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the celebrated laser cooling processes and magneto - optical traps ( mot ) @xcite have allowed the study of many exciting topics in atomic physics . among them , much interest was devoted to the study of bose - einstein condensates @xcite , which had a profound impart in our understanding of condensed matter . apart from this , a new trend has recently begun that rekindled the interest on basic properties of mot physics . this is related to the increasing number of astrophysical phenomena that we can simulate and study using ultra cold atomic clouds . in particular , we can refer to a new mechanism associated with the laser cooling process , which can lead to the formation of static and oscillating photon bubbles inside the gas @xcite . photon bubbles have been considered in the astrophysical context @xcite where huge photon densities are required to have any significant impact on high energy particles . moreover , kaiser _ et al _ @xcite were recently able to achieve random lasing in a cloud of ultra cold atoms under laboratory conditions . a similar effect was first seen decades ago in stellar clouds @xcite and in some planetary atmosphere @xcite , when random lasing was first proposed to explain why certain specific emission lines in the stellar gas are more intense than theoretically expected @xcite . finally we refer to a recent work by teras _ et al _ @xcite , where the hydrodynamic equilibrium and normal modes of cold atomic traps are investigated , combining the effects of multiple photon scattering with the thermal fluctuations inside the system , cast in the form of a polytropic equation of state . this analysis results in a generalized lane - emden equation , describing the equilibrium density profiles of the atomic cloud , similar to that describing astrophysical fluids @xcite . the process of multiple scattering of light , which typically becames significant for a number of atoms above @xmath0 , is responsible for the rich and complex behaviour of ultra cold atomic vapours . this mechanism has been described , since the early stages of mots , as the principal limitation for the compressibility of the cloud @xcite . in this regime , the atoms in the cloud are strongly correlated due to the presence of a coulomb type long - range interaction @xcite , and the description of the system as a one component plasma becomes feasible and very fruitful @xcite .    in the present paper , we present a significant extension of the previously mentioned work by teras _ et al _ @xcite , to include the case of a rotating cloud of ultra cold atoms . in section ii , we begin with the derivation of the equilibrium density profiles of such systems . numerical analysis predict the existence of a special set of solutions containing satellite rings . such satellite rings have indeed been observed in rotating clouds since the early nineties @xcite , although their nature was not completely elucidated . in section iii , we will investigate the stability of such systems , by computing the normal oscillation modes that can be excited in the cloud . finally , in section iv , we discuss the validity of our model and state some conclusions . the starting point to compute the equilibrium profiles of the ultra cold cloud of atoms confined in a mot corresponds to the setting of the fluid equations , where the collective force due to multiple scattering of photons by the atoms is included : @xmath1 @xmath2 the fluid description of the system requires some relation between the pressure @xmath3 and the atom density @xmath4 , and for that purpose we assume the existence of a generic polytropic equation of state for the mot , of the form @xmath5 as we are dealing with rotating clouds , the total force acting on an fluid element is @xmath6 , with @xmath7 , and @xmath8 the collective force determined by @xmath9 notice that in the expression for the collective force @xmath8 , the quantity @xmath10 represents the square of an effective atomic charge @xcite , where @xmath11 is the speed of light and @xmath12 the total intensity of the six cooling laser beams . the terms @xmath13 and @xmath14 represent the emission and absorption cross sections , respectively @xcite . the force @xmath15 includes the doppler cooling force , with an equivalent damping coefficient @xmath16 , and the trapping force , with an equivalent spring constant @xmath17 . the difference with respect to the non rotating case is the presence of a new force term @xmath18 . rotation in the system can easily be achieved by a slight misalignment in four of the six laser beams , and can be described by @xcite @xmath19 for this reason , we will consider from now on a cylindrically symmetric system . assuming equilibrium conditions , @xmath20 and @xmath21 , and a - dimensioning the system as @xmath22 and @xmath23 with @xmath24 we then get , in a dimensionless form @xmath25 with @xmath26 the effective plasma frequency , and @xmath27 , where @xmath28 is the angular velocity of the fluid element . the parameter @xmath29 gives the ratio between the rotation angular frequency and the frequency associated with the confinement trap , @xmath30 . the first is equivalent to an expansion force , and the second one to a contraction force , whereby this constant will be related with the stability of the cloud . in particular , the system will become unstable for @xmath31 , as it will become