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American Journal of Physics

Finite-dimensional matrix representations as calculational tools in quantum optics

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

Published Aug 1, 1993• 14 citations
Publisher

Abstract

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.

DOI
10.1119/1.17149

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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.
Mathematical theory of infinite dimensional Hilbert spaces and the theory of operator algebras acting in such spaces (or C algebras in a more abstract approach) provide a standard setting for the formulation of modern quantum mechanics. On the other hand, experimental and theoretical progress achieved in the field of quantum information theory in the last two decades has indicated the practical and technological importance of low-dimensional quantum systems, where only a few basic modes play a significant role. Such modes can often be effectively decoupled from the rest of the system and controlled separately, providing physical realizations of qubits, qutrits and other basic information carriers. Regardless of concrete physical realization, be it photon polarization, electron or nuclear spin, charge in Josephson junctions to name just a few, the mathematical description of such systems requires only finite-dimensional Hilbert space language and finite-dimensional matrix algebras. Such structures are in principle computationally manageable in sharp contrast to the infinite dimensional ones.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
This chapter aims to develop an appropriate notation and to facilitate the subsequent discussion of the role played by the Lie algebras. It is unlikely that one manages to express a sufficiently realistic model only in terms of the operators that span a finite-dimensional Lie algebra, but it may be possible to simulate most of the relevant features of many physical phenomena by means of such simple models. The chapter discusses well-known approximate methods for the treatment of the time-evolution equation. If the Hamiltonian operator belongs to a finite-dimensional Lie algebra, then the differential equation for the time-evolution operator is easier to solve, and even the calculation of transition probabilities and expectation values of observables may be straightforward. The most relevant feature of this application of the Lie algebraic method is the occurrence of a matrix equation of finite dimension in contrast with the expansion in a basis set of state vectors, which typically leads to matrices of infinite dimension.
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 ].