Bose polaron and the Efimov effect: A Gaussian-state approach

Since the Efimov effect was introduced, a detailed theoretical understanding of Efimov physics has been developed in the few-body context. However, it has proven challenging to describe the role Efimov correlations play in many-body systems such as quenched or collapsing Bose-Einstein condensates (BECs). To study the impact the Efimov effect has in such scenarios, we consider a light impurity immersed in a weakly interacting BEC, forming a Bose polaron. In this case, correlations are localized around the impurity, making it more feasible to develop a theoretical description. Specifically, we employ a variational Gaussian state Ansatz in the reference frame of the impurity, capable of capturing both the Efimov effect and the formation of a polaron cloud consisting of a macroscopic number of particles. We find that the Efimov effect entails cooperative binding of bosons to the impurity, leading to the formation of large clusters. These many-particle Efimov states exist for a wide range of scattering lengths, with energies significantly below the polaron energy. As a result, the polaron is not the ground state, but rendered a metastable excited state which can decay into these clusters. While this decay is slow for small interaction strengths, it becomes more prominent as the attraction increases, up to a point where the polaron becomes completely unstable. We show that the critical scattering length where this happens can be interpreted as a many-body shifted Efimov resonance, where the scattering of two excitations of the bath with the polaron can lead to polaron-cloud assisted bound-state formation. Compared to the few-body case, the resonance is shifted to weaker attraction due to the participation of the polaron cloud in the cooperative binding process. This represents an intriguing example of chemistry in a quantum medium [A. Christianen et al., Phys. Rev. Lett. 128, 183401 (2022)], where many-body effects lead to a shift in the resonances of the chemical recombination, which can be directly probed in state-of-the-art experiments.

Figure 1

Illustration of the main results of this work. Here we show the wave number $\kappa =-\frac{\sqrt{M\left|E\right|}}{\mathrm{\Lambda }}$ of the polaron found using Gaussian states (blue solid line) and mean-field theory (red dash-dotted line) as a function of the inverse scattering length $1/a\mathrm{\Lambda }$. In gray the wave number of the Efimov trimer (solid line) and larger Efimov clusters (dashed lines) is shown. Note that the dashed lines are only representative and not all possible Efimov clusters are displayed. For small scattering lengths (green area), the polaron is the ground state of the system and a stable quasiparticle. For $\left|a\right|>|a_{-}^{\left(\text{min}\right)}|$, the smallest scattering length at which an Efimov cluster can be formed, the polaron turns into a metastable excited state, which can decay into the large Efimov clusters via many-body scattering processes. These scattering processes become more likely as the absolute value of the scattering length increases. When a critical scattering length $a^{*}$ is reached, the polaron ceases to exist as a well-defined quasiparticle. At this point scattering of the polaron with one or two more bath particles can lead to the rapid decay into an Efimov cluster. In contrast, in mean-field theory, the polaron remains stable up to a positive scattering length $a_0$, at which an infinite number of particles piles up in two-body bound states formed with the impurity.

Figure 2

Contour plot of the energy per particle $E/\langle N\rangle$ as a function of the number of particles $\langle N\rangle$ and the scattering length $a$ in terms of the three-body parameter $\mathrm{\Lambda }$ for $M/m=6/133$. The energy is given in units of ${\mathrm{\Lambda }}^2/M$. The bold line indicates the critical scattering length at which a bound state first appears. The figure is also shown in our related work [45].

Figure 3

Spatial extent $R$, defined by Eq. (33), of an Efimov cluster as a function of the number of particles $\langle \widehat{N}\rangle$ for different scattering lengths $a\mathrm{\Lambda }$ (indicated by color). The lines terminate when the bound states disappear into the continuum (see Fig. 2). The mass ratio is $M/m=6/133$.

Figure 4

Kinetic energy of (a) the bosons and (b) the impurity compared to the total interaction energy, as a function of the scattering length $a$ (in terms of $\mathrm{\Lambda }$), for several values of $\langle N\rangle$ and $M/m=6/133$. The lines terminate when the bound states disappear into the continuum (see Fig. 2). The legend in (a) corresponds also to (b).

Figure 5

Illustration of the energy landscape of the extended Fröhlich model (1) as a function of the number of excitations $N_{\text{ex}}$ surrounding the impurity at a fixed density and scattering length in the (a) stable polaron, (b) metastable polaron, and (c) unstable polaron regimes, which correspond to the regimes shown in Fig. 1. In the stable polaron regime, there are no Efimov clusters to decay into, in the metastable regime there is a barrier protecting the polaron from this decay, and in the unstable regime this barrier has completely disappeared. The black circle and square in (b), connected by the dashed line, are of relevance for Fig. 7.

Figure 6

Polaron quasiparticle weight $Z$ as a function of the interparticle distance $n_0^{-1/3}\mathrm{\Lambda }$ and scattering length $a$ for mass ratio $M/m=6/133$. Two different interboson scattering lengths are chosen: (a) $a_B\mathrm{\Lambda }=0.01$ and (b) $a_B\mathrm{\Lambda }=0.1$. The black solid line indicates the critical scattering length $a^{*}$ at which the polaron breaks down. The dotted line in (b) is a guide to highlight the difference between (a) and (b) and corresponds to the solid line in (a). The red region close to the solid line in (a) is a region of dynamic instability. The dashed lines indicate contours of the quasiparticle weight $Z_{\text{MF}}$ obtained from mean-field theory.

Figure 7

Quantitative analysis of the polaronic instability mechanism. The number of excitations $\langle {\widehat{N}}_{\text{ex}}\rangle$ contained in the polaron cloud (Gaussian state, dashed lines; coherent state, dotted lines) compared to the critical number of particles needed for bound-state formation (solid lines) in the presence of a background BEC is plotted versus the scattering length $a\mathrm{\Lambda }$. The interboson scattering length is given by (a) $a_B\mathrm{\Lambda }=0.01$ and (b) $a_B\mathrm{\Lambda }=0.1$. The black solid line indicates the critical particle number for bound-state formation in the absence of a background BEC, corresponding to the line of $a_c$ displayed in Fig. 2. Crosses indicate where the polaron is destabilized in the Gaussian state Ansatz. Diamonds indicate the position beyond which a dynamical instability occurs. The legend in (a) also applies to (b).

Figure 8

Relative energy difference $(E-E_{\text{MF}})/E_{\text{MF}}$ between the energy $E$ obtained from the Gaussian state and mean-field theory $E_{\text{MF}}$ as a function of the density $n_0$ and scattering length $a$ for mass ratio $M/m=6/133$. Two different interboson scattering lengths are chosen: (a) $a_B\mathrm{\Lambda }=0.01$ and (b) $a_B\mathrm{\Lambda }=0.1$. The black solid lines indicate $a^{*}$. The dotted line in (b) corresponds to the solid line of (a) and is shown to more clearly indicate the difference between (a) and (b). Note that the color scale is not linear but cubic.

Figure 9

(a) Quasiparticle weight $Z$ and (b) fractional energy difference $(E-E_{\text{MF}})/E_{\text{MF}}$ as a function of the density $n_0$ and scattering length $a$ for mass ratio $M/m=6/23$ and $a_B\mathrm{\Lambda }=0.1$. Solid lines indicate the critical scattering length $a^{*}$.