Sonja Franke-Arnold & Aidan S Arnold. American Scientist. Volume 96, Issue 3. May/Jun 2008.
At room temperature, atoms zing around at random, with average speeds of about 1,000 miles an hour. In order to study such single atoms, physicists need to slow them down. Simply condensing clouds of atom-filled gases into solids doesn’t solve the problem because the atoms are then packed too closely and interact too strongly to easily study their individual properties. The trick is to slow atoms down while keeping their density low.
Such a feat can be accomplished by cooling the atoms, and an effective way to reduce atomic temperature is with lasers. There are two main ways in which light can exert mechanical forces on atoms: the scattering force and the optical dipole force. In 1933, Otto R. Frisch performed early experiments related to the former and showed that radiation pressure by light from a sodium lamp was able to deflect a beam of sodium atoms. Generally speaking, the scattering force (and the associated light pressure) results from photons “bumping” into atoms, thereby changing their momentum-an effect described by Albert Einstein in 1917. If an atom absorbs a photon it gets a velocity kick in the direction of the laser beam. This interaction both cools and slows the atom (equivalent concepts in physics, as they both involve decreasing the energy level). When many, many photons slam into an atom in this fashion, the effect is significant. Although each individual photon’s momentum is minuscule, the rapid, repeated transfer of small amounts of momentum can still lead to atom accelerations 10,000 times stronger than gravity. But the laser must be tuned to a very specific frequency, or the photons will pass right through the atoms as if they were invisible. The frequency needed also depends on the type of atom and how fast it’s moving.
The scattering force is not surgically precise, as every absorbed photon is subsequently re-emitted from the atom in a random direction. The average force therefore acts in the direction of the light beam, slowing down (or cooling) atoms that are moving toward the laser beam; however, it is always accompanied by a random (Brownian) force that heats the atoms. The Brownian heating sets a fundamental limit below which atoms cannot be cooled with the scattering force. In order to create ultracold atoms, other cooling techniques, such as evaporative cooling, have to be employed. Evaporative cooling works in essentially the same way that blowing on hot coffee helps it cool: The hotter atoms are selectively removed and thus the average energy (and hence temperature) of the remaining atoms (or coffee) drops. If the atoms collide enough to redistribute their energy, then more “hot” atoms are created and the process can be repeated.
A different light force that does not heat the atoms is the optical dipole force, which is related to the refractive index of an atom. If the frequency of light is slightly below the atomic resonance (the frequency of maximum absorption), atoms can be attracted to bright regions of a light pattern. The light is called “red detuned” because shifting toward the red end of the visible light spectrum decreases frequency. Conversely, increasing frequency (or blue detuning) above resonance causes atoms to be repelled by the light and seek darkness. Storing atoms in the dark regions of blue-detuned light helps to minimize heating caused by photon scattering. In general the optical dipole force will dominate the scattering force if the light is far from resonance and has a high intensity.
Using careful arrangements of several lasers and magnetic fields, researchers have cooled atoms to temperatures a few millionths of a degree above absolute zero, at which point the atoms are moving at manageable speeds of around half a mile an hour. The setup can also keep the atoms in a confined space for several seconds. The cooling force in these atom traps has been dubbed “optical molasses” because of the way the atoms appear to be slogging through a viscous fluid. Work in this area, ongoing since the 1970s, won the Nobel prize in 1997 for Steven Chu of Stanford University in California, Claude Cohen-Tannoudji of the Collège de France in Paris and William D. Phillips of the National Institute of Standards and Technology in Maryland. It’s now pretty routine for physicists to stop atoms in their tracks.
With their speed under such precise control, supercold atoms are studied for more than just their intrinsic properties. Fountains of cooled cesium atoms are the basis of extremely accurate atomic clocks. The ultracold atoms might themselves be made into a type of laser for atomic lithography, where they could etch out computer chips at line widths tinier than is possible with conventional methods. Perhaps the best-known application, optical tweezers, was in fact developed in parallel with laser cooling. Arthur Ashkin of Bell Laboratories pioneered this work in the 1980s. A laser is focused to a narrow point, called the beam waist, which has a strong electric field. Micron-sized particles and atoms are attracted along the field to the point at the waist. Moving and adjusting the beam allows researchers to manipulate the particle.
