We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors, as suggested by Littlewood and Varma in 1982. We analytically continue the RPA equation on the collective mode energy through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. For ∆/µ > 1.210 (very close to unitarity in a superfluid Fermi gas), where ∆ is the order parameter and µ the chemical potential, the real part of the branch is wholly within the band gap [0, 2∆]. In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically. This contradicts the result of Littlewood and Varma that prevailed so far.

We study the phononic collective modes of the pairing field ∆ in a superfluid Fermi gas at all temperatures below T c . We deal with the coupling of these modes to the fermionic continuum of quasiparticle-quasihole excitations by performing a non-perturbative analytic continuation of the pairing field propagator. At low temperature, we recover the know exponential temperature dependence of the damping rate and velocity shift of the Anderson-Bogoliubov branch. In the vicinity of T c , and in the BCS regime, our calculations reveal two phononic branches; the first one has a velocity that tends to a finite non-zero value at T c , while the second one has a velocity that vanishes with a critical exponent of 1/2 (in contradiction with Ohashi et al., J. Phys. Soc. Jap. 66, 2437), and a quality factor that diverges logarithmically with T c − T . At temperatures close to T c , this results in a double peak structure in the response function of the phase of ∆, well resolved in the BCS regime. Away from T = 0 and T c , we develop a semi-numerical method to perform the analytic continuation. This confirms the existence of two branches, and allows us to follow the disappearance of the second branch as the temperature is lowered. Our results generalize to pure fermionic condensates the double peak structure observed by Carlson and Goldman in dirty superconductors (Phys. Rev. Lett. 31, 880).

We study the interactions among phonons and the phonon lifetime in a pair-condensed Fermi gas in the BEC-BCS crossover in the collisionless regime. To compute the phonon-phonon coupling amplitudes we use a microscopic model based on a generalized BCS Ansatz including moving pairs, which allows for a systematic expansion around the mean field BCS approximation of the ground state. We show that the quantum hydrodynamic expression of the amplitudes obtained by Landau and Khalatnikov apply only on the energy shell, that is for resonant processes that conserve energy. The microscopic model yields the same excitation spectrum as the Random Phase Approximation, with a linear (phononic) start and a concavity at low wave number that changes from upwards to downwards in the BEC-BCS crossover. When the concavity of the dispersion relation is upwards at low wave number, the leading damping mechanism at low temperature is the Beliaev-Landau process 2 phonons ↔ 1 phonon while, when the concavity is downwards, it is the Landau-Khalatnikov process 2 phonons ↔ 2 phonons. In both cases, by rescaling the wave vectors to absorb the dependence on the interaction strength, we obtain a universal formula for the damping rate. This universal formula corrects and extends the original analytic results of Landau and Khalatnikov [ZhETF 19, 637 (1949)] for the 2 ↔ 2 processes in the downward concavity case. In the upward concavity case, for the Beliaev 1↔ 2 process for the unitary gas at zero temperature, we calculate the damping rate of an excitation with wave number q including the first correction proportional to q 7 to the q 5 hydrodynamic prediction, which was never done before in a systematic way.