We present a general-purpose, open-source software suite, SQUIRREL (Streamlined Quantum Unified Interface for Researching Real-time Excitations with Light), for propagating the time-dependent Schrödinger equation on complex geometries in the presence of time-dependent electric and/or magnetic fields. To handle large systems that can be executed on a conventional desktop computer, the SQUIRREL software suite uses a suite of efficient propagation methods for various quantum dynamics applications, including a new perturbation-based element-dropping algorithm that improves computational performance with minimal loss of accuracy. We analyze the efficacy of these optimizations for Crank-Nicolson, scaled Taylor series approximation, and split-operator propagation methods and discuss the range of their applicability to a variety of quantum dynamics problems. In addition, we provide several examples of time-dependent dynamics calculations and extensive documentation for generating custom geometries, potentials, and time-propagation approaches. Our numerical benchmarks and results demonstrate the versatility of the SQUIRREL software suite for efficiently calculating quantum dynamics in complex nanoscale geometries, particularly in the presence of time-dependent magnetic fields, which have received less attention in previous quantum dynamics studies. Program summary Program Title: SQUIRREL CPC Library link to program files: https://doi.org/10.17632/kfvs5s88sj.1 Licensing provisions: GNU General Public License 3 Programming language: MATLAB Supplementary material: Additional details on calculations with an anisotropic effective mass, determination of timesteps, propagation timings, and element-dropping timings. Nature of problem: The SQUIRREL software suite solves the time-dependent Schrödinger equation for electronic states in the presence of time-dependent electric and/or magnetic fields. The code is well-suited for calculating electron dynamics of complex nanostructures in the effective mass approximation, using various propagation and sparse-matrix techniques to reduce the computational complexity of these problems. The program provides a user-friendly interface for testing these optimizations and generating custom geometries and potentials for systems of interest. Solution method: The SQUIRREL software suite uses Crank-Nicolson, split-operator, Padé approximant, and scaled-Taylor-series-based propagation methods to calculate electron dynamics on a finite-element basis. The code includes a novel perturbation-based element-dropping method, which enables sparse representations of the typically dense finite element Hamiltonian matrix. This capability enables a highly efficient and flexible approach for calculating driven dynamics in quantum systems with complex geometries and time-dependent electric/magnetic fields.
We present a general-purpose, open-source software suite, SQUIRREL (Streamlined Quantum Unified Interface for Researching Real-time Excitations with Light), for propagating the time-dependent Schrödinger equation on complex geometries in the presence of time-dependent electric and/or magnetic fields. To handle large systems that can be executed on a conventional desktop computer, the SQUIRREL software suite uses a suite of efficient propagation methods for various quantum dynamics applications, including a new perturbation-based element-dropping algorithm that improves computational performance with minimal loss of accuracy. We analyze the efficacy of these optimizations for Crank-Nicolson, scaled Taylor series approximation, and split-operator propagation methods and discuss the range of their applicability to a variety of quantum dynamics problems. In addition, we provide several examples of time-dependent dynamics calculations and extensive documentation for generating custom geometries, potentials, and time-propagation approaches. Our numerical benchmarks and results demonstrate the versatility of the SQUIRREL software suite for efficiently calculating quantum dynamics in complex nanoscale geometries, particularly in the presence of time-dependent magnetic fields, which have received less attention in previous quantum dynamics studies. Program summary Program Title: SQUIRREL CPC Library link to program files: https://doi.org/10.17632/kfvs5s88sj.1 Licensing provisions: GNU General Public License 3 Programming language: MATLAB Supplementary material: Additional details on calculations with an anisotropic effective mass, determination of timesteps, propagation timings, and element-dropping timings. Nature of problem: The SQUIRREL software suite solves the time-dependent Schrödinger equation for electronic states in the presence of time-dependent electric and/or magnetic fields. The code is well-suited for calculating electron dynamics of complex nanostructures in the effective mass approximation, using various propagation and sparse-matrix techniques to reduce the computational complexity of these problems. The program provides a user-friendly interface for testing these optimizations and generating custom geometries and potentials for systems of interest. Solution method: The SQUIRREL software suite uses Crank-Nicolson, split-operator, Padé approximant, and scaled-Taylor-series-based propagation methods to calculate electron dynamics on a finite-element basis. The code includes a novel perturbation-based element-dropping method, which enables sparse representations of the typically dense finite element Hamiltonian matrix. This capability enables a highly efficient and flexible approach for calculating driven dynamics in quantum systems with complex geometries and time-dependent electric/magnetic fields.
