Linear transformations of field operator

Masashi Ban

The Lindblad operator of the Markovian quantum master equation for an oscillator system is diagonalized by means of linear transformations of annihilation and creation operators in non-equilibrium thermo field dynamics. When the linear attenuation and amplification processes are considered, the left and right eigenstates of the Lindblad operator are constructed in terms of the transformed operators. In the linear attenuation process, the results are compared with those obtained by the damping base of the Lindblad superoperator. The base describing the time-evolution of the linear amplification process is derived and compared with the damping base. The method for diagonalizing the Lindblad operator is useful for calculating average values of physical quantities in the quantum Markovian process.

Brian P. Anderson

Un Cig Ji, Nobuaki Obata

We study an implementation problem for quadratic functions of annihilation and creation operators on a boson field in terms of quantum white noise calculus. The implementation problem is shown to be equivalent to a linear differential equation for white noise operators containing quantum white noise derivatives. The solution is explicitly obtained and turns out to form a class of white noise operators including generalized Fourier–Gauss and Fourier–Mehler transforms, Bogoliubov transform, and a quantum extension of the Girsanov transform.

Ralph E. Christoffersen

J. Capmany, Daniel Pérez

Integrated multi-port interferometers are the second step in programmable photonics. Initially conceived and designed to implement fixed linear optics unitary transformations between N input and N output ports to support mode transformations in quantum optics, the rapid development of this field and the possibility of applying these structures to other areas stimulated development of devices capable of supporting arbitrary linear transformations by suitable programming and surpassing use in square matrix transformations and unitary operators. This chapter reviews some basic concepts on unitary N×N matrices and then describes in detail the basic design principles and integrated optics implementations of triangular and rectangular designs for multi-port interferometers. It then considers more specific interferometer designs, which do not provide possible implementation of general linear transformations but are optimised for performing important operations, e.g. the Fourier transform. Lastly, it covers the linear universal component, which integrates both universal beam couplers and integrated multi-port interferometers.

V. Balakrishnan