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## Bose-Einstein condensates with an internal degree of freedom

Bose-Einstein condensation in an atomic gas is a phase transition defined by the macro-scopic accumulation of atoms in the same single-particle state: if the gas is confined by an external potential, the atoms condense in the ground-state of the trap due to their bosonic statistics. They are then described by a giant wavefunction. The first experimental real-izations of Bose-Einstein condensates in laser-cooled gases has been achieved in 1995 with alkali atoms [2, 3]. Even though these atomic species have an hyperfine structure with non-zero total spin in their electronic ground state, the use of magnetic traps imposed that all the atoms remain in low-field seeking states. For atoms with a nuclear spin I = 3/2 such as 87Rb and 23Na, only the |F = 1, mF = −1 state of the lower hyperfine manifold and the |F = 2, mF = 1, 2 states of the upper one can be trapped. The condensate was actually polarized in a single magnetic state: the spin degree of freedom was frozen 1. The condensation is characterized in this case by a scalar wavefunction, describing the external degrees of freedom .

However the development of optical trapping techniques since 1997 allows to equally trap all the sublevels of an hyperfine manifold [48, 49], releasing the constraint on the internal degree of freedom and thus opening the possibility of creating Bose-Einstein con-densates with several distinguishable components. Such condensates where the atoms are allowed to occupy any of the magnetic sublevels of a single hyperfine manifold F are called spinor condensates and are represented by a (2F + 1)-component wavefunction. Multi-component condensates are realized in other systems. The mixture of condensates from diﬀerent bosonic species is also described by a multi-component wavefunction, as well as the mixture of condensates occupying N diﬀerent hyperfine states of the same isotope, for instance the |F = 1, mF = −1 and |F = 2, mF = +1 states of 87Rb, which realizes a pseudo-spin (N − 1)/2 system. Yet spinor condensates diﬀer in major ways from other kinds of multi-component condensate. The key feature of spinor condensates is the vec-torial nature of their wavefunction which expresses the possibility of population transfers between their diﬀerent components. Mixtures of condensates do not possess this property. Associated with the internal rotational symmetry of the system, the vectorial transformation of the wavefunction dramatically aﬀects the inter-atomic interactions. In particular, a characteristic of spinors that is not shared by other multi-component systems is the coher-ent internal-state dynamics driven by spin-exchange collisions. This coherent spin-mixing makes spinor condensates diﬀerent from an uncoherent overlap of 2F + 1 condensates and gives access to a rich variety of phenomena. For instance, spin-1 condensates constitute a suitable non-linear medium to create non-classical state of matter, similarly to the creation of non-classical states of light in quantum optics. Using this similarity, parametric spin amplification [33] and spin squeezing [50] have been experimentally demonstrated. The coherence of the spin collisions also appears in the Josephson junction dynamics of spin oscillations [32, 51].

Bose-Einstein condensation is a purely quantum phase transition in the sense that it is a consequence of the quantum statistics of the atoms and not of their interactions. Still in spinor gases interactions between diﬀerent spin states lead the atoms to condense in several possible phases. This is possible in spite of the weakness of the spin-dependent interaction (which contributes roughly to 1nK of energy per atom, much less than the typ-ical 100nK temperature of the condensate) because of the bosonic enhancement provided by the Bose-Einstein condensate. Because of these enhanced interactions, spinor Bose gases become magnetically ordered below the condensation threshold. The interplay be-tween spin-interactions and magnetic fields in the condensate give rise to magnetic phase transitions. We demonstrate in the following the existence of such a phase transition in spin-1 condensates with antiferromagnetic interactions.

In this chapter, we give first elements of theory of the spinor Bose gas that consti-tute the basis of the experimental and theoretical developments presented in the rest of this work. We first recall some basic properties of the Bose-Einstein condensation in a scalar gas confined in an harmonic trap. In particular we discuss the ground-state and the low-lying excited states of a weakly interacting gas. In a second part we turn to spinor condensates. We derive the interaction Hamiltonian and introduce two important approx-imations to simplify its treatment: the single-mode and the mean-field approximations. We then combine them to discuss the ground-state of the antiferromagnetic spin-1 Bose gas, which we experimentally studied in 23Na condensates. The validity of the single-mode approximation is investigated. Finally we show that new modes of excitations associated to the internal degree of freedom, analogous to spin waves in magnetic materials, arise.

