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{{Quantum book backlink|Open quantum systems}}
The '''quantum master equation''' describes the time evolution of the density operator <math>\rho</math> of an open quantum system interacting with an environment. Unlike closed systems, whose dynamics are governed by the Schrödinger equation, open systems exhibit dissipation and decoherence.<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>

Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena.

[[File:Quantum_master_equation_diagram.svg|thumb|400px|Open quantum systems evolve under the influence of an environment, leading to decoherence and dissipation described by master equations.]]

=== Density operator dynamics ===

The state of a quantum system is described by a density operator <math>\rho</math>, which evolves according to

<math>
\frac{d\rho}{dt} = \mathcal{L}(\rho),
</math>

where <math>\mathcal{L}</math> is a linear superoperator called the '''Liouvillian'''.

For a closed system, this reduces to the von Neumann equation:

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

=== Reduced dynamics ===

In an open system, one considers a combined system + environment with total density operator <math>\rho_{\text{tot}}</math>. The system state is obtained by tracing out the environment:

<math>
\rho = \mathrm{Tr}_{\text{env}}(\rho_{\text{tot}}).
</math>

This leads to effective non-unitary evolution for the system.

=== Markovian approximation ===

A common simplification assumes that the environment has no memory, so the dynamics are approximately local in time.<ref name="MIT_OCW" />

=The Lindblad form=

The most general form of a Markovian quantum master equation that preserves trace and complete positivity is the '''Lindblad equation''':

<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 system Hamiltonian
* <math>L_k</math> are '''Lindblad operators''' describing environmental interactions
* <math>\{\cdot,\cdot\}</math> denotes the anticommutator

This structure was established in the mathematical theory of quantum dynamical semigroups.<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>

=== Physical interpretation ===

The Lindblad terms represent:

* dissipation
* decoherence

Each operator <math>L_k</math> corresponds to a physical process such as spontaneous emission or dephasing.<ref name="MIT_OCW" />

=== Example: spontaneous emission ===

For a two-level atom:

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

=Decoherence and dissipation=

=== Decoherence ===

Off-diagonal density matrix elements decay:

<math>
\rho_{ij}(t) \rightarrow 0 \quad (i \neq j).
</math>

This effect limits quantum coherence in practical systems such as superconducting qubits.<ref name="Kjaergaard2020">{{cite journal |last=Kjaergaard |first=M. |last2=Schwartz |first2=M. E. |last3=Braumüller |first3=J. |last4=Krantz |first4=P. |last5=Wang |first5=J. I.-J. |last6=Gustavsson |first6=S. |last7=Oliver |first7=W. 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>

=== Dissipation ===

Energy exchange with the environment leads to relaxation toward equilibrium.

=== Timescales ===

* decoherence time
* relaxation time

=Non-Markovian dynamics=

=== Memory effects ===

Non-Markovian systems exhibit memory and possible information backflow.<ref name="Breuer2016">{{cite journal |last=Breuer |first=H.-P. |last2=Laine |first2=E.-M. |last3=Piilo |first3=J. |last4=Vacchini |first4=B. |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>

A general form is

<math>
\frac{d\rho}{dt} = \int_0^t K(t-s)\rho(s)\,ds.
</math>

=== Physical systems ===

Appears in strongly coupled and structured environments.<ref name="Breuer2016" />

=Applications=

Used in:

* quantum optics
* quantum information
* condensed matter physics
* quantum thermodynamics

These applications rely on controlled decoherence modeling.<ref name="MIT_OCW" />

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

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Master equation|1}}

Physics:Quantum Master equation - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template