As explained in a previous section taking advantage of squeezed light is very difficult since any optical loss tends to reduce or even cancel the benefit. It means that one must use very good, almost perfect, optical components. In the case of gravitational wave detectors, which are extraordinarily complex and expensive instruments, it is worth spending efforts and money to increase the sensitivity. Indeed, an improvement of the sensitivity by a factor of two means an increase by two to the third power, that is to say, a factor of eight of the volume of the universe at which an event emitting a gravitational wave of a given strength can be detected. The detection of gravitational waves with multi kilometer size interferometers in 2015 was a triumph of optical technologies after decades of efforts to reach the sensitivity permitted by the standard quantum limit. The first detection did not use squeezed light since somewhere in the universe an event happened strong enough to be detected with the sensitivity of that time. It was far from obvious however, that there would be signals large enough to be detected at this sensitivity. This is why they had already been a big effort to prepare an increase in sensitivity based on the use of squeezed light. This technique has then been tested on the existing interferometer and shown to increase the sensitivity of the detection beyond the standard quantum limit. This is a wonderful conclusion for this lesson. In this conclusion, I will present few facts about gravitational wave detection and show you how one can use squeezed light to operate the big interferometers beyond the standard quantum limit. A gravitational wave is a propagating modification of the gravitational field responsible for a change in the metric of spacetime. It is equivalent to a change of distance between freely suspended masses. Since gravitational waves are quadripolar, the changes of length along two orthogonal directions are out of phase by Pi. When the length of one arm increases the length of the other arm decreases. A Michelson interferometer with two orthogonal arms is used to detect the variation in length difference provoked by the passage of the gravitational wave. The figure shows one of the three interferometers of the collaboration LIGO Virgo which are situated in the East and West Coast of the United States for the two LIGO antennas and in Italy for the Italian-French Virgo antenna. Each arm is four kilometers long and the apparatus is able to detect a relative variation in length of the order of 10 to the minus 21. That is to say an absolute variation of a few 10 to the minus 18 meters. In order to appreciate this extraordinary result whose achievement demanded almost half a century of technological advances, remember that the size of an atomic nucleus is of the order of 10 to the minus 15 meters. That is to say, 1000 times bigger than what can be detected with this interferometer. Remarkably, in spite of their gigantic size, three orders of magnitude larger than usual interferometers, these apparatuses have reached the ultimate standard quantum limit that corresponds to the shot noise associated with a perfectly stable laser source. This is a result of decades of efforts in particular, to hang the mirrors with suspensions that decouples them from the vibrations of the ground while allowing them to react as free test masses sensitive to gravitational waves pulling or pushing them perpendicular to the mirror. Many different branches of engineering have been required to provide specs well beyond the previous state-of-the-art in mechanics, vacuum, servo-loops, and last but not least in optics. In particular, lasers were at the standard quantum limit in spite of unusually large power. The power circulating in the arms was increased thanks to resonant cavities with enhancement factor never achieved before permitted by specially developed thin films treatments on the mirrors. The result of this outlandishing engineering is shown on the plot of the measured noise as a function of the frequency for the two interferometers of LIGO. What is measured here is the standard deviation of the difference signal between the two arms in a one hertz bandwidth. This signal is expressed in units of gravitational waves strain that would give a signal equal to the measured noise. It means, that the relative variation in length of 10 to the minus 23rd could be measured in a bandwidth of one hertz around 200 hertz. You may be puzzled by the unit, strain amplitude per root hertz. If it is the case, go back to the lesson on quasi-classical states where we have calculated the shot noise power per hertz and indicated that we could also give the square root of that quantity. That is to say, the short noise amplitude per root hertz. We have a similar presentation here. The plot shows that if you filter the signal with the band-pass filter between 100 and 300 hertz. The noise is equivalent to 10 to the minus 23rd times root of 200, that is to say, of the order of 1.5 times 10 to the minus 22nd. One could thus detect a gravitational wave provoking a relative variation of length equal or larger than that quantity in that bandwidth. Actually, in the first detection on September 14th, 2015, a signal was detected with a peak gravitational waves strain of 10 to the minus 21 at a frequency chirp from 100 hertz to 300 hertz in a few 10s of milliseconds. A calculation of the expected shot noise associated to the laser yields the result equal to what is measured beyond 300 hertz. If you remember that the shot noise is a white noise you may ask why the plot is not flat and rather increases both towards low frequency and high frequencies. The answer is totally different for the two ends of the spectrum. At low frequency, it is because the suspensions of the mirrors which act as low-pass filters for the vibrations due to seismic noise are less efficient at lower frequencies and the observed noise is a residual seismic noise. At high frequency, the measured noise is indeed equal to the shot noise. But when one converts it into the strain of a gravitational wave one must take into account a reduction factor when the period of the gravitational wave signal becomes of the order of the time it takes for light to make a round trip in one arm: then the oscillating signal is averaged out. A round trip of four kilometers takes only a few 10s of microseconds but one must take into account the fact that there is a resonant cavity which increases the effective length by a factor of 300. The result is an equivalent length of 1,200 kilometers. Divided by C, it is a round-trip time of four millisecond, which means a significant reduction of sensitivity for frequencies above a few 100 hertz. You can now understand why the smallest detectable strain increases with the frequency as shown here. As early as 1981, when only a few people dreamed of detecting gravitational waves with large optical interferometers, it was suggested that using non-classical light might reduce the noise below the shot noise and thus increase the sensitivity. In fact, the concept of squeezed light was precisely invented in the context of gravitational waves detection. This is a vision of the Michelson interferometer envisaged in that theoretical paper. Compared to standard Michelson interferometer that you may know, each of the two orthogonal arms has in it a resonant cavity to increase the round-trip time. Another significant difference is the fact that one has access to two complementary outputs where one can have two detectors to perform a balanced detection. Finally, the scheme is such that one has access to a second input in which it will be possible to inject a squeezed vacuum. A proof of principle has been obtained that the sensitivity of LIGO can be increased by use of squeezed light. On the plot here, you see in red the measured noise without squeezed light. At frequencies higher than 500 hertz, it is equal to the standard quantum limit indicated with a dashed black line. When squeezed light is added, the noise is reduced and the sensitivity is increased by 28 percent. This is shown on the blue curve which is the measured noise with squeezed light introduced in the second input of the interferometer. You may find the improvement tiny but you must realize that it means an increase by a factor of more than two of the volume of the universe in which phenomena can be detected with the ultimate resolution. To better appreciate that result, one must know that the interferometer was not meant to accommodate squeezed light. With dedicated improvements, one should have a more spectacular increase in sensitivity. In fact, demonstrating that it is possible to pass the standard quantum limit on the existing apparatus was already a terrific achievement.