Strongly interacting Fermi gases are of great interest. Interacting fermions are involved in some... more Strongly interacting Fermi gases are of great interest. Interacting fermions are involved in some of the most important unanswered questions in condensed matter physics, nuclear physics, astrophysics, and cosmology. In 2008, we presented a systematic comparison of strong-coupling theories, predicted the finite-temperature phase diagram of a polarized system and investigated the collective modes in multicomponent attractive Fermi gas .
We theoretically investigate collective modes of a one-dimensional (1D) interacting Bose gas in h... more We theoretically investigate collective modes of a one-dimensional (1D) interacting Bose gas in harmonic traps at finite temperatures, by using a variational approach and local density approximation. We find that the temperature dependence of collective mode frequencies is notably different in the weakly and strongly interacting regimes. Therefore, the experimental measurement of collective modes could provide a sensitive probe for different quantum phases of a 1D trapped Bose gas, realized by tuning the interatomic interaction strength and temperature. Our prediction on the temperature dependence of the breathing mode frequency is in good qualitative agreement with an earlier experimental measurement for a weakly interacting 1D Bose gas of rubidium-87 atoms in harmonic traps [Moritz et al., Phys. Rev. Lett. 91, 250402 (2003)].
We theoretically investigate a three-dimensional Fermi gas with Rashba spin-orbit coupling in the... more We theoretically investigate a three-dimensional Fermi gas with Rashba spin-orbit coupling in the presence of both out-of-plane and in-plane Zeeman fields. We show that, driven by a sufficiently large Zeeman field, either out-of-plane or in-plane, the superfluid phase of this system exhibits a number of interesting features, including inhomogeneous Fulde-Ferrell pairing, gapped or gapless topological order and exotic quasi-particle excitations known as Weyl fermions that have linear energy dispersions in momentum space (i.e., massless Dirac fermions). The topological superfluid phase can have either four or two topologically protected Weyl nodes. We present the phase diagrams at both zero and finite temperatures and discuss the possibility of their observation in an atomic Fermi gas with synthetic spin-orbit coupling. In this context, topological superfluid phases with an imperfect Rashba spin-orbit coupling are also studied.
Topological superfluids usually refer to a superfluid state which is gapped in the bulk but metal... more Topological superfluids usually refer to a superfluid state which is gapped in the bulk but metallic at the boundary. Here we report that a gapless, topologically nontrivial superfluid with an inhomogeneous Fulde-Ferrell pairing order parameter can emerge in a two-dimensional spin-orbit coupled Fermi gas, in the presence of both in-plane and out-of-plane Zeeman fields. The Fulde-Ferrell pairing-induced by the spin-orbit coupling and in-plane Zeeman field-is responsible for this gapless feature. This exotic superfluid has a significant Berezinskii-Kosterlitz-Thouless transition temperature and has robust Majorana edge modes against disorder owing to its topological nature.
We briefly review recent progress on ultracold atomic Fermi gases with different types of synthet... more We briefly review recent progress on ultracold atomic Fermi gases with different types of synthetic spin-orbit coupling, including the one-dimensional (1D) equal weight Rashba-Dresselhaus and twodimensional (2D) Rasbha spin-orbit couplings. Theoretically, we show how the single-body, twobody and many-body properties of Fermi gases are dramatically changed by spin-orbit coupling. In particular, the interplay between spin-orbit coupling and interatomic interaction may lead to several long-sought exotic superfluid phases at low temperatures, such as anisotropic superfluid, topological superfluid and inhomogeneous superfluid. Experimentally, only the first type -equal weight combination of Rasbha and Dresselhaus spin-orbit couplings -has been realized very recently using a two-photon Raman process. We show how to characterize a normal spin-orbit coupled atomic Fermi gas in both non-interacting and strongly-interacting limits, using particularly momentumresolved radio-frequency spectroscopy. The experimental demonstration of a strongly-interacting spin-orbit coupled Fermi gas opens a promising way to observe various exotic superfluid phases in the near future.
Using a variational approach, we present the full solutions of the simplified one-dimensional two... more Using a variational approach, we present the full solutions of the simplified one-dimensional twofluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonic potential, which is recently derived by Stringari and co-workers [Phys. Rev. Lett. 105, 150402 (2010)]. We calculate the discretized mode frequencies of first and second sound along the weak axial trapping potential, as a function of temperature and the form of superfluid density. We show that the density fluctuations in second sound modes, due to their coupling to first sound modes, are large enough to be measured in current experimental setups such as that exploited by Tey et al. at the University of Innsbruck [Phys. Rev. Lett. 110, 055303 (2013)]. Owing to the sensitivity of second sounds on the form of superfluid density, the high precision of the measured second sound frequencies may provide us a promising way to accurately determine the superfluid density of a unitary Fermi gas, which so far remains elusive.
Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is normally described b... more Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is normally described by the atom-molecule theory, which treats the closed channel as a noninteracting point boson. In this work we present a theoretical description of the resonant superfluidity in analogy to the two-band superconductors. We employ the underlying two-channel scattering model of Feshbach resonance where the closed channel is treated as a composite boson with binding energy ε 0 and the resonance is triggered by the microscopic interchannel coupling U 12 . The binding energy ε 0 naturally serves as an energy scale of the system, which has been sent to infinity in the atom-molecule theory. We show that the atom-molecule theory can be viewed as a leading-order low-energy effective theory of the underlying fermionic theory in the limit ε 0 → ∞ and U 12 → 0, while keeping the phenomenological atom-molecule coupling finite. The resulting two-band description of the superfluid state is in analogy to the BCS theory of two-band superconductors. In the dilute limit ε 0 → ∞, the two-band description recovers precisely the atom-molecule theory. The two-band theory provides a natural approach to study the corrections because of a finite binding energy ε 0 in realistic experimental systems. For broad and moderate resonances, the correction is not important for current experimental densities. However, for extremely narrow resonance, we find that the correction becomes significant. The finite binding energy correction could be important for the stability of homogeneous polarized superfluid against phase separation in imbalanced Fermi gases across a narrow Feshbach resonance. PHYSICAL REVIEW A 91, 023622 (2015) Distance Energy open channel closed channel Zeeman splitting binding energy coupling
We report site-resolved radiofrequency spectroscopy measurements of Bose-Einstein condensates of ... more We report site-resolved radiofrequency spectroscopy measurements of Bose-Einstein condensates of 87 Rb atoms in about 100 sites of a one-dimensional 10 µm-period magnetic lattice produced by a grooved magnetic film plus bias fields. Site-to-site variations of the trap bottom, atom temperature, condensate fraction and chemical potential indicate that the magnetic lattice is remarkably uniform, with variations in the trap bottoms of only ±0.4 mG. At the lowest trap frequencies (radial and axial frequencies 1.5 kHz and 260 Hz, respectively), temperatures down to 0.16 µK are achieved in the magnetic lattice and at the smallest trap depths (50 kHz) condensate fractions up to 80% are observed. With increasing radial trap frequency (up to 20 kHz, or aspect ratio up to ∼ 80) large condensate fractions persist and the highly elongated clouds approach the quasi-1D Bose gas regime. The temperature estimated from analysis of the spectra is found to increase by a factor of about five which may be due to suppression of rethermalising collisions in the quasi-1D Bose gas. Measurements for different holding times in the lattice indicate a decay of the atom number with a half-life of about 0.9 s due to three-body losses and the appearance of a high temperature (∼1.5 µK) component which is attributed to atoms that have acquired energy through collisions with energetic three-body decay products.
We calculate the frequency of collective modes of a one-dimensional repulsively interacting Fermi... more We calculate the frequency of collective modes of a one-dimensional repulsively interacting Fermi gas with high-spin symmetry confined in harmonic traps at zero temperature. This is a system realizable with fermionic alkaline-earth-metal atoms such as 173 Yb, which displays an exact SU(κ) spin symmetry with κ 2 and behaves like a spinless interacting Bose gas in the limit of infinite spin components κ → ∞, namely high-spin bosonization. We solve the homogeneous equation of state of the high-spin Fermi system by using Bethe ansatz technique and obtain the density distribution in harmonic traps based on local density approximation. The frequency of collective modes is calculated by exactly solving the zero-temperature hydrodynamic equation. In the limit of large number of spin-components, we show that the mode frequency of the system approaches to that of a one-dimensional spinless interacting Bose gas, as a result of high-spin bosonization. Our prediction of collective modes is in excellent agreement with a very recent measurement for a Fermi gas of 173 Yb atoms with tunable spin confined in a two-dimensional tight optical lattice.
We theoretically investigate first and second sound of a two-dimensional (2D) atomic Bose gas in ... more We theoretically investigate first and second sound of a two-dimensional (2D) atomic Bose gas in harmonic traps by solving Landau's two-fluid hydrodynamic equations. For an isotropic trap, we find that first and second sound modes become degenerate at certain temperatures and exhibit typical avoided crossings in mode frequencies. At these temperatures, second sound has significant density fluctuation due to its hybridization with first sound and has a divergent mode frequency towards the Berezinskii-Kosterlitz-Thouless (BKT) transition. For a highly anisotropic trap, we derive the simplified one-dimensional hydrodynamic equations and discuss the sound-wave propagation along the weakly confined direction. Due to the universal jump of the superfluid density inherent to the BKT transition, we show that the first sound velocity exhibits a kink across the transition. Our predictions can be readily examined in current experimental setups for 2D dilute Bose gases. PACS numbers: 67.85.De, 03.75.Kk, 05.30.Jp
Strongly interacting Fermi gases are of great interest. Interacting fermions are involved in some... more Strongly interacting Fermi gases are of great interest. Interacting fermions are involved in some of the most important unanswered questions in condensed matter physics, nuclear physics, astrophysics, and cosmology. In 2008, we presented a systematic comparison of strong-coupling theories, predicted the finite-temperature phase diagram of a polarized system and investigated the collective modes in multicomponent attractive Fermi gas .
