5

Introduction

The Keldysh path integral can be thought of as a reformulation of the quantum optical master equation, which describes the markovian time evolution of the density operator of an open quantum system in the Schrödinger-picture.

The Keldysh-formalism allows one to calculate Green's functions. But fundamentally the Green's functions should be defined in the Heisenberg-picture, because they contain operators evaluated at different times.

But the formulation of the Heisenberg-picture for open quantum systems requires the introduction of noise terms into the equations of motion. Without the noise terms, the product rule does not hold for the time derivative.

Question

My question is, that does anyone know any good references for linking these Heisenberg picture calculations based on quantum stochastic calculus to the Green's functions in the Keldysh-formalism?

Note

If the enviromental variables are explicitly taken into account then the original system + the environment can be regarded as a larger closed system. One can then use the unitary time evolution operator to switch between the Schrödinger and the Heisenberg picture. My question was about linking these different formalisms at the level of the smaller system, where the enviromental degrees of freedom are not present. For example, the Keldysh path integral can be derived from the markovian master equation directly, without going back to the system + environment formulation.

Community
  • 1
DanielTuzes
  • 371

2 Answers2

2

I am not sure I understand your question, but this might help.

If the Lindblad master equation in the Schrödinger picture is

$$ \partial_t \hat \rho = \mathcal L(\hat \rho) = -i [\hat H, \hat \rho ] + \sum_k \gamma_k (\hat L_k \hat \rho \hat L_k^\dagger - \frac{1}{2} \{ \hat L_k^\dagger \hat L_k, \hat \rho\}),$$ you can obtain the Heisenberg representation $\partial_t \hat O = \mathcal L^*(\hat O)$ for the operator $\hat O$ by requiring that its expectation value is the same in both representations: $$ Tr[\hat O \mathcal L(\hat \rho)] = Tr[\mathcal L^*(\hat O) \hat \rho].$$

This gives $$ \mathcal L^* (\cdot) = i[\hat H, \cdot] + \sum_k \gamma_k (\hat L_k^\dagger \cdot \hat L_k - \frac{1}{2} \{\hat L_k^\dagger \hat L_k, \cdot \}).$$

There is no need of introducing noise terms in the Lindblad formalism. Maybe you can construct your path integral from this expression.

kito
  • 91
0

Green's function formalism is a general quantum mechanical formalism, unlike the master equation approach, which relies on markovian assumption/regression of fluctuations/etc.

There are many excellent textbooks on quantum field theory or quantum field theory in condensed matter physics, where the Green's function representation is derived based on the Heisenberg picture. These are usually the background for approaching the Keldysh formalism, see the references here, including some concerning decoupling the system and the environment. Specifically, Rammer&Smith's review derives the kinetic equation - a continuous form of the master equation where the bath is due to phonons or electron-electron collisions, whereas Meir, Wingreen and co-authors deal with a fermionic bath.

I could also suggest this article and the references in it, where quantum Langevin equations are obtained starting with the Heisenberg equations-of-motion.

Roger V.
  • 68,984