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226 American Scientist, Volume 96
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, physi-
cists 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 keep-
ing their density low.
Such a feat can be accomplished by
cooling the atoms, and an effective way
to reduce atomic temperature is with la-
sers. There are two main ways in which
light can exert mechanical forces on at-
oms: the scattering force and the opti-
cal 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 scat-
tering force (and the associated light
pressure) results from photons “bump-
ing” into atoms, thereby changing their
momentum—a property described by
Albert Einstein in 1917. If an atom ab-
sorbs 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 en-
ergy level). When many, many photons
slam into an atom in this fashion, the
effect is significant. Although each in-
dividual photon’s momentum is mi-
nuscule, 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 sub-
sequently 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 towards the laser beam;
however, it is always accompanied by a
random (Brownian) force that heats the
atoms. The Brownian heating sets a fun-
damental limit below which atoms can-
not be cooled with the scattering force.
In order to create ultracold atoms, other
cooling techniques, such as evaporative
cooling, have to be employed. Evapo-
rative cooling works in essentially the
same way that blowing on hot coffee
helps it cool: The hotter atoms are se-
lectively 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, at-
oms 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 de-
creases frequency. Conversely, increas-
ing 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-de-
tuned 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 abso-
lute 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 “op-
tical 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 Univer-
sity in California, Claude Cohen-Tan-
noudji of the College 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 prop-
erties. Fountains of cooled cesium at-
oms are the basis of extremely accu-
rate 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 twee-
zers, was in fact developed in parallel
with laser cooling. Arthur Ashkin of
Twisting Light to Trap Atoms
Photons carry a type of angular momentum that can
guide, trap and rotate ultracold atoms and particles
Sonja Franke-Arnold and Aidan S. Arnold
Sonja Franke-Arnold is a Research Councils UK
Fellow and since 2005 a lecturer at the University of
Glasgow. She received her doctorate from the Univer-
sity of Innsbruck in 1999. She has previously held a
Royal Society of Edinburgh and a Dorothy Hodgkin
Research Fellowship. Her interests include both
theoretical and experimental aspects of atom optics,
quantum optics and orbital angular momentum.
Aidan Arnold is a lecturer in atom optics at the Uni-
versity of Strathclyde in Glasgow. He received his
D.Phil. from the University of Sussex in 1999 for his
magnetic focusing experiments with the first Bose-
Einstein condensate (BEC) in the United Kingdom.
Aided by his Royal Society of Edinburgh Fellowship,
his team created one of the first storage rings for
BECs in 2005. Address for Franke-Arnold: Depart-
ment of Physics and Astronomy, Kelvin Building,
University of Glasgow, Glasgow G12 8QQ, Scot-
land. Internet: s.franke-arnold@physics.gla.ac.uk
2008 May–June 227www.americanscientist.org
Bell Laboratories pioneered this work
in the 1980s. A laser is focused to a nar-
row 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. Mov-
ing and adjusting the beam allows re-
searchers 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 atom-
ic 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 dif-
ferent from solids, liquids or gases. At-
oms are trapped and cooled down to
temperatures just nano-Kelvins above
absolute zero, making BECs the cold-
est substance in the known universe.
Whereas atoms at room temperature
move at an average speed of approxi-
mately 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 coher-
ent or clone-like color, phase and direc-
tion 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 Wolf-
gang Ketterle of the Massachusetts In-
stitute of Technology.
Figure 1. The intense beam of a laser may not conjure up the idea of supercooling, but in fact lasers are now routinely used to cool atoms to
temperatures very near absolute zero. Precise arrangements of lasers and magnetic fields are also used to trap and contain ultracold atoms. This
laboratory setup is used to create a type of matter called a Bose-Einstein condensate, where the atoms have been cooled into the same quantum
energy state and are indistinguishable from each other, acting as one coherent mass. In a similar setup, the authors intend to use patterns of
light interference to create bright and dark nodes, holding atoms in traps that rotate like microscopic Ferris wheels.
Pascal Goetgheluck/Photo Researchers, Inc.
