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a bose - einstein condensate ( bec ) in an optical trap  @xcite exhibits a rich variety of spin - related phenomena , such as various magnetic phases  @xcite and spin domain formation  @xcite . the nuclear spin of @xmath2 and @xmath3 is @xmath4 , and combined with the spin @xmath5 of the outermost electron , the possible hyperfine spins of these atomic species are @xmath6 and @xmath7 . the @xmath6 bec was first realized at mit with @xmath8  @xcite , for which the antiferromagnetic behavior was observed  @xcite . on the other hand , @xmath2 in the @xmath6 hyperfine state was found to be ferromagnetic  @xcite . the @xmath7 @xmath3 condensate was also realized by the mit group  @xcite , whose ground state phase at zero magnetic field is predicted to be antiferromagnetic  @xcite . however , the ground state phase and the spin dynamics of the @xmath7 @xmath3 bec have not been studied because of its very short lifetime ( a few milliseconds ) , due to the fact that the energy of the @xmath9 state is higher than that of the @xmath6 state . the @xmath7 @xmath2 bec also lies energetically higher than its @xmath6 counterpart , but its lifetime is much larger ( @xmath10 ms ) due to a fortuitous coincidence of the singlet and triplet scattering lengths  @xcite . the coherent spin dynamics of the @xmath7 bec have been observed  @xcite . the ground state of the @xmath7 bec of @xmath2 is predicted to be very close to the phase boundary between the antiferromagnetic and cyclic phases  @xcite . according to calculations performed by klausen _ et al_.  @xcite , the ground state phase of this bec at zero magnetic field is barely in the antiferromagnetic phase . recent experiments performed by the hamburg group  @xcite and by the gakushuin group  @xcite appear to support this prediction . however , due to the experimental uncertainties , the possibility that the ground state phase is cyclic has not been excluded . in the hamburg experiment  @xcite , the @xmath11 mixture was shown to be stable , which is the ground state of the antiferromagnetic phase . however , a magnetic field of 340 mg was applied to the system in this experiment . since the magnetic field lowers the energy of the @xmath11 states due to the quadratic zeeman effect , the observed stability of the @xmath11 mixture may be due to the presence of the magnetic field . that there is no spatial separation between the @xmath11 states is a necessary condition for a bec to be antiferromagnetic , but it is not sufficient . in the gakushuin experiment @xcite , the spin dynamics starting from the @xmath12 state were investigated for various values of the magnetic field . it was found that the population of the @xmath11 components after 70 ms evolution increased with a decrease of the magnetic field , which might imply that the @xmath7 bec is antiferromagnetic at zero magnetic field . however , the initial @xmath12 state is a highly excited state and the resultant state after 70 ms is far above the ground state . thus , the experimental observations do not exclude the possibility that the ground state phase at zero magnetic field is cyclic . in fact , the hamburg group observed that the spin configuration of the cyclic phase has a long lifetime  @xcite . there are two experimental difficulties that hinder the determination of the ground state phase . one is the short lifetime of @xmath13 100 ms of the upper hyperfine manifold of @xmath2 , which makes the equilibrium spin state hard to reach . in order to realize the ground state phase , we must equilibrate the system at least for a few seconds , as in the @xmath6 case  @xcite . the other difficulty is the small energy scale ( @xmath14 0.1 nk  @xcite ) of the spin - singlet - pair term whose sign determines the ground state phase . in order to determine the sign of the spin - singlet pair energy , the magnetic field must be reduced so that the quadratic zeeman energy is less than the spin - singlet pair energy . however , in such a low magnetic field the system may be affected by stray ac magnetic fields  @xcite .    to circumvent these difficulties , we propose a method to determine the value of the spin - singlet pair energy by spin exchange dynamics . we show that by controlling the initial population and phase in each magnetic sublevel , we can measure the value of the singlet - pair energy from the initial spin dynamics . this method therefore has the advantage that equilibrating the system , which takes a long time , is not necessary . furthermore , the quadratic zeeman energy is allowed to exceed the singlet - pair energy and extreme suppression of stray magnetic fields is not required . this paper is organized as follows . section  [ s : bogo ] formulates the mean field and bogoliubov theories of a spin-2 bec in the presence of the quadratic zeeman effect . section  [ s : homo ] studies the spin dynamics in a homogeneous system and proposes a method to determine