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arXiv:1901.02847 · Physical Review A

Bose polaron in spherically symmetric trap potentials: Ground states with zero and lower angular momenta

Kano Watanabe, Eiji Nakano, Hiroyuki Yabu

Published Mar 28, 2019• 5 citations
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Abstract

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.

DOI
10.1103/physreva.99.033624

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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 J\ifmmode^\elseˆ\fi2{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}^{\phantom{\rule{0.5pt}{0ex}}2}J\ifmmode^\elseˆ\fi​2 and J\ifmmode^\elseˆ\fiz{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}_{z}J\ifmmode^\elseˆ\fi​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⟨J\ifmmode^\elseˆ\fi2\ensuremath⟩\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}^{\phantom{\rule{0.5pt}{0ex}}2}\ensuremath{\rangle}\ensuremath⟨J\ifmmode^\elseˆ\fi​2\ensuremath⟩ is shown to be shared in impurity and bosons, but the value \ensuremath⟨J\ifmmode^\elseˆ\fiz\ensuremath⟩\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{J}}_{z}\ensuremath{\rangle}\ensuremath⟨J\ifmmode^\elseˆ\fi​z​\ensuremath⟩ 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.
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.
In this thesis we investigate the properties of impurities immersed in a dilute Bose gas at zero temperature using quantum Monte-Carlo methods. The interactions between bosons are modeled by a hard sphere potential with scattering length a, whereas the interactions between the impurity and the bosons are modeled by a short-range, square-well potential where both the sign and the strength of the scattering length b can be varied by adjusting the well depth. We calculate the binding energy, the effective mass and the pair correlation functions of a impurity along the attractive and the repulsive polaron branch. In particular, at the unitary limit of the impurity-bosons interaction, we find that the binding energy is much larger than the chemical potential of the bath signaling that many bosons dress the impurity thereby lowering its energy and increasing its effective mass. We characterize this state by calculating the bosons-boson pair correlation function and by investigating the dependence of the binding energy on the gas parameter of the bosonic bath. We also investigate the ground-state properties of M impurities in a Bose gas at T=0. In particular, the energy and the phase diagram by using both quantum Monte-Carlo and mean field methods.
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Quantum impurities interacting with quantum environments offer unique insights into many-body systems. Here, we explore the thermometric potential of a neutral impurity immersed in a harmonically trapped bosonic quantum gas below the Bose-Einstein condensation critical temperature TcT_cTc​. Using ab-initio Path Integral Monte Carlo simulations at finite temperatures, we analyze the impurity's sensitivity to temperature changes by exploiting experimentally accessible observables such as its spatial distribution. Our results, covering a temperature range of −1.1⩽T/Tc⩽0.9-1.1 \leqslant T/T_c \leqslant 0.9−1.1⩽T/Tc​⩽0.9, reveal that the impurity outperforms estimations based on a one-species bath at lower temperatures, achieving relative precision of 3-4\% for 1000 measurement repetitions. While non-zero boson-impurity interaction strength gBIg_{BI}gBI​ slightly reduces the accuracy, the impurity's performance remains robust, especially in the low-temperature regime T/Tc≲0.45T/T_c \lesssim 0.45T/Tc​≲0.45 withing the analyzed interaction strengths 0⩽gBI/g⩽50 \leqslant g_{BI}/g \leqslant 50⩽gBI​/g⩽5, where ggg is the boson-boson coupling. We confirm that quantum optical models can capture rather well the dependence of the temperature sensor on the impurity-gas interaction. Although our findings are in qualitative agreement with previous studies, our Monte Carlo simulations offer improved precision. We find that the maximum likelihood estimation protocol approaches the precision comparable to the limit set by the Quantum Fisher Information bound. Finally, using the Hellinger distance method, we directly extract the Fisher information and find that, by exploiting the extremum order statistics, impurities far from the trap center are more sensitive to thermal effects than those close to the trap center.
We investigate the properties of an impurity immersed in a dilute Bose gas at zero temperature using quantum Monte Carlo methods. The interactions between bosons are modeled by a hard-sphere potential with scattering length aaa, whereas the interactions between the impurity and the bosons are modeled by a short-range, square-well potential where both the sign and the strength of the scattering length bbb can be varied by adjusting the well depth. We characterize the attractive and the repulsive polaron branch by calculating the binding energy and the effective mass of the impurity. Furthermore, we investigate the structural properties of the bath, such as the impurity-boson contact parameter and the change of the density profile around the impurity. At the unitary limit of the impurity-boson interaction, we find that the effective mass of the impurity remains smaller than twice its bare mass, while the binding energy scales with ${\ensuremath{\hbar}}^{2}{n}^{2/3}/m$, where nnn is the density of the bath and mmm is the common mass of the impurity and the bosons in the bath. The implications for the phase diagram of binary Bose-Bose mixtures at low concentrations are also discussed.
For a Bose-condensed gas in an axially symmetric harmonic trap at zero temperature, a new trial wave function of the ground and single vortex states is considered based on the mean-field energy functional and variational method. Once the wave function is known, we can explore the particle-number dependence of relevant quantities, such as energy, chemical potential, aspect ratio of the velocity distribution and critical angular velocity for production of vortices. These results agree well with previous numerical solutions of the G-P equation given by Dalfovo et al.