Number and spin densities in the ground state of a trapped mixture of two pseudospin-1/2 Bose gases with interspecies spin-exchange interaction
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Number and spin densities in the ground state of a trapped mixture of twopseudospin- Bose gases with interspecies spin-exchange interaction
Jinlong Wang and Yu Shi ∗ Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
We consider the ground state of a mixture of two pseudospin- Bose gases with interspecies spinexchange in a trapping potential. In the mean field approach, the ground state can be describedin terms of four wave functions governed by a set of coupled Gross-Pitaevskii-like (GP-like) equa-tions, which differ from the usual GP equations in the existence of an interference term due tospin-exchange coupling between the two species. Using these GP-like equations, we calculate suchground state properties as chemical potentials, density profiles and spin density profiles, which aredirectly observable in experiments. We compare the cases with and without spin exchange. It isdemonstrated that the spin exchange between the two species lowers the chemical potentials, tendsto equalize the wave functions of the two pseudospin components of each species, and thus homog-enizes the spin density. The novel features of the density and spin density profiles can serve asexperimental probes of this novel Bose system.
PACS numbers: 03.75.Mn, 03.75.Gg
I. INTRODUCTION
Multicomponent Bose-Einstein condensation (BEC) has been an active subject of research in recent years. Peoplehave considered BEC of a mixture of two different spinless species [1–8] and of spinor gases such as spin-1 [9–13] andpseudospin- gases [14–16]. More recently, a mixture of two distinct species of pseudospin- gases with interspeciesspin exchange was investigated theoretically [17–23]. In such a mixture, there are N a atoms of species a and N b atomsof species b , while each atom has a pseudospin degree of freedom with basis state σ = ↑ , ↓ . The particle number ofeach species N α = N α ↑ + N α ↓ , ( α = a, b ), is conserved, but the particle number of each pseudospin component ofeach species N ασ , ( σ = ↑ , ↓ ), is not conserved because of the spin-exchange coupling between the two species. Notethat for a pseudospin- gas, the total spin of each species is always a constant S α = N α /
2, while its z -component is S αz = ( N α ↑ − N α ↓ ) / α incoming with pseudospin σ and outgoing with pseudospin σ and an atom of species β incoming withpseudospin σ and outgoing with pseudospin σ , the scattering length is denoted ξ αβσ σ σ σ . Correspondingly, theeffective interaction is g αβσ σ σ σ δ ( r − r ′ ), where g αβσ σ σ σ ≡ π ¯ h ξ αβσ σ σ σ /µ αβ , where µ αβ is the reduced mass ofthe two atoms. For convenience, we define the shorthands g αασσ ≡ g αασσσσ for the intraspecies scattering of the samepseudospin σ , g αασ ¯ σ ≡ g αασ ¯ σ ¯ σσ for the intraspecies scattering of different pseudospins σ = ¯ σ , g abσσ ′ ≡ g abσσ ′ σ ′ σ for theinterspecies scattering without spin exchange, g e ≡ g abσ ¯ σσ ¯ σ for the interspecies spin-exchange scattering. We shall alsouse the shorthands ξ αασσ for ξ αασσσσ , ξ αασ ¯ σ for 2 ξ αασ ¯ σσ ¯ σ , ξ abσσ ′ for ξ abσσ ′ σ ′ σ . In this paper, all these scattering lengths ξ ’s andthus the effective interaction strengths g ’s are considered to be positive. Thus the many-body Hamiltonian densityis [17, 18] ˆ H ( r ) = X ασ ˆ ψ † ασ [ − m α ∇ + U ασ ( r )] ˆ ψ ασ + 12 X ασσ ′ g αασσ ′ | ˆ ψ ασ | | ˆ ψ ασ ′ | + X σσ ′ g abσσ ′ | ˆ ψ aσ | | ˆ ψ bσ ′ | + g e ( ˆ ψ † a ↑ ˆ ψ † b ↓ ˆ ψ b ↑ ˆ ψ a ↓ + ˆ ψ † a ↓ ˆ ψ † b ↑ ˆ ψ b ↓ ˆ ψ a ↑ ) , (1)where U ασ ( r ) is the external trapping potential, ˆ ψ ασ ≡ ˆ ψ ασ ( r ) is the Bose