Operator Algebra Approach to Optics

Maciej Horowski, Anatol Odzijewicz, Agnieszka Tereszkiewicz

Ravinder Rupchand Puri

Pierre Meystre, Murray Sargent

A. Mufti, H. A. Schmitt, M. Sargent

A powerful operator-algebra technique is used to calculate several useful relations encountered in quantum optics, such as the disentangling of the squeezing operator. The technique uses a faithful finite dimensional matrix representation of a Lie algebra to perform representation-independent calculations and often requires considerably less computation than other methods. While it has had little exposure within the quantum optics community, the method is quite popular with mathematical physicists in field theory.

Maxim Sukharev

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 ].

Pierre Meystre, Murray Sargent