Strong-coupling Bose polarons in one dimension: Condensate deformation and modified Bogoliubov phonons

We discuss the interaction of a quantum impurity with a one-dimensional degenerate Bose gas forming a Bose polaron. In three spatial dimension, the quasiparticle is typically well described by the extended Fröhlich model, in full analogy with the solid-state counterpart. This description, which assumes an undepleted condensate, fails, however, in 1D, where the backaction of the impurity on the condensate leads to a self-bound mean-field polaron for arbitrarily weak impurity-boson interactions. We present a model that takes into account this backaction and describes the impurity-condensate interaction as coupling to phononlike excitations of a deformed condensate. A comparison of polaron energies and masses to diffusion quantum Monte Carlo simulations shows very good agreement already on the level of analytical mean-field solutions and is further improved when taking into account quantum fluctuations.

Figure 1

Mean-field solution for different interactions and various couplings. All other parameters are as in Ref. [15], i.e., $M/m=0.47$, the peak density $n_0=7/\mu \mathrm{m}$, and $g_{\mathrm{BB}}=2.36\times {10}^{-37}\mathrm{Jm}$. For the phase, we fixed $u=0.01c$, which fixes the total momentum on the mean-field level.

Figure 2

Polaron mass calculated in mean-field approximation with periodic boundary conditions (PBCs) or constant phase far away from the impurity compared to DMC calculated in Ref. [23]. While the solution using periodic boundary conditions agrees very well with the DMC results, the constant phase solution (or nonperiodic boundary conditions) yields a nonsensical result.

Figure 3

Polaron energy (a) and effective mass (b) for the repulsive polaron. The curves are obtained using different theoretical methods; all parameters are as in Ref. [15], where $\gamma \approx 0.438$. The DMC, RG, and MF (both based on the extended Fröhlich model) curves were calculated in Ref. [23]. We find exceptional agreement with the DMC results for the energy as well as the effective mass when expanding around the right mean-field solutions and including quantum fluctuations. Only for the very strong coupling regime we do not predict a saturation of the effective mass. The condensate deformation becomes relevant for $\eta /n_0\overline{\xi }>1$ Eq. (7), corresponding here to $\eta >1.9$, where predictions from the extended Fröhlich model start to deviate from the full model.

Figure 4

Polaron energy (a) and mass (b) calculated using different methods for attractive impurity couplings. All parameters are as in Ref. [15] and the RG, MF (based on the extended Fröhlich model), and DMC curves were calculated in Ref. [23]. For the polaron energy, a surprisingly good agreement is achieved with the DMC results, while the agreement is less good for the mass. We explain this by the collapse of the solution to a multiparticle bound state, where we do not expect the mean-field solution to be a good approximation. Furthermore, we do not observe the transition from the attractive to the repulsive polaron observed in the Fröhlich model, signaled by the breakdown of the RG treatment. For more detail on this transition, we refer to Refs. [23, 24].