Landau Phonon-Roton Theory Revisited for Superfluid Helium 4 and Fermi Gases Yvan Castin, Alice Sinatra, Hadrien Kurkjian

Home , Roton

Landau phonon-roton theory revisited for superfluid helium 4 and Fermi gases Yvan Castin, Alice Sinatra, Hadrien Kurkjian

To cite this version:

Yvan Castin, Alice Sinatra, Hadrien Kurkjian. Landau phonon-roton theory revisited for superfluid helium 4 and Fermi gases. Physical Review Letters, American Physical Society, 2017, 119, pp.260402. 10.1103/PhysRevLett.119.260402. hal-01570314v2

HAL Id: hal-01570314 https://hal.archives-ouvertes.fr/hal-01570314v2 Submitted on 9 Jan 2018 (v2), last revised 25 Nov 2019 (v3)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Landau phonon-roton theory revisited for superﬂuid helium 4 and Fermi gases

Yvan Castin and Alice Sinatra Laboratoire Kastler Brossel, ENS-PSL, CNRS, Sorbonne Universit´es, Coll`ege de France, 75005 Paris, France

Hadrien Kurkjian TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerp, Belgium

Liquid helium and spin-1/2 cold-atom Fermi gases both exhibit in their superﬂuid phase two dis- tinct types of excitations, gapless phonons and gapped rotons or fermionic pair-breaking excitations. In the long wavelength limit, revising and extending Landau and Khalatnikov’s theory initially de- veloped for helium [ZhETF 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.

