Vector Product Approach of Producing Non-Gaussian States

Abstract

Two new configurations of superposition of quantum states corresponding to the angular momentum of spin-1 quantum system is proposed in this article. One is a trivial state, |β, α⊥〉≡ (I −|α〉〈α|)|β〉, which consists of two Klauder’s type of spin coherent state |α〉 and |β〉 that is also perpendicular to |α〉, and the other is a non-trivial case composed of vector product of two different copies of spin states, |α〉 and |β, α⊥〉, i.e. |vps〉 := |α〉×|β, α⊥〉. Some properties of such quantum states are discussed. For instance, depending on the particular choice of parameters, we show that they minimize both the Heisenberg and Robertson-Schr$\ddot {o}$dinger uncertainty relations (RSUR) of each pair of the angular momentum components while the standard spin-coherent states do not satisfy. We have also shown that preparing the system state as a |β, α⊥〉 and choosing one of the mentioned orthogonal states, |vps〉, one can find a minimum lower bound for the Maccone-Pati uncertainty relations (MPUR) which is stronger than the other uncertainties. We also establish an irreducible and unitary representation of Lie algebra su(2) through the introduced states.