If you asked members of the public to name a concept or idea from modern physics, the most popular answer would probably be E = mc². However, the runner-up could well be Heisenberg’s uncertainty principle. This principle states that the position and linear momentum of a particle cannot be known simultaneously with arbitrary precision. More precisely, it sets a lower bound for the product of the uncertainties of these quantities: Δ xΔp ≥ h-bar/2, where Δx is the uncertainty in the position, Δp is the uncertainty in the momentum and h-bar is Planck’s constant divided by 2π.

One consequence of the uncertainty principle is that quantum objects such as electrons do not follow classical trajectories, which is in clear conflict with our everyday experience. A similar, but less well known, uncertainty relation also exists between energy and time. Now, researchers at the universities of Glasgow and Strathclyde in the UK have studied another manifestation of the uncertainty principle. In a beautiful experiment involving light beams, the team has verified the uncertainty relation between angular position and angular momentum for the first time (S Franke-Arnold et al. 2004 New J. Phys. 6 103).

Uncertain world

The angular position of a particle, Φ is simply the angle at which it is located in 2D “polar” co-ordinates. The angular momentum, L, is the rate of change of this angle with time multiplied by the mass of the particle and the square of its polar radius. A light beam consists of many photons, each of which has an angular position and an angular momentum, so the beam is best described by a probability distribution of angles. The uncertainty in the angular position, ΔΦ, is the typical spread of the angles among the photons.

In its general form, the uncertainty principle applies to every pair of “conjugate” variables (in quantum mechanics two variables, A and B, are said to be conjugate if AB – BA = ih-bar). It might therefore seem that the uncertainty relation between angular position and angular momentum is exactly the same as that between linear position and momentum. But things are not quite this simple because the angular position is restricted to values between -π and +π, whereas linear position can take any value. This has two interesting consequences. First, the uncertainty in the angular position is always finite. Second, the angular momentum is quantized and can only have certain values known as eigenvalues, whereas linear momentum, like linear position, can take any value. When the angular momentum assumes only one of these eigenvalues, the system is said to be in an angular-momentum eigenstate.

Since the uncertainty in the angular momentum, ΔL, is zero for any angular-momentum eigenstate, the product of ΔΦ and ΔL is also zero. It therefore seems as if these two conjugate variables can beat the uncertainty relation. However, when a system is in an angular-momentum eigenstate, we know absolutely nothing about its angular position. Conversely, if the angular position is confined to a narrow range of values, then the angular momentum must be distributed over a broad range of values (i.e. the system must be in a superposition of many different angular-momentum eigenstates). Mathematically, this is expressed as ΔΦΔL ≥ (h-bar/2)[1 – 2πP(π)], where P(π) is the probability that the angular position has a value of π.

It is natural to ask, which states satisfy the equality in this relation? Sonja Franke-Arnold and Stephen Barnett at Strathclyde have now calculated that the only distribution of angular position that satisfies this “minimum uncertainty” state is a Gaussian that is truncated at ±π. When the uncertainty in Φ is large, the probability of obtaining one eigenvalue of L is large and the probability for all other eigenvalues of L is small. In this case, the product of the two uncertainties is small. Conversely, for small uncertainties in angular position, the distribution of Φ is narrow and P(π) is small. Here, the angular-momentum distribution is broad and closely resembles a normal Gaussian. This situation is reminiscent of the normal position-momentum uncertainty relation, and the value of the uncertainty product ΔΦΔL is indeed close to h-bar/2.

Experimental certainty

Guided by the Strathclyde group’s theoretical work, Eric Yao and colleagues at Glasgow prepared a beam in which the angular position followed a Gaussian distribution, and then measured the resulting distribution of angular momentum. To do this, they inserted an absorber in the path of a broad light beam that let different amounts of light through depending on the light’s angular position. The absorber has a wedge shape that looks like a cake slice, and it imprints the desired angular distribution onto the intensity profile of the beam. The easiest way to make such a spatial light absorber is to use an array of liquid-crystal pixels that can be programmed with a computer. Diffraction from the absorber changes the angular-momentum distribution.

The angular momentum can then be measured using an appropriate holographic grating followed by a lens and a pinhole (the light intensity passing through the pinhole indicates the probability of finding a specific eigenvalue of the angular momentum). For a series of absorbers with different uncertainties in angular position, the observed angular-momentum distributions agree well with theory and are consistent with the uncertainty relation derived by the Strathclyde group.

This work beautifully illustrates that uncertainty relations for quantities other than position and momentum are not just theoretical constructs but can also be explored in the lab. The Glasgow experiment illustrates this uncertainty relation for the wave nature of light, which can be fully described by classical electrodynamics. But a lot of the relevant quantum physics is already captured in the present set-up because the wave description of light and the de Broglie picture are similar. The Glasgow group is now repeating the experiment for single photons, in order to investigate the angular-uncertainty relation for a genuine quantum system.