Realizing analogues of color superconductivity with ultracold alkali atoms

A degenerate three-component Fermi gas of atoms with identical attractive interactions is expected to exhibit superfluidity and magnetic order at low temperature and, for sufficiently strong pairwise interactions, become a Fermi liquid of weakly interacting trimers. The phase diagram of this system is analogous to that of quark matter at low temperature, motivating strong interest in its investigation. We describe how a three-component gas below the superfluid critical temperature can be prepared in an optical lattice. To realize an SU(3)-symmetric system, we show how pairwise interactions in the three-component atomic system can be made equal by applying radiofrequency and microwave radiation. Finally, motivated by the aim to make more accurate models of quark matter, which have color, flavor and spin degrees of freedom, we discuss how an atomic system with SU(2)⊗SU(3) symmetry can be achieved by confining a three-component Fermi gas in the p-orbital band of an optical lattice potential.


Introduction
At densities present inside neutron stars, matter may be compressed to the point where quarks are no longer confined in hadrons but instead form a degenerate Fermi liquid. Due to attractive interactions between quarks of unlike color, Cooper pairing occurs near the Fermi surface. In forming Cooper pairs, the color gauge symmetry SU(3) is spontaneously broken. While this color superconducting phase is relatively well understood at extremely high quark density (where interactions are weak due to asymptotic freedom), the nature of the color superconducting phase at intermediate densities relevant to neutron stars is unknown [1]. Furthermore, the crossover between quark matter and hadronic matter that occurs at still lower densities is poorly understood [1]. This low-temperature, high-density region of the phase diagram of quantum chromodynamics (QCD) is not accessible to terrestrial experiments and cannot be calculated with lattice QCD due to the fermion sign problem [1].
A degenerate three-component Fermi gas of atoms provides an experimentally accessible system in which to explore similar phenomena. A three-component Fermi gas with identical attractive pairwise interactions, for example, becomes superfluid at low temperature, breaking the global SU(3) symmetry of the original Hamiltonian [2][3][4][5][6][7][8][9][10][11]. The symmetry breaking pattern is SU(3) → SU (2) [4,5]. This parallels the symmetry breaking pattern [SU (3)] → [SU (2)] in two-flavor color superconductivity-a competitive candidate model of color superconductivity at intermediate densities. Here, the brackets denote gauge (rather than global) symmetries. In addition to superfluidity, a three-component Fermi gas also exhibits spontaneous magnetization driven by pairing [5,12,13]. The Ginzburg-Landau (GL) free energy for a three-component Fermi gas therefore contains two coupled order parameters for superfluidity and magnetization, respectively [11,12]. The structure of the GL free energy is identical to that of dense quark matter where the two coupled order parameters in this case describe the quark-quark condensate and chiral condensate [11]. The phase diagram of a three-component Fermi gas thus shares many similarities with the phase diagram of QCD. Indeed, a three-component Fermi gas confined in a lattice also has a transition that parallels the deconfinement-hadronization transition in 3 QCD [13,14]. In the limit of strong coupling, the three-component Fermi gas becomes a Fermi liquid of weakly interacting three-body cluster states. Since many parameters, including density, interaction strength and population imbalance, can be independently controlled in ultracold atomic gases, the entire phase diagram can be explored systematically.
In this paper, we review the properties of three-component Fermi gases. We describe how, in spite of significant loss rates due to three-body recombination, superfluidity can be achieved by confining 6 Li atoms at low density in a large-period optical lattice. We then discuss how scattering properties can be tuned, by applying radiofrequency (RF) and microwave (MW) fields, to achieve nearly perfect SU(3) symmetry in the atomic system. Finally, motivated by the aim to make more accurate models of quark matter, which have color, flavor and spin degrees of freedom, we discuss how an atomic system with SU(2) ⊗ SU(3) symmetry can be realized by confining a three-component Fermi gas in the p-orbital band of an optical lattice.

Multi-component alkali gases
Atomic alkali gases containing more than two components can be attained since the total angular momentum F (which is the sum of the nuclear I and electron S spin) allows for 2F + 1 degenerate internal states in zero magnetic field. For atoms with I > 0, F can be greater than 1/2. Furthermore, in the presence of a large magnetic field (i.e. the Paschen-Back limit) 1 , the good quantum numbers are m S and m I and there are two multiplets of (2I + 1) closely spaced levels with a given multiplet having the same electron spin projection, either m S = ±1/2 (see, for example, figure 1(a)). The pairwise interactions between atoms in a given multiplet are nearly identical since the molecular interaction potential is independent of the nuclear spin. At ultralow temperatures, the interactions are purely s-wave and parameterized by an s-wave scattering length a. For pairwise interactions among states within a multiplet, the atoms interact predominantly via the triplet molecular potential and accordingly the pairwise scattering lengths are all given approximately by the triplet scattering length a t .
In this paper, we will consider a gas of fermionic 6 Li atoms, which has a nuclear spin I = 1. The energies of the Zeeman states in 6 Li are shown in figure 1(a) as a function of magnetic field B. For B 100 G, the states separate into multiplets with |m s , m i = |− 1 2 , {1, 0, −1} in the lower-energy multiplet and |m s , m i = | 1 2 , {−1, 0, 1} in the higher-energy multiplet. We label these states |1 -|6 in order of increasing energy. We will consider a three-color Fermi gas comprised of atoms in states |1 -|3 . The scattering lengths for the three possible pairwise interactions between these states (a 12 , a 23 and a 13 ) are shown in figure 1(b). The scattering lengths a 12 , a 23 and a 13 each exhibit a Feshbach resonance where the scattering length diverges at respective fields of 834, 811 and 690 G. Of central relevance to this paper, however, is the fact that at high field, the scattering lengths all asymptote to the triplet scattering length for 6 Li, a t = −2140 a 0 . 2 Thus, in the limit of large fields, the pairwise interactions in the gas are attractive and of approximately equal magnitude. At a field of 2000 G, for example, the three scattering lengths are all within 2% of their average [15]. In section 5, we will show how the difference between the scattering lengths can be reduced even further if RF radiation is applied to dress the internal states of the atoms.  6 Li atoms as a function of magnetic field B. The three lowest-energy states (labeled |1 , |2 and |3 ) are the three internal states of 6 Li that will represent the color degree of freedom. For B 200 G, the z-projections of the electron and nuclear spin (m s and m i , respectively) are good quantum numbers and |1 ≈ | − 1 2 , 1 , |2 ≈ | − 1 2 , 0 and |3 ≈ | − 1 2 , −1 in the |m s , m i -basis. (b) The s-wave scattering length for each of the three possible pairwise interactions between states |1 -|3 as a function of B [15]. Magnetically tunable Feshbach resonances occur at B = 690, 811 and 834 G. For B > 1000 G, all three scattering lengths asymptote to the triplet scattering length a t = −2140a 0 .

