Intelligent states for a number-operator–annihilation-operator uncertainty relation

Abstract

Recently a new uncertainty relation was found as an alternative to a number-phase uncertainty relation for a harmonic oscillator. In this paper we determine numerically, via the discrete-variable-representation method known from quantum chemistry, the exact states that saturate this new uncertainty relation. We analyze the physical properties of the states and compare them to the intelligent states of the Pegg-Barnett uncertainty relation. We find that for a given set of expectation values of the physical parameters, which are the particle number and the two quadratures, the two kinds of intelligent states are equivalent. The intelligent states are the eigenstates corresponding to the lowest eigenvalue of a Hermitian operator, which, when interpreted as a Hamiltonian of a physical sytem, describes a nonlinear driven harmonic oscillator, for example, a Duffing oscillator for a certain parameter range. Hence, our results can be interpreted as the determination of the ground state of such physical systems in an explicit analytic form as well. As the Pegg-Barnett intelligent states we use are expressed in terms of a coherent-state superposition facilitating experimental synthesis, we relate the states determined here to experimentally feasible ones.