Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons

Abstract

We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N−1)-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.

Dynamics of quantum double dark-solitons and an exact finite-size scaling of Bose–Einstein condensation

“…We note that the density profile shown in panel (a) of figure 2 is identical to the upper-left panel of figure 9 for c = 0.05 in [22]. In the latter panel it was shown that the density profile of the quantum double dark-soliton state completely coincides with the density profile of the elliptic double dark-soliton solution of the GP equation.…”

“…A quantum state that has two notches in both profiles of density and square amplitude of the matrix element of the field operator was proposed in [22]. We call it the quantum double darksoliton state, and it is given by the superposition of 'two-hole' excitation states as follows:…”

“…The density profile of this state ⟨X 1 , X 2 , N| ψ † (x) ψ(x)|X 1 , X 2 , N⟩ shows the two density notches at the positions x = X 1 , X 2 , which coincides with the squared amplitude of the elliptic soliton [22]. Here, by the determinant formula for the norms of Bethe eigenstates [35,36] we can effectively evaluate the matrix element Reproduced from [22]. CC BY 4.0.…”

“…The integer J is called the winding number [17,18]. In the previous study [22], we constructed the quantum single dark-soliton with a nonzero-winding number.…”

“…Furthermore, several significant properties in the non-equilibrium dynamics of a quantum single dark-soliton have been exactly investigated [19] and the generic and the ideal Gaussian weights have been introduced [20,21]. Moreover, the density and phase profiles of quantum states of double dark-solitons have been explicitly constructed [22], and the phase shift has numerically been estimated in the scattering of two quantum dark-solitons [23].…”