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one of the long - sought goals in low - temperature physics is the creation of two interpenetrating superfluids . early efforts were directed at mixtures of helium isotopes . more recently , following the experiments with bose - einstein condensates ( bec ) of atomic gases @xcite , considerable efforts have been made to create systems where two species of atoms condense simultaneously . this goal was partially achieved for two different hyperfine spin states of @xmath1rb , which were condensed in the same trap by the technique of sympathetic cooling @xcite . later the dynamics of the complex relative motion of the condensates has been studied @xcite . the possibility of the measurement of the relative phase between the two condensates has also been demonstrated @xcite . in these experiments the two condensates have a substantial overlap , although they do not completely interpenetrate each other in the stationary state . a similar state called `` spinor condensate '' has been observed for sodium gas @xcite . theoretical investigation of two - component bose systems has started many decades ago with the construction of the phenomenological hydrodynamic equations in the spirit of the landau - khalatnikov two - fluid model for the one - component bec @xcite . later , this construction has been put onto a microscopic basis @xcite . recent experiments with alkali atoms have revived the interest in the theory of such systems . hartree - fock theory has been successfully tested on the two - component @xmath1rb system @xcite . the stability @xcite , ground - state properties @xcite and collective excitations @xcite have been studied theoretically by using the gross - pitaevskii equations . many properties of two - component , or binary , bec can be understood from symmetry arguments . compared to one - component bose superfluids , two - component systems have more interesting pattern of symmetry and symmetry breaking . condensation in binary bose systems corresponds to the spontaneous breaking of _ two _ ( instead of one ) global u(1 ) symmetries . these symmetries are related , by noether s theorem , to the separate conservation of the number of atoms of each of the two species . the quantum state of the binary bose system , therefore , is characterized by two phases of the two condensates . correspondingly , the physics of binary bec is also richer than of usual one - component systems . the effects of a symmetry are often best exposed by violating the symmetry explicitly in a controlled fashion . a very interesting feature , specific to systems consisting of atoms of the same isotope in different spin states , is that it is possible to couple two condensates by a driving electromagnetic field tuned to the transition frequency . in this case atoms can be interconverted between the two spin states and the numbers of atoms of each species are not conserved separately anymore ; only the total number of atoms is constant . this implies that , in the presence of the coupling drive , only one u(1 ) symmetry remains exact , the other one is explicitly violated . the preserved u(1 ) symmetry obviously comes from the conservation of the total number of atoms , and corresponds to changing the phases of the two condensates by the same amount ( i.e. , leaving the relative phase unchanged ) . the violated u(1 ) corresponds to changing the relative phase between the two condensate . the presence of the coupling drive lifts the degeneracy of the ground state with respect to the relative phase .    in this work , we show that a sufficiently small violation of the u(1 ) symmetry corresponding to the relative phase leads to the existence of a nontrivial static configuration  a domain wall inside which the relative phase changes by @xmath0 . this configuration is a local minimum of the energy . however , the domain wall is _ not _ topologically stable and can `` unwind '' itself . to unwind , however , the system must overcome an energy barrier . thanks to this fact , the rate of the spontaneous decay of the domain wall is exponentially suppressed . our paper is organized as follows . section [ sec : l ] introduces the field - theoretical description of binary bec . in sec.[sec : solution ] we describe the domain wall configuration , whose physical interpretation is given in sec . [ sec : interpretation ] . section [ sec : boundary ] deals with the boundary of finite domain walls and the related phenomenon of `` vortex confinement '' . section [ sec : conclusion ] contains concluding remarks . in appendix [ sec : stability ] we find the metastability condition for the domain wall in the particular case when the densities of the two components are equal , and in appendix [ sec : decay ] two different mechanisms for the decay of the domain wall , operating at different temperature regimes , are considered . in this section , we use field theory to describe general properties of binary bec . our goal is to introduce notations and the formalism to lay the ground for the discussion of the domain walls in the next section . a binary dilute bose system is described by a quantum field theory of two complex scalar fields @xmath2 and @xmath3 . these fields have the meaning of the wave functions of the two condensates . the dynamics of these fields is governed by the following lagrangian , @xmath4 where the hamiltonian @xmath5 has the form @xmath6 in eq . ( [ l ] ) @xmath7 are the chemical potentials of the two species , and @xmath8 are functions of coordinates . we assume here that the trapping potentials are sufficiently wide so that these chemical potentials can be put to constants . ] @xmath9 is the scattering amplitude , in the zero momentum limit , between an atom of the @xmath10-th species and that of the @xmath11-th species , and are proportional to the scattering lengths @xmath12 , @xmath13 and @xmath14 is the rabi frequency arising from the coupling drive .    