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similar to monopole and quadrupole breathing modes of a gaseous bose - einstein condensate , scissors modes were first studied theoretically and subsequently observed experimentally . the scissors modes are , however , rather special since they directly manifest the superfluid behavior of these atomic gases . moreover , the recent experimental studies appear to show a resonance behavior between two coupled scissors modes @xcite . from a theoretical point of view this is interesting because a linear - response calculation can neither account for the coupling nor for the resonance behavior . therefore , a first step towards an explanation of these experimental observations is to perform a calculation that goes beyond linear - response theory and accurately takes into account the mean - field interaction that couples the scissors and breathing modes . in this paper we present a simple variational method for calculating the frequencies of these various modes and their couplings beyond the linear - response . we perform our calculation at zero temperature and therefore do not consider the damping rates of the scissors modes .    the main idea behind our method is to use a time - dependent gaussian _ ansatz _ for the groundstate wave function to derive the equations of motion of the breathing modes and the scissors modes . then we expand the resulting equations of motion in deviations from equilibrium . in first order , i.e. , linear - response , we recover the expected uncoupled set of equations . the second - order calculation produces a set of coupled equations which show that we need to excite all three scissors modes in order to get a nonzero coupling . at higher orders we , however , find that we only need to excite two modes to get coupling . furthermore , we actually find under certain conditions a resonance behavior between these two modes . the layout of the paper is as follows . first , we rederive in sec . [ freqs - section ] the frequencies of the scissors modes in the linear - response limit . in sec . [ heigher ] we extend the calculation first to second , and then also to higher orders , which ultimately lead to a resonant coupling . in sec . [ solution ] we solve the equations of motion analytically near the resonance using an envelope function approach . in sec . [ conclusion ] we end with a discussion of our results . we start by considering a bose - einstein condensate trapped by the following harmonic potential @xmath0 where @xmath1 , @xmath2 , and @xmath3 are the angular frequencies of the trap , and @xmath4 is the atomic mass . a scissors mode in a bose - einstein condensate is associated with an irrotational flow with a velocity field of the form @xmath5 , if the motion is taking place in the @xmath6-plane @xcite . similar expressions hold for the two other cartisian planes . these kind of modes can be excited by a sudden rotation of the equilibrium axes of the trap . to such a perturbation the condensate will respond by oscillating around the new equilibrium axes . for example , to excite a scissors mode in the @xmath6-plane , we rotate the @xmath7 and @xmath8-axes of the trap slightly around the @xmath9-axis . if the angle of rotation is sufficiently small , the scissors mode can be approximated by a simple oscillation of the condensate around the new equilibrium axes . on the other hand , if the axes change through a large angle this method excites the @xmath10 quadrupole mode , where @xmath4 labels the projection of the angular momentum along the axis of symmetry . the maximum angle for which the scissors mode is defined increases with deformation of the trap @xcite .    to account for all three scissors modes in the three cartisian planes we employ the following trial function for the condensate order parameter @xmath11 where @xmath12 and @xmath13 , are complex time - dependent variational parameters and @xmath14{c_{xy , r}c_{xz , r}c_{yz , r}+4b_{x , r}b_{y , r}b_{z , r}-(b_{z , r}c_{xy , r}^2+b _ { y , r}c _ { xz , r}^2+b_{x , r}c_{yz , r}^2 ) } \label{a}.\ ] ] this value of the prefactor @xmath15 guarantees the normalization of the square of the wave function @xmath16 to the total number of condensed atoms @xmath17 . here @xmath18 and @xmath19 are the real parts of @xmath12 and @xmath13 , respectively . the first set of parameters , @xmath12 , give rise to the well - studied breathing modes which , for axially symmetric traps , are called the monopole and quadrupole modes depending on the value of @xmath4 being equal to zero or two , respectively . the parameters @xmath13 on the other hand determine the three scissors modes . the equations of motion for these variational parameters can be derived from the lagrangian @xmath20={1\over2}i\hbar\int { { \rm d}{\bf r } \left(\psi^*({\bf r},t){\partial\psi({\bf r},t)\over\partial t } -\psi({\bf r},t){\partial\psi^*({\bf r},t)\over\partial t}\right ) } -e[\psi,\psi^ * ] \label{lagrangian},\ ] ] where @xmath21 $ ] is the usual gross - pitaevskii energy functional given by @xmath22=\int d{\bf r } \left [ { \hbar^2\over 2m}|{\mbox{\boldmath $ \nabla$}}\psi({\bf r},t)|^2 + v({\bf r})|\psi({\bf r},t)|^2 + { 1\over2}t^{2b}|\psi({\bf r},t)|^4 -\mu|\psi({\bf r},t)|^2 \right ] \label{functional}.