The simplest model in quantum optics deals with a single two‐level atom interacting with a single mode of the radiation field. This ideal situation is implemented in cavity quantum electrodynamics (QED) experiments, using high‐quality microwave or high‐finesse optical cavities as photon boxes. It provides a fruitful test bench for fundamental quantum processes and a promising ground for quantum information processing.

29.1 Introduction

Most experiments in atomic and optical physics dealing with light–matter interactions involve a large number of atoms interacting with laser fields made up of a large number of photons. The simplest situation, however, involves a single atom interacting with one or just a few photons. Achieving this situation and making it available for applications are the aims of cavity QED.

The history of cavity QED started, about 50 years ago, with a seminal remark by Purcell (1). He realized that the radiation properties of an atom are not a fundamental property of the atom itself. Instead, they can be changed by controlling the boundary conditions of the electromagnetic field with mirrors or cavities. Cavity QED experiments initially measured modifications of spontaneous emission rates or spatial patterns in low‐quality cavities. They evolved to higher and higher atom–cavity couplings and photon storage times. Most of them are now in the so‐called regime of “strong coupling,” in which the coherent interaction of a single atom with one photon stored in a very high‐quality cavity, a modern equivalent of Einstein's photon box, overwhelms the incoherent dissipative processes. The most prominent effect in this regime is that a photon emitted into the cavity can be reabsorbed by the atom. The usually irreversible process of spontaneous emission therefore becomes reversible – a remarkable cavity QED effect!

In principle, cavity QED experiments implement a situation in a very simple manner that their results can be cast in terms of the fundamental postulates of quantum theory. They are thus appropriate for tests of basic quantum properties: quantum superposition, complementarity, or entanglement. In the context of quantum information processing, the atom and the cavity are long‐lived qubits, and their mutual interaction provides a controllable entanglement mechanism – an essential requirement for quantum computing or teleportation applications. Cavity QED is therefore a fertile ground for quantum information processing. In addition, the ability to manipulate mesoscopic fields, containing a few to a few tens of photons, made it possible to explore the fuzzy boundary between the quantum and the classical worlds, unveiling the decoherence mechanisms that confine the quantum weirdness at a microscopic scale.

Cavity QED comes in two flavors: microwave and optical. The novel regime of strong coupling was first achieved in microwave cavities, but is now also achieved in optical cavities. Although both situations share a similar underlying physics, they are nevertheless quite different and, in fact, have complementary features. In the microwave domain, highly excited “Rydberg” states interact with rather large superconducting millimeter‐wave cavities. Dissipation is extremely low, and the pace of the atom–field entanglement process is slow. An exquisite degree of control is reached, making it possible to tailor complex multiqubit entangled states. In the optical domain, low‐lying atomic levels interact with submillimeter‐sized optical cavities at room temperature. The interaction is much faster, as is the dissipation. This, however, turns out to be an asset: optical photons can efficiently be coupled in or out of the cavity. Optical cavity QED thus provides a natural and essential interface between flying photonic qubits for the transmission of quantum information and stationary atomic qubits for the storage of quantum information.

This chapter gives a brief introduction into cavity QED, both microwave and optical. It highlights the basic properties, discusses some selected achievements, and mentions a few perspectives for quantum information processing. It does not aim to give full account of all experimental developments and theoretical concepts. Instead, it concentrates mainly on some recent experimental work performed at the Laboratoire Kastler Brossel and the Max Planck Institute of Quantum Optics. More information can be found in broader reviews (2–5) and popular accounts (6).

29.2 Microwave Cavity Quantum Electrodynamics

In order to reach the strong coupling regime, a cavity QED experiment must combine large atom–field couplings with long atomic and field lifetimes. The longest photon storage times, in the 1 ms to 1 s range, are obtained in the millimeter‐wave domain (a few tens of gigahertz) with photon boxes made of superconducting materials cooled down to cryogenic temperatures. They have sizes comparable to the wavelength and provide a high‐field confinement, essential to increase the atom–field coupling. The ideal tools for cavity–field manipulations are Rydberg atoms (7). Here, the valence electron of an alkali atom is promoted to a very excited level, with a large principal quantum number N. The diameter of these giant atoms is 0.2 µm for N = 50, the typical size of a large virus or a small bacterium! Such atoms are huge antennas, strongly coupled to the millimeter‐wave fields. The “circular” atoms, realizing Bohr model's circular orbits, are particularly interesting due to their very long lifetimes (30 ms for N = 50). These levels can be counted with a large efficiency by field ionization. This detection is, moreover, state‐selective, measuring with precision the final quantum number.

Figure 29.1 presents the scheme of a cavity QED experiment using circular Rydberg atoms. For a review and additional information, see (8). Laser and microwave excitation of an atomic beam, effusing from oven O, prepares in box B one of the states |e〉 or |g〉 (N = 51 or N = 50). A reasonable approximation of single‐atom samples is obtained by preparing much less than one atom on average. When an atom is finally detected, it was most probably travelling alone. Before entering B, the atoms are velocity‐selected by standard laser techniques. The state preparation being pulsed and performed at a precise location, the position of an atom at any time during its 20 cm transit through the apparatus is well determined. Selective transformations can thus be applied at will on all atoms crossing the setup during an experimental sequence. This individual addressing is essential for quantum information‐processing experiments. The atoms, very sensitive to microwave fields, are in a cryogenic environment, cooled below 1 K and shielded from the room‐temperature blackbody background. They interact with the superconducting cavity C, nearly resonant with the transition between |e〉 and |g〉 at 51 GHz. An electric field is applied across the cavity mirrors. It can be used to tune, via the Stark effect, the atomic frequency in or out of resonance with the cavity mode, with an excellent time resolution. The atom–field evolution can be “freezed” suddenly by a large field. With moderate field amplitudes, the interaction can be tuned at will from the resonant to the dispersive regimes. The atoms are finally detected in the field ionization counter D, whose efficiency, greater than 80%, provides a nearly ideal qubit readout (9). A classical source S, coupled to C, can be used to fill the cavity with a mesoscopic quasiclassical field, with a well‐controlled phase. Its amplitude can be adjusted from a microscopic value, corresponding to a fraction of a photon on average, to a macroscopic one, with a few tens of photons. The resonant interaction of the atom with classical fields in the zones R ₁ and R ₂, sandwiching C and fed by the source S′, realizes single‐qubit gates in quantum information terms. The atom can thus be prepared in any state before entering C. The detection by D, in the {|e〉, |g〉} basis, after the gate operation performed by R ₂, amounts to measuring the atomic qubit in a completely adjustable basis. This provides a full analysis of the final atomic state, an essential ingredient to assess the fidelity of the quantum processes taking place in C.

