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Exotic Superfluidity in Imbalanced Fermi Mixtures

K. B. Gubbels

Published Feb 1, 2006• 0 citations
OpenAlex

Abstract

We consider ultracold imbalanced Fermi gases with strong attractive interactions, for which Cooper-pair formation plays an important role. The two-component mixtures consist either of identical fermionic atoms in two different hyperfine states, or of two different atomic species. In both cases, the number of atoms for each component is allowed to be different, leading to a spin imbalance or a polarization. In the second case, the two different species also give rise to a mass imbalance for the particles. Imbalanced Fermi mixtures play a fundamental role in condensed-matter, nuclear and astroparticle physics. Experimentally, the atomic mixtures are made strongly interacting with the use of two-channel Feshbach resonances. We start out by treating these in more detail. We review the relevant scattering theory, and apply it to the well-studied case of resonant s-wave collisions with zero angular momentum. Furthermore, we discuss the less-studied case of p-wave collisions with nonzero angular momentum, and the inhomogeneous case due to a trapping potential. Having discussed the two-body physics, we turn to the many-body physics. The pioneering experimental studies of Cooper pairing in lithium-6 mixtures with a spin imbalance were performed by Zwierlein et al. [Science 311, 492 (2006)] and Partridge et al. [Science 311, 503 (2006)]. We study the mean-field theory for the polarized Fermi mixture to obtain a qualitative understanding of these experiments. In the unitarity limit, where the scattering length of the interatomic interaction diverges, the theory gives rise to two exotic superfluid phases that are absent in the balanced mixture. These are the gapless superfluid Sarma phase and the phase-separated phase. By using renormalization group techniques to account for the effects of strong interactions, we obtain also quantitative agreement with Monte-Carlo calculations and experiments. We then generalize our knowledge of the solely spin-imbalanced mixture to the mass-imbalanced case, where in particular the mixture consisting of lithium-6 and potassium-40 is experimentally promising. We study the mean-field phase diagram for this mixture in the unitarity limit, where we not only find phase separation and Sarma superfluidity, but also a Lifshitz point. The latter signals an instability towards a supersolid phase. We also include the effects of fluctuations, which do not alter the topology of the phase diagram. However, fluctuation effects lower the critical temperatures with an experimentally relevant factor of three. Finally, we study superfluidity in a fully polarized gas of potassium-40 atoms near a p-wave Feshbach resonance. In particular, we consider coherent Josephson oscillations between the superfluid components in the two channels of the resonance, whose frequencies contain a clear signature of the quantum phase transition that occurs as a function of applied magnetic field.

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We consider imbalanced Fermi gases with strong attractive interactions, for which Cooper-pair formation plays an important role. The two-component mix-tures consist either of identical fermionic atoms in two different hyperfine states, or of two different atomic species both occupying only a single hyperfine state. In both cases, the number of atoms for each component is allowed to be differ-ent, which leads to a spin imbalance, or spin polarization. Two different atomic species also lead to a mass imbalance. Imbalanced Fermi gases are relevant to condensed-matter physics, nuclear physics and astroparticle physics. They have been studied intensively in recent years, following their experimental realization in ultracold atomic Fermi gases. The experimental control in such a system allows for a systematic study of the equation of state and the phase diagram as a function of temperature, spin polarization and interaction strength. In this re-view, we discuss the progress in understanding strongly-interacting imbalanced Fermi gases, where a main goal is to describe the results of the highly controlled experiments. We start by discussing Feshbach resonances, after which we treat the imbalanced Fermi gas in mean-field theory to give an introduction to the relevant physics. We encounter several unusual superfluid phases, including phase-separation, gapless Sarma superfluidity, and supersolidity. To obtain a more quantitative description of the experiments, we review also more sophisti-cated techniques, such as diagrammatic methods and the renormalization-group theory. We end the review by discussing two theoretical approaches to treat the inhomogeneous imbalanced Fermi gas, namely the Landau-Ginzburg theory and the Bogoliubov-de Gennes approach. Key words:

