Beliaev damping of the Goldstone mode in atomic Fermi superfluids

Beliaev damping in a superfluid is the decay of a collective excitation into two lower-frequency collective excitations; it represents the only decay mode for a bosonic collective excitation in a superfluid at $T=0$. The standard treatment for this decay assumes a linear spectrum, which in turn implies that the final-state momenta must be collinear to the initial state. We extend this treatment, showing that the inclusion of a gradient term in the Hamiltonian yields a realistic spectrum for the bosonic excitations; we then derive a formula for the decay rate of such excitations and show that even moderate nonlinearities in the spectrum can yield substantial deviations from the standard result. We apply our result to an attractive Fermi gas in the BCS–Bose-Einstein condensate crossover: Here the low-energy bosonic collective excitations are density oscillations driven by the phase of the pairing order field. These collective excitations, which are gapless modes as a consequence of the Goldstone mechanism, have a spectrum that is well established both theoretically and experimentally and whose linewidth, we show, is determined at low temperatures by the Beliaev decay mechanism.

Figure 1

Collective mode spectrum for $y={\left(k_F{a}_s\right)}^{-1}=0.0,y=0.5$, and $y=1.0$, from left to right. The black line represents the real part of the spectrum, the dashed lines represent $\pm$ the imaginary part, and the bold red line represents the threshold energy $E_{th}$.

Figure 2

Pair fluctuation spectral function $A_{\eta \eta }(\mathbf{k},\omega )$ for $y={\left(k_F{a}_s\right)}^{-1}=1$. The dashed red line shows the corresponding spectrum and the dashed blue lines correspond to the imaginary part of the spectrum, like in Fig. 1. For $\frac{p}{k_f}\gtrsim 1$ the line broadening due to the Beliaev decay effectively destroys a collective excitation; this is also approximately the scale marking the end of the validity of the perturbative approach. For comparison here $E_{th}>3{\epsilon }_F$.

Figure 3

Beliaev decay width calculated from Eq. (16) divided by the original Beliaev result (linear approximation), for different values of $y=1/k_F{a}_s$. The inset shows the evolution of $k_c/k_F$ as defined in Eq. (43) in the unitary to moderate BEC regime we investigate.