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arXiv:1107.2613 · Physical Review A

Controlling the pair momentum of the Fulde-Ferrell-Larkin-Ovchinnikov state in a three-dimensional Fermi gas through a one-dimensional periodic potential

Jeroen P. A. Devreese, Michiel Wouters, Jacques Tempere

Published Oct 14, 2011• 21 citations
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Abstract

The question whether a spin-imbalanced Fermi gas can accommodate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state has been the subject of intense study. This state, in which Cooper pairs obtain a nonzero momentum, has hitherto eluded experimental observation. Recently, we demonstrated that the FFLO state can be stabilized in a 3D Fermi gas, by adding a 1D periodic potential. Until now it was assumed that the FFLO wave vector always lies parallel to this periodic potential (FFLO-P). In this contribution we show that, surprisingly, the FFLO wave vector can also lie skewed with respect to the potential (FFLO-S). Starting from the partition sum, the saddle-point free energy of the system is derived within the path-integral formalism. Minimizing this free energy allows us to study the different competing ground states of the system. To qualitatively understand the underlying pairing mechanism, we visualize the Fermi surfaces of the spin-up and spin-down particles. From this visualization, we find that tilting the FFLO wave vector with respect to the direction of the periodic potential can result in a larger overlap between the pairing bands of both spin species. This skewed FFLO state can provide an additional experimental signature for observing FFLO superfluidity in a 3D Fermi gas.