PACS numbers: 03.75.Kk, 67.85.Lm, 47.37.+q

Introduction – Homogeneous superfluids with short- sion to the hydrodynamic regime ωqτγ . 1 or ωqτφ . 1 range interactions exhibit, at sufficiently low tempera- may be obtained from kinetic equations [2]. An experi- ture, phononic excitations φ as the only microscopic de- mental test of our results seems nowadays at hand, either grees of freedom. In this universal limit, all superfluids in liquid helium 4, extending the recent work of Ref.[3], of this type reduce to a weakly interacting phonon gas or in homogeneous cold Fermi gases, which the break- with a quasilinear dispersion relation, irrespective of the through of flat-bottom traps [4] allows one to prepare statistics of the underlying particles and of their inter- [5] and to acoustically excite by spatio-temporally mod- action strength. Phonon damping then only depends on ulated laser-induced optical potentials [6, 7]. the dispersion relation close to zero wavenumber (namely, Landau-Khalatnikov revisited – We recall the reasoning its slope and third derivative) and on the phonon non- of Ref.[1] to get the phonon-roton coupling in liquid he- linear coupling, deduced solely from the system equation lium 4, extending it to the phonon-fermionic quasipar- of state through Landau-Khalatnikov quantum hydrody- ticle coupling in unpolarised spin-1/2 Fermi gases. We namics [1]. first treat in first quantisation the case of a single roton In experiments, however, temperatures are not al- or fermionic excitation, considered as a γ-quasiparticle ways low enough to make the dynamics purely phononic. of position r, momentum p and spin s = 0 or s = 1/2. Other elementary excitations can enrich the problem, In a homogeneous superfluid of density ρ, its Hamilto- such as spinless bosonic rotons in liquid helium 4 and nian is given by ǫ(p,ρ), an isotropic function of p such spinful fermionic BCS-type pair-breaking excitations in that p 7→ ǫ(p,ρ) is the γ-quasiparticle dispersion rela- spin-1/2 cold-atom Fermi gases. These excitations, de- tion. In presence of acoustic waves (phonons), the super- noted here as γ-quasiparticles, exhibit in both cases an fluid acquires position-dependent density ρ(r) and veloc- energy gap ∆ > 0. Remarkably, as shown by Landau and ity v(r). For a phonon wavelength large compared to Khalatnikov [1], the phonon-roton coupling, and more the γ-quasiparticle coherence length [8], here its thermal ~2 1/2 generally phonon coupling to all gapped excitations as wavelength (2π /m∗kBT ) [42], and for a phonon an- we shall see, depend to leading order in temperature gular frequency small compared to the γ-quasiparticle only on a few parameters of the dispersion relation of “internal” energy ∆, we can write the γ-quasiparticle the γ-quasiparticles, namely the value of the minimum Hamiltonian in the local density approximation [9, 10]: ∆ and its location k0 in wavenumber space, their deriva- H = ǫ(p,ρ(r)) + p · v(r) (1) tives with respect to density, and the effective mass m∗ close to k = k . We have discovered however that the 0 The last term is a Doppler effect reflecting the energy φ − γ coupling of Ref.[1] is not exact, a fact apparently difference in the lab frame and in the frame moving with unnoticed in the literature. Our goal here is to com- the superfluid. For a weak phononic perturbation of the plete the result of Ref.