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Annalen der Physik

Quantenmechanische Beschreibung der Interferenz zwischen unabhängigen nichtmonochromatischen Lichtstrahlen

Henryk Paul

Published Jan 1, 1967• 5 citations
Publisher

Abstract

Abstract The concept of nonmonochromatic modes of the radiation field introduced recently by TITULAER and GLAUBER, is employed to give a quantum mechanical description of the interference between nonmonochromatic light beams produced by different sources (e.g. lasers). The result is, in fact, that previous calculations by the author et al. in which each light beam was idealized as an excited state of one monochromatic mode, practically unchanged apply also to nonmonochromatic modes. The previous formulae describing the interference, on the other hand, are reproduced for the special case of (nonmonochromatic) modes corresponding to finitely extended beams of plane wave type, and, thus, are founded more exactly.

DOI
10.1002/andp.19674740312

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Zur Interferenz unabhängiger Lichtstrahlen geringer Intensität

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Abstract We have observed the interference between two independent single mode He‐Ne laser beams. Our special equipment allowed us to record many times the interference pattern, when it was standing for a short time at the same position on the recorder. In this way we have obtained an interference picture of two independent light beams, which was (a) visible directly for the unaided eye for large intensities, (b) recorded on a photographic film for low intensities. The result of the experiments were in agreement with the statement of the quantummechanical description: We found a classical interference pattern down to very small intensities of the superposed beams. This result is discussed with regard to the quantum structure of light.

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Saturated Multi‐Photon‐Absorption of a Nonmonochromatic Quantized Field

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Abstract By extending a previously developed method [9] to the case of saturation, the interaction of a nonmonochromatic light beam with a saturable m‐photon‐absorber (m = 1, 2,…) is theoretically investigated for frequency widths of the incident light being small compared to the bandwidth of the absorber. It turns out that by averaging the exact Heisenberg equation of motion of the operator for the slowly varying field amplitude with respect to the atomic relaxation time and by taking the expectation value over the absorbing atomic system a reduced single‐mode equation can be derived, which refers to a mode volume precisely defined in physical terms. The resulting reduced Heisenberg equation of motion for the radiation field is compared to the single‐mode density matrix equation following from the method of SCULLY and LAMB. For ideal laser light and for chaotic light it is used to study the change of photon statistics in dependence on the degree of saturation and on the ratio of inhomogeneous and homogeneous linewidth.

Annalen der Physik•Jan 1, 1981•4 citations
Abstract The concept of nonmonochromatic modes of the radiation field introduced recently by TITULAER and GLAUBER, is employed to give a quantum mechanical description of the interference between nonmonochromatic light beams produced by different sources (e.g. lasers). The result is, in fact, that previous calculations by the author et al. in which each light beam was idealized as an excited state of one monochromatic mode, practically unchanged apply also to nonmonochromatic modes. The previous formulae describing the interference, on the other hand, are reproduced for the special case of (nonmonochromatic) modes corresponding to finitely extended beams of plane wave type, and, thus, are founded more exactly.
Abstract We have observed the interference between two independent single mode He‐Ne laser beams. Our special equipment allowed us to record many times the interference pattern, when it was standing for a short time at the same position on the recorder. In this way we have obtained an interference picture of two independent light beams, which was (a) visible directly for the unaided eye for large intensities, (b) recorded on a photographic film for low intensities. The result of the experiments were in agreement with the statement of the quantummechanical description: We found a classical interference pattern down to very small intensities of the superposed beams. This result is discussed with regard to the quantum structure of 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.
A comprehensive review is given of investigations, both theoretical and experimental, that demonstrate interference to take place even when the fields involved are produced by independent sources. The physical situation is rather different for coherent and incoherent fields. In the first case, where the sources are lasers, a conventional interference pattern can be observed which is adequately described by the classical theory. In the present paper, special attention is paid to the question of whether this interference persists when the two laser beams become strongly attenuated. In the second case, the sources are individual atoms excited by a pumping mechanism, that emit spontaneously and, hence, independently from each other. In those circumstances, no interference pattern can show up. However, it becomes evident from both the classical and the quantum-mechanical theory that interference effects can still be established by observing intensity correlations rather than the intensity itself. This point is discussed in greater detail. The pioneering experiments of Forrester, Gudmundson and Johnson, and Brown and Twiss are reviewed in this context. Especially interesting from the theoretical point of view is the case of two emitting atoms, since then the classical and the quantum-mechanical description differ significantly, the quantum theory predicting the intensity correlations to be distinctly stronger than those following from classical considerations. This specific quantum-mechanical feature is shown to be intimately connected with the corpuscular aspect of light.
IT is well known from experiment, as well as from classical theory, that strictly monochromatic light waves originating from different sources do interfere.' An example in which the waves are nearly monochromatic is found in the case of radio waves, and it is known that radio waves from two stations that have the same frequency show an interference effect. We may describe the phenomenon quantum mechanically if we represent the light quanta emitted by both sources by waves in Schroedinger configuration space. So long as we consider the quanta emitted by one source as different from those emitted by the other, the total wave function will be the product of the wave functions corresponding to the quanta emanating from both sources, and there will be no interference effect. In the present paper, we will see how things are changed when we symmetrize or antisymmetrize the wave function, i.e. assume either Bose or Fermi statistics for the particles. In the simple quantum mechanical picture that we will use here, we will assume a definite number of photons coming out from each source, and this implies that the phase relations will be undetermined.2 As a result, even the symmetrized wave functions must be expected to give, at any point in space, the same density of quanta that classical theory determines, when the intensity is averaged over the phase differences between the two sources. This implies that no interference is observed between the waves from the two sources. If we want to look for an interference effect when dealing with a quantum picture in which we have a definite number of photons, we must look for a phenomenon whose classical description should be independent of the phase. We shall make use of the fact that, in classical theory,
Abstract By extending a previously developed method [9] to the case of saturation, the interaction of a nonmonochromatic light beam with a saturable m‐photon‐absorber (m = 1, 2,…) is theoretically investigated for frequency widths of the incident light being small compared to the bandwidth of the absorber. It turns out that by averaging the exact Heisenberg equation of motion of the operator for the slowly varying field amplitude with respect to the atomic relaxation time and by taking the expectation value over the absorbing atomic system a reduced single‐mode equation can be derived, which refers to a mode volume precisely defined in physical terms. The resulting reduced Heisenberg equation of motion for the radiation field is compared to the single‐mode density matrix equation following from the method of SCULLY and LAMB. For ideal laser light and for chaotic light it is used to study the change of photon statistics in dependence on the degree of saturation and on the ratio of inhomogeneous and homogeneous linewidth.