Variational study of polarons in Bose-Einstein condensates

We use a class of variational wave functions to study the properties of an impurity in a Bose-Einstein condensate, i.e., the Bose polaron. The impurity interacts with the condensate through a contact interaction, which can be tuned by a Feshbach resonance. We find a stable attractive polaron branch that evolves continuously across the resonance to a tight-binding diatomic molecule deep in the positive scattering length side. A repulsive polaron branch with finite lifetime is also observed and it becomes unstable as the interaction strength increases. The effective mass of the attractive polaron also changes smoothly across the resonance connecting the two well-understood limits deep on both sides.

Figure 1

The solid line represents the attractive self-energy ${\Sigma }_{\mathrm{att}}\left(0\right)$. The dotted-dashed line in the negative scattering length side represents the Hartree approximation ${\Sigma }_H=g_{bf}n$ that diverges at the resonance. The dashed curve is the asymptotic bound-state energy calculated from the two-body physics $E_b=-1/\left(m_{\mathrm{red}}{a}_{bf}^2\right)$, where $a_{bf}>0$.

Figure 2

The real part (solid) and the imaginary part (dotted-dashed) of the repulsive polaronic self-energy. The thin dashed line is the asymptotic behavior of the Hartree approximation in the weakly repulsive regime $\left(a_{bf}\to 0^{+}\right)$. In the strongly interacting regime where $k_F{a}_{bf}\gtrsim 1$, the imaginary part becomes comparable to the real part of the self-energy. The repulsive polaron is no longer a well-defined quasiparticle beyond this point.

Figure 3

The rescaled dimensionless effective mass renormalization for different mass ratios $\alpha =m_f/m_b$. Three continuous curves between ${\left(k_F{a}_{bf}\right)}^{-1}=-3$ and ${\left(k_F{a}_{bf}\right)}^{-1}=3$ are of the attractive polarons. The solid, dashed purple, and dotted-dashed brown curves represent $\alpha =1,\alpha =0.2,$ and $\alpha =5$, respectively. A polaron consists of a heavier impurity has a larger rescaled effective mass renormalization. The green curve that appears in the positive side only represents the effective mass of the $\alpha =1$ repulsive polaron. The repulsive polaron becomes unstable beyond the ending point.

Figure 4

The effective mass of the attractive and repulsive polarons in noninteracting (solid) and weakly interacting (dashed, $k_F{a}_{bb}=0.2)$ BEC backgrounds. The continuous curves across the resonance are the attractive polarons. The repulsive polarons only exist in the positive scattering length side of the resonance.