Dr Lincoln Carr, a genial fedora-and-scarf-wearing physics professor here at Mines, visited the applied math department to deliver a talk on his research into the fascinating and complex world of ultra-cold physics. While he is an engaging speaker, and the things he discussed were cutting-edge, the complexity and esoteric nature of the information involved ensured that a layman was all but lost for the majority of the talk. Herein the reporter attempts to make approachable a sort of wizardry which only a mathematician or a theoretical physicist can ever truly grasp. A mathematician looks at one of the nonlinear differential equations presented during the talk and sees its pieces laid out before her in real-world terms, just as a musician sees sheet music and hears a tune; but a layman looks at the same equations and might as well be trying to read cuneiform.
The broad subject of Carr’s talk was the nature of matter itself. As most people learned in high school science class, the universe and everything in it, other than a true vacuum, is made up of ever-smaller particles: molecules, then atoms, then protons and neutrons, and finally bosons, quarks, neutrinos, and the like. Because matter and energy are the same – the fundamental message of Einstein’s famous equation, E=mc2 – the smallest particles are in fact small “wave packets.” Light was the first substance to show off this dual nature: light is both an electromagnetic wave and a particle called a photon. Technological advances have since allowed for the phenomenon to be studied in other particles as well. As the temperature of a particle, T, approaches absolute zero, the particle’s wave nature becomes measurable, then begins to overlap and merge with the wave nature of the particles around it. In other words, as T goes to zero, the particles slow their vibration, appearing as wave packets, then as overlapping “matter waves” (called Einstein-Bose condensates for the scientists who first proposed their existence), and finally merging into a single giant “matter wave”, which is known as a pure Bose condensate. Carr and his colleagues work with these condensates.
The existence of Einstein-Bose condensates was first theorised in 1925, but it was not until 1995 that this existence was finally realised, as the technology to induce the ultra-cold temperatures required to create such a condensate did not exist until then. The field has since taken off, even producing several Nobel Prize winners, thanks to further technological advances and a great deal of multidisciplinary collaboration. This area of research has many applications to the fast-growing field of quantum computing, in which special computers kept at these same ultra-cold temperatures are able to process exponentially larger quantities of data than conventional supercomputers.
Since, at its most fundamental level, all matter consists of waves, and a wave is describable as a mathematical equation, all particles can be described by probabilistic wave functions. (The probability aspect comes in thanks to the theory of quantum superposition, which states that the properties of a particle simultaneously exist in all theoretically possible states until a measurement is made on the particle.) For this talk, Carr highlighted the Nonlinear Dirac Equation, or NLDE. The NLDE has many possible solutions, with varied and surprising physical expressions, each of which represents a different variety of particle, with exotic names like “semions” and “skyrmions”. A soliton, for instance, corresponds to a kinked curve such as that seen in the bend of a DNA strand or a curl of ribbon. This “zoology” of the NLDE is not a menagerie of the imagination. The strange shapes that the NLDE can create can be modelled, and when the models are compared to real-world data, they match almost without flaw.
At temperatures in the sub-micro Kelvins – orders of magnitude colder than the vacuum of space – the speed of light falls to 0.272 cm/s. To put this in perspective, the speed of light in a vacuum at normal temperatures is a little over 29 billion cm/s. At such glacial speeds, Einstein-Bose condensates can be (and have been) photographed, including in 3D; these photographs are the proof to the theory that Carr and his colleagues deal with. In order to photograph, for instance, a solition, a scientist must first trap the particle using electromagnetic traps or a complex optical lattice made of intersecting phase-locked lasers. Where the laser beams interfere with each other, micro-traps are formed, each catching a single particle. Here, instead of using math to give insights into nature, Carr did the reverse: he used nature to aid in his math.
Graphene is a remarkable substance consisting of molecule-thick sheets of pure carbon. These sheets can be folded into any number of shapes, including the unique folded structures described by solutions of equations such as the NLDE. Carr therefore used graphene as the inspiration for his optical lattice, helping him to trap particles better. Ultimately, when the theoretical conclusions of “math world” (“a spectacular land where our imaginations can roam”, as Carr put it) are translated into reality – something actually possible, thanks to modern technology – they exhibit very specific real-world constraints which do not reveal themselves in math world. For instance, at hyper-cold temperatures a gas (such as used in these experiments) “wants” to be a solid, making the system unsustainable… at least theoretically. Carr found, however, that many solutions of the NLDE, which are unstable in infinite time – that is, in math world – were perfectly sustainable in the finite timescales of reality. In this way, nature can shine light on theory, just as theory can illuminate nature’s inner workings.