Complementarity, quartic polynomials and one-photon-added coherent and squeezed states

Complementarity (expressed through the x-p commutation relationship) yields, by using suitable non-negative polynomials, quantum inequalities. The latter become strict equalities only for restricted sets of quantum states, which display genuine quantum features. We have shown in previous studies that: (i) a certain non-negative quartic polynomial f(4) led to a new quantum inequality, expressed through the discriminant (D-r') of the equation f(4) = 0 as D-r' <= 0; (ii) D-r' = 0 gave rise to an interesting class of non-Gaussian quantum states vertical bar psi(4)> and (iii) in quantum optics, a subset of those vertical bar psi(4)> were the displaced one-photon states, previously proposed by other authors through different motivations and already generated experimentally in 2002. The extension of our previous research studies will be reported here. We shall characterize the general class of non-Gaussian quantum states vertical bar psi(4,g)> such that D-r' = 0. The class of the vertical bar psi(4,g)> is larger than and includes that of the former vertical bar psi(4)>. The Wigner function for the vertical bar psi(4,g)> is obtained and is negative in some domains. In quantum optics, a subclass of the vertical bar psi(4,g)> is formed by the single-photon-added coherent states (non-Gaussian) generated experimentally, in turn, in 2004. Another subclass of the vertical bar psi(4,g)> includes single-photon-added squeezed states and single-photon-added squeezed coherent states (both being non-Gaussian). Mandel's parameters are studied for those states: in particular, its negativity for the single-photon-added squeezed states is established for a certain interval of the squeeze factor. The (Hilbert-Schmidt) non-Gaussianities for both the displaced one-photon states and the single-photon-added squeezed states are obtained explicitly: both of them are numerically equal to 5/12. The possibility of generating the single-photon-added squeezed states experimentally is discussed briefly, by extending previous studies by other authors on producing related quantum states; in particular, we treat in outline one possible generation using optical parametric oscillators and parametric amplifiers.