Spinor BoseEinstein gases
Abstract
In a spinor BoseEinstein gas, the nonzero hyperfine spin of the gas becomes an accessible degree of freedom. At low temperature, such a gas shows both magnetic and superfluid order, and undergoes both density and spin dynamics. These lecture notes present a general overview of the properties of spinor BoseEinstein gases. The notes are divided in five sections. In the first, we summarize basic properties of multicomponent quantum fluids, focusing on the specific case of spinor BoseEinstein gases and the role of rotational symmetry in defining their properties. Second, we consider the magnetic state of a spinor BoseEinstein gas, highlighting effects of thermodynamics and BoseEinstein statistics and also of spindependent interactions between atoms. In the third section, we discuss methods for measuring the properties of magnetically ordered quantum gases and present newly developed schemes for spindependent imaging. We then discuss the dynamics of spin mixing in which the spin composition of the gas evolves through the spindependent interactions within the gas. We discuss spin mixing first from a microscopic perspective, and then advance to discussing collective and beyondmeanfield dynamics. The fifth section reviews recent studies of the magnetic excitations of quantumdegenerate spinor Bose gases. We conclude with some perspectives on future directions for research.
Back in 1998, an “Enrico Fermi” summer school on BoseEinstein Condensation in Atomic Gases was organized and held at Varenna. One of the topics touched lightly upon was the novelty of optically trapped BoseEinstein condensates, which allowed one to study ultracold atoms occupying several spin states, including those that were not amenable to magnetic trapping. And one of the applications of that new capability was the creation of spinor BoseEinstein condensates. Results from the first experiment on spinor BoseEinstein gases [1] were presented by Wolfgang Ketterle in one short segment of a several lecture series [2]. In the intervening 16 years, the study of spinor BoseEinstein gases has broadened to the point that the topic of just a few overhead transparencies grew into the focus of a threelecture series given at the 2014 “Enrico Fermi” summer school on Quantum Matter at Ultralow Temperatures by one of the authors of the present manuscript.
The present document and the lectures they summarize are meant to share with newcomers to the coldatom community some of the central concepts that spinor BoseEinstein gases bring to the fore, and some of the experimental techniques by which these concepts have been, and continue to be, studied. A lack of depth in parts of our presentation does not reflect a shortage of interesting ideas and work by the community exploring these topics, but rather reflects our attempt to present material in an accessible, pedagogical style. We hope those interested will turn their attention to several review articles [3, 4, 5] and book chapters [6, 7, 8] that have been written about spinor BoseEinstein gases. We have chosen to make two of the sections in these notes – Sec. LABEL:sec:imaging on the topic of imaging spinor gases, and Sec. LABEL:sec:excitations magnetic excitations – more technical and detailed than the others, for the reason that the material in those sections has not been covered deeply in previous reviews.
Before 1995, the list of quantum fluids that were available to study experimentally was short. Superfluid He is the oldest item on that list, first created when Kamerlingh Onnes liquified helium in 1908. However, at the time Onnes did not recognize that he had produced a new state of matter. It was not until 1937 that it was “discovered” that, at low temperatures, the He liquid was indeed a superfluid [9, 10]. This fluid is an example of a “scalar superfluid” because its order parameter is a single complex number: the ground state is described by a single complex number everywhere, while its dynamics are described by a complex number assigned to each point in space (a scalar field). The simplicity of this order parameter reflects the fact that He is a spinless atom: its electrons are paired into a spin singlet state, and so are the neutrons and protons within its nucleus.
Electronic superconductors are the second oldest on the list. These were discovered by Onnes. The simplest and best understood superconductors are described by the theory of Bardeen, Cooper and Schrieffer as having an swave order parameter. This order parameter can be understood as describing the centerofmass wavefunction of pairs of electrons, the Cooper pairs, that have formed a spinless composite boson. So, again, the quantum fluid is described by a scalar complex field, though one that it is quite different from liquid He owing to the longrange Coulomb interaction among Cooper pairs. There are examples of superconductors in which the crystal structure environment of the electrons favors their adopting an electronpair state of different symmetry. In pwave superconductors, Cooper pairs are formed between electrons of like spin [11]. In this case, the electron pair carries both orbital and spin angular momentum. In dwave superconductors, the Cooper pairs are spinless but have a nonzero orbital angular momentum. However, the possible states of both the pwave and dwave superconductors are highly constrained by the broken rotational symmetry of the crystal lattice. Thus very few – usually just one – of the conceivable angular momentum states of the Cooper pairs are accessible, and the order parameter is greatly simplified.
The last quantum fluid to be discovered before 1995 is superfluid He. The description of this superfluid is far from simple. Like the electrons in a superconductor, He atoms are fermions. At temperatures down to a few mK and pressures below 30 atmospheres, He forms a Fermi liquid. Like electrons in a metal, the distribution of helium atoms is characterized by a Fermi surface. Lowenergy excitations involve the correlated motion of helium atoms near the Fermi surface and their neighbors. Below 3 mK, these quasiparticles undergo Cooper pairing. It is found that the shortrange repulsion among helium atoms causes pwave pairing to be favored, similar to the case of pwave superconductors. However, unlike the electronic superconductor, here the helium fluid is free of the anisotropy of an underlying crystal. The physics that ensues is fascinating [12, 13, 14].
The realization of BoseEinstein condensation in atomic gases has produced a panoply of new quantum fluids. Quantum fluids have already been formed of more than a dozen atomic species, in one, two, or manyelectron atoms, some in their electronic ground state and others in metastable states. Molecular quantum gases have been produced by associating ultracold atoms into high lying molecular states, and efforts are underway to repeat the feat with molecules in their internal ground state as well. Some of these are Bose fluids while other are Cooperpaired Fermi gases; the distinction between these two fluids is blurred by experiments span the crossover from a Cooperpaired Fermi gas to a Bose fluid of bound molecules. The strength of interactions among atoms, and thereby of correlations among them, is broadly tunable [15, 16]. So, rather than studying only specific instances of broad theories of possible quantum fluids, we can now study many of them experimentally. The quantum fluids landscape is newly beckoning, inviting our exploration.
One of the appealing areas for investigation is the behavior that comes about when several quantum fluids are mixed together. Such a system will be represented by a multicomponent order parameter, for example, with one complex field for each component of the quantum fluids. In choosing which mixtures to study, it is clear that some are going to be more interesting than others. For example, if we immerse a superconductor in a cup of superfluid He, we have indeed mixed quantum fluids, but we haven’t produced something that seems very interesting; there is no motion or mixing between the fluids, and the only exchange is the (boring) thermodynamic exchange of energy. The same perhaps can be said of mixtures of He, He, and He that were considered previously [17, 18].
