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arXiv:0705.3361 · Journal of the Physical Society of Japan

Imbalanced Superfluid Phase of a Trapped Fermi Gas in the BCS–BEC Crossover Regime

Takeshi Mizushima, Masanori Ichioka, Kazushige Machida

Published Oct 10, 2007• 34 citations

Abstract

We theoretically investigate the ground state of trapped neutral fermions with population imbalance in the BCS-BEC crossover regime. On the basis of the single-channel Hamiltonian, we perform full numerical calculations of the Bogoliubov-de Gennes equation coupled with the regularized gap and number equations. The zero-temperature phase diagram in the crossover regime is presented, where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state governs the weak-coupling BCS region of a resonance. It is found that the FFLO oscillation vanishes in the BEC side, in which the system under population imbalance turns into a phase separation (PS) between locally binding superfluid and fully polarized spin domains. We also demonstrate numerical calculations with a large particle number O(10^5), comparable to that observed in recent experiments. The resulting density profile on a resonance yields the PS, which is in good agreement with the recent experiments, while the FFLO modulation exists in the pairing field. It is also proposed that the most favorable location for the detection of the FFLO oscillation is in the vicinity of the critical population imbalance in the weak coupling BCS regime, where the oscillation periodicity becomes much larger than the interparticle spacing. Finally, we analyze the radio-frequency (RF) spectroscopy in the imbalanced system. The clear difference in the RF spectroscopy between BCS and BEC sides reveals the structure of the pairing field and local ``magnetization''.

DOI: 10.1143/jpsj.76.104006

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We theoretically investigate the ground state of trapped neutral fermions with population imbalance in the BCS-BEC crossover regime. On the basis of the single-channel Hamiltonian, we perform full numerical calculations of the Bogoliubov-de Gennes equation coupled with the regularized gap and number equations. The zero-temperature phase diagram in the crossover regime is presented, where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state governs the weak-coupling BCS region of a resonance. It is found that the FFLO oscillation vanishes in the BEC side, in which the system under population imbalance turns into a phase separation (PS) between locally binding superfluid and fully polarized spin domains. We also demonstrate numerical calculations with a large particle number O(10^5), comparable to that observed in recent experiments. The resulting density profile on a resonance yields the PS, which is in good agreement with the recent experiments, while the FFLO modulation exists in the pairing field. It is also proposed that the most favorable location for the detection of the FFLO oscillation is in the vicinity of the critical population imbalance in the weak coupling BCS regime, where the oscillation periodicity becomes much larger than the interparticle spacing. Finally, we analyze the radio-frequency (RF) spectroscopy in the imbalanced system. The clear difference in the RF spectroscopy between BCS and BEC sides reveals the structure of the pairing field and local ``magnetization''.

We explore the zero-temperature phase behavior of a two-dimensional two-component atomic Fermi gas with population and mass imbalance in the regime of the BEC-BCS crossover. Working in the mean-field approximation, we show that the normal and homogeneous balanced superfluid phases are separated by an inhomogeneous superfluid phase of Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) type. We obtain an analytical expression for the line of continuous transitions separating the normal and inhomogeneous FFLO phases. We further show that the transition from the FFLO phase to the homogeneous balanced superfluid is discontinuous leading to phase separation. If the species have different masses, the superfluid phase is favored when the lighter species is in excess. We explore the implications of these findings for the properties of the two-component Fermi gas in the atomic trap geometry. Finally, we compare and contrast our findings with the predicted phase behavior of the electron-hole bilayer system.

We study the angular Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in which the rotation symmetry is spontaneously broken, in population imbalanced fermion gases near the BCS-BEC crossover. We investigate the superfluid gases at low temperatures on the basis of the Bogoliubov--de Gennes equation and examine the stability against thermal fluctuations using the

T

-matrix approach beyond the local-density approximation. We find that the angular FFLO state is stabilized in the gases confined in the toroidal trap but not in the harmonic trap. The angular FFLO state is stable near the BCS-BEC crossover owing to the formation of pseudogap. Spatial dependences of number density and local population imbalance are shown for an experimental test.

