It is time to conclude. What have you learned today? That there are situations where one must use quantum optics, there is no option. In order to be able to appreciate one of these situations, you have learned an important element of the quantum optics formalism, the expressions of single and double photodetection signals. The most important feature of these expressions, which you must remember, is that the electric field operators appear in the normal order. That is to say that the annihilation operators are on the right hand side and the creation operators are on the left. This is the so-called normal ordering of these operators. It has far reaching consequences. In particular, the description of phenomena that cannot be understood in the semi classical model, where light is not quantized. It is because of such phenomena, that quantum optics cannot be ignored, that one must use it and that you must learn it. You have seen an emblematic example of such a fully quantum phenomenon: the absence of double photo-detection in the case a one-photon wave-packet. You have seen that such a phenomena is in contradiction with the predictions of the semi-classical model. In fact, this fully quantum property can be used in one of the simplest examples of quantum technology, that is quantum cryptography with one photon wave packets. A quantum technology is a technology enabled by the fundamental laws of quantum physics, and it would fail if nature was described by classical physics. The simplest scheme was invented by Charles Bennett and Giles Brassard in 1984. But it took a long time before it was recognized as an interesting scheme. This happened in 1991 when Artur Ekert published another quantum cryptography scheme based on entanglement and drew attention onto the initial scheme now known as BB84. Quantum cryptography is nowadays available as commercial devices, and it has already been used in official tasks such as transmitting the results of "votations" in Switzerland. The schemes are more sophisticated than the one I will describe now, but the fundamental idea remains the same. Alice and Bob, two friends, want to exchange bits of information and be sure that no spy, the awful eavesdropper, traditionally called Eve, has read the information. You might object that if Eve has collected the information, it is too late to avoid him, but in fact, cryptography is more subtle than that. What Alice and Bob want to exchange securely is not the message containing information but a random series of zeroes and ones, a cryptographic key which will be used later for encoding and decoding the real message. That real message will be transmitted on an open channel after being encoded. Claude Shannon, a pioneer and a genius of the information theory, has shown that it is then impossible to break the code and decipher the message provided that the message is not longer than the key. Finally, this cryptography scheme, known as one time pad, is fully secure provided that a new encoding key long enough is generated and shared before sending each new secret message. Note that Shannon's theorem does not involve any quantum physics, it is pure mathematics. Now comes quantum physics. It allows Alice and Bob to share the key, a series of random 0 and 1, being sure that nobody else has been able to make a copy of it. This is possible thanks to quantum key distribution, QKD. The scheme involves a quantum channel, which should remain protected from the spy, and a classical channel which can be listened to by everybody including the spy. The security of this quantum channel relies on the fact that Eve will necessarily leave a trace, a footprint, if he looks at the information encoded on quantum objects used as the support of the information. For instance, Alice sends 0s and 1s using one photon wave packets with two different frequencies or two different polarizations. If Eve makes any measurement involving a photodetector, there will be missing photons in what Bob receives, and he will warn Alice on the classical channel, that the sequence she has sent, should not be used as an encoding key. In contrast, if no bit is missing, Bob can be sure that no bit has been subtracted by a spy and he can tell Alice on the open channel, that the key was safely transmitted. But wait a minute, the spy could be smart enough to re-send a photon identical to the one he has intercepted after doing his measurement. Then no photon would be missing. You will see in a future lesson that Alice and Bob can detect such a maneuver using another quantum property, the existence of superpositions of quantum states. That is to say the fact that the quantum bit, a qubit, can be not only in state zero or in state one, but also in a superposition of the state zero and the state one. Such superpositions give rise to the phenomenon of one-photon interference, which we will discover in the next lesson. In fact, what is at stake here is the famous wave-particle duality of light. Let us come back to what I claim is a most intriguing result of this lesson, the null value of the double photo-detection signal for the one photon wave packet. You might argue that if words have a meaning, the fact that a single photon cannot be detected twice should not come as a surprise since the first photodetector destroys the photon. This is very true, but light is more than a beam of particles. One should also remember that light behaves as a wave, as convincingly argued by Thomas Young and Augustin Fresnel in the early 19th century. Otherwise, how could we understand the phenomena of interference and diffraction. In fact, we are evoking here something you have already heard of, wave-particle duality. Light is both a wave and a particle as Einstein recognized it as early as 1909. Louis de Broglie later extended this notion of wave-particle duality to material particles. I'm sure that you have already heard these words, but are you comfortable with them? When I was a student, I was not satisfied with just invoking the words wave-particle duality which sounded like a magic incantation. I wanted to know how the theory could consistently describe these two contradictory behaviors of light. I also wanted to know if there were experiments showing, unambiguously, both behaviors for the same light beam. Are you like me? Do you ask yourself questions about wave-particle duality? To get some answers come again to the next lesson, bye bye.