A unified theory of strong coupling Bose polarons: From repulsive polarons to non-Gaussian many-body bound states

TL;DR

This work addresses the strong-coupling Bose polaron problem by developing a variational framework that couples Gaussian correlations in the scattering Bogoliubov modes with exact non-Gaussian correlations in the bound impurity–boson sector, enabling a nonperturbative treatment near a Feshbach resonance. By exploiting a large separation of energy scales, the authors decompose the problem into a bound-mode part described exactly (via a many-body bound-state basis) and a scattering part treated through a coherent state, yielding a tractable energy functional whose minimization reveals a discrete set of metastable many-body bound states with energies between the repulsive and attractive polaron branches. These states exhibit strong non-Gaussian correlations, antibunching in dimer occupations, and significant molecular spectral weight, predicting observable signatures in molecular spectroscopy and a unified picture in which the remnant of the attractive polaron on the repulsive side corresponds to the lowest bound state. The approach provides a general, all-coupling perspective on Bose polarons, with implications for impurity–BEC systems across dimensions and potential extensions to other bosonic environments and spectroscopic probes.

Abstract

We address the Bose polaron problem of a mobile impurity interacting strongly with a host Bose-Einstein condensate (BEC) through a Feshbach resonance. On the repulsive side at strong couplings, theoretical approaches predict two distinct polaron branches corresponding to attractive and repulsive polarons, but it remains unclear how the two are related. This is partly due to the challenges resulting from a competition of strongly attractive (destabilizing) impurity-boson interactions with weakly repulsive (stabilizing) boson-boson interactions, whose interplay is difficult to describe with contemporary theoretical methods. Here we develop a powerful variational framework that combines Gaussian correlations among impurity-boson scattering states, including up to an infinite number of bosonic excitations, with exact non-Gaussian correlations among bosons occupying an impurity-boson bound state. This variational scheme enables a full treatment of strong nonlinearities arising in the Feshbach molecule on the repulsive side of the resonance. Within this framework, we demonstrate that the interplay of impurity-induced instability and stabilization by repulsive boson-boson interactions results in a discrete set of metastable many-body bound states at intermediate energies between the attractive and repulsive polaron branches. These states exhibit strong quantum statistical characteristics in the form of non-Gaussian quantum correlations, requiring non-perturbative beyond mean-field treatments for their characterization. Furthermore, these many-body bound states have sizable molecular spectral weights, accessible via molecular spectroscopy techniques. This work provides a unified theory of attractive and repulsive Bose polarons on the repulsive side of the Feshbach resonance.

Figures (9)

• Figure 1: Schematic illustration of the Bose polaron spectrum across an impurity-boson Feshbach resonance for repulsively interacting bosons. In the presence of inter-boson interactions, the attractive polaron persists to the repulsive side as a well-defined resonance, while other metastable many-body bound states appear in addition to the repulsive polaron. These many-body bound states emerge due to the competition of multiple impurity-boson binding and inter-boson repulsion. The structure of the main component of each many-body bound state is shown schematically.
• Figure 2: (a) Energy of polaron states, including attractive and repulsive polaron, and metastable states $ms_1$ to $ms_6$ (see text), across an impurity-boson Feshbach resonance. On the attractive side ($a<0$), an impurity resonance exists corresponding to the attractive polaron branch (green dashed line), which extends to the repulsive side and remains the well-defined stable saddle point across the resonance. On the repulsive side, the repulsive polaron branch emerges as the unstable saddle point solution with a bound state, as well as two many-body bound states $ms_1$ and $ms_2$ (red and blue solid lines). The red dotted line indicates the bare dimer energy. Beyond a critical scattering length (denoted by a vertical black dotted line), further metastable many-body bound states $ms_3$ to $ms_6$ emerge in the spectrum (grey shaded solid lines). Note that the normalized energy is rescaled to show all bound states compactly. The grey-shaded region (2) on the repulsive side is bounded by $1/k_n a \simeq 1.2$ where $\mu/\varepsilon_{\mathrm{B}} \simeq 9 \times 10^{-3}$, providing a conservative bound for the validity of our theory. (b) The energy landscape over the phase space of the bound Bogoliubov mode, around the saddle points corresponding to different regions in (a). The real and imaginary parts of the coherent state variable $\alpha_{\mathrm{B}}$ serve as coordinates for the phase space of the bound Bogoliubov mode. In (1), the attractive polaron (green shaded point) is a stable saddle point, with all the fluctuation modes having positive energy. Within region (2), a dynamical instability occurs as a precursor to the formation of the repulsive polaron, signified by a single unstable phase mode with a corresponding stable amplitude mode. In (3), the repulsive polaron (purple shaded dot) is a saddle point solution with a single unstable Bogoliubov mode. The energy and particle number of many-body bound states in (a) are depicted qualitatively on the energy surfaces. The radius of each circle denotes the mean bound state occupation number, while its position on the surface denotes the energy of the state. Repulsive inter-boson interaction increases the energy of the many-body bound state with a higher particle number. By increasing $1/k_n a$, further many-body bound-states enter the atom-dimer continuum (grey shaded solid lines). Increasing the binding energy increases the number of bound bosons in the lowest many-body bound state. The vertical black dashed lines mark the level crossings between many-body bound states.
• Figure 3: Density profile of the repulsive polaron (solid purple line), attractive polaron (green dashed line), and $ms_1$ state (solid red line), as a function of the radial distance from the impurity, for (a) $1/k_n a = 2.0$ and (b) $1/k_n a = 3.61\,$. The density profiles of the attractive polaron and the $ms_1$ state are qualitatively similar.
• Figure 4: Energy in units of the dimer binding energy (a) and mean bound state occupation number (b) of the many-body bound states (red, blue and grey solid lines for $ms_1$, $ms_2$ and $ms_3$ respectively), attractive polaron (green dashed line), and repulsive polaron (purple solid line). Initially, the $ms_2$ state has higher mean bound state occupation number and energy than the $ms_1$ state, indicating the dominant effect of the inter-boson interaction on the energy of the states. Beyond the first level crossing, the mean occupation number of the $ms_1$ state increases above the $ms_2$ state due to the gain in energy from binding more bosons. The $ms_3$ state enters the dimer-boson continuum at the critical scattering length indicated by vertical dotted line in panel (b) and maintains an almost constant $N_{\mathrm{B}}\simeq3$. For increasing $1/k_n a$, the mean bound state occupation number of $ms_1$ and $ms_2$ states approach integer values. At the level crossing between $ms_1$ and $ms_3$, the states strongly mix, resulting in sharp spikes in $\langle N_{\mathrm{B}}\rangle$ in panel (b).
• Figure 5: Energy of the many-body bound states including the effect of condensate distortion obtained by fully solving Eqs. \ref{['eq:var-eq']} (dotted lines), compared to the energies obtained by setting $\alpha_{\bm{\mathrm{x}}}=0$. Including condensate distortion effects results in marginal changes in the energy (denoted by $\Delta E_{ms_i}, i= 1,2,3\,$), and wave function of many-body bound states.