Green Function in Open Quantum Systems

Imagine an open quantum system interacting with an environment that admits a density matrix (Markovian) description in terms of Lindbladians ($$c$$ and $$c^\dagger$$). Is there a meaningful way to define a single particle Green function for this system up to the time the steady state is achieved, but without knowing any more details about the environment-system interaction?

Finally, how do steady state Green functions eventually become meaningful quantities to look at within a reduced density matrix description? For example, see here.

Background confusion: Single particle Green functions could be thought of as propagation amplitudes of particles/holes from one time to another. In a closed quantum system, one would sum all amplitudes leading from the initial to the final state, which is indeed the right thing to do. However, in an open quantum system with just a reduced density matrix (Lindbladian) description and one that exchanges matter, it feels like there is no way to sensibly decide how to coherently sum up amplitudes for different processes (eg. a process in which particle vanishes, then comes back from the environment etc.), esp. when starting from some initial random density matrix. But then to have a meaningful steady state Green function, some things must be magically falling into place in the steady state $$-$$ what am I missing? Any pointers would be greatly appreciated.

There is rich literature on using Green's functions for non-equilibrium systems - this typically implies resorting to the Keldysh formalism, see here for the list of sources.

In this list it is worth paying special attention to the series of papers by Meir, Wingreen and Jauho - these focus on excluding the external degrees of freedom from a Green's function for a subsystem with discrete levels. They deal specifically on a multilevel quantum dot coupled to electron reservoirs, but the formalism is equally applicable to any of the problems usually treated with Lindbladian formalism.

To quote a few results:

• If the subsystem is non-interacting, the exact solution is possible - the effect of the environment is then indeed to introduce level broadening, although the broadening might be non-diagonal matrix.
• In case of interactions one sometimes faces a rather difficult problem of calculating $$G^<$$ green's function. Meir&Wingreen use a rather clever trick to reduce their problem to $$G^r$$, but this is not always possible.
• In case of interacting subsystem the bath may introduce non-trivial effects - in the case of the papers mentioned it is Kondo effect. In quantum optics such effects are usually neglected, since they are of higher order.
• Dear Vadim, thanks very much for your answer. I think the missing link I was looking for is the Quantum Regression Theorem. Since correlation functions require propagating not just a density matrix, but possibly a string of creation/annihilation operators multiplied to the density matrix (from the left or right), I now understand that one requires more than the quantum master equation to recover the full Keldysh structure of single particle Green functions. Jun 14 '20 at 8:58

@Vadim has given a good answer that certainly goes in the right direction. However, the formalism they refer to is still for closed quantum systems. Even though it may be applicable or extendible to open quantum systems, this constitutes a non-trivial task.

I would like to fill this gap by the following review:

• L. M. Sieberer, M. Buchhold, S. Diehl, Keldysh Field Theory for Driven Open Quantum Systems, Rep. Prog. Phys. 79, 096001 (2016), doi, arxiv:1512.00637

and references therein.

The formalism presented there can be roughly thought of as the "path integral integral"-approach to open quantum systems. The Keldysh Green's functions are the central objects (see e.g. Equation 34 in the arxiv version) and related to correlation functions of the open quantum system in a similar way to usual quantum field theory.

With regards to the OP's concrete questions, I believe the above answers

Is there a meaningful way to define a single particle Green function for this system up to the time the steady state is achieved, but without knowing any more details about the environment-system interaction?

The second question

Finally, how do steady state Green functions eventually become meaningful quantities to look at within a reduced density matrix description?

is essentially related to thermalization and equilibration processes in such systems; a rich topic which is discussed in detail in the review.

With regards to the "Background confusion", it can be said that the problem is resolved by working on the level of the density matrix, which is able to distinguish coherent and incoherent processes. This also seems to be an aspect that is not covered by the reference pointed out by Vadim.

• Thank you very much for the answer and the reference. Indeed, I have been using the Diehl-Sieberer paper for a while. But in their version of the Keldysh functional integral, I believe, they focus on steady states. Furthermore, I was not sure how functional differentiation was producing all correlation functions in one fell swoop. I understand now that the missing link is the Quantum Regression Theorem that allows us to propagate not just the density matrix but possibly a string of creation/annihilation operators multiplied to the density matrix (from the left or right). Jun 14 '20 at 9:02
• @Vivek Nice! It would probably be helpful for other readers if you wrote an answer yourself! Jun 14 '20 at 9:41
• Will add an answer at some point. :) Jun 14 '20 at 9:42

Certain open system processes can be described within the Greens function formalism by adding imaginary terms to the Hamiltonian, namely all processes where something only exits the open system and nothin comes back from the environment. So, basically decay processes. Other non-unitary processes such as dephasing cannot be directly accounted for by a Green function.

I suppose that in your case in the steady-state no such non-unitary evolution remains and thus the dynamics can be described by Green functions.

• Two questions: 1) Is it legit to say that no non-unitary evolution remains in the steady state? 2) Do transient single particle Green functions have any meaning, eg. could one do a Kadanoff-Baym kind of time evolution leading up to the steady state with just the Lindblad equation? May 25 '20 at 8:50
• 1) Not really. Think for instance of a one-dimensional open system with on the one side an (infinite) reservoir of particles and on the other an (infinite) sink of particles. You can then reach a non-equilibrium steady-state with a current of particle from reservoir to sink. See e.g. [journals.aps.org/prb/abstract/10.1103/PhysRevB.100.035126] or [iopscience.iop.org/article/10.1088/1367-2630/aa5ae8] 2.) Sorry, I do not know what Kadanoff-Baym time evolution is. May 25 '20 at 9:15