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In this thesis, we study two-component Fermi mixtures in the presence of both a mass and a population imbalance. For a large part we focus on the experimentally available 6Li-40K mixture and on the phase transitions that can occur in this mixture. In the first part of this thesis, we study the extremely polarized two-component Fermi gas with a mass imbalance in the strongly interacting regime. Specifically, we focus on the mixture of 6Li and 40K atoms. In this interaction regime spin polarons, i.e., dressed minority atoms, form. We consider the spectral function for the minority atoms, from which the lifetime and the effective mass of the spin polaron can be determined. Moreover, we predict the radio-frequency spectrum and the momentum distribution for the spin polarons for experiments with 6Li and 40K atoms. Subsequently we study the relaxation of the motion of the Fermi polaron due to spin drag. In the second part, we develop an accurate theory of resonantly interacting Fermi mixtures with both spin and mass imbalance. We consider Fermi mixtures with arbitrary mass imbalances, but focus in particular on the 6Li-40K mixture. We determine the phase diagram of the mixture for different interaction strengths that lie on the BCS side of the Feshbach resonance. We also determine the universal phase diagram at unitarity. We find for the mixtures with a sufficiently large mass imbalance, that includes the 6Li-40K mixture, a Lifshitz point in the universal phase diagram that signals an instability towards a supersolid phase. In the last part of the thesis, we show that the ultracold three-dimensional 6Li-40K mixture at unitarity can exhibit the highly exotic Larkin-Ovchinnikov superfluid phase. We determine the phase diagram for majorities of 40K atoms within mean-field theory taking the inhomogeneities of the fermion states into account exactly. We find two different inhomogeneous superfluid phases in mixtures with a majority of 40K atoms, namely the Larkin-Ovchinnikov (LO) phase with one inhomogeneous direction and a cubic phase (LO3) where three spatial translational symmetries are broken. We determine the transition between these two phases by solving the Bogoliubov-de Gennes equations in the superfluid LO phase. Subsequently, we calculate the atomic density modulation of the atoms in the LO phase and show that it is sufficiently large to be visible in experiment.

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Population and mass imbalance in atomic Fermi gases

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We develop an accurate theory of resonantly interacting Fermi mixtures with both spin and mass imbalance. We consider Fermi mixtures with arbitrary mass imbalances but focus, in particular, on the experimentally available $^{6}\mathrm{Li}$-$^{40}\mathrm{K}$ mixture. We determine the phase diagram of the mixture for different interactions strengths that lie on the BCS side of the Feshbach resonance. We also determine the universal phase diagram at unitarity. We find, for the mixtures with a sufficiently large mass imbalance, which includes the $^{6}\mathrm{Li}$-$^{40}\mathrm{K}$ mixture, a Lifshitz point in the universal phase diagram that signals an instability toward a supersolid phase.

