Noncommutative Quantum Math: When Subtraction Gets You More Than You Started With!

October 11, 2007

Researchers have demonstrated something startling. Eliminating a photon from a laser beam can lead to a beam having more photons than before. This appears to be the first practical demonstration of a basic principle of quantum physics — the quantum noncommutative effect. The result may point toward novel methods of controlling light at the quantum level and to alternative methods of achieving quantum cryptography.

In mathematics, the commutative law states that, in essence, changing the order doesn't change the result. In ordinary arithmetic, for example, addition is commutative: 3 + 2 gives the same result as 2 + 3.

Marco Bellini of the National Institute of Applied Optics, in Florence, and colleagues began by sending a laser beam through a spinning glass plate, making the ordered, directional laser light behave more chaotically — like ordinary sunlight. Even though they didn't know the number of photons in the beam, they could determine the mean number of photons, using a quantum-state detector. The researchers then added, and later subtracted, a photon from the beam before calculating its mean number of photons. They also did the opposite: They subtracted first and then added a photon to the beam.

The researchers found that, under certain conditions, subtracting a photon changed the beam's quantum state, increasing the mean number of photons. "You can start with normal light, and by adding or subtracting photons you can generate any sort of light you want," Bellini told Nature.

"It's the most direct demonstration of lack of commutativity that I am aware of," said Robert Boyd of the Institute of Optics, University of Rochester. Further, "this allows you to produce different quantum states at will."

"This experiment is beautiful," Boyd added. "Maybe it is even of practical importance."

The experiment is described in the paper "Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field," published in the Sept. 28 Science.

Source: Nature, Sept. 27, 2007; Science, Sept. 28, 2007.