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a spinor bose - einstein condensate is a system that possesses both superfluid and magnetic order @xcite . experiments are able to explore transitions between phases of different magnetic order by controlling the quadratic zeeman energy ( @xmath0 ) that shifts the magnetic sublevels @xcite . such a system presents a rich playground for studies of non - equilibrium phenomena , including defect formation related to the rate that the phase transition is crossed @xcite , through to the late - time coarsening dynamics describing how the small domains produced by the quench anneal towards the equilibrium state @xcite . studies have largely focused on the spin properties of the system , leaving the superfluid dynamics unexplored . we note that in binary condensates , which can be regarded as a pseudo spin-@xmath1 system , there has been work on both spin coarsening @xcite and aspects of the superfluid behavior ( e.g.  analyzed via kinetic energy spectra @xcite ) . the development of various techniques for measuring spin @xcite and superfluid correlations ( e.g.  see @xcite ) demonstrate that it is possible to study the evolution of both types of order in spinor experiments . the relationship between spin and superfluid order has been considered in the equilibrium properties of spinor condensates . for example , in ref .  @xcite it was found that for a large spin - dependent interaction , ferromagnetism emerges at a higher temperature than condensation .    here we study the superfluid and spin ordering dynamics of a ferromagnetic spin-1 condensate quenched into a easy - axis ferromagnetic phase , where the spin order preferentially aligns ( or anti - aligns ) along the quantization axis defining the quadratic zeeman energy shift . initially small domains form , but grow as time passes . at sufficiently late times this coarsening dynamics can enter a universal scaling regime @xcite : the correlation function of the order parameter collapses to a universal ( time - independent ) form when space is scaled by a characteristic length @xmath2 , which yields the dynamic critical exponent @xmath3 . here we find that the spin and superfluid order exhibit universal scaling , but do not develop identically . the spin order spreads more rapidly across the system than the superfluid order , with different dynamic critical exponents for each type of ordering .    in this paper we consider two types of quenches . the first ( standard ) quench ( e.g.  see @xcite ) is to start with an equilibrium non - magnetic ( polar ) condensate at large @xmath0 and then to suddenly quench to a negative @xmath0 value where the easy - axis magnetic phase is the ground state . we also propose a second type of quench that utilizes the atomic physics toolbox of coherent manipulations : this differs from the first quench in that a @xmath4 spin rotation is applied to the initial state immediately prior to @xmath0 being quenched . both quenches produce the same late time coarsening behavior , but we show that the second quench has less heating and the early - time dynamics ( where local order develops ) concludes more rapidly , suggesting this may be more suitable for experiments where long times are difficult to access . we finally consider the areas of the individual spin domains that form after the quench and how these evolve in time . our results show that once the domains develop in the system their areas vary over many orders of magnitude . at sufficiently long times the domain area distribution decays as a power law @xmath5 for sufficiently large @xmath6 , where @xmath6 is the domain area . this can be related to ideas of percolation theory @xcite , as has also been pointed in in recent work on the immiscibility phase transition in a binary condensate @xcite . the outline of the paper is as follows . in sec . [ secformalism ] we introduce the dynamical formalism used to simulate the quench dynamics . we introduce the two different quench types considered and discuss the role and nature of the unstable excitations that drive the initial phase transition dynamics . our main results are presented in sec . [ secresults ] . first we consider the early - time dynamics to emphasize the differences in how local spin order emerges for the two quench types . we then turn to considering the spin and superfluid order , as described by the relevant two - point correlation functions . we demonstrate that in the late time dynamics these correlation functions exhibit dynamical scaling , albeit with different growth laws . then we compute the spin domain areas , extract the domain size distribution and investigate the scaling of this distribution as a function of domain size and time . finally we conclude in sec . [ secconclude ] . the system we consider is a homogeneous quasi - two - dimensional ( quasi-2d ) spin-1 condensate described by the hamiltonian  @xcite@xmath7\!.