\ ] ] here @xmath23 is the two - body @xmath24-matrix , which for the atomic bose - einstein condensates of interest is related to the @xmath25-wave scattering length @xmath26 through @xmath27 . inserting our trial wave function into the lagrangian and scaling frequencies with @xmath28 and lengths with @xmath29 , it takes the dimensionless form @xmath30/n&=&\left(\alpha_{x}{\dot b}_{x , i}+\alpha_{y}{\dot b}_{y , i}+\alpha_{z}{\dot b}_{z , i}\right)/q\\\nonumber & -&{1\over2}\left [ \alpha_{x}(4|b_{x}|^2+|c_{xy}|^2+|c_{xz}|^2)\right.\\\nonumber & + & \alpha_{y}(4|b_{y}|^2+|c_{xy}|^2+|c_{yz}|^2)\\\nonumber & + & \left.\alpha_{z}(4|b_{z}|^2+|c_{xz}|^2+|c_{yz}|^2)\right]/q\\\nonumber & -&{1\over2}\left[\alpha_{x}\omega_{x}^2+\alpha_{y}\omega_{y}^2+\alpha_{z } \omega_{z}^2\right]/q\\\nonumber & -&{1\over2\sqrt{\pi}}\gamma\sqrt{q } \label{scaled lagrangian } , \end{aligned}\ ] ] where @xmath31 , @xmath32 , @xmath33 , @xmath34 , and the dot corresponds to a time derivative . in addition @xmath35 is the dimensionless parameter that represents the strength of the mean - field interaction . minimizing the lagrangian with respect to the 12 variational parameters , we get a set of 12 coupled equations of motion . the resulting equations of motion are rather lengthy and complicated . a significant simplification takes place if we expand these equations in the deviation of the variational parameters from their equilibrium values , i.e. , in @xmath36 and @xmath37 , where @xmath38 is the yet unknown eigenfrequency of the modes , and @xmath39 and @xmath40 denote the equilibrium values of @xmath12 and @xmath13 , respectively . these latter values can simply be obtained by setting the time derivatives in the equations of motion to zero . for large condensates in the so - called thomas - fermi regime we can ignore contributions from the kinetic energy @xcite and the equilibrium variational parameters take the simple form @xmath41 it is required that @xmath42 for the last equation to be valid . to first order in the deviations the equations of motion read simply @xmath43 where the vector @xmath44 contains all the possible fluctuations , and the matrix @xmath45 is given by @xmath46 where @xmath47 and @xmath48 are given explicitly in appendix [ app ] . it is clear from the last equation that to linear order the breathing modes and scissors modes are uncoupled . the dispersion relation of these modes can be obtained by setting the determinant of @xmath49 to zero . this results in @xmath50 where @xmath51 , @xmath52 , @xmath53 , @xmath54 , @xmath55 , and @xmath56 . the zeros of the first three factors in the left - hand side of the last equation give the frequencies of the scissors modes , whereas the zeros of the sixth - order polynomial give the frequencies of the breathing modes . so far we have not produced a new result , since these frequencies have been calculated previously . however , we have demonstrated a new way of calculating the frequencies of scissors modes in a relatively simple manner . the most important part of this paper is therefore contained in the next section , where we consider also the nonlinear effects produced by the mean - field interaction . in this section we consider the equations of motion for the variational parameters by taking into account several higher - order terms in the deviations @xmath57 and @xmath58 . we have calculated these equations analytically up to second order , but they turn out to be rather lengthy and contain terms that couple the breathing modes and the scissors modes . as we discuss below , in the present experiments with axially symmetric traps the coupling of the scissors modes with the quadrupole mode is always of importance , but we leave the treatment of this more complicated situation to future work . for simplicity , therefore , we focus here on triaxial traps , in which case we can ignore the breathing modes . this is achieved by simply setting @xmath59 in the full equations of motion .    up to second order , the remaining 6 first - order equations for @xmath60 can be reduced to 3 second - order equations for @xmath61 , @xmath62 , and @xmath63 by eliminating the imaginary parts . we find in detail @xmath64 \label{c1},\ ] ] @xmath65 \label{c2},\ ] ] and @xmath66 \label{c3}.