system sustain multistable hysteresis loops that can be easily modulated by changing the parameters of the pumping lasers . finally , we believe that , due to the wide range of parameters for which the system is multistable , and due to the increased number of degrees of freedom with respect to the spin - dependent case , multiatability and hysteresis loops should be within an experimental reach . therefore , the novel system of two - component fluid is a promising candidate for the realization of optical switches and memories . we thank i. carusotto and d. whittaker for discussions . this research has been supported by the spanish mec ( mat2008 - 01555 , qoit - csd2006 - 00019 ) and cam ( s-2009/esp-1503 ) . f.m.m . acknowledges financial support from the program ramn y cajal .
we study the stability of a multicomponent exciton - polariton fluid under resonant excitation within the linear response approximation of a generalized gross - pitaevskii equation . 
 we show that , two spatially homogeneous and independently tunable pumping lasers produce , for the same values of the system parameters , up to three stable solutions . 
 three - stability is understood by noting that the cavity can be either little or highly populated and , in this second case , the largest part of the population lies in either one of the two components . 
 moreover , we discuss the different kinds of instabilities appearing at different pumps intensities and compare them with the case of one - component fluids . finally , we show that easily tunable multistable hysteresis loops can be performed by the system .