Since the first observation of Bose–Einstein condensation and Fermi superfluid phenomenon in ultracold atoms system,^[1–9] tremendous attention in condensed matter community has been paid to this system both theoretically and experimentally^[10–17] because of its high controllability. In particular, using the Feshbach resonance technique,^[18,19] one can tune the two-body interaction at will. This unique feature of controlling interaction continuously provides an ideal platform to investigate the crossover from the weakly interacting BCS^[20] regime with loosely bound cooper pairs to the strongly attractive region where tightly bound molecules form a Bose–Einstein condensate (BEC)^[21,22] (BCS–BEC crossover). Furthermore, by adding a trapping potential in selective directions, the system can evolve from higher dimension to lower one in a continuous way. Among many interesting phenomena related with this cold atoms system, the vortex state is one of the most important, which receives much attention in both cold atoms^[23–25] and condensed matter communities.^[26–32] The vortex state in the BCS–BEC crossover has been investigated in great detail^[33–36] by numerically solving the Bogoliubov–de Gennes (BdG) equation^[37] and mean-field equations.^[38–40] However, the vortex state and its properties in the dimensional crossover scenario have largely been untouched, which is the main focus of this paper.

We investigate the evolution of the vortex structure in both the BCS–BEC and the three dimensions (3D) to two dimensions (2D) crossover problems in a two-component Fermi gas trapped in an optical lattice potential at zero temperature. The main results are composed of two parts. First, the atom number density and the pair density (gap profile) are calculated for different interaction strengths and lattice potential parameters. We identify the BCS and BEC regimes by the existence of the bound state and the boundary can be given by a critical interaction strength which is a function of the lattice potential parameter. In the BCS regime, the Fermi atoms can penetrate into the vortex core, which is characterized by the nonzero atom density inside the vortex core where the pair density is zero. But in the BEC regime, the Fermi atoms cannot penetrate into the vortex core. The second part deals with the vortex core size as a function of the interaction strength and the lattice potential parameters. In the BCS regime where the chemical potential is positive in the bulk, the vortex core size is proportional to the square-root of the chemical potential determined by the atom number density equation. However, in the BEC limit with a negative bulk chemical potential, the vortex core size is determined by the gap equation.

The outline of the rest of this paper is as follows. In Section 2, we introduce the grand canonical partition function for a two-component Fermi gas in the presence of an optical lattice potential. Using the Hubbard–Stratonovic transformation, we rewrite the partition function as functional integral over pairing fields. Choosing the saddle point solution suitable for the description of the vortex state, we obtain the ground state energy and consequently the atom number and gap equations. Section 3 presents a detailed numerical evaluation of the atom number and gap equation profiles of the vortex state in the BCS–BEC crossover. We also discuss the dimensional effects introduced by the optical lattice potential on the vortex structure. In Section 4, the vortex core size is calculated in the BCS–BEC crossover and the effect of the lattice potential is discussed. Finally, we summarize our results in Section 5.

We consider a 3D two-component Fermi gas with an attractive two body interaction confined in an optical lattice potential. In the path integral formalism, the system can be described by the following grand-partition function at finite temperature T (we use the convention ħ = k_B = 1 throughout this paper):

By transforming the Grassmann integral into a Bosonic one, this path integral formalism provides a convenient framework to study the pairing phenomena. By properly choosing the saddle point solution, one can reduce the current formalism into the familiar mean-field BCS theory for different pairing symmetries. For example, in the absence of vortex state, we have the trivial saddle point solution Δ (r,τ) = Δ₀ and the thermodynamic potential becomes

In the presence of a quantized vortex considered in this paper, the saddle point ansatz can be chosen as Δ (r,τ) = Δ_re^iθ, where is the distance to the vortex line and θ the angle around the vortex in the cylindrical coordinates r = (r,θ,z). With this ansatz, the effective action has the same form as Eq. (4) with the inverse Green’s function being