Superfluidity, magnetism and trimer formation in a three-component Fermi gas
In this section, we will review the recent theoretical predictions regarding the phase diagram of a three-component Fermi gas with attractive interactions. We will then highlight several striking analogies that can be drawn between superfluidity in the atomic gas and color superconductivity in quark matter and also between the phase diagram of QCD and that of the atomic system.
The Hamiltonian for a homogeneous three-component Fermi gas in a volume V with identical scattering lengths a t is given by where ψ † α,k is the creation operator of a particle in state α = 1, 2, 3 with momentumhk, µ α is the chemical potential for each color, and g = 4πh 2 a t 5 is invariant under global SU(3) transformations of the three component field operator ψ (i.e. ψ σ → U σ ,σ ψ σ where U ∈ SU(3)) as well as a global U(1) transformation of the overall phase.

Superfluidity in a three-component Fermi gas
For attractive contact interactions, fermionic atoms in different internal states form s-wave Cooper pairs at low temperature, giving rise to a pairing mean field αβ ∝ k ψ kα ψ −kβ with nonzero components. The symmetry breaking pattern for the superfluid transition can be determined from group theoretical considerations, as was first done in [4]. Since pairing is in the s-wave channel, the 3 × 3 matrix αβ is antisymmetric and therefore has three independent complex components (e.g. 12 , 23 and 13 ). Thus, αβ transforms according to a three-dimensional (3D) representation. Using the decomposition 3 ⊗ 3 =3 ⊕ 6 for the SU(3) group (where 3 denotes the irreducible representation under which ψ α is transformed and3 is the complex conjugate representation of 3), we see that αβ transforms under the complex conjugate representation3. This indicates that αβ transforms nontrivially under SU(3) and takes on different values depending on the global gauge of the fermions [14]. Thus, Cooper pairing spontaneously breaks the SU(3) symmetry of the model Hamiltonian equation (1). This is in stark contrast to the situation realized in a two-component Fermi gas symmetric under global U(1) ⊗ SU(2) transformations. Here, the antisymmetric 2 × 2 matrix has a single independent complex component and therefore transforms as a scalar under SU(2) (consistent with the decomposition 2 ⊗ 2 = 1 ⊕ 3 for the SU(2) group). Thus, in the two-component case, Cooper pairing does not break SU(2) symmetry. Rather, the symmetry breaking pattern is SU(2) ⊗ U(1) → SU (2). We refer to superfluidity in a three-component Fermi gas where the SU(3) symmetry is spontaneously broken as 'color' superfluidity to distinguish it from the superfluid phase in a two-component Fermi gas (or a three-component Fermi gas without SU(3) symmetry), where only the U(1) symmetry is spontaneously broken.
To determine the symmetry breaking pattern in an SU(3) ⊗ U(1) symmetric threecomponent Fermi gas, we follow [11,14] and consider the fermionic pairing order parameter, where αβγ is the total antisymmetric tensor. It is straightforward to show that this pairing order parameter transforms under the complex conjugate representation3 such that α → U * αβ β when ψ α → U αβ ψ β for U ∈ SU(3) [11]. Since U * is also a unitary matrix, it is always possible to preserve the inner product α | α | 2 = 0 but transform to a gauge where = (0, 0, 0 ). In this gauge, only two components are paired with one component left unpaired and the unbroken symmetries are made explicit. The system is invariant under SU(2) transformations in the subspace of the two paired components and under U(1) transformations of the global phase of the unpaired component. Thus, the symmetry breaking pattern is SU(3) ⊗ U(1) → SU(2) ⊗ U(1). Spontaneous symmetry breaking in this system yields five broken generators and one expects five corresponding Nambu-Goldstone modes. Again, this can be contrasted with the situation in a superfluid two-component Fermi gas where a single generator is broken, giving rise to a single Nambu-Goldstone mode-the Anderson-Bogoliubov mode. 6 In the gauge where only two-components of the Fermi gas are paired, it is straightforward to apply a mean-field BCS theory to find the superfluid critical temperature T c . This was first done by Modawi and Leggett [2], who found that where T F is the Fermi temperature, k F is the Fermi wavenumber and γ is Euler's constant.
Assuming a t = −2140a 0 and the density of each spin state is 3 × 10 12 atoms cm −3 , the critical temperature T c = 0.05 T F . At higher densities, the gas enters the strong coupling regime for which k F a t 1 and BCS theory is invalid. In the unitarity limit (k F a t 1), one expects T c to be a universal fraction of T F . For a two-component Fermi gas, T c 0.15 T F in the unitarity limit and one might expect a comparable T c for a three-component Fermi gas in the strong coupling limit. While temperatures T < 0.05 T F at densities 3 × 10 12 atoms cm −3 have been experimentally achieved in a two-component Fermi gas, large loss rates in three-component Fermi gases due to three-body recombination have precluded the simultaneous achievement of low temperatures and high densities. In section 4, we describe how superfluidity in a three-component Fermi gas can be achieved by confining a low-density gas in an optical lattice potential.

