Analytical study of a finite-range impurity in a one-dimensional Bose gas

TL;DR

This work addresses the divergence of polaron properties in a strongly coupled 1D Bose polaron when using a contact impurity potential. By replacing the interaction with a finite-range square-well potential and solving the Gross-Pitaevskii equation in the Lee–Low–Pines frame, the authors obtain finite polaron energy and effective mass and reveal universal strong-coupling behavior governed by the range-to-coherence-length ratio $\bar{w}= w/\xi$, with $E_{pol}\sim -1/\bar{w}^3$ for attractive polarons and $m_{eff}\sim 1/\bar{w}$. The analysis also shows $m_{eff}$ saturates at finite values for finite-range impurities, while repulsive and attractive branches exhibit distinct but finite strong-coupling limits. These results provide a physically transparent, analytically tractable description of 1D Bose polarons beyond the zero-range approximation and offer a route to study excited GP states and long-range regimes.

Abstract

One-dimensional Bose gases present an interesting setting to study the physics of Bose polarons, as density fluctuations play an enhanced role due to reduced dimensionality. Theoretical descriptions of this system have predominantly relied on contact pseudopotentials to model the impurity-bath interaction, leading to unphysical results in the strongly coupled limit. In this work, we analytically solve the Gross-Pitaevskii equation, using a square well potential instead of a zero-range potential, for the ground-state wave function of a static impurity. We compute perturbative corrections arising from infinitesimally slow impurity motion. The polaron energy and effective mass remain finite in the strongly coupled regime, in contrast to the divergent behavior obtained using a contact potential. In this limit, we characterize the polaron properties in terms of the dimensionless ratio $\bar{w}\equiv w/ξ$ between the interaction range $w$ of the impurity-bath potential and the coherence length $ξ$ of the Bose gas. The effective mass exhibits a $1/\bar{w}$ scaling. The energy of the attractive polaron scales as $-1/\bar{w}^3$, whereas the repulsive polaron features subleading corrections to the dark soliton energy at the order $\bar{w}^3$.

Figures (5)

• Figure 1: The ground-state wave function $\phi_0({x})$ (upper panels) and the corresponding phase function $\theta(x) \equiv \arctan[\tilde{v}\phi_1({x})/\phi_0({x})]$ (lower panels) of the (a) repulsive and (b) attractive interactions for various coupling strengths $\eta\equiv g_{\mathrm{IB}}/g$. The well size is fixed as $w = 0.1\tilde{\xi}$. The velocity is fixed as $\tilde{v} = 0.01$. The thin dashed line in the upper panel of (a) represents the soliton-like profile $\tanh(|x|/(\sqrt{2}\tilde{\xi}))$, shown for comparison.
• Figure 2: The polaron energy and effective mass of an impurity assuming a (a) repulsive and (b) attractive square-well potential (blue solid line), and contact potential (dashed red line) as a function of the coupling strength $\eta \equiv g_{\mathrm{IB}}/g_{\mathrm{BB}}$. The width is again $w = 0.1\xi$. The magenta points show the DMC results obtained for a system of 100 particles with a contact impurity-bath potential 2020_PRR_Fleisheur_Exact_1D_Bose_Polaron.
• Figure 3: The polaron energy and the effective mass of the (a) repulsive, and (b) attractive square-well impurity for various values of the well size $w$ as a function of the coupling strength $\eta$.
• Figure 4: The polaron properties of (a) repulsive, and (b) attractive impurity at the strong coupling limit $\eta \to \infty$ as a function of the well size $w/\tilde{\xi}$. The blue lines represent the exact numerical results, while the red lines correspond to the analytical expressions derived from expansions valid in the limit of a small interaction range, $w/\tilde{\xi} \ll 1$.
• Figure 5: The mapping between the parameters $z = \sqrt{\frac{|V_0|2m_r}{\hbar^2}} \frac{w}{2}$ of the square-well potential and the corresponding zero-energy scattering length $a_{\mathrm{IB}}$ (blue lines), as well as the coupling constant $g_{\mathrm{IB}} = -\frac{\hbar^2}{m_ra_{\mathrm{IB}}}$ (red lines), is shown for (a) attractive, and (b) repulsive impurity. The gas parameter $\gamma = 1/(2n_0^2\xi^2)=0.438$, and the impurity-bath mass ratio $m_I/m = 0.47$ are used to obtain the coupling strength $\eta$.