field operator for species α with pseudospin σ ( α = a, b , σ = ↑ , ↓ ), σ and σ ′ may or may not be equal. Expanded in terms of an orthonormal set of single-particleorbital basis states { φ ασ,i ( r ) } , ˆ ψ ασ ( r ) = X i ˆ a ασ,i φ ασ,i ( r ) , (2) ∗ [email protected] where ˆ a ασ,i is the annihilation operator corresponding to φ ασ,i ( r ). In ˆ H , the first two summations of interaction termsare the density-density interactions without spin exchange, as studied in previous models of Bose mixtures, the lastterm is the spin-exchange interaction, which causes spin correlation or entanglement between the two species, and isalso responsible for the novel features discussed in the present paper.Under the usual single orbital-mode approximation, Bose statistics and energetics governs that all atoms of eachspecies α and with the pseudospin state σ occupy the lowest-energy single-particle orbital mode hereby denoted as φ ασ ( r ), hence in the expansion (2) of the field operator ˆ ψ ασ ( r ), one only needs to consider one term ˆ a ασ φ ασ ( r ), whereˆ a ασ is the annihilation operator corresponding to φ ασ ( r ). Consequently, in each term of the many-body Hamiltonian R d r ˆ H ( r ), there is an integration of a product of single-particle wave functions, which now becomes an effectivecoefficient. Therefore, under the single orbital-mode approximation, the details of φ ασ ( r ) are not needed in describingthe many-body ground state in terms of creation and annihilation operators or, equivalently, the collective spinoperators, although such simplification is lost when one goes beyond single orbital-mode approximation. In a broadparameter regime, the two species are quantum entangled in the particle numbers of the two pseudospin states orcollective spins, that is, the two species do not undergo BEC separately, hence the ground state was dubbed entangledBEC.However, these four wave functions and the corresponding elementary excitations are important physical properties.For a uniform system, φ ασ ( r ) is simply the constant 1 / √ Ω, where Ω is the volume of the system. For convenience, wewrite the mean field value of the Bose field operator h ˆ ψ ασ i as ψ ασ e iγ ασ , where ψ ασ >
0. This is the so-called condensatewave function. The condensate wave function for pseudospin σ component of species α is ψ ασ = η ασ p N ασ / √ Ω, where N ασ is the corresponding particle number in the many-body ground state of the system, determined by the many-bodyHamiltonian, η ασ = ± is a sign. To minimize the energy, the signs of three components can be chosen to be + whilethat of the other one is chosen to be − .In a trapping potential, which is an experimental necessity for BEC of cold atoms, the wave function ψ ασ ( r )is dramatically different from a constant. Moreover, the details of ψ ασ ( r ) provide experimentally very importantinformation, as their modular square is just the particle density, which is directly measurable and is a key observable.In this paper, we consider such a pseudospin- mixture in a trapping potential, and find some interesting properties,especially the density and spin density profiles of the four lowest-energy orbital modes { φ ασ } . There had been manycalculations on such properties in other types of BEC mixtures [3, 24], which demonstrated that a trapping potentialbrings significant features absent in a homogeneous system.The GP-like equations can be obtained by using the Euler-Lagrange equation ddt (cid:18) ∂ L ∂ ˙ ψ ασ (cid:19) − ∂ L ∂ψ ασ = 0 , (3)with L = i P ασ ψ ∗ ασ ∂ t ψ ασ − h ˆ Hi , and then substituting i∂ t as µ ασ . One obtains (cid:18) − ¯ h m α ∇ + U ασ ( r ) (cid:19) ψ ασ + g αασσ | ψ ασ | ψ ασ + g αασ ¯ σ | ψ α ¯ σ | ψ ασ + g α ¯ ασσ | ψ ¯ ασ | ψ ασ + g α ¯ ασ ¯ σ | ψ ¯ α ¯ σ | ψ ασ − g e ψ ∗ ¯ α ¯ σ ψ ¯ ασ ψ α ¯ σ = µ ασ ψ ασ . (4)where g e term is due to the spin-exchange interaction, and is a new feature absent in previous models of Bosemixtures. The minus sign comes from the requirement that the phases γ ασ of the four components should satisfycos( γ a ↑ − γ a ↓ − γ b ↑ + γ b ↓ ) = − g e >
