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arXiv:1809.10601 · Physical Review A

Analytical study of static beyond-Fröhlich Bose polarons in one dimension

Ben Kain, Hong Yuan Ling

Published Sep 27, 2018• 30 citations
arXivHTMLPDF

Abstract

F. Grusdt et al. [New J. Phys. 19, 103035 (2017)] recently made a renormalization-group study of a one-dimensional Bose polaron in cold atoms. Their study went beyond the usual Fr\"ohlich description, which includes only single-phonon processes, by including two-phonon processes in which two phonons are simultaneously absorbed or emitted during impurity scattering [Y. E. Shchadilova et al., Phys. Rev. Lett. 117, 113002 (2016)]. We study this same beyond-Fr\"ohlich model, but in the static impurity limit where the ground state is described by a multimode squeezed state instead of the multimode coherent state in the static Fr\"ohlich model. We solve the system exactly by applying the generalized Bogoliubov transformation, an approach that can be straightforwardly adapted to higher dimensions. Using our exact solution, we obtain a polaron energy free of infrared divergences and construct analytically the polaron phase diagram. We find that the repulsive polaron is stable on the positive side of the impurity-boson interaction but is always thermodynamically unstable on the negative side of the impurity-boson interaction, featuring a bound state, whose binding energy we obtain analytically. We find that the attractive polaron is always dynamically unstable, featuring a pair of imaginary energies which we obtain analytically. We expect the multimode squeezed state to help with studies that go not only beyond the Fr\"ohlich paradigm but also beyond Bogoliubov theory, just as the multimode coherent state has helped with the study of Fr\"ohlich polarons.