clear in section * iii * , when we derive the frequency of the allowed oscillation modes . for non rotating clouds we would simply have @xmath32 . ( [ eq : lane0 ] ) therefore depends on the rotation state of the system . we can now realize that , by redefining the parameter @xmath33 as @xmath34 and introducing a redefined plasma frequency as @xmath35 we get @xmath36 this new equation becomes independent of rotation parameter , which is incorporated now inside the definitions of @xmath33 and @xmath37 . it remarkably implies that rotating and non rotating systems share the some kind of solutions for the equilibrium density profiles , differing only by a scale factor @xmath38 . ( [ eq : lane ] ) can be seen a generalized lane - emden equation @xcite , which was first derived for astrophysical fluids with gravitational confinement . the corresponding density profiles will only depend on the polytropic exponent @xmath39 , and on the constant @xmath37 , which accounts for the three forces in play . the latter corresponds to the _ ratio _ of the collective force , determined by @xmath40 , and the confining force ( @xmath41 ) , _ subtracted _ by the centripetal force due to rotation ( @xmath42 ) . the confining force is then directly contour - posed by the centripetal one . this ratio is sufficient to determine the density profiles .    with this interpretation of @xmath37 in mind , we can easily realize that a stable cloud of atoms can only exist if @xmath43 which implies an overall attractive force greater than the repulsive one ( multiple scattering ) . since we are interested in associating systems with different rotation states , we realize that degenerate density solutions only exist if @xmath44 , with @xmath45 for a stable solution . each of these sets of solutions ( for each value of the constant @xmath46 ) differ only by the scale factor @xmath47 . let us now compute some analytical solutions of eq . ( [ eq : lane ] ) , for some important limiting cases .      for traps with a small number of atoms , typically @xmath48 multiple scattering effects can be neglected , which allows us to take the limit @xmath49 . ( [ eq : lane ] ) then reduces to @xmath50 which takes the solution @xmath51 $ ] , or equivalently , in dimensional form @xmath52 the isothermal case , which corresponds to @xmath53 , simply yields a gaussian profile @xmath54 = n(0 ) \ , e^{-r^{2 } / 4 r^{2 } } \ , , \ ] ] which results from taking the limit @xmath55 in eq . ( [ eq : gama ] ) . ] this limit approximately describes a trap in the temperature limited regime , corresponding to a small number of particles , where the thermal effects determine the cloud dynamics and multiple scattering effects are negligible . for a larger number of atoms , as typical of present state - of - the - art traps , this no longer applies .      in the opposite case of a large number of trapped atoms , the collective force dominates over temperature effects , and we can take the limit @xmath56 in eq . ( [ eq : lane ] ) , giving rise to the solution @xmath57 or , in dimensional form @xmath58 where @xmath59 is the cloud radius and @xmath60 is the heaviside function . this is usually referred to as the _ water - bag _ profile . the formation of stable orbital modes in the density profiles is one of the most interesting features observed in rotating clouds of atoms @xcite . it was widely believed that the existence of such modes was intrinsically related with the rotation of the cloud @xcite . the investigation presented here indicates that this is not entirely true , since rotating and non rotating clouds share the same density solutions . in fact , numerical integration of the generalized lane - emden equation ( [ eq : lane ] ) , reveals a special set of solutions for @xmath61 . these appear as central cores with rings of atoms around , as depicted in figure [ fig : orbital ] .    . the black line corresponds to @xmath62 , and for the blue and red dashed lines we have @xmath63 and @xmath64 respectively . the differential equations were integrated using an explicit runge - kutta method , of fourth order . ]    in these solutions , there is a central core of atoms surrounded by a ring , typically with higher densities than the central region . as we increase the number and density of atoms in the system or , equivalently , higher rotation angular velocities - higher @xmath65 - the central core density increases . this behaviour has been experimentally observed , where a single ring structure can turn into a ring with a central core , for increasing intensity of the laser beams ( capturing more atoms in the trap ) or by increasing the laser misalignments , making the cloud to rotate faster @xcite . in our picture this corresponds to an increase of @xmath65 . the novelty of the present analysis is the description of the system in terms of an atom density function , @xmath66 , which is the experimentally accessible quantity , making possible the direct comparison of theory and experiments . although in references @xcite we can find a theoretical explanation for the rings , it is based on single atom trajectories , not allowing for a detailed comparison between theory and experiments . since we relate the atom density profiles with the equation of state of the cloud , the results presented in this paper will allow , for the first time , and using already available absorption imaging techniques , to measure the equation