by varying the action @xmath15 with respect to @xmath16 , the familiar gross - pitaevskii equations are directly obtained : @xmath17 let us start by finding the ground state when the coupling drive is off , @xmath18 . in the superfluid ground state , both @xmath2 and @xmath3 have nonzero expectation values . these can be found by minimizing the potential energy part in eq . ( [ l ] ) with respect to @xmath2 and @xmath3 . this minimization procedure gives the equations determining the densities @xmath19 , @xmath20 in terms of the chemical potentials @xmath21 and @xmath8 , @xmath22 more conveniently , one could view eq . ( [ nmu ] ) as the equations fixing the chemical potentials for given values of the densities .    strictly speaking , eqs . ( [ nmu ] ) only correspond to an _ extremum _ of the potential energy . for it to be a local minimum , the quadratic form @xmath23 needs to be positive definite : @xmath24 in fact , eq . ( [ stability ] ) is the condition for the mixture of the two bose superfluids to be thermodynamically stable against segregation @xcite . in this paper we shall assume that ( [ stability ] ) is satisfied .    in principle , the constants @xmath25 , @xmath26 and @xmath27 can be arbitrary . in this paper we limit ourselves to the regime when all three scattering lengths are close to each other , @xmath28 . in the case when the two species are rubidium atoms in different hyperfine states , these lengths were found experimentally to differ by no more than a few percent . the assumption that @xmath29 also introduces considerable technical simplifications in our treatment . we introduce the `` average '' scattering amplitude @xmath30 and the deviations from the average @xmath31 so that @xmath32 . the stability condition ( [ stability ] ) implies that @xmath33 . analogously , we introduce the average scattering lengths @xmath34 and the deviations @xmath35 and @xmath36 . note that in the limit @xmath37 the hamiltonian has an su(2 ) symmetry which leads to interesting implications @xcite .    with the lagrangian at hand , the discussion of symmetry in the introduction can be made concrete . in the absence of the coupling drive , @xmath18 , the lagrangian ( [ l ] ) possesses a u(1 ) @xmath38 u(1 ) symmetry with respect to independent phase rotations of the fields , @xmath39 the corresponding conservation laws are those of the numbers of particles of each species , @xmath40 and @xmath41 . that @xmath42 and @xmath43 at @xmath18 are conserved separately is actually a basic assumption made when we wrote down the lagrangian ( [ l ] ) . this assumption is not automatically satisfied : it requires that only elastic scattering between atoms is allowed ; inelastic scattering is forbidden . for binary bose systems made of rubidium atoms , this appears to be a good approximation @xcite .    once the coupling drive is turned on ( @xmath44 ) , the lagrangian is invariant only under a subset of the original u(1 ) @xmath38 u(1 ) rotations ; namely , those which rotate both @xmath2 and @xmath3 by the same angle , @xmath45 therefore one of the u(1 ) symmetries the system enjoyed at @xmath18 is explicitly violated . applying the goldstone theorem , we conclude that , at @xmath18 , there are two gapless excitations and only one of these modes remain gapless at @xmath44 . the gapless modes at @xmath18 are the phonons of the two types of sounds . one corresponds to the ordinary density wave ( b mode in our paper , see below ) , and another to the concentration wave ( a mode ) in which the densities of the two species oscillate relative to each other in such a way that the total density remains constant . if we view the two components as being made of the same atoms , but in different hyperfine levels , then the mode a can be alternatively interpreted as a spin polarization wave , in which the density of nuclear spin is a function of space and time . when the coupling drive is on , only the density wave remains gapless ; the phonon of the concentration ( spin density ) wave is gapped . let us compute the sound speeds at @xmath18 . to this end we write @xmath2 and @xmath3 as @xmath46 and expand eq . ( [ l ] ) to second order of @xmath47 and @xmath48 ( we will see that @xmath49 ) . we find @xmath50 ( we have thrown away total derivatives ) . the density fluctuations @xmath51 can be `` integrated out '' and replaced by the saddle point values @xmath52 . as a result , ( [ lnphi ] ) becomes @xmath53 thus , the dispersion relations for the phonons are linear , @xmath54 , and the sound speed @xmath55 satisfies the characteristic equation @xmath56 when @xmath29 , the solutions are @xmath57 we see that when @xmath58 the speed of the concentration wave ( a ) is much smaller than that of the density wave ( b ) , @xmath59 . the system is `` stiffer '' towards density fluctuations than towards fluctuations of concentration . that modes a and b are indeed concentration and density fluctuations respectively is seen from the corresponding eigenvectors . the a mode corresponds to such fluctuations in which @xmath60 while the b mode corresponds to @xmath61 therefore , in the a sound @xmath62 and @xmath63 fluctuate in such a way that the overall density remains constant ( @xmath64 ) , while the b sound corresponds to density waves in which @xmath65 , or concentration , is unchanged ( @xmath66 ) . the lagrangian ( [ lphi ] ) , in terms of the normal modes @xmath67 has the form @xmath68 +        \left [          u_\b^{-2}(\d_t\phib)^2-(\nabla\phib)^2\right ] \biggr\}\,.\ ] ]    when the coupling drive is on , @xmath44 , one should add to eq . ( [ lphi ] ) the potential energy term @xmath69 . the b sound is not affected by this term since it corresponds to such fluctuations in which @xmath70 . the phonons of the a sound acquire a gap @xmath71 for small values of @xmath14 .    in our further discussion we shall need the formulas for the healing , or correlation , lengths , which are defined via the response of the system to a static source coupled locally to the particle density , @xmath72 . as in the case of the sounds , there are also two healing lengths , @xmath73 as seen from eq . ( [ coh_lengths ] ) , @xmath74 . this is because @xmath75 is the correlation length of fluctuations of the overall density , while @xmath76 is the correlation length of the relative density .    if we take the values typical for present day experiments : @xmath77 @xmath78 , @xmath79  , and @xmath80 , then @xmath81 @xmath82 m , and @xmath83 @xmath82 m . these lengths are smaller than the typical system size in experiments . the existence of the domain wall of relative phase in binary bec with a coupling drive can be shown in a rather simple way . let us first focus only on fluctuations of the fields on length scales much larger than the largest healing length @xmath76 . in this case , the amplitudes @xmath62 and @xmath63 of @xmath16 can be regarded as `` frozen '' and the only important degrees of freedom are the phases , i.e. , @xmath84 and @xmath85 . the energy of the system is a functional of @xmath84 and @xmath85 , @xmath86 =     \int\ ! d^3\x\,\biggl [ { \hbar^2\over2m}(n_1(\nabla\varphi_1)^2        + n_2(\nabla\varphi_2)^2 )         - \hbar\omega\sqrt{n_1n_2}\cos(\varphi_1-\varphi_2 ) \biggr]\ , . \label{h}\ ] ] the potential energy term @xmath87 has its minimum at @xmath88 . the configuration in which @xmath89 over the whole space is obviously the global minimum of the total energy , and hence is the ground state . the domain wall solution that we shall describe is , in contrast , a _ local _ minimum of the energy . to find the profile of the domain wall , we vary eq . ( [ h ] ) with respect to @xmath84 and @xmath85 and obtain the following equations @xmath90 for domain walls , @xmath84 and @xmath85 are functions of only one coordinate , say , @xmath91 , and at @xmath92 both approach constant values . a nontrivial solution to eq . ( [ walleq ] ) satisfying these conditions is @xmath93 the characteristic width of the domain wall is @xmath94 . the tension of the domain wall is @xmath95    the relative phase @xmath96 changes from 0 to @xmath0 as @xmath91 runs from @xmath97 to @xmath98 . note that @xmath96 is defined modulo @xmath0 so it goes a full circle as one passes through the wall . therefore from the point of view of the energy functional ( [ h ] ) the domain wall ( [ profile ] ) is a topologically nontrivial configuration , which can not be continuously deformed into the @xmath89 configuration . in fact , one can prove that the domain wall is a configuration with minimal energy defined in eq . ( [ h ] ) among those where @xmath96 changes by @xmath0 from @xmath99 to @xmath100 , and hence , can not decay away , _ as long as eq . ( [ h ] ) applies_. the domain wall is similar to the soliton of the sine - gordon model . there is a small difference : the ground states on two sides of the domain wall are _ different _ : at @xmath99 @xmath101 , while the ground state at @xmath100 is @xmath102 , @xmath103 . since @xmath104 are defined mod @xmath0 the latter is equivalent to @xmath105 .    in reality , eq . ( [ h ] ) is not the full hamiltonian of the system : it is only an effective description valid at length scales larger than both healing lengths . the full theory ( [ h - full ] ) contains , besides @xmath84 and @xmath85 , also the density fluctuations @xmath106 and @xmath107 . as a consequence , in the full theory , the domain wall ( [ profile ] ) _ can _ be continuously deformed into the trivial configuration @xmath89 . such deformations necessarily pass through field configurations where either @xmath62 or @xmath63 vanishes at some points : at these points @xmath96 is ill - defined . thus , the domain wall is not truly topological and can `` unwind '' , i.e. , decay away . the fact that the ground states on the two sides of the wall are different does not prevent the decay : it is possible to construct a field configuration interpolating between @xmath101 and @xmath105 with arbitrarily small energy per unit area . although the domain wall is not a global minimum of the energy functional , it can still be a _ local _ minimum . in this case , to deform the domain wall into a `` topologically '' trivial configuration with @xmath89 one has to overcome an energy barrier . the wall is in this case metastable . from our previous discussion one can conclude that , roughly speaking , the wall is metastable when eq . ( [ h ] ) applies and is unstable when eq . ( [ h ] ) is not applicable . for eq.