You know what is a beam splitter, also called a semi reflecting mirror. It is a central component in many interferometers, for instance a Michelson interferometer In this section you will discover how it is used to demonstrate experimentally the typically quantum behavior of a one photon state. But let us first see how to describe its effect in classical optics. On a beam splitter, an input mode 1 is partially reflected into mode 3 and partially transmitted in mode 4. Here, the partially reflecting layers are deposited on the front face of a plate and we ignore the shift due to the plate to simplify the drawing. There is, obviously, another input mode that is coupled to the output mode 3 and 4: it is mode 2, symmetric of mode one , in the beam splitter. In classical optics, one describes the effect of the beam spliter by expressing the output electromagnetic fields in mode 3 and 4, as a function of the input fields in modes 1 and 2. More precisely, considering modes that are plane monochromatic polarized waves, all with the same frequency and with the same polarization perpendicular to the figure, it suffices to give the relations between the complex amplitudes at point O. In order to make this expression easier to read we take the fields at point <b>r</b> = 0, so that all the complex exponential space factors, exponential of i <b>k.r</b> are equal to one. Similarly, all the fields have the same frequency, and the time factors, exponential of -i omega t can be factored and ignored. To simplify we assume that the coefficients r and t are real, which is always possible by choosing the origin of the phases of the various waves. But we could have taken complex coefficients, submitted to a condition we will see below. The notations, small r and small t, obviously stand for reflection and transmission. We consider an ideal, lossless, beam splitter. This implies conservation of energy or more precisely of the sum of the Poynting vectors, that is to say the transported power. For a single input wave in channel one this conservation demands that the squared modulus of E3+ added to the squared modulus of E4+ equals the squared modulus of E1+. This is ensured by the condition r squared plus t squared equals one. One uses sometimes capital R and T instead of small r squared and small t squared. The low case coefficients are the amplitude coefficients. Upper case letters are intensity coefficients. If now we have two input waves, E1 and E2, the opposite signs of the terms with the r coefficient ensures the energy conservation if capital r plus capital t equals one. You can check it by adding the squared modulus of E3+ to the squared modulus of E4+ expressed as a function of E1+ and E2+. Formally one can write the relation between the input complex amplitudes and the output complex amplitudes using a matrix S with complex coefficients of which the relations above, are a particular case with real coefficients. Conservation of energy demands that the matrix S be unitary, since a unitary matrix conserves the norm, here the sum of the squared modulus of the amplitudes in the output space or in the input space. A unitary matrix is such that its conjugated transposed is equal to its inverse. A sufficient condition for unitarity is written here. You can check that the coefficients of the matrix, small r, small t, etc, fulfill these relations. In the general case, the matrix S describing a lossless beam splitter has complex coefficients fulfilling these relations. Here, the sum of the squared modulus of the amplitude in the input space or in the output space. Let us now switch to quantum optics. In quantum optics, one usually describes a radiation state by a state vector |psi_12> in the input space, That is the state space associated with the two modes 1 and 2. This radiation can be as well described by a state vector |psi_34> in the space spanned by the two modes 3 and 4. It would appear natural to look for the transformation U expressing |psi_34> as a function of |psi_12>. This transformation must be unitary since |psi_34> must have a norm equal to 1, as | psi_12>. In a sense, going from the input space to the output space to describe the same radiation state can be considered a change of basis. This approach where one transforms the state representation turns out to be intractable, because the matrix describing that transformation in the basis of number states, |n> is unreasonably large in general, as I will show you now. Let us start with a simple case of a state with n photons in mode 1. The state vector in the input space is just n photons in mode 1 and zero photons in mode 2. Notice the way I write that state with n photons in mode 1 and zero in mode 2. From a mathematical point of view, we have a tensor product which should be written with this symbol. But, when there is no ambiguity, we use a simpler notation without the tensor product symbol. How do we represent that state in the output space? Conservation of the number of photons tells us that the output state is a priori composed of a component with zero photon in state 3 and n photons in state 4, plus a component with one photon in state 3 and (n-1) photons in state 4 etc. We need n+1 coefficients to describe the transformation of this very simple input state. Try now to imagine the case of a two modes input state with up to a maximum of n photons in each input mode. We need (n+1) to the square coefficients to describe the most general two input modes state, which can be expanded over all possible n1, n2 states, with n1 and n2 less than or equal to n. For each input state, the output state involves up to 2n