In Week 3 of Quantum Optics 1, I explained how to deal with a beam splitter, an optical device that plays an essential role in many important quantum optics experiments. One distinguishes the input modes one and two and the output modes three and four, and there is a transformation that links the expressions of the state of radiation in the input space and in the output space. In fact, it is the same state expressed on modes one and two in the input space while in the output space, it is expressed on modes three and four. This transformation can be understood as a change of basis expressing the same state either with modes one and two or with modes three and four. In Quantum Optics 1, I insisted that in general the matrix transforming the state is unreasonably large as soon as there are more than a very small number of photons in the input modes. This is because for each mode, the dimension of the state space is as large as the maximum number of photons involved, and one must consider all the matrix elements associated with non-zero number of photons in the input modes and the output modes. There is an elegant solution to get around this problem. Transforming the electric field observables rather than the states. Indeed, the transformation of the electric field involves only a two-by-two matrix. Moreover, that matrix is exactly the same as the one for classical fields. I have written it here in the case of real and positive coefficients r and t, such that r squared plus t squared is one. Then energy is conserved for classical fields provided that the fourth coefficient is minus r. This transformation applies as well to the destruction operators associated with the operators E^(+) hat. The unitarity of the matrix entails the fact that the commutators in each output modes are one, while the cross commutator is zero as in the input modes. This allows us to express output quantities directly as a function of the input state which is known. For instance, the single detection probability in point r_4, which is in the output space is readily expressed as a function of the input space. These expression allows you to calculate the result for any input state. For a single photon in the input mode one, you obtain a detection probability proportional to t squared, the intensity transmission coefficient. For a quasi-classical state alpha, the probability of photon detection is proportional to the intensity transmission coefficient times the square modulus of alpha, that is to say the average number of photons. This kind of method can be used to calculate the average value of any observable in the output space, knowing the expression of the radiation state in the input space. We will show it on the example of the electric field observable. You have, in the previous section of this lesson, all the elements necessary to calculate the average output fields for a quasi-classical field in input one. In section three, I reminded you how the electric field observable in mode ℓ is expressed as a function of a_ℓ and a_ℓ^†. We can then use the transformation of the a operators to calculate the average electric fields in output three or four. For instance, the electric field in output three is expressed as a function of the a_1 and a_1^† operators. While the operators a_2 and a_2^† are omitted because we take vacuum in mode two. Finally, the average electric field in output mode three is equal to the one corresponding to a quasi-classical state r times alpha_1. A similar result is readily obtained for output four in which the average electric field is the one corresponding to a quasi-classical state t alpha_1. You should do the calculation to be sure that you have understood. The average values are thus transformed exactly as the classical fields would be transformed on the beam splitter. At this stage, however, we cannot conclude that the quasi-classical state alpha_1 is transformed into quasi-classical states corresponding to the transformed classical fields. In fact, this statement is true and we are going to show it now. The transformation of the operators can be used to find the output state. This is very interesting in cases where the output state assumes a simple form, but even in the general case where it is quite complicated and involves many components, the method you are going to learn now is quite efficient. The idea is to express the state in the output space using the action of the creation operator, a^†, on the vacuum. For instance, one photon in mode one and zero in mode two is obtained by applying a^†_1 to the vacuum. Now, it is easy to express the creation operators of the input modes as a function of the creation operators of the output modes. We start from the relation we know for destruction operators, invert it, and takes the hermitian conjugate. In these operations, you should have the complex conjugates of r and t, but we have chosen them real so we do not mind. Now, you can use these relations to express the state in the output space. For instance, in the case of one photon in mode one and no photon in mode two, one finds a linear superposition of one photon in mode three and one photon in mode four with coefficients r and t. Simple, isn't it? Notice that I have written the state as |psi> without any label to emphasize that it is the same state of radiation expressed in different basis, either the input states basis or the output states basis. If there are many photons in the input modes, one obtains in general complicated expressions with many terms in the basis of the output modes. But there is a remarkable exception that is a case of a quasi-classical state. To apply the method I've just shown you, You must first express a quasi-classical state as a function of the vacuum using creation operators. This is quite easy starting from the decomposition of alpha over number states. Can you figure out how to continue? Remember that each n state can be expressed as a function of the vacuum by applying n times the creation operator to the vacuum. You then get immediately an expression of the state alpha_ℓ as a function of the vacuum and of the creation operator. You can obtain a yet simpler expression remembering that an exponential can be expressed as an infinite series of terms, and then recognizing exponential of alpha times a^†, hence the final expression of alpha_ℓ which is simple and nice. You may feel uncomfortable with the notion of an exponential of an operator. In fact, just think of it as a notation to express an infinite sum of terms. If you had in the exponent several operators that do not commute, then it would be delicate, but here there is only one operator, a_ℓ^†, and you can apply the same rules as for ordinary numbers. With this expression of a quasi-classical state at the input mode one, we can readily use the method presented in the case of one photon at the input. Using the expression of a_1^†, you will obtain an expression of the state in the output space. Here again, you can use exponential of operators as if there were ordinary numbers, because the operators a_3^† and a_4^† do commute. A simple manipulation, taking into account the fact that r squared plus t squared equals one yields a remarkable result. A quasi-classical state in one input channel yields one quasi-classical state in each output channel with amplitudes same as they would be for a classical field.