Lindblad Equation

Lindblad Equation

In quantum mechanics, the Kossakowski–Lindblad equation (after Andrzej Kossakowski and Göran Lindblad) or master equation in the Lindblad form is the most general type of Markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix that is trace preserving and completely positive for any initial condition.

The Lindblad master equation for an -dimensional system's reduced density matrix can be written:

where is a (Hermitian) Hamiltonian part, the are an arbitrary orthonormal basis of the operators on the system's Hilbert space, and the are constants which determine the dynamics. The coefficient matrix must be positive to ensure that the equation is trace preserving and completely positive. The summation only runs to because we have taken to be proportional to the identity operator, in which case the summand vanishes. Our convention implies that the are traceless for . The terms in the summation where can be described in terms of the Lindblad superoperator, .

If the terms are all zero, then this is the quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem.

Note that is not necessarily equal to the self-Hamiltonian of the system. It may also incorporate effective unitary dynamics arising from the system-environment interaction.