Editing Open quantum system (section)

=== Markovian equations ===
When the interaction between the system and the environment is weak, a time-dependent [[perturbation theory]] seems appropriate for treating the evolution of the system. In other words, if the interaction between the system and its environment is weak, then any changes to the combined system over time can be approximated as originating from only the system in question. Another typical assumption is that the system and bath are initially uncorrelated <math> \rho(0)=\rho_{\rm S} \otimes \rho_{\rm B} </math>. This idea originated with [[Felix Bloch]] and was expanded upon by Alfred Redfield in his derivation of the [[Redfield equation]]. The Redfield equation is a Markovian master equation that describes the time evolution of the density matrix of the combined system. The drawback of the Redfield equation is that it does not conserve the [[Positive element|positivity]] of the density operator.

A formal construction of a local equation of motion with a [[Markovian property]] is an alternative to a reduced derivation. The theory is based on an axiomatic approach. The basic starting point is a [[completely positive map]]. The assumption is that the initial system-environment state is uncorrelated <math> \rho(0)=\rho_{\rm S} \otimes \rho_{\rm B} </math> and the combined dynamics is generated by a [[unitary operator]]. Such a map falls under the category of [[Kraus operator]]. The most general type of a time-homogeneous master equation with the [[Markovian property]] describing non-unitary evolution of the density matrix ρ that is trace-preserving and completely positive for any initial condition is the Gorini–Kossakowski–Sudarshan–[[Lindblad equation]] or GKSL equation:
:<math>\dot\rho_{\rm S}=-{i\over\hbar}[H_{\rm S},\rho_{\rm S}]+{\cal L}_{\rm D}(\rho_{\rm S}) </math>

<math> H_{\rm S}</math> is a ([[Hermitian]]) [[Hamiltonian (quantum mechanics)|Hamiltonian]] part and <math>{\cal L}_{\rm D}</math>:
:<math>{\cal L}_{\rm D}(\rho_{\rm S})=\sum_n \left(V_n\rho_{\rm S} V_n^\dagger-\frac{1}{2}\left(\rho_{\rm S} V_n^\dagger V_n + V_n^\dagger V_n\rho_{\rm S}\right)\right)</math>
is the dissipative part describing implicitly through system operators <math> V_n </math> the influence of the bath on the system.
The [[Markov property]] imposes that the system and bath are uncorrelated at all times <math> \rho_{\rm SB}=\rho_{\rm S} \otimes \rho_{\rm B} </math>.
The GKSL equation is unidirectional and leads any initial state <math> \rho_{\rm S}</math> to a steady state solution which is an invariant of the equation of motion <math> \dot \rho_{\rm S}(t \rightarrow \infty ) = 0 </math>.
The family of maps generated by the GKSL equation forms a [[Quantum dynamical semigroup]]. In some fields, such as [[quantum optics]], the term [[Lindblad superoperator]] is often used to express the quantum master equation for a dissipative system. E.B. Davis derived the GKSL with [[Markovian property]] master equations using [[perturbation theory]] and additional approximations, such as the rotating wave or secular, thus fixing the flaws of the [[Redfield equation]]. Davis construction is consistent with the Kubo-Martin-Schwinger stability criterion for thermal equilibrium i.e. the [[KMS state]].<ref>{{cite book|first=Heinz-Peter|last=Breuer|author2=F. Petruccione|title=The Theory of Open Quantum Systems|publisher=Oxford University Press|year=2007|isbn=978-0-19-921390-0}}</ref> An alternative approach to fix the Redfield has been proposed by J. Thingna, J.-S. Wang, and P. Hänggi<ref>{{Cite journal|last1=Thingna|first1=Juzar|last2=Wang|first2=Jian-Sheng|last3=Hänggi|first3=Peter|date=2012-05-21|title=Generalized Gibbs state with modified Redfield solution: Exact agreement up to second order|journal=The Journal of Chemical Physics|language=en|volume=136|issue=19|pages=194110|doi=10.1063/1.4718706|pmid=22612083|issn=0021-9606|arxiv=1203.6207|bibcode=2012JChPh.136s4110T|s2cid=7014354}}</ref> that allows for system-bath interaction to play a role in equilibrium differing from the KMS state.

In 1981, [[Amir Caldeira]] and [[Anthony J. Leggett]] proposed a simplifying assumption in which the bath is decomposed to normal modes represented as harmonic oscillators linearly coupled to the system.<ref>A. Caldeira and A. J. Leggett, ''Influence of dissipation on quantum tunneling in macroscopic systems'', Physical Review Letters, vol. 46, p. 211, 1981.</ref> As a result, the influence of the bath can be summarized by the bath spectral function. This method is known as the [[Caldeira-Leggett model|Caldeira–Leggett model]], or harmonic bath model. To proceed and obtain explicit solutions, the [[path integral formulation]] description of [[quantum mechanics]] is typically employed.  A large part of the power behind this method is the fact that harmonic oscillators are relatively well-understood compared to the true coupling that exists between the system and the bath.  Unfortunately, while the Caldeira-Leggett model is one that leads to a physically consistent picture of quantum dissipation, its [[Ergodicity|ergodic]] properties are too weak and so the dynamics of the model do not generate wide-scale [[quantum entanglement]] between the bath modes.

An alternative bath model is a spin bath.<ref>{{Cite journal|last1=Prokof'ev|first1=N. V.|last2=Stamp|first2=P. C. E.|date=2000|title=Theory of the spin bath|journal=Reports on Progress in Physics|language=en|volume=63|issue=4|pages=669|doi=10.1088/0034-4885/63/4/204 | arxiv=cond-mat/0001080|bibcode=2000RPPh...63..669P |s2cid=55075035|issn=0034-4885}}</ref> At low temperatures and weak system-bath coupling, the Caldeira-Leggett and spin bath models are equivalent. But for higher temperatures or strong system-bath coupling, the spin bath model has strong ergodic properties. Once the system is coupled, significant entanglement is generated between all modes. In other words, the spin bath model can simulate the Caldeira-Leggett model, but the opposite is not true.

An example of natural system being coupled to a spin bath is a [[N-V center|nitrogen-vacancy (N-V) center]] in diamonds. In this example, the color center is the system and the bath consists of [[carbon-13]] (<sup>13</sup>C) impurities which interact with the system via the magnetic dipole-dipole [http://iopscience.iop.org/article/10.1088/1367-2630/18/9/093001/meta interaction].

For open quantum systems where the bath has oscillations that are particularly fast, it is possible to average them out by looking at sufficiently large changes in time. This is possible because the average amplitude of fast oscillations over a large time scale is equal to the central value, which can always be chosen to be zero with a minor shift along the vertical axis. This method of simplifying problems is known as the secular approximation.