arXiv:0912.4205•Physical Review A•Jul 19, 2010•62 citations
We consider ultracold imbalanced Fermi gases with strong attractive interactions, for which Cooper-pair formation plays an important role. The two-component mixtures consist either of identical fermionic atoms in two different hyperfine states, or of two different atomic species. In both cases, the number of atoms for each component is allowed to be different, leading to a spin imbalance or a polarization. In the second case, the two different species also give rise to a mass imbalance for the particles. Imbalanced Fermi mixtures play a fundamental role in condensed-matter, nuclear and astroparticle physics. Experimentally, the atomic mixtures are made strongly interacting with the use of two-channel Feshbach resonances. We start out by treating these in more detail. We review the relevant scattering theory, and apply it to the well-studied case of resonant s-wave collisions with zero angular momentum. Furthermore, we discuss the less-studied case of p-wave collisions with nonzero angular momentum, and the inhomogeneous case due to a trapping potential. Having discussed the two-body physics, we turn to the many-body physics. The pioneering experimental studies of Cooper pairing in lithium-6 mixtures with a spin imbalance were performed by Zwierlein et al. [Science 311, 492 (2006)] and Partridge et al. [Science 311, 503 (2006)]. We study the mean-field theory for the polarized Fermi mixture to obtain a qualitative understanding of these experiments. In the unitarity limit, where the scattering length of the interatomic interaction diverges, the theory gives rise to two exotic superfluid phases that are absent in the balanced mixture. These are the gapless superfluid Sarma phase and the phase-separated phase. By using renormalization group techniques to account for the effects of strong interactions, we obtain also quantitative agreement with Monte-Carlo calculations and experiments. We then generalize our knowledge of the solely spin-imbalanced mixture to the mass-imbalanced case, where in particular the mixture consisting of lithium-6 and potassium-40 is experimentally promising. We study the mean-field phase diagram for this mixture in the unitarity limit, where we not only find phase separation and Sarma superfluidity, but also a Lifshitz point. The latter signals an instability towards a supersolid phase. We also include the effects of fluctuations, which do not alter the topology of the phase diagram. However, fluctuation effects lower the critical temperatures with an experimentally relevant factor of three. Finally, we study superfluidity in a fully polarized gas of potassium-40 atoms near a p-wave Feshbach resonance. In particular, we consider coherent Josephson oscillations between the superfluid components in the two channels of the resonance, whose frequencies contain a clear signature of the quantum phase transition that occurs as a function of applied magnetic field.
We consider imbalanced Fermi gases with strong attractive interactions, for which Cooper-pair formation plays an important role. The two-component mix-tures consist either of identical fermionic atoms in two different hyperfine states, or of two different atomic species both occupying only a single hyperfine state. In both cases, the number of atoms for each component is allowed to be differ-ent, which leads to a spin imbalance, or spin polarization. Two different atomic species also lead to a mass imbalance. Imbalanced Fermi gases are relevant to condensed-matter physics, nuclear physics and astroparticle physics. They have been studied intensively in recent years, following their experimental realization in ultracold atomic Fermi gases. The experimental control in such a system allows for a systematic study of the equation of state and the phase diagram as a function of temperature, spin polarization and interaction strength. In this re-view, we discuss the progress in understanding strongly-interacting imbalanced Fermi gases, where a main goal is to describe the results of the highly controlled experiments. We start by discussing Feshbach resonances, after which we treat the imbalanced Fermi gas in mean-field theory to give an introduction to the relevant physics. We encounter several unusual superfluid phases, including phase-separation, gapless Sarma superfluidity, and supersolidity. To obtain a more quantitative description of the experiments, we review also more sophisti-cated techniques, such as diagrammatic methods and the renormalization-group theory. We end the review by discussing two theoretical approaches to treat the inhomogeneous imbalanced Fermi gas, namely the Landau-Ginzburg theory and the Bogoliubov-de Gennes approach. Key words:
In this thesis, we study two-component Fermi mixtures in the presence of both a mass and a population imbalance. For a large part we focus on the experimentally available 6Li-40K mixture and on the phase transitions that can occur in this mixture. In the first part of this thesis, we study the extremely polarized two-component Fermi gas with a mass imbalance in the strongly interacting regime. Specifically, we focus on the mixture of 6Li and 40K atoms. In this interaction regime spin polarons, i.e., dressed minority atoms, form. We consider the spectral function for the minority atoms, from which the lifetime and the effective mass of the spin polaron can be determined. Moreover, we predict the radio-frequency spectrum and the momentum distribution for the spin polarons for experiments with 6Li and 40K atoms. Subsequently we study the relaxation of the motion of the Fermi polaron due to spin drag. In the second part, we develop an accurate theory of resonantly interacting Fermi mixtures with both spin and mass imbalance. We consider Fermi mixtures with arbitrary mass imbalances, but focus in particular on the 6Li-40K mixture. We determine the phase diagram of the mixture for different interaction strengths that lie on the BCS side of the Feshbach resonance. We also determine the universal phase diagram at unitarity. We find for the mixtures with a sufficiently large mass imbalance, that includes the 6Li-40K