**Bose-Einstein condensation in scalar gases**

**Bose-Einstein transition in an ideal gas**

We consider a gas of N non-interacting atoms confined in an harmonic potential: Vext(r) = 1 m(ωx2x2 + ωy2y2 + ωz2z2) (1.1) where m is the mass of one atom and ωi=x,y,z are the trapping frequencies along the three directions of space. The ground state wavefunction is obtained from the time-independent Schr˝odinger equation: φ(r) = mω¯ 3/4 exp − m (ωxx2 + ωyy2 + ωzz2) (1.2) where we defined the averaged frequency ω¯ = (ωxωyωz)1/3. The wavefunction φ satisfies the normalization condition d3r|φ(r)|2 = 1. The size of the condensate is independent of N and is set by the harmonic oscillator length 1/2 aho = (1.3) mω¯

The standard procedure to introduce the Bose-Einstein condensation is to consider the bosonic occupation numbers of all the excited states, sum them and show that this sum has an upper bound independent of N [52]. The maximum number of atoms that can populate the excited states decreases with the temperature T of the gas, so that below a critical temperature Tc that depends on the atom number N , the atoms have no choice but to occupy the ground-state. This point marks the onset of the Bose-Einstein condensation. As the temperature is further lowered the atoms accumulate in the ground-state whose population N0 becomes a macroscopic fraction of N . The fraction of atom in the ground-state N0/N is denoted as the condensed fraction. A semi-classical approximation valid when the thermal energy is large compared to the level spacings set by the harmonic oscillator (kB T ≫ ωx,y,z) gives a critical temperature N 1/3 kB Tc = ω¯ ≃ 0.94 ωN¯ 1/3 , (1.4) g3(1) where we introduced the family of Bose functions gα(z) = ∞ z k . When we let α k=0 k N → +∞, the proper thermodynamic limit is to let at the same time ω¯ → 0, while keeping the product N ω¯3 constant, so that the critical temperature (1.4) stays well de-fined.

The semi-classical approximation also allows one to calculated the spatial density of the thermal atoms populating the excited states. One finds: nT (r) = λT−3g3/2(eβ(µ−VEXT (R))) (1.5) where λT = h/ 2πmkB T is the De Broglie wavelength and µ is the chemical potential.

**Eﬀect of the interactions: ground-state**

Even though we consider dilute gases of typical densities ranging from 1013 to 1015cm−3, the interactions between atoms can not be neglected. We recall in this section how they modify the ground-state of the condensate [52].

**Two-body interactions**

Because of the low density, the atoms interact almost exclusively through binary collisions. Furthermore, the temperature of the gas is low enough to allow for the ”cold collisions” approximation: collisions are well described by s-wave scattering only. In this conditions, the interactions are characterized by a single parameter, the s-wave scattering length a. As the details of the two-body scattering potential are irrelevant, instead of the real (generally unknown) potential, one typically uses a simple model potential with the same scattering length. A popular choice for short range interactions is the so-called Fermi potential: Vˆ (r, r′) = 4π 2a δ(r − r′) (1.6) where δ is the Dirac distribution. In the following we note g = 4π 2a/m. Using such a contact potential to build a many-body theory is valid as long as the s-wave scattering length is much smaller than the mean inter-particle distance, which gives the condition n|a|3 ≪ 1 where n is the total density. In this dilute limit, two-boy scattering is essentially unaﬀected by the presence of other atoms and proceeds as if the two colliding atoms were alone. Positive and negative values of a correspond respectively to an eﬀective repulsion and attraction between the atoms. From now on we only consider positive values of a.

**Gross-Pitaevskii equation**

In an ideal gas the many-body ground state is simply obtained by putting all the particles in the single-particle ground-state. This is not true any more for interacting particles, and the exact many-body particles ground-state is usually hard to calculate exactly. A very successful approach to describe the interacting Bose gas is the Hartree-Fock approxima-tion which assumes that all the atoms share the same single-particle wavefunction. In this case we can derive an equation for the ground-state wavefunction φ by minimizing the free energy H − µN . We obtain the Gross-Pitaevskii equation: −2m ∇2 + Vext(r) + gN |φ(r)|2 φ(r) = µφ(r) (1.7)

The chemical potential µ corresponds mathematically to the Lagrange multiplier associ-ated with the conservation of the total atom number N . In equation (1.7) φ is normalized to unity: dr|φ(r)|2 = 1.

