The Many Wonders of Uranium Ditelluride

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Published: Monday, February 26 2024 09:24

In the menagerie of exotic materials, superconductors boast their own vibrant ecosystem.

All superconductors allow electricity to flow without any resistance. It’s their hallmark feature. But in many cases, that’s where the similarities end.

Some superconductors, like aluminum, are conventional—run-of-the-mill, bread-and-butter materials that are well understood and hold no surprises. Others are deemed unconventional: They are not yet fully understood, but that seem to follow a known pattern. But one material—uranium ditelluride (UTe2)—defies classification, continuously baffling scientists with a plethora of unexpected behaviors. 

“At first, we thought this was going to be another interesting superconductor like some other uranium compounds that have been studied in the past,” says Johnpierre Paglione, a professor of physics at the University of Maryland (UMD) and the director of the Quantum Materials Center (QMC). “But at this point, it's gone beyond that. And it's become a much richer example of how crazy a superconductor can get.”

Most superconductors start doing their resistance-less thing when they get super cold. But temperature is only one of the knobs available to researchers studying a material in the lab. Some materials slip into superconductivity when you dial in other aspects of their environment, like the pressure they’re subjected to or the strength of a magnetic field they’re bathed in. UTe2 isn’t fussy about these properties, and it happily hosts superconductivity in all kinds of different situations. And as researchers continue studying the material, they are finding more questions than answers. 

“This one material seems to do 100 different things,” says Nicholas Butch, who is a physicist at the National Institute of Standards and Technology (NIST) and a member of QMC. “Somebody asked me after one of my talks ‘What right does one material have to do all these things?’ and I said ‘Right?’”

Butch and Paglione, together with colleagues at UMD, NIST, QMC and elsewhere, have been at the forefront of exploring the many wonders of UTe2. Postdocs Shang Ran and Corey Frank, working at both NIST and QMC have spearheaded many of the efforts, from discovering superconductivity in the material to testing samples at National High Magnetic Field Laboratory facilities around the country and experimenting with different preparation techniques. And the buzz around UTe2 is catching on: QMC has been sharing the samples they synthesize with researchers at other universities, including the University of Illinois at Urbana-Champaign and Cornell University, and further study by these groups resulted in the discovery of yet more unexpected behaviors. Uranium ditelluride (UTe2)

A serendipitous discovery

Back in 2018, UMD and NIST postdoc Shang Ran was trying to synthesize U7Te12—a mixture of uranium and tellurium that’s predicted to have intriguing magnetic properties. Instead, Ran kept accidentally making UTe2. He found some literature from the 1960s suggesting UTe2 might have some interesting magnetic properties as well, and after consulting Butch, the two decided to cool it down anyway to see what would happen. Ran stuck the sample into a special helium-powered refrigerator. To his surprise, superconducting currents started to flow.

“We accidentally synthesized this uranium ditelluride, and it turned out it’s a superconductor. So, miracle!” Ran says. “That certainly brought excitement to the community and to our research.”

Ran became captivated with UTe2, and the team went on to poke and prod at it to try to understand its superconducting properties. To start, they set out to explore one of the key behaviors for any superconductor—its response to a magnetic field.  

In superconductors, electrons floating around in the material couple up, forming what’s known as Cooper pairs. These pairs act in concert with each other, and with the other pairs around them, allowing the electrons to flow without resistance. However, a strong magnetic field can break up the pairs, destroying the superconducting magic. One of the main signatures of a superconductor is how much magnetic juice it can withstand, and Ran and his collaborators set out to find this landmark for uranium ditelluride. 

To their surprise, uranium ditelluride remained a superconductor as they turned the field all the way up to the maximum power they has access to in the lab—20 tesla. That’s the combined magnetic strength of about two thousand fridge magnets, or ten times the magnetic field in an MRI machine. “I was shocked when [graduate student Chris Eckgerg] showed me the data,” says Ran. “I asked him ‘Did you measure correctly?’ We measured again and it was all correct. So, we realized, okay, there's some very strange thing going on.”

It wasn’t until they brought the material to the National High Magnetic Field Laboratory in Tallahassee, Florida that they finally found a magnet strong enough to tear apart UTe2’s Cooper pairs: It took an astounding 35 tesla to break the bond. For comparison, the first superconductor ever discovered—mercury—loses its superconductivity at a mere 0.1 tesla. This tipped off Ran, Butch, and the others that UTe2 was no conventional superconductor. They guessed that the electrons inside UTe2 form Cooper pairs in an unusually resilient way.

