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Understanding the deep connection between these -- sometimes apparently unrelated -- methods can be a powerful tool, allowing us to derive effective dynamics in unconventional systems or regimes. This tutorial aims at providing quantum theorists across multiple fields (e.g., quantum and atom optics, optomechanics, or hybrid quantum systems) with a self-contained practical toolbox to derive effective quantum dynamics, applicable to systems ranging from N<\/mml:mi><\/mml:math>-level emitters to mechanical resonators. First, we summarize the projector approach to open quantum systems and the derivation of the fundamental Nakajima-Zwanzig equation. Then, we show how three common effective equations, namely the Brownian master equation, the Born-Markov master equation, and the adiabatic elimination used in atom and molecular optics, can be derived from different perturbative expansions of the Nakajima-Zwanzig equation. We also solve in detail four specific examples using this formalism, namely a harmonic oscillator subject to displacement noise, the effective equations of a mechanical resonator cooled by an optical cavity, the Purcell effect for a qubit coupled to an optical cavity, and the adiabatic elimination in a Lambda system.<\/jats:p>","DOI":"10.22331\/q-2024-08-29-1454","type":"journal-article","created":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T12:04:01Z","timestamp":1724933041000},"page":"1454","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Tutorial: projector approach to master equations for open quantum systems"],"prefix":"10.22331","volume":"8","author":[{"given":"C.","family":"Gonzalez-Ballestero","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria"}]}],"member":"9598","published-online":{"date-parts":[[2024,8,29]]},"reference":[{"key":"0","doi-asserted-by":"crossref","unstructured":"Francesco Francesco Campaioli, Jared H Cole, and Harini Hapuarachchi. ``A tutorial on quantum master equations: Tips and tricks for quantum optics, quantum computing and beyond'' (2023). url: https:\/\/arxiv.org\/abs\/2303.16449.","DOI":"10.1103\/PRXQuantum.5.020202"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Piotr Sza\u0144kowski. ``Introduction to the theory of open quantum systems''. 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