To refine our sense of when something interesting will arise, we might look for three things (these are personal preferences, not strict rules):

Cases where the components of the quantum fluids can be transformed into one another, where we’re not considering apples and oranges (or He and He liquids), but rather situations where the experimentalist can drive transitions that convert one component of the quantum fluids into the other.

Cases where the system is not “trivially” ordered, but rather where order is established from a set of degenerate or at least nearly degenerate possibilities. The degeneracy of the ground state can be brought about either by fine tuning, or else by symmetries, continuous or discrete, possessed by the system Hamiltonian. For a bulk fluid (outside of a crystal lattice), the symmetries most likely at play are spatial and/or spinspace rotations.

Cases where the system can undergo dynamics that explore the different choices of order parameter implied above.
Spinor BoseEinstein gases, which are ones comprised of atoms (and potentially, some day, molecules too) whose internal state is free to lie anywhere within a single manifold of angular momentum states, satisfy all three of these criteria. (1) Converting one component of the gas into another is accomplished, for example, by driving magnetic dipole transitions between Zeeman levels. (2) In the absence of conditions that explicitly break rotational symmetry (such as anisotropic trapping containers or applied fields), we are certain that any manybody state with nonzero angular momentum is perforce degenerate. A ground state with angular momentum zero would not necessarily be degenerate, but we assert that such a state (discussed in Sec. Spinor BoseEinstein gases) would already be interesting enough to satisfy the “nontrivial” requirement. (3) As we will see, interactions quite generically allow for “spinmixing” dynamics where the composition of the gas among accessible Zeeman states can vary.
We would be remiss if we did not mention another class of quantum fluids that is appearing on the landscape, namely an assortment of BoseEinstein condensates of quasiparticle excitations in nonequilibrium settings. This class includes BoseEinstein condensation of excitons, excitonpolaritons, photons, and magnons (probably others as well). These quasiparticle condensates occur in systems that are pumped with a population of excitations that is larger than would exist at thermal equilibrium. Under favorable conditions, this excess number of quasiparticles decays over a time that is sufficiently long that the quasiparticles can undergo enough collisions amongst themselves or with the medium in which they propagate in order to reach a nearequilibrium distribution in position and momentum space. There are fundamental questions about whether these driven and decaying quasiparticle gases can form quantum fluids that are the same as those achieved in thermal equilibrium systems in which the particle number is truly conserved. For lack of space, we do not elaborate on the properties of such quasiparticle quantum gases here and simply refer the interested reader to the literature for further reading [19].
Experiments on spinor Bose gases have taken place using either alkali atoms (notably sodium and rubidium) or highspin atoms (notably the transition metal chromium). The selections are summarized in Table I. In all cases, the spin manifold that is occupied by the atoms of the spinor gas is one of the hyperfine spin manifolds of the electronic ground state.
Stable  Unstable  
conserved  not conserved  
Li, (f)  Cr, (not f)  Li, 
Na, (af)  Dy, (unknown)  Na, 
K, (f)  Er, (unknown)  K 
Rb, (f)  Rb  
Rb, (af or cyc)  Cs  
Rb pseudospin:  Tm, (unknown)  
,  Ho, (unknown)  
, 
To identify atomic properties that come up later in this document, let us remind the reader briefly of the structure of an alkali atom. The alkali atom is a oneelectron atom in the sense that, in the electronic ground state and accessible excited states, all but one of the electrons in the atom remain in closed atomic shells while only the state of the last electron varies. Focusing on the electronic ground state, that one electron is unpaired in an sorbital, so that the atom has no internal orbital angular momentum. The atom does have two sources of spin angular momentum in this electronic ground state: the 1/2 spin of the unpaired electron, and the spin of the nucleus denoted by the angular momentum quantum number . These two spins are energetically coupled by the hyperfine interaction.
At zero magnetic field, the atomic energy eigenstates are also eigenstates of the total angular momentum of the atom, denoted by the vector operator and known as the hyperfine spin^{1}^{1}1for notational convenience, we will make the spin vector and spin quadrupole operators dimensionless, and include factors of explicitly where needed. The angular momentum quantum number for the hyperfine spin, , takes one of two values: . We observe that for bosonic alkali atoms is a halfinteger number^{2}^{2}2We note that the quantum statistics of a neutral atom, i.e. whether it is a boson or fermion, depend on whether the number of neutrons in the nucleus is even (boson) or odd (fermion), because the sum of the number of electrons and protons is always even. The nucleus of a bosonic alkali atom will contain an even number of neutrons and an odd number of protons; hence the nuclear spin will be nonzero and halfinteger. As a counterexample, for alkaliearth atoms, the bosonic isotopes have an even number of nucleons. Nuclear stability most often then implies that the nuclear spin is , so that the bosonic atom will have zero net spin in its ground state and will not be a candidate for spinor physics.. These two hyperfine spin states are separated by an energy on the order of GHz. Each of these energy states is fold degenerate.
Thus, an alkali atom supports, in principle, two different species of spinor BoseEinstein gases: an spinor gas, and an spinor gas. In reality, one finds often that the gas of atoms in the higherenergy hyperfine spin manifold is unstable against collisions that relax the hyperfine energy, limiting the amount of time one can allow such a gas to evolve; Rb is an exception in that both its hyperfine manifolds are long lived. The remaining choices are listed in Table I.
The identification of an atomic state with a hyperfine spin quantum number is strictly valid only at zero magnetic field. A magnetic field breaks rotation symmetry, and introduces coupling and energy repulsion between the two states of the same magnetic quantum number that characterizes the spin projection along the field axis. Strong mixing between states is obtained when the magnetic field is strong enough so that the magnetic energy, roughly , is greater than the zerofield hyperfine energy splitting. With MHz/G being the Bohr magneton ( is the electron charge, is the electron mass) and using a typical hyperfine energy, this strongfield regime applies for G. So, if we restrict the magnetic field to be less than 100 G or so in strength, we can still take the atomic states as comprising spin manifolds of welldefined spin.
