Tying photonic knots in space and time

Leafing through the SPIE Photonics West program, you might have raised an eyebrow when you see the title of the OPTO Plenary talk by the University of Rostock’s Professor Dr Alexander Szameit: “Topology in space, time, and space-time.” This is because topology is traditionally regarded as a somewhat abstract branch of mathematics. You might even ask, what on Earth has it got to do with photonics?

In topology, the big-picture aspects of objects matter most. From the perspective of a topologist, a bath is no different from a bowl, a mug is the same as a donut. But the bath/bowl is a completely different object to the mug/ donut, because they have a different number of holes, i.e. 0 and 1, respectively. Topology connects these big-picture global properties with local ones, providing the rules under which you can contort the object without transforming it into something else. For example, in 2000, Thomas Fink and Young Mao treated necktie knots as inherently topological structures and in doing so found 85 different ways to tie a necktie — some of which were new to fashion.

Arguably more interesting and useful has been applying a topologist’s mindset to physics problems. Topological arguments have been used to derive properties of black holes, describe various physical phenomena like superfluidity, and even discover new (topological) phases of matter in electronic materials. Due to topological phenomena having an intrinsic robustness, qubits consisting of theoretical quasiparticles known as anyons have even been proposed as a route to fault-tolerant topological quantum computing.

Where topology has had less impact and, indeed, seems to have little bearing is in photonics. “In the end, photonics is an electromagnetic wave and interference,” says Szameit. “So, one simply does not expect topological behavior — this just seems too crazy.”

However, since their first demonstration in 2009, topological states of electromagnetic waves can and have been achieved, in the process giving birth to the field of topological photonics and revolutionizing fundamental understanding of how light can be manipulated. Now for something completely different Topological photonics is entirely different from other types of physics utilizing topology. Much of this boils down to the basic nature of the fundamental particles under study. Photons are bosons, whereas electrons are fermions. They therefore exhibit profoundly different behaviors. For example, where electrons naturally anti-bunch, photons are the opposite, as Szameit explains in more detail: “If you shoot two electrons at a beamsplitter, they never leave it as a pair, but if you have two photons going towards a beamsplitter, they leave the beamsplitter only as a pair,” he says. “It’s really, really, fundamentally different, and these concepts have to be taken into account when one thinks about topology.”

In a recent joint effort with collaborators, Szameit took advantage of this fundamental difference to reproduce the Hong–Ou–Mandel effect where perfect interference between two photons entering a beamsplitter results in them having a 50:50 chance of exiting together in either output mode.

The Hong–Ou–Mandel effect was first demonstrated in 1987 and has since become a critical physical mechanism for logic gates in cutting-edge optical quantum computing. However, it is delicate, with the beamsplitter requiring extreme precision assembly. Szameit’s version topologically protects the photons to always yield a 50:50 beamsplitting ratio, even with a far from perfect setup.

Alongside taking advantage of the specific features of electromagnetic waves, Szameit and others are also transferring concepts from the topological states in condensed-matter physics to light, which is no mean feat. “Many, many of the algorithms and concepts were implemented or invented for fermions, so one really has to think about the fundamentals and adjust everything, because we now have to apply them to electromagnetic waves,” he says.

Some of Szameit’s earliest world-firsts came from exploring this research direction, i.e. transferring ideas from topological states in condensed-matter physics to light. Based on laser-written photonic waveguide lattices they made themselves, his group was the first to demonstrate the photonic version of topological insulation, topological creation and destruction of edge modes, and a number of other phenomena.

It’s about time

However, in his OPTO Plenary, Szameit took a step beyond these research areas, focusing instead on a much more fundamental subject that “challenges the mind”: the new physics, new phenomena, and new applications made possible when considering topological effects in photonic systems not just in space, but also in time, and even spacetime. Szameit says that, generally, topology is considered in real space, with time just a parameter: “99% of physics takes place in space. Wouldn’t it be cool to really think about time?”

Yet doing so is fraught with difficulty. Up to now, topological features have been observed as robust, long-lived states residing within an energy gap localized at a spatial topological interface. Time- or spacetime-topologically protected states require an equivalent temporal interface. “And now comes the problem,” says Szameit. “In space, reflection is intuitive: you have a forward-propagating wave, an interface, and a backward-propagating one. In time, that’s tough — one cannot really have waves that move backwards in time.”

Although often considered as two sides of the same coin, unlike space, time has a unique unidirectionality, often called the arrow of time, which is intrinsically bound to the notion of cause and effect, i.e. causality. And this places a number of new constraints on time and spacetime topology. In fact, never being able to move backwards in time means the entire concept of an interface changes, creating a new notion of a spacetime-topological event, where a topological state localizes at a single point in spacetime.

Szameit attempts to elucidate this further by picturing such a state: “Suddenly you see in front of you a bright spark. It’s localized and it’s topologically protected in space and time, so it’s robust. But it disappears again and when it’s gone, it’s gone.”

Though this idea and the concepts surrounding it seem abstract, Szameit’s OPTO Plenary delved into the progress his team has made in not only developing theory around time and spacetime topology, but also conducting experiments demonstrating time-topological states and even spacetime-topological events. What is more, he described the implications and new possibilities such studies open, including applications such as spatiotemporal wave control for imaging and communication, and topological lasers.

Though he acknowledges some of the content of the talk might be unfamiliar to the audience, Szameit hopes the “cool physics” involved was fascinating to Photonics West attendees. “This is my message: when you think about conventional phenomena in space, what would happen if you transferred them to time? In many cases, it would be really different, utterly interesting, and would open the door to really new physics, new phenomena, and probably new applications.”

Benjamin Skuse is a science and technology writer with a passion for physics and mathematics whose work has appeared in major popular science outlets. This article originally appeared in the 2025 SPIE Photonics West Show Daily.