Ground-state description of a single vortex in an atomic Fermi gas: From BCS to Bose Einstein condensation

One of the most exciting developments in atomic and condensed matter physics has been the observation of superfluid ity in trapped fermionic systems [1, 2, 3, 4]. In these system s, the presence of a Feshbach resonance provides a means of tuning the attractive pairing interaction with applied mag netic field. In this way the system undergoes a continuous evolutio n from BCS to Bose-Einstein condensed (BEC) superfluidity. The most conclusive demonstration of the superfluid phase has been the experimental observation of vortices [5]. Particularly interesting from a theoretical viewpoint is the way vortices evolve from BCS to BEC. This evolution is associated, not just with a decrease in vortex size but with a complete rearrangement of the fermionic states which make up the core. As a result, there is a continuous evolution of the particle den sity within a vortex, thereby affecting the visibility of vortic es in the laboratory. In this paper we discuss the behavior of a (si ngle) vortex as the system crosses from BCS to BEC. Our work is based on simplest BCS-like ground state first introduced by Leggett [6] and Eagles [7] to treat BCS-BEC crossover. With this choice of ground state inhomogeneity effects are readily incorporated as in generalized Bogoliubov-de Gennes (BdG) theory. Here we demonstrate analytically that the BdG strong coupling description of the T = 0 vortex state coincides with the usual Gross-Pitaevskii (GP) treatment of vortices in bosonic superfluids. A fermionic theory based on BdG is, thus, very inclusive, and within this approach one expects a smooth evolution of vortices from the BCS to BEC limit as the statistics effectively change from fermionic to bosoni c. Previous studies of vortices in these fermionic superfluids addressed the BCS limit at T = 0 [8] and T ≈ Tc [9]. There is also work [10] on the T = 0 strict unitary case where a BdG approach was used with Hartree-Fock contributions included. In the present work, by contrast, we discuss the entire crossover regime and, importantly, present a detailed analysis of the energy and spatial structure within the core and how it evolves from BCS to BEC. A very different path integral approach was introduced in Ref. [11] to address vortices with BCS-BEC crossover, but here the authors note that density depletion effects appear to be unphysically large in the BCS regime. Our analytical approach builds heavily on previous work [12] which showed a general connection between GP theory and BdG. From this one can conclude that a generalized BCS theory [6] treats the bosonic degrees of freedom at the same level as GP theory. Different ground states can be contemplated, (with incomplete condensation, say) but they will not be compatible with BdG theory. In a similar way, once T 6 0 one has to incorporate noncondensed pairs, and associated pseudogap physics [13] which are not present in a finite temperature BdG theory. For the most part, BdG approaches require detailed numerical solution [8, 14, 15, 16], so it is particularly useful to have analytical tools in the BEC limit. We present this nonnumerical description first. Our general self consistent eq uations [17] are � h − µ �(r) � ∗ (r) −h ∗ + µ �� u n vn � = En � u n vn