Pure Goldstone mode in the quench dynamics of a confined ultracold Fermi gas in the BCS-BEC crossover regime

We present a numerical study of the dynamic response of a confined superfluid Fermi gas to a rapid change of the scattering length (i.e., an interaction quench). Based on a fully microscopic time-dependent density-matrix approach within the full Bogoliubov–de Gennes formalism that includes a 3D harmonic confinement we simulate and identify the emergence of a Goldstone mode of the BCS gap in a cigar-shaped ${}^6\mathrm{Li}$ gas. By analyzing this Goldstone mode over a wide range of parameters, we show that its excitation spectrum is gapless and that its main frequency is not fixed by the trapping potential but that it is determined by the details of the quench. Thus we report the emergence of a pure Goldstone mode of the BCS gap that—in contrast to situations in many previous studies—maintains its gapless excitation spectrum predicted by the Goldstone theorem. Furthermore, we observe that the size-dependent superfluid resonances resulting from the atypical BCS-BEC crossover have a direct impact on this Goldstone mode. Finally, we find that the interaction quench-induced Goldstone mode leads to a low-frequency in-phase oscillation of the single-particle occupations with complete inversion of the lowest-lying single-particle states which could provide a convenient experimental access to the pure gapless Goldstone mode.

Figure 1

Ginzburg-Landau free energy of a BCS phase of a homogeneous Fermi gas for $T<T_C$ (schematic). The rotational $\text{U}\left(1\right)$ symmetry is broken, when the gap $\mathrm{\Delta }$ of the system gains a certain phase inside the rim of the potential. The resulting excitation modes are the Higgs-amplitude mode and the Goldstone mode, i.e., a phase oscillation of the gap.

Figure 2

Phase dynamics for the full BdG solution (solid lines) and the Anderson approximate solution (dashed lines) for $\delta \left[1/\left(k_{F}a\right)\right]=-0.1$ and different final coupling strengths: $1/\left(k_F{a}_f\right)=-0.5, 1/\left(k_F{a}_f\right)=-0.9$ (scaled by a factor of 3), and $1/\left(k_F{a}_f\right)=-1.4$ (scaled by a factor of 10); parameters: $f_{\parallel }=96\mathrm{Hz}, f_{\perp }=1\mathrm{kHz}$, and $N_P=120$.

Figure 3

Frequency of the Goldstone mode resulting from the full BdG solution (solid line) and from Anderson's approximate solution (dashed line) for the same system as in Fig. 2 but varying $1/\left(k_F{a}_f\right)$ (to reduce the numerical effort we set $\mathrm{\Delta }\epsilon =0.58E_F$).

Figure 4

Frequency of the Goldstone mode $f_G$ for quenches with varying strength to $1/\left(k_F{a}_f\right)=-0.9$; parameters: $f_{\parallel }=96\mathrm{Hz}, f_{\perp }=1\mathrm{kHz}$, and $N=1000$.

Figure 5

Frequency of the Goldstone mode $f_G$ for different excitations $1/\left(k_F{a}_i\right)\to 1/\left(k_F{a}_f\right)$ corresponding to phase II of the quantum quench phase diagram (see main text); quenches corresponding to phase I are not shown; parameters: $f_{\parallel }=96\mathrm{Hz}, f_{\perp }=1\mathrm{kHz}$, and $N=1000$.

Figure 6

Frequency of the gapless Goldstone mode $f_G$ for fixed $f_{\parallel }=96\mathrm{Hz}, N_P=1000$, and $1/\left(k_{F}a\right)=-0.8\to -0.9$; upper label: system parameter $S=\mu /\hslash {\omega }_{\perp }$.

Figure 7

Single-particle energies for a strongly confined Fermi gas in a BCS phase with four atomic subbands crossing the chemical potential; ${\mathrm{\Delta }}_i^S$ denotes the gap of the subbands with $m_x+m_y=i$ and $(m_x,m_y)$ denotes the subband index (see main text); the marked states (1), (2), and (3) are discussed below; parameters: $f_{\parallel }=56\mathrm{Hz}, f_{\perp }=4\mathrm{kHz}, N=1700$, and $1/\left(k_{F}a\right)=-0.9$.

Figure 8

(a) Dynamics of three particular single-particle occupations, one state of higher energy (3) and two from subband minima (1), (2) (see Fig. 7). (b) Fourier transform of the functions in (a).

Figure 9

Single-particle occupations at different times after the interaction quench; parameters: see Fig. 7.