One of the driving forces behind the development of quantum mechanics at the start of the last century was the need to understand why atoms only emit light at certain wavelengths. Shortly afterwards quantum mechanics was applied to molecules and then to solids. Moving in the other direction, it was also applied to predict the properties of fundamental particles, notably the electron.
Quantum mechanics has been remarkably successful in all these realms. Indeed, quantum electrodynamics - the theory of how light and matter interact - is the most powerful and accurate theory in all of physics. But even more remarkable is the fact that quantum theory still continues to fascinate researchers. It might be thought that 100 years after it was developed, there would be little that we did not know about quantum mechanics. Nothing could be further from the truth. Interest in quantum mechanics - both theoretical and experimental - is probably greater now that it ever has been.
In this article we will concentrate on just one aspect of the ongoing love affair between physicists and quantum mechanics - experiments in which single atoms are trapped inside a small box or cavity containing, on average, just one photon. Atomic physicists are now able to observe the motion of a single atom in real time with high spatial and temporal resolution, to reconstruct its trajectory and to explore hitherto unknown light forces. The realization of such "single-photon optical tweezers" is opening up new possibilities in the control of the internal and external quantum states of atoms, the cooling of molecules and quantum information processing.
Atomic entrapment
The idea that an atom can be trapped by a single photon in a cavity was put forward as early as 1991 by Serge Haroche and colleagues at the Ecole Normale Supérieure in Paris and independently by Berthold-Georg Englert, then at the Max Planck Institute for Quantum Optics in Garching, and co-workers (see Haroche et al. and Englert et al.in further reading).
Both groups proposed dropping an atom into a microwave cavity where it might be trapped by the field produced by a single photon. Trapping can occur when the potential depth is larger than the kinetic energy of the atom. The depth of the potential is related to the square root of the photon's energy density in the cavity. But the energy of microwave photons is small and the volume of the cavity, which is determined by the wavelength, is large. It immediately became clear that a trap created by microwaves would be too shallow to hold an atom falling through the cavity under gravity.
The key to creating traps that are smaller and deeper is to replace the microwaves with optical photons, which have much shorter wavelengths. High-intensity visible light is now routinely used to manipulate the motion of colloidal particles, living cells and atoms, for example. These "optical tweezers" can trap objects in the focal region of a laser beam.
In addition, lasers have also been employed to slow down or "cool" atoms - an approach that has found widespread applications in fundamental and applied research. For example, exotic quantum states known as Bose-Einstein condensates, high-precision atomic clocks, and ultra-sensitive rotation and gravity sensors all employ cold atoms. Laser-cooled trapped ions are also prime candidates for an optical frequency standard or a scalable quantum computer that could, in principle, outperform conventional computers for some tasks.
All of these experiments, however, use large numbers of photons to manipulate the atomic motion, since the field of a single photon is not strong enough, in general, to trap an atom. And none of the experiments is sensitive enough to track the motion of a single atom in real time.
However, this situation has changed recently thanks to the combination of laser-cooling and trapping techniques with methods from cavity quantum electrodynamics (QED). Considerable progress has been made over the last decade in manipulating the optical properties of atoms by using cavities made of high-quality mirrors. Now, a light field inside a tiny optical cavity with highly reflective walls can ensnare a slow-moving atom.
Earlier this year Jeff Kimble at the California Institute of Technology (Caltech) and co-workers from Caltech and the University of Auckland in New Zealand, and, independently, the author's group at the Max Planck Institute for Quantum Optics (MPQ) in Garching, Germany, reported that this unique combination of techniques makes it possible to trap and track a single moving atom in an optical cavity (see Hoodet al. and Pinkse et al. in further reading). Both groups employed highly reflective mirrors to form a high-finesse optical cavity in which the light completed an almost record-breaking number of round trips. In these experiments, the cavity contained, on average, just one photon and thereby acted as a set of single-photon optical tweezers.
