Polaron formation in the vicinity of a narrow Feshbach resonance

The polaronic system consisting of an impurity in a dilute Bose-Einstein condensate is considered in the presence of a narrow Feshbach resonance. For this purpose a coupled-channel model is used, which at the mean-field level predicts the formation of quasiparticles that are a superposition of the impurity and the molecular states. The impurity-boson interactions and the coupling between the open and closed channels are then considered weak and a perturbative calculation of the corrections to the mean-field results is presented. This allows us to examine the properties of the quasiparticles, such as the lifetime and the effective mass. The relation between the model parameters and the experimental parameters is determined by identifying the low-energy $T$ matrix and applying a proper renormalization scheme.

Figure 1

The Feynman diagrams corresponding to the expressions (25) for the self-energies of the two branches. The dashed lines denote a Bogoliubov excitation and the full lines are the unperturbed upper ($+$) and lower ($-$) quasiparticles.

Figure 2

The quasiparticle dispersions ${\varepsilon }^{\pm }$ at the mean-field level [Eq. (17)] due to the mixing of the energy levels of the molecule and the ${}^6\mathrm{Li}$ impurity in the presence of a Na condensate as a function of the magnetic field $B$ at $k=0$. The dotted lines are the energies of the impurity and the molecule which are shifted due to the presence of the condensate: $E_I^{\text{MF}}=n_0{U}_P^{\uparrow \downarrow }$ and ${\varepsilon }^M=-{\varepsilon }_0^M-n_0{U}^{\uparrow \uparrow }$. The dash-dotted line gives the single-channel result.

Figure 3

(a) The perturbative correction $\Delta E^{\pm }$ to the mean-field quasiparticle energies from Fig. 2 as a function of the magnetic field $B$ at $k=0$ for a ${}^6\mathrm{Li}$ impurity in a Na condensate. The dotted line is the background result (45) and the dot-dashed line is the perturbative single-channel result. (b) The corresponding lifetime of the upper branch.

Figure 4

The lifetimes of the lower (a) and the upper (b) quasiparticle branches are presented as a function of the wave number $k$ for a ${}^6\mathrm{Li}$ impurity in a Na condensate. The inset in (a) shows the lifetime in the absence of a molecule. The dotted lines indicate the landau critical values.

Figure 5

The effective masses are presented for the two branches as a function of the magnetic field $B$ for a ${}^6\mathrm{Li}$ impurity in a Na condensate. In (a) the mean-field result (18) is presented and in (b) the perturbative correction to the inverse effective mass is shown. The dotted line is the background result (48) and the dash-dotted line is the perturbative singe-channel result.