228 American Scientist, Volume 96
Spin It Up
Trapping atoms in just one spot can be
limiting, however. If atoms could be
subject to controlled movement, shunt-
ing 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 the
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 am-
plitude but at 90-degree angles to each
other, and also exactly out of phase, they
create an electric field that travels heli-
cally 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 thereby 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, to not only trap them but spin
them in a highly predictable fashion.
The discovery of light’s angular mo-
mentum dates back almost precisely
a century, to 1909, when the physicist
John Henry Poynting identified the
momentum and energy flux of a light
beam (the rate at which energy flows
through a medium). To do this calcula-
tion, Poynting applied Maxwell’s theo-
ry of electromagnetism, which was still
young at the time. In fact, the direction
of energy flux was named in his honor
as the “Poynting vector.” Poynting also
reasoned that circularly polarized light
should carry angular momentum, an
idea that was confirmed 25 years later
in the painstaking experiments of Rich-
ard A. Beth of Princeton University.
But it was only recently, in 1992, that a
group of physicists in Han Woerdman’s
lab at Leiden University in the Neth-
erlands 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, orbital angular momentum
of light has been investigated in many
experiments, initially with classical
light beams, and increasingly on the
quantum level. Scientists worldwide
are studying it in various contexts from
optical tweezing and light-atom inter-
actions to applications in quantum in-
formation processing.
All light carries linear momentum—
each photon can be thought of as having
a linear momentum that is a small frac-
tion of its frequency. Orbital angular mo-
mentum 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 direc-
tion of the light. Whereas the linear mo-
mentum 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 si-
multaneously orbiting the sun once a
year. For light, the same terminology
was introduced, identifying circular po-
larization with spin angular momentum
and twisted phasefronts with orbital
angular momentum. However, optical
spin and orbital angular momentum
have a very different physical origin.
Circular polarization, or spin, is char-
acterized by the rotation of the electric
field vector around the beam axis. This
rotation may be anticlockwise or clock-
wise, usually described as “left-handed”
or “right-handed” circularly polarized
light, respectively. The electric field ro-
tates 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 associat-
ed with the phase structure of the light.
Orbital angular momentum arises if
the phasefronts 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 sec-
tion, 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 mil-
lion billion waves per second.
Light carrying orbital angular momen-
tum immediately looks very different; its
intensity profile, or the pattern it makes
when it hits a surface, is shaped like a
ŏ
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Figure 2. Light with orbital angular momen-
tum has phasefronts—places of constant
phase, such as wave crests—that are twisted
around the beam’s direction of propagation.
The phasefronts for light beams with l units
of orbital angular momentum are shown at
center over one wavelength (positive and nega-
tive values of l correspond to clockwise and
anticlockwise directions of twist, respectively).
The phase of these light beams changes l times
from 0 degrees to 360 degrees around the beam
profile, as depicted in the rainbow images at
right (crests are red, troughs are light blue).
At the beam center, phase is not defined and
the intensity vanishes, leading to the donut-
shaped intensity profiles at left. The radius of
the ring increases with orbital angular momen-
tum as the square root of the absolute value of
l. (Unless otherwise indicated, all images are
courtesy of the authors.)
2008 May–June 229www.americanscientist.org
ring instead of a solid circle. 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 phasefronts
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 at the same angle after a full
rotation by 360 degrees. This means that
after one wavelength, the phasefront can
wind around the center of the beam once
clockwise, or once counterclockwise, or
twice clockwise, and so forth.
The associated orbital angular mo-
mentum per photon turns out to be
based on the number of twists of the
phasefronts per wavelength of the light
(abbreviated as l). This relationship was
first realized in 1992 by Les Allen and
his coworkers at Leiden University.
Common examples of such beams are
Laguerre-Gauss beams (with the ring-
shaped 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 de-
fined and the beam contains a singular-
ity or vortex around which the helical
phasefronts swirl with ever-increasing
velocity toward the core region. Phys-
ics 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 phas-
es 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 straight-
forward way to achieve this is to pass it
through a glass plate that refracts light,
which has a varying thickness that de-
pends on the angle around the center
of the plate, thus delaying the phase at
one azimuthal position with regard to
that at a different angle. Alternatively
one can use a type of filter made of light-
bending 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 cut-
ting the edges of the slits to very precise
angles, allows most light to be directed
into the first order of the resulting dif-
fraction spectra, transforming incoming
laser light without orbital angular mo-
mentum into light with l units of orbital
angular momentum. The required pat-
tern can be calculated as the interfer-
ence pattern of the incoming light with
the desired orbital angular momentum
beam. Diffraction gratings can be simple
photographic films with the correct pat-
tern, or more conveniently written by
spatial light modulators (SLMs), pixelated
liquid-crystal devices that can be ad-
dressed and reconfigured by computers.