We study

s

-wave superconductivity in the two-dimensional attractive Hubbard model in an applied magnetic field, assume the extreme Pauli limit, and examine the role of spatial fluctuations in the coupling regime corresponding to BCS-BEC crossover. We use a decomposition of the interaction in terms of an auxiliary pairing field, retain the static mode, and sample the pairing field via a Monte Carlo approach. The method requires iterative solution of the Bogoliubov--de-Gennes equations for amplitude- and phase-fluctuating configurations of the pairing field. We establish the full thermal phase diagram of this strong-coupling problem. At low field we observe the magnetized but homogeneous ``breached pair'' superfluid phase. It reveals that

{T}_{c}

scales an order of magnitude below the mean-field estimate, spontaneous inhomogeneity in the field-induced magnetization, and a strong nonmonotonicity in the temperature dependence of the low-energy density of states. We compare our results to the experimental phase diagram of the imbalanced Fermi gas at unitarity. At higher field we obtain the modulated Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases. The thermal transition from the FFLO phases to the normal state is strongly first order. We track the fermionic momentum distribution, the density of states, and the pairing structure factor deep into the normal state. The pairing structure factor retains weak signature of finite momentum pairing to a high temperature despite the low

{T}_{c}

itself, while the spin-resolved density of states changes from the ``pseudogapped'' FFLO character to gapless and pseudogapped again with increasing temperature.

We present a numerical study of the one-dimensional BCS-BEC crossover of a spin-imbalanced Fermi gas. The crossover is described by the Bose-Fermi resonance model in a real space representation. Our main interest is in the behavior of the pair correlations, which, in the BCS limit, are of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type, while in the Bose-Einstein condensate limit, a superfluid of diatomic molecules forms that exhibits quasi-condensation at zero momentum. We use the density matrix renormalization group method to compute the phase diagram as a function of the detuning of the molecular level and the polarization. As a main result, we show that FFLO-like correlations disappear well below full polarization close to the resonance. The critical polarization depends on both the detuning and the filling.

It is shown by microscopic calculations for trapped imbalanced Fermi superfluids that the gap function always has sign changes, i.e., the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state, up to a critical imbalance P(c), beyond which normal state becomes stable, at temperature T=0. A temperature-versus-pressure phase diagram is constructed, where the BCS state without sign change is stable only at T not equal to 0. We reproduce the observed bimodality in the density profile to identify its origin and evaluate P(c) as functions of T and the coupling strength. These dependencies match with the recent experiments.

By introducing spatially varying profiles of pairing interaction or spin polarization to quasi one-dimensional two-component atomic Fermi gases confined in box potentials, we analyze the ground state structures and properties when multiple phases coexist in real space by implementing the Bogoliubov--de~Gennes equation suitable for describing inhomogeneous fermion systems. While the BCS, Fulde--Ferrell--Larkin--Ovchinnikov (FFLO), and normal phases occupy different regions on the phase diagram when the parameters are uniform, a spatial change of pairing strength or spin polarization can drive the system from the FFLO phase to a normal gas or from a BCS superfluid to the FFLO phase in real space. The FFLO phase exhibits its signature modulating order parameter at the FFLO momentum due to population imbalance, and the pair correlation penetrates the polarized normal phase and exhibits proximity effects. Meanwhile, the BCS phase tends to repel population imbalance and maintain a plateau of pairing. Interestingly, a buffer FFLO phase emerges when the spatial change attempts to join the BCS and normal phase in the presence of spin polarization. By analyzing the pairing correlations, interfacial properties, and momentum-space spectra of the inhomogeneous structures, relevant length- and momentum- scales and their interplay are characterized. We also briefly discuss implications of inhomogeneous multi-phase atomic Fermi gases with population imbalance.