0, this spin-exchange interaction is likean attractive interaction in some way, counteracting the other interaction terms if the latter is repulsive. However, itis a new effect, as it depends on the wave functions rather than the densities.When g e term is negligible, the system behaves like the usual Bose mixtures with repulsive interactions. When g e term is dominant, the system behaves in a way similar to an attractive mixture with intraspecies interac-tion negligible. Moreover, to minimize the spin-exchange interaction energy g e h ˆ ψ † a ↑ ˆ ψ † b ↓ ˆ ψ b ↑ ˆ ψ a ↓ + ˆ ψ † a ↓ ˆ ψ † b ↑ ˆ ψ b ↓ ˆ ψ a ↑ i = − g e ψ a ↑ ψ a ↓ ψ b ↓ ψ b ↑ cos( γ a ↑ − γ a ↓ − γ b ↑ + γ b ↓ ) = − g e ψ a ↑ ψ a ↓ ψ b ↓ ψ b ↑ , ψ a ↑ ψ a ↓ ψ b ↓ ψ b ↑ should be maximized, as a kind ofinterference effect, which means that for each species α = a, b , ψ α ↑ ( r ) = ψ α ↓ ( r ), hence the density profiles for the twopseudospin components of each species tends to be the same. To see this, note that each wave function can be takento be real and positive [25]. The other terms in the Hamiltonian of course break this equality, as will be studied laterin this paper.Below we shall describe the finding that the larger the interspecies spin exchange g e is, the stronger the overlapbetween the density profiles of the two pseudospin components of each species is. Hence density profiles are very goodexperimental probes of the underlying interspecies correlations. On the other hand, by comparing the experimentaland theoretical results on the number density and spin density profiles, one may estimate the spin-exchange interactionstrength g e . Experimentally, by studying the the effect of g e on the density profiles, one can obtain the informationon such a mixture.Up to now, there is not yet a report on experimental studies of such a spin-exchange mixture between differentspecies. However, the interspecies spin-exchange interaction is determined by the difference between the interspeciestriplet and singlet scattering lengths, which has been found to be quite a few nanometer (nm) [26]. In this paper, thetheoretical investigation using this parameter value clearly indicates interesting new features. Hence our work alsoprovides some motivation and methodology for experimental exploration of such a mixture.In Sec. II, the numerical method is described. In Sec. III, we calculate the ground state properties, comparing thecases with and without spin exchange, and demonstrating the experimentally observable effects of interspecies spinexchange. Then we make a summary in Sec. IV. II. NUMERICAL METHOD
We assume the trapping potential to be U ασ ( r ) = 12 M α ω α ( ρ + λ z ) , (5)where ρ = p x + y , λ represents the trap anisotropy, M α and ω α are the mass of α -atom and trap frequency,respectively. U α ↑ = U α ↓ . For a magnetic trap, M α ω α = γ α µ B B , with γ α being the g factor of α -atom, B being the central magnetic field multiplied by a normalized factor. In order that the parameter values are close tothe experimental data, we imagine species a as Rb and species b as Na, then M a ω a /M b ω b = γ a /γ b = 1, i.e. U aσ = U bσ . Define κ ≡ ω b /ω a p M a /M b = p /
23. In our calculation, we use the parameter values ω a = 2 π × ω b = κω a and λ = √
8. The values of scattering lengths are set to be ξ aa ↑↑ = 8 nm, ξ aa ↓↓ = 6 . ξ aa ↑↓ = 3 . ξ bb ↑↑ = 4nm, ξ bb ↓↓ = 2 nm, ξ bb ↑↓ = 1 . ξ ab ↑↑ = 1 . ξ ab ↓↓ = 0 .
