Elastic and inelastic transmission in guided atom lasers: A truncated Wigner approach

Julien Dujardin, Arturo Argüelles, and Peter Schlagheck

Phys. Rev. A 91, 033614 – Published 11 March 2015

Abstract

We study the transport properties of a Bose-Einstein condensate formed by an ultracold gas of bosonic atoms that is coupled from a magnetic trap into a one-dimensional waveguide. Our theoretical approach to tackling this problem is based on the truncated Wigner method for which we assume the system to consist of two semi-infinite noninteracting leads and a finite interacting scattering region with two constrictions modeling an atomic quantum dot. The transmission is computed in the steady-state regime and we find a good agreement between truncated Wigner and matrix-product state calculations. We also identify clear signatures of inelastic resonant scattering by analyzing the distribution of energy in the transmitted atomic-matter wave beam.

Received 11 December 2014

DOI:https://doi.org/10.1103/PhysRevA.91.033614

Authors & Affiliations

Julien Dujardin^*, Arturo Argüelles, and Peter Schlagheck

Département de Physique, University of Liege, 4000 Liège, Belgium

^*julien.dujardin@ulg.ac.be

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Vol. 91, Iss. 3 — March 2015

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Images

Figure 1
(a) A trapped BEC depicted by a (green) circle is loaded into a waveguide with two constrictions modeling an atomic quantum dot. The bold (black) lines represent the isopotentials of the waveguide. (b) One-dimensional infinite Bose-Hubbard (BH) chain for the quantum dot model [see Eqs. (4)]. The condensate is prepared within a trap represented by the green circle $(S)$ and coupled to the infinite BH chain (dashed green line). The big (red) circles represent a nonvanishing on-site atom-atom interaction. The two displaced sites enclosing the interaction region represent two sites where the on-site potential is nonzero.
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Figure 2
Transmission across the quantum dot configuration versus $μ / 2 J$ for ${N | κ |}^{2} = J^{2}, g = 0.2 J, V = J$ . The smooth black curve corresponds to the mean-field (GP) calculation, the jagged red curve to the TW method, and the green dots to the MPS method. We can see that for the GP method, bistability occurs for the first resonance. The dashed black line depicts states that are not accessible during a time-dependent loading of the waveguide. The TW and MPS curves are in good agreement and exhibit an imperfect transmission at resonances. The incoherent part of the transmission is represented by a light orange line. This curve shows that an appreciable amount of incoherent atoms are generated at the resonances, demonstrating a departure from of the GP model. The MPS method becomes numerically inefficient near the band edges, i.e., for $μ \approx - 2 J$ .
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Figure 3
(a) Transmission spectrum of the quantum dot geometry with $V = 4 J, g = 0.05 J, {N | κ |}^{2} = J^{2}$ computed by the GP (black line, following the same color convention as in Fig. 2) and the TW prescription (red line for the total transmission and orange line for the incoherent part of the transmission). The panels below show the energy distribution of the outgoing flux according to the TW calculation for an incident beam energy of $μ / 2 J = - 0.86$ for (b), $μ / 2 J = - 0.52$ for (c), $μ / 2 J = - 0.37$ for (d), $μ / 2 J = - 0.04$ for (e), $μ / 2 J = 0.47$ for (f), and $μ / 2 J = 0.87$ for (g). The black lines correspond to the coherent and the red lines to the total part of outgoing atoms. The peaks designated by a black arrow arise from collective oscillations about a single resonance as discussed in Sec. 4c. The peaks designated by blue (gray) arrows arise from atoms that have undergone a transition between two single-particle levels of the atomic quantum dot.
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Figure 4
Average number of atoms $〈 n_{E} 〉$ at energy $E$ for $g / E_{0} = 0.02, γ / E_{0} = 0.001, \sqrt{N} κ / E_{0} = 0.05, E_{0} T = 500 ℏ$ , and $ξ_{0} / E_{0} = 0.5$ . The presence of collective oscillations is clearly manifested in the form of two side peaks appearing at $E - μ \approx \pm 0.36 E_{0}$ . The (red) dots show the results obtained by numerically integrating Eq. (40) and applying a Laplace transform according to Eq. (45). They are in perfect agreement with the theoretical prediction (black line) of Eq. (53).
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Figure 5
Energy distribution of the transmitted beam for two different values of the chemical potential: $μ / 2 J = - 0.86$ for the upper panel and $μ / 2 J = - 0.04$ for the lower panel. The black vertical lines correspond to the expected Bogoliubov energies. They are in good agreement with the TW calculations. The grey zones correspond to the expected width of the peaks given by $2 Im (ε_{n})$ . For the upper panel, we see two side peaks around the chemical potential creating a collective oscillation inside the quantum dot. For the lower panel, we observe that, on top of the collective oscillation, inelastic collisions occur transferring atoms in the second or fourth energy level as depicted in the sketch on the top right side. The TW results are well reproduced by the Bogoliubov theory.
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