DOI
10.1103/physreva.84.043623

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The question whether a spin-imbalanced Fermi gas can accommodate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state has been the subject of intense study. This state, in which Cooper pairs obtain a nonzero momentum, has hitherto eluded experimental observation. Recently, we demonstrated that the FFLO state can be stabilized in a 3D Fermi gas, by adding a 1D periodic potential. Until now it was assumed that the FFLO wave vector always lies parallel to this periodic potential (FFLO-P). In this contribution we show that, surprisingly, the FFLO wave vector can also lie skewed with respect to the potential (FFLO-S). Starting from the partition sum, the saddle-point free energy of the system is derived within the path-integral formalism. Minimizing this free energy allows us to study the different competing ground states of the system. To qualitatively understand the underlying pairing mechanism, we visualize the Fermi surfaces of the spin-up and spin-down particles. From this visualization, we find that tilting the FFLO wave vector with respect to the direction of the periodic potential can result in a larger overlap between the pairing bands of both spin species. This skewed FFLO state can provide an additional experimental signature for observing FFLO superfluidity in a 3D Fermi gas.
The question whether an imbalanced Fermi gas can accommodate polarized superfluidity has been the subject of intense study. One prominent candidate for a polarized superfluid state is the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. In a one-dimensional (1D) Fermi gas, indirect evidence for this state has recently been found, but in the three-dimensional (3D) case, this exotic state has hitherto remained elusive. In this brief review, we study the influence of a 1D periodic potential on the FFLO state in an imbalanced 3D Fermi gas. We present a path-integral derivation of the free energy of the system, where the possibility of the FFLO state is included through the saddle-point approximation. Using the free energy, the phase diagram of the system, both at fixed chemical potentials and at fixed densities, is constructed, showing that the stability region of the FFLO state is significantly enlarged by the 1D potential. Subsequently, the influence of the potential on the FFLO pairing mechanism is investigated, leading to a qualitative understanding of the shape of the FFLO region in the different phase diagrams. We conclude by investigating the possibility of an FFLO state with a wave vector that lies skewed with respect to the direction along which the periodic potential varies. Our results provide an additional way to facilitate the experimental observation of the FFLO state in a 3D Fermi gas.
In ultracold Fermi gases, the effect of spin imbalance on superfluidity has been the subject of intense study. One of the reasons for this is that spin imbalance frustrates the Bardeen-Cooper-Schrieffer (BCS) superfluid pairing mechanism, in which fermions in different spin states combine into Cooper pairs with zero momentum. In 1964, it was proposed that an exotic superfluid state called the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in which the Cooper pairs have nonzero momentum, could exist in a spin-imbalanced Fermi gas. At the saddle-point (mean-field) level, it has been shown that the FFLO state only occupies a very small sliver in the ground-state phase diagram of a three-dimensional (3D) Fermi gas. However, a question that remains to be investigated is as follows: What is the influence of phase fluctuations on the FFLO state? In this work, we show that phase fluctuations only lead to relatively small quantitative corrections to the presence of the FFLO state in the saddle-point phase diagram of a 3D spin-imbalanced Fermi gas. Starting from the partition function of the system, we calculate the effective action within the path-integral adiabatic approximation. The action is then expanded up to second order in the fluctuation field around the saddle point, leading to the fluctuation free energy. Using this free energy, we calculate corrections due to phase fluctuations to the BCS-FFLO transition in the saddle-point phase diagram. At temperatures at which the FFLO state exists, we find only small corrections to the size of the FFLO area. Our results suggest that fluctuations of the phase of the FFLO order parameter, which can be interpreted as an oscillation of its momentum vector, do not cause an instability of the FFLO state with respect to the BCS state.
We describe an imbalanced superfluid Fermi gas in three dimensions within the path-integral framework. To allow for the formation of the Fulde-Ferell-Larkin-Ovchinnikov state (FFLO state), a suitable form of the saddle point is chosen, in which the pairs have a finite center-of-mass momentum. To test the correctness of this path-integral description, the zero-temperature phase diagram for an imbalanced Fermi gas in three dimensions is calculated, and compared to recent theoretical results. Subsequently, we investigate two models that describe the effect of imposing a one-dimensional (1D) optical potential on the three-dimensional (3D) imbalanced Fermi gas. We show that this 1D optical potential can greatly enlarge the stability region of the FFLO state, relative to the case of the 3D Fermi gas without 1D periodic modulation. Furthermore, it is shown that there exists a direct connection between the center-of-mass momentum of the FFLO pairs and the wave vector of the optical potential. We propose that this concept can be used experimentally to resonantly enhance the stability region of the FFLO state.
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase was first predicted in two-dimensional superconductors about 50 years ago, but so far unambiguous experimental evidence is still lacking. The recently experimentally realized spin-imbalanced Fermi gases may potentially unveil this elusive state, but they require very stringent experimental conditions. In this Rapid Communication, we show that FFLO phases may be observed even in a three-dimensional (3D) degenerate Fermi gas with spin-orbit coupling and an in-plane Zeeman field. The FFLO phase is driven by the interplay between the asymmetry of the Fermi surface and the superfluid order, instead of the interplay between magnetic and superconducting order in solid materials. The predicted FFLO phase exists in a giant parameter region, possesses a stable long-range superfluid order due to the 3D geometry, and can be observed with an experimentally already achieved temperature ($T\ensuremath{\sim}0.05{E}_{F}$), thus opening a fascinating avenue for exploring FFLO physics.
We show that the combination of spin-orbit coupling and in-plane Zeeman field in a two-dimensional degenerate Fermi gas can lead to a larger parameter region for Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases than that using spin-imbalanced Fermi gases. The resulting FFLO superfluids are also more stable due to the enhanced energy difference between FFLO and conventional Bardeen-Cooper-Schrieffer (BCS) excited states. We clarify the crucial role of the symmetry of Fermi surface on the formation of finite momentum pairing. The phase diagram for FFLO superfluids is obtained in the BCS-BEC crossover region and possible experimental observations of FFLO phases are discussed.
The recent development of experimental techniques in ultracold atomic Fermi gases is extremely helpful in the progress of the realization of the unconventional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase in quasi-one dimensional systems (Liao et al 2010 Nature 467 567). Due to a Fermi surface nesting, which is enhanced in 1D, the low-dimensional systems are particularly good candidates to find the FFLO phase stable. We investigate the influence of a dimensional crossover (from one dimension (1D) to two dimensions (2D) or three dimensions (3D)) on the stability of the FFLO state in the spin-imbalanced attractive Hubbard model.