67 nm, ξ ab ↑↓ = 1 . ξ e and study itseffect on the number densities and the spin densities.We expand ψ ασ in terms of N basis eigenfunctions of the non-interacting Schr¨odinger equation in an anisotropicharmonic potential (5), that is, ψ ασ ( r ) = N basis X r =0 A rασ R m r n r ( ρ )Φ m r ( ϕ ) Z w r ( z ) , (6)where R m r n r ( ρ ), Φ m r ( ϕ ) and Z w r ( z ) correspond to the cylindrical coordinates ρ , ϕ and z , respectively, A rασ is theexpanding coefficient under the condition P N basis r ( A rασ ) = N ασ . For the ground state, only eigenfunctions with m r = 0 are relevant.Therefore, GP-like equation (4) is transformed to the following nonlinear equation( E lασ − µ ασ ) A lασ + g αασσ X ijk A iασ A jασ A kασ I ( iα, jα, kα, lα )+ g αασ ¯ σ X ijk A iα ¯ σ A jα ¯ σ A kασ I ( iα, jα, kα, lα ) + g α ¯ ασσ X ijk A i ¯ ασ A j ¯ ασ A kασ I ( i ¯ α, j ¯ α, kα, lα )+ g α ¯ ασ ¯ σ X ijk A i ¯ α ¯ σ A j ¯ α ¯ σ A kασ I ( i ¯ α, j ¯ α, kα, lα ) − g e X ijk A i ¯ α ¯ σ A j ¯ ασ A kα ¯ σ I ( i ¯ α, j ¯ α, k ¯ α, l ¯ α ) = 0 , (7)where E lασ = ((2 n l + m l + 1) + ( w l + 1 / λ )¯ hω α , I ( iα, jα, kβ, lβ ) ≡ π Z ∞ R m i n i ( η ρα ρ ) R m j n j ( η ρα ρ ) R m k n k ( η ρβ ρ ) R m l n l ( η ρβ ρ ) ρdρ × Z + ∞−∞ Z w i ( η zα z ) Z w j ( η zα z ) Z w k ( η zβ z ) Z w l ( η zβ z ) dz, where m r ( r = i, j, k, l ) should be 0. There are various algorithms to determine the solutions of the nonlinear equationset, such as fixed-point iteration and Newton method. We use Broyden method to obtain the solutions of Eq. (7)because of its high speed and precision. a, a, b, b, / N FIG. 1: Reduced chemical potential µ ασ / ¯ hω α varying with the atom number N of each species, at a generic parameter pointwithout interspecies spin exchange, i.e. ξ e = 0. µ ασ as well as µ aσ / ¯ hω a − µ bσ / ¯ hω b increase with N . III. CALCULATIONS
We shall use the GP-like equations (4) to study the effect of interspecies spin exchange on the number densities n ασ = | ψ ασ ( r ) | , and the spin densities S αz ( r ) = 12 [ | ψ α ↑ ( r ) | − | ψ α ↓ ( r ) | ] . For simplicity, we assume that the atom numbers of the two species are equal, i.e. N a = N b = N . Although thecases of N ≤ N . A. The case without interspecies spin exchange
First we consider the case of ξ e = 0, in which the system reduces to a mixture of four spinless condensates. Theresults for this case are summarized in Figures 1, 2, 3, 4, 5, 6. They are all consequences of minimizing the repulsiveinteractions among the four components, as detailed in the following.We have µ α ↑ > µ α ↓ under the present parameter values, as indicated in Fig. 1, which shows the chemical potential µ ασ as a function of N . µ ασ as well as µ aσ / ¯ hω a − µ bσ / ¯ hω b increase with N . The difference between µ α ↑ and µ α ↓ is due to spin dependence of various scattering lengths. This result can be confirmed by a calculation based on theThomas-Fermi approximation, which leads to µ ασ = [ R U ασ d r + g αασσ N ασ + g αασ ¯ σ N α ¯ σ + g α ¯ ασσ N ¯ ασ + g α ¯ ασ ¯ σ N ¯ α ¯ σ ] . In asense, µ ασ represents the average energy of one atom of species α with pseudospin σ . We can observe that µ ασ / ¯ hω α decreases with N , towards the single particle value 1 + λ/ . n ασ ( r ) = | ψ ασ ( r ) | . The density profilesfor several different values of N are shown in Fig. 2, where the distributions along ρ and z directions are depictedrespectively. These plots display some interesting features. Obviously n α ↑ and n α ↓ are complementary, because ofthe normalization condition R ( n α ↑ + n α ↓ ) d r = N . When N is small, n α ↑ and n α ↓ are close to each other, becausethe interaction energy is small. But with the increase of N , the difference between n α ↑ and n α ↓ increases in order tolower the interaction energy. When N is large enough, two or more peaks may appear in some density profiles, dueto the inclusion of higher order eigenfunctions in