Mean field ground state of a spin-1 condensate in a magnetic field

We revisit the topic of the mean field ground state of a spin-1 atomic condensate inside a uniform magnetic field ($B$) under the constraints that both the total number of atoms ($N$) and the magnetization ($\cal M$) are conserved. In the presence of an internal state (spin component) independent trap, we also investigate the dependence of the so-called single spatial mode approximation (SMA) on the magnitude of the magnetic field and ${\cal M}$. Our result indicate that the quadratic Zeeman effect is an important factor in balancing the mean field energy from elastic atom-atom collisions that are known to conserve both $N$ and $\cal M$.


Introduction
Atomic Bose-Einstein condensates (BEC) have provided a successful testing ground for theoretical studies of quantum many-body systems [1]. In earlier BEC experiments, atoms were spatially confined with magnetic traps, which essentially freeze the atomic internal degrees of freedom [2]. Most studies were thus focused on scalar models, i.e. single component quantum degenerate gases [3]. More recently, the emergence of spin-1 condensates [4,5,6] (of atoms with hyperfine quantum number F = 1) has created opportunities for understanding degenerate gases with internal degrees of freedom [7,8,9,10,11,14].
In this paper, we investigate the mean field ground state structures of a spin-1 atomic condensate in the presence of an external magnetic field (B). We focus on several aspects of the ground state properties strongly affected by the requirement that elastic atom-atom collisions conserve both the total number of atoms (N) and the magnetization (M). Several earlier studies have focused on the global ground state structures when the conservation of M was ignored, or in the limiting case of a vanishingly small magnetic field (B = 0) [6,7,8,9,10,11,14]. As we show in this study, in the presence of a nonzero magnetic field, the conservation of M leads to ground state population distributions significantly different from that of the global ground state.
Our system is described by the Hamiltonian (repeated indices are summed) [7] where ψ j ( r) is the field operator that annihilates an atom in the j-th (j = +, 0, −) internal state at location r, L ij ≡ −h 2 ∇ 2 /2M + V ext ( r) δ ij with M the mass of each atom and V ext ( r) an internal state independent trap potential. Terms with coefficients c 0 and c 2 of Eq. (1) describe elastic collisions of the spin-1 atom (|F = 1, M F = +, 0, − ), expressed in terms of the scattering length a 0 (a 2 ) for two spin-1 atoms in the combined symmetric channel of total spin 0 (2), c 0 = 4πh 2 (a 0 + 2a 2 )/3M and c 2 = 4πh 2 (a 2 − a 0 )/3M. F η=x,y,z are spin-1 matrices with The external magnetic field B is taken to be along the quantization axis (ẑ), it induces a Zeeman shift on each atom given by According to the Breit-Rabi formula [12], the individual level shift can be expressed as where E HFS is the hyperfine splitting [12], and g I is the Lande g-factor for the atomic nuclei with nuclear spin I. µ I is the nuclear magneton and α = (g I µ I B + g J µ B B)/E HFS with g J the Lande g-factor for the valence electron with total angular momentum J. µ B is the Bohr magneton.