photons. And we need 2n+1 to the square coefficients, to describe the most general output state, with up to 2n photon in each output mode. It means that we need a matrix with 2n+1 to the square rows, and n+1 to the square columns, to express |psi_34> as a function of |psi_12>. Of course, in our example, many coefficients would be zero. For instance, because you cannot have more than 2n photons in |psi_34>, so a component with 2n photons in 3 and 2n photons in 4, cannot exist. In fact, the dimension of the output space should be equal to the dimension of the input space, because we have a unitary transformation. So there is a maximum of n+1 to the square output states to consider. But writing that transformation would anyway be a nightmare, as soon as n is larger than a few units. Fortunately there is a simpler method to express quantum quantities expressed in the output space, the space of modes 3 and 4, knowing the state of radiation expressed in the input space, the space of modes one and two. I'm going to sketch a somewhat formal justification of the method. If you feel too uncomfortable with it, you can just wait for the result, which you must know since you will use it quite often. This is the method, let us assume we want to calculate the quantum average of an observable O, defined in the output space, for instance, the number of photons in mode 3. It is expressed as a function of the state of radiation in the output space. But in general, we do not know the expression of the radiation state in the output space. What we know is the expression of psi in the input space. But even if we cannot write explicitly the matrix representing the transformation that relates |psi_in> and |psi_out>, we can anyway formally write |psi_out> equals U |psi_in>. I have replaced notation |psi_12> and |psi_34> by |psi_in> and |psi_out> to emphasize the generality of the method. We do not need to know explicitly the transformation U. It suffices that it exists. The transformation U allows us to express the average as a function of |psi_in> using the transformation U and its hermitian conjugate U_dagger. We can slightly modify the way it is written and now the average of O out can be considered an average in the input space, provided that we consider U_dagger O U, as the expression of the observable O in the input space. In fact, it is exactly how operators are transformed in a change of basis. Finally, the average of an observable O expressed in the output space is equal to the average of the transform of O in the input space. So if we know the radiation state |psi> in the input space the only thing we need to know is how to express O in the input space which we note Oin. It turns out that this is often quite simple as we will see now in the case of the beam splitter. Let us use the formula we have just established in the case where we want to calculate a quantity O-output that depends on the electric field operators E3 and/or E4. For instance we want to calculate the simple photo detection signals, w1 in the output channel 4. We need thus to evaluate the average of E4- times E4+ in the output state, but what we know is the input state. All we have to do is express E4+ and E4- as a function of the operators E1 and E2 and calculate the average in the input state. Since the field operators in each input or output space have only two components, the transformation of the field is a simple matrix, 2 by 2. This is much simpler than transforming the states. But there is more. Do you know what is the matrix S in the case of a beam splitter? Nothing else than the matrix transforming the classical complex amplitudes We know that matrix, so we can express the output field operators as a function of the input operators. In fact, this result can be generalized to any optics system in which light is propagating. So transformation between the input and output field operators is the same as the transformation between the input and output classical fields. This is an extremely important result that you must be sure to understand because it is the key of many calculations in quantum optics which turn out to be quite simple when using that method. We will see examples of applying it in the rest of this lesson. But before doing it, let me give you two justifications for the unbelievably simple recipe that the matrix transforming the operators is the same as the matrix transforming the classical amplitudes. Firstly, note that the same relations hold for the quantum averages of the field. You will learn in a future lesson that there are states of the quantized radiation called quasi classical states whose averages evolve exactly as classical fields. The relation here applied to quasi classical states makes a consistent connection between classical and quantum optics. The second justification for these relations is that they imply the conservation of the fundamental commutation relations, between a and a_dagger. To show it you can first notice that the same relations hold for the E+ fields and for the annihilation operators, since the frequency and polarization are the same. And the fields are taken at r = 0. From this, using the commutation relations in the input space, you will easily demonstrate that the commutation relations in the output space, are what you expect if you want to have a consistent formalism. More precisely the commutator of a and a_dagger in an output mode, has the value of one, and all the other commutators are null in particular between different modes. But enough of dry formalism It's time to apply it to a one-photon state impinging on a beam splitter. (music plays)