MCTDH-X: The multiconfigurational time-dependent Hartree method for indistinguishable particles software
Authors: Lin Rui , Molignini Paolo , Papariello Luca , Tsatsos Marios C , Lévêque Camille , Weiner Storm E , Fasshauer Elke , Chitra R , Lode Axel U J 
Published: February 2020

Abstract
We introduce and describe the multiconfigurational time-depenent Hartree for indistinguishable particles (MCTDH-X) software, which is hosted, documented, and distributed at http://ultracold.org. This powerful tool allows the investigation of ground state properties and dynamics of interacting quantum many-body systems in different spatial dimensions. The MCTDH-X software is a set of programs and scripts to compute, analyze, and visualize solutions for the time-dependent and time-independent many-body Schrödinger equation for indistinguishable quantum particles. As the MCTDH-X software represents a general solver for the Schrödinger equation, it is applicable to a wide range of problems in the fields of atomic, optical, molecular physics, and condensed matter systems. In particular, it can be used to study light–matter interactions, correlated dynamics of electrons in the solid state as well as some aspects related to quantum information and computing. The MCTDH-X software solves a set of nonlinear coupled working equations based on the application of the time-dependent variational principle to the Schrödinger equation. These equations are obtained by using an ansatz for the many-body wavefunction that is a expansion in a set of time-dependent, fully symmetrized bosonic (X = B) or fully anti-symmetrized fermionic (X = F) many-body basis states. It is the time-dependence of the basis set that enables MCTDH-X to deal with quantum dynamics at a superior accuracy as compared to, for instance, exact diagonalization approaches with a static basis, where the number of basis states necessary to capture the dynamics of the wavefunction typically grows rapidly with time. Herein, we give an introduction to the MCTDH-X software via an easy-to-follow tutorial with a focus on accessibility. The illustrated exemplary problems are hosted at http://ultracold.org/tutorial and consider the physics of a few interacting bosons or fermions in a double-well potential. We explore computationally the position-space and momentum-space density, the one-body reduced density matrix, Glauber correlation functions, phases, (dynamical) phase transitions, and the imaging of the quantum systems in single-shot images. Although a few particles in a double well potential represent a minimal model system, we are able to demonstrate a rich variety of phenomena with it. We use the double well to illustrate the fermionization of bosonic particles, the crystallization of fermionic particles, characteristics of the superfluid and Mott-insulator quantum phases in Hubbard models, and even dynamical phase transitions. We provide a complete set of input files and scripts to redo all computations in this paper at http://ultracold.org/data/tutorial_input_files.zip, accompanied by tutorial videos at https://tinyurl.com/tjx35sq. Our tutorial should guide the potential users to apply the MCTDH-X software also to more complex systems.