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Spectral decomposition of single-tone-driven quantum phase modulation

Electro-optic phase modulators driven by a single radio-frequency tone Ω can be described at the quantum level as scattering devices where input single-mode radiation undergoes energy changes in multiples of ℏΩ. In this paper, we study the spectral representation of the unitary, multimode scattering operator describing these devices. The eigenvalue equation, phase modulation being a process preserving the photon number, is solved at each subspace with definite number of photons. In the one-photon subspace the problem is equivalent to the computation of the continuous spectrum of the Susskind–Glogower cosine operator of the harmonic oscillator. Using this analogy, the spectral decomposition in is constructed and shown to be equivalent to the usual Fock-space representation. The result is then generalized to arbitrary N-photon subspaces, where eigenvectors are symmetrized combinations of N one-photon eigenvectors and the continuous spectrum spans the entire unit circle. Approximate normalizable one-photon eigenstates are constructed in terms of London phase states truncated to optical bands. Finally, we show that synchronous ultrashort pulse trains represent classical field configurations with the same structure as these approximate eigenstates, and that they can be considered as approximate eigenvectors of the classical formulation of phase modulation.