By displaying different diffraction pat-
terns, the experimenter can use the same
SLM to generate any desired orbital an-
gular momentum beam.
As there are two spin polarization
states, left- and right-handed circular
polarized light, the polarization is of-
ten 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 an-
gular 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 normal 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 momen-
tum can be made visible by transferring
it to microscopic particles. Small dielec-
tric particles can be trapped in the bright
regions of light fields—a miniature ver-
sion of a Star Trek tractor beam. The gra-
dient force “pulls” the particle into the
bright regions of red-detuned light, and
particles can be held in place and manip-
ulated at the focal position. In 1995, Ha-
lina Rubinsztein-Dunlop and coworkers
at the University of Queensland in Aus-
tralia transferred orbital angular momen-
tum from a helically phased laser beam
to a small ceramic particle held suspend-
ed 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 birefrin-
gent particle. Shortly afterwards, Miles
Padgett and coworkers extended these
experiments in a setup that allowed the
transfer of both spin and orbital angular
Figure 3. Light can carry orbital angular momentum if its phasefronts are twisted around the direc-
tion that the light is moving. The linear momentum of the light (perpendicular to the phasefronts;
shown as colored arrows) has a small component around the propagation direction, leading to an
orbital angular momentum. This image has been stretched along the beam axis to show detail; in
reality the vectors are almost straight lines This light beam has five twists, or five-fold rotational
symmetry, and resembles fusilli pasta (bottom left) with a similar three-fold rotational symmetry.
230 American Scientist, Volume 96
momentum to a birefringent particle,
termed an “optical wrench” (or in Brit-
ain, 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 or-
bital angular momentum, an l value, of
one. 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 ro-
tational forces imparted by spin and
orbital angular momenta.
It may be worth noting that the tweez-
ing force that leads to the trapping of
particles has a different physical origin to
the force that causes rotation. The tweez-
ing force results from the fact that light is
refracted by the particles, thus transfer-
ring linear momentum from the light to
the particle, and is acting in the radial
direction. The force causing rotation in-
stead arises from a transfer of linear mo-
mentum in the azimuthal direction, the
direction without an intensity gradient.
Optical Crystals
In recent experiments twisted light car-
rying orbital angular momentum has
been used to trap and manipulate atoms
and even to transfer orbital angular mo-
mentum to cold atoms and BECs. Ini-
tial 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 have
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 anoth-
er 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 elec-
tromagnetic (light) waves which span
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 will
add together (interfere): Two wave crests
at the same place and time are in phase
and add constructively to a larger wave
crest. If, however, one wave’s crest and
another equal-sized wave’s trough add
together, the waves will cancel. When
the light from two horizontally separat-
ed 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 op-
posite 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 angle. The two Laguerre-
Gauss beams will therefore be in phase
at some angles and not at others.
To some extent we can illustrate this
effect with an analog clock. The min-
ute hand rotates around the clock face
12 times faster than the hour hand. The
minute and the hour hand 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
Figure 4. Angular momentum can be transferred from light to matter. This can be demonstrat-
ed by holding a micron-sized dielectric particle (green ball) in the bright ring of a laser beam.
If the laser is circularly polarized, its electric field rotates around the beam axis; it can transfer
this spin angular momentum to the particle and make it spin around its own axis (left). If the
laser has twisted phase fronts, it can transfer orbital angular momentum, making the particle
orbit around the beam axis (right).
