Quantum master equation

A quantum master equation is a generalization of the idea of a master equation. Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements. A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.

The Lindblad equation was a primitive example of a quantum master equation. Another primitive example of a quantum master equation is the Nakajima-Zwanzig equation which is also sometimes known as the "generalized quantum master equation" or the "exact master equation" because it is non-Markovian. However, it is extremely difficult to solve, and has therefore been replaced by more sophisticated approaches.

More modern quantum master equations which show better agreement with exact numerical calculations include the polaron transformed quantum master equation, and the variational polaron transformed quantum master equation.^[1]

Alternatives to quantum master equations include numerically exact approaches such as those based on numerical Feynman integrals, DMRG, MCTDH, and Tanimura and Kubo's hierarchical equations of motion.

See also

• Open quantum system
• Quantum dynamics
• Quantum coherence
• Differential equation
• Master equation
• Lindblad equation
• Nakajima-Zwanzig equation
• Feynman integral

References