the spin - singlet pair energy from the spin exchange dynamics . section  [ s : trap ] numerically verifies the proposed method for a trapped system with two - body loss . section  [ s : exp ] discusses the initial spin preparation and sec . [ s : conc ] concludes the paper . detailed derivations of the analytic solutions are given in the appendix . from rotational symmetry , the low - energy interaction between two atoms with hyperfine spin @xmath15 can be classified in terms of the total spin @xmath16 , and each scattering channel can be described by the corresponding @xmath17-wave scattering length @xmath18  @xcite . in the case of @xmath7 there are three scattering channels @xmath19 , @xmath20 , @xmath21 , and the interaction is described by @xmath22 , where @xmath23 is the mass of the atom and @xmath24 projects the state of two colliding atoms into the state of total spin @xmath25 . the interaction hamiltonian can be rewritten  @xcite as @xmath26 , where @xmath27 and @xmath28 are the spin operators for atoms 1 and 2 and the interaction coefficients are given by    @xmath29    for the linear and quadratic zeeman effects , the energy of the atom depends on the magnetic field @xmath30 as @xmath31 , where @xmath32 and @xmath33 with @xmath34 the bohr magneton and @xmath35 the hyperfine splitting between the states @xmath6 and @xmath7  @xcite . we assume @xmath36 throughout this paper , which is the case for spin-2 @xmath2 . the mean - field energy of the entire system is given by @xmath37 denotes the magnetic sublevels , @xmath38 is an external potential , @xmath39 is the total particle - number density , @xmath40 is the spin vector with @xmath41 spin-2 matrix @xmath42 , and @xmath43 is the singlet - pair amplitude . minimizing the action , @xmath44 gives the multicomponent gross - pitaevskii ( gp ) equations for the spin-2 bec as    [ gp ] @xmath45    where @xmath46 . we note that the linear zeeman term only rotates the hyperfine spin about the @xmath47 axis , and hence the spin dynamics are essentially independent of the linear zeeman term . in fact , setting @xmath48 and noting that the phase factor also arises in the @xmath49 terms as @xmath50 , we can completely eliminate the linear zeeman terms in eqs . ( [ gp1 ] ) and ( [ gp2 ] ) . this property is due to the rotational symmetry of our system with respect to the @xmath47 axis , which results in the conservation of the projected angular momentum on the @xmath47 axis , @xmath51 .      in an ultracold spinor bec which is isolated from the environment , the total spin angular momentum in the direction of the magnetic field is conserved for a long time ( @xmath52 1 sec )  @xcite . we therefore minimize the energy of the system in the subspace of a given @xmath53 ; we shall refer to the resulting minimized state as the `` ground state '' . for simplicity , we restrict ourselves to the subspace of @xmath54 in this paper , i.e. , we seek the ground state in the subspace of @xmath55 . we consider the uniform case by setting @xmath56 and @xmath57 , where the density @xmath58 is constant and @xmath59 satisfies @xmath60 . the energy per atom is given by @xmath61 where @xmath62 , @xmath63 , and @xmath64 for @xmath65 , @xmath66 , @xmath20 . the linear zeeman term is absent in eq . ( [ ene ] ) because @xmath54 .    in order to find the ground state , we compare the energies of the stationary states given in ref .  @xcite . the energy of the antiferromagnetic state , @xmath67 where @xmath68 is the phase of each component , is given by @xmath69 this energy is independent of the phase factor @xmath70 . for the cyclic state , @xmath71 the energy of the system takes the form @xmath72 where @xmath73 . when @xmath74 , eq . ( [ en ] ) is minimized by @xmath75 , which corresponds to the antiferromagnetic state @xmath76 . when @xmath77 , eq . ( [ en ] ) becomes minimal with @xmath78 and @xmath79 substituting this into eq . ( [ en ] ) , we obtain @xmath80 the states @xmath76 and @xmath81 satisfy @xmath82 . even if the local spin @xmath83 is nonzero , the total spin @xmath84 can be zero by the formation of the spatial spin structure  @xcite . for example , a staggered domain structure of the ferromagnetic state gives @xmath55 . the kinetic energy at the domain wall can be neglected when the region of the wall is much smaller than the region of the domain . hence , we also consider the case of @xmath85 , bearing in mind that the total spin is maintained as zero , @xmath55 , by some structure formation . the energy of the ferromagnetic state , @xmath86 or @xmath87 , is given by @xmath88 we also consider a state found in ref . @xcite , @xmath89 the energy of this state is obtained to be @xmath90 this is minimized by @xmath91 for @xmath92 , and the energy then becomes @xmath93    comparing the energies @xmath94 , @xmath95 , @xmath96 , and @xmath97 , we obtain the phase diagrams shown in fig .  [ f : diagram ] .     