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Figure 5. The best way to create a steady interference pattern between two lasers is to start out with
one laser beam and split it in two, creating a pair of identical beams. The beams are then focused
onto a spatial light modulator (top right). This liquid crystal display is programmed with a pair
of different holographic patterns, one for each beam, that diffract the beams by different amounts
and give them different amounts of orbital angular momentum (here, an l of 3 on the left and
11 on the right). Overlapping the beams creates an interference pattern, here with 11 - 3 = 8 dark
regions, inside a bright intensity ring, which rotates due to the frequency difference between the
two beams. This optical Ferris wheel can then trap and rotate atoms.
2008 May–June 231www.americanscientist.org
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 ex-
actly out of phase, so that the combined
beam is darkest. To continue the anal-
ogy for more general Laguerre-Gauss
beams one would require a funky clock
where one hand rotates l1times for every
l2 rotations of the other hand (negative
l-values imply counter-clockwise rota-
tion). 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 lat-
tices suitable for confining atoms at ei-
ther 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 lat-
tices could provide a physical realization
of a quantum register, where atoms in
each light cell correspond to one quan-
tum 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 artifi-
cial crystal at will. Very recently we have
investigated an optical setup that will
be used to trap cold atoms in a ring lat-
tice. A standard optical lattice is a “cube”
with sides of about 100 sites, but pure
crystals in 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-dimen-
sional lattice, which is particularly inter-
esting as quantum effects are strongest at
low dimensions.
We have realized our optical ring
lattice experimentally by superimpos-
ing two light beams that carry orbital
angular momentum. Overlapping two
co-propagating Laguerre-Gauss beams
with opposite values of l, the beams in-
terfere constructively at angles where
their phases match and destructively in
between, where they are exactly out of
phase. The resulting interference pat-
tern is a ring of 2l bright regions. Using
red-detuned light, atoms can be trapped
at these lattice sites by the optical di-
pole force. Alternatively, lattices with
dark intensity regions surrounded by
bright light can be generated by choos-
ing appropriate pairs of Laguerre-Gauss
beams with different orbital angular
momenta. The radius of the bright in-
tensity rings of Laguerre-Gauss beams
increases with the square root of the ab-
solute 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 or-
bital 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
containing dark regions that number 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 phasefronts 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 at-
oms 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 evapo-
ration. Although evaporative cooling
loses a lot of atoms (typically only 0.1
percent of the initial atoms remain), it
JOUFOTJUZJOUFOTJUZ QIBTFQIBTF
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Figure 6. The interference of two light beams with different orbital angular momenta results
in special light patterns with a modulated intensity around the resulting ring lattice. The ab-
solute value of l (the orbital angular momentum) for the first beam, subtracted from that for
the second, gives the number of bright (top) or dark (bottom) fringes. The black and white im-
ages show beam intensity profiles, whereas the rainbow images are phase profiles. Traveling
through the rainbow once is equivalent to the phase change over one wavelength. Experimen-
tal results showing these bright and dark ring lattices are at right.
232 American Scientist, Volume 96
could be used to make individual Bose-
Einstein condensates at the sites of a
magneto-optical ring lattice.
Optical Ferris Wheels
So far we have only considered the pos-
sibility 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 violin-
ists play a similar note on untuned in-
struments, the result is a sound that has
the average frequency of the two notes,
but the volume of the sound swells and
dims at a rate equivalent to the differ-
ence 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 with
synchronized sound crests and troughs.
The acoustic beat note describes a mov-
ing 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 sta-
bility 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 identi-
cal parts using a beamsplitter, and then
tune the frequency of each beam inde-
pendently by almost (but not quite) the
same amount. In this way we can make
ring lattices that are static, or that can ro-
tate 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 quan-
tum 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. Low-
ering 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 remain-
ing 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 ver-
satile tool to trap and rotate atoms at
bright or dark lattice positions, and it al-
lows 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 ma-
nipulate an existing Bose-Einstein con-
densate at Strathclyde University in a
dark optical Ferris wheel.
