In Lesson seven of Quantum Optics 1, I introduced the notion of polarization of a photon to show how polarized single photon can be used as a quantum bit, a qubit, in the BB84 quantum cryptography protocol. In fact, single photon polarization is a remarkable example of a two-level system, that is to say, a quantum system whose quantum state belongs to a two-dimensional space. Many basic courses of quantum mechanics present spin one-half as an emblematic example of two-level systems. Polarized single photons are described with an equivalent formalism, but from an experimental point of view they are more convenient than spin one-half particles. Indeed, photons are immune to ambient noise, if one uses light in the visible or near visible range. If they propagate in vacuum or in isotropic and homogeneous media, polarization is maintained along propagation. It suffices then to use optical components of good-quality, whose action onto polarization is well-controlled, to implement various transformations of the polarization, that is to say, of the quantum state. Finally, photon counting technique can perform almost ideal measurements. In this section, I will rapidly remind you what is a polarized single photon wave-packet and establish notations that will be used all along this lesson. Be sure that you fully master these notations, it is crucial for understanding the rest of the lesson. Remember, that a one-photon wave packet is a superposition of elementary one-photon states. If they have all the same direction and polarization, the state can be factorized: the first part describes the polarization while the second part describes the spatial part of the wave packet. The two-dimensional space associated with the polarization is a plane perpendicular to the direction of propagation and one can take the two linear polarizations along x and y as a basis to describe any polarization. The corresponding space of quantum states has dimension 2, and a useful basis to describe the most general polarized state is constituted by the two-state vectors associated with the linear polarizations x and y. There are other bases of the polarization state space. For instance, right and left circular polarizations, Epsilon plus and Epsilon minus. They correspond to the circular polarization of classical electromagnetism, or optics, described by complex unit vectors Epsilon plus and Epsilon minus, associated with positive or negative rotation senses in relation to the z axis. In the description above we have restricted ourselves to the case of all the k vectors along the same direction, which means a priory plane waves infinitely extending in the transverse direction. But we want to be able to have a wave packet with a well-defined number of photons, for instance one photon. So we use the approximation of a cylindrical beam with a finite transverse surface S, large enough that we can neglect diffraction. This approximation can be generalized to the case of beams that have a small divergence, that is to say, a dispersion in the directions of the wave vector components. One can then make the paraxial approximation in which the longitudinal component of the polarization is neglected, so that polarization still belongs to a two-dimensional space. A well-known and useful example is the Gaussian modes used, for instance, to describe laser beams, which can as well be used to describe the propagation of a wave packet. In the factorized form of the polarized single photon state, the second factor describes the spatial part of the state. One can associate it with the classical spatial function phi of the coordinates. It is a classical electromagnetic mode with the transverse extension S, which we take uniform to simplify. The evolution of phi with time is given by a sum of complex exponentials. In the Schrödinger point of view, the corresponding one-photon state vector evolves similarly. In fact, the function phi of x, y, z, and t, yields the spatial temporal profile of the one-photon wave packet in the following sense: if we put at position z a small detector of transverse extension dx dy, and repeat the measurement many times, we obtain a probability distribution of single detection, which is the squared modulus of phi, times the velocity of light c and the quantum efficiency eta. Now, if we have a photodetector whose transverse dimensions are larger than the size of the mode, and if we integrate the signal over a time long enough to encompass the whole wave packet, the total probability of single photodetection is one times the quantum efficiency eta. In the rest of the lesson, in order to have simple formulae, we will assume that eta is equal to 1, unless specified differently. In this lesson, the important quantum observable is polarization, and keeping the whole expression including the spatial temporal profile is not useful, provided that you remember that we are dealing with one-photon wave packets, whose total probability of photo-detection is one, with the detection event happening in a well-defined interval of time. So we can use simplified notations where we explicitly write the polarization but the wave packet to which this polarization belongs is just indicated by a label which is explicit. This kind of notation can be adapted to each particular situation. For instance, if we consider a polarized wave packet propagating in the positive direction along z, we can write Epsilon plus z, Epsilon minus z describes a wave packet of polarization Epsilon propagating along minus z. In fact, this kind of notation gives a lot of flexibility and is usually unambiguous, provided that one remembers that it refers to a one-photon wave packet. For instance, you should be able to tell what represents each term in the relation that defines the state Epsilon plus with a subscript plus z. Be sure that you fully understand these notations, which represent a big simplification and which we will use intensively. In lesson 7 of quantum optics I, you already encountered a wonderful device: a polarization splitter, followed by two photo-detectors. When a single photon wave packet falls on it, it will be detected in one of the two channels, respectively labeled plus one or minus one. The channel labeled plus one is associated with the polarization along the axis of the polarizer, which we assume aligned along x, so that a photon with polarization x is detected with a 100 percent probability in channel plus one. Similarly, a photon with polarization y is detected with certainty in channel minus one. This measurement corresponds to a quantum observable A hat, formed as the sum of the two projectors on x or y, with coefficients plus one and minus one, which are the eigenvalues of the observable. In the basis (x,y) of the polarization space, the observable A hat is a diagonal matrix, with plus 1 and minus 1 on the diagonal. This should remind you of the formalism of spin one-half, which is indeed formally equivalent. A classical beam with a polarization at an angle theta from the x-axis would be partly transmitted and partly reflected, with coefficients cosine theta squared and sine theta squared, given by the Malus's law. A single photon linearly polarized at theta from the x-axis will not be split. It will be detected either in the plus 1 channel or in the minus 1 channel. Since the state is expressed on the eigenstates of the measurable observable, the probabilities are given by the squared modulus of the coefficients of that expression. The result corresponds to the Malus's law. These probabilities can be measured by repeating the experiment many times with a single-photon prepared always in the same state of polarization. If one rotate the polarizer around the axis z by an angle alpha, what is the observable associated with it? The eigenstates of that observable are polarizations along the polarizer axis at an angle alpha from the x-axis, and along the orthogonal direction. The observable is the sum of the two associated projectors, with signs plus 1 and minus 1. Expressing these polarizations on the basis (x,y), you will obtain easily the matrix representing the corresponding observable in the (x,y) basis. This is the matrix A_alpha. You can check that its eigenvalues are plus one and minus one and that the eigenstates are Epsilon alpha and Epsilon alpha plus Pi over 2. You are now equipped with a formalism that will allow you to discover a new quantum mystery: entanglement. You will do it in the case of photon entangled in polarization.