The search for gravitational waves has been a strong stimulus for considering many different detection schemes from resonant mechanical devices to optical interferometers, able to measure the tiny length difference induced in its arms by passage of a gravitational wave. By the early 1980s, calculation of the ultimate accuracy of an optical interferometer based on the shot noise of ideal lasers showed that gravitational waves detection would require interferometers of a size and a quality unheard of at that time. Then physicists discovered the theoretical possibility to improve the situation by using squeezed light. You can read here a citation from a paper by Carlton Caves, where the notion of squeezed state is introduced for the first time to my knowledge. Stop the video if you want to read the whole citation. You can also have a look at the full paper added as a document to the lesson. You are going to see how squeezed light can improve the sensitivity of a Mach-Zehnder interferometer beyond the standard limit. It is not exactly the kind of interferometer used for gravitational wave detection, but the basic idea is the same, and it is simpler to explain. Remember what is a Mach-Zehnder interferometer, which was presented in the first quantum optics course. A laser beam, described as a quasi-classical state Alpha enters in one input here channel 2, while vacuum enters in the other channel here, channel 1. I have exchanged the roles of one and two, compared to lesson 6 of quantum optics 1 in order to have a situation fully resembling the balanced homodyne detection operators presented today in section 1. You know how to calculate the number of counts, N_5 and N_6 during a time associated with the mode of duration, capital D. During that time, the average number of photons entering channel 2 is equal to the square modulus of Alpha. To calculate the signals in channels five and six, it suffices to express the destruction operator a_5 and a_6 as a function of a_1 and a_2. The quantity Delta is a phase difference associated with the path difference in arms three and four. The expressions here are obtained with balance beam splitters. That is to say with equal reflection and transmission. If you do the full calculation, and want to compare with my expression, you must know that I have taken a minus sign for the amplitude reflection coefficient from two to four, and a minus sign for reflection from Mode 3, to Mode 5. Taking the average of a dagger five, a_5, and of a dagger six, a_6, we see input state vacuum times Alpha, you obtain in each channel the usual sinusoidal variation as a function of the phase difference, Delta. The path difference and thus the phase difference can be adjusted with a piezo transducer acting on mirror M_3. Suppose now, you want to measure a small variation in the difference between the arm's length. It is a good idea to use both signals more precisely to subtract them, and work around the working point, where the variation with delta is maximum. It is obviously the case around delta equals Pi over two where the derivative of the difference is largest. This is a balanced Mach-Zehnder interferometer. In the balanced situation, one works around zero. Any variation Epsilon in the dephasing yields a signal equal to Epsilon times the number N_2 of photons entering the interferometer for the measurement time, T. It looks pretty sensitive if N_2 is large, but in fact, such a claim is meaningless, until we know the magnitude of noise on the difference N_6 minus N_5. In a classical description of shot noise for a perfectly stable laser at the input, the fluctuations on both detectors are due to the randomness in the photoelectric effect yielding N_5 and N_6. These two quantities are independent Poisson random variables and their difference is a Poisson random variable with a variance equal to the sum of the variances. The fluctuation is thus found equal to root of the sum of N_5 and N_6, that is to say, N_2. The signal to noise ratio is equal to Epsilon times root of N_2. It means that one can detect a dephasing of Epsilon equals 1 over root N_2 with a signal-to-noise ratio of one. If N_2 is large, it is indeed a pretty sensitive measurement. But if it is not sufficient, the only mean to increase it is to increase the number of photons, N_2, that is to say to have either a larger laser power, or a longer duration of the measurement. The laser power can not be increased without limit, and increasing the measurement time amounts to decreasing the bandwidth of frequencies where one can detect the phase difference. That is annoying for a signal which can happen at any frequency. It is to surmount these problems that Carlton Caves introduced the idea of using squeezed light. Understanding it demands to revisit the description of the noise in a fully quantum formalism as we do now. In the fully quantum point of view, we must calculate the average of the observable N6 minus N5 and the standard deviation of that observable for the input state with vacuum in channel