Tutorial: projector approach to master equations for open quantum systems
arXiv:2305.19704 · doi:10.22331/q-2024-08-29-1454
Abstract
Most quantum theorists are familiar with different ways of describing the effective quantum dynamics of a system coupled to external degrees of freedom, such as the Born-Markov master equation or the adiabatic elimination. 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-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.
30 pages, 5 figures
References in corpus (38)
- Cavity Optomechanics
- Long-distance quantum communication with atomic ensembles and linear optics
- Quantum state transfer and entanglement distribution among distant nodes in a quantum network
- A short introduction to the Lindblad Master Equation
- Entanglement of two qubits mediated by one-dimensional plasmonic waveguides
- Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps
- Dissipation and Ultrastrong Coupling in Circuit QED
- Sideband Cooling Beyond the Quantum Limit with Squeezed Light
- Adiabatic Elimination in a Lambda System
- Observation of non-Markovian micro-mechanical Brownian motion
- Open Quantum Dynamics: Complete Positivity and Entanglement
- A Brief History of the GKLS Equation
- Cavity-Assisted Back Action Cooling of Mechanical Resonators
- Bloch-Redfield equations for modeling light-harvesting complexes
- Nonperturbative master equation solution of central spin dephasing dynamics
- Dissipation and Thermal Noise in Hybrid Quantum Systems in the Ultrastrong Coupling Regime
- The Rotating-Wave Approximation: Consistency and Applicability from an Open Quantum System Analysis
- A Tutorial on Quantum Master Equations: Tips and tricks for quantum optics, quantum computing and beyond
- Correlated projection operator approach to non-Markovian dynamics in spin baths
- Theory for Cavity Cooling of Levitated Nanoparticles via Coherent Scattering: Master Equation Approach
- An introduction to operational quantum dynamics
- Non-Markovian dynamics of a single electron spin coupled to a nuclear spin bath
- Generalized Schrieffer-Wolff Formalism for Dissipative Systems
- Macroscopic Quantum Superpositions via Dynamics in a Wide Double-Well Potential
- Fast Quantum Interference of a Nanoparticle via Optical Potential Control
- A recipe for Hamiltonian of system-environment coupling applicable to ultrastrong light-matter interaction regime
- Canonically consistent quantum master equation
- A Lindblad master equation capable of describing hybrid quantum systems in the ultra-strong coupling regime
- Initial Correlations in Open Quantum Systems: Constructing Linear Dynamical Maps and Master Equations
- Degenerated Liouvillians and Steady-State Reduced Density Matrices
- Non-Markovian decoherence dynamics of the hybrid quantum system with a cavity strongly coupling to a spin ensemble: a master equation approach
- Introduction to the theory of open quantum systems
- Quantum tight-binding chains with dissipative coupling
- Taming the Bloch-Redfield equation: Recovering an accurate Lindblad equation for general open quantum systems
- Uncovering nonperturbative dynamics of the biased sub-Ohmic spin-boson model with variational Matrix Product States
- Nontrivial eigenvalues of the Liouvillian of an open quantum system
- Multiple-scale analysis of open quantum systems
- Theory of waveguide-QED with moving emitters