Driven impurity in an ultracold one-dimensional Bose gas with intermediate interaction strength

Claudio Castelnovo, Jean-Sébastien Caux, and Steven H. Simon

Phys. Rev. A 93, 013613 – Published 14 January 2016

Abstract

We study a single impurity driven by a constant force through a one-dimensional Bose gas using a Lieb-Liniger based approach. Our calculation is exact in the interaction among the particles in the Bose gas, and is perturbative in the interaction between the gas and the impurity. In contrast to previous studies of this problem, we are able to handle arbitrary interaction strength for the Bose gas. We find very good agreement with recent experiments [Palzer et al., Phys. Rev. Lett. 103, 150601 (2009)].

Received 6 June 2015

DOI:https://doi.org/10.1103/PhysRevA.93.013613

Physics Subject Headings (PhySH)

Impurities

Bose gases Ultracold gases

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Claudio Castelnovo^1,2,3, Jean-Sébastien Caux⁴, and Steven H. Simon¹

¹Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom
²SEPnet and Hubbard Theory Consortium, Department of Physics, Royal Holloway University of London, Egham TW20 0EX, United Kingdom
³TCM group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
⁴Institute for Theoretical Physics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands

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Vol. 93, Iss. 1 — January 2016

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Images

Figure 1
Position of the center of mass as a function of time from the numerical simulations in comparison with the experimental data from Fig. 3 in Ref. [1] (blue open squares). The free-fall analytic solution $x (t) = g t^{2} / 2$ is also shown for comparison (red solid dots). Top panel: numerical results at fixed density $n ≃ 1.2 μ m^{- 1}$ , for $γ = 1.9$ (cyan open circles), $γ = 2.6$ (cyan solid dots), and $γ = 4.0$ (cyan open triangles). Bottom panel: numerical results at fixed $γ = 2.6$ , for $n ≃ 0.84$ (cyan open circles), $n ≃ 1.2$ (cyan solid dots), and $n ≃ 1.56 μ m^{- 1}$ (cyan open triangles).
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Figure 2
Behavior starting from Gaussian distributed initial conditions over $N_{hist} = 100 000$ histories, for $γ = 1.1, 1.9, 2.1$ (black, red, and blue, respectively) with a finite trap of uniform density (see text for size and density details). The results are expressed as histograms of the position of the particles after they have been falling for $18.6$ ms [31]. The corresponding experimental results in Fig. 5 of Ref. [1] are shown as thin dotted lines.
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Figure 3
Different (static) density profiles for the 1D condensate used to assess how the shape affects the results of our simulations: (i) square (red), (ii) square with a square depletion at the center of the condensate (magenta), (iii) parabolic (blue), and (iv) parabolic with a Gaussian-smoothed Wigner function depletion at the center of the condensate (green).
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Figure 4
Density profiles at different times starting from the initial conditions discussed in the main text, averaged over $N_{hist} = 100 000$ histories. From left to right: $t = 1.5, t = 10$ , and $t = 18.7$ ms. The colors correspond to the four cases discussed in the text: (i) a finite uniform condensate (black); (ii) a finite uniform condensate with a square depletion at the center (red); (iii) a parabolic condensate (blue); and (iv) a parabolic condensate with a Gaussian-smoothed Wigner function depletion at the center (green).
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Figure 5
Position of the center of mass as a function of time from our simulations in the TG limit, considering a uniform 1D gas with $n = 1.2 μ m^{- 1}$ and $γ = 4.1, 5, 6, 8, 10, 25, 100$ (red, blue, green, magenta, cyan, yellow, and black, respectively). The black dashed line represents the free-fall curve. The black solid line corresponds to the expected behavior in the $γ \to \infty$ limit (i.e., terminal velocity $v_{F}$ ). Blue open squares (joined by a dotted line) represent the experimental data from Fig. 3 in Ref. [1].
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