and @xmath36 ( a ) with respect to @xmath98 and @xmath99 and ( b ) with respect to @xmath100 and @xmath101 . the symbols af , c , f , and m indicate the states @xmath76 , @xmath81 , @xmath102 , and @xmath103 , respectively . in ( b ) , @xmath104 is assumed . , width=317 ]    we have assumed here that one of the states @xmath76 , @xmath81 , @xmath102 , and @xmath103 is the ground state , based on the numerical results of ref . we have confirmed that this assumption is correct using the monte carlo method .    in the case of @xmath2 , @xmath105 is positive and @xmath106 is at most of the same order of magnitude as @xmath105  @xcite . hence , the ground state of the spin-2 @xmath2 bec is antiferromagnetic or cyclic depending on whether @xmath74 or @xmath77 . therefore , in order to determine the sign of @xmath107 from the ground state , we must suppress the magnetic field so that the condition @xmath108 is met . in the experiment of the hamburg group  @xcite , a magnetic field of @xmath109 mg was applied , which corresponds to @xmath110 nk . the mean atomic density was @xmath111 , and if @xmath112 , we have @xmath113 nk . thus , @xmath114 , which is consistent with the observed stability of the antiferromagnetic state @xmath76 . in the gakushuin experiment  @xcite , @xmath115 mg and @xmath116 , corresponding to @xmath117 nk and @xmath118 nk . hence , the reported parameters of ref .  @xcite belong to the cyclic phase . the antiferromagnetic ( or stretched ) state , @xmath119 was found in experiments to be stable in the presence of a magnetic field  @xcite . to examine this result , we will investigate the stability of this state using bogoliubov analysis . substituting eq . ( [ polar ] ) into eqs . ( [ gp ] ) , we find that it evolves as @xmath120 with @xmath121 , and hence eq . ( [ polar ] ) is stationary in the rotating frame @xmath122 . we can therefore perform bogoliubov analysis with respect to the state ( [ polar ] ) in the rotating frame , which is described by the gp equations ( [ gp ] ) without the linear zeeman terms . we set the wave function as @xmath123 and substitute it into the gp equations in the rotating frame . taking the linear terms with respect to @xmath124 , we obtain    [ bogo ] @xmath125    these equations can be reduced to the eigenvalue problem by expansion of @xmath124 as @xmath126 and the eigenenergies are obtained as    [ eigen ] @xmath127^{1/2 } , \label{mode1 } \\ & & \left [ \varepsilon_k \left ( \varepsilon_k + 8 \tilde c_1 - 2 \tilde c_2 / 5 \right ) \right]^{1/2 } , \label{mode2 } \\ & & \left [ ( \varepsilon_k - 3q ) \left ( \varepsilon_k - 3q + 2 \tilde c_1 - 2 \tilde c_2 / 5 \right ) \right]^{1/2 } , \label{mode3 } \\ & & \left [ ( \varepsilon_k - 4q ) \left ( \varepsilon_k - 4q - 2 \tilde c_2 / 5 \right ) \right]^{1/2 } , \label{mode4}\end{aligned}\ ] ]    where @xmath128 .    if the eigenenergy is complex , the corresponding mode is dynamically unstable against exponential growth . the eigenvectors @xmath129 and @xmath130 of the first mode  ( [ mode1 ] ) are proportional to @xmath131 , which is the same form as @xmath132 , indicating that a collapse occurs if @xmath133 . the eigenvectors of the second mode  ( [ mode2 ] ) are proportional to @xmath134 . since this eigenmode transfers the @xmath11 component to the @xmath135 component , excitation of the mode  ( [ mode2 ] ) gives rise to exchange of atoms between the @xmath11 components . this implies that phase separation between the two components occurs for @xmath136 . therefore , the fact that no phase separation is observed in experiments  @xcite only indicates that @xmath137 ; it is not sufficient to conclude that @xmath138 . the third mode  ( [ mode3 ] ) is two - fold degenerate and the eigenvectors are proportional to @xmath139 and @xmath140 . therefore , when @xmath141 , the state ( [ polar ] ) is dynamically unstable against the growth of the @xmath142 components . similarly the last mode  ( [ mode4 ] ) is proportional to @xmath143 and describes the growth of the @xmath12 component . dynamical instability arises in this mode for @xmath77 , consistent with the fact that the ground state has the @xmath12 component for @xmath77 , as shown by eq . ( [ cyclic ] ) .    for later use , we also perform bogoliubov analysis for the stationary state @xmath144 . the eigenenergies are obtained to be    [ eigen2 ] @xmath127^{1/2 } , \label{mode2_1 } \\ & & \left [ ( \varepsilon_k + q ) \left ( \varepsilon_k + q + 6 \tilde c_1 - 2 \tilde c_2 / 5 \right ) \right]^{1/2 } , \label{mode2_2 } \\ & & \left [ ( \varepsilon_k + 4q ) \left ( \varepsilon_k + 4q - 2 \tilde c_2 / 5 \right ) \right]^{1/2}. \label{mode2_3}\end{aligned}\ ] ]    the eigenvector of the first mode ( [ mode2_1 ] ) is proportional to @xmath145 , implying that the state @xmath146 collapses if @xmath147 . we note that this condition for the collapse of the bec is the same as that for @xmath132 . the second and third modes , eqs . ( [ mode2_2 ] ) and ( [ mode2_3 ] ) , are both two - fold degenerate . the eigenvectors of the mode ( [ mode2_2 ] ) are proportional to @xmath148 and @xmath140 , and therefore the dynamical instability in this mode increases the @xmath149 components . similarly , the mode ( [ mode2_3 ] ) has eigenvectors proportional to @xmath150 and @xmath151 . in contrast to the excitations for @xmath132 , there are always dynamically unstable modes in eqs . ( [ mode2_2 ] ) and ( [ mode2_3 ] ) because of @xmath36 , and therefore the @xmath12 state is always dynamically unstable for @xmath152 in a homogeneous system . in order to determine the value of @xmath107 from the spin dynamics , we consider the spin dependent interaction that is sensitive only to the spin - singlet term . the elementary processes in the spin - singlet channel are @xmath153 , @xmath154 , @xmath155 , and @xmath156 . on the other hand , the @xmath105 term consists of elementary processes such as @xmath157 . the spin independent interaction does not flip the spin . thus , the elementary process appearing only in the @xmath107 term is @xmath155 . we therefore focus on this process and consider the initial state of the form @xmath158 since the overall phase and spin rotation about the @xmath47 axis do not affect the physics , the relevant phase appears only in the combination of @xmath159 . we therefore assume that @xmath160 and @xmath161 without loss of generality . if @xmath162 are exactly zero in the initial state , we find from eq . ( [ gp2 ] ) that @xmath163 are always zero within the mean field approximation . we therefore assume @xmath164 in the following analysis . in experiments @xcite , this assumption holds at least for an initial period of @xmath10 ms . although the @xmath149 components exponentially grow in the presence of the dynamical instability , the assumption @xmath164 is still valid in the early stage of time evolution . the gp equations then reduce to    [ gp02 ] @xmath165    differentiating eq . ( [ gp02_0 ] ) with respect to time and using eq . ( [ gp02_1 ] ) , we can eliminate @xmath166 to obtain @xmath167 where the energy per atom , @xmath168 is a constant of motion . it is interesting to note that the form of eq . ( [ eqzeta0 ] ) coincides with that describing a 1d bec on a rotating ring if the time derivative is replaced with the spatial derivative  @xcite . the solution is obtained ( see appendix for derivation ) as @xmath169 where @xmath170 , @xmath171 , @xmath172 , @xmath173 , @xmath174 , and @xmath175 are given by eqs . ( [ phi0 ] ) , ( [ a0org ] ) , ( [ eqb0 ] ) , ( [ alphorg ] ) , ( [ beta0org ] ) , and ( [ nuorg ] ) , respectively . the function sn is a jacobian elliptic function  @xcite . if we assume @xmath176 in the initial state , i.e. , @xmath177 the constants in the solution ( [ sol1 ] ) become @xmath178 , @xmath179 , and @xmath180 , \label{a0 } \\ \alpha & = & \sqrt{\frac{8q}{5 \hbar^2 } \left\ { \tilde c_2 \left [ 1 - 2 \zeta_0 ^ 2(0 ) \right ] + 10 q \right\ } } , \label{alpha } \\ \nu & = & \frac{\tilde c_2 ^ 2 \zeta_0 ^ 2(0 ) \left [ 1 - \zeta_0 ^ 2(0 ) \right]}{10 q \left\ { \tilde c_2 \left [ 1 - 2 \zeta_0 ^ 2(0 ) \right ] + 10 q \right\}}. \label{m}\end{aligned}\ ] ] the sign of @xmath171 is opposite to that of @xmath107 because @xmath36 . the constants @xmath173 and @xmath175 are real and positive if @xmath181 + 10 q < 0 $ ] . we suppose that the initial state ( [ init ] ) is prepared and the magnetic field satisfies @xmath182 . under this condition , we find that @xmath173 and @xmath175 in eqs . ( [ alpha ] ) and ( [ m ] ) are real and positive . since the atomic density is typically @xmath183 in experiments  @xcite and @xmath184 is of order of the bohr radius at most  @xcite , the condition @xmath182 requires @xmath185 mg . at this magnetic field , the ground state in the subspace of @xmath54 is the stretched state ( [ polar ] ) irrespective of the sign of @xmath107 . the jacobian elliptic function @xmath186 initially increases for positive @xmath173 and @xmath175 , and hence the @xmath12 component @xmath187 in eq . ( [ sol1 ] ) initially decreases for @xmath188 and increases for @xmath189 . therefore , we can determine the sign of @xmath107 from the sign of the initial change in the spin population , which is the main result of this paper . this method is suitable for atomic species with short lifetimes , since only the initial stage of time evolution is needed for the determination of @xmath107 .    figure  [ f : unifev ] shows time evolution of the spin populations which are obtained by numerically solving eq . ( [ gp02 ] ) . @xcite and a magnetic field of 250 mg . the scattering lengths are assumed to be @xmath190 in ( a ) and @xmath191 in ( b ) , where @xmath192 is the bohr radius . the initial state is chosen to be @xmath193 , @xmath194 , and @xmath195 . the dashed curves are drawn according to eq . ( [ sin ] ) . , width=317 ]    the results confirm the relation between the sign of @xmath107 and the sign of the initial change in the spin populations . in fig . [ f : unifev ] , the @xmath149 components are assumed to have small initial values ( @xmath195 ) to assess imperfect population transfer in realistic situations . we find that the dynamical instability in the @xmath149 components have little effect on the initial dynamics of the @xmath12 , @xmath196 components .    