Spectra and Generalized Eigenfunctions of the One- and Two-Mode Squeezing Operators in Quantum Optics

Abstract

In recent years the concept of squeezed state has become central in quantum optics both from the theoretical and experimental point of view [1]. In the simplest one-mode case a squeezed state is defined here as a displaced squeezed vacuum of the form |α, ς) = D(α)S(ς)|0> obtained by applying the squeezing operator S(ς) = exp[(ς *a ² — ς a ⁺² ) 12] [2] and the displacement operator D(α) = exp( α a ⁺ — α * a) on the vacuum state |0> (i e the ground state of the one-dimensional harmonic oscillator), ς = re ^12θ and α = |α| e ^iß are complex numbers, and a = (q + ip)/√2, a ⁺ =(q-ip)/ √ 2 in terms of the normalized coordinate and momentum operators q and p , where [q, p] = iI The term “squeezing” comes from the fact that for cos 2θ > tanh r (> 0) the dispersion Δq of the coordinate operator in the squeezed state is smaller than the vacuum state value 1/√2 : the uncertainty of the value of q is squeezed compared to the vacuum value. If one performs a rotation in the qp-plane (“phase space”) to new canonically conjugate operators q _θ = cos θ q + sin θ p , p _θ = sin θ q + cos θ p the dispersions Δ q _θ and Δ p _θ will be equal to e ^-r / √2 and e ^r √2, respectively, and the correlation Δ (q _θ p _θ ) [3] is zero. This implies that the variance matrix of the original set qp (a symmetric 2x2 matrix with the variances (Δq)² and ( Δ p) ² as diagonal elements and Δ (qp) as off diagonal elements) has its determinant equal to the minimum allowed value 1/4 . This can actually be taken as a characteristic property of squeezed states: they are the states (amongst a priori even mixed states) that give equality in the relation ( Δ q) ² ( Δ p) ² — [ Δ (qp)] ² ≥ 1 / 4 (sometimes called the Schrödinger-Robertson uncertainty relation).