Five-body recombination, in which five identical atoms form a tetramer molecule and a single free atom, could be the largest contributor to loss from ultracold atom traps at specific “Efimov resonances”, according to calculations done by physicists in the US. The process, which is less well understood than three- and four-body recombination, could be useful for building molecules, and potentially for modelling nuclear fusion.

A collision involving trapped atoms can be either elastic – in which the internal states of the atoms and their total kinetic energy remain unchanged – or inelastic, in which there is an interchange between the kinetic energy of the system and the internal energy states of the colliding atoms.

Most collisions in a dilute quantum gas involve only two atoms, and when physicists were first studying Bose-Einstein condensates (the ultralow-temperature state of some atomic gases), they suppressed inelastic two-body collisions, keeping the atoms in the desired state and preserving the condensate. A relatively small number of collisions, however, involve three or more bodies colliding simultaneously.

“They couldn’t turn off three body [inelastic collisions], and that turned out to be the main reason atoms leaked out of the condensate,” says theoretical physicist Chris Greene of Purdue University in the US.

Something remarkable

While attempting to understand inelastic three-body collisions, Greene and colleagues made the connection to work done in the 1970s by the Soviet theoretician Vitaly Efimov. He showed that at specific “resonances” of the scattering length, quantum mechanics allowed two colliding particles that could otherwise not form a bound state to do so in the presence of a third particle. While Efimov first considered the scattering of nucleons (protons and neutrons) or alpha particles, the effect applies to atoms and other quantum particles.

In the case of trapped atoms, the bound dimer and free atom are then ejected from the trap by the energy released from the binding event. “There were signatures of this famous Efimov effect that had never been seen experimentally,” Greene says. This was confirmed in 2005 by experiments from Rudolf Grimm’s group at the University of Innsbruck in Austria.

Hundreds of scientific papers have now been written about three-body recombination. Greene and colleagues subsequently predicted resonances at which four-body Efimov recombination could occur, producing a trimer. These were observed almost immediately by Grimm and colleagues. “Five was just too hard for us to do at the time, and only now are we able to go that next step,” says Greene.

Principal loss channel

In the new work, Greene and colleague Michael Higgins modelled collisions between identical caesium atoms in an optical trap. At specific resonances, five-body recombination – in which four atoms combine to produce a tetramer and a free particle – is not only enhanced but becomes the principal loss channel. The researchers believe these resonances should be experimentally observable using today’s laser box traps, which hold atomic gases in a square-well potential.

“For most ultracold experiments, researchers will be avoiding loss as much as possible – they would stay away from these resonances,” says Greene; “But for those of us in the few-body community interested in how atoms bind and resonate and how to describe complicated rearrangement, it’s really interesting to look at these points where the loss becomes resonant and very strong.” This is one technique that can be used to create new molecules, for example.

In future, Greene hopes to apply the model to nucleons themselves. “There have been very few people in the few-body theory community willing to tackle a five-particle collision – the Schrödinger equation has so many dimensions,” he says.

Fusion reactions

He hopes it may be possible to apply the researchers’ toolkit to nuclear reactions. “The famous one is the deuterium/tritium fusion reaction. When they collide they can form an alpha particle and a neutron and release a ton of energy, and that’s the basis of fusion reactors…There’s only one theory in the world from the nuclear community, and it’s such an important reaction I think it needs to be checked,” he says.

The researchers also wish to study the possibility of even larger bound states. However, they foresee a problem because the scattering length of the ground state resonance gets shorter and shorter with each additional particle. “Eventually the scattering length will no longer be the dominant length scale in the problem, and we think between five and six is about where that border line occurs,” Greene says. Nevertheless, higher-lying, more loosely-bound six-body Efimov resonances could potentially be visible at longer scattering lengths.

The research is described in Proceedings of the National Academy of Sciences.

Theoretical physicist Ravi Rau of Louisiana State University in the US is impressed by Greene and Higgins’ work. “For quite some time Chris Greene and a succession of his students and post-docs have been extending the three-body work that they did, using the same techniques, to four and now five particles,” he says. “Each step is much more complicated, and that he could use this technique to extend it to five bosons is what I see as significant.” Rau says, however, that “there is a vast gulf” between five atoms and the number treated by statistical mechanics, so new theoretical approaches may be required to bridge the gap.

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In his debut book, Einstein’s Tutor: the Story of Emmy Noether and the Invention of Modern Physics, Lee Phillips champions the life and work of German mathematician Emmy Noether (1882–1935). Despite living a life filled with obstacles, injustices and discrimination as a Jewish mathematician, Noether revolutionized the field and discovered “the single most profound result in all of physics”. Phillips’ book weaves the story of her extraordinary life around the central subject of “Noether’s theorem”, which itself sits at the heart of a fascinating era in the development of modern theoretical physics.

Noether grew up at a time when women had few rights. Unable to officially register as a student, she was instead able to audit courses at the University of Erlangen in Bavaria, with the support of her father who was a mathematics professor there. At the time, young Noether was one of only two female auditors in the university of 986 students. Just two years previously, the university faculty had declared that mixed-sex education would “overthrow academic order”. Despite going against this formidable status quo, she was able to graduate in 1903.