in fig . [ f : unifev ] , @xmath197 . for @xmath198 , the jacobian elliptic function can be approximated by the trigonometric function as  @xcite @xmath199 and the spin evolution is then approximated by    [ sin ] @xmath200    the amplitude of this oscillation is @xmath201 , showing explicitly that whether the spin population initially increases or decreases depends on the sign of @xmath107 . we can also determine the magnitude of @xmath107 by measuring the amplitude of the oscillations . in fig . [ f : unifev ] , eq . ( [ sin ] ) is plotted as dashed curves , which are in good agreement with the numerical solutions ( solid curves ) in the early stages of the time evolution .    the differences between the numerical and analytic curves in fig . [ f : unifev ] arise mainly from the growth of the @xmath149 components in the numerical result due to the dynamical instability . we find that the growth of the @xmath149 components in fig . [ f : unifev ] ( b ) is larger than that in fig . [ f : unifev ] ( a ) . this can qualitatively be understood from the bogoliubov energy ( [ mode2_2 ] ) , which describes the growth of the @xmath149 components from the state @xmath144 . the imaginary part of eq . ( [ mode2_2 ] ) at @xmath202 , @xmath203^{1/2}$ ] , is larger for @xmath189 than for @xmath188 since @xmath36 and @xmath204 in the present case . we note that the relative phase @xmath205 in the initial state ( [ initz ] ) is crucial for determining the value of @xmath107 by the above method . for example , when the initial state is given by eq . ( [ initz ] ) with @xmath206 , or without loss of generality , @xmath207 the time evolution becomes eq . ( [ sol1 ] ) , where @xmath171 , @xmath175 , and @xmath173 are given by eqs . ( [ samea0 ] ) , ( [ samenu ] ) , and ( [ samealph ] ) , respectively ( see appendix for details ) . these constants satisfy @xmath208 and @xmath209 , and @xmath173 is real . using the relation  @xcite @xmath210 and noting that the jacobian elliptic function @xmath211 initially increases , we find that @xmath187 initially decreases for @xmath189 and increases for @xmath188 . this behavior is opposite to that in fig . [ f : unifev ] and therefore control of the relative phase @xmath212 in the initial state is crucial for determining the sign of @xmath107 . figure  [ f : phase ] shows the initial evolution of @xmath213 with @xmath212 varying from @xmath0 to @xmath214 , which indicates that the phase of the oscillation depends on @xmath212 .     with the same parameters as in figs .  [ f : unifev ] ( a ) and ( b ) . the initial state is given by @xmath193 , @xmath215 , @xmath216 , and @xmath217 , where @xmath218 is an integer ranging from 0 to 4 . , width=317 ]      if @xmath219 , the stretched state ( [ polar ] ) is dynamically unstable against growth of the @xmath12 component , as shown by eq . ( [ mode4 ] ) . this is because the ground state is in the cyclic phase given by eq . ( [ zeta ] ) if @xmath219 and @xmath137 . we can therefore conclude that @xmath220 and hence @xmath221 , if we observe the growth of the @xmath12 component from the initial stretched state . suppose that @xmath222 in the initial state given by eq . ( [ init ] ) and @xmath223 , then @xmath175 is negative and @xmath173 is pure imaginary from eqs . ( [ alpha ] ) and ( [ m ] ) . in this case , the jacobian elliptic function can be rewritten as @xmath224 since @xmath225 and hence @xmath226 , we can make the approximation  @xcite @xmath227 where @xmath228 . when @xmath229 , eq . ( [ sdeq ] ) reduces to @xmath230 equation  ( [ sol1 ] ) is thus approximated to give @xmath231 the dashed curve in fig . [ f : cyclic ] shows eq . ( [ tanh ] ) . the initial state is given by @xmath232 , @xmath233 , @xmath234 . the dashed curve is drawn according to eq . ( [ tanh ] ) . , width=317 ]    a comparison of this with the corresponding numerical result ( solid curve ) shows that eq . ( [ tanh ] ) is in good agreement with the numerical result , with a small discrepancy arising mainly from the growth of the @xmath235 components in the numerical result due to the dynamical instability . the @xmath12 component reaches a maximum at @xmath236 . we note that a characteristic time scale of the dynamics in fig . [ f : cyclic ] is @xmath10 ms , which is much larger than the @xmath237 ms seen in fig . [ f : unifev ] . this is because the former time scale is set by the dynamical instability , while the latter is set by the time evolution from a non - stationary state . this method can thus determine the value of @xmath107 , provided that ( i ) @xmath107 is positive , ( ii ) the magnetic field is suppressed so that @xmath238 , and ( iii ) the time scale of the dynamics is much faster than the lifetime of the @xmath7 state . if @xmath107 is negative , the present method can not determine the value of @xmath107 .    