The microscopic particles used in optical tweezing are still gigantic on the scale of single atoms. However, it is possible to use techniques related to optical tweezers even at the atomic level. Ashkin was also in the team that created the first all-optical trap for atoms. In atom optics laboratories around the world, optical, magnetic and other forces are used to generate gaseous Bose-Einstein condensates (BECs), a form of matter entirely different from solids, liquids or gases. Atoms are trapped and cooled down to temperatures just nanokelvins above absolute zero, making BECs the coldest substance in the known universe. Whereas atoms at room temperature move at an average speed of approximately the speed of sound, atoms in a BEC advance just a few millimeters in one second. Most importantly, all the atoms in a BEC are in exactly the same quantum state (the lowest one possible), have the same energy and oscillate together (much like the coherent or clone-like color, phase and direction of photons in a laser beam). BECs offer immense promise for precision measurements, quantum computation and nanofabrication. Pioneering BEC experiments led to the award of the 2001 Nobel prize in physics to Eric A. Cornell and Carl E. Wieman, both of the University of Colorado, and Wolfgang Ketterle of the Massachusetts Institute of Technology.
Spin It Up
Trapping atoms in just one spot can be limiting, however. If atoms could be subject to controlled movement, shunting them around could be the basis of, say, the memory in an atomic computer, as well as numerous other applications. Light gives us some options here as well, in the form of polarization. The direction, or vector, of the electric field of a light beam always oscillates in a plane perpendicular to the direction of the light’s motion. If the light is linearly polarized, the electric field vector moves up and down, tracing out a straight line when the light wave is viewed head-on. Many sunglasses have polarizing filters which block the horizontally polarized light reflected from water or snow.
If the light is made up of two linearly polarized waves, with the same amplitude but at 90-degree angles to each other, and also exactly out of phase, the waves create an electric field that travels helically along the direction of the light’s movement. In cross-section, this field looks like a circle, so the light is said to be circularly polarized. The polarization of light can easily be changed by inserting a filter called a quarter-wave retardation plate into the light beam, which slows down one component of the electric field vector and transforms linearly polarized light into circularly polarized light or vice versa. Circularly polarized light carries spin angular momentum, and its photons can impart this force to atoms, not only to trap them but also to spin them in a highly predictable fashion.
The discovery of light’s angular momentum dates back almost precisely a century, to 1909, when the physicist John Henry Poynting calculated the momentum of a light beam and also its and energy flux (the rate at which energy flows through a medium). The direction of energy flux was named in his honor as the Poynting vector. To do this calculation, Poynting applied Maxwell’s theory of electromagnetism, which was still young at the time. Poynting also reasoned that circularly polarized light should carry angular momentum, an idea that was confirmed 25 years later in the painstaking experiments of Richard A. Beth of Princeton University. But it was only recently, in 1992, that a group of physicists in Han Woerdman’s laboratory at Leiden University in the Netherlands realized that not all of light’s angular momentum is in the form of circular polarization: Apart from “spin” angular momentum, a light beam may also have “orbital” angular momentum. Since then, the orbital angular momentum of light has been investigated in many experiments, initially with classical techniques, and increasingly on the quantum level. Scientists worldwide are studying it in various contexts from optical tweezing and light-atom interactions to applications in quantum information processing.
All light carries linear momentum-each photon can be thought of as having a linear momentum that is a small fraction of its frequency. Orbital angular momentum arises if the light’s wave fronts are bent in space in such a way that the local energy flow (the Poynting vector) spirals around the propagation direction of the light. Whereas the linear momentum is associated with the “push” of light, its orbital angular momentum results in a “twist.”
