Beam splitter Compared with other fundamental experiments in physics, optical tests of quantum mechanics are often distinguished by their simplicity. Most quantum optical experiments do not require a whole industry - an optical table of equipment and a few people are often sufficient. Good ideas, good research problems are more important. “Research is to see what everybody has seen and to think what nobody has thought” (Jammer, 1989). A simple optical beam splitter, for instance, is already a nice device to demonstrate the quantum nature of light. Quite a number of puzzling quantum effects have been seen by splitting or recombining photons at a small cube of glass. Additionally, the beam splitter serves as a theoretical paradigm for other linear optical devices. Interferometers, semitransparent mirrors, dielectric interfaces, wave guide couplers, and polarizers are all described sufficiently well by a simple beam splitter model. The quantum effects of almost all passive optical devices can be understood assuming appropriate beam splitter models. (It's all done with mirrors.) Heisenberg picture An ideal beam splitter is a reversible, lossless device in which two incident beams may interfere to produce two outgoing beams. For instance, a dielectric interface inside a cube or plate of glass splits a light beam into two, see Fig. 5.1. We may reverse this situation by sending the two beams back to the cube, where they interfere constructively to restore the original beam.
Beam splitter Compared with other fundamental experiments in physics, optical tests of quantum mechanics are often distinguished by their simplicity. Most quantum optical experiments do not require a whole industry - an optical table of equipment and a few people are often sufficient. Good ideas, good research problems are more important. “Research is to see what everybody has seen and to think what nobody has thought” (Jammer, 1989). A simple optical beam splitter, for instance, is already a nice device to demonstrate the quantum nature of light. Quite a number of puzzling quantum effects have been seen by splitting or recombining photons at a small cube of glass. Additionally, the beam splitter serves as a theoretical paradigm for other linear optical devices. Interferometers, semitransparent mirrors, dielectric interfaces, wave guide couplers, and polarizers are all described sufficiently well by a simple beam splitter model. The quantum effects of almost all passive optical devices can be understood assuming appropriate beam splitter models. (It's all done with mirrors.) Heisenberg picture An ideal beam splitter is a reversible, lossless device in which two incident beams may interfere to produce two outgoing beams. For instance, a dielectric interface inside a cube or plate of glass splits a light beam into two, see Fig. 5.1. We may reverse this situation by sending the two beams back to the cube, where they interfere constructively to restore the original beam.
Simple optical instruments are linear optical networks where the incident light modes are turned into equal numbers of outgoing modes by linear transformations. For example, such instruments are beam splitters, multiports, interferometers, fibre couplers, polarizers, gravitational lenses, parametric amplifiers, phase-conjugating mirrors and also black holes. The paper develops the quantum theory of simple optical instruments and applies the theory to a few characteristic situations, to the splitting and interference of photons and to the manifestation of Einstein–Podolsky–Rosen correlations in parametric downconversion. How to model irreversible devices such as absorbers and amplifiers is also shown. Finally, the paper develops the theory of Hawking radiation for a simple optical black hole. The paper is intended as a primer, as a nearly self-consistent tutorial. The reader should be familiar with basic quantum mechanics and statistics, and perhaps with optics and some elementary field theory. The quantum theory of light in dielectrics serves as the starting point and, in the concluding section, as a guide to understand quantum black holes.
