The Lindblad superoperator is often used to express the quantum master equation for a dissipative system.
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In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that phase coherence is maintained throughout the process, and is a consequence of the fact that all participating degrees of freedom are considered. However, any real physical system is not absolutely isolated, and will interact with its environment. This interaction with degrees of freedom external to the system results in dissipation of energy into the surroundings, and randomization of phase. This latter effect is the reason quantum mechanics is difficult to observe on a macroscopic scale. More so, understanding the interaction of a quantum system with its environment is necessary to understanding many commonly observed phenomena like the spontaneous emission of atoms, or the performance of many quantum technological devices, like the laser.
Certain mathematical techniques have been introduced to treat the interaction of a quantum system with its environment. One of these is the use of the density matrix, and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the Schrödinger picture or Heisenberg picture, it allows more easily for the inclusion of incoherent processes, which represent environmental interactions. The density operator has the property that it can represent a classical mixture of quantum states, and is thus vital to accurately describe the dynamics of so-called open quantum systems.
For a collapse operator
Such a term is found regularly in the Lindblad equation as used in quantum optics, where it can express absorption or emission of photons from a reservoir. For example, the master equation for a single mode optical resonator (e.g. a Fabry–Perot cavity) coupled to a thermal bath is
where
Derivation from Liouvillian dynamics
The derivation assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a Hamiltonian written in terms of operators acting only on the respective subspace of the total Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath. The most general form of this Hamiltonian is
The dynamics of the entire system can be described by the Liouville equation of motion,
where the Hamiltonian
This implicit equation for
We proceed with the derivation by assuming the interaction is initiated at
tracing over the bath degrees of freedom,
This equation is exact for the time dynamics of the system density matrix but requires full knowledge of the dynamics of the bath degrees of freedom. A simplifying assumption called the Born approximation rests on the largeness of the bath and the relative weakness of the coupling, which is to say the coupling of the system to the bath should not significantly alter the bath eigenstates. In this case the full density matrix is factorable for all times as
The equation is now explicit in the system degrees of freedom, but is very difficult to solve. A final assumption is the Born-Markoff approximation that the time derivative of the density matrix depends only on its current state, and not on its past. This assumption is valid under fast bath dynamics, wherein correlations between bath and system variables are lost extremely quickly, and amounts to replacing
This is the final form of the master equation we need.
Linear Coupling
If the interaction Hamiltonian is assumed to have the form
for system operators
which can be expanded as
The expectation values