([h ] ) to be valid the wall has to be wider than the largest healing length , @xmath76 . since the width of the wall decreases as one increases the rabi frequency @xmath14 , the wall is metastable only when @xmath14 is smaller than some critical value @xmath108 . let us define @xmath109 as the value of the rabi frequency at which the width of the wall @xmath94 , as defined in eq . ( [ profile ] ) , is equal to to the longer healing length @xmath76 in eq.([coh_lengths ] ) : @xmath110 the wall is metastable when @xmath14 is less than some critical value @xmath108 of order @xmath109 , @xmath111 to find the exact value of @xmath108 one needs to perform a more refined stability analysis . we present such an analysis for the special case @xmath112 ( i.e. , when the densities of the two species are equal ) in appendix [ sec : stability ] . parametrically , the result is consistent with eq . ( [ omegac ] ) , and the ratio @xmath113 is found to be @xmath114 . using the numerical values typical for experiments with rb : @xmath115 a few @xmath116 @xmath78 , @xmath117  , one finds @xmath118 hz . therefore to have a stable wall the rabi frequency needs to be smaller than about 100 hz . the domain wall can not be thinner than the longer correlation length @xmath76 , which was estimated above to be a few @xmath82 m . even when @xmath119 , a metastable wall can still spontaneously decay ( burst ) . such a decay , as we have said , requires overcoming a potential barrier . at sufficiently large temperature , the mechanism for the decay is thermal activation ( over the barrier ) . at zero or small temperatures , the mechanism of the decay is quantum tunneling under the energy barrier . the decay through thermal activation , which is relevant for the temperatures achieved in current experiments , is considered in appendix [ sec : decay ] , where it is shown that the decay rate is exponentially suppressed , since a global ( macroscopic ) fluctuation is required , unless @xmath14 is very close to @xmath108 . the domain wall solution found in sec .  [ sec : solution ] allows an interesting physical interpretation which suggests a possible way for their creation in experiments @xcite . we first note that in a bec the superfluid velocity is proportional to the gradient of the phase . in a two - component bec , there are two such velocities . @xmath120 equation ( [ walleq ] ) , in particular , implies that the total particle number current vanishes , @xmath121 . individually , however , the particle number current of each species @xmath122 and @xmath123 are nonzero . from eq.([profile ] ) we find @xmath124 thus the domain wall is a configuration where the two components flow in opposite directions . the flow is illustrated in fig.[fig : flow ] . the velocities of the components are largest at the center of the wall ( @xmath125 ) and decrease as one moves toward the edge of the wall . the flow is concentrated on the wall ; outside the wall ( @xmath126 ) there is essentially no flow .    equation ( [ walleq ] ) can be rewritten in terms of the currents as @xmath127 for a stationary configuration as the one we are considering , eq.([conversion ] ) means that the number of particles in each species is not conserved . it also implies that there is a _ conversion _ of atoms between the two energy levels due to the coupling drive . in the left half of the wall @xmath128 and there is a conversion of atoms of the second type to atoms of the first type . in the right half @xmath129 and the conversion goes the opposite way ( fig.[fig : flow ] ) . the rate of conversion is @xmath130 as is the flow , the conversion rate is also maximal near the wall . far from the wall ( @xmath126 ) there is essentially no conversion . the conversion rate ( [ conversionrate ] ) changes sign at @xmath131 . since different species correspond to different energy levels of the atom , energy is absorbed in one half of the wall and released in the other . plane accompanied by spin flips on both sides of it . ] the interpretation of the domain walls given above suggests a possible method for their creation in experiments @xcite . one starts with the coupling drive off ( @xmath18 ) and prepares a state where the two condensates flow in opposite directions ( for example , by manipulating the traps ) . in such a state the relative phase @xmath132 is a linear function of the coordinate along the direction of motion ( say , @xmath91 ) . one then slowly increases @xmath14 . the domain walls will be created and the centers of the walls are located at the points where @xmath96 was an odd multiple of @xmath133 ( @xmath134 , @xmath135 , @xmath136 , etc . ) before @xmath14 was turned on . by changing