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arXiv:1308.1304 · Journal of Physics B Atomic Molecular and Optical Physics

Density and spin linear response of atomic Fermi superfluids with population imbalance in the BCS–BEC crossover

Hao Guo, Yang Li, Yan He, Chih-Chun Chien

Published Mar 28, 2014• 4 citations
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Abstract

We present a theoretical study of the density and spin (representing the two components) linear response of Fermi superfluids with tunable attractive interactions and population imbalance. In both linear response theories, we find that the fluctuations of the order parameter must be treated on equal footing with the gauge transformations associated with the symmetries of the Hamiltonian so that important constraints including various sum rules can be satisfied. Both theories can be applied to the whole BCS–Bose–Einstein condensation crossover. The spin linear responses are qualitatively different with and without population imbalance because collective-mode effects from the fluctuations of the order parameter survive in the presence of population imbalance, even though the associated symmetry is not broken by the order parameter. Since a polarized superfluid becomes unstable at low temperatures in the weak and intermediate coupling regimes, we found that the density and spin susceptibilities diverge as the system approaches the unstable regime, but the emergence of phase separation preempts the divergence.

DOI
10.1088/0953-4075/47/8/085302

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We present a theoretical study of the density and spin (representing the two components) linear response of Fermi superfluids with tunable attractive interactions and population imbalance. In both linear response theories, we find that the fluctuations of the order parameter must be treated on equal footing with the gauge transformations associated with the symmetries of the Hamiltonian so that important constraints including various sum rules can be satisfied. Both theories can be applied to the whole BCS–Bose–Einstein condensation crossover. The spin linear responses are qualitatively different with and without population imbalance because collective-mode effects from the fluctuations of the order parameter survive in the presence of population imbalance, even though the associated symmetry is not broken by the order parameter. Since a polarized superfluid becomes unstable at low temperatures in the weak and intermediate coupling regimes, we found that the density and spin susceptibilities diverge as the system approaches the unstable regime, but the emergence of phase separation preempts the divergence.
We investigate a two-component Fermi gas with unequal spin populations along the BCS-BEC crossover. By using the extended BCS equations and the concept of off-diagonal-long-range-order we derive a formula for the condensate number of Cooper pairs as a function of energy gap, average chemical potential, imbalance chemical potential and temperature. Then we study the zero-temperature condensate fraction of Cooper pairs by varying interaction strength and polarization, finding a depletion of the condensate fraction by increasing the population imbalance. We also consider explicitly the presence of an external harmonic confinement and we study, within the local-density approximation, the phase separation between superfluid and normal phase regions of the polarized fermionic cloud. In particular, we calculate both condensate density profiles and total density profiles from the inner superfluid core to the normal region passing for the interface, where a finite jump in the density is a clear manifestation of this phase-separated regime. Finally, we compare our theoretical results with the available experimental data on the condensate fraction of polarized 6Li atoms [Science 311, 492 (2006)]. These experimental data are in reasonable agreement with our predictions in a suitable range of polarizations, but only in the BCS side of the crossover up to unitarity.
Recent experimental progresses allow for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger-than-BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC) crossover, especially below the superfluid transition temperature TcT_cTc​. We adapt a particular ttt-matrix approach which is close to the G0GG_0GG0​G formalism to incorporate non-condensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities, QQQ-limit Ward identity are indeed satisfied.
Recent experimental progress allows for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity, and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger than BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC) crossover, especially below the superfluid transition temperature Tc . We adapt a particular t -matrix approach which is close to the G0G formalism to incorporate noncondensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities and Q -limit Ward identity are indeed satisfied.
We study the finite temperature TTT behavior of trapped Fermi gases as they undergo BCS--Bose-Einstein condensation (BEC) crossover, in the presence of a population imbalance. Our results, in qualitative agreement with recent experiments, show how the superfluid phase transition is directly reflected in the particle density profiles. We demonstrate that at $T\ensuremath{\ne}0$ and in the near-BEC and unitary regimes, the polarization is excluded from the superfluid core. Importantly, a substantial polarization fraction is carried by a normal region of the trap having strong pair correlations, which we associate with noncondensed pairs or the ``pseudogap phase.''
In this paper we present a very general theoretical framework for addressing fermionic superfluids over the entire range of BCS to Bose Einstein condensation (BEC) crossover in the presence of population imbalance or spin polarization. Our emphasis is on providing a theory which reduces to the standard zero temperature mean-field theories in the literature, but necessarily includes pairing fluctuation effects at nonzero temperature within a consistent framework. Physically, these effects are associated with the presence of preformed pairs (or a fermionic pseudogap) in the normal phase, and pair excitations of the condensate, in the superfluid phase. We show how this finite TTT theory of fermionic pair condensates bears many similarities to the condensation of point bosons. In the process we examine three different types of condensate: the usual breached pair or Sarma phase and both the one- and two-plane-wave Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) states. The last of these has been discussed in the literature albeit only within a Landau-Ginzburg formalism, generally valid near Tc{T}_{c}Tc​. Here we show how to arrive at the two-plane-wave LOFF state in the ground state as well as at general temperature TTT.
We study the zero temperature phase diagram of an imbalanced bilayer of dipolar fermions. We consider perpendicularly aligned identical dipoles in two layers and investigate the effect of population imbalance on the ground state phase at different layer spacings and average densities. The attractive part of the interlayer interaction could lead to the BEC-BCS crossover and the Fermi surface mismatch between two layers results in interesting uniform and nonuniform superfluid phases, which we have investigated here using the BCS mean-field theory together with the superfluid-mass density criterion. The density imbalance reduces the pairing gap. At low densities, where the system is on the BEC side of the crossover, this reduction is quite smooth while a dense system rapidly becomes normal at intermediate density polarizations. Stable homogeneous superfluidity is predicted to appear on the phase diagram when the dipolar length exceeds both the layer spacing and the average intralayer distance between dipoles, a regime which should be readily accessible experimentally. This homogeneous superfluid phase becomes unstable at intermediate densities and layer spacings. We have also examined that these uniform and inhomogeneous superfluid phases survive when the effects of intralayer screenings are also incorporated in the formalism.