# 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](https://arxiv.org/abs/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](https://numbl.org) 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`](Example1.m) and [`Example2.m`](Example2.m) files.

## Closed quantum systems (article §2)

| Script | Article | What it computes |
| --- | --- | --- |
| [`Example1.m`](Example1.m) | §2.1 | Magnetization dynamics of the closed two-spin Ising model via the time propagator `expm(-iHt)` |
| [`IsingModelTransitionPhase_new.m`](IsingModelTransitionPhase_new.m) | §2.1 | Dynamical quantum phase transition of the Ising model (6 spins): rate function and magnetization |
| [`CavityQEDTransitionPhase.m`](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`](Example2.m) | §3.1 | Markovian master equation for two interacting qubits (open Ising), solved by diagonalizing the Lindbladian |
| [`TwoLevelSystemCoupledLight.m`](TwoLevelSystemCoupledLight.m) | §3.2 | A two-level atom coupled to a photon reservoir; numerical solution compared to the exact one |
| [`NonMarkovianDynamicsPureDephasing.m`](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`](getSci.m) | `IsingModelTransitionPhase_new.m` | the many-body Pauli operator on site *i* |
| [`acav.m`](acav.m) | `CavityQEDTransitionPhase.m` | the cavity annihilation operator on site *i* |
| [`sigmap.m`](sigmap.m) | `CavityQEDTransitionPhase.m` | the atomic raising operator on site *i* |
| [`QuantumSimulationCavityArray.m`](QuantumSimulationCavityArray.m) | `CavityQEDTransitionPhase.m` | the Jaynes–Cummings/Rabi Hamiltonian, time evolution, and observables |
| [`sortingEigenvalues.m`](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`](.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](https://arxiv.org/abs/1911.04906).