in the case of fig . [ f : cyclic ] ( @xmath219 ) , the @xmath239 population initially increases for @xmath188 , which is opposite to the case of fig .  [ f : unifev ] ( @xmath182 ) . nevertheless , the method to determine the sign of @xmath107 in sec . [ ss : main ] is still applicable , since we can easily find whether we are in a region of @xmath240 or @xmath219 . if the amplitude of the initial oscillation monotonically decreases without changing sign with an increase in the magnetic field , we can conclude that @xmath240 . we investigate the spin dynamics in a trap potential to verify the results in sec . [ s : homo ] in a realistic situation . we use an axisymmetric trap potential with radial and axial frequencies @xmath241 hz , which was used in the experiment of ref . the two - body loss rate was found to be roughly independent of the magnetic field and therefore it is considered to be insensitive to the spin - mixing dynamics  @xcite . we therefore take into account the two - body loss by adding the term @xmath242 to the right - hand side of eq . ( [ gp ] ) with @xmath243 . we prepare the ground state @xmath244 for the @xmath245 state by the imaginary - time propagating method  @xcite and transfer it to each spin component as @xmath246 , @xmath247 , and @xmath248 . we use this state as an initial state , where the initial number of atoms is assumed to be @xmath249  @xcite .    the time evolution of each spin population obtained by numerically solving the gp equation  ( [ gp ] ) with the two - body loss terms is shown in fig .  [ f : trap ] .     of each spin component @xmath250 in the trap with @xmath251 hz and @xmath252 hz  @xcite . the scattering lengths are assumed to be @xmath253 in ( a ) and @xmath254 in ( b ) , where @xmath192 is the bohr radius . the strength of the magnetic field is 250 mg , and the two - body loss coefficient is @xmath255  @xcite . the @xmath245 ground state @xmath244 is prepared , and is initially transferred to each spin component as @xmath246 , @xmath256 , and @xmath257 . the insets show the column density distributions of the @xmath258 components from top to bottom at @xmath259 ms , each having a @xmath260 field of view . , width=317 ]    we find that the aforementioned dependence of the initial spin evolution on the sign of @xmath107 is valid in the trapped system , i.e. , the quantity @xmath261 first decreases for @xmath188 and increases for @xmath189 . the growth in @xmath262 due to the dynamical instability and the two - body loss does not affect this dependence . thus , we have confirmed that the sign of @xmath107 can be determined from the spin dynamics in a realistic situation . we find that the growth of the @xmath149 components is larger for @xmath263 than for @xmath188 , which is similar to the homogeneous case shown in fig . [ f : unifev ] .    the insets in fig . [ f : trap ] show the column density distribution of each spin component at @xmath259 ms . the initial distribution is almost preserved at @xmath259 ms and no spin domains are formed . as shown in fig .  [ f : phase ] , our method to determine the sign of @xmath107 hinges on the relative phase @xmath264 in the initial state ( [ initz ] ) . therefore we must control the phase experimentally . in the experiments of refs . @xcite , the rapid - adiabatic - passage technique  @xcite was employed to prepare the initial state . this method can control the phase in the initial state in principle , if the relative phases between the applied rf fields are controlled . we propose here a new method to prepare the initial state using the raman transition . we consider the raman transition , for example , between the @xmath7 and @xmath265 @xmath266 states by circularly polarized photons , as illustrated in fig . [ f : level ] .     and @xmath267 @xmath266 states . the circularly polarized light @xmath268 changes @xmath250 by @xmath269 and hence starting from the @xmath7 , @xmath12 state , the system undergoes transitions only to @xmath267 , @xmath149 and @xmath7 , @xmath11 states with the clebsch - gordan coefficients given by @xmath270 and @xmath271 , respectively . , width=317 ]    the initial state is assumed to be the @xmath12 lower state . hence , the relevant states are the @xmath12 , @xmath196 lower states and the @xmath149 upper states , whose amplitudes are denoted by @xmath272 , @xmath166 , and @xmath273 , respectively . the equations of motion are given by    [ raman1 ] @xmath274    where @xmath275 and @xmath276 are the coupling constants and frequencies of the @xmath268 photons , @xmath277 is the transition energy of the @xmath266 line , and @xmath278 and @xmath279 are the linear and quadratic zeeman energies for the upper hyperfine manifold . the coupling constants @xmath275 are given by the product of the amplitude of the circularly polarized light and the dipole