**Thomas-Fermi approximation**

To estimate the importance of the eﬀect of the interactions on the ground state wavefunc-tion we compare the interaction energy Eint to the kinetic energy Ekin of the system. One finds that the ratio of the two energies is given by: int ≈ N a (1.8) Ekin aho

If this number is large, the ground state is essentially determined by the interaction and the kinetic term can be neglected compared to the interaction one in the Gross-Pitaevskii equation. This is the Thomas-Fermi approximation. From (1.7) we find the density of the condensate:

n(r) = |φ(r)|2 = g−1 max [µ − Vext(r), 0] (1.9)

Due to the shape of the trapping potential, the density of the condensate is parabolic. (For non-interacting atoms it was gaussian). The chemical potential is calculated by integrating (1.9) over space and equaling it to N :

ω¯ a 2/5

µ = 15N (1.10)

We then obtain the radius of the condensate Ri = 2µ/mωi2:

ω¯ a ) 1/5 (1.11)

Ri = aho (15N

for i = (x, y, z). Since we assume N a/aho ≫ 1, the size of the condensate is much larger than the size of the non-interacting ground state aho. This is an eﬀect of the repulsive interaction between the atoms.

### Eﬀect of the interactions: excited states

We now consider the eﬀect of the interactions on the excited states. In a non-interacting condensate, elementary excitations consist in the promotion of atoms of the condensate to single-particle excited states of the trap. Their spectrum is then the one of an har-monic oscillator. Interactions modify this spectrum. Low-lying excitations can be derived following a procedure introduced by Bogoliubov [53]. For simplicity we here consider the case of an homogeneous gas of N atoms in a volume V .

In second quantization a generic many-body Hamiltonian reads: Hˆ = drΨˆ†(r) −2m ∇2 Ψ(ˆ r) + drdr′Ψˆ†(r)Ψˆ†(r′)Vˆ (r − r′)Ψ(ˆr′)Ψ(ˆr) (1.12)

In the following we replace the two-body interaction operator V by expression (1.6).

For uniform systems it is convenient to expand the atomic-field operator in the basis of plane waves. We assume that a macroscopic fraction of the atoms are in the conden-sate, which is defined by the k = 0. We note N0 the number of atoms in this mode.The commutator of the associated creation and annihilation operators is much smaller than their action on the state of the condensate (on order √ N0), so that these operators can be approximated by c-numbers:

aˆ0† ≃ aˆ0 ≃

N0 (1.13)

Separating the k = 0 mode from the others, and retaining only terms which are at least quadratic in aˆ†0 and aˆ0, the Hamiltonian becomes: ˆ gN 2 + K=0 (ǫk + gN † K gN † † K K

H ≈ 2V V )ˆaKaˆ + 2V K=0 (ˆaKaˆ−K + aˆ aˆ− ) (1.14)

where ǫk = 2k2/2M . The first term is the energy of the condensate (due to interaction since the kinetic energy is zero). The factor in front of aˆ†KaˆK has two terms: the kinetic energy and the interaction with the condensate. The last term comes from processes where two atoms of the condensate are scattered into states of momenta +k and −k, and the inverse process where two atoms with momenta +k and −k are scattered into the con-densate.

The Hamiltonian (1.14) can be diagonalized by a Bogoliubov transformation. We in-troduce new operators αˆk defined by: αˆK† = uKaˆK† + vKaˆ−K (1.15) where uK and vK are amplitudes to determine. We impose that these new operators obey the bosonic commutation relations:

[αˆK, αˆK†′ ] = δK,K′ , [αˆK, αˆK′ ] = [αˆK†, αˆK†′ ] = 0 (1.16)

Figure 1.1: Spectrum of Bogoliubov excitations. The solid line shows the energy εk, the dashed line shows ǫk + gn.

Inserting (1.15) into the Hamiltonian (1.14), we determine uK and vK such that the Hamil-tonian becomes diagonal. We obtain: ˆ (N/V ) + † K (1.17) H=E0 εkαˆKαˆ K=0

where E0(N/V ) is the energy of the ground-state. We note n = N/V the density. The excited states correspond to creations of quasiparticles with an energy εk given by: εk = ǫk(ǫk + 2gn) (1.18)

For long wavelength excitations (k ≪ mgn/ 2) we find the phonon dispersion law ε ≈ cs k, where cs = gn/m is the sound velocity, whereas in the opposite limit the short wavelength excitations (k ≫ mgn/ 2) are free particles excitations with energy ε k ≈ ǫk + gn. The spectrum at high energies resemble the one of a non-interacting system, and excitations are there single-particle excitations (the amplitude vK goes to zero for large k). On the other hand, the lowest modes are collective modes involving more than one particle.