The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation . They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summary Program Title: CoOMBE CPC Library link to program files: https://doi.org/10.17632/5wsg9d52dk.1 Developers' repository link: https://github.com/durham-qlm/CoOMBE Licensing provisions: GPLv3 Programming language: Fortran 90 Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes , absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields , this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.). Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations , which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.
In this work we present TDStool, a general-purpose easy-to-use software tool for the solution of the time-dependent Schrödinger equation in 2D and 3D domains with arbitrary time-dependent potentials. The numerical algorithms adopted in the code, namely Fourier split-step and box-integration methods, are sketched and the main characteristics of the tool are illustrated. As an example, the dynamics of a single electron in systems of two and three coupled quantum dots is obtained. The code is released as an open-source project and has a build-in graphical interface for the visualization of the results.
The time-dependent Schrödinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials science. However, solving the TDSE for complex fermionic systems remains a significant challenge, particularly due to the need to capture the time-evolving many-body correlations, while the antisymmetric nature of fermionic wavefunctions complicates the function space in which these solutions must be represented. We propose a general-purpose neural network framework for solving the real-space TDSE, Fermionic Antisymmetric Spatio-Temporal Network, which treats time as an explicit input alongside spatial coordinates, enabling a unified spatiotemporal representation of complex, antisymmetric wavefunctions for fermionic systems. This approach formulates the TDSE as a global optimization problem, avoiding step-by-step propagation and supporting highly parallelizable training. The method is demonstrated on five benchmark problems, achieving excellent agreement with reference solutions across all cases. These results demonstrate the method's accuracy and flexibility within the bound-state manifold across various dimensions and interaction regimes. While the current localized Ansatz inherently restricts the description of extensive ionization and continuum states, the method demonstrates the capability to stably simulate coherent multi-electron dynamics over extended time windows. Our framework offers a highly expressive alternative to traditional basis-dependent or mean-field methods, opening new possibilities for ab initio simulations of time-dependent quantum systems, with applications in quantum dynamics, molecular control, and ultrafast spectroscopy.
wavepacket is a program to simulate the dynamics of a wave packet interacting with a time-dependent potential, with the time-dependent Schrödinger equation being solved on a one-, two-, or three-dimensional spatial grid using the split operator method. This new version fixes bugs present in the original program. Program Title: wavepacket CPC Library link to program files: https://doi.org/10.17632/gnxmkygxrw.1 Licensing provisions: GPLv3 Programming language: C (iso C99) Journal reference of previous version: Comput. Phys. Commun. 185 (2014) 407–414. Does the new version supersede the previous version?: Yes Reasons for the new version: Bug fixes. Summary of revisions: This new version fixes bugs in the program wavepacket [1]. In the original version, the function print_wf_bin, used (optionally) to write out the wave function as a binary file, contained a bug that would result in only part of the full wave function being written out when the program was run in parallel. Additionally, negative values of the parameters nt and/or nprint could lead to the program being stuck in an infinite loop. Additional checks have been added to ensure that both these parameters are positive (or zero). Nature of problem: Solves the time-dependent Schrödinger equation for a single particle interacting with a time-dependent potential. Solution method: The wave function is described by its value on a spatial grid and the evolution operator is approximated using the split-operator method [2,3], with the kinetic energy operator calculated using a Fast Fourier Transform. Additional comments including restrictions and unusual features: Simulation can be in one, two, or three dimensions. The fftw [4] implementation of the fast Fourier transform is used, so the corresponding library must be installed. Serial and parallel versions are compiled from the same source files. The parallel version is based on the Message-Passing Interface (mpi) [5]. C.M. Dion, A. Hashemloo, G. Rahali, Comput. Phys. Commun. 185 (2014) 407–414. M.D. Feit, J.A. Fleck, Jr., A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47 (1982) 412–433. M.D. Feit, J.A. Fleck, Jr., Solution of the Schrödinger equation by a spectral method II: Vibrational energy levels of triatomic molecules, J. Chem. Phys. 78 (1) (1983) 301–308. http://www.fftw.org http://www.mpi-forum.org
