Lindblad equation

The Lindblad equation describes continuous-time, Markovian evolution of an open quantum system’s density matrix while preserving complete positivity and trace.

Density matrix: A matrix representing a quantum system’s state, including statistical mixtures and coherences. The equation governs how this representation changes over time.
Completely positive map: A linear map that remains positive when extended by the identity operation on any auxiliary system. Complete positivity is required of the evolution generated by the equation.
Göran Lindblad: A Swedish mathematician whose 1976 work characterized generators of completely positive quantum dynamical semigroups. His theorem gives the equation its name and general form.
Quantum optics: The study of how light interacts with matter using quantum theory. It models emission, absorption, and loss in optical systems.
Redfield equation: A perturbative master equation for open quantum systems that can fail to preserve positivity. It can retain more microscopic detail but lacks Lindblad’s general positivity guarantee.
Hamiltonian (quantum mechanics): The operator corresponding to a quantum system’s total energy and generating its unitary time evolution. Its commutator supplies the equation’s coherent evolution term.
Positive map: A linear map that sends positive operators to positive operators. Complete positivity strengthens this basic preservation requirement.
Vittorio Gorini: An Italian mathematical physicist who co-authored a 1976 characterization of quantum dynamical semigroup generators. His independent result established the same generator structure.
Quantum decoherence: The loss of observable quantum coherence through entanglement with uncontrolled degrees of freedom. Dissipative terms describe how environmental coupling suppresses coherence.
Non-Markovian dynamics: Dynamics in which a system’s future evolution depends on environmental memory or its history. Such memory effects generally fall outside the equation’s Markovian framework.

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Hierarchical equations of motionCompared with: Its memoryless framework is simpler but cannot generally reproduce the hierarchy’s non-Markovian effects.