Energy, decay rate, and effective masses for a moving polaron in a Fermi sea: Explicit results in the weakly attractive limit

We study the properties of an impurity of mass M moving through a spatially homogeneous three-dimensional fully polarized Fermi gas of particles of mass m. In the weakly attractive limit, where the effective coupling constant g → 0 and perturbation theory can be used, both for a broad and a narrow Feshbach resonance, we obtain an explicit analytical expression for the complex energy ∆E(K) of the moving impurity up to order two included in g. This also gives access to its longitudinal and transverse effective masses m∗‖(K), m ∗ ⊥(K), as functions of the impurity wave vector K. Depending on the modulus of K and on the impurity-to-fermion mass ratio M/m we identify four regions separated by singularities in derivatives with respect to K of the second-order term of ∆E(K), and we discuss the physical origin of these regions. Remarkably, the second-order term of m∗‖(K) presents points of non-differentiability, replaced by a logarithmic divergence for M = m, when K is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold atom experiments. Introduction. – Recent cold atom experiments have reached an unprecedented accuracy in measuring the equation of state of an interacting Fermi gas [1–4]. This has allowed to confirm that in strongly spin-polarized configurations the minority atoms dressed by the Fermi sea of the majority atoms form a normal gas of quasiparticles called Fermi polarons [5–7]. While the problem of a single impurity at rest in a Fermi sea has been thoroughly studied [5–15], previous works on a moving impurity have focused only on its decay rate [10, 16]. However, the real part of the polaronic complex energy is also of theoretical interest and is experimentally accessible [17]. Whereas most theories for the impurity at rest successfully used a variational ansatz [6], this ansatz is not reliable anymore for a moving polaron with momentum ~K since at low K it wrongly predicts a zero decay rate in contrast to the K-law found in [10,16]. In this work, we focus on the weakly attractive regime kFa → 0, where a is the s-wave scattering length of a minority atom and a fermion and kF is the Fermi wave number of the majority atoms. Using a systematic expansion in powers of kFa, we go beyond the Fermi liquid description of [16]: We determine not only the decay rate, but also the real part of the polaronic complex energy, and being not restricted to low momenta, we have access to momentum-dependent effective masses of the polaron. Within our microscopic approach we have a complete description of the system: We determine regions of parameters separated by singularities in derivatives of the secondorder term of the polaronic complex energy, as shown in Fig.1. We discuss the physical origin of the singularities and the relevance for current cold atom experiments. The model. – At zero temperature, we consider in three dimensions an ideal Fermi gas of particles of same spin state and mass m perturbed by the presence of a moving impurity of mass M and momentum ~K. While we assume no interactions among the fermions (contrary to [18]), the impurity interacts with each fermion through a s-wave interaction of negligible range. The system is enclosed in a quantization volume V with periodic boundary conditions, and is described by the Hamiltonian Ĥ = Ĥ0 + V̂ : Ĥ0 = ∑