A quantum theory of the lossless beam splitter is given in terms of the quasi mode theory of macroscopic canonical quantization used to treat problems in cavity quantum electrodynamics and quantum optics. A Heisenberg picture approach to quantum scattering theory is applied, in which the input and output operators that are related via the scattering operator are linked to quantum optical measurements described via multi-time quantum correlation functions. In the application to the beam splitter the Heisenberg equations of motion for the input operators associated with the quasi mode annihilation operators are formally solved in a rotating picture to show that the unitary transform of the incident quasi mode annihilation operator (via the scattering operator) is just a linear combination of the incident and reflected quasi mode annihilation operators, in accordance with assumptions made in previous treatments of the beam splitter. The results depend on conservation of the transverse component of the wave vector, which follows from the form of the quasi mode-quasi mode coupling constants, and on conservation of the unperturbed energy, which follows from scattering theory. The applicability of quantum scattering theory to the beam splitter is justified in the usual situation where integrated one photon and two photon detection rates are finite for incident light field states of interest.
A quantum theory of the lossless beam splitter is given in terms of the quasi mode theory of macroscopic canonical quantization used to treat problems in cavity quantum electrodynamics and quantum optics. A Heisenberg picture approach to quantum scattering theory is applied, in which the input and output operators that are related via the scattering operator are linked to quantum optical measurements described via multi-time quantum correlation functions. In the application to the beam splitter the Heisenberg equations of motion for the input operators associated with the quasi mode annihilation operators are formally solved in a rotating picture to show that the unitary transform of the incident quasi mode annihilation operator (via the scattering operator) is just a linear combination of the incident and reflected quasi mode annihilation operators, in accordance with assumptions made in previous treatments of the beam splitter. The results depend on conservation of the transverse component of the wave vector, which follows from the form of the quasi mode-quasi mode coupling constants, and on conservation of the unperturbed energy, which follows from scattering theory. The applicability of quantum scattering theory to the beam splitter is justified in the usual situation where integrated one photon and two photon detection rates are finite for incident light field states of interest.
A macroscopic, canonical quantization of the EM field and radiating atom system in quantum optics and cavity QED involving classical, linear optical devices, based on expanding the vector potential in terms of quasi mode functions is presented. The quasi mode functions approximate the true mode functions for the device, and are obtained by solving the Helmholtz equation for an idealized spatially dependent electric permittivity function describing the device. The Hamiltonian for the EM field and radiating atom system is obtained in multipolar form and the quantum EM field is found to be equivalent to a set of quantum harmonic oscillators, one oscillator per quasi mode. However, unlike true mode theory where the quantum harmonic oscillators are uncoupled, in the quasi mode theory they are coupled and photon exchange processes can occur. Explicit expressions for the coupling constants are obtained. The interaction energy between the radiative atoms and the quantum EM field depends on the amplitudes of the quasi mode functions at the positions of the radiating atoms, similar to that for the true mode approach. The simpler forms for the quasi mode functions enable the atom-field interaction energy to be written in a form in which the atoms are only coupled to certain types of modes—for example cavity quasi modes, which are large inside the optical cavity. In such cases the escape of energy from excited atoms in the cavity can be pictured in quasi mode theory as a two step process—the atom de-excites and creates a photon in a cavity quasi mode, the photon in the cavity quasi mode is then lost and appears as a photon in an external quasi mode. In this process the first step occurs via the atom-cavity quasi mode interaction, the second through coupling between cavity and external quasi modes. This may be contrasted with the true mode approach, where the excited atom loses its energy and the photon is created in one of the true modes. As all true modes have non-zero amplitudes outside as well as inside the cavity, the escape of energy from excited atoms in the cavity is seen as a one step process. An application of the quasi mode theory to the quantum theory of the beam splitter is outlined. The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.
The generalization of the quasi mode theory of macroscopic quantization in quantum optics and cavity QED presented in the previous paper, is applied to provide a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes. The derivation of the laws of reflection and refraction is achieved through the dual application of the quasi mode theory and a quantum scattering theory based on the Heisenberg picture. Formal expressions from scattering theory are given for the reflection and transmission coefficients. The behaviour of the intensity for a localized one photon wave packet coming in at time minus infinity from the incident direction is examined and it is shown that at time plus infinity, the light intensity is only significant where the classical laws of reflection and refraction predict. The occurrence of both refraction and reflection is dependent upon the quasi mode theory coupling constants between incident and transmitted region quasi modes being non-zero, and it is seen that the contributions to such coupling constants come from the overlap of the mode functions in the boundary layer region, as might be expected from a microscopic theory.
The generalization of the quasi mode theory of macroscopic quantization in quantum optics and cavity QED presented in the previous paper, is applied to provide a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes. The derivation of the laws of reflection and refraction is achieved through the dual application of the quasi mode theory and a quantum scattering theory based on the Heisenberg picture. Formal expressions from scattering theory are given for the reflection and transmission coefficients. The behaviour of the intensity for a localized one photon wave packet coming in at time minus infinity from the incident direction is examined and it is shown that at time plus infinity, the light intensity is only significant where the classical laws of reflection and refraction predict. The occurrence of both refraction and reflection is dependent upon the quasi mode theory coupling constants between incident and transmitted region quasi modes being nonzero, and it is seen that the contributions to such coupling constants come from the overlap of the mode functions in the boundary layer region, as might be expected from a microscopic theory.
The quasi mode theory of macroscopic quantization in quantum optics and cavity QED developed by Dalton, Barnett and Knight is generalized. This generalization allows for cases in which two or more quasi permittivities, along with their associated mode functions, are needed to describe the classical optics device. It brings problems such as reflection and refraction at a dielectric boundary, the linear coupler, and the coupling of two optical cavities within the scope of the theory. For the most part, the results that are obtained here are simple generalizations of those obtained in previous work. However the coupling constants, which are of great importance in applications of the theory, are shown to contain significant additional terms which cannot be ‘guessed’ from the simpler forms. The expressions for the coupling constants suggest that the critical factor in determining the strength of coupling between a pair of quasi modes is their degree of spatial overlap. In an accompanying paper a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary is given as an illustration of the generalized theory. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes.
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.
Using conditional measurement on a beam splitter, we study the transformation of the quantum state of the signal mode within the concept of two-port non-unitary transformation. Allowing for arbitrary quantum states of both the input reference mode and the output reference mode on which the measurement is performed, we show that the non-unitary transformation operator can be given as an s-ordered operator product, where the value of s is entirely determined by the absolute value of the beam splitter reflectance (or transmittance). The formalism generalizes previously obtained results that can be recovered by simple specification of the non-unitary transformation operator. As an application, we consider the generation of Schrödinger-cat-like states. An extension to mixed states and imperfect detection is outlined.