Until the beginning of the 1980s the fluctuations associated with the shot noise of a perfectly stable beam were considered the ultimate limit to the accuracy of optical measurements. Since shot noise is related to the quantum nature of photodetection, the associated limit was named the standard quantum limit, SQL. In quantum optics, a perfectly stable beam is described by a quasi-classical state whose probability of detection per unit time is constant, and the standard quantum limit is thus associated with fluctuations in quasi-classical states. But in the early 1980s Carlton Caves, a theorist working on the possibility of using optical interferometers to detect gravitational waves realized that it would be possible to pass the standard quantum limit by using a new kind of states of light, which he called squeezed states of light. The name squeezed indicates that fluctuation on some observables can be reduced, compressed. We have already seen in the previous section that for some values of the time of observation, the electric field observable can have a dispersion less than the standard quantum dispersion associated with a quasi-classical state. But you know that there is no detector fast enough to observe the oscillating electric field of visible or infrared light. So what we have seen in the previous section may seem a purely academic feature. In fact, as we saw earlier in this lesson, there are observables that do not depend on time and can be measured even for a rapidly oscillating field. These are the quadrature components of the field, which can be measured using a balanced homodyne detection scheme. For squeezed states of light, some quadrature have a dispersion below the standard quantum limit. As a result, it is possible to determine either the amplitude or the phase of an optical wave with an accuracy better than the standard quantum limit. Look again at the plot of the electric field observable in the case of a squeezed state with R positive. The dispersion is reduced at Omega t equals Pi over 2, 3 Pi over 2, etc. that is to say at times when the average amplitude is maximum. It means that making measurements at these times would allow us to determine the amplitude of the field with an accuracy better than the standard quantum limit. In fact, we know that it is possible to sample the field exactly at these times. It suffices to make a balanced homodyne measurement of the Q or the minus Q quadrature. We want to calculate the average value and the dispersion of the Q quadrature in a squeezed state with R positive. Remember that the squeezed state is an eigenstate of the generalized destruction operator A_R. So we express Q as a function of A_R. Since the squeezed state is an eigenstate of A_R, the average of Q is given by a simple expression. To obtain the dispersion of Q, you must calculate the average of its square and subtract the square of the average. Once again, the calculation is analogous to the one done in the case of quasi-classical states using the generalized destruction and creation operators A_R, of which the squeezed state Alpha is an eigenstate. You can use again the trick consisting of isolating the normal form and keeping only the remaining terms stemming from the non-null commutator. You should do that calculation to realize how convenient it is. Taking the square root of the variance, you will find a dispersion equal to the standard quantum limit reduced by a factor exponential of minus R. A similar calculation would yield a dispersion larger than the standard quantum limit for the P quadrature, and the product of the dispersions is equal to h bar over 2. That is to say, exactly the minimum value allowed by Heisenberg relations. Once again, that squeezed state is a minimum dispersion state. It is instructive to represent the quadratures in the complex plane as we did for quasi-classical states. We can associate to the squeezed state an ellipse such that the projection on the horizontal axis yields the Q quadrature with its dispersion. The projection on the vertical axis yields the P quadrature with its dispersion. A projection onto an axis at an angle Theta yields the average and the dispersion of Q Theta. Although this diagram has some similarity with the diagram used to describe the field evolution, it should not be confused with it. Here, the ellipse represents the ensemble of the results of many measurements of Q, of P, and of all possible Q Theta. Remember that these observables are time independent and we know how to measure them. It has been proposed to use amplitude squeezed light to improve measurements of extremely small absorption of samples that cannot endure large light power, for instance, biological objects. Indeed, with a standard beam described by a quasi-classical state, we know that in order to increase the accuracy of the measurement for a given duration of the experiment, the only solution is to increase the intensity of the beam. There is obviously a limit to the power of a laser beam and moreover, if a power increase is technically possible, it may lead to the destruction of the sample. With squeezed light, it is possible to increase the accuracy of the amplitude measurement and thus of absorption without increasing the average power of the beam. Consider an amplitude measurement based on homodyne balanced detection of the Q quadrature of a squeezed state with R and Alpha positive. The relative uncertainty of the measurement is reduced by a factor exponential of minus R, compared to the case of a measurement on a quasi-classical state. The relative accuracy of the Q measurement is thus improved. But you may ask the question, why compare with a quasi-classical state characterized by Alpha prime? The reason is that the power of the beam is the same in both cases. To compare the power transported by different states of light, it is sufficient to compare the average number of photons in the same quantization volume used to define the modes. We know that in the case of a quasi-classical state Alpha prime, the average number of photons is the square of the modulus of Alpha prime. We want now to calculate it for a squeezed state. Once again, you must express a and a dagger as a function of the destruction and creation operators A_R and A dagger R. This is the result. Did you find that? Maybe you have a different form, but try to transform it to have only terms in normal order with A dagger R on the left and A_R on the right. You know how to do it using the commutator of AR and A dagger R. It yields a supplementary term hyperbolic sine of R squared. Now, it is trivial to complete the calculation since Alpha R is an eigenstate of A_R. Using the fact that Alpha is real, the expression simplifies to Alpha prime squared plus a term which is negligible if Alpha prime is large compared to one. We can thus conclude that the average power of the squeezed state Alpha R is almost the same as the average power of a quasi-classical state Alpha prime. These are the states to compare. If we demanded the powers are the same, there is definitely an improvement in the relative accuracy of the Q quadrature when a squeezed state is used. This is a measurement below the standard quantum limit. If we consider now the case of R negative with Alpha real, what is reduced is the dispersion around null values of the field. Determining the value of Omega t for which the field is null is in fact a measurement of the phase of the field. For such squeezed states, the measurement has a better accuracy than full quasi-classical state with the same amplitude. Such a state is phase squeezed. A good manner to understand phase squeezing is to use a complex plane representation of the quadratures. For R negative, the squeeze quadrature is P. Its dispersion is reduced by a factor exponential of R, which is smaller than one. In term of measurements of parameters characterizing a wave, it corresponds to phase squeezing. Indeed, if you think of measuring the phase of a complex number inside the green ellipse, the uncertainty on its phase is given by delta P over its modulus. Reducing delta P amounts to reducing the phase dispersion. Phase squeezed states have a phase determined with an uncertainty below the standard quantum limit. As a consequence, they offer the possibility to improve interferometric measurements of small phase variations as we will see soon.