Mean field approximation
At near zero temperatures and when the total number of condensed atoms is large, the ground state is essentially determined by the mean field term Φ i = ψ i . Neglecting all quantum fluctuations we arrive at the mean field energy functional from Eq. (1) [8,13] where the symmetric part is invariant under the exchange of spin component indices, thus is independent of the external B field. The Zeeman shift as given by the Breit-Rabi formula (2) can be described by two positive parameters [13] 2η 0 = E − − E + , which measure approximately the linear and quadratic Zeeman effects. The B-field dependence of η 0 and δ for a 87 Rb atom are displayed in Fig. 1. The elastic atomic collisions as described by the c 0 and c 2 parts of the Hamiltonian (1) conserve both N and M, which in the mean field approximation are given by Before continuing our discussion of the ground state structures, we shall first briefly comment on the importance of the above two constraints. In a typical experiment, the last stage before condensation consists of atomic evaporations, during which neither N nor M is conserved. For a scalar condensate, typically the ground state is obtained from a minimization of Eq. (3) subjected to the constraint of only N conservation. This gives rise to the Gross-Pitaevskii equation (GPE) and the associated condensate chemical potential, which mathematically is simply the Lagrange multiplier of the constrained minimization. A spin-1 condensate requires the introduction of two Lagrange multipliers during the minimization subjected to both the N and M conservation constraints, as was first performed in [13].
When atomic interactions are ferromagnetic (c 2 < 0 as for 87 Rb atoms) and when the external B-field is negligible, we have shown previously that the ground state structure is simply a state where all individual atomic spins are aligned in the same direction [14]. In this case, the conservation of M can be simply satisfied by tilting the quantization axis away from the direction of the condensate spin. This can always be done if a system described by (1) is rotationally symmetric, and thus contains the SO(3) symmetry [7]. The presence of a nonzero B field, on the other hand, breaks the rotational symmetry, [e.g. the linear Zeeman shift, reduces the SO(3) to SO(2) symmetry], thus the conservation of M has to be included in the minimization process directly.
The global ground state phase diagram including both linear and quadratic Zeeman effect was first investigated by Stenger et al [6]. In this early study, although the M conservation was included in their formulation, it was not separately discussed, consequently their results do not easily apply to systems with fixed values of M. The ground state structures as given in Ref. [6] correspond to the actual ground state as realized through a M non-conserving evaporation process (e.g. in the presence of a nonzero B-field) that serves as a reservoir for condensate magnetization. Our study to be presented here, on the other hand, would explicitly discuss the phase diagram for fixed values of M, which could physically correspond to experimental ground states (with/without a B-field) due to a M conserving evaporation process. Although more limited, as our results can be traced to linear trajectories of M = const. in the phase diagram of Ref. [13], we expect them to be useful, especially in predicting ground state structures when a ready-made spinor condensate is subjected to external manipulations that conserve both N and M.
When atomic interactions are anti-ferromagnetic (c 2 > 0), the global ground state was first determined to be a total spin singlet [8]. More elaborate studies, including quantum fluctuations, were performed by Ho and Yip [17] and Koashi and Ueda [18]. Unfortunately, these results [17,18] do not correspond to actual ground states as realized in current experiments, because of the presence of background magnetic fields. For instance, the states as found in Ref. [17] are only possible if the magnetic field B is less than 70µG at the condensate density as realized in the MIT experiments [13]. The Zeeman shift (see Fig. 1) due to the presence of even a small magnetic field can overwhelm atomic mean field interaction and typical atomic thermal energy, thus if it were not for the conservation of M, the ground state would simply correspond to all atoms condense into the lowest Zeeman sublevel of |M F = 1 . We It is easy to check that the phase convention of ferromagnetic/anti-ferromagnetic interactions as obtained previously [10] in the absence of a B-field still remains true, i.e.