of state of the system . in the next section , along with the stability analysis of the solutions , we will also introduce an alternative way of measuring the equation of state , through the normal modes of oscillation in the system . in fact , the density profiles for @xmath61 present higher order orbital modes , with two or more distinguishable rings around the central core in spatially extended solutions . as the system is limited by the width of the laser beams and strongly depends on the number of trapped atoms , only a central core should be observed , eventually with one or two rings . experimentally , the transition from one to two rings has been observed @xcite when the number of trapped atoms increases . in figure [ fig : density ] we show a numerical solution where a second ring of atoms is included .     and two rings are displayed , which should be possible for a larger number of trapped atoms . ] we now evaluate the nature of localized oscillations , or normal modes of the system , using a perturbative analysis on the above fluid description . it will be useful to notice that the collective force can be derived from a potential , @xmath67 , such that @xmath68 . this is a coulomb type of potential , due to the existence of an effective electric charge for the neutral atoms , and resulting from the exchange of scattered photons between nearby atoms . it can be seen as a mean - field potential associated with multiple photon scattering . we now introduce small perturbations in the equilibrium quantities , labelled with the subscript @xmath69 ,    @xmath70    with the perturbations varying as @xmath71 . this prescription allows us to linearize the set of fluid equations , neglecting quadratic terms of the form @xmath72 , which yields    @xmath73 \\ +   & \frac{1}{m } \boldsymbol{\nabla } \cdot \left [ n_{0}(r ) \boldsymbol{\nabla } \delta \phi_{c } \right ] \ , , \end{split}\ ] ]    @xmath74    where we have used @xmath75 , from the polytropic equation of state . it is worth noticing that this equation for the perturbation @xmath76 is formally identical to that derived for non rotating clouds @xcite . this is to be expected , since we only consider radial perturbations , of the form @xmath77 , and @xmath78 . the existence of a finite angular velocity is put in play by the equilibrium density profiles @xmath79 from which the previous equation depends . recall that , due to the geometry of the problem , we have used cylindrical coordinates . as a last remark , notice that the term @xmath80 in the previous equation corresponds to a damping of the modes , which is to be expected since it results from the cooling doppler force , which is a viscous force , @xmath81 . in fact , along with the mode damping , this term also introduces a small correction in the mode frequency . but , as it turns out , in typical experimental conditions , the frequency associated with this term is small in comparison with the trapping frequency @xmath30 , and the plasma frequency @xmath82 , for what we can neglect this correction from now on @xcite .    in order to combine equations ( [ eq : norm1 ] ) and ( [ eq : llap ] ) , we can now introduce the auxiliary quantity @xmath46 , defined as @xmath83 . making proper substitutions , the linearized equations can finally be put together , resulting in a single expression @xmath84 \eta ( r ) = -\frac{\gamma c_{\gamma}}{m } \left [ r n_{0 } ^{\gamma - 1 } ( r ) \frac{d}{d r } \left ( \frac{1}{r } \frac{d \eta } { dr } \right ) \right ] \ , , \ ] ] with @xmath85 . we have then reduced the problem of finding the normal modes of oscillation to an eigenvalue problem . despite the linear form of the differential equation this is a non - trivial problem , and general solutions involve numerical simulations . for that reason we will only examine some limiting cases .      for small clouds , typically with @xmath48 trapped atoms , the effects of multiple scattering can be neglected , and we can set @xmath86 , or equivalently @xmath87 in eq . ( [ eq : norm1 ] ) , which simplifies to @xmath88 = 0 \ , .\ ] ] replacing the equilibrium density profile @xmath89 , given by eq . ( [ eq : gama ] ) , into eq . ( [ eq : tlr ] ) and writing the result in terms of the adimensional variable @xmath90 , as defined before , we obtain @xmath91 = 0 \ , , \ ] ] where we performed another change of variables , namely @xmath92 . for this differential equation we can try out solutions in the form of a power series @xmath93 where we can distinguish between even ( @xmath94 ) and odd ( @xmath95 ) contributions , because they lead to slightly different types of oscillations . for the lowest radial modes , @xmath96 , the solutions correspond to surface excitations @xcite , and the even perturbations solutions , @xmath97 , correspond to breathing modes . deriving the ansatz ( [ eq : ansatz ] ) , and replacing in eq . ( [ eq : tlr1 ] ) , we obtain a recursive expression for the coefficients @xmath98 , and the allowed modes of the system are @xmath99 as mentioned before , setting @xmath96 we get the surface modes @xmath100 breathing modes , @xmath97 , are also possible even in small traps , with a spectrum given by @xmath101 similar results are found in non - rotating spherical clouds @xcite . as anticipated in section * ii * , the cloud becomes unstable for @xmath31 , when the centrifugal force due to rotation is higher than the confining force . these results , along