Most quantum entanglement manipulations realized so far with this setup rely on the resonant atom–cavity interaction. The simplest situation is an atom in the upper state, |e〉, entering the empty resonant cavity (vacuum state |0〉). The initial quantum state |e, 0〉 is degenerate with |g, 1〉 representing an atom in the lower state with one photon in C. The atom–field dipole interaction couples these states, and the atom–cavity system thus oscillates between them in a “vacuum Rabi oscillation.” Note that no evolution takes place when the initial state is |g, 0〉 (the atom in the ground state and empty cavity) since there is no excitation to exchange. Figure 29.2 presents an experimental vacuum Rabi oscillation. The probability P_e for detecting the atom in |e〉 is plotted as a function of the atom–cavity interaction time, t _i. The observation of four 20 µs periods shows that the coherent atom–cavity interaction dominates dissipative processes, fulfilling the strong coupling condition. This oscillation is a reversible spontaneous emission process. The atom in |e〉 emits a photon. When the emission occurs in free space, the photon escapes at light velocity and is lost. Ordinary spontaneous emission is irreversible. Here, the emitted photon remains trapped in C, ready to be absorbed again by the atom. In the strong coupling regime, spontaneous emission is a reversible process!

Oscillatory spontaneous emission is at the heart of an interesting quantum device, studied mostly in Munich: the micromaser (see the chapter by Raithel et al. in (2) and (10–12)). A continuous stream of single Rydberg atoms crosses the cavity. The competition between cumulative atomic emissions and cavity damping results in the build up of a mesoscopic field, containing up to a few tens of photons. The cavity damping time is so long that the maser action is sustained, close to threshold, even though the average time interval between atoms is much greater than their transit time through C. The micromaser operates thus with much less than a single excited atom at a time, a regime in which quantum effects are of paramount importance (13,14). The Rydberg atom–cavity coupling is so large that these remarkable micromasers can even operate on a two‐photon transition, a rather exotic kind of quantum oscillator (15).

The vacuum Rabi oscillation provides elementary stitches to knit complex entangled states. Three atom–cavity interaction times are particularly interesting. They are depicted by black circles in Figure 29.2. After a quarter of a period (π/2 pulse), the atom and the cavity are in a coherent superposition of |e, 0〉 and |g, 1〉 with equal weights. This is an entangled state, similar to that of the spin pair used to discuss the EPR (Einstein–Podolski–Rosen) situation illustrating quantum nonlocality. The atom–cavity entanglement lives as long as the photon, about a millisecond. This time is much longer than the mere 5 µs required for the entanglement creation, making complex entanglement‐knitting sequences possible. Half a period of the quantum Rabi oscillation (π pulse) corresponds to an atom–cavity state exchange. An atom entering the empty cavity in a superposition of its energy states always ends up in |g〉, leaving in C a coherent superposition of the zero‐ and one‐photon states. In quantum information terms, the qubit carried by the atom is copied onto the cavity. The process is reversible. An atom entering C in the lower level |g〉 for a half‐period interaction acquires the quantum information stored in the cavity, which is left in the vacuum state. This operation does not create atom–cavity entanglement, but is essential since the cavity field is not directly accessible in these experiments, in contrast to optical cavity QED situations (see below). The cavity mode is initialized with the help of properly prepared atoms. All the information on the cavity mode is retrieved by probe atoms. Finally, a full Rabi oscillation period (2π pulse) drives the atom–cavity system back to its initial state, albeit with a change in the state sign, reminiscent of the π‐phase shift experienced by a spin‐1/2 system undergoing a 2π rotation in real space. The same phase shift occurs when the initial state is |g, 1〉, the atom transiently absorbing the photon and releasing it. Note again that |g, 0〉 remains invariant. The state phase shift is thus conditioned on the states of the atom and the cavity. The 2π pulse implements a conditional dynamics, the building block for a quantum gate.

Combining these transformations, it is now possible to realize quantum information‐processing sequences of increasing complexity (8). In a quantum memory experiment (16), a qubit carried by a first atom is copied onto C by a π quantum Rabi pulse, stored for a while as a superposition of zero‐ and one‐photon states, and later acquired by a second atom undergoing another π‐pulse. An EPR atomic pair is created by entangling a first atom with C (π/2 quantum Rabi pulse). The cavity state and, hence, its entanglement with the first atom is then copied onto a second atom by a π‐pulse. Quantum correlations observed between the atomic states for different detection basis settings assess the coherence of the process (17). Two nondegenerate modes of the cavity are entangled in a similar way, through their successive resonant interactions with a single atom (18). A full‐fledged quantum gate uses the full Rabi period (19). The atomic qubit is coded onto the transition between |g〉 and another level |i〉 (circular level with N = 49). When the system is initially in |g, 1〉, it undergoes a π‐phase shift. All other levels remain unchanged (|i〉 is not in resonance with the cavity and |g, 0〉 does not evolve). This is the truth table of a quantum phase gate acting on the atom and the field mode. Two single‐qubit gates performed in R ₁ and R ₂ transform it into a CNOT gate, conditioning the atomic state on the cavity mode. In this interesting situation, the outgoing atomic state reveals the presence of a single photon in the cavity. When the photon is “seen” by the atom, it stays in the cavity (it is first absorbed by the atom, but then re‐emitted). This quantum nondemolition detection (20) is quite different from a standard photodetection in which the photon is destroyed. In other words, the same photon can be detected twice or more!