Magnetism in a three-component Fermi gas
In addition to superfluidity, a three-component gas also exhibits spontaneous magnetic ordering (i.e. population imbalance) [5,12] characterized by a secondary order parameter φ ≡ N 1 V − N 3 V [11]. Here, N 1 is the number of particles in state |1 and N = 2N 1 + N 3 is the total number of particles in the gauge where states |1 and |2 pair and state |3 is unpaired. Physically, population imbalance occurs since the energy of the system can be minimized for a fixed total number of atoms by transferring population from the unpaired state |3 into the pairing states |1 and |2 where their energy is reduced by the condensation energy. Magnetization is observed in mean-field Bardeen, Cooper and Schrieffer (BCS) theory by fixing the total number of atoms but allowing the number of atoms in each state to vary. Since superfluidity drives magnetization in the three-component system, φ > 0 only when > 0 [5,11,12]. Meanfield theory indicates that the fractional population of one of the paired states grows from 33% in the limit of weak interactions to 41% at unitarity (where a → ∞) to 50% in the BEC limit where dimer molecules form [5,11]. In an experimental system where the population in each state is individually conserved, population imbalance manifests itself as the formation of domains where each domain contains an equal number of paired species but a smaller number of unpaired species. The species that pair in different domains are not the same and therefore the three-component order parameter points in different directions for different domains.

Trimer formation of strongly interacting fermions in an optical lattice
The phenomena of color superfluidity and magnetic ordering also occur if the three-component Fermi gas is confined in the lowest band of a three-dimensional (3D) cubic lattice potential. However, in this case, Rapp et al [13] showed that, in the limit of strong interactions, the ground state becomes a Fermi gas of repulsively interacting three-body clusters (i.e. trimers).

7
The Hamiltonian for an SU(3) symmetric three-component Fermi gas confined in the lowest Bloch band of a 3D cubic lattice is given by where c † i,α is a fermionic creation operator for a particle on site i in state α, t is the tunneling matrix element for nearest neighbor hopping, i j indicates that the sum is over nearest neighbors, U is the strength of the on-site interaction, and n i,α = c † i,α c i,α . Note that only fermions in different internal states interact and U < 0 for attractive interactions. For weak interactions, |U |/t 1, the ground state is a color superfluid that spontaneously breaks SU(3) symmetry [3,4]. However, in the limit of strong interactions, |U |/t 1, the tunneling term in equation (4) can be neglected and the on-site interaction term is minimized when three atoms in different internal states occupy the same lattice site [13]. Thus, the ground state in this limit consists of three-atom trimers localized on individual sites of the lattice. If the hopping term is introduced as a perturbation, it allows, in second order, the trimers to reduce their kinetic energy by an amount ∼ t 2 |U | by the virtual tunneling of individual atoms to adjacent unoccupied lattice sites [13]. However, virtual tunneling to an adjacent site is forbidden by the exclusion principle if it is occupied by another trimer. Thus, only isolated trimers can fully reduce their kinetic energy by virtual fluctuations, giving rise to an effective repulsion between the trimers [13]. To third order in t, an entire trimer can tunnel to an adjacent site with an effective tunneling matrix element t tri = t 3 |U | 2 [13]. Thus, the ground state in this limit is a gas of trimers with large effective mass and weak repulsive interactions. Note that this phase is invariant under global SU(3) transformations since the three-body color wavefunction of a trimer must be an (antisymmetric) color-singlet state. Therefore, a quantum phase transition must separate the color superfluid phase and the trimer phase since they have different symmetries [13].

Analogies to color superconductivity and the chromodynamics (QCD) phase diagram
There are several striking analogies between the three-component Fermi gas described above and matter at densities found inside neutron stars. In neutron stars, nuclei may be compressed to the point that quarks become deconfined from hadrons and form a degenerate Fermi liquid. At sufficiently high density, the Fermi liquid becomes weakly interacting as a consequence of asymptotic freedom. In this weak coupling limit, the quark-quark interaction is attractive for the color anti-triplet configuration [16]. Thus, in the low-temperature high-density limit, one expects quark matter to be a degenerate liquid of quarks with Cooper pairing between quarks of unlike color. Since pairs of quarks cannot be color singlets, the quark-quark condensate by necessity breaks the SU(3) color gauge symmetry. By an essentially identical argument, we showed in section 3.1 that the SU(3) symmetry in a three-component Fermi gas is broken by Cooper pairing. In fact, the symmetry breaking pattern in two-flavor color superconductivity is SU(3) → SU(2). A notable difference between the atomic gas and quark matter is that a global SU(3) symmetry is broken in the former whereas a gauge SU(3) symmetry is broken in the latter.
The difference between global and gauge SU(3) symmetries notwithstanding, a threecomponent Fermi gas can provide an analogue of dense quark matter that is experimentally accessible. For example, Ozawa and Baym [11] recently showed that the GL free energy for a three-component Fermi gas is identical in structure to that for quark matter. In the GL free energy for dense quark matter, two coupled order parameters appear: one for quark-quark condensation and a second for chiral symmetry breaking [17,18]. In the GL free energy for a three-component Fermi gas, there is a one-to-one correspondence between these two order parameters and the order parameters for pairing and population imbalance, respectively. While the structure of the GL free energy is identical to fourth order in the order parameters, the coefficients of two terms in the order parameter expansion can differ in sign [11]. Physically, the difference between the two systems arises from the fact that population imbalance in a three-component Fermi gas is only driven by pairing, whereas chiral symmetry breaking in dense quark matter can occur even in the absence of quark-quark pairing [11].