\end{aligned}\ ] ] here @xmath8 is a three component spinor describing the condensate amplitude in the three spin hyperfine sublevels ( @xmath9 ) and @xmath0 is the quadratic zeeman shift arising from the presence of an external field along @xmath3 . the interactions are described by a density dependent term @xmath10 and a spin - density dependent term @xmath11 , where @xmath12 and @xmath13 are density - dependent and spin - dependent coupling constants , @xmath14 is the total number density , and @xmath15 is the spin density , with @xmath16 being the spin-1 matrices . the dynamics of the system is described by the spin-1 gross - pitaevskii equation ( gpe ) @xmath17    for the system to be mechanically stable we require @xmath18 , and we additionally restrict our attention here to the case of ferromagnetic interactions , i.e.  @xmath19 , as realized in @xmath20rb condensates @xcite . in spinor condensate experiments the quasi-2d regime has been realized by using a trapping potential with tight confinement in one direction ( e.g.  see @xcite ) . our interest is in homogeneous systems where the phase transition dynamics are simpler , noting that recent experiments have realized flat - bottomed optical traps for this purpose @xcite ( also see  @xcite ) . magnetic phase diagram for a ferromagnetic spin-1 condensate with zero @xmath3-magnetization . the spheres show the direction of magnetization in the three states . ( a ) for @xmath21 the magnetization lies along the @xmath22 axis and the state is termed _ easy - axis_. ( b ) for @xmath23 the magnetization lies in the transverse ( @xmath24-@xmath25 ) plane and the state is termed _ easy - plane_. quenches to this phase are not analyzed here but can be found in refs . ( c ) for @xmath26 the @xmath27 levels are unoccupied and the system is unmagnetized . this state is termed _ polar_. the directed line on the phase diagram indicates the quench we consider in this paper , where the zeeman energy is suddenly reduced from an initial value @xmath28 where the ground state is polar , to a value @xmath21 where the system favours easy - axis spin ordering.,scaledwidth=50.0% ]    the ground state of eq . ( [ spinh ] ) depends on the value of @xmath0 ( e.g.  see @xcite ) relative to the characteristic spin energy @xmath29 , where @xmath30 is the ( uniform ) condensate density . a schematic phase diagram and representation of the ground states important to this paper are shown in fig . [ phasediag ] . there are two ferromagnetic phases ( a ) and ( b ) , which differ in their symmetries , and a non - magnetized polar phase ( c ) . of the two ferromagnetic phases we only consider the easy - axis phase here as this phase admits well - defined spin domains . we briefly review the relevant order parameters for our simulations , which we express in the form @xmath31 where @xmath32 is a global ( superfluid ) phase , and @xmath33 is a normalized spinor ( i.e.  @xmath34 ) . for high values of the quadratic zeeman energy ( @xmath26 ) the the ground state is the non - magnetized polar phase , with normalized spinor @xmath35 for which @xmath36 . this state , or a spin rotated form of this state , is the initial condition for our quench . our interest is how the system then reorders when @xmath0 is quenched to a negative value , where the ground state is an easy - axis ferromagnetic phase with normalized spinor @xmath37 corresponding to magnetization of + 1 or -1 along @xmath3 , respectively . these states break the @xmath38 ( @xmath3-reflection ) symmetry of the hamiltonian . we now introduce the two types of quench we use to transition the system from the polar to the easy - axis phase . + * quench 1 ( q1 ) * : in this quench the quadratic zeeman energy is simply set to a negative value at @xmath39 . here the polar state is dynamically unstable and decays , initially by developing transverse magnetization . in this case an energy per particle of @xmath40 is liberated ( i.e.  the excess energy that the polar state has over the easy - axis ferromagnetic state for @xmath21 ) , and available to heat the system . + * quench 2 ( q2 ) * : in this quench a spin rotation of @xmath41 , is performed producing a so - called anti - ferromagnetic state : @xmath42 after the spin rotation has been preformed the quadratic zeeman energy is immediately set to a negative value . we note that this rotation ( using rf pulses ) and quench to negative @xmath0 in a spin-1 experiment has been reported in ref . @xcite .    