A weak classical pulse of light, is a light pulse produced by a standard source of light such as a bulb, a light emitting diode, or even a laser, and passed through a neutral density to attenuate it. If the attenuation is large enough, we can reach a regime where the average number of photons in the pulse. Is less than one. So we may think that it is a good approximation of the situation with one photon only. In fact, we have already addressed this question, in a previous lesson and found a major difference. In contrast with a one photon pulse, a weak classical pulse can lead to more than one photodetection. It means that sometimes, we can have a double detection. And we cannot decide whether we should take a zero or a one. Such results must then be eliminated. It means that in order to have an efficient quantum random number generator based on weak light pulses, we must be in a situation where the probability to get double photodetection is small compared to the probability of only one photodetection. To make this statement quantitative, I'm going now tell you a little bit more about the quantum optics description of weak classical pulses as an anticipation of a future lesson in our second course. In quantum optics, light pulses emitted by classical sources can be described by quasi-classical states of radiation. They are usually noted alpha. And the state describing radiation at the input of the beam splitter, is thus alpha in input one and vacuum in input two. We have taken here a simple model, where we assume a single mode quasi-classical state wave packet, so only the mode l is involved. This is analogous to the toy model we introduced about one photon states. You know that a more rigorous description should be multi mode. But the toy model with constant amplitude for a finite duration in a single mode, allows us to capture interesting physical properties. Quasi-classical states, will be studied in a future lesson. A characteristic property of these states is that they are eigenstates of the destruction operator, a_ell hat, with eigenvalue, the complex number, alpha l. One can check, and you should do it, that quasi-classical states are not, eigen states, of the number operator N_ell hat equals a_dagger a. The number of photons is not well defined. In other words, if one measures the number of photon in such a state. One has a distribution of possible results, P(n) with some dispersion. Obtaining that distribution demands some effort. But it is quite simple to obtain the average. and the dispersion of the number of photons associated with this distribution. The average number of photons is given by a calculation, using the characteristic property of the state alpha l. It is found equal to the squared modulus of alpha_ell. To obtain the dispersion of the number of photons, we must first calculate the average of n squared. Using the commutator of a and a dagger. The calculation is once again, straightforward. And leads to a variance, delta n. Squared, equal to the average number n bar. Does this ring a bell? Yes, this is a property of a Poisson distribution. And in fact, one can demonstrate that the distribution of photons for a quasi-classical state, is a Poisson distribution. You may know, that the Poisson distribution is associated with the distribution of particles independent from each other and with a uniform probability. We can thus think of a quasi-classical light pulse, as made of independent photons. We have now a quantitative description, of the photon distribution in a monochromatic quasi-classical wave packet. It can be produced for instance by an almost monochromatic LED, excited by a constant electric pulse of duration T; or by a single mode laser with a switch open for a short duration T. We describe it as a single mode quasi-classical state, characterized by the parameter alpha_ell. If we measure the number of photons with an ideal detector and repeat the experiment many times, we obtain the distribution P(n). It is a random variable described by a Poisson law, fully characterized by its average n bar, which is equal to the square models of alpha. Remember, that its variance is equal to the average so the standard deviation is the square root of the average. The shape of a Poisson distribution, dramatically depends on its mean value. For mean values large compared to one, it is well approximated by a Gaussian centered on the mean value, and with a standard deviation equal to the square root of the mean value. But, for a mean value of the order of one or less, it is highly asymmetric with the probability of getting more than one count small but not null. This is the approximation of the Poisson distribution for an average number of photons, smaller than one. It is clear that to suppress the double detections, we must make the average number of photons as small as possible. More precisely, the ratio of cases with two photons to cases with one photon, scales as the average number of photons. For instance, for a mean number of photon of .1, we have a probability of 10% to have one photon. and only half a percent to have two photons per pulse. It means that we will reject only one case out of 20. But the price to pay, is that almost 90% of the pulses have zero photons. So nine out of ten pulses are useless. We can now understand the limitations of a quantum random number generator, based on weak classical pulses of light, compared with a quantum noise random generator based on one-photon pulses. Remember, that it is desirable to produce random bits as fast as possible. Usually, the ultimate technical limitation is the smallest duration T permitted by the electronics. With one-photon pulses, the maximum rate is 1/T while with weak classical pulse it is significantly less. Most of the quantum random number generators commercially available, are based on weak classical pulses much simpler to produce. But, progress ib one-photon sources will lead to faster quantum random noise generators.