matrix element between the @xmath17 and @xmath280 states . using the new variables @xmath281 and @xmath282 , the coefficients on the right - hand side of eq . ( [ raman1 ] ) become time independent . assuming @xmath283 and @xmath284 , and eliminating @xmath273 from eq . ( [ raman1 ] ) , we obtain    [ raman2 ] @xmath285    where @xmath286 . if the amplitudes of the @xmath268 fields are the same , i.e. , @xmath287 is met , the coefficients of the first terms on the right - hand side of eq . ( [ raman2 ] ) become equal , i.e. , @xmath288 . the solutions are thus obtained as    @xmath289 , which is independent of the phase of the circularly polarized light and the applied magnetic field . this method is thus suitable for preparing the initial state ( [ init ] ) . we can check whether the prepared state has the correct relative phase @xmath290 . rotating the spin state ( [ initz ] ) by @xmath291 around the axis on the @xmath292-@xmath293 plane as @xmath294 with @xmath295 , we find that the population of each component becomes    @xmath296    where @xmath297 . let us consider the quantity @xmath298 which can be measured by the spin - dependent non - destructive imaging reported in ref . we note that if @xmath290 , eq .  ( [ fz2 ] ) is independent of @xmath299 and is constant , while if @xmath300 , eq .  ( [ fz2 ] ) oscillates , since the phase @xmath301 rotates in the magnetic field . we can therefore confirm that the desired initial state ( [ init ] ) is obtained if the result of the polarization - dependent phase - contrast imaging  @xcite is constant in time . we have proposed a method to experimentally determine the sign of the singlet - pair energy of a spin-2 bec from spin exchange dynamics , which determines whether the @xmath7 @xmath2 bec at zero magnetic field is antiferromagnetic or cyclic . we have obtained analytic solutions of the multicomponent gp equations and found that if we prepare the initial state in the magnetic sublevels @xmath12 , @xmath196 with the appropriate relative phase relationship , we can determine the sign of @xmath107 from the sign of the initial change in the spin populations . for example , in the case of the initial state in eq . ( [ initz ] ) , we can conclude that @xmath107 is positive or negative if the @xmath12 component first decreases or increases . we can also determine the magnitude of @xmath107 from the amplitude of the oscillation in the spin populations . since only the initial evolution for a period of @xmath302 ms is needed , we can use this method in the presence of atom loss by inelastic collisions and even in the presence of dynamical instabilities . we have numerically confirmed that this method is applicable for a trapped system with a realistic two - body loss coefficient @xmath303 . the required condition for the magnetic field is @xmath240 , so our method works even when the quadratic zeeman energy exceeds the singlet - pair energy . the initial spin state in which the populations and relative phases are controlled as eq . ( [ initz ] ) can be prepared by , e.g. , the raman technique as shown in sec . [ s : exp ] . we have shown that the coherent spin dynamics serve as an efficient probe for investigating spinor properties especially when the atomic species have a short lifetime and therefore the equilibrium state can not be reached . the coherent spin dynamics may also reveal other equilibrium spinor properties as well as nonequilibrium spinor properties that can not be obtained from the static equilibrium . we would like to thank t. kuwamoto and t. hirano for fruitful discussions . this work was supported by grant - in - aids for scientific research ( grant no . 17740263 and no . 15340129 ) and by a 21st century coe program at tokyo tech `` nanometer - scale quantum physics , '' from the ministry of education , culture , sports , science and technology of japan . in this appendix , we solve eq . ( [ gp02 ] ) for the initial condition ( [ initz ] ) with @xmath160 and @xmath304 , and derive the solution ( [ sol1 ] ) . defining a new variable @xmath305 we can rewrite eq . ( [ eqzeta0 ] ) as @xmath306 where @xmath307 with the spin - dependent energy @xmath308      we write the complex variable as @xmath312 with real functions @xmath313 and @xmath314 , where @xmath315 and @xmath316 . substituting this into eq . ( [ eqz0 ] ) and taking the real and imaginary parts , we obtain @xmath317 these equations can be integrated to give @xmath318 where @xmath319 and @xmath320 are constants of integration . the time derivative of @xmath309 at @xmath321 , @xmath322 , gives @xmath323 and @xmath324 . the constant @xmath320 is then written as @xmath325 the constant @xmath319 is similarly obtained as @xmath326    we introduce a new variable @xmath327 through @xmath328 where @xmath171 and @xmath172 are constants . substituting eq . ( [ rho ] ) into eq . ( [ rdot2 ] ) , we obtain @xmath329 we determine @xmath172 so that the @xmath330 term in eq .  ( [ rhoeq ] ) vanishes : @xmath331 the right - hand side ( rhs ) of eq . ( [ rhoeq ] ) then becomes quartic with respect to @xmath327 . we determine @xmath171 so that the rhs of eq . ( [ rhoeq ] ) becomes proportional to @xmath332 ; the result is @xmath333 \nonumber \\\end{aligned}\ ] ] with @xmath334 equation ( [ rhoeq ] ) thus becomes @xmath335 this equation can be integrated to give @xmath336 we rewrite this equation as @xmath337 where @xmath338 the left - hand side of eq . ( [ intrho ] ) is an elliptic integral of the first kind and hence @xmath339  @xcite . thus , @xmath340 from eqs . ( [ z0 ] ) and ( [ phidot ] ) , the phase @xmath341 of @xmath272 is given by @xmath342    similarly , we can obtain the solution for @xmath343 . from the form of eq . ( [ gp02 ] ) and the initial condition @xmath344 , we find that the relation @xmath345 always holds . hence , we consider only @xmath346 . eliminating @xmath347 from eq . ( [ gp02 ] ) , we have the equation of motion for @xmath346 as @xmath348 \zeta_2 - i \frac{2 \tilde c_2}{5 } \hbar \dot{\zeta}_2.\ ] ] the new variable @xmath349 reduces eq . ( [ ddotz2 ] ) to @xmath350 where @xmath351 and @xmath352 . the initial conditions for @xmath353 are @xmath354 and @xmath355 - \frac{4 i q}{\hbar } \zeta_2(0).\ ] ] equation ( [ z2eq ] ) has the same form as eq . ( [ eqz0 ] ) , in which @xmath356 and @xmath357 are replaced by @xmath358 and @xmath359 , and therefore the derivation from eq . ( [ rphi1 ] ) to eq . ( [ phi0 ] ) can similarly be applied . we consider the case of @xmath176 in the initial condition given in eq . ( [ initz ] ) . in this case , eq . ( [ zdot0 ] ) is pure imaginary , and eq . ( [ eqb0 ] ) can be solved to give @xmath360 . equation ( [ a0org ] ) then reduces to @xmath361 , \end{array } \right.\ ] ] where the upper and lower expressions correspond to the plus and minus signs of @xmath362 in eq . ( [ a0org ] ) . using this @xmath171 , @xmath175 and @xmath173 become @xmath363 + 10 q \right\}}{\displaystyle \tilde c_2 ^ 2 \zeta_0 ^ 2(0 ) \left [ 1 - \zeta_0 ^ 2(0 ) \right ] } , \\ \displaystyle \frac{\tilde c_2 ^ 2 \zeta_0 ^ 2(0 ) \left [ 1 - \zeta_0 ^ 2(0 ) \right]}{\displaystyle 10 q \left\ { \tilde c_2 \left [ 1 - 2 \zeta_0 ^ 2(0 ) \right ] + 10 q \right\ } } , \end{array } \right . \\ \label{appalph } \alpha & = & \left\ { \begin{array}{l } \displaystyle \frac{2 |\tilde c_2|}{5 \hbar } \sqrt{\zeta_0 ^ 2(0 ) \left [ 1 - \zeta_0 ^ 2(0 ) \right ] } , \\ \displaystyle \sqrt{\frac{8q}{5 \hbar^2 } \left\ { \tilde c_2 \left [ 1 - 2 \zeta_0 ^ 2(0 ) \right ] + 10 q \right\}}. \end{array } \right.\end{aligned}\ ] ] it follows from eq . ( [ rho ] ) and @xmath364 that @xmath365 and hence @xmath178 . using the relation @xmath366 we can show that the two expressions in eqs . ( [ appa0 ] ) , ( [ appnu ] ) , and ( [ appalph ] ) are equivalent to each other . we used the lower expressions in sec . [ s : homo ] . the @xmath11 components @xmath367 have the same form as eq . ( [ zeta0 ] ) , in which @xmath171 , @xmath174 , and @xmath172 are replaced by @xmath368 , @xmath369 , and @xmath172 . these constants are given by @xmath370 , \end{array } \right.\ ] ] @xmath371 , and @xmath372 . finally , we consider the case of @xmath373 in the initial condition ( [ initz ] ) , i.e. , all the components are real and positive . we can solve eq . ( [ eqb0 ] ) also in this case to give @xmath364 and hence @xmath178 . the constants @xmath171 , @xmath175 , and @xmath173 reduce to @xmath374 + 10 q \right\ } - |q| \sqrt{(\tilde c_2 + 10 q)^2 - 40 \tilde c_2 q \zeta_0 ^ 2(0 ) } \right\ } , % % % \nonumber \\ \\ \label{samenu } \nu & = & \frac{q \left\ { \tilde c_2 [ 1 - 2 \zeta_0 ^ 2(0 ) ] + 10 q \right\ } - |q| \sqrt{(\tilde c_2 + 10 q)^2 - 40 \tilde c_2 q \zeta_0 ^ 2(0)}}{q \left\ { \tilde c_2 [ 1 - 2 \zeta_0 ^ 2(0 ) ] + 10 q \right\ } + |q| \sqrt{(\tilde c_2 + 10 q)^2 - 40 \tilde c_2 q \zeta_0 ^ 2(0 ) } } , \\ \label{samealph } \alpha & = & \sqrt{\frac{8 \tilde c_2 q a_0}{5 \hbar^2 \nu}},\end{aligned}\ ] ] where we take the lower sign in eq . ( [ a0org ] ) . since @xmath375 + 10 q \right\}^2 < ( \tilde c_2 + 10 q)^2 - 40 \tilde c_2 q \zeta_0 ^ 2(0)$ ] , we find @xmath376 and @xmath209 .
we propose a method to determine the singlet - pair energy of a spin-2 bose - einstein condensate ( bec ) . by preparing the initial populations in the magnetic sublevels @xmath0 and @xmath1 with appropriate relative phases , we can obtain the coefficient of the spin singlet - pair term from the spin exchange dynamics . 
 this method is suitable for hyperfine states with short lifetimes , since only the initial change in the population of each magnetic sublevel is needed . 
 this method therefore enables the determination of the ground state phase of a spin-2 @xmath2 bec at zero magnetic field , which is considered to lie in the immediate vicinity of the boundary between the antiferromagnetic and cyclic phases . 
 we also show that the initial state in which relative phases are controlled can be prepared by raman processes .