In quantum mechanics the position and momentum operators are related to each other via the Fourier transform. In the same way, here we show that the so-called Pegg-Barnett phase operator can be obtained by the application of the discrete Fourier transform to the number operators defined in a finite-dimensional Hilbert space. Furthermore, we show that the structure of the London-Susskind-Glogower phase operator, whose natural logarithm gives rise to the Pegg-Barnett phase operator, is contained in the Hamiltonian of circular waveguide arrays. Our results may find applications in the development of new finite-dimensional photonic systems with interesting phase-dependent properties.

Abstract:We study the entropy of a quantized field in interaction with a two-level atom (in a pure state) when the field is initially in a mixture of two number states. We then generalise the result for a thermal state; i.e., an (infinite) statistical mixture of number states. We show that for some specific interaction times, the atom passes its purity to the field and therefore the field entropy decreases from its initial value.

In this paper, we show that the Pegg–Barnett formalism accepts coherent states constructed as eigenstates of the annihilation operator, considering both the number and the phase. These operators are defined within a ( s + 1 ) -dimensional Hilbert space H s and with periodic conditions. The coherent states that we find are determined by the eigenvalue of the annihilation operator, which leads to a discrete spectrum. This approach allows calculation of the discrete-finite counterpart of the Wigner function in a phase space defined by the variables of number and phase.