Companion to Closed & Open Quantum Systems#

A companion to the tutorial article "Coding closed and open quantum systems in MATLAB: applications in quantum optics and condensed matter" by Ariel Norambuena, Diego Tancara, and Raúl Coto — arXiv:1911.04906.

The article presents MATLAB code for simulating many-body quantum systems: magnetization dynamics of the closed and open Ising model, a dynamical quantum phase transition in cavity-QED arrays, Markovian dynamics of interacting two-level systems, and the non-Markovian pure-dephasing spin-boson model. This project collects those programs as runnable numbl scripts so you can execute and edit them in your browser — no MATLAB, no install. Pick a script in the panel on the left and click Run. Edits stay in memory for your session.

This README is a guide to which program to open, not a replacement for the article. Read the article for the physics, the equations, and the derivations; come here to run the code. The scripts are reproduced verbatim from the article's source (arXiv ancillary files) and run unmodified. The only repackaging: the article presents Example 1 and Example 2 as a sequence of code listings, which are concatenated here into single runnable Example1.m and Example2.m files.

Closed quantum systems (article §2)#

Script	Article	What it computes
Example1.m	§2.1	Magnetization dynamics of the closed two-spin Ising model via the time propagator expm(-iHt)
IsingModelTransitionPhase_new.m	§2.1	Dynamical quantum phase transition of the Ising model (6 spins): rate function and magnetization
CavityQEDTransitionPhase.m	§2.2	Dynamical phase transition in a cavity-QED array — Jaynes–Cummings vs. Rabi (order parameter and rate function)

Open quantum dynamics (article §3)#

Script	Article	What it computes
Example2.m	§3.1	Markovian master equation for two interacting qubits (open Ising), solved by diagonalizing the Lindbladian
TwoLevelSystemCoupledLight.m	§3.2	A two-level atom coupled to a photon reservoir; numerical solution compared to the exact one
NonMarkovianDynamicsPureDephasing.m	§3.4	Non-Markovian pure-dephasing spin-boson model: time-dependent rates, coherence, and a non-Markovianity measure

Helper functions#

These are called by the scripts above; they are functions, not standalone programs, so there is nothing to "Run" directly.

File	Used by	Builds
getSci.m	IsingModelTransitionPhase_new.m	the many-body Pauli operator on site i
acav.m	CavityQEDTransitionPhase.m	the cavity annihilation operator on site i
sigmap.m	CavityQEDTransitionPhase.m	the atomic raising operator on site i
QuantumSimulationCavityArray.m	CavityQEDTransitionPhase.m	the Jaynes–Cummings/Rabi Hamiltonian, time evolution, and observables
sortingEigenvalues.m	Example2.m, TwoLevelSystemCoupledLight.m	the sorted left/right eigenmatrices of the Lindbladian

Notes#

• Time evolution of closed systems uses the matrix exponential U = expm(-1i*H*dt). Open systems are solved by diagonalizing the Lindblad superoperator with eig and reconstructing the density matrix from its eigenmodes (sortingEigenvalues).
• CavityQEDTransitionPhase.m is the heaviest example — it sweeps 25 detunings for two models with long time evolutions, so it takes a couple of minutes in the browser. The others run in a few seconds.
• The article uses parfor in the cavity-QED sweep; numbl runs it as an ordinary (serial) loop, which gives the same result.

How it's deployed#

On every push to main, the workflow in .github/workflows/deploy.yml bundles these files with the numbl browser IDE and publishes the result to GitHub Pages — no server, nothing to build by hand. Edit numbl-project.json to change the title or which file opens first.

Reference#

Ariel Norambuena, Diego Tancara, Raúl Coto, Coding closed and open quantum systems in MATLAB: applications in quantum optics and condensed matter, arXiv:1911.04906.