Math

Physicists Realized an Infinite 'Hilbert Hotel' with Beams of Light

Infinity doesn't get much realer.

Daniel Oberhaus

Daniel Oberhaus

Hilbert Hotel realized with light. Image: Václav Potoček et al.

It's a dark and stormy night as you and your friends pull up to the front of the Infinite Hotel. You've been driving for hours and all you want is a shower and a warm bed. The sign out front of the hotel said NO VACANCY, but it's the only place around so you figured you try your luck. You glance at the lemniscate decal adorning the entrance to the hotel before pushing the door open. A bell tinkles softly, announcing your arrival.

There's an old man at the front desk and his name tag tells you he is called Hilbert.

"I am Hilbert," says the old man. "This is my hotel."

You tell Hilbert that you need a room for you and your friends. "How many rooms do you have?" you ask.

"An infinite number of rooms," replies Hilbert. "But unfortunately they're all full."

You turn to leave, but Hilbert stops you. "Wait a minute – how many friends do you have?"

"I have an infinite number of friends," you reply.

The old man takes a moment to look through his register and then picks up the phone and mumbles something into the speaker. He turns back to you with a smile. "You're in luck," he says. "We have just enough vacancies for you and your friends after all."

If you're puzzled by old man Hilbert's response, and what he possibly could've said into the phone to procure that many vacancies, you're probably not alone.

This illustration, developed by the mathematician David Hilbert in 1924, was meant to demonstrate the paradoxical nature of infinity. More recently, a team of physicists experimentally realized this thought experiment using light. Their work is described in the Physical Review Letters.

Put more concisely, the story goes like this: A hotel with an infinite number of rooms, all of which are occupied, shouldn't be able to accommodate more guests. Full house.

Yet, due to the nature of infinity, the hotel manager will actually be able to accommodate more guests. All he has to do is move all the current guests to a different room according to some rule. For example, say everyone moves to the room number that is twice the number of the room they are currently occupying (guest in room n moves to room 2n). This will open up all the odd numbered rooms to the guests and voila, you've created an infinite number of unoccupied rooms, all of them with odd numbers. And there is an quantity number of odd numbers.

Image: Miatto et al

The physicists detail two approaches to modeling Hilbert's paradoxical hotel, one of which is theoretical (on paper) and one of which is experimental (real-life). Both approaches make use of the quantum mechanical principle stating that a quantum system can be comprised of an infinite number of quantum states. These states are to represent hotel rooms.

In the theoretical model, the team uses the infinite number of potential energy levels of a given particle confined to an infinite square well. Infinite square wells are hypothetical constructs mostly meant to demonstrate the difference between classical and quantum systems. You can imagine it as an enclosure with infinitely high walls, such that a particle confined within can increase in energy forever without having enough to actually escape the well. A confined infinity.

Since the team couldn't construct a real-life infinite box to test their hypotheses, they had to get creative and find alternative ways to create the same effects as this hypothetical structure.

In order to bring their hotel to life the team made use of photons and the infinite number of orbital angular momentum (OAM) states of light. OAM is associated with the rotation of an object around a fixed axis and in the case of light beams, it refers to how OAM modes twist in space as they propagate—the OAM simply tells you by how much. By manipulating light's OAM, the team created an effect that could be interpreted as a square well that wraps around itself.

"This kind of manipulation of OAM states of light has never been done before," Filippo Miatto, a postdoc fellow at the University of Waterloo's Institute for Quantum Computing and a coauthor of the study, told Motherboard. "It could open the way for non-linear operations, which are at the core of quantum information processing."


Image: Miatto et al

Each time the team ran the experiment, they multiplied the OAM superposition by three—much like in the above story where old man Hilbert multiplied all the guest's room number by two, creating vacancies—although they also demonstrated that this multiplication could be carried out with factors of 2, 4, and so on.

In effect, the team was taking a given number of states and multiplying the amplitude of these states by three. This created an infinite number of states that weren't occupied and could thus carry information (the gaps between the red petals in the image above representing the empty states).

"Philosophically, after our findings, one could argue that the real world can accommodate for the mathematical notion of infinity in the sense of Hilbert's paradox,"Miatto added in a press release. "Or at least, the formalism of quantum mechanics allows for that."

While the ability to physically realize Hilbert Hotels with light is pretty mind blowing, the applications of this discovery are likely to remain esoteric for a while. As the team details in the study, this advance could one day prove useful in quantum and classical information processing by creating frequency gaps between several channels before combining them into a single communication channel.