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{{Quantum book backlink|Open quantum systems}}
The '''Lindblad equation''' is the standard equation describing the Markovian time evolution of the density operator <math>\rho</math> of an open quantum system. It gives the most general linear dynamical law that preserves trace and complete positivity for a quantum dynamical semigroup.<ref name="Lindblad1976">{{cite journal |last=Lindblad |first=Göran |title=On the generators of quantum dynamical semigroups |journal=Communications in Mathematical Physics |volume=48 |issue=2 |pages=119–130 |year=1976 |url=https://link.springer.com/article/10.1007/BF01608499 |doi=10.1007/BF01608499}}</ref><ref name="MIT_OCW">{{cite web |url=https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/resources/mit22_51f12_ch8/ |title=22.51 Course Notes, Chapter 8: Open Quantum Systems |website=MIT OpenCourseWare |access-date=2026-04-12}}</ref>
It is fundamental in quantum optics, quantum information, decoherence theory, and the study of dissipative quantum systems.<ref name="MIT_OCW" />

[[File:Quantum_Lindblad_equation.svg|thumb|400px|The Lindblad equation gives the most general Markovian, completely positive, trace-preserving evolution of an open quantum system.]]

=Lindblad equation=
== General form ==
The Lindblad equation is usually written as

<math>
\frac{d\rho}{dt}
=
-\frac{i}{\hbar}[\hat{H},\rho]
+
\sum_k
\left(
L_k \rho L_k^\dagger
-\frac{1}{2}\{L_k^\dagger L_k,\rho\}
\right).
</math>

Here:

* <math>\hat{H}</math> is the Hamiltonian of the system,
* <math>L_k</math> are the '''Lindblad operators''' or '''jump operators''',
* <math>\{\cdot,\cdot\}</math> denotes the anticommutator.

The first term gives the usual unitary evolution, while the additional terms describe dissipation, decoherence, and irreversible coupling to the environment.<ref name="MIT_OCW" />

=== Hamiltonian part ===

The commutator term

<math>
-\frac{i}{\hbar}[\hat{H},\rho]
</math>

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 <math>L_k</math> 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.<ref name="Lindblad1976" />

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.<ref name="Lindblad1976" />

=== 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.<ref name="MIT_OCW" />

=== Trace preservation ===

The density operator must satisfy

<math>
\mathrm{Tr}(\rho)=1.
</math>

The Lindblad structure is constructed so that

<math>
\frac{d}{dt}\mathrm{Tr}(\rho)=0,
</math>

so probability is conserved during the evolution.<ref name="Lindblad1976" />

== Jump operators ==

The operators <math>L_k</math> 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.<ref name="MIT_OCW" />

=== Example: spontaneous emission ===

For a two-level atom with decay rate <math>\gamma</math>, spontaneous emission can be modeled by

<math>
L = \sqrt{\gamma}\,\sigma_-,
</math>

where <math>\sigma_-</math> 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.<ref name="MIT_OCW" />

=== Example: pure dephasing ===

Pure dephasing can be represented by an operator such as

<math>
L = \sqrt{\gamma_\phi}\,\sigma_z.
</math>

This causes the off-diagonal elements of <math>\rho</math> 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 <math>\rho(t)</math>, not on the detailed past history. This leads to a time-local differential equation of Lindblad form.<ref name="MIT_OCW" />

=== 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.<ref name="Breuer2016">{{cite journal |last=Breuer |first=Heinz-Peter |last2=Laine |first2=Elsi-Mari |last3=Piilo |first3=Jyrki |last4=Vacchini |first4=Bassano |title=Colloquium: Non-Markovian dynamics in open quantum systems |journal=Reviews of Modern Physics |volume=88 |issue=2 |pages=021002 |year=2016 |url=https://link.aps.org/doi/10.1103/RevModPhys.88.021002 |doi=10.1103/RevModPhys.88.021002}}</ref>

== Properties ==

The Lindblad equation has several important mathematical and physical properties.

=== Linearity ===

The equation is linear in the density operator <math>\rho</math>, which makes it compatible with statistical mixtures.

=== Hermiticity preservation ===

If <math>\rho</math> 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.<ref name="Lindblad1976" />

=== Stationary states ===

A stationary state <math>\rho_{\mathrm{ss}}</math> satisfies

<math>
\frac{d\rho_{\mathrm{ss}}}{dt}=0.
</math>

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.<ref name="QubitReview">{{cite journal |last=Kjaergaard |first=Morten |last2=Schwartz |first2=Michael E. |last3=Braumüller |first3=Jochen |last4=Krantz |first4=Philip |last5=Wang |first5=J. I.-J. |last6=Gustavsson |first6=Simon |last7=Oliver |first7=William D. |title=Engineering high-coherence superconducting qubits |journal=Nature Reviews Materials |volume=5 |pages=309–324 |year=2020 |url=https://www.nature.com/articles/s41578-021-00370-4 |doi=10.1038/s41578-021-00370-4}}</ref>

=== 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.<ref name="MIT_OCW" />

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

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Lindblad equation|1}}

Physics:Quantum Lindblad equation - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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