the expansion (6) when the interaction energy becomes important.The profiles of the total density of each species, n α = n α ↑ + n α ↓ , are shown in Fig. 3. We can see that not allcomponents co-exist in every region, in analogy with the two-component BEC [3]. This is in order to minimize thetotal energy under the given parameter values.The density profiles are more extended in ρ direction than in z direction, as exhibited in Fig. 2 and in Fig. 3. Thereason is that the trapping in z direction is stronger than that in ρ direction. ρ ( µ m) N=100 ρ ( µ m) N=1000 ρ ( µ m) N=5000 ρ ( µ m) N=10000 µ m) D en s i t y ( a t o m s / c m ) N=100 µ m) N=1000 µ m) N=5000 µ m) N=10000 a, ↑ a, ↓ b, ↑ b, ↓ FIG. 2: Density n ασ = | ψ ασ | for each pseudospin component of each species, at a generic parameter point with ξ e = 0, asdefined in the text, for N = 100 , , , N is atom number of each species. The upper plots are profiles along ρ direction with z = 0. The lower plots are profiles along z direction with ρ = 0. More than one peak appear in some plots, dueto the inclusion of higher order eigenfunctions in the expansion (6) when the interaction energy becomes important. We have also plotted the two-dimensional density profiles with ρ and z as the two coordinates, as shown in Fig. 4for λ = √
8, and in Fig. 5 for λ = 1, both with N = 10000. The complementarity between the two pseudospincomponents of each species is very clear. In the former case, as λ = 1, the profiles are asymmetric between ρ and z directions. In the latter case, as λ = 1, the profiles are symmetric between ρ and z directions.Now we consider the spin density S αz ( r ) = [ | ψ α ↑ ( r ) | − | ψ α ↓ ( r ) | ] . The spin density profiles are shown in Fig. 6for N = 100 , , , ρ direction, the spin density increases from a negative valueat the center to a positive value at a certain radius, and then gradually decreases to zero. Both the radius with thepositive maximal spin density and the radius where the spin density becomes zero increase with N . This feature canbe understood in terms of the difference between the densities of ↑ and ↓ atoms shown in Fig. 2. That the spin densityis negative in inside regimes while positive in outside regime is because we have assumed ξ αα ↑↑ > ξ αα ↓↓ , consequentlymore ↑ atoms tend to stay in the outside regime where the density is lower because of trapping potential, while more ↓ atoms tend to stay in the inside regime, in order to lower the total energy. Of course the spin density approacheszero for large enough value of ρ , as the densities of ↑ and ↓ atoms both approach zero. Along z direction with ρ = 0,the spin density is mostly negative, also because more ↑ atoms than ↓ atoms stay in the larger ρ regime. This effectweakens with the increase of z , because the trapping potential increases, consequently the difference between thenumbers of ↓ and ↑ atoms decreases. There is only a very small regime where the spin density becomes positive butthe values are too small to be visible on the plots. This feature is unlike the that of the profiles along ρ direction, asthere must be some regime of ρ with more ↑ atoms. B. The case with interspecies spin exchange
Now we come to the effect of interspecies spin exchange, i.e. the case of ξ e = 0. We have chosen ξ e =0.53 nm, 1.07nm, 2.03 nm, 4.27 nm. As stated in the Introduction, the mean-field spin-exchange interaction energy − g e ψ a ↑ ψ a ↓ ψ b ↓ ψ b ↑ (8) ρ ( µ m) N=100 ρ ( µ m) N=1000 ρ ( µ m) N=5000 ρ ( µ m) N=10000 µ m) D en s i t y ( a t o m s / c m ) N=100 µ m) N=1000 a b0 5 100123456 z( µ m) N=5000 µ m) N=10000