with those from the previous section , introduce the possibility of _ measuring _ the equation of state for the ultra cold gas , simply determined by the polytropic exponent @xmath39 . in fact , experimental techniques allow for very precise measurements of oscillation frequencies , and these results relate the polytropic exponent @xmath39 with the modes frequencies and the angular velocity of the cloud . in contrast with the temperature limited regime for small traps , for a large number of trapped atoms , the system is dominated by the collective effects , and we can neglect the temperature effects by setting @xmath102 in eq . ( [ eq : eq1 ] ) , yielding @xmath103 introducing the water - bag solution for the equilibrium density profile given by eq . ( [ eq : wb ] ) , we obtain @xmath104 again , and as expected , we have an instability for @xmath31 , for the same reasons as before . remember that the multiple scattering regime corresponds to the limit @xmath105 with @xmath106 , which implies @xmath107 . we then have a breathing mode in the system at the plasma frequency @xmath82 . the latter result is formally analogous to that well - known in plasma physics , where uncompressional monopole oscillation of a plasma cloud takes place at the classical plasma frequency . the same solution is also found for neutral atoms with spherical symmetry @xcite . these normal modes are oscillations of the cloud , which should not be confused with the hybrid sound waves discussed in ref . @xcite . in contrast with the present normal modes , which are global oscillations associated with the existence of boundaries , the hybrid sound waves are elementary excitations of the gas , which can also exist in an infinite medium . the corresponding frequencies are different , and can therefore be identified experimentally . we shall , however , notice that the solution ( [ eq : plasma_mode ] ) is not unique . a simple manipulation of eq . ( [ eq : norm1 ] ) and ( [ eq : llap ] ) and introducing the equilibrium water - bag profile yields @xmath108 = 0 \ , , \ ] ] with @xmath109 , the effective dielectric function . this equation holds for the internal region of the cloud , while in the outside we simply have @xmath110 , @xmath111 . this is quite appropriate for the water - bag density profile . the breathing mode obtained earlier can be derived from this alternative formulation by setting the particular solution to eq . ( [ eq : eqp ] ) as @xmath112 , which implies @xmath113 as before . the general solution to eq . ( [ eq : eqp ] ) is given by a simple power series in spherical coordinates . in cylindrical coordinates , the solution is more complicate but it can be given in terms of the bessel functions . we consider solutions valid in the two different regions , the inside region of the cloud , @xmath114 , and the outside region , @xmath115 , where @xmath59 stands for the cloud radius . the resulting solutions of the above laplace equation then read @xcite @xmath116 and @xmath117 where @xmath118 and @xmath119 are the modified bessel functions , and @xmath120 and @xmath121 are constants . we have also included the possibility of a perturbation propagating in the @xmath122 direction with wavevector @xmath123 , different from the radial oscillations introduced by eq . ( [ eq : oscillation ] ) . these solutions have to satisfy regular continuity conditions at the surface of the cloud ( @xmath124 ) , such that @xmath125 @xmath126 the first continuity condition implies @xmath127 , and the second condition yields the allowed modes as @xmath128^{-1/2 }   \ , .\ ] ] in figure [ fig : dispersao ] , we represent the dispersion relation given by eq . ( [ eq : omegap ] ) for different values of @xmath95 . notice that a similar result can be found in the context of a cylindrical pore surface - plasmon modes @xcite .    . ]    we easily realize that , in the long wavelength limit , @xmath129 , we recover the breathing mode derived earlier at @xmath130 , for @xmath97 . in this same limit , all the remaining modes ( @xmath131 ) collapse in a single frequency at @xmath132 , characteristic of a cylindrical system . it is worth mentioning that for a spherical system , the same problem also yields a breathing mode at @xmath133 and , solving the laplace equation in spherical coordinates @xcite , we would get @xmath134 which collapses into our second breathing mode in the limit @xmath135 , where we get @xmath136 . notice that by reformulating the problem in terms of an effective dielectric function - see eq . ( [ eq : eqp ] ) - we introduced the possibility for propagating modes in the boundless @xmath122 direction , and determined by radial boundary conditions , in eq . ( [ eq : bound1 ] ) and eq . ( [ eq : bound2 ] ) . these modes have some similarities with the plasma waves considered in @xcite , which take the form @xmath137 ( @xmath138 is the sound speed ) . in the low wavenumber limit , the dispersion relation of eq . ( [ eq : omegap ] ) , and figure ( [ fig : dispersao ] ) , takes the same form of these plasma waves , @xmath139 , where @xmath140 is some constant for each @xmath95 solution , clarifying plasma like nature of these oscillations . in contrast , in the high wavenumber limit , the radial boundary conditions would considerably alter the dispersive relation , as indicated above . such modes could be also be excited in bound extended systems , if the size in the @xmath122 direction is large