The Lindblad equation is the standard equation describing the Markovian time evolution of the density operator $ρ$ of an open quantum system. It gives the most general linear dynamical law that preserves trace and complete positivity for a quantum dynamical semigroup.^[1]^[2] It is fundamental in quantum optics, quantum information, decoherence theory, and the study of dissipative quantum systems.^[2]

Lindblad equation

General form

The Lindblad equation is usually written as

$\frac{d ρ}{d t} = - \frac{i}{ℏ} [\hat{H}, ρ] + \sum_{k} (L_{k} ρ L_{k}^{†} - \frac{1}{2} {L_{k}^{†} L_{k}, ρ}) .$

Here:

$\hat{H}$ is the Hamiltonian of the system,
$L_{k}$ are the Lindblad operators or jump operators,
${\cdot, \cdot}$ denotes the anticommutator.

The first term gives the usual unitary evolution, while the additional terms describe dissipation, decoherence, and irreversible coupling to the environment.^[2]

Hamiltonian part

The commutator term

$- \frac{i}{ℏ} [\hat{H}, ρ]$

is the same as in the von Neumann equation for a closed quantum system. It generates reversible evolution.

Dissipative part

The sum over the operators $L_{k}$ adds non-unitary effects caused by the environment. These terms allow one to model processes such as

spontaneous emission,
dephasing,
amplitude damping,
thermal relaxation.

GKSL structure

The Lindblad equation is often called the GKSL equation, after Gorini, Kossakowski, Sudarshan, and Lindblad, because the general structure of Markovian quantum master equations was established in parallel by these authors in 1976.^[1]

Its importance is that it characterizes the generators of completely positive trace-preserving semigroups, which are the physically acceptable Markovian evolutions of open quantum systems.^[1]

Complete positivity

A density operator must remain positive under time evolution. In quantum theory, this requirement is stronger than ordinary positivity because the system may be entangled with another system. The Lindblad form guarantees complete positivity, which ensures that the evolution remains physical even when extended to larger Hilbert spaces.^[2]

Trace preservation

The density operator must satisfy

$T r (ρ) = 1 .$

The Lindblad structure is constructed so that

$\frac{d}{d t} T r (ρ) = 0,$

so probability is conserved during the evolution.^[1]

Jump operators

The operators $L_{k}$ encode specific channels through which the environment influences the system.

Physical meaning

Each jump operator represents a distinct dissipative process. For example:

photon emission from an excited atom,
loss of phase coherence,
coupling to a thermal bath.

The form of the Lindblad operators depends on the microscopic interaction between the system and its environment.^[2]

Example: spontaneous emission

For a two-level atom with decay rate $γ$ , spontaneous emission can be modeled by

$L = \sqrt{γ} σ_{-},$

where $σ_{-}$ is the lowering operator.

The master equation then describes decay from the excited state to the ground state together with the associated loss of coherence.^[2]

Example: pure dephasing

Pure dephasing can be represented by an operator such as

$L = \sqrt{γ_{ϕ}} σ_{z} .$

This causes the off-diagonal elements of $ρ$ to decay while leaving the populations unchanged.

Relation to open quantum systems

The Lindblad equation arises when a quantum system interacts weakly with an environment and the environment can be treated as memoryless.

Markovian approximation

In the Markovian approximation, the future evolution depends only on the present state $ρ (t)$ , not on the detailed past history. This leads to a time-local differential equation of Lindblad form.^[2]

Born approximation

A common derivation assumes weak system-environment coupling and factorization of the total state into system and bath parts. Together with the Markov approximation, this leads to an effective reduced dynamics for the system alone.

Beyond Markovian dynamics

If the environment has memory, the evolution is no longer exactly Lindbladian. In that case one must use more general non-Markovian master equations, often involving memory kernels or time-dependent generators.^[3]

Properties

The Lindblad equation has several important mathematical and physical properties.

Linearity

The equation is linear in the density operator $ρ$ , which makes it compatible with statistical mixtures.

Hermiticity preservation

If $ρ$ is Hermitian initially, it remains Hermitian during the evolution.

Positivity and complete positivity

The Lindblad form ensures that the eigenvalues of the density operator remain non-negative, and more strongly, that the map is completely positive.^[1]

Stationary states

A stationary state $ρ_{s s}$ satisfies

$\frac{d ρ_{s s}}{d t} = 0 .$

Such states are important in dissipative state preparation, laser theory, and driven open quantum systems.

Applications

The Lindblad equation is widely used in modern quantum physics.

Quantum optics

It describes radiative decay, cavity loss, resonance fluorescence, and atom-photon interactions.

Quantum information

It is used to model noisy qubits, decoherence, error channels, and dissipative control in quantum computers.^[4]

Condensed matter and thermodynamics

It is also used for transport, thermalization, driven-dissipative systems, and nonequilibrium statistical mechanics.

Physical significance

The Lindblad equation provides the standard mathematical language for irreversible quantum dynamics. It extends the unitary formalism of closed systems to realistic situations where quantum systems are noisy, dissipative, and coupled to external degrees of freedom.^[2]

It is therefore one of the central equations of open quantum theory.