Three-component Fermi gases also offer the possibility of studying an analogue of the QCD deconfinement-hadronization transition. As discussed in section 3.3, Rapp et al [13] have shown that Cooper pairing in a three-component Fermi gas should eventually give way to the formation of a weakly interacting gas of bound three-body cluster states. This is similar to the formation of baryons from quark matter as the density of quarks is reduced. Thus, the phase diagram for a three-component Fermi gas with attractive identical interactions closely mimics the conjectured phase diagram of QCD. A color superconducting ground state exists in the limit of weak interactions (high quark chemical potential) and gives way to the formation of color singlet trimers (baryons) in the limit of strong interactions (low quark chemical potential).

Achieving color superfluidity in an optical lattice
In section 3.1, we noted that the critical temperature T c in a homogeneous gas can be made as large as 0.05T F -0.15T F for a t = −2140 a 0 and a density in each spin state n α 3 × 10 12 cm −3 . However, degenerate three-component Fermi gases at densities this large are difficult to achieve due to a large rate of inelastic three-body recombination events [19]. In a three-body recombination event, the collision of three atoms results in the formation of a tightly bound molecular dimer and a free atom. The binding energy released in forming the dimer is far greater than the depth of the confining potential and both the atom and the molecule are lost. Threebody recombination is not a significant problem in two-component Fermi gases as the event rate is suppressed by the exclusion principle since two of the three fermions involved in the s-wave collision must be identical [20]. However, this suppression mechanism does not exist in a three-component Fermi gas and the recombination event rate can be quite high [19]. The density of atoms in a homogeneous system decreases according toṅ α = −K 3 n 1 n 2 n 3 , where K 3 is the recombination rate constant.
For three fermions interacting via short-range potentials, all three-body observables, including the recombination rate constant K 3 , can be expressed in terms of universal functions of the scattering length a and a complex valued three-body parameterκ = κ * + i η * (as long as the scattering length is large compared with the characteristic range of the interaction potential) [21]. The recombination rate is given by a) is a universal function that is log-periodic in a [21]. The three-body parameterκ has been determined experimentally for the 6 Li system in the magnetic field regime of interest for color superfluidity [19]. At a field of 2000 G where a t −2222a 0 , the recombination rate constant K 3 = 5.8 +2.1 −3.6 × 10 −22 cm 6 s −1 . Thus, for a three-component gas with equal densities n = n 1,2,3 = 3 × 10 12 cm −3 , the fractional atom loss rateṅ/n = −K 3 n 2 −5220 s −1 , suggesting that the density is reduced by 1/e in only 190 µs. Since the fractional atom loss rate is proportional to the square of the density, the loss rate can be dramatically reduced by significantly lowering 9 the density. However, the BCS critical temperature (equation (3)) becomes exponentially suppressed in this case 3 . For example, at a density n = 10 10 cm −3 , the fractional atom loss rateṅ/n = 0.06 s −1 is significantly reduced but the critical temperature T c ∼ 10 −7 T F has been exponentially suppressed.
By confining atoms in an optical lattice potential, the BCS critical temperature can be exponentially increased even though the gas remains at low density. This is done by quenching the kinetic energy of the atoms (i.e. giving them a large effective mass) so that attractive interactions become strong in comparison [22]. To create a 3D cubic optical lattice, six laser beams are configured such that pairs of beams counterpropagate along each of three orthogonal axes. This generates a potential of the form where k L is the laser wavenumber and V 0 is the depth typically measured in units of the photon recoil energy (E R =h 2 k 2 L /2 m). For atoms confined in the lowest Bloch band of this periodic potential, the system can be described by the Hubbard Hamiltonian equation (4). For an optical lattice of depth V 0 = s E R , the tunneling matrix element t = E R (2/ √ π)s 3/2 exp(−2 √ s) and on-site interaction energy U = E R a t k L √ 8/π s 3/2 in the tight-binding limit. Note that t, which determines the width of the lowest band (12t for a 3D cubic lattice) and is inversely proportional to the effective mass, is exponentially reduced as the depth V 0 is increased.
Quite generically, the BCS critical temperature in the weak coupling limit can be estimated as is the density of states at the Fermi surface and U is the interaction strength. For a fixed interaction strength, the critical temperature can be exponentially increased by increasing the density of states at the Fermi surface. For a homogeneous system in the absence of an optical lattice, the critical temperature is exponentially increased by increasing the density of the gas and thereby increasing E F and D(E F ) (since D(E) ∝ √ E). In the case of an optical lattice, on the other hand, the density of states is increased by exponentially reducing the width of the lowest band with increasing lattice depth so that the density of states at the Fermi energy D(E F ) is made large even though E F remains small. For a 3D cubic lattice at half-filling, for example, the Fermi energy E F = 6 t and D(E F ) = 1/7t (where t is the tunneling matrix element) and the critical temperature where U is the on-site interaction energy. The tunneling matrix element is exponentially reduced as the depth of the lattice is increased, allowing for a dramatic increase in T c with only a modest increase in density.
When D(E F ) U becomes of the order of unity or larger, the weak coupling BCS picture breaks down and cannot be used to predict T c . In order to provide an estimate of experimental lattice and gas parameters that maximize T c , we provide here a heuristic argument. We will consider a three-component Fermi gas confined in a 3D cubic lattice at one-third filling and make the crude approximation that pairing is so strong that the gas becomes completely magnetized (see figure 2(a)) forming domains that each contain only two species. In each of these domains, the lattice is now at half-filling for the two species. The critical temperature for a two-component Fermi gas in a 3D cubic lattice at half-filling can be found exactly from quantum Monte Carlo calculations [23]. Figure 2(b) shows the critical temperature in units of t as a function of |U |/t. The black dots are a quantum Monte Carlo calculation of the Neél temperature for the repulsive Hubbard model at half-filling [23], which, through the repulsive-attractive transformation of the Hubbard model at half-filling, is identical to T c . The (c) The reduction in energy due to Cooper pairing ( 0.6 t) will not be sufficient to overcome the increase in local Fermi energy (≈t) required for complete phase separation and only partial polarization will be achieved. Thus, domains will form in which the numbers of particles of one color will be fewer than those of the other two, but will not be zero.