In mechanics, any rotation can be split into its spin and orbital parts: Spin refers to the rotation of the particle around its own axis, whereas the orbital part relates to the rotation around a fixed reference axis. It’s the same concept as the Earth spinning on its axis once a day and simultaneously orbiting the sun once a year. For light, the same terminology was introduced, identifying circular polarization with spin angular momentum and twisted phase fronts with orbital angular momentum. However, optical spin and orbital angular momentum have a very different physical origin. Circular polarization, or spin, is characterized by the rotation of the electric field vector around the beam axis. This rotation may be anticlockwise or clockwise, usually described as “left-handed” or “right-handed” circularly polarized light, respectively. The electric field rotates once around the beam axis over a wavelength of the light, much faster than could be discerned by our eyes, which moreover are insensitive to the polarization of light.
Unlike spin angular momentum, the orbital angular momentum is associated with the phase structure of the light. Orbital angular momentum arises if the phase fronts are twisted around the direction of light propagation, looking like variations on a spiral staircase, a DNA double-helix or fusilli pasta. Light produced by lasers usually does not carry orbital angular momentum. The beam profile has a bright center and its brightness falls off with a bell-shaped intensity distribution, so in cross section the beam looks like a circle that fades out towards the edges. All crests (and troughs) of the light waves arrive uniformly across the beam profile, a bit like the waves rolling in on a long straight beach-except at a rate of a million billion waves per second.
Light carrying orbital angular momentum looks very different; its intensity profile, or the pattern it makes when it hits a surface, is shaped like a ring instead of a disk. However, the ring-shaped intensity is the result of the beam’s particular phase profile: All around the ring of light, the light waves are arriving at slightly different times relative to each other. The phase fronts cannot be twisted at any arbitrary angle of steepness, because at any point of a light wave its phase must be uniquely defined; mathematically speaking, the phase at any given angle must be the same as that after a full rotation by 360 degrees. This means that after one wave-length, the phase front can wind around the center of the beam once clockwise, or once counterclockwise, or twice in either direction, and so forth.
The associated orbital angular momentum per photon turns out to be based on the number of twists of the phase fronts per wavelength of the light (abbreviated l). This relationship was first realized in 1992 by Les Alien and his coworkers at Leiden University. Common examples of such beams are Laguerre-Gauss beams (with the ringshaped intensity profile) or Bessel beams (which look like targets in cross-section). Because of their ringlike appearance, Laguerre-Gauss modes are sometimes also called “donut” modes. At the center of these light beams the phase is not defined and the beam contains a singularity or vortex around which the helical phase fronts swirl with ever-increasing velocity toward the core region. Physics does not allow undefined phases or infinite velocities, so the intensity of any physical light beam with orbital angular momentum vanishes at the center (and you can’t tell if you are at a wave crest or trough if you are in a dead calm). At the dark core, all waves with different phases overlap and cancel each other out.
In order to convert a laser beam to a Laguerre-Gauss mode, we must modify its phase structure. The most straightforward way to achieve this is to pass it through a glass plate that refracts light and that has a varying thickness that depends on the angle around the center of the plate, thus delaying the phase at one azimuthal position with respect to that at a different angle. Alternatively one can use a type of filter made of lightbending slits, called a diffraction grating, which in this case contains forked slits with l number of prongs at the beam center. Light that is diffracted from such gratings is twisted and has the typical ring shape. Blazing the grating, or cutting the edges of the slits to very precise angles, allows most light to be directed into the first order of the resulting diffraction pattern, transforming incoming laser light without orbital angular momentum into light with l units of orbital angular momentum. The required pattern can be calculated as the interference pattern of the incoming light with the desired orbital angular momentum beam. Diffraction gratings can be simple photographic films with the correct pattern, or more conveniently written by spatial light modulators (SLMs), pixelated liquid-crystal devices that can be addressed and reconfigured by computers. By displaying different diffraction patterns, the experimenter can use the same SLM to generate any desired orbital angular momentum beam.
