In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the environment or a bath. In general, these interactions significantly change the dynamics of the system and result in quantum dissipation, such that the information contained in the system is lost to its environment. 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.
Techniques developed in the context of open quantum systems have proven powerful in fields such as quantum optics, quantum measurement theory, quantum statistical mechanics, quantum information science, quantum thermodynamics, quantum cosmology, quantum biology, and semi-classical approximations.
\Psi
\Psi
\rho
A
(\rho ⋅ A)=\rm{tr}\{\rhoA\}
In general, the time evolution of closed quantum systems is described by unitary operators acting on the system. For open systems, however, the interactions between the system and its environment make it so that the dynamics of the system cannot be accurately described using unitary operators alone.
The time evolution of quantum systems can be determined by solving the effective equations of motion, also known as master equations, that govern how the density matrix describing the system changes over time and the dynamics of the observables that are associated with the system. 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. As such, the theory of open quantum systems seeks an economical treatment of the dynamics of the system and its observables. Typical observables of interest include things like energy and the robustness of quantum coherence (i.e. a measure of a state's coherence). Loss of energy to the environment is termed quantum dissipation, while loss of coherence is termed quantum decoherence.
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. A common objective is to derive a reduced description wherein the system's dynamics are considered explicitly and the bath's dynamics are described implicitly. 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 the combined system bath scenario the global Hamiltonian can be decomposed into:
H=H\rm+H\rm+H\rm
where
H\rm
H\rm
H\rm
\rho\rm(t)=\rm{tr}\rm\{\rhoSB(t)\}
Another common assumption that is used to make systems easier to solve is the assumption that the state of the system at the next moment depends only on the current state of the system. in other words, the system doesn't have a memory of its previous states. Systems that have this property are known as Markovian systems. This approximation is justified when the system in question has enough time for the system to relax to equilibrium before being perturbed again by interactions with its environment. For systems that have very fast or very frequent perturbations from their coupling to their environment, this approximation becomes much less accurate.
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
\rho(0)=\rho\rm ⊗ \rho\rm
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
\rho(0)=\rho\rm ⊗ \rho\rm
| \rho |
\rm=-{i\over\hbar}[H\rm,\rho\rm]+{\calL}\rm(\rho\rm)
H\rm
{\calL}\rm
{\calL}\rm(\rho\rm)=\sumn\left(Vn\rho\rm
| ||||
| V | ||||
| n |
\left(\rho\rm
| \dagger | |
| V | |
| n |
Vn+
| \dagger | |
| V | |
| n |
Vn\rho\rm\right)\right)
Vn
\rho\rm=\rho\rm ⊗ \rho\rm
\rho\rm
| \rho |
\rm(t → infty)=0
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.[5] 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, 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 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.[6] 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 nitrogen-vacancy (N-V) center in diamonds. In this example, the color center is the system and the bath consists of carbon-13 (13C) impurities which interact with the system via the magnetic dipole-dipole 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.
Open quantum systems that do not have the Markovian property are generally much more difficult to solve. This is largely due to the fact that the next state of a non-Markovian system is determined by each of its previous states, which rapidly increases the memory requirements to compute the evolution of the system. Currently, the methods of treating these systems employ what are known as projection operator techniques. These techniques employ a projection operator
l{P}
l{P}
\rho
l{P}\rho
\rho
l{Q}
l{P}+l{Q}=l{I}
l{I}
l{Q}
\rho
l{Q}\rho
\rho
l{P}\rho
One such derivation using the projection operator technique results in what is known as the Nakajima–Zwanzig equation. This derivation highlights the problem of the reduced dynamics being non-local in time:
\partialt{\rho}S=l{P}{\calL}{{\rho}S
Here the effect of the bath throughout the time evolution of the system is hidden in the memory kernel
\kappa(\tau)
\partialt\rhoS={\calL}\rhoS
In some cases, the projection operator technique can be used to reduce the dependence of the system's next state on all of its previous states. 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. Because these equations can neglect more of the history of the system, they are often easier to solve than things like the Nakajima-Zwanzig equation.
Another approach emerges as an analogue of classical dissipation theory developed by Ryogo Kubo and Y. Tanimura. This approach is connected to hierarchical equations of motion which embed the density operator in a larger space of auxiliary operators such that a time local equation is obtained for the whole set and their memory is contained in the auxiliary operators.