Correspondence between dark solitons and the type II excitations of the Lieb-Liniger model

A one-dimensional model of bosons with repulsive short-range interactions, solved analytically by Lieb and Liniger many years ago, predicts the existence of two branches of elementary excitations. One of them represents Bogoliubov phonons; the other, as suggested by some authors, might be related to dark solitons. On the other hand, it has been already demonstrated within the framework of the classical field approximation that a quasi-one-dimensional interacting Bose gas at equilibrium exhibits excitations which are phonons and dark solitons. By showing that for lower temperatures statistical distributions of dark solitons obtained within the classical field approximation are well represented by the distributions of quasiparticles of the second kind derived from fully quantum description, we demonstrate that type II excitations in the Lieb-Liniger model are, indeed, quantum solitons.

Figure 1

Average number of type II excitations as a function of quantized momentum at thermal equilibrium for the system of $N=1000$ atoms and $\gamma =0.02$ calculated within the Yang-Yang model. Dots correspond, from the top to bottom, to the temperatures ${10}^5,7\times {10}^4,4\times {10}^4$, and ${10}^4$ in units of ${\hslash }^2/m{L}^2{k}_B$. Maximal excitation momentum equals to $N\pi /L$. For the lowest considered temperature, $T={10}^4$, the Lieb-Liniger and Yang-Yang models give practically the same curves.

Figure 2

Number of solitons moving to the right as returned by the soliton-counting procedure for a system consisting of two solitons having velocities $u=\pm 0.5$ (in units of speed of sound, $s)$. The strength of the nonlinear term in the nonlinear Schrödinger equation describing the evolution of solitons equals the strength of the term describing the interactions in the mean-field approximation for the system of $N=1000$ atoms and $\gamma =0.04$ (red squares), $\gamma =0.02$ (blue triangles), and $\gamma =0.01$ (green diamonds). The results were obtained based on the cut for $k=10\times 2\pi /L$.

Figure 3

Relative velocity shift as returned by the soliton-counting procedure for a system consisting of two solitons as a function of soliton velocity. Different nonlinearity strengths $\gamma =0.04,0.02$, and $0.01$ are considered. Red squares (blue triangles) were obtained by analytical (numerical) calculations.

Figure 4

Relative velocity shift (upper frame) and the number of solitons moving to the right (lower frame) as returned by the soliton-counting procedure for the system consisting of 2 (red square), 8 (blue triangles), and 14 (green diamonds) solitons. Here, $\gamma =0.02$.

Figure 5

Fourier transform, in the position and time, of the density of the classical field, first squared and next averaged over 12 realizations (left panel). The number of atoms $N=1000$ and the temperature is ${10}^4$. Blue solid lines are drawn according to $\omega =\pm sk$, where $s$ is the speed of sound. The right panel shows ${\left|f(k,u)\right|}^2$ for $k=10\times 2\pi /L,20\times 2\pi /L$, and $30\times 2\pi /L$.

Figure 6

Comparison between the Yang-Yang model and the classical field approximation for the temperatures $T={10}^4$ and $4\times {10}^4$. Frame shows the average number of type II excitations (from the Yang-Yang model) and of dark solitons (from the classical field approximation) vs the momentum. The data are additionally averaged over the direction of the momentum. For the lowest considered temperature, $T={10}^4$, the agreement between numerical (dark line) and analytical (black line) results is good. For higher temperature, $T=4\times {10}^4$, a disagreement appears.

Figure 7

Comparison between the Yang-Yang model and the classical field approximation for the temperatures (from the top to bottom) ${10}^5,7\times {10}^4,4\times {10}^4$, and ${10}^4$. Frame shows the average number of type II excitations (from the Yang-Yang model: dark lines) and of dark solitons (from the classical field approximation and soliton-counting procedure corrected for higher temperatures due to the presence of solitonic collisions: light lines) vs the momentum. The data are additionally averaged over the direction of the momentum. Now, the agreement between numerical and analytical results is partially restored for all temperatures. The quantitative disagreement for higher temperatures is still present because, in fact, both the relative velocity shift and the collisional factor slightly depend on the solitonic velocity (see Fig. 4) and two-parameter correction is insufficient.