In this paper we describe in detail a graphical method allowing the computation of field operator transformations in quantum optics (QO). Its applications include beam splitters (BS), Mach-Zehnder interferometers (MZI), optical resonators (Fabry-Perot etc) as well as non-linear crystals featuring the process of spontaneous parametric down-conversion (SPDC). Its main advantage compared to the traditional computation step-by-step method is its visual and intuitive approach, somehow similar to Feynman's diagrammatic approach in Quantum Field Theory. It also seems adapted to computerbased implementations since calculations mainly consist on complex additions and multiplications, not matrix operations.
IntroductionThe quantum optical description [1-3] of optical devices and systems allows one to compute the transformation of classical and non-classical states of light.Linear passive optical devices feature, among others, the beam splitter (BS). In the quantum-optical description [2-5], field operators replace classical fields. Beam splitters have been extensively discussed [6-9], sometimes in a rather mathematical fashion [6,7]. A Mach-Zehnder interferometer (MZI) is a device composed of two beam splitters and two mirrors [2][3][4]. Its popularity has led to its use in countless important experiments [10][11][12][13].Sometimes optical devices contain resonators, the most widely used being the Fabry-Perot cavity [14,15]. It can be described classically or quantum mechanically. Although resonators can change very abruptly their input-output characteristics, they are linear optical devices.Following Burnham and Weinberg's [16] introduction of non-linear crystals featuring SPDC, Mandel's group set a string of experiments that revolutionized the field of quantum optics [17][18][19]. Ou, Wang, Zou and Mandel (OWZM) proposed [20] an experimental setup with two non-linear crystals. What they suggested, and then experimentally confirmed [18], was a 'phase memory' if two crystals are pumped by the same laser.Zou, Wang and Mandel (ZWM) [19] used the same two-crystal setup but made the idler mode of the first crystal pass through the second one, making them indistinguishable. A 'mind boggling' experiment resulted, that puzzled many physicists. Two decades later, the same ZWM experimental principle triggered a new field of research: quantum imaging [21][22][23][24].The computation of input-output operator relations for optical devices comprising linear optical components is done through a cascade of successive operator (i.e. matrix) transformations. The complexity of these operations rapidly grows with the number of inputs/outputs as well as the complexity of system, obscuring the physics at work behind the used models. For example a single Mach-Zehnder interferometer poses no particular calculation problems, however a double Mach-Zehnder [25] or two nested Mach-Zehnder configurations [26,27] demand longer and tedious calculations.The situation gets worse for two or more non-linear crystals. Using a complete Hamiltonian approach ...