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We consider in dimension 3 a homogeneous superfluid at very low temperature $T$ having two types of excitations, (i) gapless acoustic phonons with a linear dispersion relation at low wave number, and (ii) gapped $\gamma $ quasiparticles with a quadratic (massive) dispersion relation in the vicinity of its extrema. Recent works [Nicolis and Penco (2018), Castin, Sinatra and Kurkjian (2017, 2019)], extending the historical study by Landau and Khalatnikov on the phonon–roton interaction in liquid helium 4, have explicitly determined the scattering amplitude of a thermal phonon on a $\gamma $ quasiparticle at rest to leading order in temperature. We generalize this calculation to the case of a $\gamma $ quasiparticle of arbitrary subsonic group velocity, with a rigorous construction of the $S$ matrix between exact asymptotic states, taking into account the unceasing phonon–phonon and phonon–$\gamma $ interaction, which dresses the incoming and emerging phonon and $\gamma $ quasiparticle by virtual phonons; this sheds new light on the Feynman diagrams of phonon–$\gamma $ scattering. In the whole domain of the parameter space (wave number $k$, interaction strength, etc.) where the $\gamma $ quasiparticle is energetically stable with respect to the emission of phonons of arbitrary wavevector, we can therefore characterize the erratic motion it performs in the superfluid due to its unceasing collisions with thermal phonons, through (a) the mean force $F(k)$ and (b) longitudinal and transverse momentum diffusion coefficients $D_{\sslash }(k)$ and $D_{\perp }(k)$ coming into play in a Fokker–Planck equation, then, at long times when the quasiparticle has thermalized, (c) the spatial diffusion coefficient $\mathcal{D}^{\mathrm{spa}}$, independent of $k$. At the location $k_0$ of an extremum of the dispersion relation, where the group velocity of the quasiparticle vanishes, $F(k)$ varies linearly with velocity with an isotropic viscous friction coefficient $\alpha $ that we calculate; if $k_0=0$, the momentum diffusion is also isotropic and $F(k_0)=0$; if $k_0>0$, it is not ($D_{\sslash }(k_0)\ne D_{\perp }(k_0)$), and $F(k_0)$ is non-zero but subleading with respect to $\alpha $ by one order in temperature. The velocity time correlation function, whose integral gives $\mathcal{D}^{\mathrm{spa}}$, also distinguishes between these two cases ($k_0$ is now the location of the minimum): if $k_0=0$, it decreases exponentially, with the expected viscous damping rate of the mean velocity; if $k_0>0$, it is bimodal and has a second component, with an amplitude lower by a factor ${\propto } T$, but with a lower damping rate in the same ratio (it is the thermalization rate of the velocity direction); this balances that. We also characterize analytically the behavior of the force and of the momentum diffusion in the vicinity of any sonic edge of the stability domain where the quasiparticle speed tends to the speed of sound in the superfluid. The general expressions given in this work are supposedly exact to leading order in temperature (order $T^8$ for $F(k)$, order $T^9$ for $D_{\sslash }(k)$, $D_{\perp }(k)$ and $F(k_0)$, order $T^{-7}$ for $\mathcal{D}^{\mathrm{spa}}$). They however require an exact knowledge of the dispersion relation of the $\gamma $ quasiparticle and of the equation of state of the superfluid at zero temperature. We therefore illustrate them in the BCS approximation, after calculating the stability domain, for a fermionic $\gamma $ quasiparticle (an unpaired fermion) in a superfluid of unpolarized spin $1/2$ fermions, a system that can be realised with cold atoms in flat bottom traps; this domain also exhibits an interesting, unobserved first order subsonic instability line where the quasi-particle is destabilized by emission of phonons of finite wave vectors, in addition to the expected sonic instability line resulting from Landau’s criterion. By the way, we refute the thesis of Lerch, Bartosch and Kopietz (2008), stating that there would be no fermionic quasiparticle in such a superfluid.

The superfluid, pair condensed spin-$1/2$ Fermi gases are supposed to exhibit at nonzero wave vector a still unobserved collective excitation mode in their pair-breaking continuum. Using BCS theory at zero temperature and in the long wavelength limit, we predict that this mode is quantitatively observable (in frequency, width and spectral weight) in the response of a cold atom gas to a laser Bragg excitation, if one measures the perturbation induced on the order parameter modulus rather than on the density.

The pair-condensed unpolarized spin-$1/2$ Fermi gases have a collective excitation branch in their pair-breaking continuum (V.A. Andrianov, V.N. Popov, 1976). We study it at zero temperature, with the eigenenergy equation deduced from the linearized time-dependent BCS theory and extended analytically to the lower half complex plane through its branch cut, calculating both the dispersion relation and the spectral weights (quasiparticle residues) of the branch. In the case of BCS superconductors, so called because the effect of the ion lattice is replaced by a short-range electron-electron interaction, we also include the Coulomb interaction and we restrict ourselves to the weak coupling limit $\Delta /\mu \rightarrow 0^+$ ($\Delta $ is the order parameter, $\mu $ the chemical potential) and to wavenumbers $q=O(1/\xi )$ where $\xi $ is the size of a pair; when the complex energy $z_q$ is expressed in units of $\Delta $ and $q$ in units of $1/\xi $, the branch follows a universal law insensitive to the Coulomb interaction. In the case of cold atoms in the BEC-BCS crossover, only a contact interaction remains, but the coupling strength $\Delta /\mu $ can take arbitrary values, and we study the branch at any wave number. At weak coupling, we predict three scales, that already mentioned $q\approx 1/\xi $, that $q\approx (\Delta /\mu )^{-1/3}/\xi $ where the real part of the dispersion relation has a minimum and that $q\approx (\mu /\Delta )/\xi \approx k_{\mathrm{F}}$ ($k_{\mathrm{F}}$ is the Fermi wave number) where the branch reaches the edge of its existence domain. Near the point where the chemical potential vanishes on the BCS side, $\mu /\Delta \rightarrow 0^+$, where $\xi \approx k_{\mathrm{F}}$, we find two scales $q\approx (\mu /\Delta )^{1/2}/\xi $ and $q\approx 1/\xi $. In all cases, the branch has a limit $2\Delta $ and a quadratic start at $q=0$. These results were obtained for $\mu >0$, where the eigenenergy equation admits at least two branching points $\epsilon _a(q)$ and $\epsilon _b(q)$ on the positive real axis, and for an analytic continuation through the interval $[\epsilon _a(q),\epsilon _b(q)] $. We find new continuum branches by performing the analytic continuation through $[\epsilon _b(q),+\infty [$ or even, for $q$ low enough, where there is a third real positive branching point $\epsilon _c(q)$, through $[\epsilon _b(q),\epsilon _c(q)]$ and $[\epsilon _c(q),+\infty [$. On the BEC side $\mu 0$, some of these new branches have a low-wavenumber exotic hypoacoustic $z_q\approx q^{3/2}$ or hyperacoustic $z_q\approx q^{4/5}$ behavior. For $\mu