Detecting single atoms
Large samples of atoms can be detected using light when the energy of the beam matches the energy difference between two electronic levels in the atoms (i.e. when the light is resonant with an atomic transition). The atoms absorb light and thus reduce the flux of photons transmitted through the sample. This effect is large and can easily be measured when the sample contains at least a few thousand atoms. But the detection of just a single atom is by no means straightforward. In particular, the attenuation of the light beam due to the presence of a single atom is too small to be visible among the fluctuations or "noise" in the intensity of the laser.
The noise problem is less severe in fluorescence imaging, when single ions or atoms at rest in a trap absorb and emit photons. While this imaging technique has become routine, it is important to note that the available signal is severely limited by the rate of photon scattering and the solid angle of the detection system. Long integration times are typically required to observe the particle, which makes the detection scheme unsuitable for tracking the motion of a single atom with high spatial and temporal resolution.
Non-resonant light, however, can get round the disadvantage of the resonant-detection scheme. In this case, the single atom does not absorb or emit light, rather it shifts the phase of the incoming light wave - an effect that can be attributed to the atom's refractive index.
Of course, the refractive index of a single atom is small, but its effect is enhanced in a high-finesse cavity simply because the light travels back and forth many times between the cavity mirrors. In the most recent experiments the finesse of the cavities has approached values as high as 500 000, which means that the mirrors reflect the light about 160 000 times. In this way, the circulating light probes the atom again and again, thereby picking up a large phase shift after many round trips.
It follows that the refractive index of even a single atom in the cavity can significantly change the optical path length between the mirrors. As a consequence, the atom is able to tune the cavity in or out of resonance with the light from an external laser. (The resonant frequency or wavelength of the empty cavity is determined by the mirror separation.) For a fixed laser frequency, a moving atom therefore induces changes in the intensity of the light transmitted through the cavity - an effect that can easily be measured when the cavity resonance is narrow.
Resonant light can also be used to observe an atom in the cavity. In this case, the refractive index does not change but the amount of absorption of the light is large. This absorption decreases the transmission of light through the cavity and increases the reflection - a surprisingly large effect due to just a single atom in the cavity. Such an effect was first observed in 1996 by Hideo Mabuchi and co-workers at Caltech for single atoms falling slowly through a high-finesse cavity (see further reading).
In our experiment at the MPQ, we first collect rubidium atoms within a trap and then cool them with a combination of magnetic and optical techniques (figure 1). The atoms are then thrown upwards into a Fabry-Perot cavity using lasers, which create a so-called moving molasses. The cavity is placed at the turning point in the "atomic fountain" and is illuminated with a weak light beam from a diode laser. This light forms a standing wave between the two mirrors due to the multiple reflections, with nodes (i.e. minima in intensity) located at the mirror surface. The slow speed with which the atom passes through the cavity means that the atom can be observed for a relatively long period of time by recording the intensity of the light transmitted through the cavity. The atom can be detected by its influence on either the absorption or refractive index (figures 1b and 1c, respectively).
Cavity quantum electrodynamics
But what is the optimal intensity of light needed to detect single atoms? Intuitively, one would expect that the signal-to-noise ratio increases with the intensity of the illuminating laser, thus making a powerful laser beam more useful than a weak one. However, a strong laser beam can easily excite the atom into a higher-energy state where it loses its ability to absorb more light - an effect known as saturation. At this stage the atomic medium becomes transparent.
Saturation also changes the refractive index of the atom. And for sufficiently high-intensity lasers, this refractive index approaches that of a vacuum. Under this condition, the atom can no longer shift the phase of the light wave. Saturation makes it harder for single atoms to be detected via absorption or changes in the refractive index of the cavity when the intensity is above a certain value.
But just how large is this upper limit on the intensity? For the caesium and rubidium atoms in the experiments at Caltech and the MPQ, saturation occurs at modest intensities. As the intensity is proportional to the number of photons per cavity volume, fewer photons are needed to saturate the atom as the size of the cavity gets smaller. In the recent experiments, the spacing between the mirrors is as small as 10 microns. An atom in such a tiny cavity can become saturated even when there is less than one photon present, on average, which explains why power levels of about 1 picowatt (10-12 W) - corresponding to about one cavity photon - are used in these experiments.