for this initial condition the energy per particle available to heat the system is smaller than for q1 and is independent of @xmath0 ( for @xmath21 ) , i.e.   @xmath43       ( color online ) bogoliubov spectra for the post - quench initial states emphasizing the dynamically unstable modes . spectra of the @xmath44 initial state produced by the q1 quench at ( a ) @xmath45 and ( b ) @xmath46 . spectra of the @xmath47 initial state produced by the q2 quench at ( a ) @xmath45 and ( b ) @xmath46 . the lines indicate the real parts of the energies , while the dashed lines indicate the imaginary part . the phonon ( @xmath48 ) and magnon ( @xmath49 ) excitation branches are labelled in the plots , noting that the magnon branches are degenerate in ( a ) and ( b ) . these results are for the case @xmath50 . , scaledwidth=50.0% ]    insight into the initial post - quench dynamics is also provided through understanding the bogoliubov quasi - particle excitations of the initial state . for a spin-1 system there are three excitation branches , which can be typically classified as a phonon branch and two magnon ( or spin - wave ) branches . in discussing these modes we adopt the naming conventions used in @xcite and refer to the phonon branch as @xmath51 and the magnon branches as @xmath52 . the instabilities of the initial state following the quench are revealed by the imaginary parts in the quasi - particle energies @xmath53 . as our system is mechanically stable ( and @xmath54 ) the phonon branch always remains stable , and exhibits a rapidly rising linear spectrum @xmath55 , where @xmath56 is the speed of sound .      for the q1 quench the initial non - equilibrium state ( p - phase ) at @xmath21 has a bogoliubov spectrum with two unstable magnon branches . for our case of zero magnetization along @xmath3 these magnon branches and a condensate with zero @xmath3-magnetization . ] are degenerate with the dispersion relation @xmath57 where @xmath58 . for @xmath21 these magnon branches have dynamically unstable modes ( i.e.  the quasi - particle energies @xmath59 are imaginary ) for the range of @xmath60 values @xmath61 these magnon modes have amplitude in the @xmath62 sub - levels and lead to the decay of the polar phase by the development of transverse magnetization . as the value of @xmath0 changes the character of the unstable modes remains qualitatively the same [ see figs . [ figspectrum](a ) and ( b ) ] , with the quantitative change that the unstable modes shift to larger @xmath60 as @xmath0 becomes more negative .      for the q2 quench the initial state ( af - phase ) has two magnon branches of qualitatively different behavior @xmath63 the @xmath64 branch consists of modes that have amplitude in the @xmath62 sublevels giving rise to longitudinal magnetic fluctuations , while the @xmath65 branch modes have amplitude in the @xmath66 sublevel giving rise to transverse fluctuations . the unstable modes for the @xmath64 branch are independent of @xmath0 and occur for the @xmath60 range @xmath67 e.g. , see figs . [ figspectrum](c ) and ( d ) . the @xmath65 branch has a finite region of unstable modes @xmath68 for @xmath69 ( noting that the lower bound is 0 for @xmath21 ) . at @xmath70 [ see figs . [ figspectrum](d ) ] , this branch becomes stable , and the condensate dynamics is driven entirely by the unstable modes in the @xmath64 branch . we simulate the dynamics using the spin-1 gpe ( [ spingpes ] ) with noise added to the initial state to seed the growth of symmetry breaking domains . for the initial state of the q1 quench we take @xmath71 where @xmath72 is a uniform ( zero - momentum ) polar condensate and @xmath73 is a small noise field used to seed the dynamical instabilities that occur after the quench . the precise form of noise used is based on the truncated wigner formalism @xcite and is described in ref . @xcite . for the q2 quench the spin rotation @xmath74 is applied to state ( [ eqpic ] ) to yield the initial condition . we use a condensate areal density of @xmath75 , where @xmath76 is the spin healing length . we evolve the spin-1 gpe for the spinor field @xmath77 on a 2d square grid with dimensions @xmath78 covered by an @xmath79 grid of equally spaced points . here most of our results are calculated on a grid of linear size @xmath80 with @xmath81 points . we evolve the spin-1 gpe   using a fourth - order symplectic method that uses fast fourier transforms to evaluate the kinetic energy operators with spectral accuracy @xcite . the quadratic zeeman energy is set at the final quench value @xmath82 for the duration of the simulation dynamics , so that the quench is