As there are two spin polarization states, left- and right-handed circular polarized light, the polarization is often employed as a model for a quantum bit, or qubit. Unlike the bits of normal computers, which can be either “1” or “0,” a qubit can be a superposition of varying amounts of “1” and “0,” which proves advantageous for solving certain computational problems. The orbital angular momentum of light can instead take on infinitely many discrete values and has become a popular model for a qudit, a higher-dimensional quantum bit. Both classical and qubit computers encode information in strings of 1s and 0s, whereas orbital angular momentum provides a larger alphabet in which to encode information. When l is 0, this could correspond to A, an l of 1 could be B, 2 could be C, and so on.
The effect of light’s angular momentum can be made visible by transferring it to microscopic particles. Small dielectric (insulating) particles can be trapped in the bright regions of light fields-a miniature version of a Star Trek tractor beam. The gradient force “pulls” the particle into the bright regions of red-detuned light, and particles can be held in place and manipulated at the focal position. In 1995, Halina Rubinsztein-Dunlop and coworkers at the University of Queensland in Australia transferred orbital angular momentum from a helically phased laser beam to a small ceramic particle held suspended in optical tweezers. Three years later, the same group used a similar setup to transfer spin angular momentum from a circularly polarized beam to a birefringent particle. Shortly afterward, Miles Padgett and coworkers extended these experiments in a setup that allowed the transfer of both spin and orbital angular momentum to a birefringent particle, termed an “optical wrench” (or in Britain an optical spanner). They trapped a micrometer-sized particle in the bright ring of a Laguerre-Gauss beam. The particle was rotated around the beam axis by applying a beam with an orbital angular momentum, an l value, of 1. This rotation could be stopped (or sped up) by imparting an additional spin angular momentum of -1 or 1, by changing the beam polarization from left to right circular, so that the total angular momentum either vanishes or adds to 2. This experiment proved the mechanical equivalence of the rotational forces imparted by spin and orbital angular momenta.
It may be worth noting that the tweezing force that leads to the trapping of particles has a different physical origin than the force that causes rotation. The tweezing force results from the fact that light is refracted by the particles, thus transferring linear momentum from the light to the particle, and acts in the radial direction. The force causing rotation instead arises from a transfer of linear momentum in the azimuthal direction, the direction without an intensity gradient.
In recent experiments twisted light carrying orbital angular momentum has been used to trap and manipulate atoms and even to transfer orbital angular momentum to cold atoms and BECs. Initial atomic experiments with Laguerre-Gauss laser beams relied on the spatial intensity structure of twisted light rather than on its phase structure-the orbital angular momentum of the light played no role in these experiments. A single Laguerre-Gauss beam with its dark axis cylindrically surrounded by a bright tube of light can form an “optical pipe.” For blue-detuned light, the optical dipole force attracts atoms to the dark center. In 2001 Klaus Sengstock and colleagues at Hannover University in Germany guided Bose-Einstein condensates along such light tubes, and a few years earlier Takahiro Kuga and his group at Tokyo University “plugged” a light tube at both ends with additional blue-detuned light in order to trap cold atoms. If a red-detuned Laguerre-Gauss beam is used instead, atoms can be stored in the long, bright “optical cylinder.”
What happens when we add another twisted laser beam to the mix? We are surrounded by innumerable waves of many different kinds: water waves, sound waves and the huge range of electromagnetic waves, which span the spectrum from radio waves through visible light, right out to gamma- and x-ray radiation. One thing all waves have in common is that when two waves overlap they interfere: Two wave crests at the same place and time are in phase and add constructively to form a larger wave crest. If, however, one wave’s crest and another equal-sized wave’s trough coincide, the waves cancel. When the light from two horizontally separated coherent light sources is combined on a distant screen, we observe a pattern of bright and dark vertical interference fringes where the light waves from the two sources combine with equal and opposite phases, respectively.
Now consider two Laguerre-Gauss beams traveling in the same direction. Unlike interfering water waves or plane light waves, the phase of each wave is not uniform over the beam profile but changes with azimuth angle. The two Laguerre-Gauss beams will thus be in phase at some angles and not at others.