The most complex quantum‐information sequence realized so far is the creation of a three‐qubit entangled state (21). The cavity is entangled with a first atom, as in the first step of the EPR pair generation above. A second atom then enters the mode and realizes a quantum phase‐gate operation, instead of a cavity mode readout. This atom gets entangled with C and, hence, with the first one, completing the three‐qubit entanglement. The quantum correlations between these qubits are then measured. A third atom is involved, which reads out the field state. Altogether, the production and analysis of this entanglement involve four qubits, three one‐qubit gates and three two‐qubit ones. It is still among the most complex sequences realized with individually addressed qubits.

In these experiments, the entanglement fidelity is mainly limited by cavity damping, reaching 54% for the three‐qubit entanglement. A promising quantum gate gets rid of this limitation (22). Two atoms, one in |e〉 and one in |g〉, interact simultaneously with the nonresonant cavity. The first atom virtually emits a photon in C, which is immediately absorbed by the other. This cavity‐induced coherent “collision,” reminiscent of the resonant van der Waals interaction in free space, creates a two‐atom entangled state and provides the conditional dynamics of a quantum gate. Since the photon is only virtually stored in C, the gate fidelity is, to first order, impervious to cavity losses or residual thermal photons. This scheme is thus very promising for the implementation of simple quantum algorithms (23) with moderate‐quality cavities at finite temperatures.

Another remarkable feature of these experiments is the ability to manipulate in C mesoscopic fields, made up of a few to a few tens of photons, which are useful tools to explore the quantum–classical boundary. These mesoscopic objects can be entangled with a single atom crossing the cavity, as the famous Schrödinger cat gets entangled with a single radioactive atom. A nonresonant atom, in the dispersive regime, cannot absorb or emit a photon, but is a piece of transparent dielectrics, whose index of refraction transiently shifts the cavity frequency. The atom–cavity interaction then results in a classical phase shift for the cavity field, with opposite values for an atom in level |e〉 or |g〉. An atom in a state superposition then prepares a mesoscopic superposition state involving two field phases at the same time, a situation closely linked with the Schrödinger cat, suspended between life and death in quantum limbs. The slow relaxation of the cavity makes it possible to study in “real time” the decoherence mechanism (8) transforming this quantum superposition into a probabilistic alternative, the transition being faster and faster when the cat's size increases.

The resonant atom–field coupling also involves such bizarre states. An atom interacting with a very large field undergoes a trivial Rabi oscillation between levels |e〉 and |g〉 and leaves the field unchanged. In a mesoscopic field, containing only a few tens of photons, the situation is much more interesting. Then, the photon number graininess results in an atom–field phase entanglement (24,25). The field is separated into two phase components, rotating in opposite directions. A phase distribution measurement, based on a homodyne method, directly reveals this separation (see Figure 29.3). In other words, an atom at resonance is in a quantum superposition of two states with very large refractive indices, an utterly nonclassical result (the index of refraction of a classical charged oscillator at resonance is 1). The resonant interaction prepares large coherent cat states, which will be fantastic tools for new decoherence studies. They are important for fundamental quantum mechanics issues and also because decoherence is the most serious obstacle on the road toward practical quantum computation. The direct determination of the cavity–field Wigner function (26), which provides a complete and pictorial insight into the cavity–field quantum state, will make it possible to put our understanding of decoherence under close scrutiny.

29.3 Optical Cavity Quantum Electrodynamics

All cavity QED experiments are characterized by three physically distinct timescales. One is the period of the oscillatory exchange of a single energy quantum between the atom and the cavity, the vacuum Rabi time; see Figure 29.2. A second time is the transit time of the atom through the cavity. The third time comes from the coupling of the combined atom–cavity system to the environment and is determined by the photon lifetime inside the cavity and the atomic lifetime due to spontaneous emission into directions not supported by the cavity.

In principle, these three timescales can be arbitrary, making the description of an experiment rather tedious. Cavity QED, however, achieves the ideal situation in which these timescales can differ by several orders of magnitude. The distinct hierarchy is the key ingredient for coherently controlling the system at the level of single atomic and photonic quanta. In the microwave domain, it ensures that different atoms passing the cavity one after the other interact with essentially the same cavity field, thereby “seeing” the previous atom. In the optical domain, the timescales follow a different hierarchy. While in the regime of strong coupling, the vacuum‐Rabi period is still shorter than the lifetimes of both the cavity and the atom, the transit time can now be many orders of magnitude longer, in particular when laser‐cooled slow atoms are employed. It follows that a single atom can interact with literally thousands and even millions of photons one after the other. This provides an excellent opportunity to make real‐time measurements on a single atom by observing the photons emitted from the cavity. In fact, the rate of information one can achieve from a single intracavity atom can significantly exceed the corresponding rate from a free‐space atom, for essentially two reasons: one is the more or less “one‐dimensional” radiation environment, with the cavity effectively covering the full solid angle. The other reason is the fast timescale provided by the short vacuum‐Rabi period in the regime of strong coupling. The loss of photons is therefore a highly useful ingredient of optical cavity QED experiments.

It follows that atoms and photons play opposite roles in microwave and optical cavity QED. This can also be understood when comparing the kind of excitation that is typically employed to drive the atom–cavity system in the two domains. In most microwave experiments, energy is provided by atoms entering the cavity in the excited state, quickly depositing a photon into the cavity. In the optical domain, atoms tend to be in their ground state, and excitation of the system is provided by an external laser. Two configurations are possible: Firstly, the laser illuminates the system from the side, driving the atom which then emits a photon into the cavity, again by virtue of the short vacuum‐Rabi time in the strong coupling regime. Secondly, laser light is coupled into the cavity whose transmission is modified by the presence of the atom. In both configurations, accurate knowledge about the atom can be obtained by observing with unprecedented time resolution the photons that escape the cavity through one (or both) of its mirrors.

