Topological Winding Readout of an Emergent Page-Wootters Clock

In the standard view of quantum mechanics, time is not a thing you can hold or measure from within a system; it is an external ruler, a silent backdrop against which particles move and change. This works well for most calculations, but it leaves a gap in our understanding of how time might emerge from the universe itself. A theoretical framework known as the Page–Wootters construction proposes a different way to see things: it treats time as an internal property of the system, like a clock built into the machinery of the world. In this view, the rest of the universe evolves relative to the reading of this internal clock, rather than against an outside timer. This idea has moved from pure theory to experiments with entangled light, but a new question has remained unanswered: if time is a property of a system, can that property carry a hidden, unchangeable signature, much like a knot in a string that cannot be untied without cutting the string?

Researchers at the University of Surrey have now proposed a way to answer this question using light and a specific type of topological invariant, a mathematical property that remains fixed even when a system is stretched or twisted. They suggest a photonic architecture where an internal clock is not just a ticking mechanism, but a structure that winds around a hidden loop. By cycling the conditions of a light wave through a closed path, they show that the clock accumulates a geometric twist that is quantized, meaning it comes in whole, discrete steps rather than a smooth, continuous flow. This twist, or winding, is robust against small errors and can be measured without needing an external reference clock, a feat that was previously thought to require a standard timekeeper from outside the system.

The experiment they propose relies on a chain of tiny, coupled optical rings made from thin-film lithium niobate, a material known for its ability to manipulate light efficiently. Inside this chain, light travels in two distinct modes, which the researchers designate as the "clock" and the "system." These two modes are generated together as a pair of photons through a process called spontaneous four-wave mixing, where a strong laser beam splits into two new photons that are inextricably linked. One photon, the idler, acts as the clock, while the other, the signal, acts as the system that evolves in time. The clock photon moves through a specific energy landscape, known as a Rice–Mele band, which is shaped by alternating couplings and detunings between the rings. By slowly and carefully cycling these couplings and detunings around a specific point where the energy gap closes, the researchers drive the clock photon through a complete loop.

When the clock photon completes this loop, it does not simply return to where it started; it acquires a phase shift, a change in its wave-like state, that depends on the path it took. The researchers found that this phase shift has two parts: a dynamical part that depends on how long the process took and the energy of the photon, and a geometric part that depends only on the shape of the path. The geometric part is the key. As the clock photon moves through the entire range of possible momenta in the crystal, this geometric phase winds around exactly once for every unit of topological charge, a property known as the Chern number. This winding is an integer, a whole number that cannot change unless the energy gap in the system closes completely. It is this integer winding that the researchers aim to measure, rather than the raw displacement of the photon, which would be a messy, fractional value for the narrow beams of light they use.

To read out this winding, the team proposes a clever interferometric setup that cancels out the messy dynamical part of the phase, leaving only the clean geometric twist. They split the entangled photon pairs into two parallel paths. In one path, the clock photon is driven through the cycle in the forward direction, while in the other, it is driven through the exact same cycle in reverse. Because the dynamical phase depends only on the path and the time, it remains the same in both directions, but the geometric phase flips its sign. When the two paths are recombined, the dynamical phases cancel each other out, and the remaining signal reveals twice the geometric phase. This allows the researchers to isolate the topological winding from the noise of the system.

A critical challenge in this measurement is that the clock photon is entangled with the signal photon, and the researchers need to scan the momentum of the clock without touching it directly. They solve this by using the signal photon as a relational reference. Because the two photons are energy-anticorrelated, filtering the signal photon to a specific frequency automatically selects a specific momentum for the clock photon. By stepping the filter on the signal side, the researchers can scan the clock photon across its entire momentum range, measuring the phase of the interference fringe at each step. This process effectively maps out the winding of the geometric phase across the Brillouin zone, the full range of momenta available to the photon.

The beauty of this approach lies in its protection. The integer nature of the winding means it is immune to smooth deformations of the device or the control loop. As long as the energy gap between the light bands remains open, the winding cannot change; it is pinned in place. This is a form of topological protection, where the information is stored in a global property of the system rather than a local detail that can be easily disturbed. Furthermore, the measurement itself is digital. The researchers do not try to measure a precise angle that could be slightly off; instead, they count the number of times the phase winds around. If the noise in the system is below a certain threshold, the probability of miscounting this integer drops exponentially, making the result incredibly reliable. This is in stark contrast to measuring a continuous delay, where noise would cause the error to grow linearly.

The proposed setup is designed to be feasible with current technology. Using thin-film lithium niobate microrings, the researchers estimate that the required cycle time is about 0.8 nanoseconds, which fits comfortably within the lifetime of the photons in the system. The quality factor of the rings, a measure of how well they trap light, needs to be around 2.5 million, a value that has already been demonstrated in similar devices. The system requires a chain of at least 100 unit cells to ensure the photons have enough space to travel and interact, but the topological nature of the measurement means that small imperfections in the fabrication of these rings will not ruin the result. The integer winding survives as long as the energy gap remains open, regardless of minor disorder in the device.

This work represents a significant step in understanding the nature of time in quantum systems. By demonstrating that a clock degree of freedom can carry a topological invariant, the researchers show that time, when treated as an internal property, can have a robust, quantized structure. The ability to measure this structure without an external reference clock challenges the traditional view of time as a mere external parameter. While the experiment has not yet been built, the detailed parameter budget and the clear theoretical framework suggest that it is within reach. The proposal offers a new way to probe the fundamental relationship between time, topology, and quantum mechanics, potentially opening doors to new types of quantum sensors and clocks that are inherently protected against noise and error. The success of this approach would not only validate the Page–Wootters construction in a new regime but also provide a concrete example of how topological protection can be applied to the very concept of time itself.