Physics:Quantum Master equation: Difference between revisions
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'''Master equation''' is a Book I topic in the Quantum Collection. The quantum master equation describes the time evolution of the density operator \rho 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. Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena. The quantum master equation describes the time evolution of the density operator \rho 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. Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena. | |||
Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena. | |||
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Revision as of 08:15, 20 May 2026
Master equation is a Book I topic in the Quantum Collection. The quantum master equation describes the time evolution of the density operator \rho 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. Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena. The quantum master equation describes the time evolution of the density operator \rho 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. Master equations provide a framework for studying irreversible processes, quantum noise, and relaxation phenomena.
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 Cite error: Invalid
<ref>tag; no text was provided for refs namedMIT_OCW - ↑ 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.
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