Landau Phonon-Roton Theory Revisited for Superfluid ${}^4\mathrm{He}$ and Fermi Gases

Liquid helium and spin-$1/2$ cold-atom Fermi gases both exhibit in their superfluid phase two distinct types of excitations, gapless phonons and gapped rotons or fermionic pair-breaking excitations. In the long wavelength limit, revising and extending the theory of Landau and Khalatnikov initially developed for helium [Zh. Exp. Teor. Fiz. 19, 637 (1949)], we obtain universal expressions for three- and four-body couplings among these two types of excitations. We calculate the corresponding phonon damping rates at low temperature and compare them to those of a pure phononic origin in high-pressure liquid helium and in strongly interacting Fermi gases, paving the way to experimental observations.

Figure 1

Phonon damping rates at angular frequency ${\omega }_{\mathbf{q}}=2\pi \times 165\mathrm{GHz}$ ($q=0.3{\AA }^{-1}$) in liquid ${}^4\mathrm{He}$ at pressure $P=20$ bar as functions of temperature. Solid line: purely phononic damping ${\mathrm{\Gamma }}_{\varphi \varphi }$ due to Landau-Khalatnikov four-phonon processes [1, 6, 21]; it depends on the curvature parameter $\gamma$ defined as ${\omega }_{\mathbf{q}}=cq[1+(\gamma /8)(\hslash q/mc{)}^2+O(q^4)]$. Interpolating measurements of $P\mapsto \gamma (P)$ in Refs. [25, 26] gives $\gamma =-6.9$. Dashed black line [dash-dotted black line]: damping due to scattering [absorption or emission] by rotons, see Eq. (12) [Eq. (15)]. Red dashed line: original formula of Ref. [1] for the damping rate due to phonon-roton scattering. The roton parameters are extracted from their dispersion relation $k\mapsto {\epsilon }_{\mathbf{k}}$ measured at various pressures [27]: $\mathrm{\Delta }/k_B=7.44\mathrm{K}$, $k_0=2.05{\AA }^{-1}$, $m_{*}/m=0.11$, $\rho k_0^{\prime }/k_0=0.39$, $\rho {\mathrm{\Delta }}^{\prime }/\mathrm{\Delta }=-1.64$, ${\rho }^2{\mathrm{\Delta }}^{\prime \prime }/\mathrm{\Delta }=-8.03$, $\rho {m_{*}}^{\prime }/m_{*}=-4.7$. In Eq. (15), parabolic approximation Eq. (9) is used (hence, ${\epsilon }_{k_{*}}/\mathrm{\Delta }\simeq 1.43$). The speed of sound $c=346.6\mathrm{m}/\mathrm{s}$, and the Grüneisen parameter $d\ln c/d\ln \rho =2.274$ entering in ${\mathrm{\Gamma }}_{\varphi \varphi }$, are taken from equation of state Eq. (A1) of Ref. [28]. The low values $\hslash q/mc=0.13$ and $k_{B}T/m{c}^2<{10}^{-2}$ justify our use of quantum hydrodynamics.

Figure 2

Phonon damping rates at wave number $q=mc/2\hslash$ in unpolarized homogeneous cold-atom Fermi gases in thermodynamic limit as functions of temperature. (a) At unitarity $a^{-1}=0$, where most parameters of the phonons and fermionic quasiparticles are measured (see text). (b) On the BCS side $1/k_{F}a=-0.389$, these parameters are estimated in BCS theory ($\mu /{\epsilon }_F\simeq 0.809$, $\mathrm{\Delta }/\mu \simeq 0.566$, $m_{*}/m=\mathrm{\Delta }/2\mu$, $\rho {\mu }^{\prime }/\mu \simeq 0.602$, $\rho {\mathrm{\Delta }}^{\prime }/\mathrm{\Delta }\simeq 0.815$, ${\rho }^2{\mathrm{\Delta }}^{\prime \prime }/\mathrm{\Delta }\simeq -0.209$, $d\ln c/d\ln \rho \simeq 0.303$). In both cases the curvature parameter $\gamma$ defined in the caption of Fig. 1 is estimated in the RPA [42]. Solid line: phonon-phonon (a) Beliaev-Landau damping $\varphi \leftrightarrow \varphi \varphi$ (for $\gamma >0$) as in Eqs. (121) and (122) of Ref. [21] (independent of $|\gamma |$) and (b) Landau-Khalatnikov damping $\varphi \varphi \leftrightarrow \varphi \varphi$ (for $\gamma \simeq -0.30<0$) [6, 21]. Dashed line [dash-dotted line]: scattering [absorption or emission] phonon-fermionic quasiparticle processes, as in Eq. (12) [Eq. (15)]. In Eq. (15), we took for ${\epsilon }_k$ (a) the form proposed in Ref. [39] (hence, ${\epsilon }_{k_{*}}/\mathrm{\Delta }\simeq 1.12$) and (b) the BCS form (hence, ${\epsilon }_{k_{*}}/\mathrm{\Delta }\simeq 1.14$). $\mu$ is the $T=0$ gas chemical potential, and the plotted quantities are in fact inverse quality factors. Here $k_{B}T/m{c}^2>0.03$ in contrast to Fig. 1 where $k_{B}T/m{c}^2<0.01$: cold atoms are effectively farther from the $T\to 0$ limit than liquid helium, hence the inversion of the ${\mathrm{\Gamma }}_{\mathbf{q}}^{\mathrm{scat}}\text{−}{\mathrm{\Gamma }}_{\mathbf{q}}^{a\text{or}e}$ hierarchy.