We study attractive fermions in an optical lattice superimposed by a trapping potential, such that fermions may form bosonic molecules. We map the model onto nonlinear field equations depending on the Nambu-Gor'kov propagator. The resulting field equations where solved numerically by a relaxation technique that allowed us to calculate the inhomogeneous densities of fermions and condensed molecules at zero temperature. When the interactions between fermions are strong there is a competition between unbound fermions and bound molecules leading to an unexpected reduction of the non-homogeneous density of fermions at the center of the trap.
Deep Dive into Spin-1/2 fermions with attractive interaction in an optical lattice.
We study attractive fermions in an optical lattice superimposed by a trapping potential, such that fermions may form bosonic molecules. We map the model onto nonlinear field equations depending on the Nambu-Gor’kov propagator. The resulting field equations where solved numerically by a relaxation technique that allowed us to calculate the inhomogeneous densities of fermions and condensed molecules at zero temperature. When the interactions between fermions are strong there is a competition between unbound fermions and bound molecules leading to an unexpected reduction of the non-homogeneous density of fermions at the center of the trap.
Ultra-cold gases present many-body systems with a remarkable tunability of its parameters, providing us with a platform for the investigation of numerous properties of complex quantum systems. For instance, it has become possible to tune the interaction strength between atoms over a wide range by Feshbach resonance [1]. The investigation of ultra-cold Fermi gases started shortly after the discovery of Bose-Einstein condensation of bosonic atoms [2] when quantum degeneracy in a gas of fermionic atoms was obtained [3]. For fermions with weak attraction we have the celebrated phenomenon of Cooper pairing [4]. By tuning to strong attraction one enters the regime of diatomic molecules, whose size is much smaller than that of Cooper pairs. These molecules then may condense into a Bose-Einstein condensate of a hard-core Bose gas. The crossover from the weakly interacting BCS regime to the strongly interacting BEC regime has been the subject of theoretical studies in [5,6]. The bosonic molecules, formed of pairs of fermionic atoms, were produced experimentally in a trapped system [7] as well as in an optical lattice [8].
Most investigations were based on continuous Fermi gases. The more recently introduced optical lattices in ultra-cold gases [9] may have a number of interesting effects on the Fermi gas. First of all, the dispersion of the atoms will be changed by the lattice. Moreover, the interaction between the atoms has a strong effect on the quantum states of the Fermi gas by allowing, for instance, to form Mott states [10]. This could mean that the molecular gas in the BEC regime does not condense but becomes a Mott state. In the BCS regime, on the other hand, the effect of an optical lattice is not so dramatic because the Cooper pair radius is much larger than the lattice spacing. Consequently, the BEC-BCS crossover can be much richer in the presence of an optical lattice.
In this paper, we study strongly attractive fermions in an optical lattice superimposed by a trapping potential. We first show that the phase diagram of tightly bound fermions contains a Bose-Einstein condensed phase and a Mott insulating phase of such molecules. Then we study the system in a harmonic trap and calculate the density of unpaired fermions in the presence of a condensate state of molecules. We show that there is a competition between paired fermions and unpaired fermions which leads to a reduction of the density of unpaired fermions at the center of the trap.
The simplest lattice model of an interacting Fermi gas is the Hubbard model [11]. It describes the competition between the kinetic energy of the fermions and a local interaction and provides a phase diagram that includes a Fermi liquid and a Mott insulator. Originally introduced for a repulsive Fermi gas (e.g. electrons in a metal), the model can also be used for neutral fermionic atoms with attractive local interaction. For very strong attraction, however, the local interaction is insufficient to describe the physics of the Fermi gas. This due to the fact that strong attraction causes pairing of fermions to local molecules. On the other hand, the kinetic term in the Hubbard model allows only individual tunneling of fermions. This means that the tightly bound molecules must dissociate into independent fermions in order to tunnel in the optical lattice. The associated energy of such a process is of the order of the attractive interaction. (Actually, the effective tunneling rate is ∼ 2t 2 /U [6], where t is the tunneling rate of individual fermions and U is the strength of the local attraction.) This means that the motion of molecules is strongly suppressed in the Hubbard model [12]. On the other hand, there is no reason for the molecules not tunnel freely in the optical lattice because they only have to obey the Pauli principle. The solution of this problem is an extension of the attractive fermionic Hubbard model that includes an additional kinetic term for the bosonic molecules [13][14][15][16]. The Hamiltonian for an attractive Fermi gas in a d-dimensional optical lattice is a molecular fermionic Hubbard (MFH) model and reads
Here, ĉr,σ (ĉ † r,σ ) is the annihilation (creation) operator for particles at lattice site r. The index σ =↑, ↓ represents two hyperfine states of fermionic atoms, e.g., 40 K or 6 Li. In this work we consider only the symmetric case where the number of fermions in each component is the same. Nearest-neighbor tunneling of the individual fermions is described by the parameter t. There is also a term with parameter J which is understood as a tunneling term of dressed fermionic pairs [14,15]. U ∼ U bg -4dg 2 /δ accounts for an effective local attractive interaction between fermions with the detuning δ, coupling between fermions and molecules g and U bg = 4πh 2 a b /m with the background scattering length a b and m is the mass of fermions. In the strong coupling regime |δ| ≫ g and a b is small and positive [15]. So that for strong attractions U < J
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