To some extent we can illustrate this effect with an analog clock. The minute hand rotates around the clock face 12 times faster than the hour hand. The minute and hour hands are aligned at 11 distinct times during a day, such as at 1:05:27, 2:10:55 and 12:00:00. Similarly, a Laguerre-Gauss beam with an l of 1 (l1) is in phase with another with an l of 12 (l2) at 11 angles, at which points the beams will interfere constructively.
The clock hands are also furthest apart at 11 different times (such as 12:32:44 and 11:27:16), corresponding to the angles at which the Laguerre-Gauss beams are exactly out of phase, so that the combined beam is darkest. To continue the analogy for more general Laguerre-Gauss beams, one would require a funky clock where one hand rotates l1 times for every l2 rotations of the other hand (negative l values imply counter-clockwise rotation). The limitation of the analogy is that for the combined Laguerre-Gauss laser beams, all bright and dark regions can be seen simultaneously. We use such interference to generate optical ring lattices suitable for confining atoms at either the bright or the dark region within the interference pattern.
Optical lattices are (ironically) a very hot, dynamic topic in cold atoms. An optical lattice confines atoms at regularly spaced positions, similar to the lattice of atoms that exist in a pure crystal of, say, diamond. Superimposing different light beams generates an interference pattern with alternating bright and dark regions-an optical crystal. Optical lattices could provide a physical realization of a quantum register, where atoms in each light cell correspond to one quantum bit of information. Optical lattices also allow the investigation of problems commonly associated with solid-state physics but enable the experimenters to change certain parameters of their artificial crystal at will. Very recently we have investigated an optical setup that will be used to trap cold atoms in a ring lattice. A standard optical lattice is a “cube” with sides of about 100 sites, but pure crystals in the solid state can be much more extensive. Because a ring has no end or beginning point, a ring lattice is a good approximation of an infinite one-dimensional lattice, which is particularly interesting as quantum effects are strongest at low dimensions.
We have realized our optical ring lattice experimentally by superimposing two light beams that carry orbital angular momentum. Overlapping two co-propagating Laguerre-Gauss beams with opposite values of l, the beams interfere constructively at angles where their phases match and destructively in between, where they are exactly out of phase. The resulting interference pattern is a ring of 2l bright regions. Using red-detuned light, atoms can be trapped at these lattice sites by the optical dipole force. Alternatively, lattices with dark intensity regions surrounded by bright light can be generated by choosing appropriate pairs of Laguerre-Gauss beams with different orbital angular momenta. The radius of the bright intensity rings of Laguerre-Gauss beams increases with the square root of the absolute value of l, so the intensity ring of the beam with the larger orbital angular momentum has a larger radius.
At the same time, the peak intensity of a beam decreases again by the same value, the square root of the absolute value of l, and for equal power, the outer intensity ring is dimmer than the other. Complete constructive or destructive interference, however, requires equal light intensities and therefore occurs at a radius where the light intensities of the two beams balance. By choosing the orbital angular momenta of the beams so that the rings are separated by one ring width and adjusting the beam power so that ideally the rings have equal peak intensity, we can generate a bright ring having a number of dark regions equal to the absolute value of l2-l1. It is worth noting that the dark regions form at positions where the phase is singular, at vortex positions (in other words, the dark cores around which the phase fronts rotate). This ensures that the dark lattice sites are really and truly dark.
Both bright and dark optical ring lattice potentials can strongly confine atoms in the transverse direction of the beam but do not sufficiently confine atoms along the beam axis. In the case of a red-detuned bright lattice, limited axial confinement can be achieved by tightly focusing the beam, but this will usually not be sufficient to confine relatively “hot” atoms. We can solve this problem by using a magnetic trap in conjunction with the optical trap. The magnetic trap provides the strong axial confinement required to prevent atoms from leaking out of the ring lattice along the beam axis. Once confined in the ring lattice, atoms could then be cooled by evaporation. Although evaporative cooling loses a lot of atoms (typically only 0.1 percent of the initial atoms remain), it could be used to make individual Bose-Einstein condensates at the sites of a magneto-optical ring lattice.
