Physics:Quantum Master equation: Difference between revisions
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=== Markovian approximation === | === Markovian approximation === | ||
A common simplification assumes that the environment has no memory, so the dynamics are approximately local in time.<ref name="MIT_OCW" /> | A common simplification assumes that the environment has no memory, so the dynamics are approximately local in time.<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> | ||
== The Lindblad form == | == The Lindblad form == | ||
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Latest revision as of 00:31, 24 May 2026
Master equation unlike closed systems, whose dynamics are governed by the Schrödinger equation, open systems exhibit dissipation and decoherence. Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena. Unlike closed systems, whose dynamics are governed by the Schrödinger equation, open systems exhibit dissipation and decoherence. Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena. For a closed system, this reduces to the von Neumann equation: The system state is obtained by tracing out the environment: This leads to effective non-unitary evolution for the system. A common simplification assumes that the environment has no memory, so the dynamics are approximately local in time.
Density operator dynamics
The state of a quantum system is described by a density operator , which evolves according to
where is a linear superoperator called the Liouvillian.
For a closed system, this reduces to the von Neumann equation:
Reduced dynamics
In an open system, one considers a combined system + environment with total density operator . The system state is obtained by tracing out the environment:
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.[1]
The Lindblad form
The most general form of a Markovian quantum master equation that preserves trace and complete positivity is the Lindblad equation:
Here:
- is the system Hamiltonian
- are Lindblad operators describing environmental interactions
- denotes the anticommutator
This structure was established in the mathematical theory of quantum dynamical semigroups.[2]
Physical interpretation
The Lindblad terms represent:
- dissipation
- decoherence
Each operator corresponds to a physical process such as spontaneous emission or dephasing.[1]
Example: spontaneous emission
For a two-level atom:
Decoherence and dissipation
Decoherence
Off-diagonal density matrix elements decay:
This effect limits quantum coherence in practical systems such as superconducting qubits.[3]
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.[4]
A general form is
Physical systems
Appears in strongly coupled and structured environments.[4]
Applications
Used in:
- quantum optics
- quantum information
- condensed matter physics
- quantum thermodynamics
These applications rely on controlled decoherence modeling.[1]
See also
Table of contents (217 articles)
Index
Full contents
References
- ↑ 1.0 1.1 1.2 "22.51 Course Notes, Chapter 8: Open Quantum Systems". https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/resources/mit22_51f12_ch8/.
- ↑ Lindblad, Göran (1976). "On the generators of quantum dynamical semigroups". Communications in Mathematical Physics 48 (2): 119–130. doi:10.1007/BF01608499. https://link.springer.com/article/10.1007/BF01608499.
- ↑ Kjaergaard, M.; Schwartz, M. E.; Braumüller, J.; Krantz, P.; Wang, J. I.-J.; Gustavsson, S.; Oliver, W. D. (2020). "Engineering high-coherence superconducting qubits". Nature Reviews Materials 5: 309–324. doi:10.1038/s41578-021-00370-4. https://www.nature.com/articles/s41578-021-00370-4.
- ↑ 4.0 4.1 Breuer, H.-P.; Laine, E.-M.; Piilo, J.; Vacchini, B. (2016). "Colloquium: Non-Markovian dynamics in open quantum systems". Reviews of Modern Physics 88 (2): 021002. doi:10.1103/RevModPhys.88.021002. https://link.aps.org/doi/10.1103/RevModPhys.88.021002.
Source attribution: Physics:Quantum Master equation