The saturation problem is particularly severe in the case when the light is in resonance with an atomic-transition frequency. For non-resonant light, more photons are needed to saturate the atom, thereby relaxing the constraints on the light intensity.
What happens when the light is intense enough to saturate the atom? In this case, the atom spends a significant fraction of its time in the excited state. It can return to the ground state either by spontaneous emission or when it is stimulated by the light field in the cavity - a much faster process. When the intensity of this light field is large, the atom is far more likely to emit a photon via stimulated emission.
In a small cavity, a single-photon field is intense enough to stimulate the decay of an excited atomic state. Amazingly, the photon does not need to be in the cavity before the emission begins. Spontaneous emission leads to a photon in the cavity, which stimulates its own emission. As a consequence, an excited atom will radiate its energy into the cavity, rather than into the free-space continuum outside the cavity.
If the finesse is large, the photon is stored in the cavity and is periodically absorbed by the atom and re-emitted into the cavity many times before being lost into the environment outside the cavity. Such novel oscillatory radiation properties are typical of the so-called strong coupling regime of cavity QED, where the coherent coupling of a single atom to a single photon makes spontaneous emission a reversible process. These radiation properties have previously been investigated by many groups worldwide, but the motion of an atom under these conditions can only now be explored with the new generation of cavity-QED experiments.
Light force
Radiation pressure is probably the best known of the forces that light can exert on an atom. In this case, an atom absorbs resonant light and receives a momentum kick in the direction of the laser beam. Although the atom's momentum changes again when it spontaneously emits a photon, this second kick is in a completely random direction and therefore averages to zero after many absorption-emission cycles.
Induced transitions, on the other hand, lead to a so-called dipole force. This force can be understood classically by noting that the electric field of the driving laser induces a mechanical oscillation of the atom's electron. The oscillating dipole moment that is produced experiences a force in a light field with an intensity gradient, such as a standing wave.
The sign of this force depends on the "detuning" of the laser with respect to the atomic-transition frequency. For example, when the laser frequency is lower than the atomic frequency, the induced atomic dipole oscillates in phase with the driving laser field, and the atom is attracted towards regions of high intensity just like a small piece of paper is attracted towards an electrically charged object. Hence, the dipole force can trap particles in the focal region of a "red-detuned" laser beam. For a "blue-detuned" laser (i.e. when the laser frequency is higher than the atomic-transition frequency), the dipole oscillates out of phase with respect to the laser, so the atom is repelled from the high-intensity regions.
Inside a cavity, the radiation properties of the atom change, with dramatic consequences for the forces that the light can produce. New effects can be expected for a moving atom because it induces position-dependent changes in the field intensity inside the cavity. In 1997, for example, Peter Horak and co-workers at the University of Innsbruck in Austria suggested that an atom could be cooled while moving through the nodes and antinodes (i.e. the minima and maxima) of a standing-wave cavity, similar to the one in figure 1a.
To explain this cooling mechanism and illustrate why the cavity plays an essential role, let's consider the situation where the strong coupling of the atom at an antinode enhances the intensity of the light field in the cavity, as it did in figure 1c. In this case, the laser is red-detuned with respect to the atom so that the dipole force attracts the atom towards the antinode. Hence, a moving atom decelerates when approaching the neighbouring node. When the atom reaches that node, its coupling with the cavity mode vanishes and the intensity of the light field decreases. Consequently, the atom moves in the dark when it approaches the next antinode and gains little kinetic energy, certainly not enough to compensate the previous loss.
As a result, the atom slows down simply because the field inside a high-quality cavity cannot adjust fast enough to the atom's motion. Unlike conventional laser cooling in which the atoms slow down by spontaneously emitting photons, the dissipative mechanism in cavity cooling involves the loss of photons from the cavity. Using this cavity-mediated "friction force", it might become possible to cool molecules, for which standard laser-cooling techniques fail, as was emphasized earlier this year by Vladan Vuletic and Steven Chu of Stanford University in the US (see further reading).
Cavity-mediated cooling is interesting because it might complement other techniques that have recently been developed to trap molecules, in particular by Hendrick Bethlem and co-workers at Nijmegen University in the Netherlands.