[1], and to quantitatively obtain superfluid, we expand the Hamiltonian to second order phonon damping rates due to the φ − γ coupling as func- in density fluctuations δρ(r)= ρ(r) − ρ: tions of temperature, a nontrivial task in the considered strongly interacting systems. We restrict to the colli- 1 H≃ ǫ(p,ρ)+ ∂ ǫ(p,ρ)δρ(r)+ p · v(r)+ ∂2ǫ(p,ρ)δρ2(r) sionless regime ωqτγ ≫ 1 and ωqτφ ≫ 1, where ωq is the ρ 2 ρ angular eigenfrequency of the considered phonon mode (2) of wavevector q, and τγ (τφ) is a typical collision time of not paying attention yet to the noncommutation of r thermal γ-quasiparticles (thermal phonons). An exten- and p. Phonons are bosonic quasiparticles connected to 2 the expansion of δρ(r) and v(r) on eigenmodes of the γ scattering is obtained by adding the contributions of quantum-hydrodynamic equations linearised around the the direct process (terms of Hˆ quadratic in ˆb), and of homogeneous solution at rest in the quantisation volume the absorption-emission or emission-absorption process V: (terms linear in ˆb) treated to second order in perturbation theory [1]: δρ(r) 1 ρq ˆ ρq ˆ† iq·r = 1/2 bq + b−q e (3) v(r) V vq −vq q=6 0 X q ′ q ′ q ′ eff ′ ′ k k k A (k, q; k , q )= ′ + ′ + ′ 1/2 2 k q k q k q with modal amplitudes ρq = [~ρq/(2mc)] and vq = 1/2 [~c/(2mρq)] q, m being the mass of a superfluid parti- A (k, q; k + q)A (k′, q′; k′ + q′) = A (k, q; k′, q′)+ 1 1 cle and c the sound velocity. The annihilation and cre- 2 ~ω + ǫ − ǫ ˆ ˆ† q k k+q ation operators bq and bq of a phonon of wavevector q ′ ′ ′ ′ A1(k − q , q ; k)A1(k − q , q; k ) and energy ~ωq = ~cq obey usual commutation relations + (7) † ǫk − ~ωq′ − ǫk q′ ˆ ˆ ′ − [bq, bq′ ]= δq,q . For an arbitrary number of γ-quasiparticles, we switch where in the second (third) term the γ-quasiparticle first to second quantisation and rewrite Eq.(2) as absorbs phonon q (emits phonon q′) then emits phonon q′ q ′ (absorbs phonon ). Up to this point this agrees with † A1(k, q; k ) † Hˆ = ǫkγˆ γˆk + (ˆγ ′ γˆk ˆbq+h.c.) Ref.[1], except that the first derivative ∂ρ∆ in Eq.(5), kσ σ V1/2 k σ σ k k k′ q thought to be anomalously small in low-pressure helium, X,σ ,X, ,σ ′ ′ was neglected in Ref.[1]. Eq. (7), issued from a local A2(k, q; k , q ) † × δk+q,k′ + γˆ ′ γˆkσδk+q,k′+q′ density approximation, holds to leading order in a low- V k σ k k′ q q′ , X, , ,σ energy limit. We then take the T → 0 limit with scaling † 1 laws × [ˆb ′ˆbq + (ˆb q′ˆbq + h.c.)] (4) q 2 − 1/2 q ≈ T, k − k0 ≈ T (8) † whereγ ˆkσ andγ ˆkσ are bosonic (rotons, s = 0, σ = 0) or fermionic (s =1/2, σ =↑, ↓) annihilation and creation reflecting the fact that the thermal energy of a phonon ~ operators of a γ-quasiparticle of wavevector k = p/~ in is cq ≈ kBT and the effective kinetic energy of a γ- spin component σ, obeying usual commutation or anti- quasiparticle, that admits the expansion commutation relations. The first sum in the right-hand ~2 2 (k − k0) 3 side of Eq.(4) gives the γ-quasiparticle energy in the un- ǫk − ∆ = + O(k − k0) (9) k→k0 perturbed superfluid, with ǫk ≡ ǫ(~k,ρ). The second 2m∗ sum, originating from the Doppler term and the term is also ≈ k T . The coupling amplitudes A and energy linear in δρ in Eq.(2), describes absorption or emission B 1 denominators in Eq.(7) must be expanded up to relative of a phonon by a γ-quasiparticle, characterised by the corrections of order T [43]. On the contrary, it suffices to amplitude expand A2 to leading order T in temperature. We hence

