A new, microscopic formulation of the second law of thermodynamics for coherently driven quantum systems has been proposed by researchers in Switzerland and Germany. The researchers applied their formulation to several canonical quantum systems, such as a three-level maser. They believe the result provides a tighter definition of entropy in such systems, and could form a basis for further exploration.

In any physical process, the first law of thermodynamics says that the total energy must always be conserved, with some converted to useful work and the remainder dissipated as heat. The second law of thermodynamics says that, in any allowed process, the total amount of heat (the entropy) must always increase.

“I like to think of work being mediated by degrees of freedom that we control and heat being mediated by degrees of freedom that we cannot control,” explains theoretical physicist Patrick Potts of the University of Basel in Switzerland. “In the macroscopic scenario, for example, work would be performed by some piston – we can move it.” The heat, meanwhile, goes into modes such as phonons generated by friction.

Murky at small scales

This distinction, however, becomes murky at small scales: “Once you go microscopic everything’s microscopic, so it becomes much more difficult to say ‘what is it that that you control – where is the work mediated – and what is it that you cannot control?’,” says Potts.

Potts and colleagues in Basel and at RWTH Aachen University in Germany examined the case of optical cavities driven by laser light, systems that can do work: “If you think of a laser as being able to promote a system from a ground state to an excited state, that’s very important to what’s being done in quantum computers, for example,” says Potts. “If you rotate a qubit, you’re doing exactly that.”

The light interacts with the cavity and makes an arbitrary number of bounces before leaking out. This emergent light is traditionally treated as heat in quantum simulations. However, it can still be partially coherent – if the cavity is empty, it can be just as coherent as the incoming light and can do just as much work.

In 2020, quantum optician Alexia Auffèves of Université Grenoble Alpes in France and colleagues noted that the coherent component of the light exiting a cavity could potentially do work. In the new study, the researchers embedded this in a consistent thermodynamic framework. They studied several examples and formulated physically consistent laws of thermodynamics.

In particular, they looked at the three-level maser, which is a canonical example of a quantum heat engine. However, it has generally been modelled semi-classically by assuming that the cavity contains a macroscopic electromagnetic field.

Work vanishes

“The old description will tell you that you put energy into this macroscopic field and that is work,” says Potts, “But once you describe the cavity quantum mechanically using the old framework then – poof! – the work is gone…Putting energy into the light field is no longer considered work, and whatever leaves the cavity is considered heat.”

The researchers new thermodynamic treatment allows them to treat the cavity quantum mechanically and to parametrize the minimum degree of entropy in the radiation that emerges – how much radiation must be converted to uncontrolled degrees of freedom that can do no useful work and how much can remain coherent.

The researchers are now applying their formalism to study thermodynamic uncertainty relations as an extension of the traditional second law of thermodynamics. “It’s actually a trade-off between three things – not just efficiency and power, but fluctuations also play a role,” says Potts. “So the more fluctuations you allow for, the higher you can get the efficiency and the power at the same time. These three things are very interesting to look at with this new formalism because these thermodynamic uncertainty relations hold for classical systems, but not for quantum systems.”

“This [work] fits very well into a question that has been heavily discussed for a long time in the quantum thermodynamics community, which is how to properly define work and how to  properly define useful resources,” says quantum theorist Federico Cerisola of the UK’s University of Exeter. “In particular, they very convincingly argue that, in the particular family of experiments they’re describing, there are resources that have been ignored in the past when using more standard approaches that can still be used for something useful.”

Cerisola says that, in his view, the logical next step is to propose a system – ideally one that can be implemented experimentally – in which radiation that would traditionally have been considered waste actually does useful work.

The research is described in Physical Review Letters.  

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When I was five years old, my family moved into a 1930s semi-detached house with a long strip of garden. At the end of the garden was a miniature orchard of eight apple trees the previous owners had planted – and it was there that I, much like another significantly more famous physicist, learned an important lesson about gravity.

