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# Spin-Statistics Theorem Survives Test

Photonics.com

Jun 2010
BERKELEY, Calif., June 28, 2010 — The most rigorous trials yet of a fundamental assumption about how particles behave on the atomic scale has been conducted at the University of California at Berkeley.

"We tested one of the major theoretical pillars of quantum field theory, the spin-statistics theorem," said Damon English, a postdoctoral fellow in the University of UC Berkeley's Department of Physics, who led the experiment. "Essentially we were asking, are photons really perfect bosons?"

A pump laser provides the energy for two dye lasers. The dye lasers are directed into a vacuum chamber where they interact with a beam of barium atoms. (Photo: Damon English/UC Berkeley)

The best theory for explaining the subatomic world got its start in 1928 when theorist Paul Dirac combined quantum mechanics with special relativity to explain the behavior of the electron. The result was relativistic quantum mechanics, which became a major ingredient in quantum field theory. With a few assumptions and ad hoc adjustments, quantum field theory has proven powerful enough to form the basis of the Standard Model of particles and forces.

"Even so, it should be remembered that the Standard Model is not a final theory of all phenomena, and is therefore inherently incomplete," said Dmitry Budker, a staff scientist in the Nuclear Science Div. of Lawrence Berkeley National Laboratory and a professor of physics at the University of California at Berkeley.

Budker has long been interested in testing widely accepted underpinnings of physical theory to their limits. In a recent issue of *Physical Review Letters,* he and his colleagues report the most rigorous trials yet of a fundamental assumption about how particles behave on the atomic scale.
#### Why we need the spin-statistics theorem

The spin-statistics theorem dictates that all fundamental particles must be classified into one of two types, fermions or bosons. (The names come from the statistics, Fermi-Dirac statistics and Bose-Einstein statistics, that explain their respective behaviors.)

No two electrons can be in the same quantum state. For example, no two electrons in an atom can have identical sets of quantum numbers. Any number of bosons can occupy the same quantum state, however; among other phenomena, this is what makes laser beams possible.

Two opposed laser beams, identical except for polarization, attempt to excite forbidden two-photon transitions in a beam of barium atoms. If photons sometimes act like fermions, every once in a while a barium atom would absorb two photons and subsequently emit a flash of light. No significant events were observed. (Illustration: Damon English/UC Berkeley)

Electrons, neutrons, protons, and many other particles of matter are fermions. Bosons are a decidedly mixed bunch that includes the photons of electromagnetic force, the W and Z bosons of the weak force, and such matter particles as deuterium nuclei, pi mesons, and a raft of others. Given the pandemonium in this particle zoo, it takes the spin-statistics theorem to tell what's a fermion and what's a boson.

The way to tell them apart is by their spin — not the classical spin of a whirling top but intrinsic angular momentum, a quantum concept. Quantum spin is either integer (0, 1, 2…) or half integer, an odd number of halves (1/2, 3/2…). Bosons have integer spin. Fermions have half integer spin.

"There's a mathematical proof of the spin-statistics theorem, but it's so abstruse you have to be a professional quantum field theorist to understand it," said Budker. "Every attempt to find a simple explanation has failed, even by scientists as distinguished as Richard Feynman. The proof itself is based on assumptions, some explicit, some subtle. That's why experimental tests are essential."

"If we were to knock down the spin-statistics theorem, the whole edifice of quantum field theory would come crashing down with it," English said. "The consequences would be far-reaching, affecting our assumptions about the structure of spacetime and even causality itself."
#### In search of forbidden transitions

English and Budker, working with Valeriy Yashchuk, a staff scientist at Berkeley Lab's Advanced Light Source, set out to test the theorem by using laser beams to excite the electrons in barium atoms. For experimenters, barium atoms have particularly convenient two-photon transitions, in which two photons are absorbed simultaneously and together contribute to lifting an atom's electrons to a higher energy state.

Dima Budker, left, is in a fermionic state, occupied by only one of himself. Many copies of bosonic Damon English, right, occupy the same state at once. (Composite of photos by Roy Kaltschmidt and Damon English)

"Two-photon transitions aren't rare," said English, "but what makes them different from single-photon transitions is that there can be two possible paths to the final excited state – two paths that differ by the order in which the photons are absorbed during the transition. These paths can interfere, destructively or constructively. One of the factors that determines whether the interference is constructive or destructive is whether photons are bosons or fermions."

In the particular barium two-photon transition the researchers used, the spin-statistics theorem forbids the transition when the two photons have the same wavelength. These forbidden two-photon transitions are allowed by every known conservation law except the spin-statistics theorem. What English, Yashchuk, and Budker were looking for were exceptions to this rule, or as English puts it, "bosons acting like fermions."

The experiment starts with a stream of barium atoms; two lasers are aimed at it from opposite sides to prevent unwanted effects associated with atomic recoil. The lasers are tuned to the same frequency but have opposite polarization, which is necessary to preserve angular momentum. If forbidden transitions were caused by two same-wavelength photons from the two lasers, they would be detected when the atoms emit a particular color of fluorescent light.

The researchers carefully and repeatedly tuned through the region where forbidden two-photon transitions, if any were to occur, would reveal themselves. They detected nothing. These stringent results limit the probability that any two photons could violate the spin-statistics theorem: the chances that two photons are in a fermionic state are no better than one in a hundred billion — by far the most sensitive test yet at low energies, which may well be more sensitive than similar evidence from high-energy particle colliders.

Two green laser beams enter a vacuum chamber from left and right and meet at the center, where they intersect a beam of barium atoms. (Photo: Damon English/UC Berkeley)

Budker emphasizes that this was "a true tabletop experiment, able to make significant discoveries in particle physics without spending billions of dollars." Its prototype was originally devised by Budker and David DeMille, now at Yale, who in 1999 were able to severely limit the probability of photons being in a "wrong" (fermionic) state. The latest experiment, conducted at UC Berkeley, uses a more refined method and improves on the earlier result by more than three orders of magnitude.

"We keep looking, because experimental tests at ever increasing sensitivity are motivated by the fundamental importance of quantum statistics," said Budker. "The spin-statistics connection is one of the most basic assumptions in our understanding of the fundamental laws of nature."

"Spectroscopic test of Bose-Einstein statistics for photons," by English, Yashchuk, and Budker, appears in the June 25 issue of *Physical Review Letters* and is available online. The research was supported by the National Science Foundation.

For more information, visit: www.berkeley.edu

GLOSSARY

- electron
- A charged elementary particle of an atom; the term is most commonly used in reference to the negatively charged particle called a negatron. Its mass at rest is me = 9.109558 x 10-31 kg, its charge is 1.6021917 x 10-19 C, and its spin quantum number is 1/2. Its positive counterpart is called a positron, and possesses the same characteristics, except for the reversal of the charge.
- photon
- A quantum of electromagnetic energy of a single mode; i.e., a single wavelength, direction and polarization. As a unit of energy, each photon equals hn, h being Planck's constant and n, the frequency of the propagating electromagnetic wave. The momentum of the photon in the direction of propagation is hn/c, c being the speed of light.
- polarization
- With respect to light radiation, the restriction of the vibrations of the magnetic or electric field vector to a single plane. In a beam of electromagnetic radiation, the polarization direction is the direction of the electric field vector (with no distinction between positive and negative as the field oscillates back and forth). The polarization vector is always in the plane at right angles to the beam direction. Near some given stationary point in space the polarization direction in the beam...

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