FIG. 3: Profiles of the total density n α = n α ↑ + n α ↓ , for each species, along ρ direction with z = 0, and along z direction with ρ = 0, at a generic parameter point with ξ e = 0, as defined in the text, for N = 100 , , , ρ direction than in z direction, as the trapping in z direction is stronger than in ρ direction. −5 0 5−505024 ρ ( µ m)z( µ m) n a ↑ , ( a t o m s / c m ) −5 0 5−505024 ρ ( µ m)z( µ m) n a ↓ , ( a t o m s / c m ) −5 0 5−505024 ρ ( µ m)z( µ m) n b ↑ , ( a t o m s / c m ) −5 0 5−505024 ρ ( µ m)z( µ m) n b ↓ , ( a t o m s / c m ) FIG. 4: Two-dimensional density profiles on a cross section including z -axis, with ρ and z as the coordinates, in a genericparameter point defined in the text, with ξ e = 0, λ = √ N = 10000. The profiles are asymmetric between ρ and z directions as λ = 1. −5 0 5−505024 ρ ( µ m)z( µ m) n a ↑ , ( a t o m s / c m ) −5 0 5−505024 ρ ( µ m)z( µ m) n a ↓ , ( a t o m s / c m ) −5 0 5−505024 ρ ( µ m)z( µ m) n b ↑ , ( a t o m s / c m ) −5 0 5−505024 ρ ( µ m)z( µ m) n b ↓ , ( a t o m s / c m ) FIG. 5: Two-dimensional density profiles on a cross section including z -axis, with ρ and z as the coordinates, in a genericparameter point defined in the text, with ξ e = 0, λ = 1 and N = 10000. The profiles are symmetric between ρ and z directionsas λ = 1. is an interference term and acts like an attractive interaction among the four orbital wave functions in some way, butit depends on the wave functions rather than the densities. This interference effect is manifested in the number andspin density profiles.The spin exchange lowers the chemical potential, as evident in Fig. 7, which is the numerical result of µ ασ as afunction of N for four values of ξ e . It can be seen from the plots that the larger ξ e is, the lower µ ασ is. However, inall these cases, dµ ασ /dN remains positive.The interference among the four orbital wave functions is also manifested in the density profiles, as shown in Fig. 8,Fig. 9 and Fig. 10. The spin exchange energy (8) is minimized when ψ α ↑ = ψ α ↓ for each species α . It is competitivewith the other interactions. If it dominates the energy, in order to lower the energy, the wave functions of the twopseudospin components of each species tend to be close to each other, compared with the case without spin exchange.Indeed, this tendency can be observed in Fig. 8 and Fig. 9, which show the density profiles for each pseudospincomponent of each species, in ρ direction with z = 0 and in z direction with ρ = 0 respectively. The larger ξ e is,the more dominant the spin-exchange interaction is, then the closer ψ α ↑ and ψ α ↓ are to each other. When ξ e is largeenough, the overlapping effect becomes very visible. Nevertheless, the overlap is not complete, because of other termsin the energy.Fig. 10 depicts the profile of the total density of each species, which can be compared with the plot for the same N in Fig. 3, it can be seen that the overlap regime of the two species is also enhanced by spin exchange, because of theeffect of spin exchange term (8).Moreover, we have also calculated the spin density profile, as shown in Fig. 11 for N = 10000. Spin density S αz = ( | ψ α ↑ | − | ψ α ↓ | ) is proportional to the difference between the densities of the two pseudospin components, soit is a quantification of the overlap between the two pseudospin components. Evidently, with the increase of ξ e , thevariation of the spin density with the radius decreases. In other words, both spin density S zα of each species and thetotal spin density S z are homogenized by the spin exchange. When ξ e is large enough, each spin density tends tovanish. The underlying reason is also because ψ α ↑ and ψ α ↓ , whose difference gives the spin density of species α , tendto be close to each other in order to lower the spin-exchange energy. Consequently the spin density of each species α tend to vanish. In both the case without spin exchange and the case with spin exchange, the location on z -axis wherethe spin density becomes zero is much smaller than that along ρ -direction. Compared with the case without spinexchange, another notable feature is that when ξ e and N are large enough, in the spin density profile along z directionwith ρ = 0, the regime with positive spin density becomes more visible (Fig. 11). This is because the negative sign ofthe spin-exchange interaction counteracts the trapping potential, even though the trapping is stronger in z directionthan in ρ direction. −3 z( µ m) S p i n den s i t y ( / c m ) N=100 µ m) N=1000 µ m) N=5000 ρ ( µ m) N=10000 µ m) N=10000 ρ ( µ m) N=5000 ρ ( µ m) N=1000 −3 ρ ( µ m) N=100 a b total
