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arXiv:1610.00421 · Physical Review A

Bose-Einstein-condensate polaron in harmonic trap potentials in the weak-coupling regime: Lee-Low-Pines–type approach

Eiji Nakano, Hiroyuki Yabu, Kei Iida

Published Feb 24, 2017• 14 citations
arXivHTMLPDF

Abstract

We have calculated the zero-temperature binding energy of a single impurity atom immersed in a Bose-Einstein condensate (BEC) of ultracold atoms. The impurity and the condensed atoms are trapped in the respective axially symmetric harmonic potentials, where the impurity interacts with bosonic atoms in the condensate via low-energy $s$-wave scattering. In this case, bosons are excited around the impurity to form a quasiparticle, namely, a BEC polaron. We have developed a variational method, a la Lee-Low-Pines (LLP), for description of the polaron that has a conserved angular momentum around the symmetric axis. We find from numerical results that the binding between the impurity and the excited bosons breaks the degeneracy of the impurity energy with respect to the total angular momentum of the polaron. The angular momentum is partially shared by the excited bosons in a manner that is similar to the drag effect on the polaron momentum by a phonon cloud in the LLP theory for the electron-phonon system.

DOI
10.1103/physreva.95.023626

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We have calculated the zero-temperature binding energy of a single impurity atom immersed in a Bose-Einstein condensate (BEC) of ultracold atoms. The impurity and the condensed atoms are trapped in the respective axially symmetric harmonic potentials, where the impurity interacts with bosonic atoms in the condensate via low-energy sss-wave scattering. In this case, bosons are excited around the impurity to form a quasiparticle, namely, a BEC polaron. We have developed a variational method, a la Lee-Low-Pines (LLP), for description of the polaron that has a conserved angular momentum around the symmetric axis. We find from numerical results that the binding between the impurity and the excited bosons breaks the degeneracy of the impurity energy with respect to the total angular momentum of the polaron. The angular momentum is partially shared by the excited bosons in a manner that is similar to the drag effect on the polaron momentum by a phonon cloud in the LLP theory for the electron-phonon system.
Single-atomic impurities immersed in a dilute Bose gas in the spherically symmetric harmonic trap potentials are studied at zero temperature. In order to find the ground state of the polarons, we present a conditional variational method with fixed expectation values of the total angular momentum operators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}^{\phantom{\rule{0.5pt}{0ex}}2}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}_{z}$ of the system, using a cranking gauge transformation for bosons to move them in the frame co-rotating with the impurity. In the formulation, the expectation value $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}^{\phantom{\rule{0.5pt}{0ex}}2}\ensuremath{\rangle}$ is shown to be shared in impurity and bosons, but the value $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}_{z}\ensuremath{\rangle}$ is carried by the impurity due to the rotational symmetry. We also analyze the ground-state properties numerically obtained in this variational method for the system of the attractive impurity-boson interaction, and find that excited boson distributions around the impurity overlap largely with impurity's wave function in their quantum-number spaces and also in the real space because of the attractive interaction employed.
The description of an impurity atom in a Bose-Einstein condensate can be cast in the form of Fr\"ohlich's polaron Hamiltonian, where the Bogoliubov excitations play the role of the phonons. An expression for the corresponding polaronic coupling strength is derived, relating the coupling strength to the scattering lengths, the trap size and the number of Bose condensed atoms. This allows to identify several approaches to reach the strong-coupling limit for the quantum gas polarons, whereas this limit was hitherto experimentally inaccessible in solids. We apply Feynman's path-integral method to calculate for all coupling strengths the polaronic shift in the free energy and the increase in the effective mass. The effect of temperature on these quantities is included in the description. We find similarities to the acoustic polaron results and indications of a transition between free polarons and self-trapped polarons. The prospects, based on the current theory, of investigating the polaron physics with ultracold gases are discussed for lithium atoms in a sodium condensate.
We investigate the ground state properties of the gaseous mixture of a single species of bosons and fermions at zero temperature, where bosons are major in population over fermions, and form the Bose-Einstein condensate (BEC). The boson-boson and boson-fermion interactions are assumed to be weakly repulsive and attractive, respectively, while the fermion-fermion interaction is absent due to the Pauli exclusion for the low energy sss-wave scattering. We treat fermions as a gas of polarons dressed with Bogoliubov phonons, which is an elementary excitation of the BEC, and evaluate the ground state properties with the method developed by Lemmens, Devreese, and Brosens (LDB) originally for the electron polaron gas, and also with a general extension of the Lee-Low-Pines theory for many-body systems (eLLP), which incorporates the phonon drag effects as in the original LLP theory. The formulation of eLLP is developed and discussed in the present paper. The binding (interaction) energy of the polaron gas is calculated in these methods and shown to be finite (negative) for the dilute gas of heavy fermions with attractive boson-fermion interactions, though the suppression by the many-body effects exists.
We investigate the effects of a nearly uniform Bose-Einstein condensate (BEC) on the properties of immersed trapped impurity atoms. Using a weak-coupling expansion in the BEC-impurity interaction strength, we derive a model describing polarons, i.e., impurities dressed by a coherent state of Bogoliubov phonons, and apply it to ultracold bosonic atoms in an optical lattice. We show that, with increasing BEC temperature, the transport properties of the impurities change from coherent to diffusive. Furthermore, stable polaron clusters are formed via a phonon-mediated off-site attraction.
We investigate the properties of a dilute gas of impurities embedded in an ultracold gas of bosons that forms a Bose–Einstein condensate (BEC). This work focuses mainly on the equation of state (EoS) of the impurity gas at zero temperature and the induced interaction between impurities mediated by the host bath. We use perturbative field-theory approaches, such as Hugenholtz–Pines formalism, in the weakly interacting regime. In turn, for strong interactions, we aim at non-perturbative techniques such as quantum–Monte Carlo (QMC) methods. Our findings agree with experimental observations for an ultra dilute gas of impurities, modeled in the framework of the single impurity problem; however, as the density of impurities increases, systematic deviations are displayed with respect to the one-body Bose polaron problem.
We consider an impurity immersed in a Bose-Einstein condensate with tunable boson-impurity interactions. Such a Bose polaron has recently been predicted to exhibit an intriguing energy spectrum at finite temperature, where the ground-state quasiparticle evenly splits into two branches as the temperature is increased from zero [Guenther et al., Phys. Rev. Lett. 120, 050405 (2018)]. To investigate this theoretical prediction, we employ a recently developed variational approach that systematically includes multibody correlations between the impurity and the finite-temperature medium, thus allowing us to go beyond previous finite-temperature methods. Crucially, we find that the number of quasiparticle branches is simply set by the number of hole excitations of the thermal cloud, such that including up to one hole yields one splitting, two holes yields two splittings, and so on. Moreover, this effect is independent of the impurity mass. We thus expect that the exact ground-state quasiparticle will evolve into a single broad peak for temperatures T>0T>0T>0, with a broadening that scales as T3/4{T}^{3/4}T3/4 at low temperatures and sufficiently weak boson-boson interactions. In the zero-temperature limit, we show that our calculated ground-state polaron energy is in excellent agreement with recent quantum Monte Carlo results and with experiments.