With the prominent progress in quantum computing and quantum information theory, nonclassical states, which have prospects for applications within these fields, have become more noticeable by researchers within the last few years. There has been much research recently and many applications of nonclassical states using their nonclassical properties have already been developed. Entanglement especially is an essential resource in quantum computing and quantum information theory. Among the huge number of nonclassical states, pair coherent states (PCS) and trio coherent states (TCS), as a three mode extension of PCS, have the special capability that they are naturally entangled.
In this thesis the nonclassical properties and generation of trio coherent states and K-dimensional trio coherent states (KTCS) are introduced and studied. KTCS are a K-dimensional generalization of TCS which are eigenstates of product of three independent mode lowering operators. These states have a three mode nature so they are expected to play an important role in various applications as GHZ state or W state are utilized. In this sense TCS and KTCS are introduced and studied with their nonclassical properties such as their sub-poissonian nature and their violation of Cauchy-Schwarz inequalities. Generation of nonclassical states is one of the interesting topics and it was initially studied in the 90``s. The generation scheme for vibrational modes of a trapped ion are presented and cases of TCS and KTCS (when K = 2) are given here. To show how the generation scheme for TCS and KTCS functions, a Monte Carlo wave-function (MCWF) approach is reviewed and a numerical simulation using this approach is also performed.