Many-Body Matter-Wave Dark Soliton

Dominique Delande and Krzysztof Sacha

Phys. Rev. Lett. 112, 040402 – Published 27 January 2014

Abstract

The Gross-Pitaevskii equation—which describes interacting bosons in the mean-field approximation—possesses solitonic solutions in dimension one. For repulsively interacting particles, the stationary soliton is dark, i.e., is represented by a local density minimum. Many-body effects may lead to filling of the dark soliton. Using quasiexact many-body simulations, we show that, in single realizations, the soliton appears totally dark although the single particle density tends to be uniform.

Received 26 July 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.040402

Authors & Affiliations

Dominique Delande¹ and Krzysztof Sacha^2,3

¹Laboratoire Kastler Brossel, UPMC-Paris6, ENS, CNRS; 4 Place Jussieu, F-75005 Paris, France
²Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, Ulica Reymonta 4, PL-30-059 Kraków, Poland
³Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, Ulica Reymonta 4, PL-30-059 Kraków, Poland

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Vol. 112, Iss. 4 — 31 January 2014

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Images

Figure 1
Temporal evolution of a dark soliton in a box. The soliton is prepared by starting from the ground state in the presence of a barrier located in the middle of the box which creates a large density hole. At $t = 0$ the barrier is turned off and a phase is imprinted. The subsequent dynamics is computed using a quasiexact many-body TEBD algorithm. Different panels show the one-body density at increasing times. At short time, the soliton notch emits density waves (visible as small modulations of the otherwise flat background) and is progressively filled at longer times. The density notch also spatially spreads producing an almost uniform density. In this figure and the following ones, $x$ is in units of the healing length $ξ$ .
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Figure 2
(a) shows the almost uniform one-body density at time $t = 25$ , while (b)–(d) display the histograms resulting from 3 independent quantum measurements of the positions of all particles. All the measurements are for the same many-body state. Each measurement observes the density notch associated with the dark soliton at a well defined position, but this position fluctuates randomly for each measurement. Each histogram can be well fitted by a mean field dark soliton profile (red thick curve), with “shot noise” fluctuations associated with the quantum measurement. Although the one-body density is almost uniform and does not display the existence of the dark soliton, a complete measurement of particle positions reveals its existence.
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Figure 3
Average density profile with respect to the soliton position $q$ . (a) Results from the averaging over 50 independent measurements of the same many-body state at time $t = 25$ . Shot noise is still present, but the density notch of the dark soliton clearly emerges. (b)–(e) are obtained by a massive averaging over $10^{5}$ independent measurements at various times. The shape does not change during the time propagation and is in excellent agreement with the mean field prediction $| ϕ_{0} (x) |^{2}$ , Eq. (1) (blue dashed curve). The dark soliton structure persists for very long times, even if the soliton position is delocalized over a large range.
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Figure 4
(a) Probability density for the position $q$ of the dark soliton, built from $10^{5}$ independent measurements of the same many-body state, at $t = 0$ (blue long-dashed), 2 (orange short-dashed), 5 (green dashed-dotted), 12 (red double-dashed-dotted), and 25 (black solid). Each curve is an almost perfect Gaussian, as predicted for the spreading of a quantum free wave packet. This proves that the effective one-body theory, where the position $q$ of the dark soliton is treated as a quantum degree of freedom, efficiently describes the many-body evolution. (b) Spreading $Δ q^{2}$ of the dark soliton position vs time. For $t > 5$ , the soliton is delocalized over a length larger than its size. The prediction of the effective one-body theory, shown by a blue dashed curve, is almost indistinguishable from the many-body calculation.
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