Quasiparticle properties of long-range impurities in a Bose condensate

TL;DR

This work analyzes Bose polarons formed by long-range impurities in a weakly interacting Bose condensate, focusing on the interplay of three length scales: $n_0^{-1/3}$, $r_{ ext{eff}}$, and $\xi$. It employs two complementary methods—the coherent-state variational approach in momentum space (including beyond-Fröhlich terms) and a perturbative Gross-Pitaevskii theory—to compute polaron energy, residue $Z$, and effective mass $m_{ ext{eff}}$, across regimes of coupling strength and bath density. A key finding is that the dimensionless coupling $|a_{ ext{IB}}|n_0^{1/3}$ remains the principal parameter, with finite $Z$ and well-defined $m_{ ext{eff}}$ at weak to intermediate coupling, while increasing $r_{ ext{eff}}n_0^{1/3}$ progressively suppresses $Z$ due to occupation of molecular states; resonances shift in density- and interaction-dependent ways, especially when $\xi \lesssim r_{ ext{eff}}$. Across ionic and Rydberg impurities, the two theories agree qualitatively and quantitatively, and bath–bath interactions (through $a_{BB}$) generally enhance quasiparticle robustness, suggesting practical avenues for observing and controlling Bose polarons in ultracold gases.

Abstract

An impurity immersed in a Bose condensate can form a quasiparticle known as a Bose polaron. When the impurity-boson interaction is short-ranged, the quasiparticle properties can be characterized in terms of the impurity-boson scattering length $a_{\mathrm{IB}}$ and the condensate coherence length $ξ$, a universal description that remains valid irrespective of the bath density $n_0$. Long-ranged interactions -- such as provided by Rydberg or ionic impurities -- introduce an effective interaction range $r_{\mathrm{eff}}$ as the third length scale. These competing length scales raise the question of whether a universal description remains valid across different bath densities. In this study, we discuss the quasiparticle nature of long-range impurities and its dependence on the length scales $n_0^{-1/3}$, $r_\mathrm{eff}$, and $ξ$. We employ two complementary theories -- the coherent state Ansatz and the perturbative Gross-Pitaevskii theory -- which incorporate beyond-Fröhlich interactions. We derive an analytical expression for the beyond-Fröhlich effective mass for a contact interaction and numerically compute the effective mass for long-range impurities. We argue that the coupling parameter $|a_{\mathrm{IB}}|n_0^{1/3}$ remains the principal parameter governing the properties of the polaron. For weak ($|a_\mathrm{IB}|n_0^{1/3}\ll 1$) and intermediate ($|a_\mathrm{IB}|n_0^{1/3}\simeq 1$) values of the coupling parameter, long-range impurities in a Bose condensate are well-described as quasiparticles with a finite quasiparticle weight and a well-defined effective mass. However, the quasiparticle weight becomes significantly suppressed as the effective impurity volume is occupied by an increasing number of bath particles ($r_{\mathrm{eff}}n_0^{1/3} \gg 1$).

Figures (6)

• Figure 1: The quasiparticle properties of the short-range impurity: (a) the polaron energy $E_{\mathrm{pol}}$, (b) the residue $Z$, and (c) the effective masses $m_{\mathrm{eff}}$, as a function of ${a}^{-1}_{\mathrm{IB}}$, calculated by the CS ansatz (red solid) and GP theory (blue solid). The black dashed line is the MF polaron energy $E_{\mathrm{MF}}$. The Fröhlich effective mass (blue dashed) is also shown. The numerical computations are executed with the momentum cut-off $\Lambda n_0^{-1/3} = 30$, the step size $d{k} n_0^{-1/3}=0.1$, and $a_{\mathrm{BB}} = 0.01n_0^{-1/3}$. The gray shaded areas indicate the parameter regime in which the density around the impurity violates the small gas parameter assumption.
• Figure 2: The quasiparticle properties of the ionic impurity calculated by the GP theory (blue solid) and CS ansatz (red solid): (a) The polaron energy $E_{\mathrm{pol}}$, (b) the quasiparticle weight $Z$, (c) the effective mass $m_{\mathrm{eff}}$, as a function of the well depth $b$ of the ionic potential. The dashed magenta (black) line in (a) represents $E_{\mathrm{zr}}$ ($E_{\mathrm{MF}}$). The blue dashed line is the Fröhlich effective mass. We set $a_{\mathrm{BB}} = 0.05 r_{\mathrm{ion}}$ and $n_0 r_{\mathrm{ion}}^3 = 1$. The numerical parameters for the momentum are $dkr_{\mathrm{ion}}= 0.1$ and $\Lambda r_{\mathrm{ion}}= 30$. The gray shaded areas indicate where the small gas parameter assumption breaks down.
• Figure 3: The quasiparticle properties of the Rydberg impurity in a weakly interacting Bose condensate, calculated by the CS ansatz (red lines) and the perturbative GP theory (blue lines): (a) The polaron energy $E_{\mathrm{pol}}$, (b) the quasiparticle weight $Z$, (c) the effective mass $m_{\mathrm{eff}}$, as a function of the electron-bath particle scattering length $a_{\mathrm{s}}$. The dashed magenta (black) line in (a) represents $E_{\mathrm{zr}}$ ($E_{\mathrm{MF}}$). The polaron energy, significantly deviating from $E_{\mathrm{MF}}$, is largely consistent with the $E_{\mathrm{zr}}$. The dashed blue line in (c) is the Fröhlich effective mass. Here we set the bath-bath scattering length $a_{\mathrm{BB}} = 100a_0$ and the condensate density $n_0^{1/3} r_{\mathrm{eff}} \simeq 0.5$ ($n_0 a_0^3 = 1.48\times 10^{-12}$) . In numerical analysis, we set $dka_0 = 10^{-5}$ and $\Lambda = 4000dk$. The gray shaded areas indicate where the small gas parameter assumption breaks down.
• Figure 4: The quasiparticle properties of (a) ionic, and (b) Rydberg impurities across various condensate density regimes as a function of the potential parameters $b$ and $a_{\mathrm{s}}$, respectively. The intraspecies interactions are set to $a_{\mathrm{BB}} = 0.05r_{\mathrm{ion}}$ and $a_{\mathrm{BB}} = 100a_0$ for the ion and Rydberg impurities, respectively. The dashed magenta (black) lines represents $E_{\mathrm{zr}}$ ($E_{\mathrm{MF}}$). The ratios adjacent to curves represent the multiplication factors of the actual values.
• Figure 5: The quasiparticle properties of (a) contact, (b) ionic, and (c) Rydberg impurities for various bath-bath scattering length $a_{\mathrm{BB}}$. The condensate density is set to ${n}_0 r_{\mathrm{ion}}^3=1$ for ionic, and $n_0a_0^3 = 1.48 \times 10^{-12}$ ($n_0^{1/3}r_{\mathrm{eff}} \simeq 0.5$) for Rydberg impurities. The dashed magenta (black) lines represents $E_{\mathrm{zr}}$ ($E_{\mathrm{MF}}$).