Simple optical instruments are linear optical networks where the incident light modes are turned into equal numbers of outgoing modes by linear transformations. For example, such instruments are beam splitters, multiports, interferometers, fibre couplers, polarizers, gravitational lenses, parametric amplifiers, phase-conjugating mirrors and also black holes. The article develops the quantum theory of simple optical instruments and applies the theory to a few characteristic situations, to the splitting and interference of photons and to the manifestation of Einstein-Podolsky-Rosen correlations in parametric downconversion. How to model irreversible devices such as absorbers and amplifiers is also shown. Finally, the article develops the theory of Hawking radiation for a simple optical black hole. The paper is intended as a primer, as a nearly self-consistent tutorial. The reader should be familiar with basic quantum mechanics and statistics, and perhaps with optics and some elementary field theory. The quantum theory of light in dielectrics serves as the starting point and, in the concluding section, as a guide to understand quantum black holes. 1 1
Beamsplitting Interference phenomena are certainly among the most exciting phenomena in the whole of physics. In the following we will concentrate mainly on interference of weak fields; i.e. the beams contain, on average, only a few photons. The principle of classical interference is as follows: a light beam is split by an optical element, for example by a semitransparent mirror or a screen with several very small apertures, into two or more partial beams. These beams will take different paths and are then reunited and form interference patterns. The first step, the splitting of the beam into partial beams, plays a decisive role; light beams coming from different sources (or from different spatial areas of the same source) do not interfere with each other! We start our discussion of interference with an analysis of the action of a beamsplitter. To form a realistic idea of this device, let us imagine a semitransparent mirror. (Our considerations apply equally well to a screen with two apertures. We could also generalize to cases of unbalanced mirrors, with reflectivity different from 1/2, or screens with apertures of different size.) The classical wave picture can describe interference phenomena without any great effort: the incoming beam is split into the reflected and the transmitted partial wave, and each of these waves contains half of the energy. The process of splitting becomes conceptually difficult only when we think of the beam as consisting of spatially localized energy packets, or photons.
The photon in the interferometer This chapter introduces many of the ideas of quantum theory by exploring three specific “case studies” of quantum systems. Each is an example of a qubit , a generic name for the simplest type of quantum system. The concepts we develop will be incorporated into a rigorous mathematical framework in the next chapter. Our business here is to provide some intuition about why that mathematical framework is reasonable and appropriate for dealing with the quantum facts of life. Interferometers In Section 1.2 we discussed the two-slit interference experiment with a single photon. In that experiment, the partial waves of probability amplitude were spread throughout the entire region of space beyond the two slits. It is much easier to analyze the situation in an interferometer , an optical apparatus in which the light is restricted to a finite number of discrete beams . The beams may be guided from one point to another, split apart or recombined as needed, and when two beams are recombined into one, the result may show interference effects. At the end of the interferometer, one or more sensors can measure the intensity of various beams. (A beam is just a possible path for the light, so there is nothing paradoxical in talking about a beam of zero intensity.) Figure 2.1 shows the layout of a Mach–Zehnder interferometer , which is an example of this kind of apparatus.
This study explores how optical beam splitters provide a conceptual and operational link between classical and quantum descriptions of light. In the classical picture, they are modeled through energy-conserving field amplitudes and a four-port scattering matrix, which successfully explain interference in systems such as Michelson and Mach–Zehnder interferometers. Moving to the quantum domain, the field amplitudes are replaced by bosonic creation and annihilation operators that obey the Weyl–Heisenberg algebra. Using the Glauber displacement operator, we define coherent states whose expectation values reproduce classical monochromatic fields while exhibiting Poissonian photon statistics. This unified approach shows that beam splitters diagnose the nature of incident light: coherent states yield classical interference, while indistinguishable single-photons reveal quantum signatures such as antibunching and Hong–Ou–Mandel interference. The analysis relies on operator formalism and correlation functions that capture both average-field behavior and nonclassical statistics. As beam-splitting technologies transition from bulk optics to integrated photonics and metasurfaces, this framework offers a consistent basis for modeling optical circuits across the classical-quantum divide. By highlighting the dual role of the beam splitter as a passive device and a quantum probe, we emphasize its importance for quantum optics, quantum information, and hybrid photonic platforms.
The familiar input-output relations for an optical beam splitter are generalized to allow for linear absorption by the medium forming the mirror. Beam-splitter losses generally affect the noise levels detectable in experiments involving nonclassical light. When employed to investigate two-photon interference effects, a lossy beam splitter can lead to apparent nonlinear absorption, which, in the most extreme case, leads to either both or neither of the photons being absorbed. The degree of second-order coherence of antibunched light can be maintained on transmission through the beam splitter but any amplitude squeezing in the incident light is degraded.