It is in 1985 that we could characterize and use one-photon wave-packets emitted by the one photon source I am going to describe now. Such sources are still widely used, and they are called sources of heralded one photons. They emit one photon wave packets at random uncontrolled times, but there is a heralding signal announcing the emission of each individual photon. I will first explain how they work, since they are based on a quantum optics phenomenon without any classical counterpart: two photon radiative cascades. At the end of this section, I will also describe sources emitting one-photon states on demand at the time decided by the operator. You can see here the scheme of our one photon source. Calcium atoms were excited by lasers to the level |u>. This direct process involving two lasers is a non-linear process of order two, called a two-photon process. We will study such process in a future lesson. An atom excited to the level |u> re-emits a pairs of photons at frequencies omega_H and omega_0. This process is called a radiative cascade. The photons were emitted in all direction in space, and we collected a fraction of them with wide aperture lenses followed by frequency filters at omega_H and omega_0 respectively. The green photons, hbar omega_H were detected by the photon detector D_H while the violet photons hbar omega_0 were detected at D. An electronic device known as a time to digital converter measured the delay between the detection at D_H and the next detection at D. Let us consider an excited atom that emits a photon hbar omega_H. The index H stands for heralding, meaning that the detection of such a photon announces the emission of a photon hbar omega_0. More precisely, the detection time of the first photon corrected for the propagation from the emitter to the detector D_H is the time of preparation of the atom in the state |e>. It is the initial time t0 in the calculation of section three that describes the emission of a spontaneous one photon wave packet. That photon is detected on detector D, at time t and the time to digital converter yields the value of the delay between the detections of the two photons. After accumulating enough data one has a histogram, which displays a number of events as a function of the observed delay. The observed signal is what we calculated in section three. There is a steep rise followed by an exponential decay. There are, however, two differences with the ideal scheme of section three. The first difference is that we do not control the time t0 of emission, we only observe its value, by detecting the heralding photon, hbar omega_H. The second difference is that we do not collect all the hbar omega_0 photons associated with the heralding photon. This means that the state in the collimated beam is, in fact, composed of a fraction of a one-photon state plus some vacuum component. Gamma is a coefficient with a modulus less than 1. Its squared modulus is obviously the fraction of one photon wavepackets collected by the lens, and it is called the collection efficiency. The fraction of vacuum component is then 1 minus the square modulus of gamma. You can check by simple inspection that the vacuum component does not change the essential of the results obtained in section three. The reason is that we have calculated quantities expressed with operators in the normal order with annihilation operators on the right hand side. The action on the vacuum of these operators yields the number 0. So the added vacuum term does not contribute, and the only correction on the probability of detection at D is the multiplying factor less than one, squared modulus of gamma. Integrating over time, we obtain the probability of detection of the photon hbar omega_0 for each detected heralding photon as a product of the quantum efficiency eta of the detector by the collection efficiency. We note epsilon this overall detection efficiency. I know we already use the letter epsilon for polarization, but in that case, it was a vector with an over arrow. So you should be able to avoid confusion. You may ask yourself why we used such a complicated scheme rather than just exciting a single atom and looking either for the photon hbar omega_H or for the photon hbar omega_0? The reason is that in 1985, nobody knew how to isolate a neutral atom and keep it at rest. Remember that in a vapour at room temperature, atoms move at velocities of several hundreds of meters per second, the velocity of sound. Moreover, even at small vapour pressure, the smallest volume that one can resolve with a microscope of a few cubic micrometers, still contains millions of atoms. It is only after the development of methods for cooling and trapping atoms with lasers, that it became possible in the 1990s to trap and observe a single atom isolated in space. So what was available to us in 1985 was a sample with millions of atoms. When we excited them with our two continuous lasers, we had several millions of radiative cascades excited each second. How could we then isolate single atom emissions? The answer is: we isolated single atom emissions in time. Remember that the radiative lifetime of the state |e>, one over gamma is about five nanoseconds. This means that when you detect a heralding photon omega_H, you are almost sure that the corresponding photon hbar omega_0 will have been emitted after 10 to 15 nanoseconds. This is much less than the average time between the emission by two different atoms, which is of the order of the inverse of the rate of cascades. For an average rate of cascades of a few millions per second, the average time between the emission of two different atoms is of several hundreds of nano seconds. This is much more than the delay between two photons from the same atom. It means that two detections at D_H and D, separated by not more than a few lifetimes of |e>, are very likely to correspond to two photons coming from the same emitter. So if we observe a