Density Matrix From Photon Number Tomography

We provide a simple analytic relation which connects the density operator of the radiation field with the number probabilities. The problem of experimentally ”sampling” a general matrix elements is studied, and the deleterious effects of nonunit quantum efficiency in the detection process are analyzed showing how they can be reduced by using the squeezing technique. The obtained result is particulary useful for intracavity field reconstruction states. PACS number(s): 03.65.Bz, 42.50.Dv On leave from Lebedev Physical Institute, Moscow, Russia. 1 After the seminal paper of Vogel and Risken [1] and the experimental realization of their result by Smithey et al. [2] it becames clear that the homodyne measurement of an electromagnetic field permits the reconstruction of the Wigner function of a quantum state, by just varying the phase of the local oscillator, since then named optical homodyne tomography. In Ref. [1] it was shown, indeed, that the rotated quadrature distribution may be expressed in terms of any s-parametrized Wigner function [3]. These papers give rise to a pletora of other papers [4, 5] where the limitations due to the detectors quantum efficiency η were also taken into account, and a fundamental limit was established showing the impossibility of the tomographic reconstruction for detectors quantum efficiency less than 0.5 [6]. Very recently it was shown [7] that, in principle, there is no such limit; however, in order to get the desired results for η arbitrarily small, one needs to employ a rather complicate mathematical procedure for loss error compensation, which in practice make still persistent a lower limit on η values because of numerical problems. Anyway, all the above mentioned papers rely on homodyne measurements, or any other phase dependent measurement process. While this work was in progress we became aware that Wallentowitz and Vogel [8] proposed the s-parametrized Wigner function reconstruction similar, in its essence, to the present one by using direct photon counting and contemporarely an analogous scheme was adopted by Banaszek and Wódkiewicz [9]; however, in the following we shall discuss the relevant differences with our work. We shall show in this letter that the state retrieval by photon counting can be used not only when the output beam is mixed with a reference field at a beamsplitter, as in Refs. [8, 9], but also in a physical situation similar to the one proposed by Brune et al. [10] for the generation and measurement of a Schrödinger cat state allowing thus the possibility of reconstructing its density matrix and, more generally, the possibility