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Nature

The Quantum Theory of Dispersion

GREGORY A. BREIT

Published Aug 30, 1924• 30 citations
Publisher
DOI
10.1038/114310a0

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Abstract This chapter covers the history of dispersion theory starting from the classical theories of the late nineteenth century, involving charged harmonic oscillators with resonance frequencies at the absorption frequencies of the dispersing materials to account for anomalous dispersion, the phenomenon that the index of refraction decreases rather than increases with frequency around the absorption frequencies. It then covers dispersion theories formulated in the context of the old quantum theory, in which the link between radiation frequencies and orbital frequencies is severed. Drawing on these efforts, especially by Ladenburg and Reiche in the early 1920s, and using Bohr’s correspondence principle and other ingredients of the old quantum theory, Kramers found a successful quantum formula in late 1923. He only published a full treatment of his new dispersion theory in a paper with Heisenberg in early 1925. This paper played a key role in laying the foundations of matrix mechanics.
We develop a quantum theory of propagation in dispersive nonlinear media via a quantized field theory and a coherent state expansion. The resulting stochastic nonlinear Schrodinger equation includes both quantum fluctuations and thermal noise due to refractive-index fluctuations in the medium. The theory can describe either cw propagation with dispersion or soliton propagation. In the cw case, enhanced squeezing occurs with anomalous dispersion. This is reduced by thermal noise, just as it is in the normal dispersion case. The noise-induced reduction in squeezing is more severe with normal dispersion owing to the intrinsic lack of phase matching in the normal dispersion regime. With anomalous dispersion, the cancellation of nonlinear by linear dispersion allows squeezing to occur over a wide bandwidth even in the presence of thermal fluctuations. Our theory is also able to treat soliton propagation in the anomalous dispersion case. A substantial degree of pulsed squeezing is obtained, and we point out that an optimal detection scheme requires a pulsed local oscillator. Results for the effects of thermal noise on soliton propagation are presented, in which case improvement is obtained by reducing the soliton time duration.