effectively instantaneous at @xmath39 from the initial condition for the case of the q1 and q2 quench , respectively . in order for the post - quench dynamics to become universal it is necessary to simulate the system over many spin times @xmath83 . indeed , here we evolve out to times of up to @xmath84 , and on such long time - scales the use of a highly accurate symplectic algorithm ensures we obtain accurate solutions that conserve energy and magnetization . all the results presented here are for @xmath85 . this choice is made to ensure our interactions are in a similar regime to the binary condensate immiscibility simulations of ref . @xcite , noting that the mapping of binary condensate parameters onto effective density and spin dependent interactions is discussed in ref . we have also performed simulations for @xmath86 to verify that our results are qualitatively unchanged with a smaller spin - dependent interaction . growth of local magnetization following the q1 and q2 quench to ( a ) @xmath45 and ( b ) @xmath46 . in all cases the ( local ) magnetization is calculated as a spatial average over the system at each time , i.e.  @xmath87 for @xmath88 ( plain lines ) and @xmath89 ( lines with symbols ) . other parameters as in fig . [ figspectrum ] . , scaledwidth=50.0% ]    for both the q1 and q2 quenches the initial state is unmagnetized , but the unstable modes lead to the spin density growing exponentially . the initial post - quench dynamics depends upon the kind and depth ( i.e.  @xmath0 ) of the quench . we take the _ early - time _ regime to be the initial time period over which the longitudinal magnetization locally develops . for our parameters ( see fig .  [ figstdynamics ] ) this regime extends up to @xmath90 . for times after this the longitudinal magnetization is the dominant component of the magnetization , and the dynamics can be then be analysed in terms of spin domain coarsening , which is the focus of the next subsection . we provide examples of the early - time dynamics of the local magnetization in fig . [ figstdynamics ] for the q1 and q2 quenches to two different final values of @xmath0 . we generally observe that for the q2 quench the longitudinal magnetization develops more rapidly and approaches a larger value at late times , compared to the q1 case . this second observation can be understood as arising from the additional heating in the q1 quench ( as discussed in sec . [ secphasesquenches ] ) which tends to reduce the ground state order . notably , for even deeper quenches ( @xmath91 ) than presented here we find that local longitudinal magnetization obtained after q1 quench decreases as the final value of @xmath0 becomes more negative , while the q2 quench is insensitive to the  value of @xmath0 . in fig . [ figdomains ] we compare the magnetized domains formed near the end of the early - time dynamics . these results emphasize that the q2 quench produces larger and less noisy domains following the quench . longitudinal spin density at a time of @xmath92 after the ( a ) q1 and ( b ) q2 quench to @xmath45 . the white line in each image indicates the spin correlation length @xmath93 defined in the text , with a value of ( a ) @xmath94 and ( b ) @xmath95.the spin density is only shown over a subregion of the simulation , with other parameters as in fig . [ figspectrum ] . , scaledwidth=50.0% ]    we can obtain qualitative insight into other aspects of the dynamics in fig .  [ figstdynamics ] through considering the unstable excitations of the initial states , discussed in sec .  [ secunstablemodes ] .    for the q1 quench to @xmath45 [ fig . [ figstdynamics](a ) ] and @xmath96 [ fig . [ figstdynamics](b ) ] the dynamics proceeds in a qualitatively similar manner . the two degenerate magnon branches [ eq . ( [ eqmagnonp ] ) ] have unstable sections , and the growth of these excitations causes the transfer of atoms into the @xmath62 sublevels ( i.e.  spin - mixing ) , leading to the exponential growth of transverse magnetization . after this process saturates ( also see @xcite ) the longitudinal magnetization then develops as the transverse magnetization decays .    for the q2 quench the dynamics depends more strongly on the final @xmath0 value . for @xmath69 [ e.g.  fig . [ figstdynamics](a ) ] both the magnon branches [ eqs .  ( [ eqmagnonaf1 ] ) and ( [ eqmagnonaf2 ] ) ] are unstable . as discussed in sec . [ secmagnonsq2 ] , these magnons have transverse and longitudinal magnetic character and the simulation results show that both components of magnetization initially grow in a similar manner until saturation . for a deeper quench to @xmath91 , only the longitudinal magnon branch [ eq . ( [ eqmagnonaf1 ] ) ] is unstable , and we see [ fig . [ figstdynamics](b ) ] that the longitudinal magnetization develops with negligible growth in transverse magnetization . this case has similarities to the immiscibility transition in a binary condensate @xcite , in that the initial state occupies the @xmath62 sublevels and the dynamics causes the condensate density in these two sublevels spatially seperate . in contrast for shallow quenches ( @xmath69 ) the unstable magnon causing transverse magnetization allows spin mixing to transfer some of the @xmath62 atoms into the @xmath66 sublevel .      in the early - time dynamics local longitudinal magnetization is established by small domains forming of microscopic ( @xmath97 ) dimensions ( cf . [ figdomains ] ) . these domains then grow by coarsening dynamics . the average spatial distribution of domains in the system is captured by computing the relevant two - point correlation function . here we consider the relevant spin correlation functions @xmath98 where @xmath99 is the transverse magnetization density and the average is taken at a time @xmath100 after the quench . the functions @xmath101 and @xmath102 characterize longitudinal and transverse magnetization , respectively . for our case of quenches to the easy - axis phase @xmath101 is the ( spin ) _ order parameter _ correlation function of interest for coarsening ( e.g.  see @xcite ) . we also consider @xmath102 to demonstrate the contrasting behavior for transverse magnetisation which does not order .    here we also address the superfluid order as a second type of order that develops in the system . this order is characterized by the global phase coherence of the atomic fields , and for the easy - axis phase this is captured by the correlation function @xmath103    in expressions ( [ gz])-([g0 ] ) we have utilized translational invariance to spatially average the correlation function . additionally , we use spatial isotropy of the correlation functions to perform an angular average over all points at a distance @xmath104 and further improve statistical sampling by averaging over 32 simulation trajectories conducted with different initial noise . correlation functions for q1 quench to @xmath45 . the longitudinal spin correlation function ( a ) at various times and ( b ) scaled by the length scale @xmath105 to reveal correlation function collapse . the superfluid correlation function ( c ) at various times and ( d ) scaled by @xmath106 . ( e ) the evolution of the length scales . results with symbols out to @xmath107 are for simulations with @xmath108 and @xmath81 points , whiles the dashed lines out to @xmath84 are from simulations with @xmath109 with @xmath81 points . , scaledwidth=50.0% ] correlation functions for q2 quench to @xmath45 . the longitudinal spin correlation function ( a ) at various times and ( b ) scaled by the length scale @xmath105 to reveal correlation function collapse . the superfluid correlation function ( c ) at various times and ( d ) scaled by @xmath106 . ( e ) the evolution of the length scales . results with symbols out to @xmath107 are for simulations with @xmath108 and @xmath81 points , whiles the dashed lines out to @xmath84 are from simulations with @xmath109 with @xmath81 points . , scaledwidth=50.0% ]    the temporal evolution of the @xmath101 and @xmath110 correlation functions is shown in figs .  [ figcorrq1 ] and [ figcorrq2 ] for both types of quench . the length scale over which the correlation function decays can be taken to define a characteristic domain size . for @xmath101 we take this length scale @xmath105 to be the first zero crossing of @xmath111 . the @xmath110 and @xmath102 ( not shown ) correlation functions do not usually have a zero crossing so we take @xmath106 to be the position where the correlation peak in @xmath110 decays to 0.25 of its maximal value ( occurring at @xmath112 ) , and using the same procedure we define the length scale @xmath113 for @xmath102 .    as time progresses the length scales @xmath93 and @xmath114 are seen to grow as the longitudinal spin order and superfluid order extends over large regions [ figs . [ figcorrq1](e ) and [ figcorrq2](e ) ] . in contrast the length scale @xmath113 , which is not associated with the ground state order of the system , does not grow and remains comparable to the microscopic length scale @xmath115 . we investigate whether the system exhibits dynamic scale invariance , i.e.  whether correlations of the order parameters at late times collapse onto a single universal curve @xmath116 when lengths are scaled by a characteristic length scale @xmath117 @xmath118 using the length scales @xmath105 and @xmath106 we demonstrate correlation function collapse for sufficiently late times in both the spin order [ see figs .  [ figcorrq1](b),[figcorrq2](b ) ] and the superfluid order [ see figs .  [ figcorrq1](d ) , [ figcorrq2](d ) ] .    for both quenches at late times ( @xmath119 ) @xmath93 is seen to grow as @xmath120 as found previously @xcite . the superfluid order ( length scale @xmath114 ) grows in a similar manner to @xmath93 for intermediate times , but eventually ( @xmath121 ) grows more slowly . our best fit to the exponent gives @xmath122 . this result is slower than the @xmath123 scaling expected from dissipative xy - model dynamics @xcite , where the logarithmic correction to the @xmath124 exponent originates from the presence of free vortices . for the case of a quasi-2d scalar condensate damle _ et al . _ @xcite found @xmath125 , albeit for a higher temperature regime ( close to the critical temperature for vortex unbinding ) and using small simulation grids . recent results of karl _ et al . _ @xcite reveal even slower dynamics ( @xmath126 ) for particular initial condition of a quasi-2d condensate . we have performed simulations for deeper quenches to @xmath46 and for the case of a larger interaction ratio @xmath127 to verify that the spin and superfluid ordering occurs with the same exponents . we noted in sec . [ secearlytime ] that local order developed faster in the early - time dynamics for the q2 quench relative to q1 . comparing figs . [ figcorrq1](e ) and [ figcorrq2](e ) we also observe that for times @xmath128 the values of @xmath93 and @xmath114 for the q2 quench are appreciably larger that those of the q1 quench at the same time , and grow at a more steady rate [ the @xmath93 correlation lengths at @xmath129 are also shown in fig . [ figdomains ] ] . the @xmath110 correlation function at @xmath130 in fig .  [ figcorrq2](c ) shows that the system has long - range superfluid order at early times ( i.e.  @xmath110 approaches a constant non - zero value for large @xmath104 ) . this order eventually decays ( @xmath121 ) and the @xmath110 behavior is similar to that observed in the q1 quench . the long - range order is established by the initial spin rotation used in q2 quench , which rotates the coherent polar state into @xmath131 sublevels . in the postquench dynamics , this coherence is rapidly destroyed by the immiscibility dynamics leading to spin domain formation .    in figs .  [ figcorrq1](e ) and [ figcorrq2](e ) we also show results for a calculation with a grid of @xmath81 and @xmath109 ( i.e.  with a grid point spacing about 60% larger than the other results ) . the greater spatial range in this case allows us to simulate longer ( we need to ensure that @xmath132 to avoid finite size effects ) . the results from these calculations are shown as dashed lines and are seen to lie on top of the main results for characteristic length scales associated with the order parameter ( i.e.   @xmath93 and @xmath114 ) , but differ appreciably from the main results for the behavior of @xmath113 . this nicely demonstrates the universality of the order parameter coarsening dynamics , while emphasizing that the transverse magnetization is a disordered degree of freedom that is sensitive to the microscopic details ( in this case it is largely determined by the thermalization of the @xmath66 spin waves , see @xcite ) . the binary character of the easy - axis spin order allows clear identification of spin domains ( e.g.  see fig . [ figdomains ] ) . in this subsection we consider the properties of the domains produced as characterized by the domain area distribution @xmath133 where @xmath134 is the average number of domains of area @xmath6 to @xmath135 per unit system area at time @xmath100 . in simulations we define positive domains as connected spatial regions where the @xmath22 spin density is positive to be part of a positive domain . ] , and all locations satisfying this are set to a value of + 1 , and all other locations are set to a value of zero . on this _ binary image _ we apply the hoshen - kopelman algorithm @xcite to label the clusters of contiguous + 1 cells . similarly , a binary image can be constructed for the negative domains ( with @xmath136 ) . analyzing the combined set of positive and negative magnetized domains at time @xmath100 we compute their areas , and bin this to determine @xmath133 . to improve statistics we use 32 trajectories to average for @xmath133 . domain area distributions for the ( a)-(b ) q1 and ( c)-(d ) q2 quenches . the domains area distribution is shown at early - times as a function of ( a ) , ( c ) area and of ( b ) , ( d ) scaled - area @xmath137 , where @xmath93 is the longitudinal spin correlation length evaluated as discussed in sec . [ seccoarsening ] . the insets in ( b ) and ( d ) use the same axis range as the main plots , and indicate results at later times for comparison . , scaledwidth=50.0% ]    our results for @xmath133 at various times are shown in fig . [ figdomainareas ] . while the late time behavior of the domain distributions is reasonably similar [ see insets to figs . [ figdomainareas](b ) and ( d ) ] , the early - time behavior is strikingly different . notably , even for times as short as @xmath138 in the q2 case , @xmath133 is seen to extend out to areas of order the system size [ i.e.  @xmath139 , see fig . [ figdomainareas](c ) ] , which indicates that the system can have a percolating domain that connects across the system . in contrast , such percolating domains for the q1 quench only develop on long time scales [ i.e.  @xmath140 in fig .  [ figdomainareas](a ) ] . this difference likely originates from the nature of the @xmath141 magnon branch for the q2 quench which has unstable wavelengths extending from @xmath97 up to the system size [ e.g.  see fig . [ figspectrum](c ) ] . in comparison the unstable modes for the q1 quench are restricted to a finite wavelength range around @xmath142 [ e.g.  see fig .  [ figspectrum](a ) ] .    we can apply dynamic scaling to the domain size distribution . since the @xmath101 correlation function grows as @xmath105 , that the domain areas should grow as @xmath143 , and if dynamic scaling holds we would expect @xmath144 to be a universal time - independent distribution . we perform this scaling in figs . [ figdomainareas](b ) and ( d ) . at late - times ( insets to those figures ) we see reasonable collapse of the area distribution , however at earlier - times ( @xmath145 ) only the q2 quench exhibits scaling collapse over a wide range of areas . we also indicate on the scaled results the @xmath146 power law , which provides a reasonable fit to the distribution for sufficiently large domains . such power law behavior in domain size ( or  cluster size " ) distribution is predicted in percolation theory near the percolation transition with @xmath147 in 2d . normally in percolation theory the occupation probability ( @xmath148 ) is varied to pass through the transition . for the spin-1 system this is controlled by the total ( conserved ) @xmath3-magnetization , and for our results ( where the total @xmath3-magnetization is zero ) the probability of a location being in a positive ( or negative ) domain is @xmath149 , which is approximately the percolation threshold condition in 2d . in order to explore the percolation transition it would be necessary to vary the total @xmath3-magnetization , which would in turn vary the relative portion of two domain types . such a study has been performed using simulations of a segregating binary condensate , where the effective magnetization is controlled by varying the relative portion of the two atomic species @xcite . this study presented strong evidence for a percolation threshold at @xmath150 . we have considered the spin and superfluid ordering of a spin-1 condensate . we have found that in the coarsening dynamics these two types of order exhibit dynamic scaling , but develop with different dynamic critical exponents . we find that ( and as previously observed in refs . @xcite ) the spin - order develops with a growth law of @xmath151 , consistent with a binary fluid in the inertial hydrodynamic regime @xcite . our new results here concern the superfluid ordering , which we find to grow much more slowly as @xmath152 . indeed , our simulations out to @xmath84 reveal that @xmath114 is almost an order of magnitude smaller than @xmath93 [ see figs .  [ figcorrq1](e ) and [ figcorrq2](e ) ] . future work will involve developing an understanding for the slower superfluid growth rate , and whether aspects of the spin domains ( e.g.  shedding of various types of vortices at the spin domain walls , or other types of topological defects ) play a role . another direction for investigation is to consider our observed dynamics in the context of pre - thermalization and turbulence theory , which have proven fruitful directions of research for the immiscibility transition in a binary condensate @xcite ( also see @xcite ) . in addition to the standard q1 quench , in which the quadratic zeeman energy is suddenly changed , we propose and simulate a second type of quench ( q2 ) in which a spin rotation is applied prior to the zeeman energy being changed . in the late - time dynamics both quenches exhibit the same ordering dynamics , demonstrating the universal nature of the coarsening dynamics . however , the q2 quench has