In a box with periodic boundary conditions, the possible values of the components of k are discrete: ki = 2π/Li, where i = x, y, z and Li is the size of the box in direction i. In an harmonic potential, the complete calculation of elementary excitations is tedious, but we can still estimate the energy of the phonon modes, approximating the harmonic trap by a box. We take for the size of the box twice the radius of the condensate in the Thomas-Fermi regime RT F = 2µ/mω2 = 2gn/mω2, where ω is here the frequency of the harmonic potential that we assume isotropic for simplicity. If we calculate the energy of the first phonon mode we get: ε1 ∼ cs ∼ ω (1.19)

We find that the first phonon mode has an energy on the order of ω, the quantum of energy associated to the harmonic potential. This is confirmed by a more rigorous analysis [52, 54].

**Spin-1 Bose-Einstein condensates: spin Hamiltonian**

We now consider a Bose-Einstein condensate with an internal degree of freedom, and in particular the case of a condensate of particles of spin 1. We first look at the single particle level, and describe the internal structure and how it couples to external fields. We then move to the level of two particles and discuss how they interact before finally deriving the many-body Hamiltonian.

**Single spin-1 particle**

**Internal structure**

We describe here the electronic structure of 23Na. The fine structure results from the coupling of the electron spin S to its orbital angular momentum L. We note the total electron angular momentum J = S + L. The ground state 32S1/2 corresponds to L = 0 so that J = 1/2. The two first excited states 32P1/2 and 32P3/2 correspond to L = 1 and have J = 1/2 and J = 3/2 respectively. The two transitions 32S1/2 → 32P1/2 and 32S1/2 → 32P1/2 form a fine structure doublet noted D1 and D2. Each of these three levels have an hyperfine structure resulting from the coupling of the total electron angular momentum J to the nuclear angular momentum I. We note the total angular momentum F = J+I. In 23Na the nucleus has an angular momentum I = 3/2 so that the ground state

J = 1/2 splits into two levels of total spin F = 1 and F = 2. The hyperfine splitting in the ground state is ∆Ehf s ≈ 1.77 GHz [55]. The 32P1/2 and 32P3/2 states split respectively in two (F = 1, 2) and four levels (F = 0, 1, 2, 3). Each of these hyperfine level contains 2F + 1 Zeeman sublevels labeled my the projection of the total angular momentum on the quantization axis, mF = −2F, −2F + 1, …, 2F . In absence of external magnetic field these levels are degenerate.

**Zeeman eﬀect**

We now consider the eﬀect of an external magnetic field oriented along the quantization axis and of amplitude B. The angular momentum of a particle couples to the magnetic field and the degeneracy of the Zeeman sublevels is then broken. In the electronic ground state the shifts of the Zeeman sub levels are given by the Breit-Rabi formula [55]: EmF = −mF gI µI B − 1 1 + mF α + α2 (1.20) where α = (gJ − gI )µB B/∆Ehf s, µI is the nuclear magneton, µB is the Bohr magneton, gI and gJ are respectively the nuclear and electronic Land´e g-factors.

Typical magnetic fields used experimentally are on the order of a few G, so that the Zeeman shifts are small compared to ∆Ehf s. One can expand the Breit-Rabi formula up to order 2 in α, and neglecting gI compared to gJ we get: EmF = mF p + mF2 q (1.21)

Figure 1.2: Fine structure and hyperfine structure of the electronic ground state of Sodium. with

(gF µB B)2

p = gF µB B , q = (1.22)

∆Ehf s

where gF = −1/2 for the F = 1 hyperfine manifold is calculated from gJ and gI . Numer-ically one has µB /2 ≈ h × 700 kHz/G, and µ2B /4∆hf ≈ h × 277 Hz/G2.

