We study the problem of a mobile impurity of mass
M interacting {\sl via} a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass
m. 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
Z, and the impurity-to-fermion pair correlation function
G(x). We show that
G(x) deviates from unity at large distances as
−(A_4+B_4cos2k_Fx)/(k_Fx)4, where
k_F is the Fermi momentum; since
A_4>0 and
B_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 and the asymptotic distance where the
1/x4 law applies, they reveal that
G(x) exhibits an intriguing multiscale structure.
Based on the recently developed interaction renormalization for the one-dimensional
p-wave interaction, we study the problem of a single impurity immersed in a highly polarized Fermi sea. They interact through a narrow
p-wave Feshbach resonance, so the effective range
r0 naturally appears in the system. We use the variational approach limited to single-particle--hole excitations on top of the unperturbed Fermi sea. The polaron exhibits two branches of solutions, namely, the attractive and repulsive branches, with varying scattering length across the resonance. We calculate the energy spectrum, residue, and effective mass for each of the branches. We compare the polaronic energy with the energy of the dressed molecule and find that the molecular state is energetically favored when increasing the interaction strength. The critical interaction strength for the polaron-to-molecule transition will shift to the BCS side of the
p-wave resonance as the effective range increases.
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 $ΔE(\KK)$ 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.
We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass
m perturbed by the presence of a single distinguishable impurity of mass
M. The interaction between the impurity and the fermions involves only the partial
s wave through the scattering length
a and has negligible range
b compared to the inverse Fermi wave number
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. 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
a. 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=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=kF, and we predict an interesting scaling law in the neighborhood of
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.
Abstract Using the variational method, we study the properties of a spin-down impurity immersed in a one-dimensional (1D) spin-up Fermi sea. With repulsive interactions between them, the impurity is dressed up by surrounding particles in Fermi sea and forms a polaron. We clearly calculate the binding energy, effective mass, momentum distribution, Tan contact, and pair correlation. Even in strong repulsive regimes, the results can agree with the exact Bethe Ansatz results. The repulsive polaron energy E + is below Fermi energy E F and no negative effective masses are found in whole interaction regimes, unequal masses polarons are also calculated. We show a clear momentum distribution and calculate the Tan contact from three different aspects. Furthermore, we explore the particle-hole excitation and find that the hole terms in Fermi sea have a great influence on the polaron energy and contact in repulsive regime. These results show that the variational method can still be used effectively in 1D repulsive polaron system .
We briefly review some current theoretical and experimental aspects of the problem of a single spinless impurity in a 3D polarised atomic Fermi gas at zero temperature where the interactions can be tuned using a wide Feshbach resonance. We show that various few-body states in vacuum composed of the impurity and background gas atoms (single impurity, dimer, trimer, tetramer) give rise to corresponding dressed states ({\em polaron}, {\em dimeron}, {\em trimeron}, {\em tetrameron}) in the gas and inherit many of their characteristics. We study the ground state focussing on the choice of wave function and its properties. We raise a few unsolved problems: whether the polaron and dimeron are really separate branches, what other few-body states might exist, the nature of the groundstate for large numbers of particle-hole pairs and why is the polaron ansatz so good. We then turn to the excited states, and to the calculation of the effective mass. We examine the bounds on the effective mass and raise a conjecture about that of composite quasiparticle states.