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.
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 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.
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.
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.
In three-dimensional Fermi gases with spin imbalance, a competition exists between Cooper pairing with zero and finite momentum. The latter gives rise to the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superfluid phase, which only exists in a restricted area of the phase diagram as a function of chemical potential imbalance and interaction strength. Applying an optical potential along one direction enhances the FFLO region in this phase diagram. In this paper, we construct the phase diagram as a function of polarization and interaction strength in order to study the competition between the FFLO phase and the spin balanced Bardeen–Cooper–Schrieffer (BCS) phase. This allows us to take into account the region of phase separation, and provides a more direct connection with experiment. Subsequently, we investigate the effects of the wavelength and the depth of the optical potential, which is applied along one direction, on the FFLO state. It is shown that the FFLO state can exist up to a higher level of spin imbalance if the wavelength of the optical potential becomes smaller. Our results give rise to an interesting effect: the maximal polarization at which the FFLO state can exist decreases when the interaction strength exceeds a certain critical value. This counterintuitive phenomenon is discussed and the connection to the optical potential is explained.
Abstract Ultracold two-component Fermi gases with a tunable population imbalance have provided an excellent opportunity for studying the exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, which have been of great interest in condensed matter physics. However, the FFLO states have not been observed experimentally in Fermi gases in three dimensions (3D), possibly due to their small phase space volume and extremely low temperature required for an equal-mass Fermi gas. Here we explore possible effects of mass imbalance, mainly in a 6Li–40K mixture, on the one-plane-wave FFLO phases for a 3D homogeneous case at the mean-field level. We present various phase diagrams related to the FFLO states at both zero and finite temperatures, throughout the BCS-BEC crossover, and show that a large mass ratio may enhance substantially FFLO type of pairing.
We study the phase diagram of an imbalanced two-component Fermi gas in optical lattices of 1–3 dimensions (1D–3D), considering the possibilities of the Fulde–Ferrel–Larkin–Ovchinnikov (FFLO), Sarma/breached pair, BCS and normal states as well as phase separation, at finite and zero temperatures. In particular, phase diagrams with respect to average chemical potential and the chemical potential difference of the two components are considered, because this gives the essential information about the shell structures of phases that will occur in the presence of an additional (harmonic) confinement. These phase diagrams in 1D, 2D and 3D show in a striking way the effect of Van Hove singularities on the FFLO state. Although we focus on population imbalanced gases, the results are relevant also for the (effective) mass imbalanced case. We demonstrate by LDA calculations that various shell structures such as normal–FFLO–BCS–FFLO–normal, or FFLO–normal, are possible in presence of a background harmonic trap. The phases are reflected in noise correlations: especially in 1D the unpaired atoms leave a clear signature of the FFLO state as a zero-correlation area ('breach') within the Fermi sea. This strong signature occurs both for a 1D lattice as well as for a 1D continuum. We also discuss the effect of Hartree energies and the Gorkov correction on the phase diagrams.