A special kind of dance

The electrons in a superconductor are kind of like a group of couples on a dance floor. In conventional superconductors, the electron pairs dance together in a straight line, a simple dance where partners mirror each other known as spin-singlet pairing. This synchronized movement allows them to glide effortlessly across the dance floor without any hindrance. However, in some unconventional superconductors, the electron pairs dance in swirly circles, spinning around each other as they glide across the dance floor. This unique dance style, known as spin-triplet pairing, gives them a different kind of coordination.

One consequence of this swirly dance pattern is that breaking up the partners with a magnetic field is much harder, which would explain the high magnetic field UTe2 could withstand. To check if that was going on inside UTe2, the QMC team collaborated with the group of Yuji Furukawa at Iowa State University. The Iowa team used their best techniques for distinguishing between the electron dance patterns, nuclear magnetic resonance spectroscopy. These studies confirmed Ran’s suspicions that UTe2 is a rare spin-triplet superconductor. 

Fewer than a dozen materials are suspected of spin-triplet pairing, and the other candidates are difficult to study—they are either hard to synthesize reliably or they only become superconducting under intense pressures or extremely low temperatures. Uranium ditelluride appears to be the most user-friendly spin-triplet superconductor to date, presenting a rare opportunity for researchers. 

“This is the only triplet superconductor I know that can be studied by so many different probes,” says Vidya Madhavan, a condensed matter physicist and professor at the University of Illinois at Urbana-Champaign (UIUC) who is a longtime collaborator of the QMC team. 

In addition to satisfying a physicist’s basic curiosity, spin-triplet superconductors might be useful as platforms for quantum computing. Spin-triplet pairing is a necessary ingredient for a yet rarer property that hasn’t been confirmed in any superconductor to date—a non-trivial topology. If spin-triplet pairing imbues electron couples with killer dance moves, a non-trivial topology warps the whole dance floor with curves and twists, radically changing the dance patterns of all the couples en masse. 

In the months following the discovery of UTe2’s swirly dance patterns, some evidence suggested that UTe2 might not only be a spin-triplet superconductor but also possess that topological special sauce. The evidence is not yet conclusive, but researchers are hard at work trying to sort this out, as well as understand more about what makes UTe2 tick. And their sleuthing keeps turning up more surprises. 

Superconductivity raised from the dead (and the never-born)

Ran and his labmates were wondering why 35 tesla seemed to be the magic number that broke superconductivity in UTe2. In search of clues, they went back to the National High Magnetic Field Lab.  They kept turning up the magnetic field even higher, looking at how the non-superconducting chunk responded. They also tilted the sample, putting the magnetic field off-kilter from UTe2’s natural crystal structure. 

Shockingly, as they kept rotating the sample, superconductivity reappeared at a field of 40 tesla. This was strange. Turning the field up really high killed the superconductivity, but if you kept going it came back to life. This phenomenon was termed Lazarus superconductivity after the biblical figure raised from the dead. Lazarus superconductivity is extremely rare, though not entirely unprecedented. It’s cropped up in a handful of materials before, and scientists think they have plausible mechanisms for explaining the effect. But none of those mechanisms seemed applicable to UTe2. 

In 2020, Ran joined the physics department at Washington University in St Louis, passing the torch of Butch’s QMC lab to a new postdoctoral researcher, Corey Frank. Frank had just completed her PhD in solid-state chemistry—the perfect background for mastering different ways of concocting the UTe2 crystal. She played with the initial concentrations of the starting materials as well as precise techniques and temperatures of preparation. Among other things, Frank developed a protocol for making UTe2 samples that are just shy of superconducting by making them intentionally just a bit dirty, peppering the crystals with purposefully introduced defects. These defects gum up the pathways by which electrons pair up and find their dance partners, preventing the development of superconductivity. “You can learn a lot about a phase by studying what kills it,” Frank says. 

Frank and her colleagues made a purposefully dirty sample and took it with them on another trip to the National High Magnetic Field Laboratory, this time in Los Alamos, New Mexico. They stuck the sample into the huge magnets and cranked up the field. Once the field was high enough and the sample had the right orientation, the resistance through the material dropped to zero—superconductivity was revived. 

