In this section, you will encounter a graphic representation of a quantum field and it's quadratures in the complex plane, which turns out to be extremely useful to get an intuitive understanding of squeezed states. To introduce it, I will take the example of quasi-classical state. You have already seen a part of the content of the present section in the lesson on single mode quasi-classical states, but, it is important to master these notions, since similar reasoning will lead to dramatically different illuminating results in the case of squeezed states of light. This section, can thus be considered a warm-up to prepare the section on squeezed states. Balanced homodyne detection, allows one to measure any quadrature of the field. Repeating the measurement many times, we can thus determine the quantum average and the dispersion for any quadrature. What do we expect for a quasi-classical states, characterized by the complex number Alpha one? The calculation of the quantum averages of Q_1 and P_1 presents no difficulty and yield the results that we will use soon. It is also interesting to calculate the average of the quadrature associated with a phase Theta of the local oscillator. The calculation is not difficult, be sure that you can do it and obtain the right result. It is convenient to represent the results we have obtained in a complex plane with axis Q and P respectively, the real and imaginary axis. We use units of root two h bar. We can then plot the averages of Q and P on the corresponding axis. It appears that the quadratures are nothing else than the real and imaginary parts of the complex number Alpha, which we then represent in the complex plane. If now we consider the average of the quadrature Q Theta associated with the phase Theta of the local oscillator, it is a projection of Alpha onto an axis at Theta from the Q axis. Once again, you must be sure to understand the meaning of this representation. It is associated with many measurements on many copies of the same state of radiation or equivalently, many successive measurements. For each observable, one needs many measurements to determine the average value. Moreover, remember that it is not possible to measure the various quadrature simultaneously, since they do not commute and one must have different series of measurements for the various quadratures. We want now to obtain the dispersion of the results of measurements of each quadrature. As you know, the variance of a random variable can be expressed as the average of the square minus the square of the average. Let us first calculate the average of the square of Q. I give here the result of the calculation without the details, since we have made several times such calculation. Do you know where the term one comes from? Would you have put the term one without hesitation? If not, do the calculation and you will see that the one appears when you put the term a times a dagger in the normal order using the commutation relation of a and a dagger. You can then obtain the variance of Q_1, which has a remarkable value. This result can be obtained doing the full calculation. But you can also use a trick I taught you for quasi-classical states using the normal ordering. A similar calculation leads to the same result for P_1 or for any other quadrature component. All standard deviations, that is to say roots of the variances, are thus equal. It is remarkable that the product of the standard deviation of the two canonically conjugate observable Q_1 and P_1 is exactly h bar over 2. That is to say the minimum value permitted by the Heisenberg relation. One says that a quasi-classical state is a minimum dispersion state. For each series of measurements of a particular quadrature, we have an ensemble of results with a dispersion characterized by a standard deviation, whose value has been found equal to root of h bar over 2. The same for any quadrature. We then represent the spread of each observable by a segment of length twice the standard deviation. That is, one in units of root two h bar. Each of these segments can reconsider the projection on to the corresponding axis of a disc of radius one-half around Alpha. To sum up, if we measure a specific quadrature, Q of Theta, we obtain results with a distribution that corresponds to projection onto the Theta axis of points in the green disk around Alpha. This statement is of course too simplified, and in fact for each quadrature there is a distribution of the results of measurements characterized by a probability law. For a quasi-classical state, it is a Gaussian, centered on the average and with the standard deviation just mentioned. We can then represent this result with a two-dimensional Gaussian center on Alpha, integrating that to the Gaussian along the direction perpendicular to any axis for instance, the axis of Q of Theta, will give a Gaussian which is a probability distribution of Q of Theta. It is indeed a well-known property of two-dimensional Gaussians with rotational symmetry that the projection on any axis is a one-dimensional Gaussian with the same standard deviation. You may remember that vacuum is a particular case of a quasi-classical state, whose Alpha equals zero. So the dispersion of any quadrature component is the same. That is to say root of h bar over 2. We can then represent it with the same disk centered at zero. This suggests that the quasi-classical state Alpha is nothing else than the vacuum displaced by the complex number Alpha. This consistent with a property that you may remember, the quasi-classical state Alpha can be obtained by application to the vacuum of the operator, exponential of Alpha times a dagger, with a normalizing factor. In fact, that operator is a simplified form of a unitary operator named displacement operator D of Alpha, whose second term yields a null contribution when applied to the vacuum but which makes that operator unitary. You should remember that expressing a quasi-classical state with a displacement operator applied to the vacuum is quite useful. The graphical representation is a useful way of remembering it. It is well-known in classical physics that the complex plane representation is useful to visualize the time evolution of any oscillating phenomenon. In fact, it is a dynamical generalization of the static representation that we have just introduced. We will use it here to represent the time evolution of the electric field average and its dispersion for a quasi-classical state. You may find it not so interesting here but wait until we use a similar presentation for squeezed states. Then, it will allow you to better understand why these states are fascinating. Remember the expression of the average of electric field of a quasi-classical state of complex number Alpha one, as a function of position and time. To have simpler expressions and concentrate on the time evolution, we take the field at r equals zero. These expression shows that we can obtain the value of the average of electric field within a factor of minus 2E_1, by letting Alpha one rotate in the complex plane at the angular frequency, minus omega and projecting it onto the imaginary axis. The result is the expected sinusoidal function. The full interest of this representation is a possibility to represent not only the average, but also the dispersion of the results of the field measurement using a disk centered on the rotating complex number. The projection of that rotating disk onto the imaginary axis yields a band around the evolving average with a half width equal to the standard deviation. In units of 2E_1, that half-width is one-half. Once again, you can check that the band has a constant height in spite of a visual impression that it is not the case. The green arrows have all the same length. To finish with this section, I would like to draw your attention on to the difference between the latter representation which allows you to describe the evolution of the field with time and the formal representation which is about the quadratures. In contrast to the field evolution which is time dependent and which is unobservable in the current state of technology, the quadrature representation is static. Each quadrature is a constant quantity whose average and dispersion can be determined by effecting a series of balanced homodyne measurements. Changing the phase of the local oscillator, it is possible to measure various quadrature corresponding to projection onto axis at various angles with a Q axis. The collection of all results leads to drawing the green disk. Notice that the quadrature representation is nothing else than the field evolution diagram, at time t equals zero. The figures shown here corresponds to the case of a quasi-classical state, but similar representations can be used for other types of states of a single mode as we will see now.
