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Optica Acta International Journal of Optics

Theory of Single-photon Interference Experiments

M. Ley, R. Loudon

Published Apr 1, 1986• 13 citations
Publisher

Abstract

Expressions are derived for the density matrix elements appropriate to single-quantum excitation of the two arms of optical interferometers where an amplifier is inserted into one of the arms. The results are used to calculate mean intensities, correlations and photon number distributions for the outputs of Michelson and Hanbury Brown and Twiss interferometers. It is concluded that the noise output of the amplifier obscures quantum effects in the interferometer responses.

DOI
10.1080/713821962

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Expressions are derived for the density matrix elements appropriate to single-quantum excitation of the two arms of optical interferometers where an amplifier is inserted into one of the arms. The results are used to calculate mean intensities, correlations and photon number distributions for the outputs of Michelson and Hanbury Brown and Twiss interferometers. It is concluded that the noise output of the amplifier obscures quantum effects in the interferometer responses.
This chapter presents experiments employing the optical devices described in Chapter 8 and the detection schemes discussed in Chapter 9. Single-photon interference — described by a first-order correlation function — is destroyed by measurements yielding “which-path” information. The Hanbury Brown-Twiss effect is described by a second-order (intensity-intensity) correlation function. Two-photon interference is observed with the Hong-Ou-Mandel (HOM) interferometer, in which a pair of photons interfere at a single beam splitter. In the Franson interferometer, the interference occurs between photons at spatially separated beam splitters. The HOM effect is used to measure the propagation speed of a single photon in a dielectric medium and the tunneling speed across an optical barrier. Similar techniques show that the mere possibility of observing which-path information destroys single-photon interference, even if no measurements are actually made.
A quantum-statistical description of noise in interferometers is given in terms of the Wigner distribution function. The interferometer may contain an amplifier in one of its arms. The input field can be in a variety of states such as a Fock state, a coherent state or a squeezed coherent state. The Wigner function of the output field at the detector is shown to have a general Gaussian form with non-zero complex field amplitude and with other parameters related to the characteristics of the input field, amplifier and beam splitters, etc. An explicit form of the photon-number distribution is given. Higher-order correlations of the output field can be obtained from the Gaussian property of the Wigner function.
The interference effects that can be observed in the two output arms of a lossless beam splitter are calculated for incident light in the form of a photon-pair excitation in the two input arms. The output state that occurs when the photon pair is excited in a single input arm resembles that expected for independent classical particles, whereas quantum interference effects occur when the photon pair is divided between the two input arms. Detailed output photocount correlation functions are calculated for two-photon input states produced by a two-atom light source, a degenerate or nondegenerate parametric oscillator in a high-Q cavity, and an atomic cascade emission light source.
An intensity-correlation experiment is proposed to test the second-order coherence of laser light from a cw-operated laser based on the Glauber formalism. The calculation utilizes the known form of the density matrix for radiation from a randomly excited source such as a discharge tube which is equivalent to filtered black-body radiation, and represents the laser radiation field by a coherent state. The intensity correlation calculated exhibits the standard Hanbury Brown-Twiss term and an extra term due to intensity interference between the Fourier components of the thermal field and the laser mode.
An assembly of noninteracting atoms may become correlated upon interaction with entangled photons, and certain elements of their joint density matrix can then show collective resonances. We explore experimental signatures of these resonances in the nonlinear response of a pair of two-level atoms. We find that these resonances are canceled out in stimulated signals such as pump-probe and two-photon absorption due to the destructive interference of two-photon-absorption and emission pathways in the joint two-particle space. However, they may be observed in photon statistics (Hanbury-Brown--Twiss) measurements through the attenuation of two-time intensity correlations.
Multiphoton interference results in modulations of output probabilities with phase shift periods that are much shorter than $2\ensuremath{\pi}$. Here, we investigate the physics behind these statistical patterns in the case of well-defined photon numbers in the input and output modes of a two-path interferometer. We show that the periodicity of the multiphoton interference is related to the weak value of the unobserved intensity difference between the two arms of the interferometer. This means that the operator relations between the photon number differences in input, path, and output can be used to determine the periodicity of the experimentally observed quantum interference, establishing an important link between the classical causality of random phase interference and quantum effects that depend on the superposition of classically distinct possibilities.