Impurities in Bose-Einstein Condensates: From Polaron to Soliton

We propose that impurities in a Bose-Einstein condensate which is coupled to a transversely laser-pumped multimode cavity form an experimentally accessible and analytically tractable model system for the study of impurities solvated in correlated liquids and the breakdown of linear-response theory. As the strength of the coupling constant between the impurity and the Bose-Einstein condensate is increased, which is possible through Feshbach resonance methods, the impurity passes from a large to a small polaron state, and then to an impurity-soliton state. This last transition marks the breakdown of linear-response theory.

Figure 1

Effective mass versus dimensionless coupling constant and (a) dissipation, (b) temperature, and (c) distance from the ordering transition. Other parameters are set to (a) $\beta =\chi =100$ and $\mathrm{\Gamma }=1$, (b) $\gamma =\chi =100$ and $\mathrm{\Gamma }=1$, and (c) $\beta =\chi =\gamma =100$. In (a) and (b) the red dots indicate the critical points where the discontinuity of the effective mass closes. The dark arrows marked by ${\alpha }_{\mathrm{th}}$ correspond to the dotted line in (c), i.e., $\gamma ,\beta \to \infty$, at $\mathrm{\Gamma }=1$.

Figure 2

Variational free energies as a function of the modulation amplitude ${\rho }_0$. The blue, red, and green (top, bottom, and middle) curves show the condensate free energy ${\mathcal{F}}_{\mathrm{B}}({\rho }_0)$, impurity free energy ${\mathcal{F}}_I({\rho }_0)$, and the total free energy $\mathcal{F}({\rho }_0)={\mathcal{F}}_{\mathrm{B}}({\rho }_0)+{\mathcal{F}}_I({\rho }_0)$, respectively. The absolute minima of total free energies are indicated by black dots in each panel. (a1),(a2) Massive impurities: the free energy of the impurity decreases linearly with ${\rho }_0$. For increasing pseudopotential, the absolute minimum shifts from the neighborhood of the Gaussian minimum to that of the vacuum of the ordered phase—corresponding to a transition from the small polaron to the impurity soliton. Panels (b1) and (b2) include the zero-point energy of a bound-state particle, which follows the red curves. The minimum at ${\rho }_0=0$ corresponds to the large polaron (b1). For increasing pseudopotentials (b2), there may be transitions from the large polaron to the small polaron and then to the soliton, or a single transition directly from the large polaron to the impurity soliton.

Figure 3

The different states of the BEC-impurity system are shown schematically. In each panel, the blue color (surrounding background) indicates the condensate modulation amplitude while the red cloud at the center indicates the particle. From left to right, large polaron, small polaron, and the soliton.