quantum Monte Carlo calculations are joined by a smoothing spline (solid blue) to Hartree-Fock theory (dashed light blue) in the weak coupling limit and a solution of the spin-1/2 Heisenberg model (dot-dashed purple) in the strong coupling limit [23]. The inset shows the same T c now plotted as a function of V 0 in units of E R assuming that a = −2222 a 0 and a lattice constant of d = 3.22 µm. Thus, for a two-component Fermi gas at half-filling, T c can be as high as T c 0.35 t. This maximum in T c corresponds to a lattice depth V 0 = 3.8 E R for the given parameters. Note that the average density of atoms in each internal state is n α 10 10 cm −3 for a lattice at one-third filling. While this argument relied on our assumption of complete magnetization, which may not be entirely fulfilled, we will use these parameters moving forward as we assess the feasibility of achieving color superfluidity in this system. For a 3D cubic lattice at one-third filling, the Fermi energy E F = 4.8 t and the superfluid critical temperature may be as high as T c 0.07 T F . To reach the superfluid transition, a reduced temperature T /T F < 0.07 must be achieved in free space prior to loading atoms into the optical lattice since T /T F remains approximately constant as the lattice potential is increasing adiabatically [24]. Given the large three-body recombination rate in three-component Fermi gases, we will now assess whether such temperatures can be achieved.
As stated previously, the state-of-the-art for cooling two-component Fermi gases is T = 0.05T F . We now show that a degenerate three-component Fermi gas can be created from a twocomponent Fermi gas at a reduced temperature T /T F = 0.05 with no additional evaporative cooling. A RF magnetic field can be used to populate all three internal states starting from a two-component Fermi gas by driving transitions between states |1 -|2 and |2 -|3 [19]. If these RF transitions are driven in the presence of a magnetic field gradient, an incoherent threecomponent mixture with equal populations in each state is created. In the absence of collisions, the three-component Fermi gas so produced would not be in a thermal state. However, since the three distinguishable fermions will collide, the mixture will equilibrate at a new reduced temperature τ = T /T F . Since the RF field neither adds nor removes kinetic energy from the system, we can determine τ by equating the total energy in the two-component gas at initial reduced temperature τ = T /T F with the total energy in the three-component gas at reduced temperature τ , Here, f ( , τ ) = [exp[( − µ/E F )/τ ] + 1] −1 is the Fermi-Dirac distribution function, is the energy in units of E F and D( ) is the density of states. For a homogeneous system, this condition reduces to 2 2/3 (τ ) 5/2 Li 5/2 (−Z(τ )) = 3 2/3 τ 5/2 Li 5/2 (−Z(τ )), where Li n is the polylogarithm function and Z(τ ) = exp[µ/(E F τ )] is the fugacity. For a given initial reduced temperature, numerical root finding gives the reduced temperature τ of the three-component gas after equilibration. For example, for τ = 0.1, we find that τ = 0.33, and for τ = 0.05, we find that τ = 0.31. Thus, a degenerate three-component Fermi gas can be produced from a degenerate two-component Fermi gas without any further evaporative cooling. However, achieving the higher degree of degeneracy required to observe color superfluidity will require a modest amount of evaporative cooling. Successful evaporative cooling requires that the ratio of the elastic collision rate to the trap loss rate is made as large as possible. As a point of reference, the necessary condition for runaway evaporative cooling of a thermal gas in a linear trapping potential is that the ratio between good and bad collision rates is 100 [25]. Here we compare the elastic collision rate in a three-component Fermi gas with the loss rate due to three-body recombination. The elastic collision rate obtained from the quantum Boltzmann equation (assuming ergodicity) is [26] Here, min = min( 1 , 2 , 3 , 4 ) and σ = 8πa 2 . This equation describes particles in states with energy 1 and 2 scattering into states with energy 3 and 4 . Note that the elastic collision rate is reduced by Pauli blocking due to the [1 − f ( )] terms, which inhibit scattering into the final states if the occupation in those states approaches unity. The collision rate (obtained from numerical integration) for a three-component Fermi gas at temperature T = 0.3 T F , density n α = 10 10 cm −3 (in each spin state) and a = −2222 a 0 is 8 s −1 . The corresponding threebody recombination rateṅ α /n α = −K 3 n 2 = −0.06 s −1 . Thus, the ratio between good and bad collisions is 140. Conditions are therefore favorable for further evaporative cooling to reduce the temperature of an already degenerate Fermi gas to T < 0.07 T F .
