Dynamics of a Mobile Ion in a Bose-Einstein Condensate

TL;DR

This work presents a co-moving-frame, mean-field treatment of a single ion impurity interacting with a Bose-Einstein condensate via a long-range potential. By applying the Lee-Low-Pines transformation and a modified Gross-Pitaevskii equation, the authors extract the polaron dispersion $E({\\bm p}_0)=E_0 + {\\bm p}_0^2 /(2 m^{\\star})$ and show that the impurity attains a nonzero stationary momentum with a mass renormalization $m^{\\star}$ due to dressing by host atoms. Nonlinear dynamics generate density waves and momentum exchange between the impurity and condensate, leading to damping-like behavior and, at strong coupling, coherent oscillations of the impurity momentum (flutter) across both 1D and 3D, with the effective mass and asymptotic momentum depending on dimensionality, initial state, and coupling strength. The results illuminate quantum transport and solvation dynamics in strongly interacting quantum mixtures and offer a versatile framework for exploring impurity dynamics in Bose gases.

Abstract

Characterization of the dynamics of an impurity immersed in a quantum medium is vital for fundamental understanding of matter as well as applications in modern day quantum technologies. The case of strong and long-ranged interactions is of particular importance here, as it opens the possibility to leverage quantum correlations in controlling the system properties. Here, we consider a charged impurity moving in a bosonic gas and study its properties out of equilibrium. We extract the stationary momentum of the ion at long times, which is nonzero due to the superfluid nature of the medium, and the effective mass which stems from dressing the impurity with the host atoms. The nonlinear evolution leads not only to emission of density waves, but also momentum transfer back to the ion, resulting in the possibility of oscillatory dynamics.

Figures (7)

• Figure 1: Schematic representation of the studied system of the ion (red circle) interacting with the background BEC (blue background density). The ionic impurity interacts with the condensate and gets dressed by this interaction forming a polaron. Due to the interactions its motion is damped by an effective force (blue arrow), transferring the initial ion's momentum (red arrow) to the condensate. In turn, the extra enery of the condensate is emitted in the form of density perturbations.
• Figure 2: Summary of the static and dynamical solutions for the 1D system. (a) Ground state density profile of the Bose gas interacting with a stationary charged impurity (the interaction strength is set to $C_4 = 1.0$); (b,c) space-time diagram showing the time evolution of the condensate density, where the initial state corresponds to that from panel (a) and for $|{\bm p}_0| = 0.5$, $|{\bm p}_0| = 1.5$ respectively; (d,e) space-time diagram showing the time evolution of the condensate density after a quench from the noninteracting system, for the same momenta as in panels (b,c); (f) a cut through the BEC density at time $t = 4$ for the case of the noninteracting protocol shown in panels (d,e), the red dashed line shows the propagation of sound with the speed of $c = 1$.
• Figure 3: The ion momentum as a function of time for (a) a quench from static ion and (b) a quench from non-interacting system. (c) Asymptotic ion momentum ${\bm p}_I(t \to \infty)$ as a function of the ion-BEC interaction strength for the same two quench protocols. Here, we present results for $|{\bm p}_0| = 1$ in our units.
• Figure 4: (a) An example of a "quantum flutter", when strong interactions lead to oscillations of the ion's momentum ($C_4 = 6$ and $g = 0.1$). (b) The effective damping force acting on the impurity calculated from the numerical results (solid line) and from Eq. (\ref{['eq.dpdt']}) (circles). Here, $|{\bm p}_0| = 1$; the effect is largely independent of the initial momentum.
• Figure 5: (a) Effective mass for a light ion ($m_I = m_B$, blue circles) and a heavy ion ($m_I = 10\, m_B$, orange circles) as a function of the interaction strength $C_4$ in 1D. The dashed lines are added as a guide for the eye. (b) Scattering length corresponding to the one-dimensional version of the potential \ref{['eq.ionpot']} as a function of the interaction strength $C_4$ in a wider range, showing the position of the scattering resonance.