Editing Open quantum system (section)

=== Non-Markovian equations ===
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 (linear algebra)|projection operator]] techniques. These techniques employ a projection operator <math>\mathcal{P}</math>, which effectively applies the trace over the environment as described previously.  The result of applying <math>\mathcal{P}</math> to <math>\rho</math>(i.e. calculating <math>\mathcal{P}\rho</math>) is called the ''relevant part'' of <math>\rho</math>.  For completeness, another operator <math>\mathcal{Q}</math> is defined so that <math>\mathcal{P}+\mathcal{Q}=\mathcal{I}</math> where <math>\mathcal{I}</math> is the identity matrix.  The result of applying <math>\mathcal{Q}</math> to <math>\rho</math>(i.e. calculating <math>\mathcal{Q}\rho</math>) is called the ''irrelevant part'' of <math>\rho</math>.  The primary goal of these methods is to then derive a master equation that defines the evolution of <math>\mathcal{P}\rho</math>.

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:

: <math>\partial_t{\rho }_\mathrm{S}=\mathcal{P}{\cal L}{{\rho}_\mathrm{S}}+\int_{0}^{t}{dt'\mathcal{K}({t}'){{\rho }_\mathrm{S}}(t-{t}')}.</math>

Here the effect of the bath throughout the time evolution of the system is hidden in the memory kernel <math> \kappa (\tau)</math>. While the Nakajima-Zwanzig equation is an exact equation that holds for almost all open quantum systems and environments, it can be very difficult to solve.  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: <math> \partial_t \rho_S = {\cal L } \rho_S </math>. Other examples of valid approximations include the weak-coupling approximation and the single-coupling approximation.

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.