More recent additions to the family of atoms that can be lasercooled and brought to quantum degeneracy are elements in the transition metal and lanthanide series. These are manyelectron atoms, with several electrons lying in the valence shell, and for which the optical spectrum is rich with different series of states corresponding to different arrangements of these many electrons. It came as a surprise that laser cooling and trapping techniques work so well for these complex atoms and their complex spectra [20, 21].
The valence shell for these atoms has a high angular momentum. When these shells are only partly filled, the electrons arrange themselves to maximize the total electron spin that can thus become large. The partly filled shells can also have a nonzero orbital angular momentum . Both and contribute substantially to the magnetic moment of the atom. Add in the possible contribution of nuclear spin, and one has the potential for creating spinor BoseEinstein gases of very high spin .
The ground state angular momentum and the magnetic moment of these atoms can be determined by applying Hund’s rules for atomic structure, an exercise that is described in standard atomic and condensedmatter physics textbooks. For bosonic chromium [22], with the configuration , one finds the quantum numbers and magnetic moment . For bosonic dysprosium [23], with the configuration , one finds the quantum numbers and magnetic moment . For bosonic erbium [24], with two extra electrons in the orbital, we find and . The point here is not only that the spin quantum number of gases formed of these atoms is large, which, for example, leads to an interesting variety of magnetically ordered states that might be realized in such gases [25, 26, 27, 28, 29, for example], but also that the effects of dipolar interactions, scaling in strength as , become increasingly dominant.
One consequence of the large magnetic moment is that the highspin gases undergo rapid dipolar relaxation whereas the alkali gases do not. Let us distinguish here between two types of collisions: swave contact collisions and dipolar relaxation collisions (see Ref. [30] for a nice review of quantum scattering in the context of ultracold atomic gases). The swave contact interaction describes lowenergy collisions where atoms interact with one another via a shortrange molecular potential, where “shortrange” means that the range of the potential is small compared with the deBroglie wavelength defined by the small relative velocity of the incident colliding atoms. In an swave contact collision, two atoms enter into and emerge from the collision with no relative, centerofmass orbital angular momentum. As we discuss further below, this situation requires that the total spin angular momentum of the two colliding bodies be conserved in the collision. In contrast, in a dipolar relaxation collision, the colliding atoms emerge with a nonzero value of the centerofmass angular momentum. This change in orbital angular momentum implies a change also in the total hyperfine spin of the colliding pair.
In alkali spinor gases, dipolar relaxation collisions occur (typically) much less frequently than swave contact interactions. Consider the evolution of such a gas in the presence of a small, uniform applied magnetic field. Singleparticle dynamics in the presence of this field conserve the populations in each of the Zeeman sublevels of the spinor gas. The swave contact interaction between two atoms conserves their total hyperfine spin, but not necessarily the spin of each atom. Generically, then, one finds that such collisions redistribute atoms among the Zeeman sublevels of the gas. However, the conservation of the total hyperfine spin implies that the net hyperfine spin along the field axis remains constant.
Remarkably, the conservation of the longitudinal (along the field axis) total hyperfine spin implies that, to first order, the Zeeman energy shifts imposed by applied magnetic fields are irrelevant to the evolution of the spinor gas. If the gas were completely free to change its spin composition, then the linear Zeeman energy from magnetic fields of typical strength – say 10’s of mG – would completely overwhelm all other energy scales of the gas (the temperature, the chemical potential, the spindependent swave interaction strength, the magnetic dipoledipole interaction, etc.). The ground state of such a spinor gas would trivially be one in which the gas is fully magnetized along the magnetic field direction. However, absent dipolar relaxation collisions, this Zeemanenergy equilibration does not occur. Rather, the gas evolves as if these Zeeman energy shifts did not exist at all, i.e. as if it were in a zerofield environment (except for residual effects of the magnetic dipole interaction). The swave contact interactions then provide a means for the gas to evolve to an equilibrium constrained to a constant longitudinal magnetization, the value of which is established by the experimentalist by the application of radio or microwave frequency fields that can drive atoms between Zeeman states.
In contrast, in the highspin spinor gases, dipolar relaxation collisions occur more rapidly, and the constraint of constant longitudinal magnetization is lifted. In an applied magnetic field of moderate strength, the highspin spinor gas will indeed evolve toward the trivial magnetically ordered state where all atomic spins are aligned with the field. The resulting quantum fluid only occupies a single magnetic state (), and so the physics is described by only a single complex number, just like a scalar fluid. If one wants to access nontrivial magnetic order in such gases, one must reduce the applied magnetic field to very small values. For example, one may want to make the applied field smaller than the field generated by the magnetization of the gas in order to realize the chiral spin textures proposed in Refs. [31, 32]. We can estimate this field as where is the atomic density (say ) and is the atomic magnetic moment (say dysprosium’s ). We arrive at a field of . So experiments on these highspin spinor gases require that the magnetic field be controlled with great care.
This distinction between “small” and “large” magnetic moments can be made quantitative by comparing two energies: the van der Waals energy, and the magnetic dipole energy. The van der Waals interaction leads to a potential of the form where is the distance between two atoms and quantifies the dc electric polarizability of the atoms. We can turn this potential into an energy scale by calculating the distance , known as the van der Waals length, where the confinement energy equals the magnitude of the van der Waals potential. We obtain , where is the atomic mass. We can now assess the magnetic dipole energy as the energy of two atomic dipoles separated by a distance ; this energy depends on the orientation of the two dipoles relative to the separation between them, but let’s just define the energy as being where is the vacuum permeability and is the atomic magnetic moment. So now the importance of the dipolar interactions in the collisions between atoms is quantified by the ratio
\hb@xt@.01(1) 
A similar ratio is obtained by taking the ratio of the meanfield contact interaction energy and the meanfield magnetic dipole energy of a spherical ball of atoms with uniform density . The meanfield contact interaction energy is given as , where is the swave scattering length, while the magnetic dipole energy is . So we obtain the ratio
\hb@xt@.01(2) 
Recall that the van der Waals potential plays a dominant role in lowenergy atomic collisions, so that is indeed on the order of except near a collisional resonance [33]. Thus, our two expressions for are the same apart for a numerical factor of order unity.
On the right hand side of the expression above, we quantify the ratio with the magnetic moment being measured in units of the Bohr magneton, being measured in amu, and the being measured in units of the Bohr radius . With this expression we can compare the relevance of dipolar interactions in the collisions of various gases. For example, for Rb, with and , we have , confirming the fact that swave interactions dominate the collisional physics of alkali gases. In contrast, for Dy, with and [34], we have so that dipolar interactions are highly significant.