**Optical trapping**

We now discuss the interaction between spin-1 particles and the electric field of a laser beam. The motivation is that the production of a spinor condensate requires that the diﬀerent spin states of an hyperfine manifold see the same trapping potential. Such a spin-independent potential is provided by an optical dipole trap. The electric field of a red-detuned focused laser beam induces in polarizable atoms a dipole moment whose interaction with the same electrical field creates the trapping potential. For a laser which is far-oﬀ resonant to the atomic transition of interest, this interaction is described by the Hamiltonian Hˆdip = −Eˆ(−).α.Eˆ(+) (1.23) where E are the electric field operators creating and annihilating photons in the laser mode, and α is the atomic polarizability tensor given by α = − PˆF dPF ′ d†PF (1.24) F,F ′ ∆FF′ F (F ′) labels the angular momentum of the ground (excited) states manifold. PˆF is the ˆ ′ = ωL−ωFF ′ is projector on the manifold F , d is the electric dipole operator,and ∆F F the detuning of the laser frequency to the atomic transition F − F ′. The polarizability tensor α can be understood in terms of a scattering interaction: an atom from the ground state manifold F is virtually promoted to the manifold F ′ by the raising dipole operator ˆ† ˆ(+) d and at the same time a photon is annihilated by E . The atom then decays back to the manifold F (possibly in an other state) and a photon is created. The polarizability α is built as the dyadic product of two vector operator, and so is a rank-2 tensor that can decomposed in the sum of irreducible operators of rank 0,1 and 2. It follows that the Hamiltonian H can as well be written:

ˆ ˆ (0) ˆ (1) ˆ (2)

Hdip = Hdip + Hdip + Hdip (1.25) ˆ (i)

where H is the component of rank i. If we restrict the ground states of the atoms to one manifold F we have [56]

ˆ (0) Hdip

ˆ (1) Hdip

ˆ (2) Hdip

∝ E(−).E(+).✶F (1.26)

∝ (E(−) × E(+)).Fˆ F(F + 1) (1.27)

∝ Ei(−)Ej(+) (FiFj + Fj Fi) − ✶F (1.28)

where F is the total angular momentum of the atoms. The weight of these diﬀerent contributions are respectively given by 1/∆F F ′ , ∆f s/∆2F F ′ and ∆hf s/∆2F F ′ , where ∆f s is the fine structure splitting of the excited states and ∆hf s the hyperfine structure splitting. If the detuning of the laser to the atomic transition is large compared to hyperfine splitting, the rank-2 contribution of the polarizability becomes negligible. Besides, if the laser light is linearly polarized the vector term vanishes. In these conditions, only the scalar part of the Hamiltonian remains, that acts as a state-independent light shift. The three spin states then see the same trapping potential. Conversely, reducing the detuning or using circularly polarized light allows to manipulate the atoms in a spin-dependent way. In the following we consider the trapping potential is independent of the spin state and note it Vext(r).

#### Two-body scattering of two spin-1 particles

We derive here the Hamiltonian describing the interaction of two atoms of spin 1. For atoms with an internal degree of freedom, collisions connect asymptotic incoming and outgoing states that are a product of an orbital and an internal state. Internal and orbital states are labeled by quantum numbers, in particular by their total angular momentum. As we explained in the previous section, in dilute gases at very low temperature, binary s-wave collisions prevail, and we can make the cold-collisions approximation: the wavefunction of the relative motion of the two atoms has then zero total angular momentum.

We assume the interaction is rotationally invariant in real and in spin space2 . This is exact in the absence of symmetry breaking due in particular to applied magnetic fields. Nonetheless, this approximation remains valid at low magnetic fields. As a consequence the total angular momentum is conserved. We additionally make the approximation that orbital and internal degrees of freedom do not couple during the collision. Then orbital and spin angular momentum are independently conserved.

The wave-function of bosons has to be symmetric under the exchange of any two particles. Because of the s-wave approximation, the orbital part of the wavefunction is symmetric under such exchange. Its internal part then also has to be symmetric. The parity of the wave-function is (−1)F , so that for two colliding spin-1 particles there only exists two scattering channels: F = 0 and F = 2. Because of the rotational symmetry of the interaction, the two atoms stay in the same channel during the collision.

