Strong-coupling Bose polarons in a Bose-Einstein condensate

We use a nonperturbative renormalization group approach to develop a unified picture of the Bose polaron problem, where a mobile impurity is strongly interacting with a surrounding Bose-Einstein condensate (BEC). A detailed theoretical analysis of the phase diagram is presented and the polaron-to-molecule transition is discussed. For attractive polarons we argue that a description in terms of an effective Fröhlich Hamiltonian with renormalized parameters is possible. Its strong-coupling regime is realized close to a Feshbach resonance, where we predict a sharp increase of the effective mass. Already for weaker interactions, before the polaron mass diverges, we predict a transition to a regime where states exist below the polaron energy and the attractive polaron is no longer the ground state. On the repulsive side of the Feshbach resonance we recover the repulsive polaron, which has a finite lifetime because it can decay into low-lying molecular states. We show for the entire range of couplings that the polaron energy has logarithmic corrections in comparison with predictions by the mean-field approach. We demonstrate that they are a consequence of the polaronic mass renormalization which is due to quantum fluctuations of correlated phonons in the polaron cloud.

Figure 1

The polaron branch of the spectral function $I\left(\omega \right)$ is shown for an impurity in a BEC at different values of the impurity-boson scattering length $a_{\mathrm{IB}}$. Its qualitative features change at the critical values $a_{\mathrm{IB},\pm }$ indicated in the phase diagram at the bottom of the figure. We neglected direct phonon-phonon interactions (Bogoliubov approximation), leading to a breakdown of the RG in the central region $④$. The same parameters were used as in Fig. 2 of Ref. [24], i.e., $M/m_{\mathrm{B}}=1$ and $n_0=0.25{\xi }^{-3}$, but for a UV cutoff ${\mathrm{\Lambda }}_0={10}^3/\xi$. Here $\xi$($c$) is the healing length (speed of sound) in the BEC, $a_{\mathrm{IB}}$ denotes the scattering length, and $M$ and $m_{\mathrm{B}}$ are impurity and boson masses, respectively.

Figure 2

Close to the Feshbach resonance the quasiparticle weight of the polaron vanishes. Polarons in this regime can be prepared by adiabatically changing the interaction strength. For comparison mean-field (MF) calculations are shown. We used the same parameters as in the recent experimental observation of Bose polarons [13], $M/m_{\mathrm{B}}=1$ and $n_0=2.3\times {10}^{14}{\mathrm{cm}}^{-3}$, and a UV cutoff ${\mathrm{\Lambda }}_0=1/60a_0$ corresponding to the inverse characteristic range $r^{*}$ of the Feshbach resonance [47] estimated in Ref. [13] ($a_0$ is the Bohr radius).

Figure 3

Our RG calculations predict a divergence of the effective polaron mass $M_{\mathrm{p}}$ on the attractive side of the Feshbach resonance. For the RG the renormalized mass $\mathcal{M}\approx M_{\mathrm{p}}$ was calculated and for comparison mean-field calculations are also shown. We used the same parameters as in the recent experimental observation of Bose polarons [13]; see also Fig. 2.

Figure 4

(a) The phase diagram of the zero-dimensional toy model Eq. (7) is shown as a function of the interaction strength $g_{\mathrm{IB}}$. Different regimes are distinguished by the properties of the polaron saddle point. (b) The spectral function is calculated for the toy model in the three different regions discussed in the text.

Figure 5

The phase diagram of an impurity in a BEC obtained from MF theory: On the attractive side the saddle-point solution describes a stable polaron where both phase (${\vartheta }_k$) and particle-number ($\delta n_k$) fluctuations cost a positive energy $E$ (region I). Beyond $1/a_{\text{IB},-}^{\text{MF}}$ an unstable mode appears which is bound to the impurity, where phase fluctuations provide an unstable direction (region II). For $1/a_{\text{IB}}>1/a_{\text{IB},+}^{\text{MF}}$ particle-number fluctuations provide another unstable direction. Together with the phase fluctuations they form a molecular bound state at energies below the polaron energy $E_0^{\text{MF}}$ (region IV). For comparison we show how the phase diagram changes when quantum fluctuations are taken into account in the RG (gray).

Figure 6

We summarize the main features of the RG flow which is obtained from analyzing divergencies of the coupling constants $G_{\pm }\left(\mathrm{\Lambda }\right)$, $\mathcal{M}\left(\mathrm{\Lambda }\right)$, and $\beta \left(\mathrm{\Lambda }\right)$.

Figure 7

The polaron energy $E_0$ is calculated using different approaches. It is dominated by the Fröhlich part of the Hamiltonian. We used the same parameters as in Fig 2 of Ref. [24], i.e., $M/m_{\mathrm{B}}=1$ and $n_0=0.25{\xi }^{-3}$, but for a UV cutoff ${\mathrm{\Lambda }}_0={10}^3/\xi$.

Figure 8

The polaron phase diagram can be obtained from the coupling constants $G_{\pm }$, as described in the text. We plot their fully converged values in the IR limit as a function of the inverse two-particle scattering length $a_{\text{IB}}$. Sufficiently far on the attractive side neither of the two coupling constants diverges in the RG and the ground state is an attractive Fröhlich-type polaron. At $a_{\text{IB},-}^{\mathrm{RG}}$ the coupling $G_{-}$ associated with phase fluctuations diverges and an unstable mode appears at energies below the polaron energy. At $a_{\text{IB},+}^{\mathrm{RG}}$ the coupling $G_{+}$ associated with particle-number fluctuations also diverges and molecular bound states appear below the polaron energy. In the shaded region the RG breaks down. On the repulsive side for $1/a_{\mathrm{IB}}>1/a_{\text{IB},+}^{\mathrm{MF}}$ we obtain a repulsive Fröhlich polaron which can decay into the molecular states. We used the same parameters as in Fig. 7; cf. Ref. [24].

Figure 9

We compare predictions for the polaron energy, obtained by the RG approach presented here, with recent measurements from radio-frequency absorption spectra. The experimental data points correspond to the center of the peak in the absorption spectrum whereas our theoretical curves correspond to the polaron energy and do not take into account incoherent excitations. In (a) we compare to the experiment by Jørgensen et al. [13] with a mixture of different hyperfine states of ${}^{39}\mathrm{K}$. The parameters used for our theoretical analysis are the same as in Figs. 3 and 2. In (b) we compare to the experiment by Hu et al. [14] with a mixture of ${}^{87}\mathrm{Rb}$ and ${}^{40}\mathrm{K}$ impurities. In this case we used $n_0=1.8\times {10}^{14}{\mathrm{cm}}^{-3}$, $a_{\mathrm{BB}}=100a_0$ and we employed a UV cutoff ${\mathrm{\Lambda }}_0={10}^3/\xi$.