ABSTRACT
When we investigate properties of a quantum optical system, we notice that many systems including a interferometer and a parametric amplifier can be described in terms of the generators of the SU(1, 1) and SU(2) Lie algebras [1–12]. Hence, the Lie algebra method [13–15] is very useful for investigating such quantum optical systems [16,17]. It is also well known that the Liouville-space method [18–21] is suitable for investigating nonequilibrium dynamics of an open quantum system which is placed under the influence of a large environmental system. Indeed, the Markovian quantum master equation of the Lindblad form [22,23] can be represented in terms of the generators of the Lie algebra. Therefore, the combination of the Lie algebra method and the Liouville-space method provides a powerful tool for investigating irreversible time-evolution of a quantum optical system. Furthermore, since the nonequilibrium thermo field dynamics [24–28], which is abbreviated as NETFD, can disentangle operator algebra in the Liouville space, it gives a useful calculational technique in the Liouville-space method. Therefore, in this chapter, we review a synthetic method of Lie algebra and NETFD and its applications to quantum optical systems [31–38].
