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arXiv:1307.6326 · Europhysics Letters (EPL)

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

C. Trefzger, Yvan Castin

Published Dec 1, 2013• 7 citations
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Abstract

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 $g\to0^-$ 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 $ΔE(\KK)$ of the moving impurity up to order two included in $g$. This also gives access to its longitudinal and transverse effective masses $m_\parallel^*(\KK)$, $m_\perp^*(\KK)$, as functions of the impurity wave vector $\KK$. Depending on the modulus of $\KK$ and on the impurity-to-fermion mass ratio $M/m$ we identify four regions separated by singularities in derivatives with respect to $\KK$ of the second-order term of $ΔE(\KK)$, and we discuss the physical origin of these regions. Remarkably, the second-order term of $m_\parallel^*(\KK)$ presents points of non-differentiability, replaced by a logarithmic divergence for $M=m$, when $\KK$ is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold atom experiments.

DOI
10.1209/0295-5075/104/50005

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We study the properties of an impurity of mass MMM moving through a spatially homogeneous three-dimensional fully polarized Fermi gas of particles of mass mmm. In the weakly attractive limit, where the effective coupling constant g→0−g\to0^-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 ΔE(\KK)ΔE(\KK)ΔE(\KK) of the moving impurity up to order two included in ggg. This also gives access to its longitudinal and transverse effective masses m∥∗(\KK)m_\parallel^*(\KK)m∥∗​(\KK), m⊥∗(\KK)m_\perp^*(\KK)m⊥∗​(\KK), as functions of the impurity wave vector \KK\KK\KK. Depending on the modulus of \KK\KK\KK and on the impurity-to-fermion mass ratio M/mM/mM/m we identify four regions separated by singularities in derivatives with respect to \KK\KK\KK of the second-order term of ΔE(\KK)ΔE(\KK)ΔE(\KK), and we discuss the physical origin of these regions. Remarkably, the second-order term of m∥∗(\KK)m_\parallel^*(\KK)m∥∗​(\KK) presents points of non-differentiability, replaced by a logarithmic divergence for M=mM=mM=m, when \KK\KK\KK is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold atom experiments.
We study the problem of a mobile impurity of mass MMM interacting {\sl via} a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass mmm. Truncating the Hilbert space to at most one pair of particle-hole excitations of the Fermi sea, we determine ground state properties of the polaronic branch other than its energy, namely the polaron quasiparticle residue ZZZ, and the impurity-to-fermion pair correlation function G(x)G(x)G(x). We show that G(x)G(x)G(x) deviates from unity at large distances as −(A_4+B_4cos⁡2k_Fx)/(k_Fx)4-(A\_4+B\_4 \cos 2 k\_F x)/(k\_F x)^4−(A_4+B_4cos2k_Fx)/(k_Fx)4, where k_Fk\_Fk_F is the Fermi momentum; since A_4>0A\_4>0A_4>0 and B_4>0B\_4>0B_4>0, the polaron has a diverging rms radius and exhibits Friedel-like oscillations. In the weakly attractive limit, we obtain analytical results, that in particular detect the failure of the Hilbert space truncation for a diverging mass impurity, as expected from Anderson orthogonality catastrophe; at distances between ∼1/k_F\sim 1/k\_F∼1/k_F and the asymptotic distance where the 1/x41/x^41/x4 law applies, they reveal that G(x)G(x)G(x) exhibits an intriguing multiscale structure.
We study the problem of a single impurity of mass MMM immersed in a Fermi sea of particles of mass mmm. The impurity and the fermions interact through an sss-wave narrow Feshbach resonance, so that the Feshbach length R∗{R}_{*}R∗​ naturally appears in the system. We use simple variational ans\"atze, limited to at most one pair of particle-hole excitations of the Fermi sea, and we determine for the polaronic and dimeronic branches the phase diagram between absolute ground state, local minimum, thermodynamically unstable regions (with negative effective mass), and regions of complex energies (with negative imaginary part). We also determine the closed-channel population which is experimentally accessible. Finally, we identify a nontrivial weakly attractive limit where analytical results can be obtained, in particular for the crossing point between the polaronic and dimeronic energy branches.
We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass mmm perturbed by the presence of a single distinguishable impurity of mass MMM. The interaction between the impurity and the fermions involves only the partial sss wave through the scattering length aaa and has negligible range bbb compared to the inverse Fermi wave number 1/kF1/{k}_{\mathrm{F}}1/kF​ of the gas. Through the interactions with the Fermi gas the impurity gives birth to a quasiparticle, which will be here a Fermi polaron (or more precisely a monomeron). We consider the general case of an impurity moving with wave vector $\mathbf{K}\ensuremath{\ne}\mathbf{0}$: Then the quasiparticle acquires a finite lifetime in its initial momentum channel because it can radiate particle-hole pairs in the Fermi sea. A description of the system using a variational approach, based on a finite number of particle-hole excitations of the Fermi sea, then becomes inappropriate around K=0\mathbf{K}=\mathbf{0}K=0. We rely thus upon perturbation theory, where the small and negative parameter ${k}_{\mathrm{F}}a\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ excludes any branches other than the monomeronic one in the ground state (as, e.g., the dimeronic one), and allows us a systematic study of the system. We calculate the impurity self-energy ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$ up to second order included in aaa. Remarkably, we obtain an analytical explicit expression for ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$, allowing us to study its derivatives in the plane $(K,\ensuremath{\omega})$. These present interesting singularities, which in general appear in the third-order derivatives ${\ensuremath{\partial}}^{3}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$. In the special case of equal masses, M=mM=mM=m, singularities appear already in the physically more accessible second-order derivatives ${\ensuremath{\partial}}^{2}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$; using a self-consistent heuristic approach based on ${\ensuremath{\Sigma}}^{(2)}$ we then regularize the divergence of the second-order derivative ${\ensuremath{\partial}}_{K}^{2}\ensuremath{\Delta}E(\mathbf{K})$ of the complex energy of the quasiparticle found in Trefzger and Castin [Europhys. Lett. 104, 50005 (2013)] at K=kFK={k}_{\mathrm{F}}K=kF​, and we predict an interesting scaling law in the neighborhood of K=kFK={k}_{\mathrm{F}}K=kF​. As a by product of our theory we have access to all moments of the momentum of the particle-hole pair emitted by the impurity while damping its motion in the Fermi sea at the level of Fermi's golden rule.
We theoretically investigate how quasiparticle properties of an attractive Fermi polaron are affected by nonzero temperature and finite impurity concentration in three dimensions and in free space. By applying both non-self-consistent and self-consistent many-body TTT-matrix theories, we calculate the polaron energy (including decay rate), effective mass, and residue, as functions of temperature and impurity concentration. The temperature and concentration dependencies are weak on the BCS side with a negative impurity-medium scattering length. Toward the strong attraction regime across the unitary limit, we find sizable dependencies. In particular, with increasing temperature the effective mass quickly approaches the bare mass and the residue is significantly enhanced. At temperature $T\ensuremath{\sim}0.1{T}_{F}$, where TF{T}_{F}TF​ is the Fermi temperature of the background Fermi sea, the residual polaron-polaron interaction seems to become attractive. This leads to a notable down-shift in the polaron energy. We show that, by taking into account the temperature and impurity concentration effects, the measured polaron energy in the first Fermi polaron experiment [Schirotzek , Phys. Rev. Lett. 102, 230402 (2009)] could be better theoretically explained.
The Fermi polaron, formed by an impurity interacting with a surrounding Fermi sea, exemplifies the canonical quasiparticle concept as a cornerstone in our description of quantum many-body systems across a wide range of energy scales. Experiments on atomic quantum gases have provided profound insights into the universal nature of the Fermi polaron. While most previous studies have focused on the case of zero impurity momentum, finite-momentum properties have remained largely uncharted. Here, we investigate the moving Fermi polaron by combining a novel Raman acceleration scheme with high-precision radio-frequency spectroscopy, exploring the quasiparticle dispersion relation over a wide range of momenta. We compare our measurements of energy shifts and spectral linewidths with a microscopic theory and reach quantitative agreement for all momenta. For low momenta, we find the energy of the moving polaron to be fully consistent with the Fermi liquid picture of a dressed particle with a constant effective mass. At high momenta, the polaron approaches the behavior of a weakly interacting bare particle, featuring small energy shifts and weak broadening. For intermediate momenta, broadening is generally larger and, most strikingly, the behavior differs for attractive and repulsive polarons. While the repulsive polaron exhibits a smooth connection between both regimes along with a monotonic change of the energy shift, the attractive case shows a peculiar non-monotonic behavior. With increasing momentum, the attractive polaron enters a regime where its energy deviates from the constant effective mass expression and broadening suddenly increases. By comparing this observation with theory, we show that this abrupt behavior coincides with the attractive polaron entering a molecule-hole continuum, where it is no longer the ground state. We interpret this as a motion-induced polaron-molecule transition.
An impurity atom immersed in an ultracold atomic Fermi gas can form a quasiparticle, so-called Fermi polaron, due to impurity-fermion interaction. We consider a three-dimensional homogeneous dipolar Fermi gas as a medium, where the interatomic dipole-dipole interaction (DDI) makes the Fermi surface deformed into a spheroidal shape, and, using a Chevy-type variational method, investigate the ground-state properties of the Fermi polaron: the effective mass, the momentum distribution of a particle-hole excitation, the drag parameter, and the medium-density modification around the impurity. These quantities are shown to exhibit spatial anisotropies in such a way as to reflect the momentum anisotropy of the background dipolar Fermi gas. We also give numerical results for the polaron properties at the unitarity limit of the impurity-fermion interaction in the case in which the impurity and fermion masses are equal. It has been found that the transverse effective mass and the transverse momentum drag parameter of the polaron both tend to decrease by $\ensuremath{\sim}10%$ when the DDI strength is raised from 0 up to around its critical value, while the longitudinal ones exhibit a very weak dependence on the DDI.