mixture, a Lifshitz point in the universal phase diagram that signals an instability towards a supersolid phase. In the last part of the thesis, we show that the ultracold three-dimensional 6Li-40K mixture at unitarity can exhibit the highly exotic Larkin-Ovchinnikov superfluid phase. We determine the phase diagram for majorities of 40K atoms within mean-field theory taking the inhomogeneities of the fermion states into account exactly. We find two different inhomogeneous superfluid phases in mixtures with a majority of 40K atoms, namely the Larkin-Ovchinnikov (LO) phase with one inhomogeneous direction and a cubic phase (LO3) where three spatial translational symmetries are broken. We determine the transition between these two phases by solving the Bogoliubov-de Gennes equations in the superfluid LO phase. Subsequently, we calculate the atomic density modulation of the atoms in the LO phase and show that it is sufficiently large to be visible in experiment.
The work described in this thesis can be divided into two parts. The first part is concerned with the production of degenerate Fermi gases and Bose-Fermi mixtures. The second part of the thesis describes studies of strongly interacting spin-mixtures of ultracold fermions. We have cooled a magnetically trapped spin-polarized 6Li Fermi gas into quantum degeneracy by sympathetic cooling with bosonic 23Na. In order to do this, we have converted our previously developed apparatus for Bose-Einstein condensation of 23Na into a two-species cooling facility. We have produced 6Li samples with up to 7 × 107 atoms at temperatures below 0.5TF , and up to 3× 107 atoms at temperatures as low as 0.05TF . Further, we have also produced stable quantum degenerate mixtures of 23Na and 6Li, with several million atoms of each species. In a second set of experiments, we have studied strongly interacting spin-mixtures of 6Li. The strength of interactions was resonantly controlled by applying a magnetic field and tuning to “Feshbach resonances”, which occur when the total energy of the colliding particles coincides with the energy of a bound molecular state. These studies are relevant for the continuing quest for a superfluid transition in a Fermi gas. By studying the inelastic decay of the cloud, we have experimentally identified two Feshbach resonances for the inter-state interactions of the two lowest ground states of 6Li. Most importantly, we found that the lifetime of the strongly interacting system is long enough to observe a superfluid transition in the future. The effects of the coherent collisions in 6Li spin-mixtures were studied using radiofrequency (RF) spectroscopy. We demonstrated the absence of mean field “clock shifts” in a two-state mixture of fermions, and have introduced a three-level method to observe the interaction energy shifts. In the strongly interacting regime, near a Feshbach resonance, we have observed universal behavior of the interaction energy, independent of the value of the scattering length. Finally, we have theoretically studied the effects of Pauli blocking during a free expansion of a degenerate Fermi gas. We have concluded that a strongly interacting normal gas will undergo a hydrodynamic expansion even at zero temperature, due to inefficiency of the Pauli suppression of collisions. Therefore, the already observed hydrodynamic expansion cannot be conclusively interpreted as a signature of superfluidity. Thesis Supervisor: Wolfgang Ketterle Title: John D. MacArthur Professor of Physics
The topic of the present lecture notes are two-species quantum mixtures composed of a deeply degenerate Fermi gas and a second component, the latter being fermionic or bosonic. A key ingredient is the possibility to tune the sss-wave interaction between the different species by means of magnetically controlled Feshbach resonances, which allow us to investigate regimes of strong interactions. In two case studies, we review our experiments on mixtures of 6^66Li fermions with fermionic 40^{40}40K or bosonic 41^{41}41K atoms and on mixtures of fermionic 161^{161}161Dy with 40^{40}40K atoms. We cover various topics of fermionic quantum many-body physics, ranging from impurity physics and quasiparticles over phase separation to the formation of ultracold molecules and progress towards novel superfluids.
We present a detailed study of the population imbalanced three-component Hubbard chain with attractive interactions. Such a system can be realized experimentally with three different hyperfine states of ultra cold 6^66Li atoms in an optical lattice. We find that there are different phases that compete with each other in this system: A molecular superfluid phase in which the three fermion species pair up to form molecules (trions), a usual pairing phase involving two species with exactly opposite momenta, and a more exotic generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase consisting of three competing pairing tendencies with different non-zero center-of-mass momenta. At large attractive interactions the system exhibits strong tendencies towards collapse and phase separation. Employing the density-matrix-renormalization-group-method (DMRG) to determine the decay exponents of the various correlators we establish the phase diagram of this model for different fillings and interactions. We also discuss the experimentally relevant situation in a trap and report the existence of an additional region where two species are dynamically balanced.
We develop an accurate theory of resonantly interacting Fermi mixtures with both spin and mass imbalance. We consider Fermi mixtures with arbitrary mass imbalances but focus, in particular, on the experimentally available 6Li^{6}\mathrm{Li}6Li-40K^{40}\mathrm{K}40K mixture. We determine the phase diagram of the mixture for different interactions strengths that lie on the BCS side of the Feshbach resonance. We also determine the universal phase diagram at unitarity. We find, for the mixtures with a sufficiently large mass imbalance, which includes the 6Li^{6}\mathrm{Li}6Li-40K^{40}\mathrm{K}40K mixture, a Lifshitz point in the universal phase diagram that signals an instability toward a supersolid phase.