Assuming that a three-component Fermi gas with T < 0.07 T F is prepared by evaporative cooling, color superfluidity can be achieved by adiabatically increasing the depth of the optical 12 lattice to a value V 0 4E R . If the lattice is increased adiabatically (i.e. at fixed entropy), the ratio T /T F remains approximately constant in a homogeneous system [24]. Determining the time scale required for adiabaticity in an interacting many-body system is beyond the scope of this paper. However, we can make several general remarks. When the lattice is applied, the dispersion relation, in the tight binding limit, becomes E(k) = −2 t[cos(k x d) + cos(k y d) + cos(k z d)] and hence is anisotropic. Thus, the application of the lattice requires a change in the shape of the Fermi surface and a redistribution of occupation numbers in k-space. The change in the Fermi surface is most dramatic when the lattice depth increases from zero to V 0 1E R [22]. The timescale required to increase the depth to V 0 = 1E R must therefore be long compared with the collision time in order to allow for a redistribution in k-space. Alternatively, the lattice can be increased to V 0 = 1E R simultaneously with the final stage of evaporative cooling. This does not produce significantly more loss due to three-body recombination during evaporation since the increase in density is very modest. As the lattice is turned on, the density at the center of each site will be increased above the mean density averaged over many sites, n 0 . For the spatially dependent density that results, the three-body recombination loss rateṄ /N = −K 3 n 2 , where n 2 is the squared density averaged over a unit cell of the lattice. At a depth of V 0 = 1E R , the averaged square density n 2 1.3n 2 0 , resulting in a 30% higher loss rate during evaporation. Once a Fermi gas at T /T F < 0.07 is prepared in a V 0 = 1E R deep lattice by evaporative cooling, the lattice depth can be increased relatively rapidly to V 0 4E R since the change in shape of the Fermi surface is much less dramatic. To maintain constant entropy, the occupation number in k-space does not need to change significantly. Changing the depth of the lattice too rapidly does not change the quasi-momentum k but can cause interband excitation. So the depth of the lattice should be raised on a timescale that is long compared with h/E R since the band gap ∼ E R . For 6 Li atoms in a lattice with d = 3.22 µm, the lattice can be raised over a time of 10 × h/E R 13 ms. Once the depth of the lattice is V 0 = 4E R , the three-body recombination rate will be increased due to the increase in density. For V 0 = 4E R , the averaged square density n 2 10 n 2 0 , and the three-body loss rate is ten times higher than that obtained in the absence of the lattice (i.e.Ṅ /N = −K 3 n 2 = −0.6 s −1 for the parameters suggested above for the 6 Li system). This loss rate should be sufficiently low to allow for Cooper pair formation, which occurs on a timescale ∼h/ 0 2 h/t (e.g. 2 h/t 40 ms for the 6 Li system).
The approach outlined above minimizes the detrimental effects of three-body loss by working with low-density atomic samples confined in a large-period optical lattice. In this case, the rate at which occupation on a triply occupied lattice site decays due to three-body recombination, γ 3 , is much smaller than t/h (i.e. the tunneling rate). Alternatively, the authors of [27] have recently proposed that loss due to three-body recombination can also be suppressed by working in the opposite limit where γ 3 t/h. Here, rapid three-body recombination provides a mechanism for continuously measuring the number of lattice sites that have triple occupancy and, by the quantum Zeno effect, constrains the evolution of the system such that lattice sites never become triply occupied. In addition to suppressing loss due to three-body recombination, the constraint that triply occupied sites are absent stabilizes the color superfluid phase against the formation of a trionic phase at strong coupling. Suppression of the trionic phase has now been predicted for 1D [27] and higher-dimensional [28] systems. Thus, the constrained case (corresponding to γ 3 t/h) is of interest for the study of the color superfluid phase in the weak, intermediate and strong coupling limits, whereas the unconstrained case (corresponding to γ 3 t/h as considered in this paper) may allow for observing the phase transition between a color superfluid and a Fermi liquid of trimers.

Realizing SU(3)-invariant interactions
Our discussion of superfluidity in a three-component Fermi gas has so far assumed that the three pairwise scattering lengths are identical and that the Hamiltonian in equation (1) is SU(3) symmetric. In reality, the three scattering lengths for a three-component alkali gas will be nearly, but not exactly, equal for a large but finite bias field. 4 In the case of 6 Li, the interaction between states |1 and |2 is the most attractive (i.e. a 12 < a 23 < a 13 ). This anisotropy can fundamentally alter the character of the superfluid phase. For example, the anisotropy of the interactions breaks the SU(3) symmetry and can lock the three-component superfluid order parameter so that only 12 = 0. In this case, the color superfluid phase is an essentially standard U(1) superfluid in the presence of a decoupled Fermi liquid of atoms in state |3 [14]. Since only the U(1) symmetry is spontaneously broken in this case, a single NG mode exists rather than five. Perhaps most dramatically, breaking the SU(3) has existential implications for vortices. In a system with perfect SU(3) symmetry, Rapp et al [14] point out that any vortex can be twisted away due to the large internal symmetry of the order parameter, whereas vortices can exist in a U(1) superfluid. Finally, anisotropic interactions may change the nature of the phase transition between superfluidity and a Fermi gas of trimers from a second-order phase transition to one of first order [14]. For these reasons, it is worthwhile exploring methods to reduce the difference between scattering lengths to arbitrarily small levels. In this section, we describe how changes in scattering lengths can be affected by applying RF radiation to reduce the difference between scattering lengths in the 6 Li system by more than one order of magnitude. In this case, the difference between on-site interaction energies U for different pairs is at least one order of magnitude smaller than any other energy scale in the system, including the pairing gap. Further, we suggest that the application of an MW field in addition to the RF field can allow for independent tuning of the three scattering lengths and the realization of perfect SU(3) symmetry.
When RF radiation is near resonant with a transition between two atomic Zeeman states (see figure 3(a)), the eigenstates of the composite atom+field system (i.e. the RF-dressed states) contain an admixture of the two coupled Zeeman states. When two such RF-dressed atoms collide, the scattering properties in the dressed channel sample the scattering properties of all of the undressed two-body collision channels that have now been coupled by the RF field [29,30]. Thus, if the scattering lengths in two or more undressed collision channels differ substantially, controlled RF coupling of these channels can be used to tune the scattering lengths in the dressed channels. Using a simple model, we calculate below how the scattering lengths in a threecomponent 6 Li mixture can be tuned by applying RF fields to achieve nearly perfect SU(3) symmetry. The use of RF or MW radiation to control scattering lengths in other atoms or in other contexts has also been considered recently in [29][30][31][32][33].
Before computing scattering matrix elements for dressed state collisions, we first identify an appropriate basis to describe the asymptotically separated two-body wavefunction (i.e. the scattering channels). In the presence of RF radiation, the RF photon field and the internal states are the atomic internal state Hamiltonian, the Hamiltonian for the RF field and the atom-photon coupling Hamiltonian, respectively. Here, A hf is the hyperfine constant, S k and I k are, respectively, the electron and nuclear spins for each atom, µ B is the Bohr magneton, B 0 is the bias magnetic field in theẑ direction, g s and g i are the g-factors for the electron and nuclear spin, ω rf is the frequency of the RF photon, a † (a) is the creation (annihilation) operator for a RF photon,ê is the polarization vector of the RF magnetic field and η s/i are the coupling constants between the electron/nuclear spin and the photon field. For a field mode of frequency ω rf in a volume V , the coupling constants η s/i = g s/i µ B hωrf 2 0 c 2 V . Eigenstates of H A + H γ + H Aγ , which are antisymmetric under exchange of the fermionic particle labels, are used as basis states to calculate the scattering matrix elements for s-wave collisions at threshold. These are identical to the basis states used in [30].
