In a previous lesson, we already encountered a Mach-Zehnder interferometer, the favorite interferometer of theorists. Classically, a wave enters in input 1, is split into reflected and transmitted components 3 and 4, which are recombined on a second beam splitter. The intensities in the two outputs 5 and 6, depend on the difference in the optical path 3 and 4, between the two beam splitters. This means that when we move mirror M3, which is mounted on a piezo transducer, one observes a modulation of the signals at detectors D5 and D6. You remember that we calculated the interference signal for the toy model, of a one photon wave packet consisting of a monochromatic wave with a finite duration. Today, we are speaking of real experiments done with one photon wave packets emitted by spontaneous emission. They are composed of many one photon states with different frequencies. We are going to do the calculations in the Heisenberg formalism where the field operators depend not only on position, but also on time. In contrast, the state does not depend on time, and the coefficients in its expansion keep their value at time t0. The real positive coefficient, K, is chosen to normalize the state. We have already encountered that state: it corresponds to a wave packet starting at t0 and decaying exponentially. To be more precise, this means that the probability of a photo detection at O in the input channel 1, is a wave packet rising suddenly at t0, and decaying exponentially. The radiation state in input 2 is the vacuum, so the input state vector should write as the product of one in input 1 times the vacuum in input 2. In fact, we will not keep that vacuum term in the calculation since here it plays no role. Note, however, that there are other calculations where the vacuum in one input channel of an interferometer does play a role. So be careful, and if you have any doubt, do not hesitate to take into account the vacuum in the second input, and to check that it does not play a role. It would be a good idea to do it when you study this lesson: you could repeat the calculation I'm going to do now, keeping the term associated with the vacuum in channel 2, and check by yourself that it does not change the result. Let us then develop our calculation of the probabilities of single detections at detectors D5 and D6. We know that what we have to do is to express the output fields E+_5 and E+_6 as a function of the input fields E+_1 and E +_2 and apply them to the input state. We are going to use a useful feature of the Heisenberg formalism: we can directly obtain the expressions of the field operators by taking the classical expressions obtained in the same situation. So let us recall the classical optics calculation which we have already encountered in a previous lesson. I recall here the expressions describing the classical fields transformations on the two beam splitters. The amplitude reflection and transmission coefficients for the complex amplitude E+ are taken equal to the square roots of capital R and capital T. You remember that the minus sign on one of the reflection terms is associated with the unitarity of the transformation. It ensures energy conservation in classical optics and the conservation of the commutation relations in quantum optics. The different choices for the position of the minus signs associated with the two different beam splitters correspond to the fact that the second beam splitter is reversed, as shown on the figure. We assume that the coefficients R and T do not depend on the frequency of the field components, otherwise, we should use the Fourier decomposition of the fields. Note also, that I ignore the polarization of the fields for simplification. You can think of polarization perpendicular to the figure and consider the values on the axis perpendicular to the figure. As just explained, we will ignore the field E_2 corresponding to an empty input and erase the corresponding term. I have not yet written explicitly the dependence on time, since all values are taken at the same point O, but we must do it as soon as fields propagate between O and O'. In the vacuum or in a non-dispersive medium, a simple retardation term is enough to take into account the propagation. This allows us to express the classical field complex amplitudes in output 5 and 6 as a function of the input field amplitude in channel 1. We can now substitute the classical complex field amplitudes with the corresponding E+ operators to obtain the output fields as a function of E1+_hat. Let us then calculate the photo-detection probability at D5. We must express E5+_hat at D5 as a function of E1+_hat applied to the input state and take the squared modulus. The propagation factor from O' to the detector D5 is an exponential with a complex exponent, and its squared modulus is 1. So it is enough to calculate the expression involving E5+_hat, at O'. Note that if we had kept the E2+_hat terms, they would give null contributions. Be sure to understand this point. In order to use the expression of E5+_hat, as a function of E1+_hat, we must calculate E1+_hat at time t-L3/c applied to the one photon wave packet, and similarly, for t-L4/c. We have already done that calculation in section one. The result depends only on the retarded times tau_3 or tau_4. In the case of a spontaneous photon wave packet, we have found the result written here within a normalization constant that I ignore, to keep only the most important factors. The operator E1+ applied to the one photon state yields the vacuum state times a function f(tau). That function f is a decaying exponential starting at tau equals 0. Remember that we have two different values of tau for channels 3 and 4. H(tau) is the step function, null for negative taus, We thus have a wave packet starting at tau equals 0, and decaying exponentially with an inverse time constant gamma over 2 for the amplitude. The phase factor evolves at the central frequency omega_0. Now, adding the two terms associated with L3 and L4, we find that the vacuum state factorizes, and we obtain the squared modulus of the sum of two complex amplitudes as would be written with a classical wave packet. So the probability of a single detection in the output 5 will exhibit the usual interference term. Let us write the expansion showing the interference term, in the case of balanced beam splitters, that is to say with R = T = one-half. To complete the calculation it is convenient to introduce the average L_bar of the two paths and the path difference delta_L. We now assume that the path difference, delta_L is small enough that the corresponding time shift is small compared to the time constant, 1 over gamma. This is the case for typical atomic transitions, where gamma is larger than one nanosecond, while delta_L is a few wavelength, that is to say, a few micrometers for visible light: delta_L over C is then of the order of a few femtoseconds, as you can check yourself. It means that the rising edge and the decaying exponential can be taken identical for the two terms, and we can use their expression as a function of tau bar which depends on the average propagation time in the interferometer. This simplification cannot be made for the phase factor which changes by 2pi when the length L changes by the one wavelength. This is why we have the different plus and minus delta L factors in f3 and f4. Inserting these approximate values of f3 and f4 in the expression of the w_1 of D5, we obtain the familiar interference term: 1-cosine of the phase difference. Remember that w_1 is the rate of detection per unit surface of the photodetector D5, and to obtain the observed signal, we must integrate over the detector. This is the result of this integration, yielding the rate of photodetection at D5. In order to obtain this expression, I have adapted the calculation of the previous lesson to obtain the exact prefactor. Epsilon is the global quantum efficiency including the collection efficiency and the detector efficiency. Note that I have also replaced 1-cosine by twice the squared sine of half the phase difference. The result is the decaying exponential characteristic of the spontaneous emission wave packet, with the retardation associated with the propagation in the interferometer. The rate of photodetection is the product of the decaying exponential by the interference term. Integrating over the gate, we obtain the probability of a photodetection per pulse, that is to say, the interference term only multiplied by the global detection efficiency epsilon. It depends only on the path difference delta_L, and it can be measured by repeating the experiment a large enough number of times at each value of the position of the mirror. A similar calculation would give the rate of photodetection at D6. Its integral is the probability per pulse to detect the photon at D6 for a given value of delta_L. Note that it is complementary of P of D5, so that the sum of P of D5 and P of D6 is constant and equal to epsilon. Now that we have obtained these results, we are ready to understand the results of the real experiments.
