# Light mimics hotel with limitless vacancies

Hilbert illustration of infinity inspires new twist in optical information storage

By Andrew Grant

A mind-bogglingly large hotel has provided inspiration for expanding the data-carrying capacity of light.

A new technique that manipulates the twistiness of light is the optical equivalent of a mathematician’s thought experiment for creating more space in a hotel with an infinite number of rooms. In research published in the Oct. 16 *Physical Review Letters*, physicists tripled the degree of twistiness of a light beam.

Because physicists can encode data into those twisted bits of light (*SN: 7/27/13, p. 11*), the scheme creates vacancies for adding more twist-encoded data to a single beam. “It’s a trick to give yourself more bandwidth,” says study coauthor Robert Boyd, an optical physicist at the University of Rochester in New York. Light already carries data over fiber-optic cables across the globe, though not via twists.

In a 1925 lecture, German mathematician David Hilbert referenced a hypothetical hotel with an infinite number of rooms, all occupied. Yet the hotel has a permanent vacancy sign. When a guest checks in, the innkeeper has every current occupant shift up one room, leaving Room 1 vacant. The hotel even has a solution when an infinite influx of weary travelers arrive: All current guests move to twice their current room number so that only even-numbered rooms are taken.

Infinity plays a limited role in our finite world, but it is relevant in quantum physics. Atoms, for example, have an infinite number of discrete energy levels that are analogous to Hilbert hotel rooms. Drawing inspiration from Hilbert, quantum physicist John Jeffers at the University of Strathclyde in Glasgow, Scotland, and colleagues, also coauthors of the new study, devised a theoretical scheme for multiplying energy levels by a particular value, shifting them upward while creating an infinite number of vacancies.

Similarly, a beam of light (or even a single photon) can be imparted with a discrete but unlimited number of twists. Using the theoretical work as a guide, Boyd and colleagues performed an optics experiment to implement the Hilbert hotel multiplication scheme with laser light. The “room number” was the number of twists per wavelength, and the researchers chose to multiply by three rather than two.

A small liquid crystal display called a spatial light modulator imparted a particular twist in the beam — say one twist. Then a series of lenses unspooled the light, making the beam easier to manipulate. After bouncing the beam off two more modulators and retwisting it, the light had become three times as twisty: the component of the beam originally in the one-twist state now had three twists.

By transforming one twist into three, the researchers ensured that no beam component existed in the one- and two-twist states. The states were vacant, like rooms in the hotel. And like the Hilbert hotel innkeeper who moves all the guests at once, the technique multiplies all the twistiness states at once.

“It’s very clever work,” says David Andrews, a quantum physicist and chemist at the University of East Anglia in Norwich, England. But he says the connection with the Hilbert hotel is a bit of a stretch: While light can technically contain infinite levels of twistiness, practical constraints may limit total twists to a few hundred.

Fortunately, infinite twistiness isn’t required for an important practical application. Boyd’s Rochester colleague Mohammad Mirhosseini envisions performing Hilbert hotel operations to manipulate one beam into carrying information on odd-numbered amounts of twistiness and another on even-numbered twists. Then the beams could be combined, potentially doubling data capacity.