“I was so excited,” Frank recalls. “You're not allowed to jump when you're on the platform of a high-field magnet, but I had to get down from the magnet so I could jump. It was amazing.”

This was completely unprecedented. In all the previous Lazarus superconducting materials, the mechanism behind the rebirth was presumed to involve recreating the conditions at a low magnetic field. Here, recreating conditions at a low magnetic field would not result in superconductivity because the samples had intentional defects, and yet there it was—superconductivity raised not from the dead, but from the never-born, a high-field superconducting phase all its own. The team reported this phenomenon last year in a preprint.

“We know how high field superconductivity works, the rules that govern that, and this one breaks those rules,” Frank says. “So the fact that we have this much more robust high-field phase is wild. I cannot overemphasize how unexpected it is.” 

The authors have some ideas of what could be causing this behavior, and they say further experiments are needed to figure out if those ideas are correct. For now, the experiments are on hold as they require even stronger magnetic fields than the National High Magnetic Field Laboratory currently offers. In the meantime, the QMC team is still studying how this superconductivity dies, comparing their revived samples to others in search of a pattern. 

Making waves 

Over many years Ran, Frank, and other members of the Butch lab have mastered the dual feats of growing pure uranium ditelluride crystals and studying their overall behavior—superconductivity, response to magnetic fields, and more. But they lacked the tools and expertise to zoom in on the microscopic, atom-by-atom behavior of UTe2. So they’ve enlisted the help of Vidya Madhavan’s team at UIUC.

In her lab, Madhavan has a scanning tunneling microscope (STM). An STM works by bringing a bit of metal tapered down to a tiny, fine point extremely close to the surface of a sample—so close that electrons from the sample can hop over to the conducting tip, or vice versa. By measuring how many electrons make the jump, scientists can learn a lot about the microscopic structure of a material, including where the electrons are on the surface of the sample.

The Butch group sent Madhavan a sample, and Anuva Aishwarya, a graduate student at UIUC who led the study, placed a sample of UTe2 into the scanning tunneling microscope. The team cooled the material just shy of its superconducting temperature, and they stumbled upon another surprise: The electrons didn’t follow the ups and downs of UTe2’s crystal structure. Instead, they clustered together and then apart, forming waves of charge frozen into the surface with a pattern all their own.

These kinds of charge density waves are uncommon but not unprecedented. However, the measurements performed by Ran, Frank or others at QMC didn’t show any indication that a charge density wave might be found in UTe2. To Madhavan and her team, this came out of nowhere.

To try to understand what they were seeing, Aishwarya and her lab mates probed the behavior of these waves in different temperatures and magnetic fields. They found that, in a magnetic field, the charge density wave seemed intimately related to superconductivity itself. As they turned up the field, the charge density wave broke down at precisely the same field strength as superconductivity. This tipped off Madhavan and her collaborators at UIUC that maybe this wave had some relationship to the superconductivity in uranium ditelluride.

If you want to pick out individual electrons, a regular STM is great. But if you want to peer inside the dance patterns of electron couples in a superconductor, you need an STM armed with a special kind of tip—one that is itself a superconductor. The team of Seamus Davis at Cornell University had just such a superconducting tip. They became intrigued by Madhavan’s results and got in on the action. They obtained another sample from the QMC team and stuck it in their specialized STM. They found that the electron pairs behaved similarly to the lone electrons. Here, too, the pairs clustered together and apart, forming a so-called pair density wave with the same beat as the charge density wave observed by Madhavan. This is the first time such a pair density wave has been found in a spin-triplet superconductor.

As with many aspects of UTe2, the origins of the charge and pair density waves remain far from clear. But, Ran comments, these waves are a fairly common feature in unconventional spin-singlet superconductors. This may provide clues for how all these different strange superconductors are connected. “We eventually need to understand unconventional superconductivity overall,” Ran says. “And having this common theme I think is very important for theorists.”

While theorists are hard at work trying to crack the puzzle of unconventional superconductivity, Ran, Butch, and other researchers are continuing to explore all that UTe2 has in store. “It's really rich. It’s a great place to explore,” says Butch. “This one material underscores how little we know about spin triplet physics. It’s as if we are writing textbooks about it right now. So that's actually very exciting.”