We present QDT (“quantum dynamics toolbox”), an open-source Matlab software package that enables users to simulate coupled quantum systems in the subsystem energy eigenbasis using modular functions. QDT requires no user knowledge of operator matrix assembly and automatically performs all necessary operator constructions and Hilbert space expansions. Density matrix propagation is performed by numerically solving the Liouville–von-Neumann equation. In order to simulate dissipation and decoherence effects, the Lindblad formalism is implemented. Furthermore, QDT supplies practical analysis and plotting functions, such as visualization of density matrix and expectation value dynamics, that facilitate the evaluation of simulation results. QDT further provides a module for the simulation of coherent multidimensional spectroscopy. Program Title: QDT — A Quantum Dynamics Toolbox CPC Library link to program files: https://doi.org/10.17632/786vb4rgy3.1 Code Ocean capsule: https://codeocean.com/capsule/4608215 Licensing provisions: MIT Programming language: Matlab R2018a (and above) with parallel computing toolbox (optional) [1] Supplementary material: Available on the QDT website [2] Nature of problem: Coupled quantum systems can be easily set up and discussed as level schemes in the subsystem energy eigenbasis. The simulation of the time evolution of these systems by numerically solving differential equations is not straightforward, however. Simulation code that is written from scratch is time-consuming to create and prone to errors during implementation. Furthermore, if code bases deviate drastically between research groups, reproducibility of results becomes challenging, and communication is hindered. Solution method: QDT addresses the above issues by providing a modular toolbox that contains, among others, elements for the simulation of multi-level quantum systems, quantum harmonic oscillators, their interaction with external electric fields, and dissipation. These elements can be combined by the user to build a specific system of interest. All operators are created and expanded automatically based on a single script file provided by the user. Simulation tasks are automated by a single function call, and parallelized in case of extensive tasks.
Quantum many-body systems are challenging targets for computational physics due to their large number of degrees of freedom. The tensor networks, particularly Tensor Product States (TPS) and Projected Entangled Pair States (PEPS), effectively represent these systems on two-dimensional lattices. However, the technical complexity of TPS/PEPS-based coding can be challenging for many researchers to manage effectively. To reduce this problem, we developed TeNeS (Tensor Network Solver). This paper introduces TeNeS -v2, which extends TeNeS with real-time and finite temperature simulations, providing deeper insights into quantum many-body systems. We detail the new algorithms, input/output design, and application examples, demonstrating TeNeS -v2's applicability to various quantum spin and Bose models on two-dimensional lattices. New version program summary Program Title: TeNeS CPC Library link to program files: https://doi.org/10.17632/psm26xxbvd.2 Developer's repository link:: https://github.com/issp-center-dev/TeNeS Licensing provisions: GPLv3 Programming language: C++11 Journal reference of previous version: Comput. Phys. Commun. (2022) 279, 108437, doi:10.1016/j.cpc.2022.108437 Reasons for the new version: To extend the capabilities of TeNeS to include real-time and finite temperature simulations, enabling deeper insights into quantum many-body systems beyond ground states. Summary of revisions: TeNeS -v2 introduces real-time evolution and finite temperature simulation capabilities, expanding the range of quantum many-body phenomena that can be studied. Nature of problem: Quantum many-body systems are extremely difficult to simulate because of the huge dimension of the Hilbert space. Conventional methods have difficulty in accurately representing these systems, especially for properties beyond small lattices and ground states. Advanced computational techniques are required to efficiently capture the dynamics and thermal properties of these systems. Solution method: TeNeS employs tensor networks, specifically Tensor Product States (TPS) and Projected Entangled Pair States (PEPS), to efficiently represent quantum many-body states on two-dimensional lattices. The real-time evolution is handled through a straightforward extension of imaginary time evolution methods, allowing the study of dynamical properties. Finite temperature simulations are conducted using an imaginary time evolution approach starting from the infinite-temperature mixed state. These methods enable the accurate calculation of various physical properties in different quantum states. Additional comments including restrictions and unusual features: TeNeS -v2 maintains ease of use by providing flexible input file configurations and supporting a variety of two-dimensional lattice models. Increasing the bond dimension to improve accuracy requires more computational resources. Finite temperature calculations may exhibit unphysical behavior due to limitations of the tensor network approximation method. Users should be aware of these potential problems and interpret the results following the suggestions made in the text.