**August 27, 2017**

A few weeks back, we had a paper out on the arXiv, which I haven’t had time to write about yet. https://arxiv.org/abs/1707.09211

The topic of the paper is quantum master equations – a somewhat technical subject, but very important for much of the other physics we study, especially small thermal machines, like the ones I have written about here and here.

When we try to describe a thermal machine, we are faced with a problem. The machine necessarily interacts with some thermal reservoirs. These are large, messy systems with many, many particles. In fact, this is true more generally. Any small quantum system interacts with the surrounding environment in some way. We may do our best to isolate it (and experimentalists typically do a good job!), but some weak interaction will always be present. The environment is big and complicated, and it is extremely cumbersome, if not impossible, to describe in detail what is going on with all the individual particles there. It would make our lives miserable if we had to try…

This is where quantum master equations come in. Instead of describing the environment in detail, one can account for the average effect it has on the system. The noiseless behaviour that an isolated system would follow is modified to include noise introduced by the environment. There are various techniques for doing so. The quantum master equation approach is one of the most important and wide spread.

They gives us a powerful computational tool, and we rely on them a lot we try to understand what is going on, for example in quantum thermal machines. They have been around for more than half a century, but there are still aspects which are not completely understood. Since it accounts for the effects of a large, complicated environment, which is not explicitly described, deriving a master equation always involves some approximations. And it can sometimes be unclear when these approximations are reliable.

In our paper, we address one such ambiguity which is particularly relevant for studying small quantum thermal machines, or more generally, energy transport in a small quantum system (this might be relevant e.g. in photosynthesis, where light energy is transported through molecules).

Imagine that the quantum system consists of two particles. Imagine that each particle is in contact with a separate environment, and that the particles also interact with each other. Now one could derive a quantum master equation for the system in two different ways. One could either first account for the noise introduced by the environments on each particle separately, and then account for the interaction between them. Or one could first account for the interaction between the particles, and then find the noise induced by the environments on this composite system. This leads to two different master equations, often referred to as ‘local’ and ‘global’, because in the former case, noise acts locally on each particle, while in the latter it acts on both particles in a collective manner.

There has been quite a bit of discussion in the community on whether the local or global approach is appropriate for describing certain thermal machines, and even results showing that employing a master equation in the wrong regime can lead to violation of fundamental physical principles such as the second law of thermodynamics. In our paper, we compare the two approaches against an exactly solvable model (that is, where the environment can be treated in detail) and study rigorously when one or the other approach holds. We find what could be intuitively expected: When the interaction between the system particles is weak, the local approach is valid and the global fails. On the other hand, when the inter-system interaction is strong, the two particles should be treated as single system, and the global approach is the valid one. For intermediate couplings, both approaches approximate the true evolution well.

This is reassuring, and provides a solid foundation for our (and others’) studies of small quantum thermal machines and other open quantum systems.

** Published paper:** https://iopscience.iop.org/article/10.1088/1367-2630/aa964f