Story by Dina Genkina

Simulations of ‘Backwards Time Travel’ Can Improve Scientific Experiments

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Published: Saturday, October 14 2023 05:40

Reference Publications
Nonclassical Advantage in Metrology Established via Quantum Simulations of Hypothetical Closed Timelike Curves, D. Arvidsson-Shukur, A. McConnell, and N. Halpern, Phys. Rev. Lett., 131, 150202, (2023)

Embracing Uncertainty Helps Bring Order to Quantum Chaos

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Published: Thursday, September 28 2023 12:40

Advocating for Quantum Simulation of Extreme Physics

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Published: Friday, September 08 2023 07:34

The Big Bang, supernovae, collisions of nuclei at breakneck speeds—our universe is filled with extreme phenomena, both natural and human-made. But the surprising thing is that all of these seemingly distinct processes are governed by the same underlying physics: a combination of quantum mechanics and Einstein’s theory of special relativity known as quantum field theory.

Theoretical nuclear and particle physicists wield quantum field theory in their efforts to understand interactions between many particles or the behavior of particles with extremely large energies. This is no easy feat: At least theoretically, quantum field theory plays out in an infinite universe with particles constantly popping in and out of existence. Even the world’s biggest supercomputer would never be able to model it exactly. Fortunately, there are many computational tricks that can make the problem more tractable—like cutting up the infinite universe into a finite grid and taking judicious statistical samples instead of tracking every parameter of every particle—but they can only help so much. 

Over the past few years, a growing group of scientists has become wise to the potential of quantum computers to approach these calculations in a completely new way. With a fully functioning quantum computer, a lot of the approximations could be avoided, and the quantum nature of the universe could be modeled with true quantum hardware. However, quantum computers are not yet big and reliable enough to really tackle these problems, and the algorithms nuclear and particle physicists would need to run on them are not yet fully developed.

“Even if we have large-scale, fully capable quantum computers tomorrow,” said Zohreh Davoudi, associate professor of physics at UMD, “we don’t actually have all the theoretical tools and techniques to use them to solve our grand-challenge problems.”

Classical computers require exponential resources to simulate quantum physics. To simulate one extra tick of the clock or include one extra particle, the amount of computing power must grow significantly. So, the classical methods resort to approximations that fall short because they leave out details and lose the ability to address certain kinds of questions. For one, they can’t keep up with the real-time quantum evolution of the early universe. Additionally, they can’t track what happens during collisions of heavy nuclei. And finally, they are forced to ignore the quantum interactions between the myriad particles in high-energy settings, like those that are emitted from an exploding star. A quantum computer, however, could tackle these problems on their own quantum turf, without needing as many resources or resorting to as many approximations.

Now, researchers want to make sure the nascent effort to use quantum computers to simulate the extreme events of the universe continues to thrive. Davoudi, along with JQI Adjunct Fellow and College Park Professor of Physics Chris Monroe and other researchers, penned a whitepaper laying out the case for funding quantum simulation research in particle physics, published in the journal PRX Quantum in May 2023. Davoudi also co-authored a similar whitepaper in the field of nuclear physics, available on the arXiv preprint server.  

“It's a responsibility of researchers to also think at a larger scale,” said Davoudi, who is also a Fellow of the Joint Center for Quantum Information and Computer Science (QuICS) and the associate director of education at the National Science Foundation Quantum Leap Challenge Institute for Robust Quantum Simulation (RQS). “If we think this field is intellectually promising, interesting, and worth investing in as a scientist, we have to make sure that it stays healthy and lively for generations to come.”

Some sub-fields of physics, including the nuclear and particle physics communities, engage in long-term planning for the future of their field. Nuclear physicists in the U.S. plan seven years ahead, and particle physicists plan a full decade ahead. Researchers from many universities and national laboratories come together in meetings, seminars, and panel discussions over the course of a year to decide what the highest priorities in the field should be. Funding agencies in the U.S. and worldwide have historically taken these conclusions seriously. The whitepapers developed by Davoudi and her co-authors are a part of those efforts. In them, they argue for the importance of studying quantum simulation for nuclear and particle physics and make specific recommendations for further development. 

“These new research directions in both nuclear physics and high-energy physics were not part of the last U.S. long-range planning processes, because the idea had simply not been introduced at the time,” Davoudi said.

