† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 51331006, 51590883, and 11204321) and the Project of Chinese Academy of Sciences (Grant No. KJZD-EW-M05-3).
We use the path-integral formalism to investigate the vortex properties of a quasi-two dimensional (2D) Fermi superfluid system trapped in an optical lattice potential. Within the framework of mean-field theory, the cooper pair density, the atom number density, and the vortex core size are calculated from weakly interacting BCS regime to strongly coupled while weakly interacting BEC regime. Numerical results show that the atoms gradually penetrate into the vortex core as the system evolves from BEC to BCS regime. Meanwhile, the presence of the optical lattice allows us to analyze the vortex properties in the crossover from three-dimensional (3D) to 2D case. Furthermore, using a simple re-normalization procedure, we find that the two-body bound state exists only when the interaction is stronger than a critical one denoted by Gc which is obtained as a function of the lattice potential’s parameter. Finally, we investigate the vortex core size and find that it grows with increasing interaction strength. In particular, by analyzing the behavior of the vortex core size in both BCS and BEC regimes, we find that the vortex core size behaves quite differently for positive and negative chemical potentials.
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 = kB = 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 (
In the presence of a quantized vortex considered in this paper, the saddle point ansatz can be chosen as Δ (
As have been pointed out in Ref. [43],
Equations (
As shown in Ref. [38], the atom density nr and pair density Δr 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
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 Gc obtained above, we can now investigate this peculiar feature for the system under consideration.
By self-consistently solving Eqs. (
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.
The vortex core size rv can be defined as the distance from the vortex core at which the order parameter becomes zero, as shown in Fig.
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. (
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 μr > 0 with Δr → 0. Therefore, the vortex core size should be determined by the number equation with the critical condition being nr = 0, which leads to μr = 0. Finally, we obtain
In the BEC limit, since the bulk chemical potential μ0 ∼ −ɛb/2 and hence μr < 0, nr = 0 is always satisfied for arbitrary r at the vortex core. Therefore, the vortex core size should be determined by the gap equation. Since μr < 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.
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.
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