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arXiv:2203.14340 · Physical Review A

Analytical approach to the Bose polaron via a q -deformed Lie algebra

Enderalp Yakaboylu

Published Sep 28, 2022• 10 citations
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Abstract

We present an approach to the Bose polaron based on the notion of quantum groups, also known as $q$-deformed Lie algebras. In this approach, a mobile impurity can be depicted as a deformation of the Lie algebra of the bosonic creation and annihilation operators of the bath, in which the impurity is immersed. Accordingly, the Bose polaron can be described as a bath of noninteracting $q$-deformed bosons, which allows us to provide an analytical formulation of the Bose polaron at arbitrary couplings. We derive its ground state energy in the phonon branch of the Bogoliubov dispersion and demonstrate that the previously observed transition from a repulsive to an attractive polaron occurs at the vicinity where the quantum group symmetry is broken. Furthermore, our approach has the potential to open up avenues in polaron physics by connecting it with seemingly unrelated research topics where quantum groups play an essential role, such as anyons.

DOI
10.1103/physreva.106.033321

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We present an approach to the Bose polaron based on the notion of quantum groups, also known as qqq-deformed Lie algebras. In this approach, a mobile impurity can be depicted as a deformation of the Lie algebra of the bosonic creation and annihilation operators of the bath, in which the impurity is immersed. Accordingly, the Bose polaron can be described as a bath of noninteracting qqq-deformed bosons, which allows us to provide an analytical formulation of the Bose polaron at arbitrary couplings. We derive its ground state energy in the phonon branch of the Bogoliubov dispersion and demonstrate that the previously observed transition from a repulsive to an attractive polaron occurs at the vicinity where the quantum group symmetry is broken. Furthermore, our approach has the potential to open up avenues in polaron physics by connecting it with seemingly unrelated research topics where quantum groups play an essential role, such as anyons.
We present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as q-deformed Lie algebras. In particular, we show that, if the symmetry of a free quantum particle corresponds to a Lie group G, in the presence of a many-body environment this particle can be described by a deformed group, G_{q}. Crucially, the single deformation parameter, q, contains all the information about the many-particle interactions in the system. We exemplify our approach by considering a quantum rotor interacting with a bath of bosons, and demonstrate that extracting the value of q from closed-form solutions in the perturbative regime allows one to predict the behavior of the system for arbitrary values of the impurity-bath coupling strength, in good agreement with nonperturbative calculations. Furthermore, the value of the deformation parameter allows one to predict at which coupling strengths rotor-bath interactions result in a formation of a stable quasiparticle. The approach based on quantum groups does not only allow for a drastic simplification of impurity problems, but also provides valuable insights into hidden symmetries of interacting many-particle systems.
We use a nonperturbative renormalization group approach to develop a unified picture of the Bose polaron problem, where a mobile impurity is strongly interacting with a surrounding Bose-Einstein condensate (BEC). A detailed theoretical analysis of the phase diagram is presented and the polaron-to-molecule transition is discussed. For attractive polarons we argue that a description in terms of an effective Fr\"ohlich Hamiltonian with renormalized parameters is possible. Its strong-coupling regime is realized close to a Feshbach resonance, where we predict a sharp increase of the effective mass. Already for weaker interactions, before the polaron mass diverges, we predict a transition to a regime where states exist below the polaron energy and the attractive polaron is no longer the ground state. On the repulsive side of the Feshbach resonance we recover the repulsive polaron, which has a finite lifetime because it can decay into low-lying molecular states. We show for the entire range of couplings that the polaron energy has logarithmic corrections in comparison with predictions by the mean-field approach. We demonstrate that they are a consequence of the polaronic mass renormalization which is due to quantum fluctuations of correlated phonons in the polaron cloud.
The mobile impurity in a Bose-Einstein condensate (BEC) is a paradigmatic many-body problem. For weak interaction between the impurity and the BEC, the impurity deforms the BEC only slightly and it is well described within the Fröhlich model and the Bogoliubov approximation. For strong local attraction this standard approach, however, fails to balance the local attraction with the weak repulsion between the BEC particles and predicts an instability where an infinite number of bosons is attracted toward the impurity. Here we present a solution of the Bose polaron problem beyond the Bogoliubov approximation which includes the local repulsion between bosons and thereby stabilizes the Bose polaron even near and beyond the scattering resonance. We show that the Bose polaron energy remains bounded from below across the resonance and the size of the polaron dressing cloud stays finite. Our results demonstrate how the dressing cloud replaces the attractive impurity potential with an effective many-body potential that excludes binding. We find that at resonance, including the effects of boson repulsion, the polaron energy depends universally on the effective range. Moreover, while the impurity contact is strongly peaked at positive scattering length, it remains always finite. Our solution highlights how Bose polarons are self-stabilized by repulsion, providing a mechanism to understand quench dynamics and nonequilibrium time evolution at strong coupling.
We study the properties of a single impurity in a dilute Bose gas, a Bose polaron, using the functional renormalization group. We use an ansatz for the effective action motivated by a derivative expansion, and we compute the energies of the attractive and repulsive branches of excitations in both two and three spatial dimensions. Three-body correlations play an important role in the attractive branch, and we account for those by including three-body couplings between two bath bosons and the impurity. Our calculations compare very favorably with state-of-the-art experimental measurements and numerical simulations.
Polaron quasiparticles are formed when a mobile impurity is coupled to the elementary excitations of a many-particle background. In the field of ultracold atoms, the study of the associated impurity problem has attracted a growing interest over the last fifteen years. Polaron quasiparticle properties are essential to our understanding of a variety of paradigmatic quantum many-body systems realized in ultracold atomic gases and in the solid state, from imbalanced Bose–Fermi and Fermi–Fermi mixtures to fermionic Hubbard models. In this topical review, we focus on the so-called repulsive polaron branch, which emerges as an excited many-body state in systems with underlying attractive interactions such as ultracold atomic mixtures, and is characterized by an effective repulsion between the impurity and the surrounding medium. We give a brief account of the current theoretical and experimental understanding of repulsive polaron properties, for impurities embedded in both fermionic and bosonic media, and we highlight open issues deserving future investigations.
Predicting the emergent properties of impurities immersed in a quantum bath is a fundamental challenge that can defy quasiparticle treatments. Here, we measure the spectral properties and real-time dynamics of mobile impurities injected into a weakly interacting homogeneous Bose-Einstein condensate, using two broad Feshbach resonances to tune both the impurity-bath and intrabath interactions. For attractive impurity-bath interactions, the impurity spectrum features a single branch, which away from the resonance corresponds to a well-defined attractive polaron; near the resonance, we observe dramatic broadening of this branch, suggesting a breakdown of the quasiparticle picture. For repulsive impurity-bath interactions, the spectrum features two branches: the attractive branch that is dominated by excitations with energy close to that of the Feshbach dimer, but has a many-body character, and the repulsive polaron branch. Our measurements show that the behavior of impurities in weakly interacting baths is remarkably universal, controlled only by the bath density and a single dimensionless interaction parameter.