Indeed, the ideas weren’t even on Davoudi’s radar six years ago when she came to UMD to join the physics faculty as a theoretical nuclear physicist. While she was busy searching for an apartment, Davoudi saw an announcement for a workshop hosted by QuICS exploring the intersection of her field with quantum computing. Instead of looking for a place to live, she spent several days at the workshop, talking to theorists and experimentalists alike. 

Davoudi was enticed by the promise of quantum simulations to solve the kinds of problems she was unable to address with classical computational tools, and it changed the course of her career. In the years since, she has developed new theoretical techniques and collaborated with experimentalists to push the boundaries of what quantum simulators can do to help uncover the basic physics of the universe.

Davoudi wants to ensure that this burgeoning field continues to thrive into the future. In the whitepapers, she and her co-authors identified specific problems where quantum computing holds the most promise. Then, they made three main recommendations to ensure the success of the field for the next seven to 10 years. 

First, they recommended funding for theoretical efforts to develop algorithms that run on quantum hardware. Even though the potential of quantum computing is clear, detailed algorithms for simulating quantum field theory on a quantum computer are still in their infancy. Developing these will require a dedicated effort by the nuclear and particle physics communities. 

Second, they advocated for greater interdisciplinary communication between the nuclear, particle and quantum physics communities. Different quantum computer architectures will have different quirks and advantages, and the field theory folks will need to have access to them to figure out how to make the best use of each one. Certain implementations may, in turn, become motivated to engineer specific capabilities for the kinds of problems nuclear and particle physicists want to study. This can only be accomplished through close interdisciplinary collaboration, the authors claim. 

“As a community, we cannot isolate ourselves from the quantum information and quantum technology communities,” Davoudi said.

Third, Davoudi and her co-authors believe it is key to bring in junior researchers, train them with a diverse set of skills, and give them opportunities to contribute to this growing effort. As with the QuICS workshop that inspired Davoudi, the community should invest in education and training for the relevant skills through partnerships between universities, national labs and the private sector. 

“This is a new field, and you have to build the workforce,” Davoudi said. “I think it's important for our field to bring in diverse talent that would allow the field to continue to intellectually grow, and be able to solve the problems that we would like to eventually solve.”

Written by Dina Genkina

Novel Quantum Speed Limits Tackle Messy Reality of Disorder

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Published: Tuesday, August 22 2023 12:40

The researchers and engineers studying quantum technologies are exploring uncharted territory. Due to the unintuitive quirks of quantum physics, the terrain isn’t easy to scout, and the path of progress has been littered with wrong turns and dead ends.

Sometimes, though, theorists have streamlined progress by spotting roadblocks in the distance or identifying the rules of the road. For instance, researchers have found several quantum speed limits—called Lieb-Robinson bounds—that are impassable caps on how quickly information can travel through collections of quantum particles. They’ve even developed protocols for quantum computers that achieve the best possible speeds for specific cases. But to make calculating the limits easier, physicists have mostly neglected the influence of disorder. In the real world, disorder can’t always be ignored, so researchers need to understand its potential effects.

JQI postdoctoral researcher Chris Baldwin, JQI Fellow and Adjunct Professor Alexey Gorshkov and other JQI researchers are facing down the impact disorder has on speed limits. In an article published on June 22, 2023 in the journal Physical Review X Quantum, they described novel methods for pulling insights from the mess created by disorder and identified new types of quantum speed limits that apply when disorder is present.

"We were motivated both by the beautiful theoretical problem of proving and saturating new speed limits and by the implications that our work would have on quantum computers that inevitably have some disorder," says Gorshkov, who is also a physicist at the National Institute of Standards and Technology and a Fellow of the Joint Center for Quantum Information and Computer Science.A chain of quantum spins can pass information down a line like a bucket brigade, but sometimes disorder (represented here by the red hand and bucket) can slow down the communication. The arrows in spheres in the buckets are a geometrical representation of a quantum state. (Credit: Sean Kelley/NIST)

Baldwin, Gorshkov and colleagues began by tackling the case of a one-dimensional line of particles, where each particle can only directly interact with its neighbors. They specifically focused on the spin—a quantum property related to magnetism—of each quantum particle. A spin is like a compass needle that wants to point along a magnetic field, but, being quantum, it can point in more than one direction at a time—a phenomenon called superposition.

