As all great physics theories, quantum physics in general and quantum optics in particular, has in it more than anticipated by founding fathers. It is only in the 1980s that some theorists realized that the formalism of quantum optics allows for intriguing states of light, which nobody had yet thought of or observed. These are squeezed states of light. Beyond the interest of novelty, these states would in principle, allow physicists and engineers to improve measurements by passing a limit that had been considered fundamental for a long time, the so-called standard quantum limit. In this section, you will learn about these intriguing states of light. I will not tell you here how to produce squeezed states of light. Let me just mention that such states are not found in usual sources of light and producing such states was an important subject of research in the early 1980s. A first success was obtained in 1985 by Dick Slusher and his team at Bell Labs. Soon followed by several other groups, they all used some non-linear optics effect. Non-linear optics will be the subject of a future lesson in this course. A simple way to define squeezed states is to generalize the definition of quasi-classical states. You remember, that a single mode quasi-classical state Alpha is defined as an eigenstate of the annihilation operator. Similarly, a single mode squeezed state is defined as an eigenstate of a generalized annihilation operator capital A hat, which is a combination of the annihilation and creation operators a and a dagger of the mode. The coefficients are the hyperbolic cosine and sine of a real number R. To simplify the writing, I have dropped the subscript ell, but you must remember that we are in a single mode. As the usual annihilation operator a, A_R is not Hermitian since it is different from its Hermitian conjugate. So you should not be surprised that it's eigenvalues are complex numbers. It is also remarkable that the commutator of A_R and A dagger R is equal to 1. This is why we can call A_R and A dagger R generalized annihilation and creation operators. Of course, the expressions of A_R and A dagger R as a function of A and A dagger, can be inverted. We will use these expressions all along this lesson. Let us first calculate the average electric field in a squeezed state Alpha R. For that, we start from the Heisenberg form of the single mode electric field. We can express the electric field as a function of A_R and its conjugate using the transformation formulae just seen. Remember, that hc means Hermitian conjugate. Since the squeezed state Alpha R is an eigenstate of A_R, you can easily find the value of the average of the electric field. Here, cc means complex conjugate. In fact, this is nothing else than the expression of a classical field with a complex amplitude Alpha prime whose expression as a function of Alpha and R is here. It can also be considered the average field for a quasi-classical state characterized by the complex number Alpha prime. In the following, we will often consider the case when Alpha is real. Alpha prime is then also real and assumes a simple form Alpha times exponential of minus R. As usual in quantum mechanics, it's not enough to calculate the average of an observable. It is also essential to know the dispersion of the results either by giving the complete probability law or at least the root mean square dispersion. This is what we will calculate here. You know that the variance, that is to say, the square of the RMS dispersion is equal to the average of the square minus the square of the average of the observable. We do that calculation at the position r=0 where the electric field depends only on time. Generalizing at position r would be trivial, you know how to do it. We have already calculated the average of the field, and we need now to calculate the average of its square. The squared observable has many terms which can be gathered into four different types of terms. The crux of the calculation is to respect the ordering of A_R and A dagger R which do not commute. As for quasi-classical states, in the expansion, you better keep unchanged the three terms in normal order with A_R on the right and A dagger R on the left, then you replace the term A_R A dagger R, which is not normally ordered, by 1 plus A dagger R A_R. Grouping this last term with the existing term, you are then left width three normally other terms plus a scalar term. When you take the average in the squeezed state Alpha R in order to calculate the variance, the normally ordered terms give a contribution exactly equal to the square of the average of E, and you have in addition, the scalar terms which are the variance. At this point, you should have recognized the same trick that you learned about quasi-classical states. To calculate the variance of an observable, you write the squared observable in the normal order, and only keep the average of the supplementary terms coming from the commutation relation to obtain the variance. Since the commutation relations as well as the action of A_R and a_dagger_R on squeezed states are similar to the ones of a_dagger and a for a quasi-classical state, the same trick applies. If you are not yet comfortable with that way of doing the calculation, do the full calculation and it will eventually become clear. This trick is very useful but you should remember that it applies only to eigenstates of annihilation operators. Now, a few lines of algebra will allow you to put the variance of E in a remarkable form. This formula expresses a fundamental property of squeezed states. In contrast to quasi-classical states whose variance is constant and equal to E1 squared, the variance of the electric field in a squeezed state varies as a function of time at a given position. Now, the variation is such that there are times where the variance may be less than E1 squared, that is to say less than the value for a quasi-classical state. Is it clear to you? Have you found the times when the variance of the field is less than E1 squared? One must distinguish the cases of R positive or negative. Let us first consider the case of R negative. Then at time such that sine Omega t is null, the variance is less than E1 squared. To be specific and have simpler expressions, we take Alpha real positive so that Alpha prime is also real and positive. The average value of the field then evolves as sine Omega t as shown in red on the plot. I have indicated with the green curves the standard deviation added to the average field for the case of R negative. There are times when the dispersion is increased, and times when it is reduced. To compare with the standard deviation that one would have for a quasi-classical state of same average value, I have added black dashed lines corresponding to a standard deviation E1. It is clear that for the squeezed state, the dispersion is bigger at points pi over two, three pi over two, and smaller at points zero by et cetera. It is illuminating to use the complex plane representation to better understand the evolution of the electric field average and dispersion. As in the case of a quasi-classical state, the average is given by the projection on the imaginary axis of the complex number minus two Alpha prime E1, times the complex exponential evolution factor, rotating at angular frequency minus Omega. But when it comes to dispersion, the situation is dramatically different from the case of quasi-classical states. Rather than the projection of a disc of diameter two E1 shown by the black dotted circle, the dispersion is a projection on the imaginary axis of an ellipse with a long axis larger than two E1, and the short axis smaller than two E1. This ellipse is rotating with the complex amplitude so that the projection is modulated between values larger and smaller than two E1. Since the short axis is tangent to the circle, it is clear that the dispersion is minimum for zero pi, two pi, et cetera, while it is maximum for pi over two, three pi over two, et cetera. The maximum and minimum values of the dispersion are E1 times exponential of minus r, and exponential of r respectively. Their product is equal to E1 two to the square. That is to say the minimum value compatible with the Heisenberg dispersion relation associated with the fact that field observables at two instants separated by one quarter of the period of oscillation have a non-null commutator. We will come back to this point in the next section. Consider now the case of positive R while Alpha is still real and positive. The plot shows the evolution of the field and its dispersion. The standard deviation is maximum at Omega t equals zero by et cetera, while it is minimum at Omega t equals pi over two, three pi over two, et cetera. Again, the dotted black lines represent the standard deviation for quasi-classical state with the same average evolution. A magnification around Omega t equals pi over two, modulo pi, clearly shows the reduction of the fluctuation below the value for that quasi-classical state. Here again, the evolution can be described using a rotating ellipse in the complex plane. It is still centered on a complex number whose modulus is Alpha prime. But its long axis is tangent to the rotation, while its short axis is along the radius. The projection on the imaginary axis yields the average and the dispersion of the field. The dotted black circle represents the standard deviation for quasi-classical state Alpha prime. Here again, the product of the dispersions has the minimum value compatible with the Heisenberg relation. This means that a squeezed state is a minimum dispersion state, as a quasi-classical state.
