When does the noise add up?

June 27, 2018

Yesterday, our paper on addition of quantum master equation generators was finally published in PRA. It has been underway for what feels like a loooong time (look at the recieved and published dates!) https://doi.org/10.1103/PhysRevA.97.062124.

When we first put this paper online, I apparently didn’t get around to writing a summary, so I’ll do one here:

In the paper, we investigate a somewhat technical question, but the context is not hard to understand.

Imagine you have a cold beer on a warm summer’s day. Clearly, if you leave you beer out in the sun, it’s going to warm up. This is because the beer is not isolated from the environment. Sunlight is hitting it and the warm air is touching it, giving off some heat to the beer. We say that the beer is an open system – it is interacting with its environment.

Now imagine that you want to describe how the beer evolves over time. How warm will it be after 10 minutes? How long does it take before it gets lukewarm and icky? In principle, to figure this out, you should track the trajectory of every air molecule hitting the bottle, to calculate exacly how much energy it gave to the beer, and how many photons of sunlight was absorbed or reflected etc. etc. But that is not very practical! The environment is huge, and keeping track of all its parts is next to impossible. And we are only really interested in what happens to the beer anyway.

Fortunately, if we are just interested in the beer, it is usually enough to account for the average effect of the environment. Instead of describing how every molecule or photon is absorbed or reflected, we can just look at average rates. How many photons arrive per second on average, for example. And that will be enough to tell us, how the beer is warming up. This gives a huge simplification of the calculations.

In quantum physics, we often deal with open systems. We may try to isolate our atoms, ions, or superconducting circuits as much as possible, but there will always be some contact to the environment. Sometimes, we may even want that, for example in quantum thermal machines. So we usually resort to averaging over the environment to get an effective description of how the system of interest evolves. In particular, we often use something called a quantum master equation.

The quantum master equation is a nice mathematical tool which allows us to find the time evolution of a quantum system in contact with a given environment. For every environment, we find the master equation, and then use that to figure out what happens to our quantum system.

The question which we investigate in the paper is this: If the system is interacting with several environments at the same time, and we know the master equation for each of them, can we then just add them up to get the effect of the total environment? For example, the beer is heating up both because of the warm air, and because of the sun shining on it. If we know the rate of heating by the air and the rate of heating by the sunlight, can we then just add them to figure out how was the beer is really heating?

Adding is easy, so calculations are much easier if the answer is yes. For the warming beer, this indeed the case. However, for quantum systems, things are a bit more complicated. Sometimes adding is ok, at other times it results in evolutions that are not correct, or even in equations that do not correspond to any possible physical evolution. In our paper, we establish conditions for when adding is allowed, or gives incorrect or non-physical results.

Published paper: https://doi.org/10.1103/PhysRevA.97.062124