Let us start with an important tool of modern quantum optics, a measurement method named balanced homodyne detection, closely associated with quantum optics observable that can be measured with present technology. You know that not all the observables of the quantized electromagnetic field can be measured in the case of visible light because the frequency is too high compared to the response bandwidth of detectors. Consider the electric field observable in a single mode quasi-classical state at the frequency of 5 x 10E14 Hz corresponding to orange light Its average value oscillates at that frequency. But even the fastest detectors take the average of that signal over a time interval not less than about one picosecond, yielding a response equal to zero. It is because of that impossibility to directly measure the electric and magnetic field observables, that experimental quantum optics is primarily based on photoelectric measurements. I leave it to you to demonstrate that for the case of a single mode quasi-classical state, the probabilities per unit time of single and joint photodetections are constant. Note that I have used here the Heisenberg forms where the observables evolve with time. In fact, there are observables directly related to the electric field which can be measured even in the case of visible light: These are quadrature observables of a mode of the field. To determine them, one uses a generalization of the scheme yielding the beat note between two lasers encountered earlier. It consist of entering the field to analyze in the input one of a beam-splitter and a large quasi-classical field in the second input of the beam splitter. That second field emitted by a laser is called the local oscillator. We describe that field by a quasi-classical state alpha LO, and the whole input radiation is described by the tensor product of the two input states. There is a detector in each output port of the beam splitter and one measures the difference between the photocurrents. For an equilibrated beam splitter with equal coefficients of transmission and reflection, this is called a balanced detection. If the local oscillator has a frequency different from the one of the mode of input one, one has a heterodyne detection scheme generalizing the scheme studied in a previous lesson. Here, we rather consider the case where the local oscillator has the same frequency as the mode of input one: that is a homodyne detection scheme. You know how to evaluate the photoelectric signal in port four by using quantities of the input space. In the lesson on multi-mode quasi-classical states, we used that method to calculate the beat node between two lasers with different frequencies entering ports one and two. Here, the modes we consider in inputs one and two have the same frequency. So, all the time dependent factors disappear in the photoelectric signal. The k-vectors of the two modes are images of each other. In particular, they have the same modulus, and the spatial dependence of the photoelectric signal also disappears. We also assume that the polarization of modes one and two are image of each other so that the polarizations disappear from the photoelectric signal, and we can use scalar expressions for the fields. At the end of the day, the probability of photodetection per unit time is constant, and its integral over the volume of quantization is the average number of photons in output four in that volume within the quantum efficiency factor eta. In this formula, the product of the transverse area S of the beam by the time T of measurement is the volume of detection divided by c. The average photocurrent, i_4 bar, is thus number of photons N_4 divided by capital T, times the electron charge and the quantum efficiency. Note that the result is independent of T since the mode volume is proportional to T. Equivalently, using the expression of a_4 as a function of a_1 and a_2, you can express i_4 as a function of the creation and destruction operators in the input space. But you must be careful to respect the order of the operators that do not commute. See what I mean? [MUSIC] Similar calculations yield the average number of detected photons in output three during capital T, all the expression of the average photocurrent i_3. You can now calculate the average photocurrent's difference, which is proportional to the average of N3- N4. Using the expressions of N3 and N4, you find a simple form for N3- N4. If now you take the average in the input state, the quasi-classical state of input 2 yields a complex number alpha LO and you obtain a result depending only on the average of the creation and annihilation operators in mode one, taken in the state psi_1. It is remarkable that this result is independent of time. As announced, we thus obtain a result that does not oscillate at high frequencies and which therefore can be measured with existing technology. You might argue that a and a dagger hat are not Hermitian operators, that is to say they are not observables and thus do not correspond to genuine measurements. This is true but the sum of the two terms is associated to a Hermitian operator. This is clearer if you express alpha LO with its modulus and argument. The operator of which one takes the average is indeed Hermitian. If you separate the real and imaginary parts of alpha LO, you obtain the sum of two Hermitian operators, which can be measured separately choosing phi LO either equal to 0 or to pi over 2. We will focus on these operators in the next section. But let us first complete this section with the calculation of the fluctuations of the photocurrent difference, that is to say the noise in the balanced homodyne detection. Consider again the signal obtained in balanced homodyne detection, that is to say the difference between the photocurrents. We have shown that its average is proportional to the average of the difference of the numbers of photon N3 and N4. These difference assumes a simple form when expressed in the input space. But what is the meaning of this average? In fact, there are two averages. The first one, indicated by an overbar over the photocurrent's difference, is associated with measurement time capital T. The other one, indicated by the brackets, is a quantum average. It is associated with the fact that if we repeat measurement for a large number of times, we find a statistical distribution of results of which the average is what we have calculated. The statistical distribution has also a dispersion, traditionally called the noise of the measurement. It is characterized by the variance delta i3- i4 squared, which is equal to the average of the square minus the square of the average of the quantity i3- i4. The square of the photocurrent difference is proportional to the square of N3- N4. So the variance of the signal is proportional to the variance of N3- N4. Hence, again, it is the difference between the average of the square and the square of the average. We want to calculate the average of the square of N3- N4 in the state Psi_in. When we expand the square of N3- N4, all terms involving a dagger 2 and a2 are in normal order except one term. We can rearrange that term in the normal order using the commutation relation of a2 and a dagger 2. Using the fact that the state 2 is a quasi-classical state, we are left with terms involving only averages of the operator a1 and a dagger 1. These averages are obviously taken for the state psi 1, which we have not written explicitly in order to keep the expressions lighter. Notice that the first four terms are greater than the fifth one by a factor equal to the squared modulus of alpha LO, that is to say by the number of photons in the beam of the local oscillator divided by the number of photon in the state psi 1 during the measurement time T. That number is large compared to one, and we can neglect the last term. It is interesting to rewrite the formula using the expression of the complex number alpha LO. One then has the average of the square of the Hermitian operator already introduced. Wrapping up, we obtain an expression of the noise in the balanced signal as a function of the Hermitian operator that we have just introduced taken in the input state that is analyzed by the balanced detection. We will see now the meaning of that observable. It is, within a factor root of h bar over 2, the quadrature component Q of phi_LO, which we will define now.