Noether continued her pursuit of advanced mathematics, travelling to the “[world’s] centre of mathematics” – the University of Göttingen. Here, she was able to sit in the lectures of some of the brightest mathematical minds of the time – Karl Schwarzschild, Hermann Minkowski, Otto Blumenthal, Felix Klein and David Hilbert. While there, the law finally changed: women were, at last, allowed to enrol as students at university. In 1904 Noether returned to the University of Erlangen to complete her postgraduate dissertation under the supervision of Paul Gordan. At the time, she was the only woman to matriculate alongside 46 men.

Despite being more than qualified, Noether was unable to secure a university position after graduating from her PhD in 1907. Instead, she worked unpaid for almost a decade – teaching her father’s courses and supervising his PhD students. As of 1915, Noether was the only woman in the whole of Europe with a PhD in mathematics. She had worked hard to be recognized as an expert on symmetry and invariant theory, and eventually accepted an invitation from Klein and Hilbert to work alongside them in Göttingen. Here, the three of them would meet Albert Einstein to discuss his latest project – a general theory of relativity.

Infiltrating the boys’ club

In Einstein’s Tutor, Phillips paints an especially vivid picture of Noether’s life at Göttingen, among colleagues including Klein, Hilbert and Einstein, who loom large and bring a richness to the story. Indeed, much of the first three chapters are dedicated to these men, setting the scene for Noether’s arrival in Göttingen. Phillips makes it easy to imagine these exceptionally talented and somewhat eccentric individuals working at the forefront of mathematics and theoretical physics together. And it was here, when supporting Einstein with the development of general relativity (GR), that Noether discovered a profound result: for every symmetry in the universe, there is a corresponding conservation law.

Throughout the book, Phillips makes the case that, without Noether, Einstein would never have been able to get to the heart of GR. Einstein himself “expressed wonderment at what happened to his equations in her hands, how he never imagined that things could be expressed with such elegance and generality”. Phillips argues that Einstein should not be credited as the sole architect of GR. Indeed, the contributions of Grossman, Klein, Besso, Hilbert, and crucially, Noether, remain largely unacknowledged – a wrong that Phillips is trying to right with this book.

Phillips makes the case that, without Noether, Einstein would never have been able to get to the heart of general relativity

A key theme running through Einstein’s Tutor is the importance of the support and allyship that Noether received from her male contemporaries. While at Göttingen, there was a battle to allow Noether to receive her habilitation (eligibility for tenure). Many argued in her favour but considered her an exception, and believed that in general, women were not suited as university professors. Hilbert, in contrast, saw her sex as irrelevant (famously declaring “this is not a bath house”) and pointed out that science requires the best people, of which she was one. Einstein also fought for her on the basis of equal rights for women.

Eventually, in 1919 Noether was allowed to habilitate (as an exception to the rule) and was promoted to professor in 1922. However, she was still not paid for her work. In fact, her promotion came with the specific condition that she remained unpaid, making it clear that Noether “would not be granted any form of authority over any male employee”. Hilbert however, managed to secure a contract with a small salary for her from the university administration.

Her allies rose to the cause again in 1933, when Noether was one of the first Jewish academics to be dismissed under the Nazi regime. After her expulsion, German mathematician Helmut Hasse convinced 14 other colleagues to write letters advocating for her importance, asking that she be allowed to continue as a teacher to a small group of advanced students – the government denied this request.

When the time came to leave Germany, many colleagues wrote testimonials in her support for immigration, with one writing “She is one of the 10 or 12 leading mathematicians of the present generation in the entire world.” Rather than being placed at a prestigious university or research institute (Hermann Weyl and Einstein were both placed at “the men’s university”, the Institute for Advanced Study in Princeton), it was recommended she join Bryn Mawr, a women’s college in Pennsylvania, US. Her position there would “compete with no-one… the most distinguished feminine mathematician connected with the most distinguished feminine university”. Phillips makes clear his distaste for the phrasing of this recommendation. However, all accounts show that she was happy at Bryn Mawr and stayed there until her unexpected death in 1935 at the age of 53.

Noether’s legacy

With a PhD in theoretical physics, Phillips has worked for many years in both academia and industry. His background shows itself clearly in some unusual writing choices. While his writing style is relaxed and conversational, it includes the occasional academic turn of phrase (e.g. “In this chapter I will explain…”), which feels out of place in a popular-science book. He also has a habit of piling repetitive and overly sincere praise onto Noether. I personally prefer stories that adopt the “show, don’t tell” approach – her abilities speak for themselves, so it should be easy to let the reader come to their own conclusions.

Phillips has made the ambitious choice to write a popular-science book about complex mathematical concepts such as symmetries and conservation laws that are challenging to explain, especially to general readers. He does his best to describe the mathematics and physics behind some of the key concepts around Noether’s theorem. However, in places, you do need to have some familiarity with university-level physics and maths to properly follow his explanations. The book also includes a 40-page appendix filled with additional physics content, which I found unnecessary.

