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Physical Review A

Semiclassical expansion for nonlinear dielectric media

Mark S. Hillery, Leonard D. Mlodinow

Published Feb 1, 1985• 44 citations
Publisher

Abstract

We employ a Holstein-Primakoff expansion to derive a collective-mode macroscopic effective Hamiltonian from a microscopic model of a nonlinear optical medium where the atoms are treated as two-level systems.
DOI
10.1103/physreva.31.797

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The microscopic operators have been investigated for radiation and polarization fields within a discrete molecular medium. The medium comprises atoms (or molecules) each containing an arbitrary number of energy levels. Explicit mode expansions have been derived for the quantized microfields in terms of normal Bose operators for polariton creation and annihilation. These microscopic operators have been demonstrated to yield the correct macroscopic and local field operators presented in part I [Phys. Rev. A 53, 3543 (1996)]. On the other hand, the commutation relations between the expanded microfields differ from the exact commutation relationships. This happens because the microfields have been described in terms of a continuous refractive index. Hence the expanded field operators do not extend to modes with extremely high frequencies ranging over the photon umklapp frequencies cG. In spite of that, the proper commutation relations hold between the resultant macroscopic (averaged) fields. This justifies separate quantization of the slowly modulated (macroscopic) part of the radiation field in linear dielectrics: such an approach is utilized in phenomenological quantization schemes. Finally, although the microscopic operators presented are not complete, the mode expansions adequately represent the quantized microfields associated with the optical modes in condensed molecular media.