Physics:Quantum Master equation

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The quantum master equation describes the time evolution of the density operator ρ 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.[1]

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

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 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 (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ 1.0 1.1 1.2 1.3 "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/. 
  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

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