′ ~ ~ ′ get our main result, the effective coupling amplitude of ′ ∂ρǫk + ∂ρǫk k + k A (k, q; k )= ρ + vq · (5) 1 q 2 2 the φ − γ scattering to leading order in temperature: ′ where q, k and k are the wavevectors of the incoming ~ ~ ′ 2 ~2 2 eff ′ ′ q 1 2 ′′ ( ρk0) k0 phonon and the incoming and outgoing γ-quasiparticles. A2 (k, q; k , q ) ∼ ρ ∆ + + T →0 mcρ 2 2m 2m Eq.(5) is invariant under exchange of k and k′. This re- ( ∗ ∗ sults from symmetrisation of the various terms, in the ′ 2 ′ ′ ρ∆ ′ ρ∆ ′ ′ ρk0 2m∗c form [f(p)eiq·r +eiq·rf(p)]/2 with r and p canonically × uu + (u + u ) uu − + w ~ck0 ~ck0 k0 ~k0 conjugated operators, ensuring that the correct form of ( ′ Eq.(2) is hermitian. The third sum in Eq.(4), originat- m∗c ′ 2 ′2 ρk0 2 ′2 ing from the terms quadratic in δρ in Eq.(2), describes + ~ (u + u )w + u u − (u + u ) (10) k0 k0 )) direct scattering of a phonon on a γ-quasiparticle, with the symmetrised amplitude Here ∆′, k′ , ∆′′ are first and second derivatives of ∆ and 0 ′ ′ q·k ′ q ·k q·q 2 2 k0 with respect to ρ; u = , u = ′ , w = ′ are ∂ ǫk + ∂ ǫk′ qk q k qq ′ ′ ρ ρ ′ A2(k, q; k , q )= ρqρq′ (6) cosines of the angles between k, q and q ; our results hold 2 for k0 = 0 provided the limit k0 → 0 is taken in Eq.(10). where the primed wavevectors are the ones of emerging In Eq.(3.17) of Ref.[1], the ∆′ terms were neglected as ′ quasiparticles. It also describes negligible two-phonon said, but the last term in Eq.(10), with the factor ρk0/k0, absorption and emission. The effective amplitude for φ− was simply forgotten. 3

