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Canadian Journal of Physics

Quantum Theory of Spectral Modulation

Andrzej Zardecki, Claude A. Delisle

Published Sep 1, 1974• 1 citation
Publisher

Abstract

Recent experiments of Delisle and Brochu, in which spectral modulation at a large path difference was observed, are analyzed in quantum mechanical terms. Starting with a stationary density operator for thermal light, a two beam superposition such as taking place in a Michelson interferometer, is investigated. It is shown that a simple interference law holds separately for each mode of radiation. Consequently, a coherent superposition of light should be observed for a time delay arbitrarily large compared to the coherence time. The latter determines merely the range of intensity modulation. Analysis of the Michelson interferometer accounting for the energy returned to the source is also presented.

DOI
10.1139/p74-223

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Recent experiments of Delisle and Brochu, in which spectral modulation at a large path difference was observed, are analyzed in quantum mechanical terms. Starting with a stationary density operator for thermal light, a two beam superposition such as taking place in a Michelson interferometer, is investigated. It is shown that a simple interference law holds separately for each mode of radiation. Consequently, a coherent superposition of light should be observed for a time delay arbitrarily large compared to the coherence time. The latter determines merely the range of intensity modulation. Analysis of the Michelson interferometer accounting for the energy returned to the source is also presented.
The temporal coherence interference properties of light as revealed by single detector intensity measurements in a Michelson-Morley interferometer (MMI) is often described in terms of classical optics. We show, in a pedagogical manner, how such features of light also can be understood in terms of a more general quantum-optics framework. If a thermal reference source is used in the MMI local oscillator port in combination with a thermal source in the signal port, the interference pattern revealed by single detector intensity measurements shows a distinctive dependence on the differences in the temperature of the two sources. A related method has actually been used to perform high-precision measurements of the cosmic microwave background radiation. The general quantum-optics framework allows us to consider any initialquantum state. As an example, we consider the interference of single photons as a tool to determine the peak angular-frequency of a single-photon pulse interfering with a single-photon reference pulse. A similar consideration for laser pulses, in terms of coherent states, leads to a different response in the detector. The MMI experimental setup is therefore an example of an optical device where one, in terms of intensity measurements, can exhibit the difference between classical and quantum-mechanical light.
We investigate how the temporal coherence interference properties of light in a Michelson-Morley interferometer (MMI), using only a single-photon detector, can be understood in a quantum-optics framework in a straightforward and pedagogical manner. For this purpose we make use of elementary quantum field theory and Glaubers theory for photon detection in order to calculate the expected interference pattern in the MMI. If a thermal reference source is used in the MMI local oscillator port in combination with a thermal source in the signal port, the interference pattern revealed by such an intensity measurement shows a distinctive dependence on the differences in the temperature of the two sources. The MMI can therefore be used in order to perform temperature measurements. A related method was actually used to carry out high precision measurements of the cosmic micro-wave background radiation on board of the COBE satellite. The theoretical framework allows us to consider any initial quantum state. The interference of single photons as a tool to determine the angular peak-frequency of a one-photon pulse interfering with a single-photon reference pulse is, e.g., considered. A similar consideration for coherent laser pulses leads to a different response in the detector. The MMI experimental setup is therefore in a sense an example of an optical device where one can exhibit the difference between classical and quantum-mechanical light using only intensity measurements.
It has been shown by Mandel that the spectral density shows a cosine modulation with relative amplitude 2(Ī1 Ī2)1/2γ12(0)/(Ī1 + Ī2) when the optical path difference is much longer than the coherence length in a quite general superposition experiment of two quasi-monochromatic, polarized, and partially coherent beams of light. It is shown here that this is equally true for path differences smaller or equal to the coherence length. This conclusion is reached by considering the properly defined visibility of the modulation of the spectral density. Both aspects of the modulation of the spectral density are considered: modulation as a function of path difference for a fixed frequency ν0 and modulation as a function of frequency for a fixed path difference cT0.
In paying tribute to the many contributions of Adolf Lohmann to optical science, we must highlight his inspiring talent as a lecturer and teacher as well as his ability to introduce original concepts and ideas.We have chosen to devote our contribution to one aspect of the teaching of physical optics. When teaching interferences, the lecturer always encounters the question:Why do two different frequencies never interfere? A two-beam interferometer illuminated by light of a different frequency on each arm does not show fringes. Or to be more precise, in the quite common case of thermal light sources,1 common experience shows that interferences between two different frequencies of the source spectrum cannot be observed. Interferences with light that is not strictly monochromatic are observed but merely result from the incoherent superposition of all component monochromatic interference intensities. Providing a convincing explanation of this fact in an elementary course is not easy because statistical tools are required that are only available to the students at a more advanced level—typically, senior undergraduate or graduate. However, a full account of the phenomenon based on solid statistical grounds is not even commonly found in textbooks designed for such advanced students. Such is the aim of the present chapter. Indeed, the superposition of two different frequencies should be expected to show a beat phenomenon in time. Nevertheless, common experience shows that interference effects are stationary in time and show no beat effect. If we start from the definition of interferences as the phenomenon whereby the intensity resulting from the superposition of two light beams differs from the sum of the two intensities, we are led to clarifying the temporal aspects in the detection of light beams. First, the common situation of describing the effect at a less advanced level is examined, together with the difficulties that automatically arise. After a review of the necessary statistical concepts such as, in particular, the Wiener-Khintchin theorem, the issue of what exactly is meant by average intensity and the associated “classical detection noise” is addressed, leading to a complete model of temporal coherence in thermal light.
The conditions are examined under which interference effects should be observable in the superposition of two light beams derived from independent sources. The quantum-mechanical description of these effects differs from the classical description principally in that it is necessarily based on expectation values of the light intensity at one or more space-time points. It is shown that pure states of the radiation field which are not energy states give rise to interference effects, but in that case the two beams cannot meaningfully be described as statistically independent. For a realistic description mixed states have to be introduced, when the expectation value of the intensity gives no indication of interference effects. On the other hand, the intensity correlation at two space-time points is a periodic function of the separation of the points and indicates the presence of transient interference effects. The effects become readily observable only when the average photon occupation number of each cell of phase space is appreciably greater than 1, and this explains why laser beams were needed for the experimental observations. It is pointed out that, even in these experiments, each photon may be regarded as interfering only with itself.