Let us now look at a typical cavity QED experiment as displayed in Figure 29.4. Here, the cavity is of the Fabry–Perot type and consists of two concave dielectric mirrors facing each other at a distance of at most a few 100 µm. The cavity supports a standing wave mode with a focus at its center. An essential requirement for achieving strong coupling is to have both the waist of the cavity mode and the distance between the two mirrors small. In this case, the electric field of a single photon confined to a small volume in space is large, typically a few 100 V m⁻¹ for the above dimensions, making the dipole interaction between the atom and the photon large, too. The small mirror spacing, however, has a pronounced disadvantage: the photon lifetime is also small. To compensate this decrease in the cavity lifetime, the reflectivity of the mirrors must be as high as possible. The best commercially available mirrors feature transmission, absorption, and scattering losses down to about 1:1 000 000 each, a value several 10 000 times smaller than that of metallic mirrors. This makes it possible to realize cavities with a finesse exceeding 1 000 000 (27). In such a good cavity, single photons are reflected to and fro between the mirrors several 100 000 times.

Single atoms are now sent between the two mirrors, either dropped from above or injected from below in fountain geometry. The velocity of the atoms is reduced to a very small value by standard laser‐cooling techniques. In the simplest situation, these atoms just pass the cavity in free fall, in which case transit times of the order of a few 10 µs are achieved with atoms moving with a speed of a few m s⁻¹. Such a setup has the advantage that the mechanical influence of the cavity field on the atomic trajectory can largely be neglected. But, the atoms can also be trapped inside the cavity by means of an auxiliary laser field (not shown in Figure 29.4). This field can either be weak and near‐resonant with the atom (28,29), or strong and far‐detuned from the atom (30–34). In the first case, trapping can be achieved with single photons in the cavity on average, but trapping times are severely limited by spontaneous emission events and quantum fluctuations of both the atomic dipole and the cavity field. In the latter case, single atoms have been observed to stay inside the cavity for several 10 s (34), limited either by collisions with atoms of the background gas in a nonperfect vacuum or, ultimately, by the cavity‐enhanced momentum diffusion in the far‐detuned dipole trap. Compared to experiments with freely falling atoms, the extended cavity dwell time for a trapped atom comes at the expense of a dramatically more complex protocol of capturing, trapping, and cooling the intracavity atom. The precise control of the atomic motion between closely spaced, highly reflecting mirrors is a subject of intense investigations in several laboratories worldwide.

Systems such as that described here have been used to perform a plethora of cavity QED experiments in the optical domain during the last few years. These include the observation of single atoms in real time (35–38), the realization of an atom–cavity microscope (29) and an atomic kaleidoscope (39–41) to track the motion of individual atoms with high spatial and temporal resolution, the counterintuitive vacuum‐stimulated generation of photons with single laser‐cooled atoms freely falling through the cavity one after the other (42,43), the optical feedback on the atomic motion based on a measurement of the atom's velocity with the goal to increase the dwell time of the atom inside the cavity (44), the cooling of the atomic motion by means of a novel technique which makes use of the atom's coupling to the dissipative cavity instead of its spontaneous emission (33,34 45–47), the realization of a continuously operated single‐atom light source exhibiting a nonclassical photon statistics (48), the repeated optical transport of single (or a short string of) atoms through a high‐finesse cavity with submicron precision (38,49), and last but not least the spectroscopic investigation of the energy‐level structure of the strongly coupled bound atom–cavity system with its characteristic vacuum‐Rabi splitting, representing in the frequency domain the time‐dependent vacuum‐Rabi oscillation already known from Figure 29.2 (50,51). While all these experiments were performed with one (laser‐cooled) atom in the cavity, other experiments employed an atomic beam, with atoms passing through the cavity at thermal speed. As mentioned above, such experiments do not require the complex trapping and cooling protocol of the single‐atom experiments. In one beam experiment, the optical analog of the micromaser, a microlaser, with atoms prepared in a metastable excited state was demonstrated (52). In other beam experiments, novel quantum effects demonstrating the nonclassical photon statistics of the light transmitted through the cavity were observed (53–55), and the conditional state of the cavity field (produced by a measurement) was stabilized by means of feedback on the driving laser (56). Moreover, the transition from photon antibunching to photon bunching occurring when the average number of fluorescing atoms inside the cavity is increased from a value well below 1 to much larger than 1 (57) was investigated. But, arguably most interesting from the point of view of quantum information processing is the demonstration of a novel light source emitting single photons on demand. These experiments make full use of the unique potential offered by cavity QED concepts, as will be described in some detail now (58–63).

A remarkable feature of this new light source is that it generates photons without spontaneous emission. In particular, the emitting atom is at no time promoted to an excited state. Instead, the atom is always in a so‐called dark state: By slowly varying the parameters of the system, a stimulated Raman process transfers the atom adiabatically from one ground state to another ground state (another hyperfine or Zeeman level) while depositing a photon into the initially empty cavity (64). Experimentally, the adiabatic passage is performed by slowly increasing the intensity of the laser driving the atom to a level where its Rabi frequency exceeds the vacuum‐Rabi frequency associated with a single intracavity photon. For continuous driving, this can be achieved by displacing the laser focus downstream from the cavity axis with respect to the free atomic motion. In such a counterintuitive configuration, the photon is produced while the atom is leaving the cavity, entering the drive laser. For pulsed driving, the decrease in the atom–cavity coupling can be replaced by cavity decay. Here, the escape of the photon from the cavity finishes the adiabatic passage. In the latter case, the atom can be pumped back to the initial state and the whole process can be repeated as long as the atom resides in the cavity. In this way, a bit stream of single‐photon pulses is generated. The stimulated adiabatic transfer process has the distinctive advantage that it is intrinsically reversible and thus ideal to interconnect flying and stationary qubits, that is, photons and atoms, respectively. It also allows one to control the time‐dependent amplitude and frequency of the emitted photon by using a suitably tailored laser pulse.

Figure 29.5 shows data from the very first experiment with atoms falling through the cavity at such a low rate that the probability of having two or more atoms in the cavity is negligible (60). The figure displays the intensity correlation function of the emitted photon stream as measured with the Hanbury Brown and Twiss setup of Figure 29.4. The pronounced peaks reflect the pulsed nature of the light source and appear at times determined by the repetition rate of the pump laser, about 200 kHz in this particular experiment. The missing peak at zero delay time proves that single photons are emitted, because single photons cannot hit simultaneously the two photon detectors behind the beam splitter. The decay of the peak height for increasing the delay time comes from the finite atom–cavity transit time in this first experiment. The decay was largely suppressed in similar experiments performed recently with a trapped atom or ion (62,63).

