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

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