A New Optical Geometric Phase

A new Optical geometric phase has been measured for the first time, by a group of physicists at Colgate University. The new geometrical phase is associated with light beams carrying orbital angular momentum. This development can be considered yet another step toward understanding and exploiting the weirdness of quantum reality for performing novel feats of computation. To see the meaning behind the new effect, we shall break the explanation into parts, considering in turn the issues of phase, orbital angular momentum in light, and then geometrical phase in light.

First, phase. Many common periodic things have phase. The orientation or phase of a minute hand on a clock is the amount by which the hand has swept around the clock face: a quarter past the hour, half past the hour, etc. Except when going into a new time zone the phase of the clock regularly returns to its original position every sixty minutes. The phase of a water wave specifies where along the wave's crest-to-trough cycle it might be at any moment. Now consider a different kind of phase. Picture a sign with an arrow on it, oriented north. Starting at the equator, and without changing its orientation, push the sign along the ground one fourth of the way around the world. Next push the sign due north until you reach the north pole, where, without changing the sign's orientation, you move directly south again to return to your starting point. Even though you will have traced a closed loop the sign will now have a westerly orientation. In other words, because of the intrinsic curved geometry of the path, a change in phase will have occurred. This kind of phase change can occur in a quantum system.

Second, orbital angular momentum. The ordinary forward momentum of a particle of light is equal to Planck's constant divided by the wavelength of the equivalent light wave. Furthermore, the light is said to possess an intrinsic angular momentum, or "spin." The spin angular momentum can be oriented by polarizers so that the electric field of the light wave is oscillating vertically up and down, or horizontally back and forth. Equivalently, if the light wave is circularly polarized (the electric field precesses in corkscrew fashion as the wave moves along) the two contrary states of the spin would then correspond to the light wave's electric field precessing clockwise (in a "right-handed" way) or anticlockwise (in a"left handed" way). For the purposes of data processing a 0 or 1 bit can be associated respectively with vertical and horizontal polarizations or, equivalently, with clockwise or anticlockwise polarizations. But what does it mean for light to have "orbital" angular momentum? What is it that orbits?

To ponder this issue, picture the electric field values for a vertical planar slice of the light beam. For vertically-polarized light, the electric field at all the points on the slice are vertically oriented. Look at the same slice at a later time and the fields are still vertically oriented. For circularly polarized light, the fields in the slice will, at a certain moment, also be oriented in the same way. A moment later, however, the electric field will have precessed a bit (from the one o'clock position, say, to the three o'clock position; another way of saying this is that the phase of the electric field will have advanced a bit) but the orientation of the field at each point on the vertical slice will be the same. With the use of special gratings one can produce an entirely different mode of light, one in which the electric field phase coils around the beam axis, and the light is said to possess an orbital angular momentum, or OAM. This condition is visualized at the following website prepared by physicists at Colgate University. This extra property of "coiled light" might be exploitable for future quantum computing. For instance, recently a group at the University of Vienna used OAM in light to create a three-dimensional entangled state, or "qutrit" (Vaziri et al., Physical Review Letters, 9 Dec 2002). Third issue: geometrical phase. When a light pulse is made to follow a closed loop path in real space, the phase of the returning beam might be slightly off from the phase of light starting off at that point. This disparity (which can result in an interference effect) can be modified by changing the path length. It can also be modified by changing the path geometry. In addition, the space does not need to be real space. When the "mode" (set of standing waves in the beam) is changed, it can also produce a phase when changing the geometry of the path in "mode space," and it is this that the Colgate physicists have measured. (see a schematic of the setup).

The change in phase that a quantum system undergoes in going around a closed path in a space of states or parameters is called a "geometrical phase," and can be measured when the light emerges from the path to form a spiral shaped interference pattern at an external detector (Galvez et al., Physical Review Letters, 23 May 2003; contact Kiko Galvez, 315-228-7205). (For further background, see Physical Review Focus item and an article on geometric phase in Physics Today, Dec 1990)