DOI
10.1103/physreva.98.033610

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F. Grusdt et al. [New J. Phys. 19, 103035 (2017)] recently made a renormalization-group study of a one-dimensional Bose polaron in cold atoms. Their study went beyond the usual Fr"ohlich description, which includes only single-phonon processes, by including two-phonon processes in which two phonons are simultaneously absorbed or emitted during impurity scattering [Y. E. Shchadilova et al., Phys. Rev. Lett. 117, 113002 (2016)]. We study this same beyond-Fr"ohlich model, but in the static impurity limit where the ground state is described by a multimode squeezed state instead of the multimode coherent state in the static Fr"ohlich model. We solve the system exactly by applying the generalized Bogoliubov transformation, an approach that can be straightforwardly adapted to higher dimensions. Using our exact solution, we obtain a polaron energy free of infrared divergences and construct analytically the polaron phase diagram. We find that the repulsive polaron is stable on the positive side of the impurity-boson interaction but is always thermodynamically unstable on the negative side of the impurity-boson interaction, featuring a bound state, whose binding energy we obtain analytically. We find that the attractive polaron is always dynamically unstable, featuring a pair of imaginary energies which we obtain analytically. We expect the multimode squeezed state to help with studies that go not only beyond the Fr"ohlich paradigm but also beyond Bogoliubov theory, just as the multimode coherent state has helped with the study of Fr"ohlich polarons.
Mobile impurity atoms immersed in Bose–Einstein condensates provide a new platform for exploring Bose polarons. Recent experimental advances in the field of ultracold atoms make it possible to realize such systems with highly tunable microscopic parameters and to explore equilibrium and dynamical properties of polarons using a rich toolbox of atomic physics. In this paper we present a detailed theoretical analysis of Bose polarons in one-dimensional systems of ultracold atoms. By combining a non-perturbative renormalization group approach with numerically exact diffusion Monte Carlo calculations we obtain not only detailed numerical results over a broad range of parameters but also qualitative understanding of different regimes of the system. We find that an accurate description of Bose polarons requires the inclusion of two-phonon scattering terms which go beyond the commonly used Fröhlich model. Furthermore we show that when the Bose gas is in the strongly interacting regime, one needs to include interactions between the phonon modes. We use several theoretical approaches to calculate the polaron energy and its effective mass. The former can be measured using radio-frequency spectroscopy and the latter can be studied experimentally using impurity oscillations in a harmonic trapping potential. We compare our theoretical results for the effective mass to the experiments by Catani et al (2012 Phys. Rev. A 85 023623). In the weak-to-intermediate coupling regimes we obtain excellent quantitative agreement between theory and experiment, without any free fitting parameter. We supplement our analysis by full dynamical simulations of polaron oscillations in a shallow trapping potential. We also use our renormalization group approach to analyze the full phase diagram and identify regions that support repulsive and attractive polarons, as well as multi-particle bound states.
Mobile impurity atoms immersed in Bose–Einstein condensates provide a new platform for exploring Bose polarons. Recent experimental advances in the field of ultracold atoms make it possible to realize such systems with highly tunable microscopic parameters and to explore equilibrium and dynamical properties of polarons using a rich toolbox of atomic physics. In this paper we present a detailed theoretical analysis of Bose polarons in one-dimensional systems of ultracold atoms. By combining a non-perturbative renormalization group approach with numerically exact diffusion Monte Carlo calculations we obtain not only detailed numerical results over a broad range of parameters but also qualitative understanding of different regimes of the system. We find that an accurate description of Bose polarons requires the inclusion of two-phonon scattering terms which go beyond the commonly used Fröhlich model. Furthermore we show that when the Bose gas is in the strongly interacting regime, one needs to include interactions between the phonon modes. We use several theoretical approaches to calculate the polaron energy and its effective mass. The former can be measured using radio-frequency spectroscopy and the latter can be studied experimentally using impurity oscillations in a harmonic trapping potential. We compare our theoretical results for the effective mass to the experiments by Catani et al (2012 Phys. Rev. A 85 023623). In the weak-to-intermediate coupling regimes we obtain excellent quantitative agreement between theory and experiment, without any free fitting parameter. We supplement our analysis by full dynamical simulations of polaron oscillations in a shallow trapping potential. We also use our renormalization group approach to analyze the full phase diagram and identify regions that support repulsive and attractive polarons, as well as multi-particle bound states.
An impurity immersed in a Bose-Einstein condensate is no longer accurately described by the Fr\"ohlich Hamiltonian as the coupling between the impurity and the boson bath gets stronger. We study the dominant effects of the two-phonon terms beyond the Fr\"ohlich model on the ground-state properties of the polaron using Feynman's variational path-integral approach. The previously reported discrepancy in the effective mass between the renormalization group approach and this theory is shown to be absent in the beyond-Fr\"ohlich model on the positive side of the Feshbach resonance. Self-trapping, characterized by a sharp and dramatic increase of the effective mass, is no longer observed for the repulsive polaron once the two-phonon interactions are included. For the attractive polaron we find a divergence of the ground-state energy and effective mass at weaker couplings than previously observed within the Fr\"ohlich model.
The Fr\"ohlich model describes the interaction of a mobile impurity with a surrounding bath of phonons which leads to the formation of a quasiparticle, the polaron. In this paper an efficient renormalization group approach is presented which provides a description of Fr\"ohlich polarons in all regimes ranging from weak to strong coupling. The extended renormalization group approach introduced here is capable of predicting ground state properties for arbitrarily small impurity masses. This allows us to obtain the full phase diagram of the Fr\"ohlich Hamiltonian, which we present concretely for the Bogoliubov-Fr\"ohlich model originally introduced to describe ultracold impurities in a Bose-Einstein condensate. For this model, whose regime of validity in ultracold quantum gases is the subject of debate, we benchmark our method by comparison of the ground state energy to recent diagrammatic quantum Monte Carlo calculations.
The system of a charged impurity in an interacting Bose gas has gained significant attention due to the long-range ion-atom interactions and the study of transport properties. Here, the ground state energy of a charged Bose polaron is calculated within the Bogoliubov approximation for both the Fröhlich and beyond-Fröhlich Hamiltonians using a generalized Feynman variational path-integral approach, which obtained accurate results for other polaron problems. The generalized approach, which was used to improve the energy result for the neutral polaron, has resulted in a minor improvement, indicating that Feynman's approach is sufficient when the impurity-boson interaction is long-range. Beyond-Fröhlich corrections results in the emergence of a divergence in the polaronic energy indicating a transition between the repulsive and attractive polaron regime. The path-integral approach with the beyond-Fröhlich Hamiltonian is also compared to a field-theory calculation from Christensen et al, 2021. The validity of the Bogoliubov approximation is investigated. The optical absorption has also been calculated within the Bogoliubov approximation for weak ion-atom interactions, and the effect of finite temperature has been studied. We show that the coupling of the ion to an oscillating external electric field offers a straigtforward experimental probe for the charged polaron in a Bose gas, different from but complementary to existing spectroscopic techniques.
When a mobile impurity interacts with a many-body system, such as a phonon bath, a polaron is formed. Despite the importance of the polaron problem for a wide range of physical systems, a unified theoretical description valid for arbitrary coupling strengths is still lacking. Here we develop a renormalization group approach for analyzing a paradigmatic model of polarons, the so-called Fröhlich model, and apply it to a problem of impurity atoms immersed in a Bose-Einstein condensate of ultra cold atoms. Polaron energies obtained by our method are in excellent agreement with recent diagrammatic Monte Carlo calculations for a wide range of interaction strengths. They are found to be logarithmically divergent with the ultra-violet cut-off, but physically meaningful regularized polaron energies are also presented. Moreover, we calculate the effective mass of polarons and find a smooth crossover from weak to strong coupling regimes. Possible experimental tests of our results in current experiments with ultra cold atoms are discussed.