However, as well as changing the intensity of the intra-cavity field, an atom that periodically exchanges energy with the cavity also causes fast fluctuations in the amplitude and phase of the light field. As the trapping potential is determined by the light field within the cavity, these variations lead to fluctuations in the light force. These, in turn, affect the momentum of the atom in a random way, typically heating a cold atom by increasing its velocity.
Atom-cavity molecules
A spectacular feature of the cavity-QED scheme is that the trap is deep enough to hold a laser-cooled atom even when the cavity contains only a single photon. Trapping with a single photon can occur in a small cavity because the electric field per photon, and hence the light force per photon, is large.
But one more trick is needed to capture the atom in the photon's dipole potential: the potential must not be turned on before the approaching atom has reached the centre of the cavity. Otherwise, the atom falling into the trap from one side would escape out the other side in the same way that a marble rolled into a bowl would just roll out again without being trapped. Switching the potential on at exactly the right moment is helped by the fact that we can now observe the atom's position in a cavity field containing, on average, less than one photon.
The experimental signature of an atom trapped successfully in a cavity by this author's group at the MPQ is shown in figure 2. As the atom enters the cavity, it causes the transmission of light from an external laser to increase, thereby triggering a switch that increases the power of the driving laser. When timed properly, the atom is trapped in an antinode of the standing-wave dipole potential for up to a few milliseconds - about ten times longer than it would remain in the cavity without switching.
The large oscillations that are evident in the transmitted intensity reflect the motion of the trapped atom. In particular, the transmission is large when the atom is at the centre of the cavity, and it decreases when the atom moves away from the cavity axis.
At first sight, trapping atoms with single photons in a cavity seems to be similar to trapping atoms with laser beams in free space - with the exception that the intensity enhancement in a cavity allows us to use weak lasers. However, the strong atom-cavity coupling requires a conceptually different description. This can be understood by borrowing a simple picture from chemistry.
Just as the two protons in a hydrogen molecule can be surrounded by a symmetric (i.e. binding) or anti-symmetric (anti-binding) electron wavefunction, in the atom-cavity system the atomic dipole moment can oscillate in phase with the light field (binding) or out of phase (anti-binding).
The two states that characterize the atom-cavity "molecule" both contain one quantum of energy that can oscillate between the atom and the cavity. This quantum is therefore shared by the atom (as electronic excitation) and the cavity (as a photon), just as the electron in the hydrogen molecule is shared by both protons.
This sharing means that atom trapping can also lead to photon trapping. In this case, the presence of an atom with a long-lived excited state can prolong the time that the photon remains in the cavity.
Reconstruction of atomic trajectories
Atomic physicists can now work backwards and calculate the classical trajectory of the atom by measuring the light passing through the cavity. This is possible because the transmitted light depends on the coupling between the atom and the cavity, which in turn depends on the atom's position.
In the Caltech experiment, the large atom-field coupling confines the atom strongly to one antinode so that its motion is restricted mainly to the plane perpendicular to the cavity axis. This motion is expected to be regular, with little perturbation expected from spontaneous emission. We can therefore assume that the angular momentum of the atom around the cavity axis hardly changes during one revolution. This conservation of angular momentum means that we can identify a constant of the motion.
The two-dimensional orbit - apart from the sign of the angular momentum and the specific antinode in which the atom is confined - can be reconstructed from the data using an algorithm based on classical equations of motion. The reconstruction algorithm has been tested by applying it to the signals obtained from a simulation of the atom's motion. Indeed, Christina Hood and co-workers at Caltech have found that the spatial resolution of such inferred trajectories is typically around 2 microns on a 10 microsecond timescale (figure 3).
Our group at the MPQ has also performed simulations to explore the motion of an atom in a cavity. The trapping potential is weaker in the MPQ experiment and, hence, the atomic motion is more strongly perturbed by spontaneous emission (figure 4).
