Spatial characterization of two-photon states

In the same way that electronics is based on measuring and controlling the state of electrons, the technological applications of quantum optics will be based on our abil ily to generate and characterize photonic stales. The generation of photonic states is traditionally associated to nonlinear optics, where the interaction of a beam and a nonlinear material results in the generation of multi-photon states. The most common process is spontaneous parametric down-conversion (SPDC), which is used as a source of pairs of photons nCl only for quantum optics applications but also for quantum information and quantum cryptography. The popularity of SPDC lies in the relative simplicity of its experimental realization, and in the variety of quantum features that down-converted photons exhibit. For instance, a pair of photons generated via SPDC can be entangled in polarization, frequency, or in the equivalent degrees of freedom of orbital angular momentum, space, and transverse momentum. Standard SPDC applications focus on a single degree of freedom, wasting the entanglement in other degrees of freedom and the correlat ions between them. Among the few configurations using more than one degree of freedom are hyperentanglement, spat ial entanglement distillation using polarization, or control of the join spectrum using the pump's spatial properties. This thes is describes the spatial properties of the two-photon state generated via SPDC, considering the different parameters of the process, and the correlations between space and frequency. To achieve this goal, I use the purity 10 quantify the correlations between the photons, and between the degrees of freedom. Additionally, I study the spatial correlations by describing the transfer of orbital angular momentum (OAM) from the pump to the signal and idler photons, taking into account the pump, Ihe detection system and other parameters of the process. This thesis is composed of five chapters. Chapter 1 introduces the mode function , used throughout the thesis to describe the two-photon state in space and frequency. Chapter 2 describes the correlations between degrees of freedom or photons in the two-photon Slate, using the purity to quantify such correlations. Chapter 3 explains the mechanism of the OAM transfer from pump to signal and idler photons. Chapter 4 describes the effect of different SPDC parameters on the DAM transfer in noncollinear configurations, both theoretically and experimentally. Finally, in analogy with the downconverted case, chapter 5 lIiscusses the two-photon slale generated via Raman transition, by describing its mode function, the correlat ions between different parts of the state, and the OAM transfer in the process. The matrix. notation, introduced here to describe the two-photon mode function, reduces the calculation time for several features of the state. In particular, this notation allows to calculate the purity of different parts of the state analytically. This analytical so lution reveals the effect of each SPDC parameter over the internal correlations, and shows the necessary conditions to suppress the correlations. or to maximize them. The description of the OAM transfer mechanism shows that the pump OAM is totally transferred to the generated photons. But if only a portion of the generated photons is detected their OAM may not be equal to the pump's DAM The experiments described in the thesis show that the amount of DAM transfer in the non-collinear case is tailored by the parameters of the SPDC. The analysis of the SPDC case can be extended to other nonlinear processes, such as Raman transitions. where the specific characteristics of the process determine the correlations and the OAM transfer mechanism. The results of this thesis contribute to a full description of the correlations inside the two-photon state. Such a description allows to use the correlations as a tool to modify the spatial state of the photons. This spati al info rmation, translated into OAM modes. provides a multidimensional and continuum degree of freedom, useful for certain tasks where the polarization, discrete and bidimensionaJ, is not enough. To make such applications possible, it will be necessary to optimize the tools for the detection of DAM states at the single photon level.