In this paper, we will limit our attention to a circularly polarized RF magnetic field since the number of channel states in this case is finite and inelastic collision rates are minimized [30]. In the absence of RF-atom coupling (i.e. for H Aγ = 0), the bare basis states |{a, b}, N , which are the product of a photon number state (|N ) and an antisymmetric combination of Zeeman states (|{a, b} = 1 √ 2 (|a |b − |b |a )), are eigenstates of H A + H γ . These states become coupled when H Aγ is nonzero. For a circularly polarized RF field, H Aγ couples the subset of bare basis states that have the same projection of the total angular momentum (for atom plus photons) along theẑ axis. The 15 bare basis states coupled in the case of σ − polarized RF radiation are shown in figure 3(b). The eigenstates of equation (7) (which, in practice, are found in terms of the bare basis states) are the dressed channel states that we use as the basis to represent the scattering matrix. When we diagonalize equation (7), we assume that a † |N √ N |N + 1 (and a |N √ N |N − 1 ) since the average number of photons in the RF field, N , is very large. Further, we assume that the classical amplitude of the RF magnetic field is given by N , as would be the case if the RF field were a coherent state with a † a = N . Having found the RF-dressed channel states, we find the scattering matrix elements by solving a coupled channel Schrödinger equation for s-wave scattering, using as our Hamiltonian where r is the relative separation of the atoms, p is the relative momentum and µ is the reduced mass. The molecular interaction between two alkali atoms is given by is the singlet (triplet) molecular potential and P (S) (P (T) ) denotes a projection operator onto the subspace of singlet (triplet) states of the valence electrons. 5 To dramatically simplify the solution of the full coupled channel problem, we make two approximations. First, we use square-well potentials to represent the singlet and triplet molecular potentials. The square-well potentials have the same radius R but different depths, V 0,S and V 0,T . The parameters R, V 0,S and V 0,T are chosen to reproduce the known singlet and triplet scattering lengths, a s and a t , and the binding energy, E b , of the highest-lying bound state of the singlet molecular potential for 6 Li 2 dimers. Our second assumption is that H scatt is diagonal in the singlet-triplet basis for r < R, where the singlet and triplet molecular potentials dominate over the hyperfine interaction and the RF coupling. This simple model reproduces, to a good approximation, all of the scattering properties for 6 Li s-wave collisions reported in the literature. For example, the location and widths of the three Feshbach resonances shown in figure 1, which all result from the presence of the highest-lying bound state in the singlet molecular potential, are reproduced by this simple model. To find the threshold scattering properties for a given RF-dressed entrance channel |α , we solve the coupled channel Schrödinger equation to first order in the relative wavenumber k of the colliding atoms. The solution of the coupled channel Schrödinger equation gives the scattering matrix elements S β,β between dressed channel states |β and |β . We extract the s-wave scattering length a α for the entrance channel |α from the diagonal S-matrix element S α,α using the relation S α,α = exp(−2ikã α ) 1 − 2ikã α , whereã α = a α − ib α is the complex scattering length. If b α > 0, there are other open channel states into which |α can scatter. The threshold rate constant for scattering into these open channels is given by The relationship between b α and K 2 is given by RF-dressing of scattering states can be used to significantly reduce the difference in pairwise scattering lengths in a three-component Fermi gas. As a demonstration, we calculate the scattering lengths and inelastic decay rate constants for RF-dressed collision channels of 6 Li atoms in a B 0 = 2000 G bias field with a σ − -polarized RF magnetic field tuned near the |2 -|3 transition (see figure 3(a)). The amplitude of the RF field is B rf = 140 mG. The RF field strongly couples the bare collision channels |{1, 2}, N and |{1, 3}, N − 1 . The scattering lengths (a 12 (ω rf ), a 13 (ω rf )) and inelastic decay rate constants for the RF-dressed collision channels associated with these bare channels are shown in figure 4. Also shown is the scattering length a 23 (ω rf ) associated with the bare scattering channel |{2, 3}, N − 2 , which is not coupled to other bare scattering channels. For the |{2, 3}, N − 2 channel, the inelastic decay rate constant is identically zero for our model since we neglect weak spin-spin dipole interactions.
At a field of 2000 G and for a RF field resonant with the |2 -|3 transition (at ω rf /2π = 79.413 MHz), the three scattering lengths a 12 (ω rf ), a 23 (ω rf ) and a 13 (ω rf ) are approximately equal (to better than 1 part in 1000), as shown in figure 4(a). On resonance, the scattering lengths a 12 (ω rf ) and a 13 (ω rf ) are equal and approximately given by the average value of the scattering lengths a 12 and a 13 in the absence of RF coupling. Because a 23 (a 13 + a 23 )/2 in the absence of RF coupling, all three scattering lengths are made nearly equal for resonant RF. Exactly on resonance, the three scattering lengths differ from their average by less than 1 part in 1000. Thus, RF dressing of the collision channels achieves in this case a reduction in the difference between scattering lengths by more than one order of magnitude. The two-body inelastic rate constant for the RF-dressed channel states associated with |{1, 2}, N and |{1, 3}, N − 1 are both K 2 = 1.64 × 10 −12 cm 3 s −1 on resonance. For a low-density gas such as that discussed in section 4, this two-body decay rate constant is sufficiently small that it will not cause significant loss of atoms over the timescale required for equilibration. This example shows that by using RF radiation resonant with the |2 -|3 transition, we can reduce the difference between scattering lengths for a three-component Fermi gas to the point where the associated difference between on-site interaction energies is an order of magnitude smaller than the pairing gap 0 .