We theoretically investigate collective modes of a one-dimensional (1D) interacting Bose gas in h... more We theoretically investigate collective modes of a one-dimensional (1D) interacting Bose gas in harmonic traps at finite temperatures, by using a variational approach and local density approximation. We find that the temperature dependence of collective mode frequencies is notably different in the weakly and strongly interacting regimes. Therefore, the experimental measurement of collective modes could provide a sensitive probe for different quantum phases of a 1D trapped Bose gas, realized by tuning the interatomic interaction strength and temperature. Our prediction on the temperature dependence of the breathing mode frequency is in good qualitative agreement with an earlier experimental measurement for a weakly interacting 1D Bose gas of rubidium-87 atoms in harmonic traps [Moritz et al., Phys. Rev. Lett. 91, 250402 (2003)].
We theoretically investigate a three-dimensional Fermi gas with Rashba spin-orbit coupling in the... more We theoretically investigate a three-dimensional Fermi gas with Rashba spin-orbit coupling in the presence of both out-of-plane and in-plane Zeeman fields. We show that, driven by a sufficiently large Zeeman field, either out-of-plane or in-plane, the superfluid phase of this system exhibits a number of interesting features, including inhomogeneous Fulde-Ferrell pairing, gapped or gapless topological order and exotic quasi-particle excitations known as Weyl fermions that have linear energy dispersions in momentum space (i.e., massless Dirac fermions). The topological superfluid phase can have either four or two topologically protected Weyl nodes. We present the phase diagrams at both zero and finite temperatures and discuss the possibility of their observation in an atomic Fermi gas with synthetic spin-orbit coupling. In this context, topological superfluid phases with an imperfect Rashba spin-orbit coupling are also studied.
Topological superfluids usually refer to a superfluid state which is gapped in the bulk but metal... more Topological superfluids usually refer to a superfluid state which is gapped in the bulk but metallic at the boundary. Here we report that a gapless, topologically nontrivial superfluid with an inhomogeneous Fulde-Ferrell pairing order parameter can emerge in a two-dimensional spin-orbit coupled Fermi gas, in the presence of both in-plane and out-of-plane Zeeman fields. The Fulde-Ferrell pairing-induced by the spin-orbit coupling and in-plane Zeeman field-is responsible for this gapless feature. This exotic superfluid has a significant Berezinskii-Kosterlitz-Thouless transition temperature and has robust Majorana edge modes against disorder owing to its topological nature.
We briefly review recent progress on ultracold atomic Fermi gases with different types of synthet... more We briefly review recent progress on ultracold atomic Fermi gases with different types of synthetic spin-orbit coupling, including the one-dimensional (1D) equal weight Rashba-Dresselhaus and twodimensional (2D) Rasbha spin-orbit couplings. Theoretically, we show how the single-body, twobody and many-body properties of Fermi gases are dramatically changed by spin-orbit coupling. In particular, the interplay between spin-orbit coupling and interatomic interaction may lead to several long-sought exotic superfluid phases at low temperatures, such as anisotropic superfluid, topological superfluid and inhomogeneous superfluid. Experimentally, only the first type -equal weight combination of Rasbha and Dresselhaus spin-orbit couplings -has been realized very recently using a two-photon Raman process. We show how to characterize a normal spin-orbit coupled atomic Fermi gas in both non-interacting and strongly-interacting limits, using particularly momentumresolved radio-frequency spectroscopy. The experimental demonstration of a strongly-interacting spin-orbit coupled Fermi gas opens a promising way to observe various exotic superfluid phases in the near future.
Using a variational approach, we present the full solutions of the simplified one-dimensional two... more Using a variational approach, we present the full solutions of the simplified one-dimensional twofluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonic potential, which is recently derived by Stringari and co-workers [Phys. Rev. Lett. 105, 150402 (2010)]. We calculate the discretized mode frequencies of first and second sound along the weak axial trapping potential, as a function of temperature and the form of superfluid density. We show that the density fluctuations in second sound modes, due to their coupling to first sound modes, are large enough to be measured in current experimental setups such as that exploited by Tey et al. at the University of Innsbruck [Phys. Rev. Lett. 110, 055303 (2013)]. Owing to the sensitivity of second sounds on the form of superfluid density, the high precision of the measured second sound frequencies may provide us a promising way to accurately determine the superfluid density of a unitary Fermi gas, which so far remains elusive.
Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is normally described b... more Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is normally described by the atom-molecule theory, which treats the closed channel as a noninteracting point boson. In this work we present a theoretical description of the resonant superfluidity in analogy to the two-band superconductors. We employ the underlying two-channel scattering model of Feshbach resonance where the closed channel is treated as a composite boson with binding energy ε 0 and the resonance is triggered by the microscopic interchannel coupling U 12 . The binding energy ε 0 naturally serves as an energy scale of the system, which has been sent to infinity in the atom-molecule theory. We show that the atom-molecule theory can be viewed as a leading-order low-energy effective theory of the underlying fermionic theory in the limit ε 0 → ∞ and U 12 → 0, while keeping the phenomenological atom-molecule coupling finite. The resulting two-band description of the superfluid state is in analogy to the BCS theory of two-band superconductors. In the dilute limit ε 0 → ∞, the two-band description recovers precisely the atom-molecule theory. The two-band theory provides a natural approach to study the corrections because of a finite binding energy ε 0 in realistic experimental systems. For broad and moderate resonances, the correction is not important for current experimental densities. However, for extremely narrow resonance, we find that the correction becomes significant. The finite binding energy correction could be important for the stability of homogeneous polarized superfluid against phase separation in imbalanced Fermi gases across a narrow Feshbach resonance. PHYSICAL REVIEW A 91, 023622 (2015) Distance Energy open channel closed channel Zeeman splitting binding energy coupling
We report site-resolved radiofrequency spectroscopy measurements of Bose-Einstein condensates of ... more We report site-resolved radiofrequency spectroscopy measurements of Bose-Einstein condensates of 87 Rb atoms in about 100 sites of a one-dimensional 10 µm-period magnetic lattice produced by a grooved magnetic film plus bias fields. Site-to-site variations of the trap bottom, atom temperature, condensate fraction and chemical potential indicate that the magnetic lattice is remarkably uniform, with variations in the trap bottoms of only ±0.4 mG. At the lowest trap frequencies (radial and axial frequencies 1.5 kHz and 260 Hz, respectively), temperatures down to 0.16 µK are achieved in the magnetic lattice and at the smallest trap depths (50 kHz) condensate fractions up to 80% are observed. With increasing radial trap frequency (up to 20 kHz, or aspect ratio up to ∼ 80) large condensate fractions persist and the highly elongated clouds approach the quasi-1D Bose gas regime. The temperature estimated from analysis of the spectra is found to increase by a factor of about five which may be due to suppression of rethermalising collisions in the quasi-1D Bose gas. Measurements for different holding times in the lattice indicate a decay of the atom number with a half-life of about 0.9 s due to three-body losses and the appearance of a high temperature (∼1.5 µK) component which is attributed to atoms that have acquired energy through collisions with energetic three-body decay products.
We calculate the frequency of collective modes of a one-dimensional repulsively interacting Fermi... more We calculate the frequency of collective modes of a one-dimensional repulsively interacting Fermi gas with high-spin symmetry confined in harmonic traps at zero temperature. This is a system realizable with fermionic alkaline-earth-metal atoms such as 173 Yb, which displays an exact SU(κ) spin symmetry with κ 2 and behaves like a spinless interacting Bose gas in the limit of infinite spin components κ → ∞, namely high-spin bosonization. We solve the homogeneous equation of state of the high-spin Fermi system by using Bethe ansatz technique and obtain the density distribution in harmonic traps based on local density approximation. The frequency of collective modes is calculated by exactly solving the zero-temperature hydrodynamic equation. In the limit of large number of spin-components, we show that the mode frequency of the system approaches to that of a one-dimensional spinless interacting Bose gas, as a result of high-spin bosonization. Our prediction of collective modes is in excellent agreement with a very recent measurement for a Fermi gas of 173 Yb atoms with tunable spin confined in a two-dimensional tight optical lattice.
We theoretically investigate first and second sound of a two-dimensional (2D) atomic Bose gas in ... more We theoretically investigate first and second sound of a two-dimensional (2D) atomic Bose gas in harmonic traps by solving Landau's two-fluid hydrodynamic equations. For an isotropic trap, we find that first and second sound modes become degenerate at certain temperatures and exhibit typical avoided crossings in mode frequencies. At these temperatures, second sound has significant density fluctuation due to its hybridization with first sound and has a divergent mode frequency towards the Berezinskii-Kosterlitz-Thouless (BKT) transition. For a highly anisotropic trap, we derive the simplified one-dimensional hydrodynamic equations and discuss the sound-wave propagation along the weakly confined direction. Due to the universal jump of the superfluid density inherent to the BKT transition, we show that the first sound velocity exhibits a kink across the transition. Our predictions can be readily examined in current experimental setups for 2D dilute Bose gases. PACS numbers: 67.85.De, 03.75.Kk, 05.30.Jp
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Papers by Xia-ji Liu