Dark solitons revealed in Lieb-Liniger eigenstates

Abstract

We study how dark solitons, i.e., solutions of one-dimensional, single-particle, nonlinear, time-dependent Schrödinger equation, emerge from eigenstates of a linear many-body model of contact-interacting bosons moving on a ring, the Lieb-Liniger model. This long-standing problem has been addressed by various groups, which presented different, seemingly unrelated, procedures to reveal the solitonic waves directly from the many-body model. Here, we propose a unification of these results using a simple ansatz for the many-body eigenstate of the Lieb-Liniger model, which gives us access to systems of hundreds of atoms. In this approach, mean-field solitons emerge in a single-particle density through repeated measurements of particle positions in the ansatz state. The postmeasurement state turns out to be a wave packet of yrast states of the reduced system.

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

“…In particular, an ansatz on how to assign the Gaussian weights to the yrast states was proposed [15]. The matrix elements of the bosonic field operator of the Lieb-Liniger model are characterized by grey solitons [11], and double elliptic dark solitons are derived in association with the quantum measurement process [16]. Here, we note that solutions of the NS equation in the periodic system are expressed in terms of Jacobi's elliptic functions.…”

Beyond Gross-Pitaevskii equation for 1D gas: quasiparticles and solitons

“…Here, following the reasoning presented in Ref. [14], we find candidates for solitons in LLGPE numerically for any interaction strength, as the minimal energy states constraint to a π jump of phase at the origin. We tested numerically that these states do not move and do preserve their shape in dynamics, as the solitons should do (cf.…”

“…These two different viewpoints were recently unified in Ref. [14]. The agreement holds, however, for weak interactions only, where the GPE is reliable.…”

“…, x m−1 , and N stands for a normalization factor. It has been shown in [9,14] that for weak interactions and large m, the functions G m (x|x 1 , . .…”

“…The above correspondence has been debated for many years before being fully justified only recently by works of different authors in Refs. [6][7][8][9][10][11][12][13][14]. Should the correspondence hold for stronger interactions, however, remains an open question.…”