\ ] ] these equations show that if we ignore the second - order terms we recover our previous three uncoupled scissors modes with frequencies @xmath67 , @xmath68 and @xmath69 . it is interesting to observe that the coupling terms couple the three scissors modes such that if one mode is not excited the other two modes will not be coupled . in other words , to get a nonzero coupling up to this order , the three modes must be excited simultaneously . we believe that this important result is not an artefact of the gaussian approximation , but also holds for an exact calculation using the bogoliubov theory .    for higher - order couplings the last conclusion is no longer true . two modes can then be coupled , even when the third is not excited . from now on , therefore , we assume without loss of generality that only the @xmath70 and @xmath71 modes are excited , while @xmath72 . moreover , to investigate the possibility of a resonant coupling between these two modes , we have considered coupling terms up to ninth order . similar to the second - order case explained above , the equations of motion can thus be expressed as @xmath73 @xmath74 where @xmath75 and the sum @xmath76 does not excced 9 , which is the order up to which we have chosen to expand the equations of motion . the coefficients @xmath77 , @xmath78 are given in terms of the trap parameters @xmath1 , @xmath2 , @xmath3 , and the interaction parameter @xmath79 . a resonance between these two modes takes place when the frequency of the coupling terms is equal to the frequency of the zeroth - order term , i.e. , the first two terms . the frequency of each coupling term in the above summations is determined by substituting for @xmath80 and @xmath81 their zeroth - order solutions . therefore , the latter frequencies will be a certain linear combination of the zeroth - order scissors modes frequencies . imposing the above resonance condition on each coupling term thus results in a relation between the two scissors modes frequencies @xmath67 and @xmath68 . inspecting all coupling terms up to the ninth order , we found a very small number of terms for which the relation between @xmath67 and @xmath68 can be satisfied by real values of @xmath82 , @xmath83 , and @xmath84 . ultimately , we find only a resonance when either @xmath85 or @xmath86 is satisfied . the first resonance condition leads in the experimentally relevant axially symmetric case , where @xmath87 , to a value for the anisotropy ratio @xmath88 that is equal to 1 . this clearly corresponds to a spherically symmetric condensate . since in this case the scissors modes are degenerate with the quadrupole breathing modes , which we have ignored here , we focus from now on only on the second resonance condition . it is interesting to mention that the second resonance condition is exactly the same as the one observed experimentally @xcite . for a resonance of this kind the resonant coupling terms turn out to be of seventh order , and the equations of motion in that case read @xmath89 @xmath90 if we neglect all nonresonant terms . here @xmath91 , @xmath92 and @xmath93 are functions of @xmath82 , @xmath83 , @xmath84 , and @xmath79 that are given explicitly in appendix [ app ] . it is important to note here that our neglect of the nonresonant terms is justified when we are close enough to resonance . this is similar to the rotating - wave approximation known from quantum optics . we see that the coupling terms indeed lead to the above - mentioned resonance condition , by inserting in them the zeroth - order solutions , i.e. , @xmath94 and @xmath95 . for example , the coupling terms in eq . ( [ c1n ] ) have a total frequency of @xmath96 . separating out @xmath97 as a prefactor for the whole equation , the coupling term oscillates thus as @xmath98 therefore , a resonance takes place when @xmath99 , i.e. , when the condition in eq . ( [ cond2 ] ) is met . similarly , the coupling term of eq . ( [ c2n ] ) is also oscillating with a frequency of @xmath100 . sufficiently close to resonance we can write @xmath70 and @xmath71 as a product of two functions . one of them describes the slow envelope and the other the fast oscillation with the uncoupled scissors - mode frequency . in particular , we have @xmath101 and @xmath102 where @xmath103 and @xmath104 are the slowly varying envelope functions . substituting these expressions into eqs . ( [ c1n ] ) and ( [ c2n ] ) , ignoring second - order time derivatives of @xmath104 and @xmath103 , and then eliminating @xmath103 , we obtain the following equation for @xmath104 @xmath105 where @xmath106 this equation has a solution of the form @xmath107^{1\over1-\varepsilon } \label{sol},\ ] ] where @xmath108 and @xmath109 are two constants of integration that are determined by the initial conditions . note that the relevant quantity here is @xmath110 , which represents the actual envelope of the oscillation and is given by @xmath111^{1\over2(1-\varepsilon)}.