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. 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.
We thank Matthias Heinz and Volker Karle for helpful comments on the manuscript; Zoran Ristivojevic for useful correspondence regarding mean-field calculations of induced impurity-impurity interactions; Fabian Grusdt for sharing with us the data for the densities presented in Ref. . This work has received funding from the DFG Project No. 413495248 [VO 2437/1-1] (F. B., H.-W. H., A. G. V.) and European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411 (A. G. V.). M. L. acknowledges support by the European Research Council (ERC) Starting Grant No. 801770 (ANGULON). H.-W.H. thanks the ECT* for hospitality during the workshop “Universal physics in Many-Body Quantum Systems – From Atoms to Quarks". This infrastructure is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 824093. H.-W.H. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 279384907 - SFB 1245.
Brauneis F, Hammer H-W, Lemeshko M, Volosniev A. Impurities in a one-dimensional Bose gas: The flow equation approach. SciPost Physics. 2021;11(1). doi:10.21468/scipostphys.11.1.008
Brauneis, F., Hammer, H.-W., Lemeshko, M., & Volosniev, A. (2021). Impurities in a one-dimensional Bose gas: The flow equation approach. SciPost Physics. SciPost. https://doi.org/10.21468/scipostphys.11.1.008
Brauneis, Fabian, Hans-Werner Hammer, Mikhail Lemeshko, and Artem Volosniev. “Impurities in a One-Dimensional Bose Gas: The Flow Equation Approach.” SciPost Physics. SciPost, 2021. https://doi.org/10.21468/scipostphys.11.1.008.
F. Brauneis, H.-W. Hammer, M. Lemeshko, and A. Volosniev, “Impurities in a one-dimensional Bose gas: The flow equation approach,” SciPost Physics, vol. 11, no. 1. SciPost, 2021.
Brauneis F, Hammer H-W, Lemeshko M, Volosniev A. 2021. Impurities in a one-dimensional Bose gas: The flow equation approach. SciPost Physics. 11(1), 008.
Brauneis, Fabian, et al. “Impurities in a One-Dimensional Bose Gas: The Flow Equation Approach.” SciPost Physics, vol. 11, no. 1, 008, SciPost, 2021, doi:10.21468/scipostphys.11.1.008.
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