As have been pointed out in Ref. [43], is irrelevant to the low energy properties of the system and will be neglected in the following treatment. However, the partition function is still not integrable because of the inhomogeneous term . Following Ref. [38] and [39], we assume that the paring field varies slowly in comparison with the relevant fermion frequencies and define a local chemical potential μ_r = μ − 1/(8mr²). Under this approximation and using the tight binding approximation for the band spectrum, we can obtain the thermodynamic potential

Equations (11) and (12) are the generalized BCS mean-field equations in the presence of a single quantized vortex and an optical lattice potential. The bulk value of the chemical potential μ₀ and gap parameter Δ₀ have to be calculated by solving Eqs. (11) and (12) far away from the vortex core. From this, one can then obtain Δ_r and n_r for different parameters across the BCS–BEC and 3D–2D crossovers.

As shown in Ref. [38], the atom density n_r and pair density Δ_r behave differently on BCS side and BEC side of the Feshbach resonance characterized by the formation of bound state in vacuum. Therefore, it is necessary to identify the resonance point in the presence of an optical lattice. This has been investigated intensively using the scattering theory.^[44,45] In this paper, we use another simpler strategy^[46] which reproduces qualitatively correct results but with much less numerical efforts. In the continuum limit, the momentum integration in the gap equation is ultra-violet divergent as a result of the use of the short range s-wave interaction, which implies a high momentum cut-off with R₀ being the effective interaction range. We now replace g in terms of the binding energy ɛ_b > 0 through , from which we obtain

Fig. 1.

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Fig. 1. (a) Binding energy as a function of the dimensionless interaction parameter G defined in the text. The solid line is the plot of Eq. (13) and dashed line is for the analytical results in the weakly interacting limit presented in Ref. [46]. The red dot denotes the critical interaction parameter beyond which a bound state will form. (b) Critical interaction parameter as a function of the lattice potential parameter t (Eq. (14).

As shown in Ref. [38], the atoms can penetrate into the vortex core on the BCS side (no bound state) while not on the BEC (with bound state) side. With the critical value G_c obtained above, we can now investigate this peculiar feature for the system under consideration.

By self-consistently solving Eqs. (11) and (12) far away from the vortex r → ∞, we can obtain Δ₀ and μ₀. The vertex profiles characterized by Δ_r and μ_r can then be obtained by solving Eqs. (11) and (12) and the numerical results are presented in Fig. 1. The interaction parameters used in the numerical calculations are chosen according to Eq. (14). Without loss of generality, the other parameters we used in the numerical evaluation are chosen as k_Fd = 0.2, t = 0.5e_F, and k₀ = 2 × 10⁴k_F, with k_F being the Fermi momentum in the uniform space defined as and the corresponding Fermi energy . The numerical results are shown in Fig. 2. It can be seen that on the BCS side with G < G_c, the atoms can penetrate into the vertex core while they cannot penetrate on the BEC side. Through detailed numerical investigations for various situations, we find that this picture happens for any optical lattice parameter t.

Fig. 2.

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	Fig. 2. Vortex profiles on (a) the BCS side with dimensionless interaction parameter G = 0.045 and (b) the BEC side with G = 0.053. The critical interaction parameter Gc = 0.049 is determined by Eq. (14) with other parameters being given as kFd = 0.2, t = 0.5eF, and k0 = 2 × 104kF. Solid and dotted lines represent the atom number density and the pair density, respectively.

However, the present path-integral formalism used to describe the vortex state amounts to a local density approximation^[38] and is invalid around the vortex core. The main reason for the breakdown is the assumption that the order parameter varies smoothly. Consequently, the vortex profiles shown in Fig. 2 are estimated to be wrong approaching the vortex core and one may resort to the BdG equation which is more accurate in the vortex core region. Despite this defect, the theory naturally gives another important quantity r_v, the vortex core size, which will be discussed in detail in the following section.