Ground state in a homogeneous system
In a homogeneous system such as a box type trap (of volume V), adopting the above phase convention, the resulting ground state energy functional becomes (+/− for c 2 < 0 and c 2 > 0 respectively) Expressing everything in terms of fractional populations and fractional magnetization n i = N i /N and m = M/N, and note that n with an interaction coefficient c = c 2 N/2V, tunable through a change of condensate density.
We now minimize Eq. (10) under the two constraints n + + n 0 + n − = 1 and n + − n − = m. We restrict our discussion to the region −1 < m < 1 as the special cases of m = ±1 are trivial. Because H S , E 0 , c, η 0 , and m are all constants for given values of B, N, and V, the only part left to be minimized is In the special case of c = 0, Equation (11) reduces to The ground state is then very simple. When δ > 0, which seems to be always the case for quadratic Zeeman shift, the minimum is reached by having as large a n 0 (thus as small a n + + n − ) as possible, namely When δ = 0, we have (in general) three condensate components with n ± = (1−n 0 ±m)/2 and 0 ≤ n 0 ≤ 1 − |m|.
For ferromagnetic interactions with c < 0, we define x = n + + n − . The ground state is then determined by the minimum of with which is the same as obtained in [11,14]. However with a nonzero δ > 0, we find in general with x 0 being the root of equation g ′ + (x) + δ = 0, it turns out that there always exists one and only one solution to the equation. The equilibrium value for n 0 is larger than the result of Eq. (15) because the quadratic Zeeman effect causes a lowering of the total energy if two |M F = 0 atoms are created when an |M F = +1 atom collides with an |M F = −1 atom. Figure 2 displays the results of Eq. (16) for a typical 87 Rb condensate, for which the atomic parameters are E HFS = (2π)6.8347GHz [12], a 0 = 101.8a B , and a 2 = 100.4a B (a B is the Bohr radius) [15]. At weak magnetic fields, typically a condensate contains all three spin components. With the increasing of Bfield, the quadratic Zeeman effect becomes important which energetically favors the |0 component, so typically only two components survive: the |0 component and the larger (initial population) of the |+ or |− component, so the ground state becomes (for m > 0) n + ≃ m and n 0 ≃ 1 − m.
Finally we consider the case of anti-ferromagnetic interactions for c > 0, we have then . For δ = 0, we again recover the standard result if m = 0. When m = 0, the ground state is under-determined as many solutions are allowed as along as they satisfy n + = n − = 1 − n 0 with n 0 ∈ [0, 1].  Figure 3. The same as in Fig. 2, but now for a spin-1 23 Na condensate.