Interesting results were also obtained in an experiment in which two cavity QED photons generated one after the other were appropriately delayed and superimposed on a beam splitter (65,66). Although these two photons were produced by means of the same atom, they were truly independent because after the emission of the first photon optical excitation in combination with spontaneous emission was required to pump the atom back to the initial state before the generation of the second photon. Consider now the situation in which the two photons have identical polarizations and frequencies and, hence, are indistinguishable. In this case, it is well known that the photons coalesce, that is, they leave the beam splitter as a pair through one of the two output ports. This effect occurs only for single‐photon fields and has been observed in countless experiments with photon pairs produced by parametric fluorescence in a nonlinear crystal. It lies at the heart of quantum computing with linear optics. But, what happens when the two incoming photons have different frequencies? As the two photons are distinguishable now, the effect of photon coalescence is expected to disappear. However, it can be restored in a time‐resolved experiment with photon detectors having a response time much shorter than the duration of the single‐photon pulses. In fact, the experiment (see Figure 29.6) shows that no coincidences at the two output ports are observed for zero detection‐time delay. It is remarkable that two different single photons impinging simultaneously on a beam splitter do not produce simultaneous “clicks” in the two detectors at the two output ports, provided that the time resolution of the detectors is better than the inverse of the frequency separation of the two photons. In addition to this observation, a novel interference effect occurs that can be described as a quantum‐beat phenomenon between the two incoming single‐photon fields. The beat arises because a photon detected behind the beam splitter could equally come from either of the two input ports. The detection event therefore projects the incoming product state onto an entangled state containing one or the other photon with equal probability in one or the other input port, respectively. The relative phase of this superposition state evolves in time with the frequency difference of the two photons, leading to a beat signal. Its duration depends on the coherence time of the two incoming photons. The effect can therefore be used to characterize the coherence properties of single‐photon wave packets or, more general, the coherence properties of the single‐photon source.

It is remarkable that already in these first experiments, a very good control over flying photonic wave packets has been achieved, demonstrating the truly impressive progress in the relatively young research field of optical cavity QED. It should be noted, however, that the experimental techniques required to control both the internal and external degrees of freedom of single strongly coupled atoms are quite demanding, making the experiments a real challenge. From the theoretical side, the dissipative coupling to the environment makes the description of optical experiments difficult. An additional challenge is to properly take into account the atomic motion and the effect of the light force on the atomic trajectory and, hence, on the precise value of the atom–cavity coupling (45–47,67,68). This force arises from the recoil kicks the atom experiences when absorbing or emitting a photon. The inclusion of the light force leads to a complex interplay among the motion of the atom, its internal dynamics, and the dynamics of the cavity field (69,70). No general solutions of the problem of a driven, open system are known even for one intracavity atom.

29.4 Conclusions and Outlook

Both in the microwave and the optical domains, more experiments in the same league as those mentioned above are now in progress or planned. For example, it will be possible to repeatedly move trapped atoms in and out of the strong‐coupling region in the near future, enabling one to address individual or pairs of qubits of an atomic quantum register with a high‐finesse cavity. First experiments in this direction, with atoms localized in a standing wave dipole‐force trap which can be displaced perpendicular to the cavity axis, have already been reported (38,49). It therefore seems possible to create in a deterministic way an entanglement of one stationary atom (individually selected from a string of several atoms) and a flying photon. Alternatively, two atoms strongly coupled by the cavity could be used for scalable quantum computation (71,72). Experiments with many atoms coupled to the cavity would also allow one to exploit collective radiation effects like superradiance. The collective emission of light from the atomic ensemble automatically generates a large entangled state involving all the intracavity atoms. The basic idea here is that photons are simultaneously emitted from all atoms, making it impossible to tell, even in principle, from which atom the photon is emitted.

In another line of experiments, setups with two spatially separated cavities are presently under construction. They will offer a much greater flexibility and new possibilities for quantum information processing. For example, photons could be exchanged between two atom–cavity systems, allowing one to transfer the quantum state of one atom to another. Using the single‐photon technique described above, entanglement between two remote locations could thus be generated (73). The technology of individually addressable atom–cavity systems is intrinsically scalable, so that a large network with stationary atoms as quantum memories and flying photons as quantum messengers could be formed (74). In such a network, the state of an atom could be teleported over a macroscopic distance like several meters or even many kilometers (75,76). Moreover, nonlocal Schrödinger cat states residing simultaneously in separated cavities could be created and studied. Such states are a completely new species of quantum monsters, allowing us to advance our understanding of decoherence and nonlocality.

It is even possible to envision experiments blending cavity QED and atom chip concepts. In the latter technology, atoms are manipulated with magnetic and/or electric fields generated by the wires of a microfabricated chip. On‐chip conveyor belts can be used to transport atoms along the surface of the chip and move them into on‐chip transmission‐line cavities. Micrometer‐sized cavities between the tips of two optical fibers are presently tested in several laboratories around the world. Such integrated experiments provide a scalable architecture for quantum information processing. Coherence preserving traps can be tailored for Rydberg atoms, holding them over superconducting chips, which block their only decay, spontaneous emission (77). In addition, the on‐chip atoms could be coupled with superconducting qubits also integrated on‐chip, opening a wealth of new possibilities. Similarly, superconducting Cooper pair boxes (instead of Rydberg atoms) coupled to microwave stripline resonators (instead of Fabry–Perot resonators) offer an interesting solid‐state alternative to atom‐based cavity QED (78,79). Last but not least, the recent advances in nanotechnology will allow one to design novel wavelength‐sized optical cavities, for example, with photonic band‐gap materials. The very strong coupling that can theoretically be achieved in such small cavities could dramatically boost the speed of quantum gates or the rate of single photons delivered on demand. A first step into this direction has already been done with the achievement of strong coupling in systems with artificial atoms, that is, quantum dots (80,81).