Einstein’s Tutor does achieve its primary goal of familiarizing the reader with Emmy Noether and the tremendous significance of her work. The final chapter on her legacy breezes quickly through developments in particle physics, astrophysics, quantum computers, economics and XKCD Comics to highlight the range and impact this single theorem has had. Phillips’ goal was to take Noether into the mainstream, and this book is a small step in the right direction. As cosmologist and author Katie Mack summarizes perfectly: “Noether’s theorem is to theoretical physics what natural selection is to biology.”

• 2024 Hachette UK £25hb 368pp

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Nonlocal correlations that define quantum entanglement could be reconciled with Einstein’s theory of relativity if space–time had two temporal dimensions. That is the implication of new theoretical work that extends nonlocal hidden variable theories of quantum entanglement and proposes a potential experimental test.

Marco Pettini, a theoretical physicist at Aix Marseille University in France, says the idea arose from conversations with the mathematical physicist Roger Penrose – who shared the 2020 Nobel Prize for Physics for showing that the general theory of relativity predicted black holes. “He told me that, from his point of view, quantum entanglement is the greatest mystery that we have in physics,” says Pettini. The puzzle is encapsulated by Bell’s inequality, which was derived in the mid-1960s by the Northern Irish physicist John Bell.

Bell’s breakthrough was inspired by the 1935 Einstein–Podolsky–Rosen paradox, a thought experiment in which entangled particles in quantum superpositions (using the language of modern quantum mechanics) travel to spatially separated observers Alice and Bob. They make measurements of the same observable property of their particles. As they are superposition states, the outcome of neither measurement is certain before it is made. However, as soon as Alice measures the state, the superposition collapses and Bob’s measurement is now fixed.

Quantum scepticism

A sceptic of quantum indeterminacy could hypothetically suggest that the entangled particles carried hidden variables all along, so that when Alice made her measurement, she simply found out the state that Bob would measure rather than actually altering it. If the observers are separated by a distance so great that information about the hidden variable’s state would have to travel faster than light between them, then hidden variable theory violates relativity. Bell derived an inequality showing the maximum degree of correlation between the measurements possible if each particle carried such a “local” hidden variable, and showed it was indeed violated by quantum mechanics.

A more sophisticated alternative investigated by the theoretical physicists David Bohm and his student Jeffrey Bub, as well as by Bell himself, is a nonlocal hidden variable. This postulates that the particle – including the hidden variable – is indeed in a superposition and defined by an evolving wavefunction. When Alice makes her measurement, this superposition collapses. Bob’s value then correlates with Alice’s. For decades, researchers believed the wavefunction collapse could travel faster than light without allowing superliminal exchange of information – therefore without violating the special theory of relativity. However, in 2012 researchers showed that any finite-speed collapse propagation would enable superluminal information transmission.

“I met Roger Penrose several times, and while talking with him I asked ‘Well, why couldn’t we exploit an extra time dimension?’,” recalls Pettini. Particles could have five-dimensional wavefunctions (three spatial, two temporal), and the collapse could propagate through the extra time dimension – allowing it to appear instantaneous. Pettini says that the problem Penrose foresaw was that this would enable time travel, and the consequent possibility that one could travel back through the “extra time” to kill one’s ancestors or otherwise violate causality. However, Pettini says he “recently found in the literature a paper which has inspired some relatively standard modifications of the metric of an enlarged space–time in which massive particles are confined with respect to the extra time dimension…Since we are made of massive particles, we don’t see it.”

Toy model

Pettini believes it might be possible to test this idea experimentally. In a new paper, he proposes a hypothetical experiment (which he describes as a toy model), in which two sources emit pairs of entangled, polarized photons simultaneously. The photons from one source are collected by recipients Alice and Bob, while the photons from the other source are collected by Eve and Tom using identical detectors. Alice and Eve compare the polarizations of the photons they detect. Alice’s photon must, by fundamental quantum mechanics, be entangled with Bob’s photon, and Eve’s with Tom’s, but otherwise simple quantum mechanics gives no reason to expect any entanglement in the system.

Pettini proposes, however, that Alice and Eve should be placed much closer together, and closer to the photon sources, than to the other observers. In this case, he suggests, the communication of entanglement through the extra time dimension when the wavefunction of Alice’s particle collapses, transmitting this to Bob, or when Eve’s particle is transmitted to Tom would also transmit information between the much closer identical particles received by the other woman. This could affect the interference between Alice’s and Eve’s photons and cause a violation of Bell’s inequality. “[Alice and Eve] would influence each other as if they were entangled,” says Pettini. “This would be the smoking gun.”

Bub, now a distinguished professor emeritus at the University of Maryland, College Park, is not holding his breath. “I’m intrigued by [Pettini] exploiting my old hidden variable paper with Bohm to develop his two-time model of entanglement, but to be frank I can’t see this going anywhere,” he says. “I don’t feel the pull to provide a causal explanation of entanglement, and I don’t any more think of the ‘collapse’ of the wave function as a dynamical process.” He says the central premise of Pettini’s – that adding an extra time dimension could allow the transmission of entanglement between otherwise unrelated photons, is “a big leap”. “Personally, I wouldn’t put any money on it,” he says.

The research is described in Physical Review Research.

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