In the seventh and final lesson, we began by describing another method of statistical reconstruction of the state of a field mode, based on the principle of maximum entropy. Whereas the maximum likelihood method described in Lesson 6 determines the density operator that maximizes the probability of obtaining the frequencies of the observed eigenvalues, the maximum entropy method follows a different logic. Among all the density operators corresponding to the mean values measured, it seeks the one with the highest entropy. This condition amounts to constructing the density operator that takes into account only the information provided by the measurements, without adding any other hypothesis. It is then natural to consider that the state of the system is that which corresponds to the maximum disorder compatible with the constraints given by the result of the measurements. The search for the field density operator (and therefore its Wigner function) boils down to a constrained variation problem, the solution to which exploits the method of Lagrange multipliers. We construct a linear combination of all the operators corresponding to the measured observables, whose coefficients are these multipliers, and express the density operator as an exponential function of this combination. The Lagrange multipliers are then determined by a least-squares method, comparing the mean values measured with those corresponding to the expression of the density operator. The Lagrange multipliers are adjusted by an iterative algorithm.
In the rest of the lesson, we illustrate these general ideas by describing the reconstruction of non-classical states in cavity quantum electrodynamics experiments. After a few reminders of the general principle of these experiments, we show how they can be used to prepare either Fock states with well-defined photon numbers, or so-called "Schrödinger cat" states, which are superpositions of coherent states of different phases. This preparation is based on non-destructive measurement of the number of photons contained in the field, using Rydberg atoms passing through the cavity one by one. This measurement is based on a Ramsey-type atom interferometry method already described in the 2007-2008 lecture. Once these non-classical states have been prepared, they are shifted by homodyne mixing with a coherent field of adjustable phase and amplitude, and non-destructive measurements of the number of photons in the shifted field are then made, again using Ramsey interferometry. From the result of these measurements, the field density operator and its Wigner function are deduced by the maximum entropy principle. Alternatively, the maximum likelihood method has also been used to reconstruct Fock states. The 2009-2010 lecture on "passive" state estimation and reconstruction will be extended in 2010-2011 by a series of lessons on active methods of controlling and protecting quantum states against decoherence.