The countless avenues cavity QED opens up for quantum information processing make research with individual atoms and photons in confined space increasingly exciting even 50 years after the first ideas were formulated!

References

1. 1 Purcell, E.M. (1946) Spontaneous emission probabilities at radio frequencies. Phys. Rev., 69, 681.
2. 2 Raithel, G. et al. (1994) Cavity quantum electrodynamics, in Advances in Atomic, Molecular and Optical Physics, Supplement 2 (ed. P.R. Berman), Academic Press, New York.
3. 3 De Martini, F. and Monroe, C. (eds) (2002) Experimental Quantum Computation and Information, Proceedings of the International School of Physics “Enrico Fermi”, IOS Press, Amsterdam.
4. 4 Mabuchi, H. and Doherty, A.C. (2002) Cavity quantum electrodynamics: coherence in context. Science, 298, 1372.
5. 5 Vahala, K.J. (2003) Optical microcavities. Nature, 424, 839.
6. 6 Rempe, G. (2000) Quantum mechanics with single atoms and photons. Phys. World, 13 (12), 37.
7. 7 Gallagher, T.F. (1994) Rydberg Atoms, Cambridge University Press, Cambridge.
8. 8 Raimond, J.‐M., Brune, M., and Haroche, S. (2001) Colloquium: manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys., 73, 565.
9. 9 Maioli, P., Meunier, T., Gleyzes, S., Auffeves, A., Nogues, G., Brune, M., Raimond, J.‐M., and Haroche, S. (2005) Nondestructive Rydberg atom counting with mesoscopic fields in a cavity. Phys. Rev. Lett., 94, 113601.
10. 10 Weidinger, M., Varcoe, B.T.H., Heerlein, R., and Walther, H. (1999) Trapping states in the micromaser. Phys. Rev. Lett., 82, 3795.
11. 11 Varcoe, B.T.H., Brattke, S., Weidinger, M., and Walther, H. (2000) Preparing pure photon number states of the radiation field. Nature, 403, 743.
12. 12 Brattke, S., Varcoe, B.T.H., and Walther, H. (2001) Generation of photon number states on demand via cavity quantum electrodynamics. Phys. Rev. Lett., 86, 3534.
13. 13 Rempe, G., Walther, A., and Klein, N. (1987) Observation of quantum collapse and revival in a one‐atom maser. Phys. Rev. Lett., 58, 353.
14. 14 Rempe, G., Schmidt‐Kaler, F., and Walther, H. (1990) Observation of sub‐Poissonian photon statistics in a micromaser. Phys. Rev. Lett., 64, 2783.
15. 15 Brune, M., Raimond, J.‐M., Goy, P., Davidovich, L., and Haroche, S. (1987) Realization of a two‐photon maser oscillator. Phys. Rev. Lett., 59, 1899.
16. 16 Maitre, X., Hagley, E., Nogues, G., Wunderlich, C., Goy, P., Brune, M., Raimond, J.‐M., and Haroche, S. (1997) Quantum memory with a single photon in a cavity. Phys. Rev. Lett., 79, 769.
17. 17 Hagley, E., Maitre, X., Nogues, G., Wunderlich, C., Brune, M., Raimond, J.‐M., and Haroche, S. (1997) Generation of Einstein–Podolsky–Rosen pairs of atoms. Phys. Rev. Lett., 79, 1.
18. 18 Rauschenbeutel, A., Bertet, P., Osnaghi, S., Nogues, G., Brune, M., Raimond, J.‐M., and Haroche, S. (2001) Controlled entanglement of two field modes in a cavity quantum electrodynamics experiment. Phys. Rev. Lett., 64, 050301.
19. 19 Rauschenbeutel, A., Nogues, G., Osnaghi, S., Bertet, P., Brune, M., Raimond, J.‐M., and Haroche, S. (1999) Coherent operation of a tunable quantum phase gate in cavity QED. Phys. Rev. Lett., 83, 5166.
20. 20 Nogues, G., Rauschenbeutel, A., Osnaghi, S., Brune, M., Raimond, J.‐M., and Haroche, S. (1999) Seeing a single photon without destroying it. Nature, 400, 239.
21. 21 Rauschenbeutel, A., Nogues, G., Osnaghi, S., Bertet, P., Brune, M., Raimond, J.‐M., and Haroche, S. (2000) Step‐by‐step engineered multiparticle entanglement. Science, 288, 2024.
22. 22 Osnaghi, S., Bertet, P., Auffeves, A., Maioli, P., Brune, M., Raimond, J.‐M., and Haroche, S. (2001) Coherent control of an atomic collision in a cavity. Phys. Rev. Lett., 87, 037902.
23. 23 Yamaguchi, F., Milman, P., Brune, M., Raimond, J.‐M., and Haroche, S. (2002) Quantum search with two atom collisions in cavity QED. Phys. Rev. A, 66, 010302.
24. 24 Auffeves, A., Maioli, P., Meunier, T., Gleyzes, S., Nogues, G., Brune, M., Raimond, J.M., and Haroche, S. (2003) Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity. Phys. Rev. Lett., 91, 230405.
25. 25 Meunier, T., Gleyzes, S., Maioli, P., Auffeves, A., Nogues, G., Brune, M., Raimond, J.M., and Haroche, S. (2005) Rabi oscillations revival induced by time reversal: a test of mesoscopic quantum coherence. Phys. Rev. Lett., 94, 010401.
26. 26 Bertet, P., Auffeves, A., Maioli, P., Osnaghi, S., Meunier, T., Brune, M., Raimond, J.M., and Haroche, S. (2002) Direct measurement of the Wigner function of a one‐photon Fock state in a cavity. Phys. Rev. Lett., 89, 200402.
27. 27 Rempe, G., Thompson, R.J., Kimble, H.J., and Lalezari, R. (1992) Measurement of ultralow losses in an optical interferometer. Opt. Lett., 17, 363.