Spinor BoseEinstein gases are not just multicomponent quantum fluids, but specifically ones in which the components of the fluid are simply different orientations of the same atomic spin. The spin operator is a vector operator, one that transforms as a vector under physical rotations. The implications of this statement for the properties of spinor gases are profound. Just as we find the notion of continuous symmetry groups powerful in explaining the nature of quantum field theories or condensedmatter systems, so too do we find the notion of rotational symmetry a powerful one in simplifying and constraining the physics of spinor gases.
Consider the constraints that rotational symmetry places on the possible interactions among atoms in the spinor gas. In the absence of applied fields, and in a spherically symmetric trapping volume, the interaction between two atoms must conserve their total angular momentum, the quantum number for which we will denote by the symbol . This total angular momentum includes both the internal angular momenta of the two atoms, and also the orbital angular momentum of their relative motion.
Let us assume that the collision between two particles occurs only in the swave channel, where the relative centerofmass orbital angular momentum of the two atoms, with quantum number , is zero. This assumption is justified if we consider lowenergy collisions between two particles with a shortrange twobody potential. As discussed below, this assumption is questionable in the case of dipolar interactions.
Under this assumption, the total angular momentum of the incoming state is just the sum of the two atomic hyperfine spins, with total spin quantum number . As we are considering the collision between two identical bosons, exchange symmetry implies that the total angular momentum quantum number of the colliding pair must be an even integer; spin combinations with an oddinteger value of the total pair angular momentum simply do not collide in the swave regime. So the total atomic spin (in this case also the total atomic angular momentum ) can take different values: . The states of the colliding incident particles can thus be written in the following basis:
\hb@xt@.01(3) 
where is the projection of the total (dimensionless) angular momentum onto a selected axis.
The conservation of angular momentum under collisions now implies that the collisional interaction between two particles can be written in the form
\hb@xt@.01(4) 
where projects the twoatom state onto the subspace with total angular momentum , and is a scalar twobody operator.
The effects of interactions are now fairly well characterized. Consider collisions where the outgoing atom pair also has zero centerofmass orbital angular momentum, and hence can be written in the same basis as above (denoted by primed symbols). Since is a scalar operator, and recalling the WignerEckart theorem, nonzero matrix elements for the twobody interaction are only obtained when the outgoing total spin angular momentum is the same as the incident total spin angular momentum:
\hb@xt@.01(5) 
The simple dependence of the matrix element on and (diagonal, with only one overall constant ) again comes from the fact that is a scalar operator. The constants are determined from different scattering lengths, . To capture this part of the interaction, we can adopt the pseudopotential approach and replace the collisional interaction with the operator
\hb@xt@.01(6) 
where is the interatomic separation. The factor of is inserted to account for doublecounting in a summation over atom pairs.
It remains to consider the effects of dipolar interactions that can couple the incident swave collision channel to an outgoing channel with nonzero centerofmass orbital angular momentum with total angular momentum quantum number (coupling to is forbidden by parity). In addition to the treatment above, we now have three more outgoing basis sets to consider:
\hb@xt@.01(7) 
Matrix elements to each of these final basis states are again independent of . The dipolar interaction does not conserve the hyperfine spin; it leads to a “spinorbit” interaction through which spin angular momentum can be converted into orbital angular momentum, a phenomenon known as the Einsteinde Haas effect [35, 36, 25, 37]. This coupling also leads to dipolar relaxation among Zeeman levels, allowing the Zeeman energy of a spinor gas be exchanged with kinetic energy.
It turns out to be advisable to treat the magnetic dipolar interactions as an explicit longrange interaction, rather than adopting a pseudopotential approach and replacing the dipolar interaction with a zerorange contact interaction [38]. The pseudopotential method is appropriate for describing the nondipolar molecular potential between two atoms, which falls off as at long distance. This rapid falloff ensures that only swave interactions will persist at low energies of the incident particles. In contrast, the slower falloff of the dipoledipole interaction allows many partial waves to remain relevant to lowenergy collisions.
Having established some of the basic properties of spinor BoseEinstein gases, we ask how they behave at low temperature. Our expectations are set by the example of scalar BoseEinstein gases: At low temperature, these gases undergo BoseEinstein condensation, a phase transition driven by BoseEinstein statistics. This phase transition is observable by the growth to a nonzero value of a local order parameter, the complex field which is the superfluid order parameter. This nonzero order parameter represents a spontaneous breaking of a symmetry obeyed by the system’s Hamiltonian. In this particular case, the broken symmetry is the gauge symmetry for which the conserved quantity is the particle number.
Following this example, we expect that order will be established in lowtemperature spinor BoseEinstein gases. We expect that this order may come about through the effects of BoseEinstein statistics, and that it may represent a spontaneous breaking of symmetries of the manybody Hamiltonian, in particular the symmetry under rotation discussed in the previous sections.
We analyze the subject of magnetic ordering in spinor gases in three stages. For simplicity, each will focus on the spin1 spinor BoseEinstein gas. First, we present the simple case of a noninteracting spinor gas that equilibrates in the presence of an applied magnetic field. This case illustrates the statistical origin of magnetic ordering in a degenerate spinor Bose gas, and also exemplifies some basic features we should expect of the thermodynamic properties of a spinor gas. However, we will show that this noninteracting model provides little guidance as to the exact nature of magnetic ordering, particularly for highly degenerate gases at very low temperature. So, second, we discuss the role of those rotationally symmetric interactions outlined in the previous section. We analyze what type of magnetic order is favored under different conditions, often through the spontaneous breaking of rotational symmetry, and summarize experiments that have revealed such order and such symmetry breaking. Finally, we find an exact solution to the spindependent manybody Hamiltonian – exact at least under the assumption of there being no spatial dynamics and no dipolar interactions. We find that the ground state of one type of spin1 spinor BoseEinstein gas may be a highly correlated, noncondensed, rotationally symmetric spin state, providing a theoretical counterexample to our prediction of local meanfield ordering and spontaneous rotational symmetry breaking.
We consider a simple model of a noninteracting spinor BoseEinstein gas in a uniform box of volume , and exposed to a uniform magnetic field [39]. We assume the gas comes to thermal equilibrium, constrained only by the constant particle number . The magnetic field shifts the singleatom energy levels according to the Hamiltonian
\hb@xt@.01(8) 
where is the atomic magnetic moment (assumed positive), and where we include only the linear Zeeman energy. This Zeeman energy causes the chemical potentials μ for atoms in the different magnetic sublevels to differ, i.e.
