Plasmonics

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 ].