As I read in the shade of the trees, an apple would sometimes fall with a satisfying thunk into the soft grass beside me. Less satisfyingly, they sometimes landed on my legs, or even my head – and the big cooking apples really hurt. I soon took to sitting on old wooden pallets crudely wedged among the higher branches. It was not comfortable, but at least I could return indoors without bruises.

The effects of gravity become common sense so early in life that we rarely stop to think about them past childhood. In his new book Crush: Close Encounters with Gravity, James Riordon has decided to take us back to the basics of this most fundamental of forces. Indeed, he explores an impressively wide range of topics – from why we dream of falling and why giraffes should not exist (but do), to how black holes form and the existence of “Planet 9”.

Riordon, a physicist turned science writer, makes for a deeply engaging author. He is not afraid to put himself into the story, introducing difficult concepts through personal experience and explaining them with the help of everything including the kitchen sink, which in his hands becomes an analogue for a black hole.

Gravity as a subject can easily be both too familiar and too challenging. In Riordon’s words, “Things with mass attract each other. That’s really all there is to Newtonian gravity.” While Albert Einstein’s theory of general relativity, by contrast, is so intricate that it takes years of university-level study to truly master. Riordon avoids both pitfalls: he manages to make the simple fascinating again, and the complex understandable.

He provides captivating insights into how gravity has shaped the animal kingdom, a perspective I had never much considered. Did you know that tree snakes have their hearts positioned closer to their heads than their land-based cousins? I certainly didn’t. The higher placement ensures a steady blood flow to the brain, even when the snake is climbing vertically. It is one of many examples that make you look again at the natural world with fresh eyes.

Riordon’s treatment of gravity in Einstein’s abstract space–time is equally impressive, perhaps unsurprisingly, as his previous books include Very Easy Relativity and Relatively Easy Relativity. Riordon takes a careful, patient approach – though I have never before heard general relativity reduced to “space–time is squishy”. But why not? The phrase sticks and gives us a handhold as we scale the complications of the theory. For those who want to extend the challenge, a mathematical background to the theory is provided in an appendix, and every chapter is well referenced and accompanied with suggestions for further reading.

If anything, I found myself wanting more examples of gravity as experienced by humans and animals on Earth, as opposed to in the context of the astronomical realm. I found these down-to-earth chapters the most fascinating: they formed a bridge between the vast and the local, reminding us that the same force that governs the orbits of galaxies also brings an apple to the ground. This may be a reaction only felt by astronomers like me, who already spend their days looking upward. I can easily see how the balance Riordon chose is necessary for someone without that background, and Einstein’s gravity does require galactic scales to appreciate, after all.

Crush is a generally uncomplicated and pleasurable read. The anecdotes can sometimes be a little long-winded and there are parts of the book that are not without challenge. But it is pitched perfectly for the curious general reader and even for those dipping their toes into popular science for the first time. I can imagine an enthusiastic A-level student devouring it; it is exactly the kind of book I would have loved at that age. Even if some of it would have gone over my head, Riordon’s enthusiasm and gift for storytelling would have kept me more than interested, as I sat up on that pallet in my favourite apple tree.

I left that house, and that tree, a long time ago, but just a few miles down the road from where I live now stands another, far more famous apple tree. In the garden of Woolsthorpe Manor near Grantham, Newton is said to have watched an apple fall. From that small event, he began to ask the questions that reshaped his and our understanding of the universe. Whether or not the story is true hardly matters – Newton was constantly inspired by the natural world, so it isn’t improbable, and that apple tree remains a potent symbol of curiosity and insight.

“[Newton] could tell us that an apple falls, and how quickly it will do it. As for the question of why it falls, that took Einstein to answer,” writes Riordon. Crush is a crisp and fresh tour through a continuum from orchards to observatories, showing that every planetary orbit, pulse of starlight and even every apple fall is part of the same wondrous story.

• 2025 MIT Press 288pp £27hb

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