We have also considered whether further reductions in the difference between scattering lengths can be achieved by applying an MW radiation field. In the case just considered, we achieved nearly identical scattering lengths by resonantly coupling the bare |{1, 2}, N and |{1, 3}, N − 1 collision channels with RF radiation and found that, on resonance, a 12 (ω rf ) = a 13 (ω rf ) = (a 12 + a 23 )/2, which is close to the value of a 23 . If it were the case that a 23 = (a 12 + a 23 )/2, all three RF-dressed scattering lengths could be made identical by this technique. We now consider whether we can tune a 12 and a 23 by applying a MW radiation field so that a 23 = (a 12 + a 23 )/2. We expect that, in this case, the application of both the RF-and MW-dressing fields will allow the realization of a system with identical scattering lengths.
To tune the a 12 and a 23 scattering lengths, we consider applying a σ + -polarized MW magnetic field nearly resonant with the |2 -|5 transition, as shown in figure 3(a). This MW field couples together the bare |{1, 2}, N and |{1, 5}, N + 1 collision channels and it also couples the bare |{2, 3}, N − 2 and |{3, 5}, N − 1 collision channels. The scattering lengths for the MW-dressed channels are shown in figure 5 along with the two-body decay rate constants. For these calculations, the bias field B 0 = 2000 G and the amplitude of the MW magnetic field is B mw = 140 mG. Figure 5(a) shows that the scattering lengths a 12 (ω mw ) and a 15 (ω mw ) can change significantly near resonance due to the fact that the scattering lengths of the two bare channels (i.e. the scattering lengths away from resonance) are substantially different. Similarly, as shown in figure 5(b), the scattering lengths a 23 (ω mw ) and a 35 (ω mw ) change significantly near resonance. Near resonance, the two-body inelastic decay rate constants can be very large. An exceptional case, however, is the dressed channel associated with the |{1, 2}, N state. In this case, a 12 (ω mw ) can be changed substantially for MW frequencies tuned below the resonance condition and K 2 remains extremely small. Thus, MW-dressing may be useful in controlling the interactions in two-component Fermi gases that have population in states |1 and |2 .
In the present context, that of making slight changes in the a 12 and a 23 scattering lengths, we find that small changes can be made using MW radiation far from resonance where K 2 is reduced from its maximum value. For example, in figure 6, we plot the scattering lengths a 12 (ω mw ), a 23 (ω mw ) and a 13 (ω mw ), along with the associated two-body decay rate constants, for MW frequencies below the |2 -|5 resonance. For a MW frequency of 5.958 GHz, the average of a 12 (ω mw ) and a 13 (ωmw) is exactly equal to a 23 (ω mw ). It is for this MW power and frequency that we expect that the application of a separate RF field resonant with the |2 -|3 transition will create a three-component mixture with identically equal scattering lengths. In this case, the largest two-body decay rate constant K 2 = 2 × 10 −11 cm 3 s −1 occurs in the MW-dressed channel associated with |{2, 3}, N − 2 . This two-body decay rate constant is sufficiently small that for a 6 Li Fermi gas with n α = 10 −10 cm −3 , as considered in section 4, the equilibration time is much shorter than the timescale for atom loss. These calculations indicate that it is possible to make a three-component 6 Li Fermi gas with an exact SU(3) symmetry.

Analogues of color superconductivity with color and spin
A marked difference between quark matter and the three-component Fermi gas described so far is that quark matter has spin and flavor degrees of freedom in addition to color. The numerous degrees of freedom available in quark matter make many different patterns of Cooper pairing possible. Deciding which of these pairing patterns are favored by nature, particularly in the presence of strong interactions and imbalanced Fermi surfaces, is a significant challenge in the The two-body inelastic rate constants K 2 associated with a 12 , a 23 and a 13 . field of color superconductivity [1]. In this section, we describe how atomic systems in optical lattices can be engineered to have degrees of freedom in addition to the internal Zeeman state of the atom. To be specific, we suggest how a system with an additional pseudo-spin-1/2 degree of freedom can be realized by confining a three-component Fermi gas in the p-orbital band of an optical lattice. Superfluidity in a system with both color and spin degrees of freedom would provide an analogue of the single-flavor color superconducting phase that may occur in quark matter at intermediate densities.
Atoms localized on individual sites of an optical lattice can occupy either the lowest eigenstate of the potential at a given site (i.e. the s-orbital) or higher-lying eigenstates, such as the p x , p y or p z orbitals. For a simple cubic lattice with equal lattice depths in each direction, the p x , p y and p z orbitals are degenerate. Atoms occupying the p-orbital band of an optical lattice therefore have an orbital degree of freedom in addition to the degree of freedom associated with their internal state. Here, we will examine a case where atoms only occupy the p x and p y orbitals in a simple tetragonal lattice. We assume that the lattice depth and/or lattice constant in the z-direction is adjusted so that the p z orbital has an energy higher than that of the p x or p y orbitals and is made energetically inaccessible. In this situation, atoms will be able to occupy either the p x or p y orbital and this two-state system will provide a pseudo-spin-1/2 degree of freedom in addition to the internal state of the atom. color and overall phase. We will also investigate whether systems can be engineered that are symmetric under SU(2) ⊗ SU(3) ⊗ U(1) transformations when interactions are included. In this case, there would be several possible pairing patterns for the superfluid state analogous to the pairing patterns that occur for single-flavor color superconductivity (i.e. the color-spin locked, planar, polar and A-phases [36]). Such a system would offer the possibility of experimentally determining which pairing pattern is favored by nature in the presence of strong interactions and imbalanced populations, providing insights into the color superconducting phases thought to occur inside neutron stars.