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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.

Quantum Master equation.

Density operator dynamics

The state of a quantum system is described by a density operator ρ, which evolves according to

dρdt=ℒ(ρ),

where ℒ is a linear superoperator called the Liouvillian.

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

dρdt=−iℏ[H^,ρ].

Reduced dynamics

In an open system, one considers a combined system + environment with total density operator ρtot. The system state is obtained by tracing out the environment:

ρ=Trenv(ρtot).

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:

dρdt=−iℏ[H^,ρ]+∑k(LkρLk†−12{Lk†Lk,ρ}).

Here:

  • H^ is the system Hamiltonian
  • Lk 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 Lk corresponds to a physical process such as spontaneous emission or dephasing.[1]

Example: spontaneous emission

For a two-level atom:

L=γσ−.

Decoherence and dissipation

Decoherence

Off-diagonal density matrix elements decay:

ρij(t)→0(i≠j).

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

dρdt=∫0tK(t−s)ρ(s)ds.

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. ↑ 1.0 1.1 1.2 Cite error: Invalid <ref> tag; no text was provided for refs named MIT_OCW
  2. ↑ 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. 
  3. ↑ 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. ↑ 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. 


Author: Harold Foppele


Source attribution: Physics:Quantum Master equation

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