[Paper Review] Exact out-of-equilibrium steady states in the semiclassical limit of the interacting Bose gas

Giuseppe Del Vecchio Del Vecchio, Alvise Bastianello+2 more|arXiv (Cornell University)|Feb 4, 2020

Cold Atom Physics and Bose-Einstein Condensates123 references1 citations

TL;DR

This paper presents an exact analytical framework for computing late-time steady states in the classical limit of the one-dimensional interacting Bose gas, using the Non-Linear Schrödinger equation. By leveraging the semiclassical limit of the quantum Lieb-Liniger model and applying the inverse scattering method, the authors derive exact expressions for the Generalized Gibbs Ensemble (GGE) and full counting statistics of the density operator, validated against ab initio numerical simulations, thus solving the classical quench problem exactly in this regime.

ABSTRACT

We study the out-of-equilibrium properties of a classical integrable non-relativistic theory, with a time evolution initially prepared with a finite energy density in the thermodynamic limit. The theory considered here is the Non-Linear Schrodinger equation which describes the dynamics of the one-dimensional interacting Bose gas in the regime of high occupation numbers. The main emphasis is on the determination of the late-time Generalised Gibbs Ensemble (GGE), which can be efficiently semi-numerically computed on arbitrary initial states, completely solving the famous quench problem in the classical regime. We take advantage of known results in the quantum model and the semiclassical limit to achieve new exact results for the momenta of the density operator on arbitrary GGEs, which we successfully compare with ab-initio numerical simulations. Furthermore, we determine the whole probability distribution of the density operator (full counting statistics), whose exact expression is still out of reach in the quantum model.

Motivation & Objective

• To characterize the late-time steady state of a classical integrable field theory after a quantum quench, specifically in the semiclassical limit of the interacting Bose gas.
• To extend exact results from the quantum Lieb-Liniger model to the classical Non-Linear Schrödinger (NLS) equation via the semiclassical limit.
• To provide a complete, semi-numerical method for computing the Generalized Gibbs Ensemble (GGE) for arbitrary initial states in the classical regime.
• To derive the full counting statistics (FCS) of the density operator, a quantity still intractable in the full quantum model.
• To benchmark analytical predictions against ab initio numerical simulations of the time evolution, establishing the validity of the approach.

Proposed method

• Utilize the semiclassical limit of the quantum Lieb-Liniger model to map it to the classical Non-Linear Schrödinger (NLS) equation for the one-dimensional interacting Bose gas.
• Apply the inverse scattering method to solve the classical NLS equation exactly, enabling the construction of the full set of conserved charges and the associated GGE.
• Derive exact analytical expressions for the root density of the NLS equation in terms of the field configuration, using the monodromy matrix and transfer matrix formalism.
• Construct the Generalized Gibbs Ensemble (GGE) from the root density, enabling the computation of exact one-point functions of the density operator.
• Derive the full counting statistics (FCS) of the density operator using the exact root density and the associated generating function.
• Implement a Hamiltonian-splitting spectral method for time evolution, preserving particle number and enabling high-precision numerical validation of analytical predictions.

Experimental results

Research questions

• RQ1 Can exact late-time steady states be derived for the classical Non-Linear Schrödinger equation after a quench, even for arbitrary initial states?
• RQ2 How does the semiclassical limit of the quantum Lieb-Liniger model enable exact results for the classical NLS in terms of the root density and GGE?
• RQ3 What is the exact form of the full counting statistics (FCS) of the density operator in the classical regime, and how does it compare to quantum results?
• RQ4 To what extent can analytical GGE predictions for the density operator's moments and distribution be validated through direct numerical time evolution?
• RQ5 How does the root density of the NLS equation depend on the initial field configuration, particularly for free thermal states?

Key findings

• The root density for a homogeneous initial field configuration is analytically derived as a semicircle distribution, ρ(λ) = 1/(2π)√(4|φ|² − λ²), which is in excellent agreement with numerical simulations.
• The full counting statistics (FCS) of the density operator is derived in closed form for arbitrary GGEs in the classical NLS model, a result not yet accessible in the full quantum model.
• The analytical predictions for the first and second moments of the density operator, derived from the GGE, are quantitatively validated against ab initio numerical simulations of the time evolution.
• For free thermal initial states, the root density at large rapidities decays as ρ(λ) ∼ 1/(βλ²), consistent with the expected classical behavior and the semiclassical limit.
• The numerical implementation of the time evolution via Hamiltonian splitting preserves particle number exactly and allows for high-precision extraction of the root density, enabling reliable comparison with analytical results.
• The method successfully solves the classical quench problem in the NLS model, providing a complete and exact framework for computing late-time steady states from arbitrary initial conditions.