Lieb-Liniger model: Emergence of dark solitons in the course of measurements of particle positions

Andrzej Syrwid and Krzysztof Sacha

Phys. Rev. A 92, 032110 – Published 9 September 2015

Abstract

The Lieb-Liniger model describes bosons with contact interactions in one-dimensional space. In the limit of weak repulsive particle interactions, there are two types of low-lying excitation spectrum. The first is reproduced by the Bogoliubov dispersion relation, and the other is believed to correspond to dark soliton excitations. While there are various evidences that the type II spectrum is related to dark solitons, it has not been shown that measurements of positions of particles reveal dark soliton density profiles. Here, we employ the Bethe ansatz approach and show that dark solitons emerge in the measurement process if the system is prepared in an eigenstate corresponding to the type II spectrum. We analyze single and double dark solitons as well as weak and strong interaction regimes.

Received 18 June 2015

DOI:https://doi.org/10.1103/PhysRevA.92.032110

Authors & Affiliations

Andrzej Syrwid¹ and Krzysztof Sacha^1,2

¹Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11 PL-30-348 Kraków, Poland
²Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11 PL-30-348 Kraków, Poland

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Vol. 92, Iss. 3 — September 2015

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Images

Figure 1
(a)–(c) Results of the particle positions measurements in the system prepared in the eigenstate corresponding to the one-hole excitation with the total momentum per particle $P / N = \frac{π}{L}$ , for $N = 8, c = 0.08$ , and $L = 1$ , i.e., $γ = 0.01$ . Panel (a) shows an example of how the conditional probability density $ρ_{j} (x)$ (with values of $j$ indicated in the panel) changes after successive measurements of the particle positions. The probability density $ρ_{1} (x)$ for the measurement of the first particle is uniform as expected for system eigenstates. However, in the course of the measurements, $ρ_{j} (x)$ starts revealing a dark soliton notch and the last density $ρ_{8} (x)$ becomes hardly distinguishable from the mean-field (MF) dark soliton profile (black solid line) corresponding to the same average particle momentum as $P / N = \frac{π}{L}$ [note that the mean-field profile has been shifted so that the soliton position $x_{0}$ matches the minimum of $ρ_{8} (x)$ ]. (b) Phase of the mean-field solution (black solid line), whose probability density is shown in panel (a), and the phase of the wave function of the last particle (red [gray] dotted line) corresponding to the measurement realization presented in panel (a). Different simulations of the detection process lead to $ρ_{8} (x)$ with the notch localized at different points $x_{0}$ . When one shifts the obtained particle positions so that $x_{0}$ always equals $\frac{L}{2}$ , the average particle density can be prepared and it is shown in panel (c) (green [light gray] dashed line) together with the mean-field dark soliton profile (black line). The average density based on $10^{4}$ simulations of the measurement process. Deviations between the average density and the mean-field prediction seem to be due to small $N$ effects. That is, before the conditional probability density $ρ_{j} (x)$ converges to a final form, particles which are chosen according to different probability distributions distort the average density. Red (dark gray) line in panel (c) shows the histogram where only positions of last particles are collected, which agrees very well with the mean-field soliton profile. Panels (d)–(f) are related to data similar to panels (a)–(c) but for the total momentum per particle $P / N = \frac{7 π}{4 L}$ .
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Figure 2
Similar results as in Figs. 1 and 1 but for the eigenstate with the total momentum $P = 0$ corresponding to two-hole excitations of the $N = 8$ system with $c = 0.08$ and $L = 1$ . Histograms in panel (b) have been prepared by shifting particle positions obtained in the simulations so that one of the two minima of $ρ_{N} (x)$ always coincides with $\frac{L}{4}$ .
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Figure 3
Similar data as in Figs. 1 and 2 but for strong particle interactions, i.e., for $c = 8, N = 8$ , and $L = 1$ . Panels (a)–(c) correspond to the one-hole excitation with $P = \frac{N π}{L}$ . Panels (d)–(f) are related to the two-hole excitation with $P = 0$ . Histograms shown in panel (c) have been prepared by shifting particle positions obtained in the simulations so that the global minimum of $ρ_{N} (x)$ always coincides with $\frac{L}{2}$ . Histograms in panel (f) are based on particle positions shifted so that one of the two minima of $ρ_{N} (x)$ , as, e.g., visible in panel (d), always coincides with $\frac{L}{4}$ . Note that black lines in panels (c) and (f) are not related to mean-field prediction as in Figs. 1 and 2 but to $ρ_{N} (x)$ averaged over all (i.e., $10^{4}$ ) measurement realizations.
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