In addition to the Ferris wheel there
are many other interesting trapping ge-
ometries that involve twisted light. A
trap consisting of a stack (or lattice) of
concentric atomic rings has been experi-
mentally realized by Daniel Hennequin’s
group at the Université des Sciences et
Technologies de Lille in France. Fran-
cesco S. Cataliotti’s group at the Univer-
sity of Firenze in Italy has suggested a
theoretical orbital angular momentum
beam system that would yield a stack
of concentric ring lattices. Note that if
Figure 7. Orbital angular momentum could be used for data encryption. If a receiver’s equipment
is not exactly aligned with a beam or does not cover an entire 360-degree section of the beam, the
dectector will see a mixture of orbital angular momenta instead of the original one. This theoretical
“angular uncertainty principle” (red curve) was experimentally confirmed (blue dots). A light beam
without orbital angular momentum transmitted through an angular mask gains additional orbital
angular momentum components (middle row of images). The more the angle is restricted, the
wider the distribution of angular momenta becomes (bottom row of graphs). For narrow angles,
the uncertainty product is half of h
–
(the spin angular momentum). Without any restricting aperture,
the angular uncertainty drops to about 1.81, and there is no angular momentum uncertainty.
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2008 May–June 233www.americanscientist.org
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 in-
stead affects the motion of particles, but
it has also been speculated that quad-
rupole transitions—transitions between
atomic states that differ by two units of
spin angular momentum—may cause
a change in the orbital angular momen-
tum of the light beam, as the polariza-
tion alone does not provide enough an-
gular momentum in order to balance
the conservation during these “forbid-
den” 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 cur-
rent in a vortex state.
Uncertain Angles
One of the most famous physical laws is
the Heisenberg uncertainty principle. It
states that the momentum and position
of a particle cannot be known with arbi-
trarily high precision at the same time. 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, howev-
er, the quantum mechanical uncertainty
can be the main factor—and moreover,
it can never be overcome. There are
several such pairs of observables that
are linked by the uncertainty principle:
where and how fast a particle is, when
an event happens and with which en-
ergy. 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. Imag-
ine that sender and receiver agree on a
code written in orbital angular momen-
tum states, and the sender transmits a
beam with a particular orbital angular
momentum. A possible eavesdropper
may try to intercept the message. How-
ever, if the eavesdropper’s equipment
is not exactly aligned with the beam or
does not cover an entire 360-degree sec-
tion of the beam, the original orbital mo-
mentum 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 equiv-
alent. The uncertainty in position can
span from infinitely small values (if
we know where the particle is) to in-
finitely large values (if we don’t know
at all), and so can the uncertainty in
momentum. Their product, however,
is given by the fixed uncertainty limit
of half of the spin angular momentum.
Instead, the angular position must al-
ways 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.
Instead, for very small angle apertures,
the uncertainty in orbital angular mo-
mentum becomes large and the uncer-
tainty product approaches half of the
spin angular momentum. Our experi-
ments 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 would for
example reveal many threads of dark-
ness around which the light’s momen-
tum rotates. As a research subject and
even more so as an optical tool, however,
the orbital angular momentum is a sur-
prisingly new addition. Orbital angu-
lar momentum can be used in optical
tweezers to rotate small particles or bio-
logical cells. It can generate exotic atom
traps encompassing rings, bottles and
dynamic ring lattices with tunable bar-
riers. Orbital angular momentum can be
transferred from light to ultracold atoms,
inducing orbital currents and vortices,
and it serves as a model for applica-
tions in quantum cryptography. Given
the myriad uses for orbital angular mo-
mentum 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).
Bibliography
Amico, L., A. Osterloh and F. Cataliotti. 2005.
Quantum many particle systems in ring-
shaped optical lattices. Physical Review Letters
95: 063201.
Courtade, E., O. Houde, J.-F. Clement, P. Verkerk
and D. Hennequin. 2006. Dark optical lattice
of ring traps for cold atoms. Physical Review A
74: 031403(R).
Franke-Arnold, S., et al. 2007. Optical Ferris
wheel for ultracold atoms. Optics Express
15:8619–8625.
Franke-Arnold, S., et al. 2004. Uncertainty prin-
ciple for angular position and angular mo-
mentum. New Journal of Physics 6:103.
Simpson, N. B., K. Dholakia, L. Allen and M. J.
Padgett. 1997. The mechanical equivalence
of the spin and orbital angular momentum
of light: an optical spanner. Optics Letters
22:5254.
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