Optical Ferns Wheels
So far we have only considered the possibility of making a static ring lattice by using two Laguerre-Gauss beams of equal frequency. If our two Laguerre-Gauss beams have different frequencies, the spacing between the wave crests in the two individual beams is different. Musicians use this effect when they are tuning their instruments. If two violinists play a similar note on untuned instruments, the result is a sound that has the average frequency of the two notes, but the volume of the sound swells and fades at a rate equivalent to the difference frequency of the two notes (called the beat frequency). One musician can eliminate the mismatch of frequencies by tightening or loosening the violin string until the beat note disappears and both instruments produce sound waves at the same frequency. The acoustic beat note describes a moving interference pattern between two sound waves. The interference of our two Laguerre-Gauss beams at different frequencies instead produces a rotation of the ring lattice from one site to the next at a rate given by the difference between the two light frequencies. The resulting interference pattern is dubbed an optical Ferris wheel as it so greatly resembles that carnival ride while it rotates.
It is quite amazing that it is possible to manually tune audible sound waves (with frequencies of 50 to 20,000 hertz) to a precision of less than 1 hertz. In the case of optical frequencies (oscillating a hundred billion times faster at about 1015 hertz) achieving a similar absolute stability is at the current experimental limit of atomic and optical clocks. Instead we actually use a single laser beam with a fixed frequency, split it into two identical parts using a beam splitter, and then tune the frequency of each beam independently by almost (but not quite) the same amount. In this way we can make ring lattices that are static, or that can rotate from one lattice site to the next with frequencies from fractions of a hertz up to millions of hertz. One could imagine that such a rotatable ring lattice could also act as a quantum memory, where the atoms at each lattice site store quantum information.
Another dynamic feature of our lattice is that it is possible to smoothly change the ring lattice to a uniform ring trap by varying the relative brightness of the two Laguerre-Gauss laser beams. Lowering the potential barriers between the individual sites allows the atoms to leave their initial lattice site and move freely around the ring. In the case of the bright lattice, one Laguerre-Gauss beam could be tuned out, so that the atoms are held in the bright intensity ring of the remaining Laguerre-Gauss beam. In the case of the dark lattice, the Laguerre-Gauss beam with the higher orbital angular momentum and therefore the outer ring could be gradually dimmed out, leaving the atoms confined from the inside by the intense region of the inner Laguerre-Gauss ring, and from the outside by the magnetic potential of the quadrupole magnetic field. If the lattice sites contain BEC (which is a superfluid, like liquid helium) the ability to rotate the lattice might enable the generation of persistent currents around the ring.
The optical ring lattice provides a versatile tool to trap and rotate atoms at bright or dark lattice positions, and it allows the transition between a ring lattice and a uniform ring trap. Estimating the required experimental parameters, we found favorable conditions to realize an optical Ferris wheel for cold rubidium atoms, and we plan to confine and manipulate an existing Bose-Einstein condensate at Strathclyde University in a dark optical Ferris wheel.
In addition to the Ferris wheel there are many other interesting trapping geometries that involve twisted light. A trap consisting of a stack (or lattice) of concentric atomic rings has been experimentally realized by Daniel Hennequin’s group at the Université des Sciences et Technologies de Lille in France. Francesco S. Cataliorti’s group at the Universitá delgi Studi di Firenze in Italy has suggested a theoretical orbital-angular-momentum beam system that would yield a stack of concentric ring lattices. Note that if one rotates traps made from counter-propagating laser beams around the beam axis, this motion is accompanied by a translation along the beam axis. There are also a variety of ways to create a single dark dipole trap, or a single dark ring trap and some of these traps can use orbital-angular-momentum light.