28. 28 Pinkse, P.W.H., Fischer, T., Maunz, P., and Rempe, G. (2000) Trapping an atom with single photons. Nature, 404, 365.
29. 29 Hood, C.J., Lynn, T.W., Doherty, A.C., Parkins, A.S., and Kimble, H.J. (2000) The atom‐cavity microscope: single atoms bound in orbit by single photons. Science, 287, 1447.
30. 30 Ye, J., Vernooy, D.W., and Kimble, H.J. (1999) Trapping of single atoms in cavity QED. Phys. Rev. Lett., 83, 4987.
31. 31 McKeever, J., Buck, J.R., Boozer, A.D., Kuzmich, A., Nägerl, H.C., Stamper‐Kurn, D.M., and Kimble, H.J. (2003) State‐insensitive cooling and trapping of single atoms in an optical cavity. Phys. Rev. Lett., 90, 133602.
32. 32 McKeever, J., Buck, J.R., Boozer, A.D., and Kimble, H.J. (2004) Determination of the number of atoms trapped in an optical cavity. Phys. Rev. Lett., 93, 143601.
33. 33 Maunz, P., Puppe, T., Schuster, I., Syassen, N., Pinkse, P.W.H., and Rempe, G. (2004) Cavity cooling of single atoms. Nature, 428, 50.
34. 34 Nußmann, S., Murr, K., Hijlkema, M., Weber, B., Kuhn, A., and Rempe, G. (2005) Vacuum‐stimulated cooling of single atoms in three dimensions. Nat. Phys., 1, 122–125.
35. 35 Mabuchi, H., Turchette, Q.A., Chapman, M.S., and Kimble, H.J. (1996) Realtime detection of individual atoms falling through a high‐finesse optical cavity. Opt. Lett., 21, 1393.
36. 36 Hood, C.J., Chapman, M.S., Lynn, T.W., and Kimble, H.J. (1998) Realtime cavity QED with single atoms. Phys. Rev. Lett., 80, 4157.
37. 37 Münstermann, P., Fischer, T., Pinkse, P.W.H., and Rempe, G. (1999) Single slow atoms from an atomic fountain observed in a high‐finesse optical cavity. Opt. Commun., 159, 63.
38. 38 Sauer, J.A., Fortier, K.M., Chang, M.S., Hamley, C.D., and Chapman, M.S. (2004) Cavity QED with optically transported atoms. Phys. Rev. A, 69, 051804.
39. 39 Horak, P., Ritsch, H., Fischer, T., Maunz, P., Puppe, T., Pinkse, P.W.H., and Rempe, G. (2002) Optical kaleidoscope using a single atom. Phys. Rev. Lett., 88, 043601.
40. 40 Maunz, P., Puppe, T., Fischer, T., Pinkse, P.W.H., and Rempe, G. (2003) The emission pattern of an atomic dipole in a high‐finesse optical cavity. Opt. Lett., 28, 46.
41. 41 Puppe, T., Maunz, P., Fischer, T., Pinkse, P.W.H., and Rempe, G. (2004) Single‐atom trajectories in higher order transverse modes of a high‐finesse optical cavity. Phys. Scr., T112, 7.
42. 42 Hennrich, M., Legero, T., Kuhn, A., and Rempe, G. (2000) Vacuum‐stimulated Raman scattering based on adiabatic passage in a high‐finesse optical cavity. Phys. Rev. Lett., 85, 4872.
43. 43 Hennrich, M., Legero, T., Kuhn, A., and Rempe, G. (2003) Counter‐intuitive vacuum‐stimulated Raman scattering. J. Mod. Opt., 50, 935.
44. 44 Fischer, T., Maunz, P., Pinkse, P.W.H., Puppe, T., and Rempe, G. (2002) Feedback on the motion of a single atom in an optical cavity. Phys. Rev. Lett., 88, 163002.
45. 45 Horak, P., Hechenblaikner, G., Gheri, K.M., Stecher, H., and Ritsch, H. (1997) Cavity‐induced atom cooling in the strong coupling regime. Phys. Rev. Lett., 79, 4974.
46. 46 Vuletic, V. and Chu, S. (2000) Laser cooling of atoms, ions, or molecules by coherent scattering. Phys. Rev. Lett., 84, 3787.
47. 47 Domokos, P., Vukics, A., and Ritsch, H. (2004) Anomalous Doppler‐effect and polariton‐mediated cooling of two‐level atoms. Phys. Rev. Lett., 92, 103601.
48. 48 McKeever, J., Boca, A., Boozer, A.D., Buck, J.R., and Kimble, H.J. (2003) Experimental realization of a one‐atom laser in the regime of strong coupling. Nature, 425, 268.
49. 49 Nußmann, S., Hijlkema, M., Weber, B., Rohde, F., Rempe, G., and Kuhn, A. (2005) Nano‐positioning of single atoms in a micro‐cavity. Phys. Rev. Lett., 95, 173602.
50. 50 Boca, A., Miller, R., Birnbaum, K.M., Boozer, A.D., McKeever, J., and Kimble, H.J. (2004) Observation of the vacuum Rabi spectrum for one trapped atom. Phys. Rev. Lett., 93, 233603.
51. 51 Maunz, P., Puppe, T., Schuster, I., Syassen, N., Pinkse, P.W.H., and Rempe, G. (2005) Normal‐mode spectroscopy of a single bound atom–cavity system. Phys. Rev. Lett., 94, 033002.
52. 52 An, K., Childs, J.J., Dasari, R.R., and Feld, M.S. (1994) Microlaser – a laser with one atom in an optical resonator. Phys. Rev. Lett., 73, 3375.
53. 53 Mielke, S.L., Foster, G.T., and Orozco, L.A. (1998) Nonclassical intensity correlations in cavity QED. Phys. Rev. Lett., 80, 3948.
54. 54 Carmichael, H.J., Castro‐Beltran, H.M., Foster, G.T., and Orozco, L.A. (2000) Giant violations of classical inequalities through conditional homodyne detection of the quadrature amplitudes of light. Phys. Rev. Lett., 85, 1855.
55. 55 Foster, G.T., Orozco, L.A., Castro‐Beltran, H.M., and Carmichael, H.J. (2000) Quantum state reduction and conditional time evolution of wave‐particle correlations in cavity QED. Phys. Rev. Lett., 85, 3149.