\hb@xt@.01(9) 
Let us pause and ask whether the assumptions of this model are appropriate. For highspin atoms, rapid dipolar relaxation collisions allow the gas atoms to redistribute themselves among the Zeeman sublevels, and thus the model should apply directly. Experiments on the thermodynamics of condensed chromium atoms in moderatestrength magnetic fields are described well by our model (though with higher spin) [40].
For the lowspin, alkali metal spinor gases, the model we have presented is still valid so long as we regard the applied magnetic field as being an effective, fictitious magnetic field, the value of which is determined by the magnetization of the gas. In these gases, collisions among atoms will redistribute the kinetic (and trap potential) energy among them, but will not change the magnetization. The linear Zeeman energy determined by the applied magnetic field is inaccessible to the gas, and is thus irrelevant to its dynamics. That is, without dipolar interactions, the gas is subject to collisions of the form
\hb@xt@.01(10) 
only so long as . These processes evolve independent of the linear Zeeman shift, and thus such shifts cannot enter into the dynamics of a spinor gas. However, spinmixing collisions do generally occur, allowing the Zeemanstate populations to redistribute and come to chemical equilibrium. For example, a spin mixing collision within a spin1 gas can be viewed as the following reaction:
\hb@xt@.01(11) 
Chemical equilibrium for this reaction (i.e. where the forward and backward reaction rates are equal) implies the chemical potentials of the Zeeman sublevels are related as follows:
\hb@xt@.01(12) 
Rearranging this equation, and considering all other possible spinmixing collisions (in case of a higher spin), we see that the chemicalpotential difference is independent of . In other words, spinmixing collisions in the absence of dipolar relaxation allow an equilibration at constant magnetization that is equivalent to the equilibration in an applied magnetic field, where now this effective magnetic field is not the actual magnetic field present in the laboratory, but rather a Lagrange multiplier that serves to enforce the constraint of constant magnetization.
We continue to explore our simple model. The number of noncondensed atoms in the gas in each of the magnetic sublevels is given by the BoseEinstein distribution,
\hb@xt@.01(13) 
where , are fugacities and is the thermal deBroglie wavelength. The BoseEinstein distribution is defined for fugacities being in the range , i.e. for chemical potentials .
We consider three generic scenarios. First, at high temperature, all atoms in the gas can be accommodated within the thermal BoseEinstein distribution of the Zeeman components, i.e.
\hb@xt@.01(14) 
In this nondegenerate gas, the chemical potentials of all Zeeman components are negative. In an applied magnetic field (assumed positive, without loss of generality), the populations in the three magnetic sublevels of a spin1 gas will differ, with the largest population being in the state (Fig. 1a). The nondegenerate gas thus acquires a nonzero magnetization.
At lower temperature, the gas undergoes BoseEinstein condensation, marked by the fact that the chemical potential of at least one of the magnetic sublevels goes to zero. In an applied, positive magnetic field, the linear Zeeman shift implies that because the chemical potential of the noninteracting Bose gas cannot be positive. Therefore, the remaining chemical potentials, and , are both negative. The number of noncondensed atoms saturates at the value
\hb@xt@.01(15) 
with . The BoseEinstein condensation transition occurs when . The transition temperature, defined implicitly by the above equation, has the following limiting values
\hb@xt@.01(16) 
with being the Riemann zeta function.
BoseEinstein condensation occurs in just one of the magnetic sublevels (Fig. 1b). The BoseEinstein condensate, containing atoms, is completely spin polarized along the direction of the applied field, while the nondegenerate fraction of the gas is only partly polarized.
A third scenario to consider is BoseEinstein condensation at zero magnetic field. In this case, the chemical potentials of all Zeeman sublevels go to zero simultaneously. The condensate number is defined as before. However, the state of the BoseEinstein condensate is not determined within this model, requiring instead that we begin accounting for the effects of spindependent interactions.
In Fig. 2b, we sketch the magnetization of the Bose gas vs. applied field at several representative temperature. At temperatures above , the magnetization varies smoothly, with finite slope at , indicating that the gas is paramagnetic. For temperatures in the range , the onset of BoseEinstein condensation at nonzero magnetic field, occurring when the paramagnetic gas is sufficiently magnetized so that the density of one of the Zeeman sublevels rises above the critical density for condensation, causes a discontinuity only in the second derivative of the magnetization.
At temperatures below , the magnetization is discontinuous at zero magnetic field. The magnetic susceptibility for this gas (slope of magnetization vs. field at zero field) is infinite. The spinor BoseEinstein gas in this regime is thus ferromagnetic, even in the absence of any interaction among the atomic spins. Magnetic ordering in this case can be thought of as a parasitic phenomenon, taking advantage of the fact that BoseEinstein condensation creates a fraction of the gas with no entropy.
Our model also provides an approximation for the thermodynamic properties of the spinor BoseEinstein gas. We can use idealgas formulae to calculate the kinetic energy and entropy within each Zeeman sublevel of the gas:
\hb@xt@.01(17)  
\hb@xt@.01(18) 
Lines of constant energy and entropy determined by these formulae are shown on Fig. 2a. In the quantumdegenerate regime, we see that, at constant temperature, the energy and entropy of the gas decrease with the magnitude of the applied magnetic field. As discussed above, there is one Zeeman sublevel for which the chemical potential is zero, and thus the normal fraction of atoms within this sublevel is saturated at an energy and entropy that is determined solely by the temperature. Therefore, as the magnitude of the magnetic field is increased, the populations of the remaining magnetic sublevels decrease, decreasing the amount of thermal energy and entropy contained in these populations while increasing the number of condensed atoms.
