Fragmentation and damping of the low-energy collective modes of a weakly paired trapped atomic Fermi gas: parametric and temperature dependence

Bruun and Mottelson (Phys. Rev. Lett. 87, 270403 (2001)) analyzed the low-energy collective-mode spectrum of a superfluid trapped atomic Fermi gas in the collisionless regime using a self-consistent random-phase approximation built on Hartree-Fock-Bogoliubov quasiparticles, and explicitly deferred several issues to future publications. The monopole ($L=0$) deferral was subsequently addressed by Bruun (Phys. Rev. Lett. 89, 263002 (2002)), but two coupled questions remain unaddressed in the later literature. First, in the weak-pairing regime where the gap is smaller than the trap level spacing and Cooper pairs form within discrete harmonic-oscillator shells, how do the fragmentation and damping of the normal-phase quadrupole and spin-dipole modes depend on the position of the chemical potential relative to the shells, the particle number, and the coupling strength? Second, how does the spectrum evolve with temperature across this parameter space, and where quantitatively is the boundary between appreciably Landau-damped and essentially undamped collective motion? Partial progress exists (a coupling scan at fixed chemical potential, temperature spectra for one parameter set, and a semiclassical bulk theory without shell structure), but the systematic parametric maps and quantitative damping boundaries within the microscopic framework do not. The problem is open in the source's own terms and has gained renewed relevance through few-fermion experiments that realize the shell-resolved pairing regime.

Background

Dilute gases of fermionic alkali atoms (for example $^6$Li) trapped in a harmonic potential of frequency $\omega_T$ can undergo a BCS (Bardeen-Cooper-Schrieffer) pairing transition when two hyperfine states (the two spin components) interact attractively through an $s$-wave scattering length $a<0$ (the low-energy two-body collision parameter, negative for attraction; $s$-wave denotes the lowest, isotropic angular-momentum channel). A key diagnostic of the paired state is the collective-mode spectrum: shape oscillations of the trapped cloud, observed as peaks in the linear-response strength function $S(F,\omega)$ to an external perturbation of multipole symmetry $L$ ($L=1$ dipole, $L=2$ quadrupole, $L=0$ monopole or breathing). In the collisionless regime (quasiparticle lifetime $\tau$ with $\omega_T\tau\gg1$), the normal gas has a quadrupole density mode at $\omega\simeq2\omega_T$ and a spin-dipole mode, in which the two spin components oscillate against each other, at $\omega\simeq\omega_T$. The microscopic framework is the self-consistent random-phase approximation (RPA, the linear-response theory that sums quasiparticle-quasiparticle interactions in both the density and pairing channels): one first solves the Hartree-Fock-Bogoliubov / Bogoliubov-de Gennes (BdG) equations for quasiparticle wave functions and energies with a regularized zero-range interaction, and then diagonalizes a response matrix built from density-density and pairing-field correlation functions (the pairing field is the local anomalous amplitude $\chi=\psi_\downarrow\psi_\uparrow$).

Superfluidity adds qualitatively new spectral objects: the Goldstone (Bogoliubov-Anderson) mode, an oscillation of the phase of the pairing field $\Delta(\mathbf{r})$ that is gapless in a uniform system but acquires a finite frequency in a trap, and pair-breaking / pair-vibration (amplitude, Higgs-type) excitations costing approximately $2\Delta$. Two regimes are distinguished by the ratio of the gap to the level spacing. For strong pairing, $\Delta\gg\hbar\omega_T$ (equivalently the coherence length $\xi_{BCS}\sim\hbar k_F/(\pi m\Delta)$ is smaller than the cloud radius $R$), the gas behaves as a quasi-homogeneous bulk superfluid and hydrodynamics predicts, for example, a quadrupole Goldstone mode at $\sqrt{2}\omega_T$. For weak pairing, $\Delta(0)\lesssim\hbar\omega_T$ (here $\Delta(0)$ denotes the pairing gap at the trap center), Cooper pairs form within single harmonic-oscillator shells and the discrete shell structure controls the spectrum; this is the intrashell regime, in many ways analogous to a large atomic nucleus. Landau damping is the decay of a collective mode through its coupling to quasiparticle excitations; in a trap with discrete levels, a collective pole lying inside the two-quasiparticle continuum fragments into many discrete peaks, which appears experimentally as damping.