We have discussed orbital-angular-momentum light’s “passive” use as a dynamic storage vessel for cold atoms. However, in some recent experiments it has even been shown how to directly transfer orbital angular momentum from photons to atoms, and vice versa. While we understand how linear momentum and spin angular momentum transfer from light to atoms, the action of orbital angular momentum is less obvious. The transfer of linear momentum is linked to radiation pressure and the scattering force. Spin angular momentum can be transferred from circularly polarized light to the atom by driving transitions between different atomic spin states. The orbital angular momentum of light instead affects the motion of particles, but it has also been speculated that quadrupole transitions-transitions between atomic states that differ by two units of spin angular momentum-may cause a change in the orbital angular momentum of the light beam, as the polarization alone does not provide enough angular momentum in order to balance the conservation during these “forbidden” processes. Kozuma and colleagues experimentally prepared atoms with a mechanical orbital angular momentum and transferred this to a light beam. The reverse process is also possible: Phillips’ group was able to transfer the orbital angular momentum of light to atoms, creating a circulating state. With Bose-Einstein condensates prepared in a ring trap, they were even able to use twisted light to make a circulating persistent current in a vortex state.
One of the most famous physical laws is the Heisenberg uncertainty principle. It states that the momentum and position of a particle cannot both be known with arbitrarily high precision. In the classical world this does not matter, as measurement errors due to inaccurate tools usually exceed the tiny quantum mechanical uncertainty by far. At the level of single atoms or photons, however, the quantum mechanical uncertainty can be the main factor-and moreover, it can never be overcome. Position and momentum are not the only observables linked by an uncertainty relation. Just as one can’t know with complete precision both where a particle is and how fast it is moving, one can’t know both when an event takes place and how much energy it involves. In 2004 one of us (Franke-Arnold) and her colleagues at the Universities of Glasgow and Strathclyde investigated another uncertainty principle, between the angular momentum of a particle and its angular position.
If the beam profile of light with a fixed orbital angular momentum, say an l of 3, is obscured by a mask-for example by blocking half of the beam-the angular momentum is no longer well defined; the beam still contains light with an l of 3, but also some light at different orbital angular momenta with l’s of 2 and 4, and less light with l’s of 1 and 5, and so on. In order to know the orbital angular momentum one needs access to the full 360-degree angle of the beam profile. In fact, the angular uncertainty relation could provide some form of security if secret information were to be encoded in orbital angular momentum states. Imagine that a sender and a receiver agree on a code written in orbital angular momentum states, and the sender transmits a beam with a particular orbital angular momentum. A possible eavesdropper may try to intercept the message. However, if the eavesdropper’s equipment is not exactly aligned with the beam or does not cover an entire 360-degree section of the beam, the original orbital momentum cannot be read out properly, and instead a mixture of orbital angular momenta will be detected.
On a more fundamental level, the angular uncertainty relation proves more complicated than its linear equivalent. The uncertainty in position can span from infinitely small values (if we know where the particle is) to infinitely large values (if we don’t know at all), and so can the uncertainty in momentum. Their product, however, is given by a fixed uncertainty limit, namely half of the constant known as h. In contrast, the angular position must always be within the finite range of 0 to 360 degrees. A result of this is that the uncertainty limit no longer has a fixed value but changes depending on the angular aperture. Even if the aperture is completely open, and the light is uniformly distributed over the 360 degrees, the uncertainty in angle has its largest value at 1.81. The orbital angular momentum is then precisely defined with zero uncertainty. The uncertainty product is therefore also zero, beating the conventional limit. For very small angle apertures, the uncertainty in orbital angular momentum becomes large and the uncertainty product approaches half of h. Our experiments measuring the orbital angular momentum spectrum of light that has passed through angular masks have confirmed the uncertainty relation.
We are literally surrounded by light carrying orbital angular momentum; a close examination of light scattered from the rough surface of a wall, for example, would reveal many threads of darkness around which the light’s momentum rotates. As a research subject and even more so as an optical tool, however, the orbital angular momentum is a surprisingly new addition. Orbital angular momentum can be used in optical tweezers to rotate small particles or biological cells. It can generate exotic atom traps encompassing rings, bottles and dynamic ring lattices with tunable barriers. Orbital angular momentum can be transferred from light to ultracold atoms, inducing orbital currents and vortices, and it serves as a model for applications in quantum cryptography. Given the myriad uses for orbital-angular-momentum light, and the relative ease with which it can be generated and optimized in real time, it seems likely that its future is bright (albeit rather dark at its core and decidedly twisted).