This thermodynamic tendency is at the root of demagnetization cooling in a spinor Bose gas, which has been demonstrated both for alkali [41] and for highspin spinor gases [42, 43]. In the work with alkali gases, a fully longitudinally magnetized spinor gas was BoseEinstein condensed in an optical trap. Within the model discussed here, the full magnetization of the gas is equivalently described by the gas’ coming to free equilibrium (without the magnetization constraint for alkali spinor gases) at an effective magnetic field . The magnetization of the gas was then deliberately lowered, simply by rotating the atomic spin slightly from the longitudinal direction, using rf fields so that the spin rotation was spatially uniform. The gas was then allowed to come to thermal equilibrium under the constraint of constant magnetization. Owing to the rotational symmetry of the swave contact interactions, the spin rotation pulse did not change the internal energy of the gas. Of course, it did change the Zeeman energy of the gas greatly, but, as stated previously, the alkali gas is impervious to such energy changes. Thereafter, the gas was allowed to equilibrate at constant magnetization. The equilibration proceeds roughly at constant energy, assuming that the optically trapped gas is completely isolated from its surroundings. As such, the gas undergoes isoenergetic demagnetization, following the yellow dashed line in Fig. 2a to an equilibrium at a lower value of . As shown in the figure, following this line implies that demagnetization (of the coherent transverse magnetization produced by the spinrotation pulse) leads to cooling. Such cooling was indeed observed and used to drive down both the temperature and, upon ejecting the spinflipped atoms, the entropy per particle of the spinor gas.
In the experiments on highspin gases, demagnetization cooling is amplified by the fact that the gas can access the Zeeman energy. A spin polarized gas was prepared with its magnetic moment oriented along an applied magnetic field. Upon reducing the strength of the magnetic field, atoms were found to populate the magnetic sublevels with higher Zeeman energy, reaching those levels by converting thermal kinetic energy into Zeeman energy in a dipolar relaxation collision. In the recent work of Ref. [43], performed with a BoseEinstein condensed chromium gas, both these demagnetization cooling effects contributed, i.e. both the conversion of kinetic energy to Zeeman energy and also the cooling effect of allowing magnetic excitations into the previously longitudinally polarized gas led to a reduction in temperature and entropy.
The above model emphasized the role of BoseEinstein statistics in causing a spinor BoseEinstein gas to become magnetically ordered. Now we turn to the role of interactions in determining what type of magnetic order will emerge in conditions that are poorly described by our simple noninteractingatom model, namely under the application of very weak magnetic fields, or, equivalently, in gases that are only weakly magnetized in the longitudinal direction.
We consider the effects of spindependent swave interactions, which we characterized in Eq. Spinor BoseEinstein gases in terms of projection operators, , onto twoparticle total spin states. It is helpful to rewrite these projection operators onto more familiar forms of spinspin interactions. These interaction terms should retain the rotational symmetry of the interaction.
For the case of spin1 gases, we find that two rotationally symmetric terms suffice to describe the swave interaction. Considering operators that act on the states of two particles, labelled by indices and , we note the following identities:
\hb@xt@.01(19)  
\hb@xt@.01(20) 
Here, the subscript “sym” reminds us that we should include only states that are symmetric under the exchange of the two particles. We can then replace the spindependent interaction of Eq. Spinor BoseEinstein gases with the following expression:
\hb@xt@.01(21) 
where the coefficients and are defined as
\hb@xt@.01(22) 
The spindependent interactions are thus distilled to a form of isotropic Heisenberg interactions (). The sign of the interaction divides spin1 spinor gases into two types. For , the spindependent interaction is ferromagnetic, favoring a state where atomic spins are all maximally magnetized, i.e. the maximum spinprojection eigenstate along some axis . For , the spindependent interaction is antiferromagnetic. We recall that in solidstate magnets, antiferromagnetic interactions between neighbors in a crystal lead to a variety of interesting magnetic states, such as Néelordered antiferromagnets, valencebond solids, and spin liquids, the selection among which requires careful examination of the specific geometry and connectivity of the lattice. Similarly, in antiferromagnetic spin1 spinor gases, identifying the specific nature of magnetic ordering requires some careful thinking.
It may be useful here to remind the reader that the spin state of a spin1 particle is characterized, at the onebody level, by a 3 3 density matrix. Such a Hermitian matrix is defined by nine real quantities: three real diagonal matrix elements plus six real numbers that define the three independent offdiagonal complex matrix elements. Corresponding to each of these real numbers, there is a Hermitian matrix that defines an observable quantity. Thus, the space of onebody observables for a spin1 particle is spanned by nine basis operators. We may select these operators to be the identity operator, the three vector spin operators (denoted by symbols ), and the five spin quadrupole operators (denoted by symbols ). In the basis of eigenstates, these operators can be written in the following matrix form:
\hb@xt@.01(23)  
\hb@xt@.01(24)  
\hb@xt@.01(25) 
All nine operators (except the identity operator) are traceless. Except for , the traceless operators also have zero determinant and the same eigenvalues: .
With these operators in mind, let us express the operator , written in Eq. Spinor BoseEinstein gases, in terms of Bose field creation and annihilation operators which are written in the basis of the projection of the atomic spin along some axis (owing to the symmetry of the spindependent interaction, it does not matter which axis we choose). We obtain the following,
\hb@xt@.01(26)  
where, noting the deltafunction spatial dependence, we have dropped the explicit dependence on position which is the same for all field operators in the integrand.
One common approach to identifying the state of the degenerate gas is to assume that a BoseEinstein condensate forms in a coherent state, i.e. a manybody state that is a product state of identical singleparticle states. Let us simplify even further by stipulating that the singleparticle state is a product state of a spatial wavefunction (normalized to unity) and a spin wavefunction , and consider only the lowestenergy choice for the spin wavefunction. This is a variational, meanfield, and singlespatialmode approximation.
We are left to minimize the following energy functional:
\hb@xt@.01(27) 
where is our variational wavefunction, includes the kinetic and potential energy terms. Let us assume that is spin independent. Next, we assume that the spatial wavefunction is fixed, and that only the spin wavefunction is to be varied. With these assumptions, the only term remaining to consider is the spindependent energy, which we conveniently normalize by the number of atoms in the condensate:
\hb@xt@.01(28)  
\hb@xt@.01(29) 
Here, is the particle number, is the average density, and .
This meanfield energy functional has two extremal values. One extremum is obtained for the state that maximizes the length of the average spin vector, i.e. the state with the maximum projection of its spin along some direction , giving . This state minimizes the energy for gases with ferromagnetic interactions, with . The state is maximally magnetized, with defining the direction of the magnetization (or the opposite of that direction, depending on the sign of the gyromagnetic ratio). In our singlemode approximation, the spinor part of the wavefunction is constant in space. So a gas in the state is “ferromagnetic” in the sense that, like in a solidstate ferromagnet, it has a macroscopic magnetization that has a constant direction across the entire volume of the material.