the velocity of the initial relative motion of the condensates and the final values of @xmath14 one can change the separation between the domain walls and their width . such controlled creation of the domain walls , hopefully , can be achieved in future experiments . so far we have always considered infinite domain walls which have no boundary . it is also interesting , and perhaps more realistic , to consider domain walls with a boundary . we shall show that the domain wall can be bounded by a vortex line . suppose we have a finite domain wall whose boundary is a closed contour @xmath137 ( fig . [ fig : boundary ] ) . we shall assume that the length of @xmath137 is much larger than the width of the wall @xmath94 so one can view the domain wall as an infinitely thin membrane stretched on @xmath137 ( we shall call this picture the `` thin - wall approximation '' ) . let us now take another , smaller contour which has a nontrivial linking with @xmath137 ( @xmath138 in fig . [ fig : boundary ] ) . as one goes along @xmath138 , one crosses the membrane once , so the relative phase @xmath96 changes by @xmath0 . this is exactly what one expects from a vortex . therefore , @xmath137 can be a vortex line . we recall that the size of the core of the vortex is the healing length @xmath76 , which is smaller or of the same order as the width of the wall . therefore , in the thin wall approximation , we have an infinitely thin membrane bounded by an infinitely thin vortex line . one should note that such a bounded domain wall will tend to shrink to reduce its energy , which comes from the wall tension and the tension of the boundary vortex . conversely , for small non - zero @xmath14 , a vortex must have a domain wall attached to it to minimize the energy due to the nontrivial phase @xmath96 winding around it . this means that the energy of a single vortex per unit length is increasing linearly with the size of the system in the transverse direction . this is in contrast to the situation at @xmath18 when the vortex tension has only a logarithmic dependence on the size of the system . note that there are two types of vortices in binary bec . those of the first type , which we shall call the @xmath84 vortices , have the condensate @xmath2 vanishing at the vortex center , while @xmath3 is nonzero . analogously the @xmath85 vortices have @xmath139 and @xmath140 at their centers . as one goes around a @xmath84 vortex , @xmath84 changes by @xmath0 , while @xmath85 does not change , and vice versa for a @xmath85 vortex . thus @xmath96 changes by either @xmath0 or @xmath141 for the two types of vortices , so the domain wall can be bounded by a vortex of either type . in contrast to an individual vortex , a pair of @xmath84 and @xmath85 vortices , placed parallel to each other , will have energy per unit length which is only logarithmically divergent . that is because the @xmath96 `` charges '' of the two vortices cancel each other , so @xmath96 is trivial at spatial infinity ( no winding ) . the same situation occurs for a pair of parallel vortices of the same type ( @xmath84 or @xmath85 ) with opposite winding ( however , such vortex - antivortex pair can annihilate , while a @xmath142 pair can not ) . in a certain sense , one can talk about the phenomenon of `` vortex confinement '' : vortices exist only in pairs . this confinement should , in principle , be observable experimentally , in a rotating two - component bec , which can be already created in a laboratory @xcite . the vortices are usually identified by the density depletion at their cores , but can be also seen as dislocations of interference fringes due to phase singularities @xcite . with the coupling drive off ( @xmath18 ) such a system contains an equal number of @xmath84 and @xmath85 vortices which are distributed in space with no particular correlation between @xmath84 and @xmath85 vortices . as one turns on @xmath14 , the vortices will start to pair up and at some point the system will become a collection of composite objects , each being a bound state of a @xmath84 and a @xmath85 vortex ( fig . [ fig : pairs ] ) . the phenomenon of vortex confinement is very similar to that of quark confinement in the theory of strong interaction ( quantum chromodynamics ) . similarly to our vortices , quarks and antiquarks do not exist as individual objects , but are confined into composite objects  hadrons . the analogy with quantum chromodynamics actually stretches further . if one places a @xmath84 vortex and a @xmath85 vortex at a distance larger than @xmath94 , then a domain wall that connects these two vortices will be formed . the tension of the domain wall is the force , per unit length , that attracts the two vortices . the attractive force between the two vortices is thus independent of their separation , given that the latter is larger than the width of the domain wall . this is analogous to the confining force between a quark and an antiquark , which is also constant at large distances . a confinement model which resembles most the confinement of the vortices is the three - dimensional compact quantum electrodynamics considered in ref .  @xcite . in this theory the worldlines of electrically charged particles are analogous to the vortices in bec . one can also imagine a system of two vortices which rotate around each other so that the confining force ( from the domain wall ) balances the centrifugal force . such a system is analogous to the high - spin meson states in hadronic physics where a quark and an antiquark rotate around each other . we have shown that in a system of two interpenetrating bec with a coupling drive , there exists a domain wall solution . the relative phase between the two condensates changes by @xmath0 as one travels through the wall . the wall solution is formally similar to the kink in the sine - gordon field theory , yet it is not topologically stable and can decay . in this respect , the wall is more similar to a soap film , which can spontaneously burst .    from the mathematical point of view , the domain walls discussed in this paper are similar to the ones which have been studied in particle physics . such domain walls appear at least in two contexts : in the theory of the hypothetical axion @xcite and in high - density quark matter @xcite . the similarity is that the domain wall solution arises from the spontaneous breaking of an approximate u(1 ) symmetry . in all cases the domain wall exists only if the explicit violation of the u(1 ) symmetry ( determined by the value of the rabi frequency in our case ) is small enough . the decay of the wall in all the examples occurs via hole nucleation .    as to the experimental realization of the domain wall , one could be optimistic since the estimated critical value of the rabi frequency , for densities and scattering lengths typical for the rubidium gas in recent experiments , is of the order 100 hz which is not too small . the width of the wall , which might be as small as a few @xmath82 m if the rabi frequency is not much smaller than critical , can be also accommodated inside condensates of the size characteristic of present - day experiments . the apparent immediate obstacle is still the creation of a system where two bec truly interpenetrate . in the recent experiments with trapped atomic gas the region of overlap between the two bec is still small . one can hope the technical problems of making a genuine two - component bec to be solved in the near future which would enable one to study the domain walls experimentally . the authors thank richard friedberg for discussions . we thank riken , brookhaven national laboratory , and u.s . department of energy [ de - ac02 - 98ch10886 ] for providing the facilities essential for the completion of this work . the work of dts is supported , in part , by a doe oji grant . in the case when the densities of the two components are equal , @xmath112 , the maximal frequency @xmath108 , below which the wall configuration is still locally stable , can be found analytically . for illustrative purposes we shall consider this particular case in details . let us recall that when the scattering lengths @xmath144 , @xmath145 , and @xmath146 are approximately equal ( assuming @xmath147 ) there are two healing lengths @xmath76 and @xmath75 . the healing length related to fluctuations of the overall density @xmath75 is much smaller than the one related to fluctuations of the relative density @xmath76 : @xmath148 . we shall be interested in the case when the width of the wall @xmath94 is much larger than @xmath75 ( but we shall not presume any relation between @xmath94 and @xmath76 ) . in this case , the total density @xmath149 can be considered as frozen , and the system can be described in terms of three variables : @xmath150 , @xmath84 and @xmath85 , @xmath151 where @xmath150 runs from @xmath152 and @xmath133 . in terms of these three variables , the lagrangian has the form @xmath153-h\ , ,    \label{lthetaphi}\ ] ] where @xmath154 is the following hamiltonian @xmath155      + { 1\over2}(\delta\mu n - \delta g ' n^2)\cos\theta \nonumber\\       & & - { 1\over4}\delta gn^2\sin^2\theta      - { 1\over2}\hbar\omega n\sin\theta\cos(\varphi_1-\varphi_2)\ , .    \label{hthetaphi}\end{aligned}\ ] ] in eq . ( [ hthetaphi ] ) @xmath156 . the ground state is found by minimizing the potential term in eq . ( [ hthetaphi ] ) . from here on we consider the special case when in the ground state the density of atoms of the two species are equal , @xmath157 ( or equivalently @xmath158 . ) this requires @xmath159 . in this case @xmath154 possesses a discrete symmetry with respect to replacing @xmath160 , @xmath161 . this symmetry is what makes it possible to find @xmath108 analytically . it is more convenient to use , instead of @xmath104 , the normal modes @xmath162 defined in eqs . ( [ phiab ] ) , which in the case @xmath157 have the form @xmath163 in terms of these variables the hamiltonian becomes @xmath164    \nonumber \\    & &   - { 1\over4}\delta gn^2\sin^2\theta      - { 1\over2}\hbar\omega n\sin\theta\cos\phia\ , .    \label{hphiab}\end{aligned}\ ] ] in order to find the domain wall configuration , we need to extremize the energy with respect to @xmath150 and @xmath165 . varying with respect to @xmath166 , one finds @xmath167 this equation determines @xmath166 for given @xmath150 and @xmath96 . the task of solving eq . ( [ phibthetaphia ] ) becomes much simpler if one assumes that all variables depend only one coordinate @xmath91 . in this case @xmath168 which can be trivially solved : @xmath169 . after eliminating @xmath166 , the energy functional one has to minimize is @xmath170         -{1\over4}\delta gn^2\sin^2\theta      - { 1\over2}\hbar\omega n\sin\theta\cos\phia\ , .    \label{hphia}\ ] ] it is easy to check that the following configuration is always a local extremum of eq . ( [ hphia ] ) : @xmath171 equation ( [ extremum1 ] ) can be guessed from the symmetry of eq.([hphia ] ) under @xmath160 . to see if the domain wall solution is a local minimum , one needs to expand @xmath154 in the vicinity of ( [ extremum1],[extremum ] ) . one writes @xmath172 to the second order in @xmath173 and @xmath174 the hamiltonian ( [ hphia ] ) is @xmath175        + { \hbar^2n\over8 m } \left[(\nabla\tilde\phia)^2 +           k^2\cos\bar\phia\cdot\tilde\phia^2\right]\ , . \label{h2ndorder}\ ] ] one has to find the eigenmodes of eq . ( [ h2ndorder ] ) : if there are no negative modes then the domain wall is a local minimum of the energy ; if there exist a negative mode then the domain wall is unstable . the second , @xmath150-independent , term in eq . ( [ h2ndorder ] ) does not have negative modes ( it has only one zero mode corresponding to the translation of the wall along the @xmath91 direction ) and does not lead to instability , and hence can be ignored . taking into account the explicit solution @xmath176 , the first term in eq.([h2ndorder ] ) is @xmath177\,.\ ] ] the well - known operator @xmath178 has the lowest eigenvalue equal to @xmath179 , corresponding to the eigenfunction @xmath180 , which implies that @xmath154 does not have a negative mode if @xmath119 , where @xmath181 when @xmath182 , the configuration ( [ extremum ] ) is not a local minimum of the energy functional : the domain wall does not exist . the value ( [ omegac_exact ] ) is of the same order as @xmath109 in eq . ( [ omega0 ] ) : @xmath183 . the numerical value @xmath184 is specific for the case @xmath112 ; if @xmath185 then @xmath113 is , in general , different . as we have seen above , as long as @xmath119 , the wall minimizes the energy of the system with respect to small _ local _ variations of the condensates . however , the global minimum of the energy is achieved when the phases of the condensates are constant in space . since , as discussed above , the wall configuration can be continuously deformed into the global minimum energy configuration , i.e. , the wall can decay . here we shall estimate the lifetime of the wall due to such a decay . since the wall minimizes the energy locally , such a deformation necessarily goes through a potential barrier . thus the decay can only occur by a quantum tunneling through this barrier or , at finite temperature , by a thermal fluctuation over the barrier . in both cases , the decay rate is exponentially suppressed : in the first case by the wkb factor @xmath186 , and in the second case by the boltzmann factor @xmath187 , where @xmath188 is the height of the barrier . the first formula applies at sufficiently low temperature , while the second one applies at higher temperatures . a crude estimate ( see below ) suggests that the corresponding crossover temperature is quite small ( nanokelvins for parameters typical for present - day experiments ) , and is lower than temperatures achieved in present - day experiments . thus we shall limit our discussion to the decay by thermal fluctuation . this case is also simpler theoretically . the quantum mechanical decay , relevant for very low temperatures , will be left beyond the scope of this paper . assuming the temperature is much smaller than the critical temperature at which one of the condensates melts ( so that most atoms are still in the condensates ) , the rate of thermal activation across the barrier is @xmath189 , where @xmath188 is the height of the barrier at zero temperature . since we are dealing with an infinite - dimensional configuration space , @xmath188 should be understood as the energy at the lowest point of the barrier . this point is a saddle point ( the energy has a single negative curvature direction ) . some information about the exponent @xmath190 can be obtained by a simple scaling argument without a detailed calculation . in terms of the dimensionless variables @xmath191 defined as @xmath192 the energy becomes @xmath193     = { \hbar^3n^{1/2}\over m^{3/2}\delta g^{1/2}}\int\!d^3\widetilde \x\ ,        & & \biggl\ { { 1\over2 } \biggl [ { 1\over4}(\tilde\nabla\theta)^2 +       \cos^2{\theta\over2}(\tilde\nabla\varphi_1)^2 +        \sin^2{\theta\over2}(\tilde\nabla\varphi_2)^2\biggr ] \nonumber\\       & & -{1\over4 } ( \sin^2\theta+2\cos\theta_0\cos\theta )       -{1\over2 } { \hbar\omega\over\delta gn }        \sin\theta\cos(\varphi_1-\varphi_2)\biggr\}\ , ,    \label{hrescale}\end{aligned}\ ] ] where @xmath194 determines the relative condensate densities in the ground state at @xmath18 : @xmath195 . consider the dependence on @xmath14 at a given @xmath194 . the energy functional eq.