compared with the typical wavelength @xmath141 . as mentioned before , a cylindrical geometry can experimentally be achieved by lowering the intensity of the laser cooling beams along the rotation axis , allowing for the cloud to expand and to take a cigar - like shape . kinetic effects are not expected to play an important role here . in fact , a kinetic description is useful when considering the resonant exchange of energy between atoms moving with velocities near the phase velocity of phonons , @xmath142 . this is not relevant for normal modes , which can be understood as phonons with infinite wavelength , @xmath143 or @xmath144 . in this work we have examined the equilibrium profiles and the normal oscillating modes of rotating clouds of ultra cold atoms confined in a mot . the investigation relies on the assumption that a polytropic equation of state for the ultra cold gas is satisfied , which phenomenologically models a large class of mot s , including the state - of - the - art ones , with a large number of atoms - in fact water - bag density profiles have been experimentally observed since the early stages of large mots @xcite . an experimental validation of this equation of state , which only depends on the polytropic exponent @xmath39 , can then be performed through the measurement of the density profiles of the atomic cloud , or by determining the spectrum of normal modes . employing direct and absorption imaging techniques , this can be done in a very precise manner and would , for the first time , allow for the measurement of an equation of state of the ultra cold matter inside the trap , which constitutes one of the novelties of the present results . on the other hand , we have shown that this simple polytropic model can account for the observed stable orbital modes , that up so far were thought to be intrinsically related with the cloud rotation @xcite . the profiles with orbital modes correspond to a special set of solutions with @xmath61 . the measurement of the normal modes provides a nice way of validating these results . the model employed here is valid for cylindrically symmetric clouds , which should be a good approximation in the equatorial plane of a spherical one . in order to determine the width of the satellite rings in spherical clouds , we would however need to develop a spherical model . in the general case , and specially for rotating clouds , the appropriate set of coordinates lies somewhere in between cylindrical and spherical . for this reason we are now trying to figure out how to employ an elliptical coordinate system that in the non - rotating case would reduce to the spherical geometry ( zero eccentricity ) . this would be a more general and appropriate description of the cloud . these different aspects of the problem will be addressed in a future work . also related to these features , and following the annunciated new trend of studying astrophysical related phenomena in the laboratory , the results present here , as well as those in @xcite , allow for an investigation of the equilibrium and dynamical properties of the lane - emden equation , establishing another interesting connection between the communities of ultra - cold matter and astrophysics . in particular we can refer to the recent observation of a new class of variable stars @xcite . these rapidly rotating main - sequence stars were not expected to pulsate , or to have any other physical characteristics that would lead to periodic luminosity variations , according to current theories . one of the hypothesis being considered is that fast rotation could alter the internal conditions of a star , enough to sustain stellar pulsations . but there is currently no stellar model that can predict whether pulsation can be sustained in very fast rotating stars . the ability to achieve conditions in laboratory , namely with ultra - cold atomic gases , with similar dynamics to stellar and astrophysical fluids , can prove being important to understand the properties of these objects . all the similarities between astrophysical and cold atoms systems comes from the concept of radiation pressure , which determines the dynamics of both systems . in astrophysics its the huge photon densities that leads to this radiation pressure . in cold atoms , the low densities are compensated by the high resonances leading to working regimes dictated by the radiation pressure - multiple scattering regime . in fact , it may be possible to tune the experimental parameters in order to achieve working conditions closer to those found in astrophysical scenarios .
the multiple scattering of light in a gas of ultra cold atoms is responsible for many exciting features observed in magneto - optical traps including the collective behaviour forced by a coulomb like potential . 
 this field also induces plasma like phenomena in the cloud which allows the treatment of the system as a one component trapped plasma . with a fluid description and casting the thermodynamical behaviour in the form of a polytropic equation of state we investigate the equilibrium profiles of rotating clouds and its dependence on the experiment characteristics . 
 numerical solutions predict the formation of stable orbital modes both in rotating and non rotating clouds . 
 we also investigate the normal modes on such rotating systems . 
 the results obtained allow for the measurement of the equation of state for the cold gas .