[Paper Review] Construction of quantum dark soliton in one-dimensional Bose gas

Eriko Kaminishi, Takashi Mori+1 more|arXiv (Cornell University)|Nov 1, 2018

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper constructs a quantum dark soliton in a one-dimensional Bose gas by interpreting it as a Bose-Einstein condensation into the single-particle wave function of a classical dark soliton. It shows that this state is well-approximated by a Gaussian superposition of yrast states, achieving excellent agreement with the classical soliton density profile and exhibiting a longer lifetime than previous constructions, with decay time inversely proportional to the coupling constant in the weak-coupling regime.

ABSTRACT

Dark soliton solutions in the one-dimensional classical nonlinear Schrödinger equation has been considered to be related to the yrast states corresponding to the type-II excitations in the Lieb-Liniger model. However, the relation is nontrivial and remains unclear because a dark soliton localized in space breaks the translation symmetry, while yrast states are translationally invariant. In this work, we construct a symmetry-broken quantum soliton state and investigate the relation to the yrast states. By interpreting a quantum dark soliton as a Bose-Einstein condensation to the wave function of a classical dark soliton, we find that the quantum soliton state has a large weight only on the yrast states, which is analytically proved in the free-boson limit and numerically verified in the weak-coupling regime. By extending these results, we derive a parameter-free expression of a quantum soliton state that is written as a superposition of yrast states with Gaussian weights. The density profile of this quantum soliton state excellently agrees to that of the classical dark soliton. The dynamics of a quantum dark soliton is also studied, and it turns out that the density profile of a dark soliton decays, but the decay time increases as the inverse of the coupling constant in the weak-coupling limit.

Motivation & Objective

• To resolve the long-standing problem of constructing a symmetry-broken quantum state corresponding to a classical dark soliton in a one-dimensional Bose gas.
• To clarify the quantum-classical correspondence between yrast states in the Lieb-Liniger model and classical dark solitons.
• To provide a theoretically justified, parameter-free construction of a quantum dark soliton state that reproduces the soliton's density profile and dynamics.
• To demonstrate that the quantum soliton state has a longer lifetime than previous constructions based on Fourier transforms of yrast states.

Proposed method

• Interprets the quantum dark soliton as a Bose-Einstein condensation into the single-particle wave function of a classical dark soliton solution.
• Constructs the quantum state as a Gaussian superposition of yrast states, with mean momentum P and variance σP² ∝ c^{1/4} in the weak-coupling regime.
• Uses the Lieb-Liniger model with periodic boundary conditions and derives the state via analytical and numerical methods in the free-boson limit and weak-coupling regime.
• Verifies the density profile of the constructed state against the classical dark soliton solution, showing excellent agreement.
• Analyzes the dynamics by computing the energy variance ΔE and lifetime τ, finding τ ∝ 1/c in the weak-coupling limit.
• Compares the new state to prior constructions (e.g., Sato et al.) by evaluating decay times under the same measurement criterion.

Experimental results

Research questions

• RQ1 How can a quantum dark soliton state be constructed that breaks translation symmetry while being composed of translationally invariant yrast states?
• RQ2 What is the precise superposition structure of yrast states that reproduces the classical dark soliton's density profile?
• RQ3 How does the lifetime of a quantum dark soliton depend on the interaction strength c in the weak-coupling regime?
• RQ4 Can the quantum soliton state be interpreted as a result of successive position measurements on an yrast state, and how does this relate to the Gaussian superposition construction?

Key findings

• The quantum dark soliton state is well-approximated by a Gaussian superposition of yrast states with mean momentum P and variance σP² = 4γ³ρ₀√(ρ₀c)/3 in the weak-coupling regime.
• The density profile of the constructed quantum soliton state shows excellent agreement with the classical dark soliton solution.
• The decay time of the quantum soliton state is inversely proportional to the coupling constant c, with τ ≈ 3π/(2γ²ρ₀c), confirming τ ∝ 1/c in the weak-coupling limit.
• Numerical results show that the new soliton state decays more slowly than the Fourier-transform-based state of Sato et al., with a decay time of ~600 for c = 0.01 compared to ~100.
• In the free-boson limit (c = 0), the quantum soliton state is analytically shown to be composed almost entirely of yrast states.
• The state constructed via Gaussian superposition of yrast states is shown to be an exact solution for the dynamics of successive position measurements, with the state after N−N′ measurements being |N′,X;P⟩ with probability one.