The simulations also indicate that the trapped atom sometimes flies to another, distant, antinode, making the motion truly three dimensional (figure 5). This motion is due to two different, but equally important, mechanisms. First, the atom is heated out of an antinode due to fluctuations in the trapping potential. Next it becomes trapped in another antinode because the cavity-mediated friction force, which is proportional to the atom's velocity, cools the moving atom.
Experimental evidence for long atom flights comes from the measurement of the intensity fluctuations of the light transmitted through the cavity. The transmission is large when an atom is near an antinode and falls when the atom is near a node, thus providing valuable information about the atom's position. In particular, an atom moving along the cavity axis will periodically modulate the transmission. We have found that the intensity is noisy, in general, but occasionally it oscillates in a periodic fashion before becoming random again. According to our interpretation of this behaviour, each peak in the light intensity is due to the strong coupling of the atom to each antinode it passes, until it settles down at a distant antinode.
Looking ahead
The same techniques that allow us to measure the trajectory of an atom in a cavity could be adapted to investigate the dynamics of single molecules as they undergo chemical reactions or biological processes. Another exciting possibility is to extend the techniques developed in different areas of science and engineering in which the state of a system is monitored and appropriate feedback loops to control the state are applied. Chemical reactions, for example, can be controlled in a coherent way using suitably tailored ultra-short laser pulses. These pulses are optimized in consecutive experiments, but are always applied to molecular systems that have been prepared in identical ways. The new generation of atom-cavity experiments, however, allows us to investigate feedback loops applied to the same system over and over again without the need to prepare the system in the same initial state each time the experiment is carried out. In addition, such feedback experiments open up the exciting possibility of being able to precisely control the motion of an atom within a cavity according to the laws of quantum mechanics.
Feedback might also allow an atom in a cavity to be cooled to low temperatures. By applying corrective forces to the atom - a variant of the "stochastic cooling" technique developed to cool particles stored in high-energy accelerators - one might be able to cool the atom into the region where the quantum-mechanical nature of atomic motion will become important. At this stage, an atom can no longer be treated as a point-like particle moving along a classical trajectory. Instead, it must be thought of as a wave packet that can be observed continuously in space. According to Heisenberg's uncertainty principle, the momentum of the wave packet will be altered each time we localize the atom. Such measurements at the quantum limit will be a challenge in future experiments.
Another interesting situation arises when two or more atoms reside in the cavity simultaneously. In this case the photon emitted by one atom is stored in the cavity, absorbed by the other atom, then re-emitted into the cavity where it can be re-absorbed by the first (or even a third) atom. Hence, the atoms are not independent of each other. Instead, the common field in the cavity establishes a long-range interaction between the atoms and one can expect co-operative effects in the motion of several atoms. For example, if the field in the cavity is turned on when one of the atoms moves from an antinode to a node, then it will affect the motion of the other atoms.
The first evidence for such a co-operative effect was observed earlier this year by Peter Münstermann and co-workers in the author's group. We measured the spatial distribution of the atoms over the nodes and antinodes of the cavity field by recording the light transmission through the cavity as a function of frequency. The observed data could only be explained by taking into account the full long-range mechanical interaction between the atoms.
A system with one or more individual atoms at rest and strongly coupled to a single mode of the electromagnetic field is ideal for testing fundamental concepts of quantum computing and quantum information processing (see Physics World 1998 March). Indeed, Scott Parkins, now at the University of Auckland, and collaborators at the JILA Laboratory in Boulder, Colorado, and Caltech first proposed this system as a highly efficient quantum interface in 1993. Using the strong coupling of an atom to a single photon, it should be possible to map a quantum bit at rest from an atomic medium onto a propagating light field, and vice versa. In other words, this scheme could allow quantum information to be sent from one place to another. The first experimental results in this direction were obtained very recently by Markus Hennrich and co-workers at the MPQ. Moreover, two atoms in the cavity should make it possible to realize a "controlled NOT gate", the elementary building block of a quantum computer.
Cavity-QED experiments with single atoms and optical photons look certain to provide a rich source of physics for many years to come, and could launch a raft of future applications in both the physical and life sciences. Quantum mechanics is assured a bright future for many more years to come.