\ ] ] in first instance we might think that the real part of @xmath104 is the relevant quantity . however , in eqs . ( [ geq ] ) and ( [ feq ] ) we should in principle have taken the real part of the right - hand side . if we do that we automatically are lead to the condition that @xmath110 is the envelope of the oscillation .    for definiteness sake let us take the initial conditions @xmath112 and @xmath113 , where @xmath114 and @xmath115 are the real and imaginary parts of @xmath104 , respectively . physically , this set of initial conditions corresponds to exciting the scissors modes in the @xmath116 and the @xmath6-planes simultaneously . this should be performed experimentally by initially rotating the condensate in the @xmath116-plane by an angle @xmath117 and around the @xmath9-axis by an angle @xmath118 and then releasing the condensate . the initial angles @xmath117 and @xmath118 are related to the constants @xmath119 and @xmath120 by @xmath121 latexmath:[\[{\dot f}_{0}={\alpha\omega_{xy}^2\over\omega_{xz } } ( 2|b_{x}^{(0)}-b_z^{(0)}|\cos{\theta_0}\;\sin{\theta_0})^3    conditions the constants @xmath108 and @xmath109 are given by @xmath123 @xmath124 using these expressions and the experimental parameters from ref . @xcite , we give in fig . [ 2a ] the real part of @xmath125 . this clearly shows how the energy is being exchanged between the two modes . an interesting property of eq . ( [ f2eq ] ) is that exactly on resonance , i.e. , @xmath99 , its solution becomes nonoscillatory . indeed we find that in this case the solution is @xmath126 where again @xmath127 and @xmath128 are constants that are determined by the initial conditions . for the above initial conditions this is a decreasing function in time . physically , this means that , unlike the case of an off - resonant oscillation , it takes an infinite time for the energy that is transferred from the scissors mode in the @xmath116-plane to the scissors mode in the @xmath6-plane to get back to the mode in the @xmath116-plane . in fig . [ 2b ] we show this resonance behavior . finally , we can show that in the limit @xmath129 the oscillatory solution given in eq . ( [ sol ] ) reduces to the nonoscillatory one at resonance given by eq . ( [ sol_dec ] ) . we have explored the role of the mean - field interaction in coupling the three scissors modes of a bose - einstein condensate . a variational approach with a gaussian trial wave function , that contains a number of variational parameters describing the scissors modes , provides a relatively simple way in which we can extract the main features of this coupling . to first order in the deviations in the variational parameters from their equilibrium values we reproduce the correct frequencies of the scissors modes . to second order we show that it is not possible to have two modes that are coupled if the third mode is not excited . instead , all three modes need to be excited for nonlinear dynamics to occur .    at higher orders we find that it is possible to couple only two modes . in this case we find a resonance behavior if @xmath130 or @xmath131 . up to the nineth order in the deviations , these are the only two cases of resonance that we have found . close to resonance the equations of motion have been solved exactly using an envelope approach . the resulting dynamics is similar to a beating between two modes with a beating frequency @xmath132 . we notice that the observed resonance behavior @xcite occurs exactly for the same condition that we have obtained , namely @xmath130 . also the fact that we did not find any other resonance condition up to the ninth order , indicates that the coupling terms that lead to this resonance are also responsible for the experimental resonance in the triaxial case . furthermore , it is important to note here that , quite generally , the down - conversion process from one excitation quantum into two excitation quanta with half the energy each , i.e. , the so - called beliaev damping @xcite , vanishes for the scissors modes . this is so because of the negative parity of the scissors modes . in terms of fluctuations of the anihilation operator @xmath133 given by @xmath134 , the beliaev damping process is accounted for by an interaction term proportional to @xmath135 . here @xmath136 is again the condensate wave function . using our variational gaussian wave function given by eq . ( [ trial ] ) , we clearly see that with only two scissors modes present this