Damping rates – A straightforward application of smaller than the k significantly contributing to Eq.(13), Eq.(10) is a Fermi-golden-rule calculation of the damp- but with no constraint on the ratio ~cq/kBT , we write scat 1/2 ing rate Γq of phonons q due to scattering on γ- A1 in Eq.(5) to leading order q in q, and integrate over quasiparticles. The γ-quasiparticles are in thermal equi- the direction of k, to obtain librium with Bose or Fermi mean occupation numbers 2s −1 2 n¯ k = [exp(ǫk/k T ) − (−1) ] . So are phonons in 2 −ǫk/kB T ~2 2 γ, B a-e (2s + 1)ρ dkk e c k ′ ′ Γ ≃ ∂ρǫk + (15) modes q 6= q, with Bose occupation numbersn ¯ q = q dǫk dǫk b, 4πmc 1+¯n q −1 D | dk | b, ρ dk [exp(~ωq′ /kBT ) − 1] ; mode q is initially excited (e.g. Z 2 2 ~ −ǫk∗ /kB T by a sound wave) with an arbitrary number nb,q of (2s + 1)ρk∗ ck∗ k B T e ′ ′ ∼ ∂ρǫk∗ + (16) phonons. By including both loss q + k → q + k and T →0 4π~2mc3 ρη 1+¯n ∗ b,q gain q′ + k′ → q + k processes [44] and summing over σ, d scat Eq.(16) is an equivalent when T → 0 at fixed ~cq/k T ; one finds that dt nb,q = −Γq (nb,q − n¯b,q) with B dǫk ~ k∗ is the element of the border of D ( dk |k=k∗ = η∗ c, 3 3 ′ 2π d kd q 2 η∗ = ±) with minimal energy ǫk (when more than one of Γscat = (2s + 1) Aeff (k, q; k′, q′) q ~ (2π)6 2 such k∗ exists, one has to sum their contributions). As Z 2s ǫk∗ > ∆, the damping rate due to scattering dominates n¯b,q′n¯γ,k′ [1 + (−1) n¯γ,k] × δ(ǫk + ~ωq − ǫk′ − ~ωq′ ) the one due to absorption-emission in the mathematical n¯b,q limit T → 0 ; we shall see however that this is not always (11) so for typical temperatures in current experiments. and k′ = k + q − q′. As our low-energy theory only To be complete, we give a low-temperature equivalent of the damping rate of the γ-quasiparticle k due to in- holds for kBT ≪ ∆, the gas of γ-quasiparticles is non- 1/2 teraction with thermal phonons. With k − k0 = O(T ) degenerate, andn ¯ k ≃ exp(−ǫk/k T ) ≪ 1 in Eq.(11). γ, B γφ 7 1/3 6 as in Eq.(8), we find ~Γ ∼ (πI/42)(kBT ) /(~cρ ) , By taking the T → 0 limit at fixed ~cq/kBT and setting k ~ωq where the factor 2s + 1 is gone (no summation over σ is Aeff = f, where the dimensionless quantity f only 2 ρ needed) but I is the same angular integral as in Eq.(12). depends on angle cosines, we obtain the equivalent γφ Here scattering dominates [46]. Using τγ ≃ 1/Γk , we −∆/kB T 2 4 checked that the figures 1 and 2 below are in the col- ~ scat e k0q c 1/2 Γq ∼ (2s + 1) (m∗kBT ) I (12) T →0 (2π)9/2 ρ2 lisionless regime ωqτγ ≫ 1. Similarly, we checked that ωqτφ ≫ 1 on the figures. 2 2 2 ′ with I = d Ωk d Ωq′ f (u,u , w) an integral over solid Application to helium – Precise measurements of the angles of direction k and q′ [45]. equation of state (relating ρ to pressure) and of the roton R R One proceeds similarly for the calculation of the damp- dispersion relation for various pressures were performed a-e ′ 4 2 ing rate Γq of phonons q due to absorption q + k → k in liquid He at low temperature (kBT ≪ mc , ∆). They ′ or emission k → q + k processes by thermal equilibrium give access to the parameters k0, ∆, their derivatives γ-quasiparticles. We obtain and m∗. The measured sound velocities agree with the thermodynamic relation mc2 = ρ dµ , where µ is the zero- 3 dρ a-e 2π d k ′ 2 temperature chemical potential of the liquid. We plot Γ = (2s + 1) [A1(k, q; k )] q ~ (2π)3 in Fig. 1 the phonon damping rates as functions of tem- Z × δ(~ωq + ǫk − ǫk′ )(¯nγ,k − n¯γ,k′ ) (13) perature, for a fixed angular frequency ωq. At the cho- sen high pressure, the phonon dispersion relation is con- with k′ = k + q. Low degeneracy of the γ-quasiparticles cave at low q, therefore the Beliaev-Landau [11–16] three- and energy conservation allow us to writen ¯γ,k − n¯γ,k′ ≃ phonon process φ ↔ φφ is energetically forbidden at low exp(−ǫk/kBT )/(1 +n ¯b,q). Energy conservation leads temperature and the Landau-Khalatnikov [1, 6, 16] pro- here to a scaling on k different from Eq.(8) as it forces cess φφ ↔ φφ is dominant. Our high yet experimen- k to be at a nonzero distance from k0, even in the low- tally accessible [17, 18] value of ωq leads to attenuation phonon-energy limit: When q → 0 at fixed k, the Dirac lengths 2c/Γq short enough to be measured in centimet- delta in Eq.(13) becomes ric cells. As visible on Fig. 1, the damping of sound is in fact dominated by four-phonon Landau-Khalatnikov dǫk ~ ′ ~ −1 dk processes up to a temperature T ≃ 0.6 K. In this regime δ( ωq + ǫk − ǫk ) ∼ ( cq) δ 1 − u ~ (14) q→0 c ! one would directly observe this phonon-phonon damping mechanism, which would be a premiere. The sound at- 1 dǫk and imposes that the group velocity ~ dk of the incom- tenuation measurements of Ref.[19] in helium at 23 bars ing γ-quasiparticle is larger in absolute value than that, and ωq =2π × 1.1 GHz are indeed limited to T > 0.8 K c, of the phonons. This condition, reminiscent of Lan- where damping is still dominated by the rotons. dau’s criterion, restricts wavenumber k to a domain D Application to fermions – In cold-atom Fermi gases, in- not containing k0. In the low-q limit, that is for q much teractions occur in s-wave between opposite-spin atoms. 4