FIG. 6: Spin density profiles for N = 100 , , , ξ e = 0. The spin density approaches zero for large enough ρ or z , as the densities of ↑ and ↓ atoms both approach zero whenthe trapping potential is large enough. IV. SUMMARY
This paper concerns the ground state properties of a mixture of two species of pseudospin- Bose gases with inter-species spin-exchange interaction in a trapping potential. We have numerically calculated the four orbital condensatewave functions, each of which corresponding to a pseudospin component of each species, by using the GP-like equa-tions. We set the atom number of each species to be N . Using these wave functions, the number and spin densitiesare obtained. When the spin-exchange scattering length is zero, this mixture reduces to a mixture of the usual type,with four components. Various features appear as consequences of minimizing the density-density interaction. Forexample, with the increase of N , the difference between n α ↑ and n α ↓ for each species α increases.If there exists interspecies spin-exchange scattering, novel features absent in the usual mixtures emerge. As thespin-exchange interaction is negative as a consequence of minimizing the energy, it acts like an attractive interaction.Nevertheless, it depends on the overlap among the four wave functions. It lowers the chemical potentials and makethe densities of the two pseudospin components of each species tend to be close to each other, and thus the spindensity tends to be homogenized, and even tends to vanish when the spin-exchange scattering length is so large thatit dominates over the density-density interaction.Therefore as experimentally measurable quantities, the number and spin density profiles of such a mixture withinterspecies spin exchange are effective probes of the novel many-body ground state of this system. Acknowledgments
We thank Li Ge for useful discussion. This work was supported by the National Science Foundation of China(Grant No. 11074048) and the Ministry of Science and Technology of China (Grant No. 2009CB929204). aaaa N e =0.53 nm e =1.07 nm e =2.03 nm e =4.27 nm e =0.53 nm e =1.07 nm e =2.03 nm e =4.27 nm N bb e =0.53 nm e =1.07 nm e =2.03 nm e =4.27 nm N bb e =0.53 nm e =1.07 nm e =2.03 nm e =4.27 nm N FIG. 7: Reduced chemical potential µ ασ / ¯ hω α varying with N for four different values of ξ e . The chemical potential is loweredby the spin-exchange energy, which is attractive. dµ ασ /dN remains positive.[1] C. J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases (Cambridge University Press, Cambridge, 2002).[2] T. L. Ho and V. B. Shenoy, Phy. Rev. Lett. , 3276 (1996).[3] H. Pu, and N. P. Bigelow, Phys. Rev. Lett. , 1130 (1998).[4] P. Ao and S. T. Chui, J. Phys. B , 535 (2000).[5] B. D. Esry et al., Phys. Rev. Lett. , 3594 (1997).[6] E. Timmermans, Phys. Rev. Lett. , 5718 (1998).[7] M. Trippenbach et al.,J. Phys. B , 4017 (2000).[8] C. J. Myatt et al. , Phys. Rev. Lett. , 586 (1997); D. S. Hall et al. , Phys. Rev. Lett. , 1539 (1998); G. Modugno et al. ,Phy. Rev. Lett. , 190404 (2002); G. Roati et al. , Phys. Rev. Lett. ,010403 (2007); G. Thalhammer et al. , Phy. Rev.Lett. , 210402 (2008); S. B. Papp, J. M. Pino and C. E. Wieman, Phy. Rev. Lett. , 040402 (2008).[9] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998); T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998).[10] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. , 5257 (1998).[11] M. Koashi and M. Ueda, Phy. Rev. Lett. , 1066 (2000);[12] T.L. Ho and S.K. Yip, Phys. Rev. Lett. , 4031 (2000).