one. You should be able to do it yourself and I will only tell you what the steps of the calculation are. Firstly, using the expressions of a5 and a6 as a function of a1 and a2, you will obtain the expressions of the observables N6 and N5 and with a little bit of trigonometry that expression of their difference. You may think that it is useless to keep a1 since there is vacuum in input one but be patient and you will see once again one of the most remarkable features of modern quantum optics, the important role of vacuum in an empty port of a beam splitter. Let us now express N6 minus N5 as a function of the small dephasing Epsilon and take its average when quasi-classical state is injected in channel two and channel one is empty. All the terms with a1 are written in normal order so that their average with vacuum in mode one is null and the only non null term is the one with a2 dagger a2. We recover the average value found above and the empty channel one plays no role in that calculation. We want now to calculate the standard deviation of the fluctuations of N6 minus N5. For that, we must return to the expression of the observable N6 minus N5, and square it. In order to keep the calculation simple, we will do it for the case of Epsilon equals zero and admit that the noise will be almost the same for Epsilon different from zero. You know enough to do the full calculation in the case of Epsilon different from zero, but I warn you that it is quite a lengthy calculation. Let us then calculate the square of N6 minus N5 in the case of Epsilon equals zero. In fact, it is the same calculation as already done for the fluctuations of the balanced homodyne result, since the interferometer exactly tuned at Epsilon equal zero is equivalent to a balanced beam splitter. Let us anyway complete the calculation the three first terms are in normal order for a1 and a dagger one and there will give zero for vacuum in channel one. The calculation of the last term gives N2 as a variance of N6 minus N5. This is the same result as found in the classical reasoning. It may look a useless effort to recover the classical result, but in fact, this fully quantum calculation gives a totally different point of view on the origin of the fluctuations in the signal. The noise stems from an interference between the laser entering in mode two and vacuum fluctuations entering in mode one. Indeed, you remember that vacuum fluctuations in a mode are associated with the term one that appears when you want to put a dagger in normal order. Here, it is that term that has combined with the input two to yield the fluctuations of the balanced signal. In other words, in contrast with the semi-classical description of the noise as due to the probabilistic character of the detection of the light entering in channel two, the fully quantum treatment describes the noise as due to vacuum fluctuations entering channel one. One now understands that injecting an appropriate state in channel one may improve the situation. Let us now consider the case when the state in input one could be any state Psi_1 but we still have a quasi-classical state in input two. To keep the calculations simple, we do them for an exactly balance interferometer. That is to say we take Epsilon equals zero for which the observable N6 minus N5 is simple. We will again calculate the average of N6 minus N5 and its standard deviation. We first take the average in Alpha two only and find that what is left is the P quadrature in input one within a factor. It means that the average of these measurements yields the average of the P quadrature of the field in channel one. This is not a surprise if we remember that a perfectly balanced interferometer is equivalent to a beam splitter. Let us now calculate the average of the squared balanced signal. The calculation is the same as already done and you can skip it if you have fully understood. I repeat it here, however, if you want to check that you are comfortable with it. Taking first the average in the channel two, we have three terms with a_dagger_2 a2 in the normal order giving Alpha squared and one term a2 a dagger two which yields 1 plus Alpha squared. In such measurements, the laser beam in channel two is intense and Alpha squared is large compared to one which can be neglected. The average of the squared balanced signal is then equal within a factor to the square of the P quadrature in channel one. So an exactly balanced Mach-Zehnder interferometer with a quasi-classical state in channel two, with Alpha real measures the P quadrature of the field in channel one. Repeating the measurement, one does obtain the average and the dispersion of that quadrature of the input state in channel one. We can thus re-interpret the measurement of N6 minus N5 as a measurement of the P1 quadrature of vacuum within a factor. This is shown in the complex plane representation of the quadratures of vacuum. When you see this result, don't you think of a method to reduce the quantum noise in the balanced interferometer? The answer