6 -2 10 10 (a) 5 -3 10 10 -4 4 10

10 q -5

ω 10

3 / -6 10 q 10 [1/s] Γ

q -7 2 Γ 10 10 -8 10 1 -9 10 10 -10 0 10 10 -2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 10 (b) -3 T [K] 10 -4 10

FIG. 1: Phonon damping rates at angular frequency ωq = q -5 − 1 4 ω 10 2π × 165 GHz (q = 0.3A˚ ) in liquid He at pressure P = 20 / -6 q 10 bar as functions of temperature. Solid line: purely phononic Γ -7 damping Γ due to Landau-Khalatnikov four-phonon pro- 10 φφ -8 cesses [1, 6, 16]; it depends on the curvature parameter γ 10 γ ~q 2 4 -9 defined as ωq = cq[1 + 8 ( mc ) + O(q )]. Interpolating mea- 10 surements of P 7→ γ(P ) in Refs.[20, 21] gives γ = −6.9. -10 10 0.02 0.04 0.06 0.08 0.1 0.15 Dashed black line/dash-dotted black line: damping due to µ scattering/absorption-emission by rotons, see Eq.(12)/(15). kBT/ Red dashed line: original formula of Ref.[1] for the damping rate due to phonon-roton scattering. The roton parameters FIG. 2: Phonon damping rates at wavenumber q = mc/2~ in are extracted from their dispersion relation k 7→ ǫk measured −1 unpolarized homogeneous cold-atom Fermi gases in thermo- at various pressures [22]: ∆/kB = 7.44K, k0 = 2.05A˚ , dynamic limit as functions of temperature. (a) At unitarity ′ ′ 2 ′′ − m∗/m = 0.11, ρk0/k0 = 0.39, ρ∆ /∆ = −1.64, ρ ∆ /∆ = 1 ′ a = 0, where most parameters of the phonons and fermionic −8.03, ρm∗/m∗ = −4.7. In Eq.(15), parabolic approxima- quasiparticles are measured (see text). (b) On the BCS side tion (9) is used (hence ǫk∗ /∆ ≃ 1.43). The speed of sound 1/kFa = −0.389, these parameters are estimated in BCS the- c = 346.6 m/s, and the Grüneisen parameter dln c = 2.274 m∗ ∆ ′ dln ρ ory (µ/ǫF ≃ 0.809, ∆/µ ≃ 0.566, m = 2µ , ρµ /µ ≃ 0.602, ′ ′′ entering in Γφφ, are taken from equation of state (A1) of ρ∆ /∆ ≃ 0.815, ρ2∆ /∆ ≃ −0.209, dln c ≃ 0.303). In both ~q kB T −2 dln ρ Ref.[23]. The low values mc = 0.13 and mc2 < 10 jus- cases the curvature parameter γ defined in the caption of tify our use of quantum hydrodynamics. Fig. 1 is estimated in the RPA [37]. Solid line: phonon- phonon (a) Beliaev-Landau damping φ ↔ φφ (for γ > 0) as in Eqs.(121,122) of Ref.[16] (independent of |γ|) and (b) Landau- Khalatnikov damping φφ ↔ φφ (for γ ≃ −0.30 < 0) [6, 16]. Of negligible range, they are characterized by the scat- Dashed line/dash-dotted line: scattering/absorption-emission phonon-fermionic quasiparticle processes, as in Eq.(12)/(15). tering length a tunable by Feshbach resonance [24–29]. In Eq.(15), we took for ǫk (a) the form proposed in Ref.[34] Precise measurements of the fermionic excitation pa- (hence ǫk∗ /∆ ≃ 1.12) and (b) the BCS form (hence ǫk∗ /∆ ≃ −1 rameters k0 and ∆ were performed at unitarity a =0 1.14). µ is the T = 0 gas chemical potential, and the [30]. Due to the unitary-gas scale invariance [31–33], k is plotted quantities are in fact inverse quality factors. Here 0 2 2 proportional to the Fermi wavenumber k = (3π2ρ)1/3, kBT/mc > 0.03 in contrast to Fig.1 where kB T/mc < 0.01: F cold atoms are effectively farther from the T → 0 limit than k0 ≃ 0.92kF [30], and ∆ is proportional to the Fermi scat a-e 2 2 liquid helium, hence the inversion of the Γq -Γq hierarchy. ~ kF energy ǫF = 2m , ∆ ≃ 0.44ǫF [30]. This also de- termines their derivatives with respect to ρ. Similarly, the equation of state measured at T = 0 is simply µ = ξǫF, where ξ ≃ 0.376 [29], and the critical tem- hydrodynamic approximation where it is independent of perature is Tc ≃ 0.167ǫF/kB [29]. For the effective mass the aforementioned third derivative. of the fermionic excitations and their dispersion relation The phononic excitation branch becomes concave in at non vanishing k − k0, we must rely on results of a − the BCS limit kFa → 0 [38]. As visible on Fig. 2b, the dimensional ǫ = 4 − d expansion, m∗/m ≃ 0.56 and 2 2 2 2 phonon-phonon damping (now governed by the Landau- ~ (k −k0 ) ǫk ≃ ∆+ 2 [34]. We also trust Anderson’s RPA 8m∗k0 Khalatnikov processes mentioned earlier) is much weaker, prediction [35, 36] that the q = 0 third derivative of the and dominates the φ − γ damping only at very low tem- phononic dispersion relation is positive [37]. The damp- peratures. At commonly reached temperatures T > ing rates of phonons with wavenumber q = mc/2~ are 0.05ǫF/kB [39], the damping is in fact dominated by plotted in Fig. 2a. The contribution of the three-phonon absorption-emission φ − γ processes which, unlike in liq- Landau-Beliaev processes φ ↔ φφ, here energetically uid helium, prevail over scattering ones because of the allowed, is dominant; it is computed in the quantum- smaller value of ǫk∗ /∆. Although the associated quality 5

