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A representation of the dynamics within the superconducting system. Image credit: Steven Burrows

When a superconducting material is cooled down below a critical temperature, something seemingly magical happens: its electrical resistivity drops abruptly to zero! Initially, before 1911, this was thought to be impossible, given that electrons, which are the particles that carry electric current, typically scatter from impurities and imperfections of a crystal lattice used in conducting materials. ��Moreover, because electrons are negatively charged particles, they typically repel each other. Yet, behind the “magic” of superconductors is the fact that two electrons, in a periodic crystalline array of atoms (a web of lasers), can attract positive charges in the lattice, whose subsequent deformation mediates an attractive interaction between the electrons. This attraction favors electrons with opposite momenta to bind together, forming ‘Cooper pairs’. These pairs can coalesce into a coherent macroscopic quantum state of matter, in which they remain paired while flowing through the crystal without any resistance. Beyond their immense practical applications, superconductors also offer a promising testbed to study the fundamental physics of matter held far away from equilibrium.

In a conventional superconductor (‘s-wave’ superconductor), the two electrons in a Cooper pair must have opposite spins. But there are unconventional superconductors with p-wave symmetry, in which electrons of the same spin pair up. ��This pairing is penalized by an energy barrier and in order to overcome the barrier and pair up, electrons need to carry a non-zero angular momentum, which means that they need to spin around each other. The net angular momentum of the Cooper pairs can give rise to rich quantum behaviors and phases of matter that are intensively sought in real materials and cold atoms, but have, so far, remained elusive. ��In particular, the dynamics of p-wave superconductors taken away from equilibrium is predicted to exhibit a variety of temporal behaviors, some of which possess interesting quantum dynamics. Observing these ‘dynamical phases’ in the lab would provide a window into the nature of non-equilibrium phases of matter and some of their properties, and potentially new p-wave superconductors. In cold gases, one of the biggest challenges that has prevented researchers from observing p-wave physics is three-body losses in energy that emerge when weak p-wave interactions are enhanced via external electromagnetic fields. However, to date, liquid 3He remains the only well-established laboratory example of a p-wave superconductor.

To overcome these challenges, JILA and NIST Fellow Ana Maria Rey collaborated with NIST (National Institute of Standards and Technology) Ion Storage Group leader John Bollinger, and researchers at the University of Innsbruck, Rutgers University and the University of Colorado Boulder, to design a trapped-ion simulator for 2D p-wave superconductors. Their work paves a way for clean observations of the predicted non-equilibrium dynamics in future experiments using the trapped-ion simulator, or Penning trap. The researchers published their findings in

Circumventing Traditional Challenges

“The natural platform to study superconductors would be solid-state systems or ultra-cold fermionic gases [to avoid three-body energy loss],” explained University of Innsbruck postdoctoral researcher Athreya Shankar. “But the problem is that p-wave interactions are rather weak, and therefore hard to control. In order to enhance these interactions, you have to manipulate the system using external fields,” Shankar added, “But that can cause other unwanted problems in your experiment. This means that your gas can become unstable before you can observe the interesting dynamics.”

To circumvent these problems, Shankar, Rey, and their colleagues asked a rather unconventional question: Do we need fermions at all to simulate the non-equilibrium behavior of superconductors? The motivation for this question came from the well-known observation that the non-equilibrium behavior can be described in terms of pairs of fermions with equal and opposite momenta, i.e., Cooper pairs. “The trick, then, is to map each Cooper pair onto a two-level system, or a spin, with states `up’ and `down’,” explained Shankar. “The `up’ state indicates the presence of a Cooper pair while the `down’ states indicates that it is absent. A two-level system can be naturally encoded into two electronic levels of each trapped ion, thus providing a non-traditional route to the quantum simulation of the non-equilibrium dynamics of superconductors.” The tremendous level of control offered by trapped-ion simulators and the absence of fermionic particles meant that most of the challenges faced by traditional approaches could be overcome, or better still, were non-existent.

Using Position to Emulate Momentum

For the experiment, the researchers considered a setup using a Penning trap, in which 2D crystals of 100 to 200 trapped ions can be stored and manipulated. They proposed to encode a two-level system in each ion, which would simulate the presence or absence of a fermionic Cooper pair. The central idea was that the momentum space of Cooper pairs was mapped on to the real space of the ion crystal. “So, if there is an ion at a certain position in the crystal plane, it would emulate a Cooper pair with a certain 2D momentum in the superconductor,” Shankar said. “The farther the ion is from the crystal center, the larger the momentum of the associated Cooper pair.” With this association, mapping became more straightforward. Rey commented, “The goal, then, is to emulate the attraction between Cooper pairs of different momenta in a p-wave superconductor. The mapping of Cooper pair momenta on to ion positions means that we need to engineer couplings between ions that depend on where each ion is located in the crystal plane. Interestingly, we discovered a clever way to accomplish that by exciting the vibrations of the crystal while simultaneously using the fact that ions in a Penning trap are rotating.”

Turning a Weakness into a Strength

A standout feature of the Penning trap is that the ion crystal is actually rotating when viewed from the laboratory frame. “The rotation is necessary; it’s the source of the ion confinement in a Penning trap,” explained Bollinger. However, the rotation is usually an inconvenience, since it becomes challenging to address the ions with laser fields to manipulate their electronic states. But for the researchers, this inconvenience could actually be a benefit. “This perceived disadvantage of the Penning trap actually turned out to be the crucial ingredient to engineer ion-position dependent couplings,” Shankar pointed out. By tilting the angle of the lasers used to address the ions and by suitably tuning the laser frequencies, the electronic states of the ions can be coupled to the crystal rotation. As a result, each ion is addressed by the lasers differently, depending on its position. Rey explained, “Furthermore, the laser illumination excites vibrations in the crystal and generates a force that depends on the electronic state of the ions. As a result, the motion of the crystal mediates an effective state and position dependent coupling between ions, which emulates the attraction between the Cooper pairs mediated by lattice vibrations in a real p-wave superconductor.” This allowed the mapping of interaction strength to be more closely associated with location in the crystal. “So now, if we look at the effective interactions between ions in the crystal, we find that the interaction strength depends on the ion positions in the frame rotating with the ions, where ions are static,” Shankar added. Because of the many variables involved in the proposed setup, the entire process was not simple. “The required level of control over the tilt angle of the laser beams used to address the ions is demanding, but it is something that we have been able to previously demonstrate in the laboratory,” stated Bollinger, who looked forward to transitioning the newly proposed theory into a laboratory protocol. “This will be very exciting to give a try in the lab!”

Using their proposed design, the team illustrated that in theory, a variety of predicted non-equilibrium dynamics could be observed in an experiment. They also showed how one could infer the topological character of the observed dynamical behavior through appropriate measurements. “Furthermore, since the number of ions is not that many (only 100 to 200), the effect of quantum noise on the non-equilibrium dynamics can potentially be observed in an experiment,” Shankar added. Rey echoed the importance of this new method by stating: “This is very exciting, since this means the experiment could be able to observe new effects not predicted before, since they are very complicated to be treated by theory.”

Expanding the Toolbox

“There is a growing interest in using Penning traps for quantum simulation,” Shankar said, “and our proposal greatly expands the simulation capabilities in these systems.” Beyond p-wave superconductors, this proposal provides a general toolbox for creating and manipulating winding spin textures known as skyrmions, which could be of broader interest, for example, in studying models of quantum magnetism and exploring better ways to build spintronics devices—a technology that exploits the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge.