Second-order squeezed states
Paulina Marian · Physical Review A · 1997
We examine the generalized squeezed states defined as eigenstates of a linear combination of the lowering and raising operators ${\mathrm{a}}^{2}$ and (${\mathrm{a}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$${)}^{2}$, respectively. This approach is entirely equivalent to the minimum-uncertainty method applied to the amplitude-squared operators. We solve the eigenvalue equation in Glauber's coherent-state representation and find two independent solutions. Their Fock-state expansions, one containing only even and the other only odd number states, reveal a strong nonclassical character. We show that the calculation of the mean photon number is sufficient to obtain the expectation values of interest. Consequently, photon statistics is investigated in both cases by using the generating function of the photon-number distribution. We find the conditions under which the second-order squeezed states display photon antibunching and quadrature squeezing. Also discussed is the preservation of their amplitude-squared squeezing by linear amplification at gains exceeding 2. Analytically, our results are simple formulas in terms of Kummer and Gauss hypergeometric functions that allow straightforward numerical calculations.