favourable properties for exploring domain formation : ( i ) the domains form more rapidly and with less heating ; ( ii ) the domains percolate at early times . these properties may make the q2 quench better suited to experiments ( e.g. ,  with @xmath20rb condensates ) , where the small spin - dependent interaction makes it challenging to explore long - time post - quench dynamics . for example , some of the short - time predictions we have made for domain formation and the behavior of the spin domain area distribution should be feasible to study in current experiments . the authors acknowledge useful discussions and assistance from h.  takeuchi , l.  williamson , and l.  symes . pbb acknowledges the contribution of nz escience infrastructure ( nesi ) high - performance computing facilities , and support from the marsden fund of the royal society of new zealand . 50ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ]  + 12$12  & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty \doibase http://dx.doi.org/10.1016/j.physrep.2012.07.005 [ * * ,   ( ) ] link:\doibase 10.1103/revmodphys.85.1191 [ * * , ( ) ] http://dx.doi.org/10.1038/nature05094 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.79.043631 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.107.195306 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.84.063625 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.89.033631 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.98.160404 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.76.043613 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.75.013621 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.99.120407 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.99.130402 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.110.165301 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.90.033604 [ * * ,   ( ) ] link:\doibase 10.1103/physrevb.76.104519 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.81.053612 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.88.013630 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.91.053609 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.116.025301 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.94.023608 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.113.095702 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.88.063615 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.95.050401 [ * * ,   ( ) ] http://dx.doi.org/10.1038/35003132 [ * * ,   ( ) ] link:\doibase    10.1126/science.1138807 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.102.170401 [ * * ,   ( ) ] link:\doibase 10.1126/science.1258676 [ * * ,   ( ) ] http://dx.doi.org/10.1038/ncomms7162 [ * * ( ) ] link:\doibase 10.1103/physreva.81.053617 [ * * ,   ( ) ] link:\doibase 10.1080/00018739400101505 [ * * ,   ( ) ] @noop _ _  ( ,  ) link:\doibase 10.1103/physreva.92.043608 [ * * ,   ( ) ] link:\doibase 10.1007/s10909 - 016 - 1543 - 7 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.81.742 [ * * , ( ) ] link:\doibase 10.1143/jpsj.67.1822 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.92.140403 [ * * ,   ( ) ] http://stacks.iop.org/0295-5075/36/i=1/a=007 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.84.023606 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.115.015301 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.90.053616 [ * * ,   ( ) ] @noop * * ,   ( ) link:\doibase 10.1103/physreve.93.053309 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.58.4836 [ * * ,   ( ) ] link:\doibase 10.1103/physreve.47.1525 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.84.1503 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.54.5037 [ * * ,   ( ) ] @noop ( ) ,   link:\doibase 10.1103/physrevb.14.3438 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.31.1103 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.93.033620 [ * * ,   ( ) ]
in this paper we study the coarsening dynamics of a spinor condensate quenched into an easy - axis ferromagnetic phase by a sudden change in the quadratic zeeman energy . 
 we show that applying a spin rotation prior to changing the zeeman energy accelerates the development of local order and reduces heating . 
 we examine the longitudinal spin ordering and the superfluid ordering of the system and show that the respective order parameter correlation functions exhibit dynamic scaling in the late time dynamics . 
 our results also demonstrate that these two types of order grow at different rates , i.e.  with different dynamic critical exponents . 
 the spin domain area distribution is calculated and is shown to have power law scaling behavior expected from percolation theory .