−1 factors ωq/Γq may seem impressive, the lifetimes Γq of [9] L. H. Thomas, “The calculation of atomic fields”, Proc. the modes do not exceed one second in a gas of 6Li with Cambridge Phil. Soc. 23, 542 (1927). [10] E. Fermi, “Un metodo statistico per la determinazione di a typical Fermi temperature TF =1µK, which is shorter than what was observed in a Bose-Einstein condensate alcune prioprietàdell’atomo”, Rend. Accad. Naz. Lincei 6, 602 (1927) [“A statistical method to evaluate some [40]. Our predictions, less quantitative than on Fig. 2a, properties of the atom”] and in Collected papers, Note e are based on the BCS approximation for the equation memorie of Enrico Fermi, volume I, edited by E. Amaldi of state and the fermionic excitation dispersion relation et al. (The University of Chicago Press, Chicago, 1962). 2 2 BCS ~ k 2 2 1/2 [11] S. T. Beliaev, “Energy-Spectrum of a Non-ideal Bose ǫk ≃ ǫk = [( 2m − µ) + ∆BCS] and on the RPA 34 7 for the q = 0 third derivative of ωq (whose precise value Gas”, Zh. Eksp. Teor. Fiz. , 433 (1958) [JETP , 299 matters here). A cutting remark on Ref.[41]: even in the (1958)]. [12] L. P. Pitaevskii, S. Stringari, “Landau damping in dilute BCS approximation to which it is restricted, we disagree a-e Bose gases”, Phys. Lett. A 235, 398 (1997). with its expression of Γq . [13] S. Giorgini, “Damping in dilute Bose gases: A mean-field Conclusion – By complementing the local density ap- approach”, Phys. Rev. A 57, 2949 (1998). proximation in Ref.[1] with a systematic low-temperature [14] B. M. Abraham, Y. Eckstein, J. B. Ketterson, M. Kuch- nir, J. Vignos, “Sound Propagation in Liquid 4He”, Phys. expansion, we derived the definitive leading order ex- Rev. 181, 347 (1969). pression of the phonon-roton coupling in liquid helium [15] N. Katz, J. Steinhauer, R. Ozeri, N. Davidson, “Beliaev and we generalized it to the phonon-pair-breaking exci- Damping of Quasiparticles in a Bose-Einstein Conden- tation coupling in Fermi gases. The ever-improving ex- sate”, Phys. Rev. Lett. 89, 220401 (2002). perimental technics in these systems give access to the [16] H. Kurkjian, Y. Castin, A. Sinatra, “Three-phonon and microscopic parameters determining the coupling and al- four phonon interaction processes in a pair-condensed low for a verification in the near future. Our result Fermi gas”, Ann. Phys. (Berlin) 529, 1600352 (2017). [17] N. A. Lockerbie, A. F. G. Wyatt, R. A. Sherlock, “Mea- also clarifies the regime of temperature and interaction surement of the group velocity of 93 GHz phonons in strength in which the purely phononic φφ ↔ φφ Landau- liquid 4He”, Solid State Communications 15, 567 (1974). Khalatnikov sound damping in a superfluid, unobserved [18] W. Dietsche, “Superconducting Al-PbBi tunnel junction to this day, is dominant. as a phonon spectrometer”, Phys. Rev. Lett. 40, 786 This project received funding from the FWO and the (1978). [19] P. Berberich, P. Leiderer, S. Hunklinger, “Investigation of EU H2020 program under the MSC Grant Agreement the lifetime of longitudinal phonons at GHz frequencies No. 665501. in liquid and solid 4He”, Journal of Low Temperature Physics 22, 61 (1976). [20] D. Rugar, J. S. Foster, “Accurate measurement of low- energy phonon dispersion in liquid 4He”, Phys. Rev. B 30, 2595 (1984). [1] L. Landau, I. Khalatnikov, “Teoriya vyazkosti Geliya-II”, [21] E. C. Swenson, A. D. B. Woods, P. Martel, “Phonon Zh. Eksp. Teor. Fiz. 19, 637 (1949) [English translation dispersion in liquid Helium under pressure”, Phys. Rev. in Collected papers of L.D. Landau, chapter 69, pp.494- Lett. 29, 1148 (1972). 510, edited by D. ter Haar (Pergamon, New York, 1965)]. [22] M. R. Gibbs, K. H. Andersen, W. G. Stirling, H. Schober, [2] I. M. Khalatnikov, D.M. Chernikova, “Relaxation phe- “The collective excitations of normal and superfluid 4He: nomena in superfluid Helium”, Zh. Eksp. Teor. Fiz. 49, the dependence on pressure and temperature”, J. Phys. 1957 (1965) [JETP 22, 1336 (1966)]. Condens. Matter 11, 603 (1999). [3] B. F˚ak, T. Keller, M. E. Zhitomirsky, A. L. Chernyshev, [23] H. J. Maris, D. O. Edwards, “Thermodynamic properties “Roton-phonon interaction in superfluid 4He”, Phys. of superfluid 4He at negative pressure”, Journal of Low Rev. Lett. 109, 155305 (2012). Temperature Physics 129, 1 (2002). [4] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. [24] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Smith, Z. Hadzibabic, “Bose-Einstein condensation of Granade, J. E. Thomas, “Observation of a strongly in- atoms in a uniform potential”, Phys. Rev. Lett. 110, teracting degenerate Fermi gas of atoms”, Science 298, 200406 (2013). 2179 (2002). [5] B. Mukherjee, Zhenjie Yan, P. B. Patel, Z. Hadzibabic, [25] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. Mag- T. Yefsah, J. Struck, M. W. Zwierlein, “Homogeneous alhães, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, C. Atomic Fermi Gases”, Phys. Rev. Lett. 118, 123401 Salomon, “Measurement of the interaction energy near a (2017). Feshbach resonance in a 6Li Fermi gas”, Phys. Rev. Lett. [6] H. Kurkjian, Y. Castin, A. Sinatra, “Landau- 91, 020402 (2003). Khalatnikov phonon damping in strongly interacting [26] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Fermi gases”, EPL 116, 40002 (2016). Chin, J. H. Denschlag, R. Grimm, “Collective excitations [7] Martin Zwierlein, private communication (September of a degenerate gas at the BEC-BCS crossover”, Phys. 2017). Rev. Lett. 92, 203201 (2004). [8] C. Cohen-Tannoudji, “Atomic motion in laser light”, [27] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. §2.3, in Proceedings of the Les Houches Summer School, Raupach, A. J. Kerman, W. Ketterle, “Condensation session LIII, edited by J. Dalibard, J.-M. Raimond, J. of pairs of fermionic atoms near a Feshbach resonance”, Zinn-Justin (North-Holland, Amsterdam, 1992). Phys. Rev. Lett. 92, 120403 (2004). 6

