Open quantum system
Because no quantum system is completely isolated from its surroundings,[1] it is important to develop a theoretical framework for treating these interactions in order to obtain an accurate understanding of quantum systems.
A complete description of a quantum system requires the inclusion of the environment.
The eventual outcome of this process of embedding is the state of the whole universe described by a wavefunction
In particular, if the combined system has quantum entanglement, the state of the subsystem is not pure.
In general, however, the environment that we want to model as being a part of our system is very large and complicated, which makes finding exact solutions to the master equations difficult, if not impossible.
Due to the difficulty of determining the solutions to the master equations for a particular system and environment, a variety of techniques and approaches have been developed.
The main assumption is that the entire system-environment combination is a large closed system.
Therefore, its time evolution is governed by a unitary transformation generated by a global Hamiltonian.
For systems that have very fast or very frequent perturbations from their coupling to their environment, this approximation becomes much less accurate.
A formal construction of a local equation of motion with a Markovian property is an alternative to a reduced derivation.
The Markov property imposes that the system and bath are uncorrelated at all times
The family of maps generated by the GKSL equation forms a Quantum dynamical semigroup.
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.
[4] An alternative approach to fix the Redfield has been proposed by J. Thingna, J.-S. Wang, and P. Hänggi[5] 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.
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 ergodic properties are too weak and so the dynamics of the model do not generate wide-scale quantum entanglement between the bath modes.
[7] At low temperatures and weak system-bath coupling, the Caldeira-Leggett and spin bath models are equivalent.
An example of natural system being coupled to a spin bath is a nitrogen-vacancy (N-V) center in diamonds.
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.
Open quantum systems that do not have the Markovian property are generally much more difficult to solve.
Currently, the methods of treating these systems employ what are known as projection operator techniques.
The primary goal of these methods is to then derive a master equation that defines the evolution of
This means that approximations generally need to be introduced to reduce the complexity of the problem into something more manageable.
As an example, the assumption of a fast bath is required to lead to a time local equation:
This method of approaching open quantum systems is known as the time-convolutionless projection operator technique, and it is used to generate master equations that are inherently local in time.
Another approach emerges as an analogue of classical dissipation theory developed by Ryogo Kubo and Y. Tanimura.
