Dynamical vanishing of the order parameter in a confined Bardeen-Cooper-Schrieffer Fermi gas after an interaction quench

We present a numerical study of the Higgs mode in an ultracold confined Fermi gas after an interaction quench and find a dynamical vanishing of the superfluid order parameter. Our calculations are done within a microscopic density-matrix approach in the Bogoliubov–de Gennes framework which takes the three-dimensional cigar-shaped confinement explicitly into account. In this framework, we study the amplitude mode of the order parameter after interaction quenches starting on the BCS side of the BEC-BCS crossover close to the transition and ending in the BCS regime. We demonstrate the emergence of a dynamically vanishing superfluid order parameter in the spatiotemporal dynamics in a three-dimensional trap. Further, we show that the signal averaged over the whole trap mirrors the spatiotemporal behavior and allows us to systematically study the effects of the system size and aspect ratio on the observed dynamics. Our analysis enables us to connect the confinement-induced modifications of the dynamics to the pairing properties of the system. Finally, we demonstrate that the signature of the Higgs mode is contained in the dynamical signal of the condensate fraction, which, therefore, might provide a new experimental access to the nonadiabatic regime of the Higgs mode.

Figure 1

Spatiotemporal dynamics of the modulus of the order parameter after a sudden change of the scattering length from $1/\left(k_F{a}_i\right)=-0.4$ to $1/\left(k_F{a}_f\right)=-1.45$ and an aspect ratio of $r=50$, where ${\tau }_{\text{trap}}=h/\delta E=1/\left(2f_{\parallel }\right)$ is the characteristic time scale of the trap [34].

Figure 2

(a) Dynamics of the spatially averaged gap after a sudden change of the scattering length from $1/\left(k_F{a}_i\right)=-0.4$ (blue solid line) and $1/\left(k_F{a}_i\right)=-0.8$ (red dash-dotted line) to $1/\left(k_F{a}_f\right)=-1.45$ and an aspect ratio of $r=50$. The gap is normalized to the spatially averaged ground-state gap of the system before the quench ${\overline{\mathrm{\Delta }}}_i$. ${\overline{\mathrm{\Delta }}}_f$ is the spatially averaged ground-state gap of the final system and ${\mathrm{\Delta }}_{\infty }$ marks the height of the plateau for $1/\left(k_F{a}_i\right)=-0.4$; (b) Fourier transform of the gap dynamics with $1/\left(k_F{a}_i\right)=-0.4$ and the BdG eigenenergies $E_k$.

Figure 3

${\mathrm{\Delta }}_{\infty }$ normalized to the ground-state gap of the initial system ${\overline{\mathrm{\Delta }}}_i$ for different quenches with $1/\left(k_F{a}_f\right)=-1.45$ and aspect ratios $r$. Inset: quench parameter $1/{\left(k_F{a}_i\right)}_t$ at the transition to the dynamical vanishing vs the aspect ratio $r$. The red solid line (fit A) shows a linear fit to data for $r\le 8$ and the black dashed line (fit B) is an inverse proportional fit to data for $r\ge 9$.

Figure 4

Time scale of the initial decay with respect to the trap time $\beta {\tau }_{\text{trap}}$ for various quenches to $1/\left(k_F{a}_f\right)=-1.45$ and ${\omega }_{\parallel }=\text{const}$ for $r=5,10$, and 20. Inset: $\beta {\tau }_{\text{trap}}$ vs the aspect ratio $r$ at fixed quench conditions; the dashed black line marks the condition $\beta {\tau }_{\text{trap}}=10$ in either figure.

Figure 5

Minimal BdG quasiparticle energy $E_{\text{min}}$ for various aspect ratios in dependence of the particle number $N$. The red dashed line is a fit $E_{\text{min}}=a{N}^{1/3}+b$ to the data for $r=5$ and the dash-dotted black line is the value of $E_{\text{min}}$ for $N=1\times {10}^6$ obtained from the fit.

Figure 6

Condensate fraction (blue solid line) and modulus of averaged order parameter (red dashed line) after a quench to $1/\left(k_F{a}_f\right)=-1.45$ from (a) $1/\left(k_F{a}_i\right)=-0.8$ and from (b) $1/\left(k_F{a}_i\right)=-1.0$, with an aspect ratio $r=50$, and a particle number $N=8428$. The data is normalized to the ground-state value of the initial system $c_i$ and ${\overline{\mathrm{\Delta }}}_i$, respectively.

Figure 7

Exemplary illustration of our fitting routines in order to obtain ${\mathrm{\Delta }}_{\infty }$ for a system with $r=10$ and a quench with $1/\left(k_F{a}_i\right)=-0.66$. In this system we find a plateau behavior before ${\tau }_{\text{trap}}$, which is characterized by the nonvanishing mean value of the plateau.