A second extremum in the meanfield energy function occurs for the state that minimizes the length of the average spin vector. This is the state , which is the eigenstate for the spin projection along the axis with eigenvalue zero. The average spin in directions orthogonal to is also zero, so one obtains . This state then minimizes the energy for gases with antiferromagnetic interactions, with . We can regard this state as being nematic, in that it is aligned with an axis (which we may call the director), but that it is oriented neither in the or directions. A condensate with a spatially uniform nematic spin wavefunction is “nematically ordered” in that the director is constant in space. The is also termed polar because of the fact that rotating the state by radians about an axis transverse to produces a final state that still is a zerovalued eigenstate of the operator ; however, the rotation results in the final state differing from the initial state by a minus sign. This property is exhibited also by the p electronic orbital of atomic and chemical physics, in which the electron density is symmetric under a rotation about, say, the axis, but where the phase of the electronic wavefunction is antisymmetric under such rotation.
In Sec. Spinor BoseEinstein gases, we discussed the thermodynamics of a noninteracting spinor gas in an applied magnetic field, or, equivalently, under the constraint of constant longitudinal magnetization. Let us extend this consideration also to the interacting spinor gas in order to understand how the application of symmetry breaking fields tears at the rotationally symmetric interaction Hamiltonian and its solutions.
We consider two symmetrybreaking external influences. The first is a linear Zeeman shift, or equivalently, as discussed before, a constraint of constant longitudinal magnetization. Keeping with nomenclature in early papers on spinor BoseEinstein condensates, we account for this energy through the term . In early experiments on spinor BoseEinstein condensates, a magnetic field gradient was applied deliberately to the gas to provide clear experimental signatures of the spindependent interactions [1]. Such a gradient can be accommodated by making spatially varying; we revisit this point in Sec. Spinor BoseEinstein gases.
Secondly, we include also a quadratic Zeeman energy. As mentioned earlier, the spinor gas can evolve by the spinmixing collision. This process is insensitive to the linear Zeeman energy, but depends on the quadratic Zeeman shift, which shifts the average energy of the states (Zeeman energy of the right side of Eq. Spinor BoseEinstein gases) by an energy with respect to that of the state (Zeeman energy of the left side of Eq. Spinor BoseEinstein gases). The quadratic Zeeman shift comes about from the fact than an applied magnetic field mixes hyperfine levels, causing energy repulsion between states in, say, the upper and lower hyperfine states with the same value of in the electronic ground state of alkali gases. A quadratic Zeeman energy can also be imposed by driving the atoms with microwavefrequency magnetic fields that are near resonant with hyperfine transitions [44]. The latter approach allows one to reverse the sign of the quadratic Zeeman energy (microwavedressed hyperfine levels are made to move closer, rather than farther, in energy). For simplicity, and to match with most experiments, we consider quadratic Zeeman energy shifts in the same basis as that determined by the linear Zeeman energy, i.e. through the term . As such, the Zeeman energies still preserve the symmetry of rotations about the longitudinal direction (the axis). More generally, the quadratic and linear Zeeman energies could, in principle, be applied along different axes, breaking rotational symmetry altogether.
We now obtain a spindependent energy functional of the form
\hb@xt@.01(30) 
Expressing the spin wavefunction in the eigenbasis, the energy functional has the form
\hb@xt@.01(31) 
This expression, while correct, hides from us the fact that some of the information in the spin wavefunction is irrelevant for calculating the energy. For one, the spinor is normalized to unity, so we need not keep track of all the populations . We’ll keep track just of two, choosing the quantities
\hb@xt@.01(32)  
\hb@xt@.01(33) 
Here, is the longitudinal magnetization of the gas, normalized to lie in the range , and is the fractional population in the state. Under a constraint of constant longitudinal magnetization, is constrained to vary in the range .
In addition, the wavefunction contains three complex phases, for example, the complex arguments of the three probability amplitudes . However, two linear combinations of these phases do not affect the energy. Clearly, the energy is unaffected by multiplying the wavefunction by a global phase. Thus, the combination is unimportant to us. Also, the meanfield energy functional is invariant under rotations of the spin about the axis. Such a rotation would change the value of the phase difference , but would not change the energy. The only linear combination of phases that is still potentially important is the combination
\hb@xt@.01(34) 
In our later discussion of spinmixing dynamics, we will see that it is this phase combination that controls the direction in which populations will flow in a spinmixing collision (Eq. Spinor BoseEinstein gases).
We now rewrite the spindependent energy as
\hb@xt@.01(35) 
The task of minimizing this energy is now much clearer. The phase controls the degree of magnetization present when the spin wavefunction contains a superposition of states with . We choose for ferromagnetic interactions () to maximize that magnetization, and for antiferromagnetic interactions () in order to minimize that magnetization. All that is left is to minimize the energy with respect to the populations in the spin wavefunction, expressed through and .
Such minimization was performed initially in Ref. [1]. The results are presented in Fig. 3. Focusing first on the noninteracting case, the phase diagram is simple: For , the lowest energy state is either of the longitudinally magnetized states, , depending on the sign of . The situation here matches that of the noninteracting gas considered in Sec. Spinor BoseEinstein gases. For , the quadratic Zeeman shift favors the state where .
The spindependent swave interaction modifies this phase diagram. We observe that, for example, the region in the  plane occupied by the ground state (this is the longitudinal polar state, with the nematic vector aligned with the longitudinal axis) is increased in the case of antiferromagnetic interactions, and decreased in the case of ferromagnetic interactions.
The diagram also shows evidence of the miscibility and immiscibility of different magnetic sublevels. The miscibility of two BoseEinstein condensed components and of a gas (with equal masses, as here) is governed by the relation between the swave scattering lengths, , and where the indices indicate the two components colliding. If , then the components are miscible in that a superposition of the two components has less interaction energy than two phaseseparated domains of the two components. The swave scattering lengths for different collisions among atoms are all determined by the values of and , as shown in Table II.
We see that for antiferromagnetic interactions, the and components are immiscible. This fact is reflected in the groundstate phase diagram (Fig. 3a) by the sharp boundary between the regions in which either of these components is the ground state. In contrast, in the case of ferromagnetic interactions, the transition between these two regions is gradual. The gray region between them in Fig. 3c indicates that the meanfield ground state is a superposition of states including the and state; such a mixture allows the gas to be partly magnetized and thus take advantage of the spindependent interaction. One also sees that, for antiferromagnetic interactions, the and states are miscible. This miscibility is indicated in Fig. 3a by the gray region that lies between the and ground state regions at , in which a superposition of these two states reduces the meanfield energy. Indeed, for and , the minimum energy state for antiferromagnetic interactions is the transversely aligned polar state, i.e. the state for which the nematic director lies in the plane transverse to (in the  plane). Such a state is a superposition of the states with equal population in the two states.
