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Journal of Physics B Atomic Molecular and Optical Physics

A particle description of the quantised field-the beam splitter

Klaus Mølmer

Published Sep 28, 1988• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/21/18/005

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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.
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.
The concept of the photon is introduced by discussion of the process of electromagnetic field quantization within a closed cavity or in an open optical system. The nature of a single-photon state is clarified by consideration of its behaviour at an optical beam splitter. The importance of linear superposition or entangled states in the distinctions between quantum-mechanical photon states and classical excitations of the electromagnetic field is emphasized. These concepts and the ideas of wave-particle duality are illustrated by discussions of the effects of single-photon inputs to Brown-Twiss and Mach-Zehnder interferometers. Both the theoretical predictions and the-confirming experimental observations are covered. The defining property of the single photon in terms of its ability to trigger one. and only one, photodetection event is discussed.
Abstract The Mach–Zehnder interferometer experiment described earlier shows why and how quantum mechanics is different from classical mechanics. A photon sent through a beam splitter behaves like a particle when it is observed by only one of the two detectors. When two beam splitters are used, the photon “interferes with itself” and behaves like a wave. This is the so-called wave-particle duality of quantum mechanics which leads to quantum entanglement. This chapter discusses quantum superpositions when two or more particles are present. Understanding and analysing entanglement is one of the most interesting directions in the field of quantum information. First, a historical background of quantum entanglement is given, followed by a discussion on Bell's inequalities, separable states that do not violate Bell's inequalities, pure states that violate Bell's inequalities, mixed states that do not violate Bell's inequalities, and entanglement in second quantisation.
We present a simple yet rigorous field theoretic demonstration of the nonlocality of a single-photon field. The formalism used allows us to calculate the electric field of a single-photon light beam sent through a beam splitter, which directly demonstrates that it is the light field, rather than the photon itself regarded as a particle, that exhibits nonlocality. Our results are obtained without using either inequalities or specific measurement apparatuses, so that they have perfectly general validity.
For optical homodyning, the matrix representation of a lossless beam splitter belongs to the SU(2) group of unimodular second-order unitary matrices. The connection between this group and the rotation group in three dimensions permits the field density operators at the input and output ports of the beam splitter to be related by means of well-known angular-momentum transformations. This, in turn, provides the joint output photon-number distribution, which may be written as a Fourier series in the relative phase shift imparted by the beam splitter, for a general joint state at its inputs. The series collapses to a single term if one of the input fields is diagonal in the number-state representation. If the inputs to both ports are further restricted to be pure number states, the joint, as well as the marginal photon-number distributions, turn out to be directly proportional to the square of Jacobi polynomials in the beam-splitter transmittance. These photon-number probabilities are invariant to a set of physical and time-reversal symmetries. When one of the input photon-number states is the vacuum, the beam splitter simply deletes photons from the other port in Bernoulli fashion, as if they were classical particles. The output photon number is then described by the binomial distribution. If the inputs at the two ports are different number states, neither of which is the vacuum, the photon-number distribution is expressible in terms of summed and weighted products of the results for photomixing with the vacuum. If the inputs at the two ports are identical number states, and a beam splitter of transmittance \ensuremath{\tau}=(1/2 is used, the photon-number distribution assumes a simple but interesting form. It vanishes for odd photon numbers, indicating that the photons assemble in pairs at each output port. Finally, it is shown that homodyning quantum fluctuations can be reduced by using a balanced photomixer for arbitrary input states.