Bose Polarons at Finite Temperature and Strong Coupling

A mobile impurity coupled to a weakly interacting Bose gas, a Bose polaron, displays several interesting effects. While a single attractive quasiparticle is known to exist at zero temperature, we show here that the spectrum splits into two quasiparticles at finite temperatures for sufficiently strong impurity-boson interaction. The ground state quasiparticle has minimum energy at $T_c$, the critical temperature for Bose-Einstein condensation, and it becomes overdamped when $T\gg T_c$. The quasiparticle with higher energy instead exists only below $T_c$, since it is a strong mixture of the impurity with thermally excited collective Bogoliubov modes. This phenomenology is not restricted to ultracold gases, but should occur whenever a mobile impurity is coupled to a medium featuring a gapless bosonic mode with a large population for finite temperature.

Figure 1

Feynman diagrams yielding the self-energy $\mathrm{\Sigma }$ within the “extended” ladder approximation. Thin blue lines represent the bare impurity propagator, solid red lines are Bogoliubov propagators, dashed red lines are condensate bosons, and wavy lines represent the vacuum scattering matrix ${\mathcal{T}}_v$. The ladder self-energy is indicated by ${\mathrm{\Sigma }}_L$, and the double blue line is an impurity dressed by the condensate only.

Figure 2

Spectral function $A(\omega )$ of the attractive polarons (in units of $1/E_n$) versus $T$ at $k_n{a}_B=0.01$ and $k_{n}a=-1$, computed using the ladder approximation (dashed lines and lighter shading) and our extended model (thick lines and darker shading). For clarity, we added an artificial tiny width of $0.01E_n$ to the spectral lines. The thick solid lines in the plane show the polaron energies, as given by Eq. (4), and the width of their shading gives the damping $\mathrm{\Gamma }$. The dash-dotted lines are the low temperature result ${\omega }_0(1\pm \sqrt{Z_0{n}_{\mathrm{ex}}/n_0})$.

Figure 3

Energies (top) and residues (bottom) of the attractive polarons for $1/k_{n}a=0$ in the ladder approximation (dashed line) and from our extended model (solid lines). The width of the shading around the energies gives the damping $\mathrm{\Gamma }$. We stop plotting ${\omega }_{\uparrow }$ when its residue is below 5%. The insets depict the spectral function $A(\omega )$ at $T=0.1T_c$ (left) and $T=0.8T_c$ (right). The red line shows $Z_{\uparrow }+Z_{\downarrow }$.

Figure 4

Energy (top) and residue (bottom) of the attractive polarons versus interaction strength, at $T=0$ (blue), $T=0.5T_c$ (yellow), and $T=T_c$ (red). The filled blue circles in the energy plot indicate the results of the ladder approximation at $T=0$, and the two black lines (filled diamonds) are the mean field results $\omega =2\pi a/m_r$ (right) and $\omega =-1/(2m_r{a}^2)+{\mu }_B$ (left), valid, respectively, for $-1/k_{n}a\gg 1$ and $-1/k_{n}a\ll -1$.

Figure 5

Real part of the self-energy $\mathrm{\Sigma }(\mathbf{p}=0,\omega )$ for $k_{n}a=-0.1$, at $T=0$ (blue dotted line, almost horizontal), $T=0.05T_c$ (dashed) and $T=0.1T_c$ (solid). The different colors correspond to various values of $k_n{a}_B$. The diagonal black line represents $\omega$, and the vertical lines denote $n_0({\mathcal{T}}_v-{\mathcal{T}}_B)$.