photo detection at D during a time window opened after the detection of the heralding photon at t0, we are pretty sure that it corresponds to a one photon wave packet with a rising time at t0. More precisely, the probability that we detect a photon hbar omega_0 emitted by another atom during our observing time window, is quite negligible. Of course, the words pretty sure and quite negligible must be made quantitative, and we will do it. At this stage I only want you to understand that the heralding method allows one to isolate single atom signals although millions of atoms are in the observation volume. One isolates single emitters in time. This is a real signal observed with our source. It is a histogram of the time delays for double detections at D_H and D. The events are gathered in time slots of 0.4 nanosecond. The time resolution is of the order of one nanosecond. The number of events at the peak is about 14,000, and the corresponding statistical accuracy is better than 1%. You clearly see a steep rise followed by an exponential decay as can be checked by fitting an exponential on the data. The fit yields the lifetime of the level |e> one over gamma equal to 4.7 nanoseconds. The fact that we have a well contrasted peak above a weak pedestal shows that we indeed isolate single emitter signals. The delays t minus t0 associated with different emitters are uniformly distributed in time. They give a flat signal that you can see as a pedestal in the plot here. This pedestal scales as the square of the pairs emission rate, while the one-photon emission rate obviously scales as the pair rate. So one must keep the pair rates not too high to keep the pedestal low. One can then work with one-photon wave packets using a technique named gating. It is widely used in particle physics experiments. It consists of enabling observation only during certain time windows, triggered by an event announcing that an interesting phenomenon is likely to happen during that time window. In our experiment, we used 10 nanosecond windows. We could then perform experiments in which the heralded one-photon wave packets were passed through an optical device, for instance an interferometer, before falling on a gated detector. I will describe such experiments in the next lesson. Let me now tell you about a variety of one-photon sources available nowadays. The possibility to produce one photon states announced by a heralding photon has been extended to pairs of photons produced by a non-linear phenomenon known as parametric down-conversion. In a future lesson, you will discover this non-linear process in which a pump photon hbar omega_P, usually in the near ultraviolet range, is split in two red photons. Here, I've represented them with the same color, but they may have different frequencies, provided that there's energy conservation. The two photons are emitted in different directions, and one can be used as heralding photon, while the other one is heralded one photon wave packet. Compared with pairs of photons, issued from atomic radiative cascades, parametric down-conversion has a major advantage. For a given direction of an heralding photon, the heralded photon is emitted in a well-defined direction. It is thus possible to capture almost all the heralded one-photon wave packets, and to feed them into an optical fiber, where they can propagate at long distances, more than 10 Km. The timing is also much sharper. More precisely, each one photon wave packet has a width typically less than one picosecond and the delay between the two photons is as small. Such a small delay cannot be determined directly. No detector is fast enough, but there is an indirect evidence of such a small delay using the famous Hong Ou and Mandel effect, which I will describe in a future lesson. As in the atomic cascade case, however, the rate of emission must remain low enough so that the probability of detecting a photon from another pair during the time window remains small enough. As for the atomic cascade scheme, one cannot decide on the moment when the pair is emitted. The time of emission is random. This is why other types of sources have been developed at the turn of the century, allowing for one-photon emissions on demand. In the last decades, progress in nanotechnologies have allowed experimentalists to isolate individual emitters as you can see on this image obtained with a scanning tunneling microscope. One can then address with a microscope one of these individual emitters, excite it specifically and observe the re-emitted radiation. But what kind of individual emitters are convenient? Emitters with a level scheme as shown here have proven to be remarkable one photon sources. A laser pulse excites a molecule to a high lying level |u> which can decay rapidly towards level |e>. Level |e> has relatively long lifetime of the order of several nanoseconds, longer than the laser pulse. It means that when the laser pulse is finished, the molecule is still in the |e> level with a very high probability and can emit a single photon with a typical exponential distribution of delays, as in the atomic cascade case. But a major difference is the fact that the exciting time, t0 is determined by the time of the laser pulse, which can be controlled by the experimentalist. The one photon wave packet can then be emitted on demand. This scheme was invented in the year 2000. One has today a variety of isolated emitters of one photon wave packets. The level scheme shown here corresponds to a defect in a crystal of diamond behaving as an atom or a molecule. It is called a NV-center an acronym for nitrogen vacancy center. Such individual emitters are much used in quantum information technology. You see here another kind of one photon source on demand. It is a semiconductor quantum dot embedded in a resonant microscopic cavity, which enhances the probability of emission along the axis of the cavity. This results into a directed emission.