Tunable Polarons of Slow-Light Polaritons in a Two-Dimensional Bose-Einstein Condensate

When an impurity interacts with a bath of phonons it forms a polaron. For increasing interaction strengths the mass of the polaron increases and it can become self-trapped. For impurity atoms inside an atomic Bose-Einstein condensate (BEC) the nature of this transition is not understood. While Feynman’s variational approach to the Fröhlich model predicts a sharp transition for light impurities, renormalization group studies always predict an extended intermediate-coupling region characterized by large phonon correlations. To investigate this intricate regime and to test polaron physics beyond the validity of the Fröhlich model we suggest a versatile experimental setup that allows us to tune both the mass of the impurity and its interactions with the BEC. The impurity is realized as a dark-state polariton (DSP) inside a quasi-two-dimensional BEC. We show that its interactions with the Bogoliubov phonons lead to photonic polarons, described by the Bogoliubov-Fröhlich Hamiltonian, and make theoretical predictions using an extension of a recently introduced renormalization group approach to Fröhlich polarons.

Figure 1

Ratio of the polaron mass $M_p$ to the bare impurity mass $M$ as a function of the dimensionless coupling constant $\alpha$ in a quasi-two-dimensional BEC for different ratios of impurity to host-atom mass $M/m$. Feynman’s approach predicts a sharp transition for $M/m\lesssim 0.01$, in contrast to predictions from mean-field (MF) theory and an extended renormalization group (RG) approach introduced in Ref. [12].

Figure 2

Setup for realizing tunable Fröhlich polarons of photons in a BEC: A quasi-two-dimensional BEC of ground state atoms (b) coupled to lasers in a $\mathrm{\Lambda }$ scheme (a). By exciting the driven atoms using a probe field $\widehat{\mathcal{E}}$, a mobile impurity (a long-lived DSP) can be created. Its interactions with the Bogoliubov phonons lead to polaron formation. The mass of the impurity, as well as the polaronic coupling constant, can be tuned by changing the Rabi frequency ${\mathrm{\Omega }}_c$ of the control laser.

Figure 3

Full phase diagram of the two-dimension Bogoliubov-Fröhlich model: For sufficiently light impurities and large enough $\alpha$, an extended regime of intermediate couplings is found (as a guide to the eye, the dashed line indicates where the crossover takes place). The color plot shows the exponent $\kappa$ of the power law $M_p/M-1\sim {\alpha }^{\kappa }$, determined from the slope of curves as in Fig. 1. Parameters used in the RG simulations were ${\mathrm{\Lambda }}_0=2000/\xi$ and $P=0.01Mc$ (for their definition, see Ref. [12]). We also plotted the maximum values ${\alpha }_{\max }$ below which the Fröhlich model is valid, for $\eta =1$, 10 as defined in the text (dash-dotted lines).