How to handle nonmarkovian dynamics in open quantum system?

Answer 1

Usually equations of motion containing a time convolution arise when a subsystem is projected out. A standard procedure is to use the Zwanzig projection operator method to obtain a generalised master equation. But it can also happen that such an equation of motion is directly given. This happens with the phenomenological Maxwell's equations for dispersive and absorptive systems. Then a "reverse" procedure can be followed. A new, auxiliary, field is introduced and the combined set of fields obey a unitary time evolution which can then be quantised. A posteriory the new field can here be identified as a polarisation field. References are

A. Tip: Phys. Rev. A 57, 1998, p. 4818.

B. Gralak and A. Tip: j.Math.Phys 51, 2010, p. 052902.

The method should also work for other linear cases with a time convolution term.

It is interesting to note that one can start from a unitary time evolution, use the Zwanzig method to obtain a time evolution with a convolution term and then use the reversed method to obtain a new unitary time evolution. The latter is not the same as the original one. It essentially picks up only stuff that is needed to describe the projected system. Note further that the method is exact. No approximations are made.

Comments

Zwanzig projection operator method works when the environment (bath) stays in equilibrium. However, there are cases where the environment is can change due to the interaction with the system. For example, if we have a light pulse as input and let it interact with a qubit, the method does not work anymore.  

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But the equation I displayed can be handled. In case we encounter non-linear terms the situation becomes more complicated and I have only made partial progress in that case.. There still exists a set of equations without time convolutions but I was not able to obtain a Lagrange-Hamilton formalism in that situation.
 

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In that case the difficulty lies in the fact we may need infinite number of equations to know the dynamics. This is really a headache.

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It is not that bad. For instance if you have in the macroscopic Maxwell's equations n non-linear susceptibilities then n additional equations are needed to convert to a set of equations without time convolutions.

Answer 2

The question is rather general. Are you thinking of an equation of motion of
the type

\(\begin{equation*} \partial _{t}f(t)=Af(t)+\int_{0}^{t}ds\rho (t-s)f(s)? \end{equation*}\)

Then I know of a procedure to convert it into a set of two equations without time-convolutions. Let me know. I used it to quantise the phenomenological Maxwell's equations for an absorptive system.

Comments

This is the form of the master equation with a memory kernel. Can you convert it to time local equations?

Answer 3

You may either add a retarded memory term to your differential equation, giving an integro-differential equation of the kind appearing in the Zwanzig projection method before taking the Markovian limit, and use some phenomenological term for the memory. (This is treated, e.g., in the book by Honerkamp). 

Or you may extend the number of variables to include variables that act as memory, and then work with a Markovian master equation in the augmented set of variables. E.g., going from a hydrodynamic description with memory to a kinetic description.

Answer 4

If you're interested in a time-local prescription, then you can use the projection operator technique to derive a time-convolutionless (TCL) master equation (see the book by Breuer & Petruccione for example). This technique works through a perturbative expansion in the system-bath coupling. At lowest order the TCL equation gives results of comparable (and often much better) accuracy than the standard Nakajima-Zwanzig equation.  If your system-bath coupling is in fact strong, then the perturbative expansion will fail. In this case it is often a good idea to redefine your system to include the slow variables of your bath, and treat only the coupling to the residual fast variables perturbatively. The polaron transformed spin-boson model and its generalisations furnish a well-known example of this idea.

It would really help if you describe the Hamiltonian properly, since the question is far too general for us to give you a useful answer at present. In essence you are currently asking for the general solution for an arbitrary dynamical system: almost all physical problems are non-Markovian if you choose your variables poorly.