We consider two laser beams in the two input modes of a beam splitter. Each laser beam is described by a single mode quasi-classical state, so that the input state is a tensor product alpha_1 alpha_2. A photodetector is placed in the output channel 4. Since we have continuous beams, we will use the formalism about freely propagating beams introduced in lesson one. The average photocurrent is proportional to the mean rate of single detection, w^(1) at the detector. To calculate w^(1) at the output, we replace the electric field operator E plus in channel 4 by each expression as a function of the input fields. The result is identical to what would be obtained for classical fields of complex amplitudes alpha one times E one and alpha two times E one, with frequencies omega 1 and omega 2. The frequencies omega 1 and omega 2 are almost the same, so we can safely take the same value for the one-photon field E one and for the sensitivity s, and put them in factor. Expanding the expression of w one, we find the rate of photo-detection with a component at the beat-note frequency omega_1 minus omega_2. The phases phi_1 and phi_2 are the arguments of alpha_1 and alpha_2. Remember this formulae where the capital S is the section of the beam totally covered by the detector. T is an arbitrary time interval and the squared modulus of alpha is the photon flux times T. Using them in the expression of the photocurrent, you finally obtain a formula which only depends on the photon flux and not on arbitrary quantities. Notice that if the two terms that interfere are equal, the modulation at the beat note frequency is complete, the visibility is 100 percent. The formula we have established is about the quantum average of the photocurrent, and this raises immediately the question of the quantum fluctuations around that average. These fluctuations are the shot noise which we have encountered in Lesson 1. We will address this question later. For the time being, we consider situations where the quantum dispersion is small enough than the quantum average of the photocurrent is a faithful representation of the beat note that we expect to observe. In fact, observing the beat note between two lasers is possible only if the laser frequencies fulfill some conditions. Remember that optical frequencies for visible light are of the order of 5 10 to the 14th hertz. The beat note between two lasers chosen without care will fall in the same range of frequencies which are too high to be detected. In fact, the beat note will be observable only if the two frequencies are closer to each other than the maximum detectable frequency. It is nowadays current to have detectors responding up to a few tens of gigahertz. But at the scale of the laser frequency, it is quite small. In practice, the usual situation consists of two different realizations of the same type of laser, based on the same atomic transition. For instance, two helium neon lasers at the red wavelength of 633 nanometers, will have frequencies differing at most by a few gigahertz. Detecting the beat note between two lasers is a current technique to measure and even control the difference between the two frequencies. This is useful in experimental atomic physics, if one wants to address various hyperfine atomic transitions, which typically differ by a few gigahertz. Once a first laser is locked to a specific transition, the obtained stability and accuracy can be transferred to other lasers addressing other transitions by using feedback techniques based on the detection of the beat note between the master laser and the other lasers. Laser cooling of atoms uses systematically that kind of technique. There is no laser with a frequency omega_0 and a phase phi perfectly fixed. In fact, fluctuations of frequency and phase both contribute to the total spread in frequency since the instantaneous frequency is a time derivative of the global phase omega t plus phi. In any laser, the phase and or frequency fluctuate for many reasons: technical and fundamental. An example of technical fluctuation is the change in the resonance frequency of the laser cavity because of small changes in the laser cavity length due, for instance, to mirror vibrations or to variations in the index of refraction of air in the laser cavity. Once such technical fluctuations are fixed, there remains fundamental phase jumps, due to spontaneous emission in the laser cavity. This is the celebrated Schawlow-Townes linewidth. At the end of the day, the laser oscillation has a frequency distribution -a spectrum, characterized by a profile which is either Lorentzian or Gaussian or a convolution of both shapes. The spectrum has a width Delta omega laser and it is clear that to observe the beat note, one must have laser line-widths significantly less than the beat note frequency. In fact, the beat note itself has a spectrum, which is the convolution of the spectra of the two lasers, its width is the sum of the width in the case of Lorentzian profiles and the quadratic sum in the case of Gaussians. Typical linewidths of current laser are over a gigahertz, but with some efforts, they can be reduced to values of the order of one megahertz or less. The ultimate Schawlow-Townes limit for lasers with many photons in the laser cavity are in the Hertz range and laboratories at the cutting edge of laser technology can reach such values. So observing the beat note between two lasers has become a current technique. In quantum optics, observing the beat note between a laser and an unknown quantum field is a very powerful method to study that quantum field. This method, called heterodyning, will be explained in detail in a future lesson. Today, I will only address the case where the unknown field is a weak quasi-classical state. We consider again the scheme allowing one to observe the beat note between two quasi-classical states, and we suppose that the state in input one is very weak, while we have a state as strong as we wish in input two. The beam two is called the local oscillator. If the photon flux Phi 2 of the local oscillator is known, it is clear that measuring the amplitude of the beat note allows us to determine the photon flux Phi 1. More precisely, if we pass the photocurrent in a bandpass filter tuned to the beat note, we are left with a component proportional to the square root of the product of the photon fluxes. If we had measured directly the small signal, we would have found a current proportional to the photon flux. The ratio between the photo currents in the two methods is proportional to the ratio of the amplitudes. The factor rt is of the order of one. For a balanced splitter, it is one-half. Since Phi 2 is much larger than Phi 1, you see that heterodyne detection amplifies the signal by a factor that can be quite large. This may seem fantastic, but there is a catch. Can you find what? In fact, to evaluate a measurement method, one must consider the noise and the ultimate quantity of interest is the signal-to-noise ratio. In Lesson 1, we have established the formula about shot noise, which is the fundamental noise in photoelectric measurements. Taking this formula for the direct measurement of the weak signal, we find a signal-to-noise ratio that depends only on the weak signal, as it should. Consider now the heterodyne measurement. The strong beam totally dominates the shot noise, which is then proportional to the square root Phi 2. Using that value, we obtain a signal to noise ratio which depends only on the weak signal. It is almost the same as for the direct measurement. Disappointing? In fact, calculating the ultimate signal-to-noise ratio amounts to assuming that one has extracted all the available information in the weak beam. So it should not be a surprise that there is no way to increase that ultimate value. Although there is no miracle regarding the ultimate theoretical signal-to-noise ratio, it turns out that heterodyne detection is very useful to overcome a background noise of detectors called dark current or dark counts. That background noise which exists even in the absence of any radiation is due to several factors, in particular, thermal excitation of the electrons. It adds to the signal photocurrent. For strong light beams, the dark current is negligible compared to the signal photocurrent, and one can ignore it. But for very weak light beams, the dark current can spoil the measurement. In that case, heterodyning is the solution. Indeed, provided that the heterodyne signal is stronger than the dark current. You can reach the ultimate signal-to-noise ratio associated with the small signal. In other words, heterodyning allows one to reach the ultimate signal-to-noise ratio even for weak signals that could not be detected directly.