Abstract
A few years ago, flow equations were introduced as a technique for calculating the ground-state energies of cold Bose gases with and without impurities[1,2]. In this paper, we extend this approach to compute observables other than the energy. As an example, we calculate the densities, and phase fluctuations of one-dimensional Bose gases with one and two impurities. For a single mobile impurity, we use flow equations to validate the mean-field results obtained upon the Lee-Low-Pines transformation. We show that the mean-field approximation is accurate for all values of the boson-impurity interaction strength as long as the phase coherence length is much larger than the healing length of the condensate. For two static impurities, we calculate impurity-impurity interactions induced by the Bose gas. We find that leading order perturbation theory fails when boson-impurity interactions are stronger than boson-boson interactions. The mean-field approximation reproduces the flow equation results for all values of the boson-impurity interaction strength as long as boson-boson interactions are weak.
Highlights
To illustrate calculations of observables based upon flow equations, we investigate a onedimensional system of N bosons and a single impurity atom
The difference between the results is more noticeable, which implies that the mean-field approximation (MFA) leads to less accurate results for c < 0, see Appendix B, where we present some additional data for the case with attractive interactions
We show that the induced interactions can be accurately calculated using the mean-field approximation, at least for weak boson-boson interactions
Summary
The flow equation approach is an ab-initio method for solving many-body problems [3]. IM-SRG was recently extended to cold Bose gases [1, 2] It was tested by calculating the ground-state energies of the Lieb-Liniger model and a one-dimensional (1D) Bose gas with an impurity atom (‘Bose polaron’) [1,2]. Those works demonstrate that flow equations allow one to go beyond mean-field approximation without relying on many-body perturbation theory. Ξ is the healing length of the condensate, and γ is the dimensionless Lieb-Liniger parameter which characterizes the boson-boson interaction strength, see Sec. 3 This argument implies that as long as e π2/γ 1 one can use the MFA to study the Bose-polaron problem. For convenience of the reader, we present some additional results for a system with attractive boson-impurity interactions in Appendix B
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