′ [28] S. Nascimbène, N. Navon, K. J. Jiang, F. Chevy, C. Sa- [46] At low T , k is close to k0, emission k ↔ q + k is forbid- lomon, “Exploring the thermodynamics of a universal den by energy conservation, and absorption k + q ↔ ′ Fermi gas”, Nature 463, 1057 (2010). k , conserving energy only for q ≥ q∗ ≃ 2m∗c/~, is −~ [29] M. J. H. Ku, A. T. Sommer, L. W. Cheuk, M. W. Zwier- O(e ωq∗ /kBT ). lein, “Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas”, Sci- ence 335, 563 (2012). [30] A. Schirotzek, Y. Shin, C. H. Schunck, W. Ketterle, “De- termination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy”, Phys. Rev. Lett. 101, 140403 (2008). [31] Tin-Lun Ho, “Universal thermodynamics of degenerate quantum gases in the unitarity limit”, Phys. Rev. Lett. 92, 090402 (2004). [32] T. Enss, R. Haussmann, W. Zwerger, “Viscosity and scale invariance in the unitary Fermi gas”, Annals of Physics 326, 770 (2011). [33] Y. Castin, F. Werner, “The Unitary Gas and its Symme- try Properties”, in BCS-BEC Crossover and the Unitary Fermi gas, Springer Lecture Notes in Physics, W. Zw- erger ed. (Springer, Berlin, 2011). [34] Y. Nishida, D. T. Son, “ǫ Expansion for a Fermi gas at infinite scattering length”, Phys. Rev. Lett. 97, 050403 (2006). [35] P.W. Anderson, “Random-phase approximation in the theory of superconductivity”, Phys. Rev. 112, 1900 (1958). [36] R. Combescot, M. Y. Kagan, S. Stringari, “Collective mode of homogeneous superfluid Fermi gases in the BEC- BCS crossover”, Phys. Rev. A 74, 042717 (2006). [37] H. Kurkjian, Y. Castin, A. Sinatra, “Concavity of the collective excitation branch of a Fermi gas in the BEC- BCS crossover”, Phys. Rev. A 93, 013623 (2016). [38] M. Marini, F. Pistolesi, G. C. Strinati, “Evolution from BCS superconductivity to Bose condensation: analytic results for the crossover in three dimensions”, European Physical Journal B 1, 151 (1998). [39] Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, W. Ketterle, “Fifty- fold Improvement in the Number of Quantum Degenerate Fermionic Atoms”, Phys. Rev. Lett. 91, 160401 (2003). [40] F. Chevy, V. Bretin, P. Rosenbusch, K. W. Madison, J. Dalibard, “Transverse Breathing Mode of an Elongated Bose-Einstein Condensate”, Phys. Rev. Lett. 88, 250402 (2002). [41] Z. Zhang, W. Vincent Liu, “Finite-temperature damping of collective modes of a BCS-BEC crossover superfluid”, Phys. Rev. A 83, 023617 (2011). [42] For a thermal phonon wavenumber, this requires kB T ≪ 2 m∗c , a meaningful condition even in the BEC limit of Fermi gases where k0 = 0. 3/2 2 [43] One expands to order T for A1 and T for energy de- ′ ′ ′ nominators, q being deduced from q, k and q /q by en- ′ ′ ′ ~(k−k0)q(u−u ) ~(k−k0)u ergy conservation, q−q = m∗c [1+ m∗c ]+ 2 ′ 2 ~q (u−u ) 5/2 ′ 2m∗c +O(T ) with u and u defined below Eq.(10). [44] We also use energy conservation and the relations 1 + (−1)2sn¯ = eǫ/kB T n¯ to transform the difference of the gain and loss quantum statistical factors. 2 2 2 2 ~ k0 2 1 4α 28 2 2β 2 4α 2 [45] I/(4π) =( 2m∗mc ) [ 25 − 15 + 45 α + 9 +A( 9 − 3 )+ 2 2 4 2 1 α 2 2 4α 2α 4β 4β 3 B A +4βB( 15 − 9 )+B ( 15 − 9 + 3 + 3 )+ 9 B + 9 ], ′ 2 ′′ ′ ρk0 m∗c m∗ρ ∆ 2 ρ∆ 2 with α = k0 , β = ~k0 , A = (~k0) + α , B = ~ck0 .