([hrescale ] ) is equal to a dimensionful constant times a dimensionless functional which depends on @xmath14 only via the ratio @xmath196 , where @xmath197 , same as in ( [ omega0 ] ) . thus the saddle point of the energy is also a function of this dimensionless ratio : @xmath198 if , moreover , we define a temperature @xmath199 as @xmath200 which is of the same order as the critical temperature of the bose - einstein condensation , then the decay rate for a given @xmath65 ( or @xmath194 ) can be written as @xmath201    = \exp\biggl[-{1\over\sqrt{4\pi \delta an^{1/3 } } }    { t_0\over t } f\biggl({\omega\over\omega_0}\biggr)\biggr]\ , . \label{decayscaling}\ ] ] the form of the function @xmath202 can not be found from scaling arguments alone . however , when @xmath203 one can expect @xmath204 , and then the exponent @xmath205 is large , since @xmath206 , and @xmath207 . the function @xmath202 can be computed in the regime @xmath208 , where the saddle point configuration can be found in the `` thin - wall '' approximation . in this approximation field configurations are described at length scales much larger than the width of the domain wall . from this point of view the domain wall is an infinitely thin membrane . the saddle point configuration is a membrane with a round hole in it ( fig . [ fig : hole ] ) . the radius of the hole @xmath209 must be much larger than the width of the wall for the thin wall approximation to be valid . as discussed in sec . [ sec : boundary ] , the rim of the hole must be a vortex , since it is the boundary of the domain wall . there are two contributions to the energy difference @xmath188 between the saddle point and the domain wall configurations . one contribution is negative and comes from the hole ( since the hole is the absence of the wall ) . another contribution is positive and comes from the rim . therefore , @xmath210 where @xmath211 is the energy per unit length of the vortex ( the vortex tension ) and @xmath212 is the domain wall tension . the energy ( [ enusigma ] ) has a maximum when the radius of the hole is @xmath213 this is the radius of the critical hole . indeed , if a hole with a radius @xmath214 is nucleated , then it will expand and eventually eat up the whole wall . if , in contrast , the radius of the nucleated hole is less than critical , then the hole will shrink and disappear .    substituting @xmath215 in eq . ( [ enusigma ] ) , we find the height of the energy barrier @xmath216 as there are two types of vortices , in eq . ( [ barrier ] ) @xmath211 refers to the vortex with the smaller tension . the tension of a straight vortex is logarithmically divergent : @xmath217 where the index @xmath218 refers to the two types of vortices , and @xmath209 is the long - distance cutoff . the role of @xmath209 is played by either the size of the critical hole @xmath219 or the width of the wall @xmath94 . we shall see at the end of this section that the two lengths differ only by a logarithm , which does not affect eq . ( [ nu_i ] ) . therefore , to logarithmic accuracy , the vortex tension is @xmath220 the argument of the logarithm is @xmath221 , and is large when @xmath208 , which justifies the use of the logarithmic approximation .    without loss of generality , we can assume that @xmath222 . then the vortex of the first type has the smallest @xmath211 ( [ nu_i ] ) . substituting eqs.([nui ] ) and ( [ tension ] ) into eq . ( [ barrier ] ) , one finds that the barrier height has the form of eq . ( [ escaling ] ) , where @xmath223 the decay rate ( [ decayscaling ] ) can be rewritten in the following form @xmath224^{2/3 }    \biggl({n\over n_2}\biggr)^{3/2 }    { 1\over\sqrt{4\pi\delta an_1^{1/3}}}{t_{c1}\over t }    \biggl({\omega_0\over\omega}\biggr)^{1/2 }    \ln^2{\omega_0\over\omega}\biggr\}\ , ,    \label{gamma}\ ] ] where we introduced @xmath225  the critical temperature for the smallest of the two condensates ( @xmath62 by our choice ) . the rate ( [ gamma ] ) is exponentially suppressed when @xmath208 and @xmath226 .    to check the consistency of our assumptions , we note that the radius of the critical hole is @xmath227 comparing with the width of the wall in eq . ( [ profile ] ) , we find that @xmath219 is larger than @xmath94 by a factor @xmath228 , which is assumed to be parametrically large . thus , the use of the thin wall approximation is justified .    to make a good estimate of the crossover temperature , below which crossover proceeds via quantum tunneling , rather than via thermal activation , one needs to estimate the action on the tunneling trajectory and compare it to the exponent in the thermal activation rate @xmath190 . for a very crude estimate we can take @xmath229 the exponents @xmath230 and @xmath190 become comparable at a temperature of order few nanokelvin , given typical parameters of present day experiments .
we consider a system of two interpenetrating bose - einstein condensates of atoms in two different hyperfine spin states . 
 we show that in the presence of a small coupling drive between the two spin levels , there exist domain walls across which the relative phase of the two condensates changes by @xmath0 . 
 we give the physical interpretation of such walls . 
 we show that the wall tension determines the force between certain pairs of vortices at large distances . 
 we also show that the probability of the spontaneous decay of the domain wall is exponentially suppressed , both at finite and at zero temperature , and determine the exponents in the regime of small rabi frequency . 
 we briefly discuss how such domain walls could be created in future experiments .