The polarization of a classical electromagnetic wave was introduced in our lesson about quantization of a single mode of the electromagnetic field. You learned more when I introduced the most general form of the electromagnetic field in the absence of charges, which can be decomposed into a sum of independent modes. In both cases, we consider modes of a particular kind, polarized traveling monochromatic plane waves. Each of these modes is characterized by a k-vector, defining the direction of propagation and the frequency. You remember that, for each mode, the electric field is perpendicular to the k-vector. It thus belongs to a two dimensional space which can be spanned by two perpendicular unit vectors, epsilon_ell1 and epsilon_ell2, such that epsilon_ell1, epsilon_ell2, and k_ell form a direct triad. These notations are quite involved. They must be understood in the following sense: there are two different modes, ell1 and ell2, which have the same wave vector, k_ell, but two different polarizations, epsilon 1 and epsilon 2, associated with the two modes. To simplify the figures and the notations, we will take k_ell along the z-axis and the polarization unit vectors along x and y. This does not reduce the generality of what we are going to do. The two modes, ell1 and ell2, associated with the same k-vector, but with the two different polarizations, can then be designated by the simplified subscripts x and y. In previous lessons, when we used modes of the classical electromagnetic field to represent quantized radiation, we usually ignored polarization, either because we were considering a single mode, so only one polarization was involved, or because we considered several modes with the same polarization. Here, we consider another special case of quantized radiation involving two modes with the same k-vector but different polarizations. We take the k-vector along z and the two polarizations along x and y. Since there is only one k-vector, the classical field is a travelling wave. How can we write the most general state of quantized radiation associated with such a travelling wave? That is to say, based on these two modes? It belongs to a space that is the tensor product of the spaces of states of the two modes, of which a convenient basis is formed by the tensor products of number states of each mode. It can thus be written as a superposition of states (nx, ny). Recall that a state (nx, ny) has nx photons in mode x, and y photons in mode y, and zero photons in all other modes. Even for such an apparently simple situation, the space of states has a dimension that increases exponentially with the number of photons in each mode. If we limit the number of photons to a maximum of one in each mode, the space of states has yet a dimension of four. Some of these states are entangled states. Entanglement is a surprising and important quantum property, which has been clearly demonstrated in quantum optics experiments. It will be central in our second course on quantum optics. Today we will restrict ourselves to the case of one-photon polarized radiation by keeping only the two components with one photon. We are left with superpositions of one photon in mode x and one photon in mode y. I've used detailed notations, but as usual we can ignore modes with zero photon and thus write it in a simpler form. Such a state is a one-photon state. Check that you can demonstrate that it is an eigenstate of the number of photons operator, N_hat. The space of states of such one-photon states has dimension two. There is a one-to-one correspondence with the two dimensional vector space in which the polarization epsilon belongs. The linear polarization basis epsilon x, epsilon y, corresponds to the basis 1_x, 1_y in the space of states. A polarization epsilon, defined by the angle theta with the x-axis, corresponds to a one-photon state cosine theta 1_x + sin theta 1_y. It describes one photon with polarization epsilon at theta from x. Polarization is a quantum observable whose measurement is based on the spatial separation of two orthogonal polarizations. It is remarkable that there are natural crystals that effect this separation. Such birefringent crystals display the double refraction phenomenon whose classical explanation demands a wave model of light, as was understood by Huygens as early as 1672 and fully modeled by Fresnel in 1821. Such crystals have been extensively used to build remarkable polarization analyzers, such as Wollaston prisms. But these natural crystals are rare and expensive. Modern thin films techniques allow industrial production of excellent polarizing beamsplitter cubes, which transmit one polarization and reflect the orthogonal one. Polarization beamsplitters allow one to measure the polarization of a photon by putting a detector in each output channel. For a one-photon wave packet, we get only one result that we arbitrarily call +1 for the transmitted polarization and -1 for the reflected polarization. But what is the quantum observable associated with such a measurement? A photon polarized along x will be transmitted, while a photon polarized along y will be reflected. These states are thus eigenstates of the polarization measurement observable with eigenvalues +1 and -1. The quantum operator associated with such measurement is the sum of the projectors associated with each polarization. In the bases, 1x, 1y, it is represented by a diagonal two by two matrix. Now, suppose that you have a photon with a polarization epsilon at an angle theta from the x-axis. Can you guess what will be the result of a measurement by a polarizing beamsplitter aligned along x? Just apply the basic principles of your elementary quantum mechanics course. Check the result, which I give now. Even if you did not get the right result, it was useful to think about the question. To calculate the probabilities of either result, it suffices to project the state of the photon onto the eigenstates associated with each result and takes the squared modulus. You'll find cosine theta squared and sine theta squared, a result that should not surprise you. Now, I ask you the following question: can we use this device to measure the polarization of a single photon? Stop the video and think about it. It is very important to appreciate this question, one of the most fundamental in quantum mechanics. A single photon can be detected only once. So we get either +1 or -1, and that's all. To determine theta, we must be able to determine the probabilities of +1 and -1. This demands to repeat the experiment many times, with each time the same state for the photon. It means that, to know the polarization of a one-photon state, you must have many identical copies of it. But if you have one photon only, there is no way to know its polarization. Some of you may argue that, with a light amplifier of the kind used in lasers, we could obtain many copies of our initial photon. It is an interesting idea which has been put forward and explored a few decades ago. It turns out that it does not work because of a very important theorem, the no cloning theorem, which I will demonstrate later in this lesson. It is sometimes useful to know the expression of the polarization observable for a beamsplitter at an angle theta from the x-axis. I show here how you can obtain the matrix representing these observable in the 1x, 1y basis, leaving it to you to do the intermediate calculations. You may want to check that the eigenvalues of the final matrix are +1 and -1, and that the associated eigenstates are 1_theta and 1_theta plus pi over 2. Quantum bits, or qubits, play a central role in quantum information transmission and processing. Quantum information can be encoded on an ensemble of qubits, and by transforming the quantum state of the qubits, one can process quantum information. Any system described by a quantum state in a two dimensional space can be used as a qubit. The polarization of a single photon-wave packet is an excellent qubit, but it has the inconvenience that it cannot be stored. So it is hard to use it to store information in a static quantum memory. They are called flying qubits. Spin one-half particles such as electrons or protons would be excellent static qubits. But experimentally, it is easier to use atoms or ions, which can be individually trapped. Using lasers with well chosen frequencies and polarizations, one can interact with only two of their levels and realize a two level system. It is remarkable that there also exists artificial qubits fabricated with the techniques of nanofabrication. For instance, the current in some circuits involving a Josephson junction is quantized, and it is possible to address only two quantized levels with well chosen microwaves. At this point, you may wonder, why such a fuss about qubits, that is to say, two-level systems? After all, we have known for decades classical bits, that is to say, systems which can be in either of two states designated by 0 or 1. There is a major difference. The quantum bit can be in state 0 or in state 1, but it can also be in a linear superposition of 0 and 1. And this is a different state, precisely defined. In contrast, it does not mean anything to have a classical bit with the value of 0 and 1 at the same time. It is the possibility to have qubits in superposition states that opens the immense potential of quantum information. This is why, when you read papers describing an experimental realization of a qubit, a crucial feature is the ability of that particular qubit to remain in a linear superposition for a duration as long as possible, the coherence time. Many of the schemes in quantum information are general and bear on qubits without specifying any particular realization. If, like me, you understand better abstract schemes by thinking of a particular realization, the polarization of photons is an excellent example, and I recommend that you study this section until you feel comfortable with photon polarization.