Ground state inside a harmonic trap
In the previous section, we investigated in detail mean field ground state structures for a spin-1 condensate in a homogeneous confinement. For the case of a harmonic trap as in most experiments, there is no reason to believe a priori that the above conclusions still hold. In fact, the structures and phase diagrams as discussed before is only meaningful if the spatial mode function φ j ( r) for different spin components is identical. Otherwise, it would be impossible to classify the rich variety of possible solutions. When the spatial mode functions are the same, the spatial confinement simply introduces an average over the inhomogeneous density profile of the mode function. The aim of this section, is therefore to determine the validity of the single mode approximation (SMA) in the presence of an external B-field and a harmonic trap. For simplicity, we assume the trap to be spherically symmetric. We employ numerical methods to directly find the ground state solutions from the coupled Gross-Pitaevskii equation subjected to the conservations of both N and M [Eqs. (6)]. H = −h 2 ∇ 2 /2M + V t ( r) + c 0 n, n j = |Φ j | 2 , V t ( r) = Mω 2 r 2 /2, and n = n + + n 0 + n − . η is the Lagrange multiplier introduced to numerically enable the conservation of M.
It was shown previously that in the absence of an external B-field, and for ferromagnetic interactions, the SMA is rigorously valid despite the presence of a harmonic trap [14]. We can also show that in the presence of a nonzero B-field, the linear Zeeman shift does not affect the validity of the SMA because it can be simply balanced by the external Lagrange multiplier η. The quadratic Zeeman effect, on the other hand, can not be simply balanced, as it favors the production of two |0 atoms by annihilating one |+ and one |− atom during a collision. Such unbalanced elastic collisions thus break the SO(3) symmetry of the freedom for an arbitrary quantization axis. Therefore, we do not in general expect the SMA to remain valid inside a nonzero B-field. Numerically, we find the ground state solutions of Eq. (20) by propagating the equations in imaginary time. We typically start with an initial wave function as that of a complex Gaussian with a constant velocity: exp[−(x 2 /2q 2 x + y 2 /2q 2 y + z 2 /2q 2 z ) − i k · r]. q x , q y , q z , and k are adjustable parameters which are checked to ensure that their choices do not affect the final converged ground state [14].
For c 2 = 0 or c = 0, it is easy to check that SMA is always valid since the energy functional is symmetric with respect to spin component index. The fractional populations for each component is therefore the same as for a homogeneous system, i.e. given by ( 1−n 0 +m For 87 Rb and 23 Na condensates, which are believed to be ferromagnetic c 2 < 0 (c < 0) and anti-ferromagnetic c 2 > 0 (c > 0) respectively, Figure 5 gives typical density distributions of spacial mode function, ρ( r) = |φ j ( r)| 2 . Both sub-figures clearly indicate that SMA is no longer valid. To get an overall idea of the validity of SMA we plot in Figure 6 the overlap integrals of our mode functions with respect to the SMA mode function φ SMA ( r) as determined from a scalar GP equation with a nonlinear coefficient ∝ c 0 (due to the symmetric H S only) [14]. For a 87 Rb condensate, we see the overlap is close to unity when B is small, therefore, SMA remains approximately applicable. But it becomes increasingly bad with the increase of B. We thus conclude that the SMA remains reasonable in a weak magnetic field while it is clearly invalid in a strong B-field. In fact, our numerical results confirm that the stronger the B-field, the  Figure 7. The same as in Fig. 6, but now comparing the spin asymmetric energy E a = c 2 F 2 /2 − (η 0 + η) F z + δ F 2 z with the spin symmetric one H s .
worse the SMA gets. For typical system parameters, the dividing line occurs at a B-field of a fraction of a Gauss when the system magnetization M is not too small or too large. For a condensate with anti-ferromagnetic interactions, it was found earlier that SMA is violated in the limit of both large N and M even without an external B-field, while the case of M = 0 presents an exception where SMA remains strictly valid for B = 0 [14]. Figure 6 shows the overlap integral for a 23 Na condensate, indeed we see SMA is invalid except at M = 0 where all atoms are in the |0 component. Remarkably, despite the seemingly large deviations from the SMA (as in Fig. 6), the spin asymmetric energy term remains very small in comparison to the spin symmetric term as evidenced in Fig.  7. Figure 8 shows the dependence of fractional populations on the fractional magnetization for a 87 Rb (left column) and a 23 Na condensate (right column) at different B-fields. For 87 Rb atoms, these curves resemble the same dependence as for a homogeneous system where SMA is strictly valid. Nevertheless, we find the densities of mode functions can become quite different, i.e. SMA is not valid in general. For   Fig. 8(e) and (f)], our numerical solutions reveal again two distinct regions; one for m < m c where all three components coexist, and another one for m > m c where only two components (|+ and |− ) coexist. We find that m c increases with the B-field, and is of course limited to m c < 1. We conclude that despite the fact a harmonic trap induces spatially inhomogeneous distribution to condensate density, thus breaks the SMA in general, the overall ground state properties as measured by the fractional component distributions follow closely the results as obtained previously for the homogeneous case. Physically, we believe the above results can be understood as fractional populations relate to integrals of wave functions over all spaces, during which differences between wave functions can be averaged out. When only the |+ and |− components coexist, in fact, the two constraints on N and M always give the fractional population n ± = (1 ± m)/2 if N 0 = 0.

Conclusion
We have revisited the question of the mean field ground state structures of spin-1 condensate in the presence of a uniform magnetic field. For a homogeneous system, when c = 0, there exists in general only two nonzero components |+ and |0 , except when B = 0 where the ground state solution becomes indefinite; for ferromagnetic interactions when c < 0, the ground state in general has three nonzero components; when c > 0 as for anti-ferromagnetic interactions, except for m = 0, there are two regions: one for δ > 2c[1 − 1 − m 2 c ] where three nonzero components coexist and one for δ ≤ 2c[1 − 1 − m 2 c ] where only two components coexist. Inside a harmonic trap, these results remain largely true, although the SMA becomes generally invalid. We find interestingly (see Fig. 9), the B field (or the δ) dependence of the critical value m c that separates the two and three component condensate regions, remains almost identical as that given by the analytical formulae δ = 2c[1 − 1 − m 2 c ] for a homogeneous system. In a sense, this also points to the validity of the use of a mean field description, as the number of atoms is really large (10 6 ).

Acknowledgement
We acknowledge interesting discussions with Prof. M. S. Chapman and Mr. M. -S. Chang. This work is supported by NSF.