**Table of contents :**

**Introduction **

**1 Spin-1 Bose-Einstein condensates **

1.1 Bose-Einstein condensates with an internal degree of freedom

1.2 Bose-Einstein condensation in scalar gases

1.2.1 Bose-Einstein transition in an ideal gas

1.2.2 Effect of the interactions: ground-state

1.2.3 Effect of the interactions: excited states

1.3 Spin-1 Bose-Einstein condensates: spin Hamiltonian

1.3.1 Single spin-1 particle

1.3.2 Two-body scattering of two spin-1 particles

1.3.3 Many-body Hamiltonian

1.3.4 Effect of applied magnetic fields

1.4 Mean-field theory of spin-1 condensates

1.4.1 Single-mode approximation

1.4.2 Mean-field approximation

1.4.3 Ground-state in the Single-mode approximation

1.4.4 Validity of the single-mode approximation

1.4.5 Excitations in a spinor condensate

1.5 Conclusion

**2 Production, manipulation and detection of a spin-1 Bose-Einstein condensate of Sodium **

2.1 Experimental methods

2.1.1 The experimental chamber and the atomic source

2.1.2 Magneto-Optical Trap

2.1.3 Resonant laser

2.1.4 Loading in a Crossed Dipole Trap and two-step evaporation

2.1.5 Condensation in the dimple trap

2.2 Diagnostic of the spinor gas

2.2.1 Application of magnetic fields

2.2.2 Stern Gerlach separation

2.2.3 Imaging set-up

2.2.4 Calibration of the scattering cross-sections

2.2.5 Imaging noise

2.3 Preparation of a controlled magnetization

2.3.1 Magnetic fields control

2.3.2 Spin-mixing

2.3.3 Spin distillation

2.4 Conclusion

**3 Mean-field study of an antiferromagnetic spinor condensate**

3.1 Nematic order in spinor condensates

3.1.1 Definition of the nematic order parameter

3.1.2 Application to mean-field states

3.1.3 Nematic order of a mean-field ground-state

3.2 Experimental study of the phase diagram

3.2.1 Experimental sequence

3.2.2 Results

3.3 Detection of spin-nematic order

3.3.1 Rotation of the spinor wavefunction

3.3.2 Experimental implementation of three-level Rabi oscillations

3.3.3 Evidence for phase-locking

3.4 Conclusion

**4 Spin fragmentation in a spin-1 Bose gas **

4.1 Fragmentation of a spinor condensate at zero field

4.1.1 Fragmented Bose-Einstein condensates

4.1.2 Spin fragmentation in an antiferromagnetic spinor BEC at T = 0

4.1.3 Spin fragmentation at finite temperatures

4.2 The broken-symmetry picture

4.2.1 Broken-symmetry picture at T = 0

4.2.2 Broken-symmetry approach at finite temperatures

4.2.3 SU(3) coherent states

4.2.4 Broken symmetry description of a spin-1 gas with constrained magnetization

4.3 Connection to spontaneous symmetry breaking

4.3.1 Spontaneous symmetry breaking in the thermodynamic limit

4.4 Conclusion

**5 Observation of spin fragmentation and spin thermometry **

5.1 Observation of spin fluctuations

5.1.1 Experimental sequence

5.1.2 Data acquisition

5.1.3 Measured moments of n0

5.2 Statistical analysis of the distributions of n0 and mz

5.2.1 Model and method

5.3 Spin temperature and condensed fraction during the evaporation

5.3.1 Temperatures at fixed trap depth

5.4 Two spinor fluids isolated from each other

5.4.1 Comparison of spin and kinetic temperatures

5.4.2 Large q: the condensate and the thermal gas are coupled

5.4.3 Low q: condensate at equilibrium but decoupled from the thermal gas

5.5 Conclusion

**Conclusion and perspectives **

A Numerical methods for the spinor Gross-Pitaevskii equations 155

A.1 Gross-Pitaevskii equations in imaginary time

A.1.1 The imaginary time propagation method

A.1.2 Dimensionless coupled Gross-Pitaevskii equations

A.2 Propagation of the finite differences scheme

A.3 Numerical implementation

B Geometrical representation of a spin-1 state

B.1 Bloch-Rabi representation

B.2 Application to the mean-field ground state

C Three-level Rabi oscillation

D Generalized coherent states

D.1 Construction of generalized coherent states

D.2 Spin coherent states

D.3 SU(3) coherent states

D.4 Diagonal representation of few-body operators in the SU(3) coherent states basis

E Spin fragmentation of Bose-Einstein condensates with antiferromagnetic interactions

**Bibliography**