Ultracold Atoms in 1D Optical Lattices: Mean Field, Quantum Field, Computation, and Soliton Formation

TL;DR

This paper explores the relationship between mean-field and quantum descriptions of ultracold bosons in 1D optical lattices, demonstrating quantum analogs of solitons and analyzing their dynamics using advanced numerical methods.

Contribution

It establishes a connection between the discrete nonlinear Schrödinger equation and the Bose-Hubbard Hamiltonian, and constructs quantum dark soliton states from mean-field solutions.

Findings

Quantum dark solitons can be constructed from mean-field states.

Dynamical properties of solitons are analyzed within the Bose-Hubbard model.

Numerical methods include time-evolving block decimation for quantum simulations.

Abstract

In this work, we highlight the correspondence between two descriptions of a system of ultracold bosons in a one-dimensional optical lattice potential: (1) the discrete nonlinear Schr\"{o}dinger equation, a discrete mean-field theory, and (2) the Bose-Hubbard Hamiltonian, a discrete quantum-field theory. The former is recovered from the latter in the limit of a product of local coherent states. Using a truncated form of these mean-field states as initial conditions, we build quantum analogs to the dark soliton solutions of the discrete nonlinear Schr\"{o}dinger equation and investigate their dynamical properties in the Bose-Hubbard Hamiltonian. We also discuss specifics of the numerical methods employed for both our mean-field and quantum calculations, where in the latter case we use the time-evolving block decimation algorithm due to Vidal.