Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum and Semiclassical Optics Journal of the European Optical Society Part B

A quantum description of the beam splitter

Alfredo Luis, Luis Lorenzo Sánchez-Soto

Published Apr 1, 1995• 68 citations
Publisher

Abstract

The unitary operator representing the action of a linear lossless passive beam splitter is presented. The derivation is based on an elementary group-theoretical approach that allows us to understand this action in terms of the Stokes operators for the incident beam. This operator is used to illustrate with some examples the coupling of noise in the beam splitter.

DOI
10.1088/1355-5111/7/2/005

Related papers

Quantum description of the beam splitter

Sudhakar Prasad, Marlan O. Scully, W. Martienssen

The lossless beam splitter is classically well understood. The purpose of this work is to enable one to describe the action of a lossless plane-parallel beam splitter on an incident light beam in purely quantum-mechanical terms. Such a description may be obtained in terms of a unitary transformation that connects the incoming radiation-field operator to the outgoing one. This transformation, which we shall derive and discuss here, provides an elegant means to address issues and concepts that are intrinsically quantum-mechanical.

Journal of the Optical Society of America B•Feb 1, 1987•0 citations

Quasi mode theory of the beam splitter a quantum scattering theory approach

Bryan J. Dalton, Stephen Mark Barnett, P. L. Knight

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.

Journal of Modern Optics•Aug 1, 1999•7 citations

A particle description of the quantised field-the beam splitter

Klaus Mølmer

The quantised electromagnetic field is represented by products, polynomials and power series of one-photon states-a particle-like description, the use of which will be demonstrated by an analysis of the beam splitter. With some examples, the 'connection' from pure to mixed states is discussed, and general relations for the action of the beam splitter upon mixed states are derived.

Journal of Physics B Atomic Molecular and Optical Physics•Sep 28, 1988•3 citations

Quantum mechanical Fresnel relations

Jagdish Rai

Several authors1,2 have studied the lossless beam splitter on a quantum mechanical basis. In this approach the action of the beam splitter is described by unitary transformations that act on the operators for the radiation fields present. In this paper we study quantum mechanically the general case of reflection and transmission from a boundary. We obtain the quantum analog of the classical Fresnel relations. We find that the general operators that describe reflection and transmission at a boundary are a series of unitary operators involving appropriate radiation fields. The generalization for the nonlinear media presents many interesting situations where the classical laws of reflection and refraction are modified. This provides an interesting description of phase conjugation.

Annual Meeting Optical Society of America•Jan 1, 1991•0 citations

Photon number conservation and photon interference

Mátyás Koniorczyk, József Janszky

The group theoretical aspect of the description of passive lossless optical four-ports (beam splitters) is revisited. It is shown through an example, that this approach can be useful in understanding interferometric schemes where a low number of photons interfere. The formalism is extended to passive lossless optical six-ports, their SU(3)-theory is outlined.

arXiv:quant-ph/0110170•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 23, 2002•0 citations

Quantum statistics of a lossless beam splitter: SU(2) symmetry in phase space

Ulf Leonhardt

A lossless beam splitter (a dielectric interface, a passive interferometer, or a linear coupler) changes the quantum state of two incident modes by an SU(2) transformation. Apart from phase shifting, the argument of the quadrature wave function of the system undergoes a rotation. Quasiprobabilities are changed by the inverse mode transformation. The use of balanced beam splitting allows the simultaneous measurement of conjugate quadrature components via homodyning the emerging beams with two strong coherent reference fields that differ in their phases by \ensuremath{\pi}/2. The measured probability distribution is given by a generalized Q function. It depends on the state of the field entering the second beam-splitter port. For a vacuum, the Q function will be obtained. The use of unbalanced beam splitting allows the measurement of a squeezed Q function without using squeezed states. Dissipation in Gaussian reservoirs corresponds exactly to a heuristic beam-splitter model. As a mathematical tool, the Fokker-Planck equation of damping in phase-sensitive reservoirs and the corresponding quantum master equation were solved. The dissipative decay of a Schr\"odinger-cat state was studied as an example. The sensitivity of quantum coherence with respect to damping can be interpreted geometrically.

Physical Review A•Oct 1, 1993•132 citations
The unitary operator representing the action of a linear lossless passive beam splitter is presented. The derivation is based on an elementary group-theoretical approach that allows us to understand this action in terms of the Stokes operators for the incident beam. This operator is used to illustrate with some examples the coupling of noise in the beam splitter.
The lossless beam splitter is classically well understood. The purpose of this work is to enable one to describe the action of a lossless plane-parallel beam splitter on an incident light beam in purely quantum-mechanical terms. Such a description may be obtained in terms of a unitary transformation that connects the incoming radiation-field operator to the outgoing one. This transformation, which we shall derive and discuss here, provides an elegant means to address issues and concepts that are intrinsically quantum-mechanical.
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.
Several authors1,2 have studied the lossless beam splitter on a quantum mechanical basis. In this approach the action of the beam splitter is described by unitary transformations that act on the operators for the radiation fields present. In this paper we study quantum mechanically the general case of reflection and transmission from a boundary. We obtain the quantum analog of the classical Fresnel relations. We find that the general operators that describe reflection and transmission at a boundary are a series of unitary operators involving appropriate radiation fields. The generalization for the nonlinear media presents many interesting situations where the classical laws of reflection and refraction are modified. This provides an interesting description of phase conjugation.
A lossless beam splitter (a dielectric interface, a passive interferometer, or a linear coupler) changes the quantum state of two incident modes by an SU(2) transformation. Apart from phase shifting, the argument of the quadrature wave function of the system undergoes a rotation. Quasiprobabilities are changed by the inverse mode transformation. The use of balanced beam splitting allows the simultaneous measurement of conjugate quadrature components via homodyning the emerging beams with two strong coherent reference fields that differ in their phases by \ensuremath{\pi}/2. The measured probability distribution is given by a generalized Q function. It depends on the state of the field entering the second beam-splitter port. For a vacuum, the Q function will be obtained. The use of unbalanced beam splitting allows the measurement of a squeezed Q function without using squeezed states. Dissipation in Gaussian reservoirs corresponds exactly to a heuristic beam-splitter model. As a mathematical tool, the Fokker-Planck equation of damping in phase-sensitive reservoirs and the corresponding quantum master equation were solved. The dissipative decay of a Schr\"odinger-cat state was studied as an example. The sensitivity of quantum coherence with respect to damping can be interpreted geometrically.