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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://journals.aps.org/pra/abstract/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 nonphysical results.
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Happy to say that our paper on thermal generation of maximal entanglement in any dimension (which I wrote about here: https://plus.google.com/u/0/+JonatanBohrBrask/posts/hpX9uZLii5Z) is now out in Quantum :). A first in Quantum for me.
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I will be moving soon to the Danish Technical University (not far from Copenhagen), and I am looking for great students to come work with me on topics of quantum information and quantum thermodynamics.

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How hard is it to measure temperatures close to absolute zero? One can easily imagine that as the temperature becomes lower and lower, it also gets more and more difficult to measure. But how difficult exactly? In a new paper, out on the arXiv today, we try to answer this question: https://arxiv.org/abs/1711.09827

A lot of experiments in quantum physics are done at very low temperature, so low-temperature thermometry is relevant in practice in many labs. And it is also a very interesting piece of the bigger puzzle of understanding thermodynamics at the quantum scale.

It is well known that cooling to absolute zero is very hard. The third law of thermodynamics implies that it takes infinitely long time to get exactly to zero. This has long been understood for classical systems, and has recently been put on rigorous mathematical footing for quantum systems as well.

So, we know it is hard to make small temperatures. Is it also hard to measure them?

Curiously, the third law itself does not limit the absolute error of thermometers at low temperatures. The uncertainty could be constant or even decrease when the temperature goes to zero, without being at odds with the third law (the relative error must increase though).

But in many cases, the answer is yes. In fact, the error in a best estimate of the temperature increases very fast when the temperature approaches absolute zero - it increases exponentially. This has been seen before, and is also confirmed by our calculations. It holds for any system of finite size - so almost any system one is likely to encounter in practice. In principle, it looks like low-temperature thermometry is indeed hard.

However, in the paper we show that there are some important exceptions where the error does not increase exponentially. It can happen for infinite systems. And it can happen when we have limited access to the system which we want to know the temperature of.

In the first case, the exponential scaling does not apply. And in the second case, for finite systems, the restrictions coming from limited access may be much more important, such that the exponential scaling irrelevant. Combining the two, we find an interesting result: even for limited access on infinite systems, the exponential error can be beaten, and low-temperature thermometry becomes 'easy' again. In principle, any precision allowed by the third law can be achieved. So one could even have constant error, or thermometers which get better closer to absolute zero.

We give several examples of physical systems where the exponential error scaling is overcome, and so thermometry is much less hard than what might have been expected. But so far, we have not found any system where the error does not increase near absolute zero (it just increases much slower than exponentially).

So is it really possible to have thermometers which get better at very low temperature? We don't know yet!
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Some two weeks ago, we had a new paper out on the arXiv, which I haven't had the time to write about until now: https://arxiv.org/abs/1710.11624 . It has been under way for quite a while, but now we finally managed to put it all on (virtual) paper and get it out there.

This work contributes to building a consistent picture of thermodynamics at the quantum scale.

Thermodynamics explains how machines like steam engines and fridges work. It describes how heat can be moved around or transformed into other useful forms of energy, such as motion in a locomotive. And it tells us the fundamental limits on how well any such machine can perform, no matter how clever and intricate. In turn, the study of ideal machines has taught us fundamental things about nature, such as the second law of thermodynamics, which says that the entropy of an isolated system can never decrease (often thought of as stating the the 'mess' of the universe can only get bigger over time). Or the third law, which says that cooling to absolute theory requires infinite resources.

Traditionally, thermodynamics deals with big systems (think, steam locomotive), whose components are well described by classical physics. However, if we would look at smaller and smaller scales, then eventually we will need quantum physics to describe these components, and quantum phenomena will start to become important. What does thermodynamics look like at this quantum scale? Do the well known laws still hold? Can we make sense of such microscopic thermal processes? These are interesting theoretical questions. And by now, experimental techniques are getting so advanced that we can actually begin to build something like steam engines and fridges on the nanoscale. So they are starting to be relevant in practice as well.

A lot is already known about quantum thermodynamics, including generalisations of the Second and Third Laws and many results about the behaviour of thermal machines. However, it is fair to say that it is still work in progress - we do not yet have a full, coherent picture of quantum thermodynamics. Different approaches have been developed and it is not always clear how they fit together.

One point where classical and quantum thermodynamics differ is on how much it 'costs' to have control over a system, in terms of work energy (think of work as energy in an ordered, useful form as opposed to disordered heat energy). In the classical world the work cost of control can usually be neglected. Not so in the quantum world. There the cost of control can be a significant part of the cost of operating a machine.

In our paper, we add a piece towards completing the puzzle of quantum thermodynamics by studying the role of control for cooling. By looking at small fridges with more or less available control, we are able to compare different paradigms, which have been developed in the field, and compare how much one can cool under each of them, and how much it costs.

