Let us look again at what is measured in a balanced homodyne detection. The average and variance of the balanced signal are proportional to the average and variance of the observable a_1 dagger, times exponential of i Phi_LO plus Hermitian conjugate. Separating the real and imaginary parts of the complex exponentials, we obtain an expression with observables that should ring a bell in your memory. Do you remember them? These Hermitian operators are within a factor, the quadrature observables, Q and P of the mode one, introduced in the process of quantization of radiation of that mode. With the factor root of h bar over 2, they are canonically conjugate observables with a commutator equal to i h bar. Each of these observables can be measured separately, by choosing the phase of the local oscillator Phi_LO equal to 0 or Pi over 2. At this stage, I suppose that many of you are intrigued by these observables, which were introduced in the process of quantization, but almost never used in quantum optics 1. In fact, they are very important quantities in modern quantum optics. Now, I have to tell you about a peculiarity of the formalism of quantum optics, which is not very elegant. It is all about a question of conventions and notations, and it is tempting to choose notations that would improve the situation. It turns out, however, that the majority of papers and books of quantum optics use these not so nice notations. So we have no choice if we want to be comfortable, when reading the usual literature. The problem arises when we want to express the electric field observable, as a function of the quadrature observables. Starting with the Heisenberg expression of the electric field observable in a mode ell, you can express it as a function of the quadrature canonical observables, Q_l and P_l, by inverting their expressions. The result is a form, where the two quadrature components are respectively multiplied by evolution factors sine or cosine of k r minus Omega t, times various multiplying constants. The constant E1 times root of two over h bar are necessary to have homogeneous quantities, but there is a minus sign, which is a nuisance, believe me. To keep that nuisance in a corner a good habit, consists of reasoning as much as possible with the quadratures, Q and P, and expressing the electric field only at the end of the calculation when necessary. This is what we will do as often as possible in the following of this lesson and of the course. If you have never used the notion of quadratures, you may find them quite abstract and have some difficulty to get an intuition about them. In fact, the notion of electric field quadratures stems from classical electromagnetism. To show it, let us consider a classical field in the mode l. The mode is characterized by its k vector and its polarization epsilon l. The state of the field in that mode is fully determined by the value of the complex amplitude equal to Alpha l, within a constant factor i times E1_l. The complex number, Alpha l, can be defined by two real numbers, its modulus and its argument Phi l, and the field assumes a familiar form as a function of these quantities. But the complex number, Alpha l, can be as well defined by its real and imaginary parts. These are equal within a constant to the classical quantities, Ql and Pl, corresponding to the observable Ql hat and Pl hat. To sum up, the state of the classical field in the mode is fully characterized by two real numbers in different manners. It can be the module and the phase of the complex amplitude, but it can as well be the quadrature components. If now we take the point of view of quantum optics, it turns out that the two quadrature components correspond to genuine observables. That is to say, Hermitian operators. While it is not the case for the module and the argument, since the phase is not a fully legitimate observable in quantum mechanics. This is why, to measure the state of the quantum fields in a mode, measurements of the quadratures constitute a privileged method. Let me finish with some remarks about vocabulary. Using the quantity Ql and Pl, we can obviously express the classical field as the sum of two components oscillating respectively as sine of Omega l t minus k l r and cosine of the same quantity. The first component has an amplitude proportional to Ql and is called the in-phase quadrature. The second component as an amplitude proportional to Pl and is called the out-of-phase quadrature. These two components are out of phase of each other by Pi over 2. But what means in phase and out of phase? In phase with what? Remember, that in order to quantize radiation, we introduced modes. That is to say, functions on which the most general classical electromagnetic field can be decomposed. In free space, a convenient basis is made of polarized monochromatic traveling waves, characterized by a k-vector and a polarization. Then, the state of a classical field in that mode is fully determined by two real numbers, for instance, the modulus and the phase of the complex amplitude. But the result has an objective meaning, only if the origin of time and space has been defined. If we change the origin of time, the same field must be described by a complex amplitude with a different phase, Phi prime. It means that we must make a choice, for instance, fix the origin of time. But when this is done, it means that the same origin of time must be used for all fields, the local oscillator and the field to analyze. More precisely as we will see, once a common origin of time is decided, we can choose a relative phase in order to select the observable which is measured, either Q or P or a combination. But before studying in detail the role of the relative phase, you must realize that controlling it demands to fulfill a crucial condition. The two fields must be phased locked. That is to say that, the two clocks that define the time evolution of each field, must have a relative drift much less than one period over the whole measurement time. Although not fully impossible with cutting edge technology, this is quite demanding. Fortunately, there is a much easier solution, which is to lock the phase of the two fields to the same clock. If that clock drifts, it does not affect the phase difference between the two fields. One can thus easily control the phase difference in homodyne detection. This is the expression of the observable that we can study with balanced homodyne detection. By choosing the phase of the local oscillator equal either to 0 or to Pi over 2, one can make measurements of either Q1 or P1, which are the quadrature either in phase or out of phase with a master clock. Note that one has to make a choice. It is not possible to measure both quadrature simultaneously. This is consistent with the fact that their commutator is not null. Of course, if one repeats many times measurements on the same field, it is possible to make measurements on both quadratures. This is the meaning of figures, where results of measurement of both quadratures are plotted on the same diagram. You will see such figures in the next sections. Do not forget that these figures demand many measurements. Repeating many times measurements on identically prepared system, one can obtain the average value and the variance of Q1 and P1. You know that since their commutator is equal to i h bar, they obey the Heisenberg dispersion relation. If rather than 0 or Pi over two, we choose a phase Theta or Theta plus Pi over 2 for the local oscillator, balanced homodyne detection allows one to measure the components Q1 of Theta or P1 of Theta. These observables have a commutator equal to i h bar and their standard deviations obey the Heisenberg relation. You can develop the expressions of Q1 of Theta and P1 of Theta, to express them as function of Q1 and P1. This transformation can be interpreted as a rotation of angle Theta. Consider again, the expression of the electric field observable as a function of the quadrature observables. It is clear that at position and time such that k r minus Omega t equals 0 modulo 2 Pi, the electric field observable is equal to the P quadrature observable, within a factor. So when we measure the P quadrature, in fact, we measure the value of the electric field at well defined times for a given position. So we can get its average value at these times, as shown in that example of a quasi-classical state in input 1. But we can also get the electric field dispersion or even though whole distribution at these times. This is called sampling the field at these times. This reasoning can be extended to any quadrature observable, which means that we can sample the field at any instant of the period.