Bruun and Mottelson (Phys. Rev. Lett. 87, 270403 (2001)) presented the first systematic self-consistent RPA calculation of this spectrum. For weak pairing ($\Delta(0)\simeq0.69\hbar\omega_T$, with the Hartree mean field lowering the $n=33$ oscillator shell so that it straddles the chemical potential $\mu_F=32\hbar\omega_T$) they found a new low-frequency quadrupole response at $\simeq0.4\omega_T$ arising from breaking weakly bound pairs, and fragmentation with partial damping of the normal-phase quadrupole ($\sim2\omega_T$) and spin-dipole ($\sim\omega_T$) modes. For strong pairing ($\Delta(0)=6\hbar\omega_T$) the quadrupole Goldstone mode at $\simeq1.4\omega_T\approx\sqrt{2}\omega_T$ dominates, the spin-dipole mode is damped away below $T_c$ (the critical temperature of the superfluid transition; a superfluid cannot sustain relative motion of the paired components without breaking pairs), and at intermediate temperatures $0<T<T_c$ the collective mode suffers appreciable Landau damping; the spectrum was found consistent with the interpretation that the collective mode becomes undamped once the temperature is low enough that the lowest quasiparticle excitation energy is at least $2\hbar\omega_T$ (about $T_c/4$ for their parameters). The paper explicitly deferred the monopole ($L=0$) analysis to a future publication and the detailed parameter dependence of the weak-pairing fragmentation and damping, and it treated the temperature dependence only for one parameter set per regime. Of these deferrals, the monopole part was subsequently carried out (Bruun, Phys. Rev. Lett. 89, 263002 (2002); Grasso, Khan and Urban, Phys. Rev. A 72, 043617 (2005)); the remaining two items define the open problem stated here.

Problem Statement

Within the self-consistent collisionless RPA (quasiparticle-RPA) framework for a two-component Fermi gas with attractive $s$-wave interaction ($k_F|a|\ll1$, the dilute weakly interacting BCS regime, with $k_F$ the Fermi wave number) in a spherical harmonic trap, determine:

1. (Weak-pairing parametric dependence) How the fragmentation and damping of the normal-phase quadrupole ($\sim2\omega_T$) and spin-dipole ($\sim\omega_T$) modes evolve with increasing pairing strength in the weak-pairing regime $\Delta(0)\lesssim\hbar\omega_T$, as a systematic function of (a) the exact location of the chemical potential relative to the (Hartree-shifted) oscillator shells / quasiparticle bands, (b) the number of trapped particles, and (c) the coupling strength. This is the parametric map of the strength function $S(F,\omega)$ that the source paper (Bruun and Mottelson, Phys. Rev. Lett. 87, 270403 (2001), DOI: 10.1103/PhysRevLett.87.270403) promised as a future publication.

2. (Temperature dependence and damping boundaries) The temperature dependence of the spectrum across this parameter space, in particular the quantitative boundary, in terms of the lowest quasiparticle excitation energy relative to the mode energy, temperature, particle number, and coupling, between appreciably Landau-damped and essentially undamped collective modes (the boundary being the locus where the lowest quasiparticle excitation energy equals the mode energy, so that a mode lying inside the two-quasiparticle continuum is appreciably damped and one outside it is essentially undamped), testing the interpretation proposed in the source that a mode becomes undamped once the lowest quasiparticle excitation energy exceeds the mode energy (there $2\hbar\omega_T$, reached near $T_c/4$ for the source's strong-pairing parameters). This item also requires, jointly with the boundary above, the microscopic explanation of the intermediate-temperature damping and double-peak structure of the quadrupole response, whose description as dynamical two-fluid behavior was explicitly left open by Grasso, Khan and Urban (Phys. Rev. A 72, 043617 (2005), DOI: 10.1103/PhysRevA.72.043617).

The monopole ($L=0$) part of the original deferral was resolved by Bruun (Phys. Rev. Lett. 89, 263002 (2002), DOI: 10.1103/PhysRevLett.89.263002) and further analyzed by Grasso, Khan and Urban (2005; DOI: 10.1103/PhysRevA.72.043617), and is therefore not part of this problem. The two requested results share one overarching objective, the quantitative microscopic characterization of how weak pairing and discrete shell structure modify the measurable collective-mode spectrum of a trapped Fermi gas; the physical interpretation of the fragmentation and damping is part of this shared objective.

References

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