integral vanishes , since the integrant is an odd function . we believe that this result is independent of our trial wave function and also true within an exact approach @xcite . in our case the seventh - order coupling terms in eqs . ( [ c1n ] ) and ( [ c2n ] ) correspond to a quadratic collisional damping process , where three excitation quanta decay into four excitation quanta with half the frequency and one excitation quantum with the same frequency . this is shown schematically in fig . [ schem ] . this makes sense physically since this is the lowest - order nonvanishing process if beliaev damping is forbidden and we are forced to apply the interaction term proportional to @xmath137 twice to accomplish the down conversion . another point worth mentioning here is that in the experiments the resonance is studied in most detail for an axially symmetric trap with an anisotropy ratio close to @xmath138 , when the @xmath6-scissors mode is degenerate with the quadrupole mode , which we have excluded in the present calculation . in that case we may argue that one scissors - mode quantum can decay into either another scissors - mode quantum and a quadrupole - mode quantum or into two quadrupole - mode quanta . these are in principle also beliaev damping processes , which are however again forbidden for similar reasons as before . nevertheless , a good understanding of these experiments requires an analysis that includes not only the @xmath6 and the @xmath116 scissors modes but also the quadrupole mode , because of the degeneracy that occures in this case . however , our theory should be directly applicable to the experiments with a triaxial trap . we hope that in the future more detailed experiments of this kind will also be performed , to make a direct comparison between our theory and experiment possible . we would like to thank c. j. foot and p. drummond for useful discussions . this work is supported by the stichting voor fundamenteel onderzoek der materie ( fom ) , which is financially supported by the nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) . d. s. jin , j. r. ensher , m. r. mathews , c. e. wieman , and e. a. cornell , phys . lett . * 77 * , 420 ( 1996 ) . mewes , m. r. andrews , n. j. van druten , d. s. durfee , c. g. townsend , and w. ketterle , phys . lett . * 77 * , 988 ( 1996 ) . s. stringari , phys . 77 * , 2360 ( 1996 ) . k. g. singh and d. s. rokhsar , phys . lett . * 77 * , 1667 ( 1996 ) . m. edwards , p. a. ruprecht , k. burnett , r. j. dodd , and c. w. clarck , phys lett . * 77 * , 1671 ( 1996 ) . y. castin and r. dum , phys . * 77 * , 5315 ( 1996 ) . v. m. perez - carcia , h. michinel , j. i. cirac , m. lewenstein , and p. zoller , phys . . lett . * 77 * , 5320 ( 1996 ) . d. gury - odelin and s. stringari , phys . lett . * 83 * , 4452 ( 1999 ) . m. o. da c. pires and e. j. v. de passos , j. phys . b : at . mol . . phys . * 33 * , 3929 ( 2000 ) . o. m. marag , s. a. hopkins , j. arlt , e. hodby , g. hechenblaikner , and c. j. foot , phys . * 84 * , 2056 ( 2000 ) . e. hodby , o. m. marag , g. hechenblaikner , and c. j. foot , phys . lett . * 86 * , 2196 ( 2001 ) . o. m. marag , g. hechenblaikner , e. hodby , and c. j. foot , cond - mat/0101213 . b. jackson and e. zaremba , cond - mat/0105465 .    c. j. foot , private communication . g. baym and c. j. pethick , phys . lett . * 76 * , 6 ( 1996 ) . f. dalfovo , s. giorgini , l. p. pitaevskii , and s. stringari , rev . phys . * 71 * , 463 ( 1999 ) . s. t. beliaev , sov . jetp * 34 * , 323 ( 1958 ) . see also g. hechenblaikner , e. hodby , s. a. hopkins , o. m. marag , and c. j. foot , cond - mat/0105394 . the matrices @xmath47 and @xmath139 are given by @xmath140 and @xmath141 the coupling coefficients @xmath91 , @xmath92 and @xmath93 , appearing first in eqs . ( [ c1n ] ) and ( [ c2n ] ) , are given by @xmath142\\\nonumber & /&\left({\pi } ^{\frac{6}{5}}\,{{{\omega } _ x}}^4\,{{{\omega } _ y}}^2\ , \omega_{xy}^4 \omega_{xz}^4\right),\end{aligned}\ ] ] @xmath143\\\nonumber & /&\left({\pi } ^{\frac{6}{5}}\ , { { { \omega } _ x}}^4\,{{{\omega } _ y}}^2\ , \omega_{xy}^4\ , \omega_{xz}^4\right),\end{aligned}\ ] ] and @xmath144 respectively .
we explore the nonlinear coupling of the three scissors modes of an anisotropic bose - einstein condensate . 
 we show that only when the frequency of one of the scissors modes is twice the frequency of another scissors mode , these two modes can be resonantly coupled and a down conversion can occur . 
 we perform the calculation variationally using a gaussian trial wave function . 
 this enables us to obtain simple analytical results that describe the oscillation and resonance behaviour of the two coupled modes . 
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