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On monday last week, we had a new paper out on the arXiv: https://arxiv.org/abs/1710.02621

In it, we consider a quantum thermal machine for generating entanglement. Entanglement is a form of quantum correlations which are essential in quantum information protocols. It can be used for ultra-precise measurements, for example of magnetic fields, and for quantum computing, among many other applications. The machine uses only differences in temperatures and interactions that do not require any active control, and so entanglement can be obtained just by turning on the machine and waiting, which is neat.

What I find particularly nice is that the paper is a good example of scientific collaboration across the globe. The new scheme improves upon a design for thermal entanglement generation which we developed with colleagues here in Geneva and in Barcelona. I wrote about that work here: https://plus.google.com/+JonatanBohrBrask/posts/ho5rbgRFdpp (have a look for a summary of what makes thermal entanglement generation exciting). Recently, colleagues in Qufu, China, who had been working on similar setups, realised that the entanglement generation could be improved by including an additional thermal bath. Zhong-Xiao Man then contacted me, and with Armin Tavakoli we confirmed their results and thought about how the new scheme could be realised in practice. After just a few emails back and forth, the paper came together. The bulk of the work was done by Zhong-Xiao and his colleagues, but Armin and I also made a significant contribution, and in the end, I think the paper is much better than what any of us might have done alone.

So thanks to Zhong-Xiao for bringing on us on board. And I am happy to do research in the age of the internet, which has made these kind of interaction much much easier, faster, and likelier to happen :).

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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 [ https://plus.google.com/u/0/+JonatanBohrBrask/posts/hpX9uZLii5Z ] and here [ https://plus.google.com/u/0/+JonatanBohrBrask/posts/8JRpYSYKadU ]

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.
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Today we have a new paper out on the arXiv in which we describe a quantum thermal machine which generates maximal entanglement in any dimension. https://arxiv.org/abs/1708.01428

Thermal machines do a lot of useful tasks for us. Power plants turn heat into electricity. Refrigerators keep our beers cold. Steam locomotives pull trains (OK, maybe they mostly don't any more - but a steam engine is the archetypical picture of a thermal machine). In general they are machines which move heat around, or transform it. Often by connecting different points of the machine to different temperatures and exploiting the heat flow between them.

Classical thermal machines are big. Think about power plants - and even a fridge is about the size of a person. At these scales, we don't need to worry about quantum physics to understand what is going on. But what happens if make such a machine smaller and smaller, to the point where quantum effects become important? Say we take a locomotive and scale it down, down, down, until the boiler and gears and so on consist of just a few atoms? Of course, it won't really be a locomotive any more - but maybe we can learn something interesting?

Indeed, we can. And the machine can still be useful.

In recent years, physicists have learned a lot about thermodynamics on the quantum scale by studying such tiny thermal machines. Looking at how fundamental concepts from classical thermodynamics, such as the 2nd law or Carnot efficiency, behave in the quantum regime, we gain new insights into the differences between the classical and quantum worlds.

In the spirit of invention of steam engines (like at the time of the industrial revolution), we can also think about whether there are new tasks that such quantum thermal machines could do.

Entanglement is an essential quantum phenomena. Objects which are entangled behave as if they are a single entity even when separated and manipulated independently. This enables new, powerful applications such as quantum computing and quantum metrology, and is at the heart of the foundations of quantum physics. So creating and studying entanglement is very interesting from both fundamental and applied points of view. Might a thermal machine be used to generate entanglement then?

Entanglement is generally very fragile, and thermal noise tends to wash it out quickly. In fact, a lot of effort in quantum physics experiments goes into keeping the systems cold and isolated, so as to be able to observe the genuinely quantum effects. So it is not at all obvious that using a thermal machine for entanglement generation would work. However, it turns out that connecting with noisy environments can indeed help to create and keep entanglement stable, in certain systems.

This was already realised and studied by other researchers. Then, a couple of years ago, we described a minimal thermal machine generating entanglement (as I wrote about here https://plus.google.com/+JonatanBohrBrask/posts/ho5rbgRFdpp). That setup was nice because it was really the simplest quantum thermal machine imaginable, using just two quantum bits and two different temperatures, and it turned out this was already sufficient to see entanglement.

However, the amount of entanglement which that machine could generate was rather limited. In our new paper, we present a new quantum thermal machine - not much more complicated - which generates maximal entanglement. And it does this, not only for two quantum bits, but also for two quantum trits, and in fact for two quantum systems of any dimension (which we prove analytically thanks to the hard work of Armin Tavakoli). The new machine again uses just two different temperatures and two quantum systems of some given dimension. One system is connected to a cold bath, one to a hot bath, and they interact with each other. When heat flows from hot to cold through the two systems, they become entangled.

So indeed, a quantum thermal machine can be useful. And maybe in the future we might see quantum computers, or sensitive quantum sensors, based on thermally generated entanglement.