56. 56 Smith, W.P., Reiner, J.E., Orozco, L.A., Kuhr, S., and Wiseman, H.M. (2002) Capture and release of a conditional state of a cavity QED system by quantum feedback. Phys. Rev. Lett., 89, 133601.
57. 57 Hennrich, M., Kuhn, A., and Rempe, G. (2005) Transition from antibunching to bunching in cavity QED. Phys. Rev. Lett., 94, 053604.
58. 58 Law, C.K. and Kimble, H.J. (1997) Deterministic generation of a bit‐stream of single‐photon pulses. J. Mod. Opt., 44, 2067.
59. 59 Kuhn, A., Hennrich, M., Bondo, T., and Rempe, G. (1999) Controlled generation of single photons from a strongly coupled atom–cavity system. Appl. Phys. B, 69, 373.
60. 60 Kuhn, A., Hennrich, M., and Rempe, G. (2002) Deterministic single‐photon source for distributed quantum networking. Phys. Rev. Lett., 89, 067901; see also (2003) Phys. Rev. Lett. 90, 249802.
61. 61 Hennrich, M., Legero, T., Kuhn, A., and Rempe, G. (2004) Photon statistics of a non‐stationary periodically driven single‐photon source. New J. Phys., 6, 86.
62. 62 McKeever, J., Boca, A., Boozer, A.D., Miller, R., Buck, J.R., Kuzmich, A., and Kimble, H.J. (2004) Deterministic generation of single photons from one atom trapped in a cavity. Science, 303, 1992.
63. 63 Keller, M., Lange, B., Hayasaka, K., Lange, W., and Walther, H. (2004) Continuous generation of single photons with controlled waveform in an ion‐trap cavity system. Nature, 431, 1075.
64. 64 Kuhn, A. and Rempe, G. (2002) Optical cavity QED: fundamentals and application as a single‐photon light source, in Experimental Quantum Computation and Information, Proceedings of the International School of Physics “Enrico Fermi” Course CXLVIII (eds F. De Martini and C. Monroe), IOS Press, Amsterdam, pp. 37–66.
65. 65 Legero, T., Wilk, T., Kuhn, A., and Rempe, G. (2003) Time‐resolved two‐photon quantum interference. Appl. Phys. B, 77, 797.
66. 66 Legero, T., Wilk, T., Hennrich, M., Rempe, G., and Kuhn, A. (2004) Quantum beat of two single photons. Phys. Rev. Lett., 93, 070503.
67. 67 van Enk, S.J., McKeever, J., Kimble, H.J., and Ye, J. (2001) Cooling of a single atom in an optical trap inside a resonator. Phys. Rev. A, 64, 013407.
68. 68 Murr, K. (2003) On the suppression of the diffusion and the quantum nature of a cavity mode. Optical bistability: forces and friction in driven cavities. J. Phys. B: At. Mol. Opt. Phys., 36, 2515.
69. 69 Münstermann, P., Fischer, T., Maunz, P., Pinkse, P.W.H., and Rempe, G. (1999) Dynamics of single‐atom motion observed in a high‐finesse cavity. Phys. Rev. Lett., 82, 3791.
70. 70 Münstermann, P., Fischer, T., Maunz, P., Pinkse, P.W.H., and Rempe, G. (2000) Observation of cavity‐mediated long‐range light forces between strongly coupled atoms. Phys. Rev. Lett., 84, 4068.
71. 71 Pellizzari, T., Gardiner, S.A., Cirac, J.I., and Zoller, P. (1995) Decoherence, continuous observation, and quantum computing: a cavity QED model. Phys. Rev. Lett., 75, 3788.
72. 72 Duan, L.M. and Kimble, H.J. (2004) Scalable photonic quantum computation through cavity‐assisted interactions. Phys. Rev. Lett., 92, 127902.
73. 73 Browne, D.E., Plenio, M.B., and Huelga, S.F. (2003) Robust creation of entanglement between ions in spatially separate cavities. Phys. Rev. Lett., 91, 067901.
74. 74 Cirac, J.I., Zoller, P., Kimble, H.J., and Mabuchi, H. (1997) Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett., 78, 3221.
75. 75 Bose, S., Knight, P.L., Plenio, M.B., and Vedral, V. (1999) Proposal for teleportation of an atomic state via cavity decay. Phys. Rev. Lett., 83, 5158.
76. 76 Lloyd, S., Shahriar, M.S., Shapiro, J.H., and Hemmer, P.R. (2001) Long distance unconditional teleportation of atomic states via complete Bell‐state measurements. Phys. Rev. Lett., 87, 167903.
77. 77 Hyafil, P., Mozley, J., Perrin, A., Tailleur, J., Nogues, G., Brune, M., Raimond, J.M., and Haroche, S. (2004) Coherence‐preserving trap architecture for long‐term control of giant Rydberg atoms. Phys. Rev. Lett., 93, 103001.
78. 78 Chiorescu, I., Bertet, P., Semba, K., Nakamura, Y., Harmans, C.J.P.M., and Mooij, J.E. (2004) Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature, 431, 159.
79. 79 Wallraff, A., Schuster, D.I., Blais, A., Frunzio, L., Huang, R.‐S., Majer, J., Kumar, S., Girvin, S.M., and Schoelkopf, R.J. (2004) Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature, 431, 162.
80. 80 Reithmaier, J.P., Sek, G., Loffler, A., Hofmann, C., Kuhn, S., Reitzenstein, S., Keldysh, L.V., Kulakovskii, V.D., Reinecke, T.L., and Forchel, A. (2004) Strong coupling in a single quantum dot‐semiconductor microcavity system. Nature, 432, 197.
81. 81 Yoshie, T., Scherer, A., Hendrickson, J., Khitrova, G., Gibbs, H.M., Rupper, G., Ell, C., Shchekin, O.B., and Deppe, D.G. (2004) Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity. Nature, 432, 200.