The vortex core size r_v can be defined as the distance from the vortex core at which the order parameter becomes zero, as shown in Fig. 2. Numerical solutions of r_v as a function of dimensionless interaction parameter G are represented in Fig. 3, with other parameters chosen the same as those used to obtain Fig. 2. One can see that, in the BCS limit, the vortex core size approaches a constant value, which agrees with the results in Ref. [38]. However, in the BEC limit, the vortex core size tends to a constant instead of growing with increasing interaction strength in the homogeneous space shown in Ref. [38].

Fig. 3.

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	Fig. 3. Vortex core size as a function of the dimensionless interaction parameter G. The lattice parameters are chosen to be the same as those used to obtain Fig. 2. The solid blue line represents the numerical results obtained from the self-consistent solutions of Eqs. (11) and (12). The dashed black line is the asymptotic behavior (Eq. (15)) of the vortex core size in the BCS regime.

In order to understand the asymptotic behaviors of the vortex core size in the BCS and the BEC limits, we analyze the properties of the gap equation (Eq. (11)) and number equation (Eq. (12)) approaching the vortex core. By setting △_r = 0, one can easily check that the momentum integration on the right side of Eq. (11) is divergent for μ_r > 0. Therefore, the only acceptable solution for the local chemical potential at the vortex core should be μ_r ≤ 0. Besides, the atom density in the vortex core also goes to zero for μ_r ≤ 0. In conclusion, the local chemical potential should satisfy μ_r ≤ 0 approaching the vortex core.

In the BCS regime, the bulk chemical potential μ₀ > 0. Since the momentum integral is divergent for positive local chemical potential, the gap equation can be satisfied for any μ_r > 0 with Δ_r → 0. Therefore, the vortex core size should be determined by the number equation with the critical condition being n_r = 0, which leads to μ_r = 0. Finally, we obtain

In the BEC limit, since the bulk chemical potential μ₀ ∼ −ɛ_b/2 and hence μ_r < 0, n_r = 0 is always satisfied for arbitrary r at the vortex core. Therefore, the vortex core size should be determined by the gap equation. Since μ_r < 0, the momentum integral is convergent and the vortex core size can be obtained analytically from the following equation:

As can be seen in Fig. 3, r_v tends to be a constant instead of growing with increasing interaction strength in the case without the optical lattice trap. As stated in Ref. [38], in the BEC limit, the vortex core size is identified as the healing length since the system can be considered as a Bose–Einstein condensate of tightly bound molecules with weak repulsive interactions. On the mean-field level, the molecular scattering length a_m = 2a₀ with a₀ being the scattering length of Fermi atoms with opposite spins. However, in the presence of the optical lattice potential, the interaction between molecules is a constant.^[46] Consequently, the vortex core size saturates to a fixed value approaching the BEC limit, as depicted in Fig. 3.

In summary, we have extended the path-integral mean-field theory for the vortex state to the case with an optical lattice trapping potential. In the low energy limit, we only considered the lowest band and used a Hubbard-type spectrum to effectively describe the effects of the periodic potential. Within the path-integral formalism, we derived the mean-field number and gap equation which were then solved self-consistently to obtain the vortex profiles and the vortex core size. For the vortex profile characterized by the atom number density and pair density, like the homogeneous case, we found that the atoms can penetrate into the vortex core in the BCS regime while it cannot occur in the BEC regime. The dimensional effect was also discussed by calculating the bound state energy as a function of the interaction and lattice parameters. We found that the bound state emerges only when the interaction strength is bigger than a critical value. From the vortex profiles, we also obtained the vortex core size as a function of the interaction parameter. By analyzing the atom number and gap equations at the vortex core, we found that the vortex core size is determined by the atom number equation in the BCS regime for positive bulk chemical potential and by the gap equation in the BEC regime for negative bulk chemical potential.

The mean-field approximation is problematic near the vortex core. Therefore, it is important to know how the vortex profile evolves in the dimensional crossover by using a beyond mean-field theory, like the BdG equation^[37] and the beyond-mean-field corrections.^[47,48] Besides, the vortex properties in the presence of other trapping potentials and spin–orbit coupling are also under investigations.