[13] J. Stenger et al. , Nature , 345 (1998); H.-J. Miesner et al. , Phy. Rev. Lett. , 2228 (1999); D. M. Stamper-Kurn etal. , Phy. Rev. Lett. , 661 (1999); H. Schmaljohann et al. , Phy. Rev. Lett. , 040402 (2004).[14] A. B. Kuklov and B.V. Svistunov, Phys. Rev. Lett. , 170403 (2002).[15] S. Ashhab and A.J. Leggett, Phys. Rev. A , 063612 (2003).[16] Z. B. Li and C. G. Bao, Phys. Rev. A , 013606 (2006).[17] Y. Shi, Int. J. Mod. Phys. B , 3007 (2001).[18] Y. Shi and Q. Niu, Phy. Rev. Lett. , 140401 (2006).[19] Y. Shi, Europhys. Lett. , 60008 (2009). ρ ( µ m) N=100, ξ e =0.53nm ρ ( µ m) N=1000, ξ e =0.53nm ρ ( µ m) N=100, ξ e =1.07nm ρ ( µ m) N=1000, ξ e =1.07nm ρ ( µ m) N=100, ξ e =2.03nm ρ ( µ m) N=1000, ξ e =2.03nm N=100, ξ e =4.27nm ρ ( µ m) N=1000, ξ e =4.27nm ρ ( µ m) N=5000, ξ e =0.53nm N=5000, ξ e =1.07nm N=5000, ξ e =2.03nm N=5000, ξ e =4.27nm ρ ( µ m) D en s i t y ( a t o m s / c m ) N=10000, ξ e =0.53nm ρ ( µ m) N=10000, ξ e =1.07nm ρ ( µ m) N=10000, ξ e =2.03nm ρ ( µ m) N=10000, ξ e =4.27nm a, ↑ a, ↓ b, ↑ b, ↓ FIG. 8: Density profile n ασ = | ψ ασ | for each pseudospin component σ of each species α along ρ direction on the plane z = 0for N = 100 , , , ξ e . Other parameter values are given in the main text. The larger ξ e , thestronger overlap among the four wave functions.[20] Y. Shi, Phys. Rev. A , 013637 (2010).[21] R. Wu and Y. Shi, Phys. Rev. A , 025601 (2011).[22] L. Ge and Yu Shi, J. Stat. Mech. P06004 (2012).[23] L. Ge and Yu Shi, J. Stat. Mech. P10020 (2012).[24] M. Edwards and K. Brunett, Phys. Rev. A , 1382 (1995); P. A. Ruprecht, M. J. Holland, K. Brunett, and M. Edwards,Phys. Rev. A , 4704 (1995); M. Edwards, R. J. Dodd and C. W. Clark, P. A. Ruprecht and K. Burnett, Phys. Rev. A , R1950 (1996).[25] R. P. Feynman, Statistical Mechanics , (Benjiamin/Cummings, Reading, 1972).[26] G. Modugno et al. , Phys. Rev. Lett. , 190404 (2002); G. Ferrari et al. , Phys. Rev. Lett. , 053202 (2002); A. Simoni etal. , Phys. Rev. Lett. , 163202 (2003); S. Inouye et al. , Phys. Rev. Lett. , 183201 (2004); A. Simoni et al. , Phys. Rev.A. , 052705 (2008); M. Gacesa, P. Pellegrini and R. Cˆot´e, Phys. Rev. A , 010701 (R) (2008). µ m) N=100, ξ e =0.53nm µ m) N=1000, ξ e =0.53nm µ m) N=100, ξ e =1.07nm µ m) N=1000, ξ e =1.07nm µ m) N=100, ξ e =2.03nm µ m) N=1000, ξ e =2.03nm µ m) N=100, ξ e =4.27nm µ m) N=1000, ξ e =4.27nm µ m) N=5000, ξ e =0.53nm µ m) N=5000, ξ e =1.07nm µ m) N=5000, ξ e =2.03nm µ m) D en s i t y ( a t o m s / c m ) N=5000, ξ e =4.27nm µ m) N=10000, ξ e =0.53nm µ m) N=10000, ξ e =1.07nm µ m) N=10000, ξ e =2.03nm µ m) N=10000, ξ e =4.27nm a, ↑ a, ↓ b, ↑ b, ↓ FIG. 9: Density profile n ασ = | ψ ασ | for each pseudospin component σ of each species α along z direction on the line ρ = 0for N = 100 , , , ξ e . Other parameter values are given in the main text. The larger ξ e , thestronger overlap among the four wave functions. ρ ( µ m) D en s i t y ( a t o m s / c m ) ξ e =0.53nm 0 5 1002468 ρ ( µ m) ξ e =1.07nm 0 5 1002468 ρ ( µ m) ξ e =2.03nm 0 5 1002468 ρ ( µ m) ξ e =4.27nm0 5 1002468 z( µ m) ξ e =0.53nm 0 5 1002468 z( µ m) ξ e =1.07nm 0 5 1002468 z( µ m) ξ e =2.03nm 0 5 1002468 z( µ m) ξ e =4.27nma b FIG. 10: Profiles of the total density n α for each species α along ρ direction on the plane z = 0 and on the line ρ = 0 for atomnumber N = 10000 and several values of ξ e . The larger ξ e , the closer the profiles of the two species. µ m) S p i n den s i t y ( / c m ) ξ e =0.53nm 0 2 4 6 8−6−4−202 z( µ m) ξ e =1.07nm 0 2 4 6 8−6−4−202 z( µ m) ξ e =2.03nm 0 2 4 6 8−6−4−202 ρ ( µ m) ξ e =4.27nm0 2 4 6 8−6−4−202 z( µ m) ξ e =4.27nm0 2 4 6 8−6−4−202 ρ ( µ m) ξ e =2.03nm0 2 4 6 8−6−4−202 ρ ( µ m) ξ e =1.07nm0 2 4 6 8−6−4−202 ρ ( µ m) ξ e =0.53nm a b total FIG. 11: Spin density profiles in a generic parameter point as defined in the main text, for N = 10000 and various values of ξ e . The larger ξ ee