Spins pass information to each other through interactions, so a line of spins can act like a bucket brigade passing quantum information: One jiggles its neighbor, which jiggles its neighbor on the other side, and the information makes its way down the line.

But if something is slightly off about a spin’s connection to a neighbor—there’s some disorder—the spin will fumble handing over the quantum data and slow things down. With their imperfect handovers, spins resemble people in a bucket brigade each working at a slightly different speed. Most people probably take a similar amount of time to pass a bucket, maybe clustered around a couple of seconds. But if enough random people are pulled in, a few speed demons may only need a second while others might take five seconds or more.

To account for the full range of possibilities in quantum systems, Baldwin and colleagues didn’t limit themselves to a fixed number of possible speeds. Instead, they analyzed distributions of speeds that extend from the quickest transfers for ideal connections between spins down infinitely to even the slightest chances of handoffs taking millennia or longer for arbitrarily bad connections.

In future quantum computers, experts expect millions of spins to work together. With so many spins, even long odds of any individual being a slowpoke can combine into a safe bet that one, or even several, will be present.

To make sense of the sea of possibilities presented by disorder’s influence on handoff speeds, Baldwin and colleagues pulled out the tools of probability theory. These tools allowed them to glean information about speed limits from the statistics of how transfer speeds are peppered throughout the line. With probability theory, they derived new speed limits for whole groups of spin chains based on the big picture without needing to know anything about the links between any particular spins.

The team was particularly interested in investigating if the speeds information can reach in different systems depend on the distance it is traveling. Some physical processes, like light travelling through space, resemble a car steadily cruising down an empty highway, where the travel time is directly proportional to the distance—it takes twice as long to move twice as far. But the speeds of other processes, like perfume defusing through a room, don’t have such a straightforward proportional behavior and can look more like a flagging runner who takes longer and longer the farther they push themselves. Knowing the relationship between speed and distance for quantum information is valuable when researchers are weighing their options for scaling up quantum computers.

With their new results, the researchers determined that information can’t always propagate at a steady speed indefinitely, and they identified the border between conditions that allow a steady speed from those that only allow a deteriorating pace.

They also found that their method allowed them to define two distinct types of limits that tell them different things. One they dubbed “almost always bounds” because the bounds hold for almost all the sections of a chain. These limits apply to any sufficiently long stretch of spins even though they might occasionally be violated for small sections of the chain—like if there is an unusual clump of speed demons in the brigade. These limits allow researchers to guarantee conditions, like that a particular spin won’t be disturbed by activity further down the line within a particular time window.

The researchers called the second type of limit “infinitely often bounds” because they are guaranteed to apply to some stretches of an infinite chain but there isn’t a guarantee that the limit will definitely hold for any particular stretch no matter how long a section is being considered. So, these limits are expected to occasionally pop up on sections of the chain and generally lower the limit from that set by the almost always bound—like a car occasionally entering a work zone on the highway. Having an idea of these lower speed limits that are likely to pop up can help researchers to judge the reasonable minimum amount of time to dedicate to getting the bucket all the way across a stretch of the brigade.

The newly defined limits allowed the team members to resolve a lingering discrepancy: The existing Lieb-Robinson bounds had set a higher ceiling than any information transfer protocol had reached. The mismatch could have been the result of either researchers not being creative enough in designing the protocols or them failing to account for something that enforced a lower limit. Accounting for disorder more carefully dropped the theoretical ceiling down to match the speed of existing protocols.

“For a while, we had this gap,” Baldwin says. “The main exciting thing of this work was figuring out how we could completely close this gap.”

The researchers say there is further work to be done exploring the applications of these limits and determining when the two types of bounds have significant impacts.

“The main direction I want to take this going forward is going beyond one dimension,” Baldwin says. “My suspicion is that the picture will end up looking very different, but I think it's still worth having this one-dimensional case in mind when we start to do that.”

Original story by Bailey Bedford: https://jqi.umd.edu/news/novel-quantum-speed-limits-tackle-messy-reality-disorder

In addition to Baldwin and Gorshkov, authors on the publications included UMD graduate student Adam Ehrenberg and former UMD graduate student Andrew Guo.

About the Research

Reference Publication
Disordered Lieb-Robinson bounds in one dimension, C. Baldwin, A. Ehrenberg, A. Y. Guo, and A. V. Gorshkov, PRX Quantum, 4, (2023) PRXQuantum.4.020349.pdf

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