is obviously: "use squeezed light". If we use a P-squeezed state, that is to say, a squeezed state with R negative, then we have a reduced fluctuation on our result. What is represented here is a special case of a P-squeezed state. It is what is called the P-squeezed vacuum. The representations suggests that introducing it in input 1, could reduce the fluctuation on the phase measurement below the standard quantum noise, below the standard quantum limit. We are going to show it now. Let us first establish some important properties of a squeezed vacuum. It is defined as a squeezed state with Alpha equals 0. It is noted ket |0,R>. Alpha prime is then equal to zero, and thus the average electric field is null as well as the average of each quadrature. The green surface representing the fluctuations of P and Q is thus centered on zero. To evaluate the fluctuations of P, you must calculate its variance which is simply the average of its square. The calculation is quite simple since all terms are null in the squeezed vacuum except for the one which is not in normal order. It is readily evaluated using the commutator of A_R and A dagger R. The result is a variance equal to the variance of standard vacuum. That is to say, h-bar over 2 multiplied by the squeezing factor exponential of 2R. If R is negative as assumed for the figure here, the fluctuations of P are indeed below the standard quantum value. Squeezed vacuum has a surprising property, in stark contrast with ordinary vacuum. Although the average field is zero and the product of the standard deviations of Q and P has a minimum value of h-bar over 2 as for ordinary vacuum, the average number of photons is not null in a squeezed vacuum. Once again, the calculation is quite simple. If you express the number of photons operator as a function of A_R, and A dagger R. In the resulting expression, the average in the state |0,R> of all terms in normal order of A_R and A dagger R is null. There is, however, a remaining term which yields an average number of photon equal to the square of the hyperbolic sinh of R. This is obviously a small number, but nevertheless, a measurement with a sensitive photoelectric detector will give a non-null result. We are now armed to understand how it is possible to measure small variations of the path difference in a Mach-Zehnder interferometer with a sensitivity better than the one associated with the standard quantum limit. We still have a perfectly stabilized laser in channel 2. But we inject a squeezed vacuum in channel 1. The interferometer is balanced. That is to say, the average path difference is set to a quarter wavelength, so that the phase difference is Pi over 2, plus a small variation epsilon. We have already expressed the observable N6 minus N5, as a function of the input operators a_1 and a_2. With the input state consider here, it is obviously convenient to express a_1 as a function of the operator, A_R1. I let you do the calculation and eliminate all terms in normal order of A_R1 whose average will be null. In the average of N6 minus N5, there are only two terms different from zero. The first one is Alpha squared, that is to say, the number N2 or photons in the laser beam. The second one is again evaluated using the commutator A dagger R1, A_R1 and yields the number of photons in the squeezed vacuum, which is negligible compared to Alpha squared. The average signal is thus almost equal to epsilon times N2 as in the case of an empty input 1. We want now to calculate the fluctuations of N6 minus N5. Here again, we will do the calculation in the case epsilon equals 0, and admit that the result remains the same at the leading order as for epsilon different from zero but small. The calculation for epsilon different from zero is tedious. But you can do it if you want. You have learned all the necessary ingredients. For the case epsilon equals 0, it is the same calculation as already done. You can jump at the end of the calculation, but I nevertheless give the steps. First, because you take the average with a quasi-classical state Alpha in input 2, you can replace all the terms a_2 or a dagger two by Alpha, including in expressions which are not in normal order. Indeed, it amounts to neglecting one compared to the number of photons N2 which is large. You are then left with the average of a quantity, which is the squared P1 quadrature within a factor. We have already done the calculation of the variance of the P quadrature for a squeezed vacuum, and found a modification of the standard value by a factor exponential of 2R. The calculation yields the expected result. That is to say, a standard deviation reduced by the squeezing factor exponential of R less than one for R negative, since the average is null, this quantity is also the variance. The signal to noise ratio, is thus increased by e to the minus R, which means that the minimum detectable dephasing for a given signal to noise ratio is reduced by a factor e to the R. The sensitivity is thus larger than the standard quantum limit.
