Abstract
We study how dark solitons, i.e. solutions of one-dimensional single-particle nonlinear time-dependent Schr\"odinger 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 was 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 post-measurement state turns out to be a wave packet of yrast states of the reduced system.
Highlights
The famous Lieb-Liniger (LL) model [1,2] describes particles moving along a circle and interacting via δ interatomic potential
How accurately does the energy of the ansatz agree with the energy of the exact LL solution, the yrast state, as the interaction strength increasing? In Fig. 2, we present the energies as functions of the MF parameter ng, where n = N/L
We studied the correspondence between the yrast states of the Lieb-Liniger Hamiltonian and the mean-field solitons from the nonlinear Schrödinger equation
Summary
The famous Lieb-Liniger (LL) model [1,2] describes particles moving along a circle and interacting via δ interatomic potential. Where the wave function φMF(x, t ) is interpreted as an orbital occupied by a macroscopic number of atoms, m is the particle mass, and g is the interaction strength The latter equation (1) is useful in many areas of physics ranging from nonlinear optics [8] to hydrodynamics [9,10]. In the case of gases, when their atoms repel each other, i.e., g > 0, a soliton is a rarefaction in the atomic density, which moves with a constant speed, preserves its shape, and is unusually robust thanks to the balance between dispersion and nonlinearity [16] In this case (g > 0), the soliton is called a dark soliton, which can be either black or gray. The LL model includes all correlations between particles in a linear Hamiltonian, while the MF approach gets rid of mutual correlations but introduces the nonlinearity in the
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