Compelling experimental evidence for the realization of such meanfield ground states for spinor gases has been obtained. The ferromagnetic case is exemplified by the spinor gas of Rb. The first experiments on this system were performed by the groups of Chapman [45] and Sengstock [46]. They examined optically trapped, BoseEinstein condensed gases that were prepared with zero longitudinal magnetization (i.e. with ), by preparing all atoms in the state. Then, the gases were allowed to evolve under an applied, uniform magnetic field, which produced an adjustable, positive value of .
At high , the state being lower than energy than the average energy of the states that would be produced by spin mixing, we expect the state to be the preferred ground state of the atoms. At low , one expects the gas to acquire magnetization. Consistent with the initial magnetization of the gas, one expects the gas to become transversely magnetized, described by the singleatom spin wavefunction
\hb@xt@.01(36) 
over the range . Here, the transverse magnetization has the amplitude , i.e. ranging from zero magnetization at the transition from the paramagnetic to the ferromagnetic phase () to full magnetization at where the spindependent interactions are the only spindependent energy term. In the experiment, the Zeemanstate populations of the gas were measured, and indeed one found the expected ratio of populations in the states.
The Zeeman state populations on their own do not confirm that the gas has adopted a state with nonzero magnetization; for this, one needs to establish that the Zeeman populations are coherent with one another, and that the phase relations among them lead to a nonzero magnetic moment. For example, compare the two spin wavefunctions
\hb@xt@.01(37) 
The first of these, , is the state magnetized along the axis. The second of these, , differing from the first only in a phase factor on one of the probability amplitudes, is a polar state with zero magnetization. Without measurements of the relative phase between Zeemanstate probability amplitudes, one cannot discern these states.
Two separate experiments confirmed that these relative phases are indeed those that maximize the magnetization in the case of ferromagnetic interactions. The Chapman group utilized spinmixing dynamics, which we discuss in Sec. Spinor BoseEinstein gases, to cause the magnetic state of the gas to reveal itself [47]. After the gas had come presumably to equilibrium, the experimenters applied a briefly pulsed quadratic Zeeman energy shift. This pulse knocked the state away from its equilibrium state and initiated a telltale oscillation of the Zeemanstate populations, the dynamics of which were in excellent agreement with numerical calculations and indicative of the initial state being ferromagnetic as expected.
More direct evidence of the groundstate structure of the Rb gas comes from direct measurements of the gas magnetization. The imaging methods used to measure such magnetization are discussed in Sec. LABEL:sec:imaging. In the work of Ref. [48], such measurements confirmed that a ferromagnetic spinor gas prepared with no initial spin coherence whatsoever will, upon being cooled to quantum degeneracy and allowed to evolve for long times, become spontaneously magnetized. For a positive quadratic Zeeman energy (), the magnetization was found to lie in the transverse plane, while for (realized by the application of microwave fields) the magnetization tended to orient along the magnetic field axis (Fig. 4).
The case of antiferromagnetic interactions is exemplified by the spinor condensate of sodium. The first experiments on spinor condensates, performed with such a gas, confirmed the increased energetic stability of the polar state by placing the trapped gas in a magnetic field gradient [1]. Under such a gradient, a gas within a net zero longitudinal magnetization would tend to become polarized, lowering its energy by adopting the strongfield seeking state ( in this case) at the highfield end of the trap, and the weakfield seeking state () at the lowfield end. However, such an arrangement maximizes the spindependent interaction of the antiferromagnetic spinor gas. A lower energy configuration is achieved by reducing the magnetization of the gas at the cloud center. Depending on the sign of the quadratic Zeeman shift, the favored nonmagnetic state is a polar state () with the director pointing either transversely (for ) or axially (). Measurements of the onedimensional spinstate distribution of the gas (along the direction of the field gradient) showed the expected magnetization pattern, in quantitative agreement with expectations.
It is clear, however, that these meanfield product states cannot be the true ground state of the interacting spinor gas. Consider specifically the case of antiferromagnetic interactions, the ones we suspect, based on solidstate magnetism, are going to be the most interesting. Meanfield theory predicts the state with all atoms in the Zeeman state to be a ground state; for a gas of atoms, we may write this state in the Fockstate basis as where we enumerate the atoms in the states sequentially. However, the manybody Hamiltonian contains a term describing spinmixing collisions, meaning that this meanfield state is coupled, for example, to the orthogonal state . The true groundstate of the system must contain a superposition of (at least) these two manybody states.
A similar scenario applies in the interacting scalar Bose gas. The meanfield ground state for a condensate in a uniform container is one with all atoms in the zeromomentum state. However, the Hamiltonian includes terms describing elastic collisions that conserve the total momentum of the colliding atom pair. One such term describes a collision in which two atoms are excited out of the zeromomentum state and promoted to states of nonzero counterpropagating momenta . Thus, the true manybody ground state must include also a population of atoms in nonzero momentum states. Yet, the number of such momentumexcited atom pairs remains small owing to the energy penalty for creating such excitations. The total number of excited pairs comprises the quantum depletion, which remains small in the case of weak interactions [49].
Is there a similar energy penalty for spinmixing collisions, which might suppress the population of atoms outside the spin state and constrain the quantum depletion atop the meanfield state? Consider the state generated from our initial meanfield ansatz by a slight geometric rotation, i.e. all atoms in the polar state that is aligned with an axis just slightly tilted from the axis. Such a state also contains a superposition of the and states, and yet the energy of this rotated state is no higher than the initial state. Thus we suspect, without rigorous proof yet, that the spinmixing interaction will mix new spin configurations into the meanfield initial state without an energy penalty, different from the situation with a weakly interacting scalar Bose gas. If this suspicion is correct, then the quantum depletion of the meanfield state will be massive, and meanfield state would no longer be a good approximation to the manybody ground state.
This vague argument was cast more rigorously and elegantly by Law, Pu and Bigelow [50]. One realizes that the exact manybody Hamiltonian, containing a sum of spindependent interactions between all pairs of atoms in the gas, treats all atoms symmetrically. We might expect, therefore, that the Hamiltonian can be written not just in the basis of spin operators