Implementation of multimode Gaussian unitaries using primitive components

In the field of continuous-variable systems a fundamental role is played by Gaussian unitaries, that is, operators that preserve the Gaussian character of the states. The paper deals with the implementation of an arbitrary multimode Gaussian unitary by means of primitive components, namely, single-mode operators and simple two-mode operators (beam splitters). A partial solution to this problem is provided by the Bloch-Messiah reduction, which gives an architecture consisting of a multimode displacement, a diagonal squeezer, and two rotation operators. In this architecture displacement and squeezing are already formed by primitive components, so that it remains to find the implementation of the rotation operators. To this end, a suitable factorization of unitary matrices allows one to complete the desired implementation with primitive components. The theory is illustrated with an example of application to a four-mode Gaussian unitary.

Figure 1

Graphical representation of the primitive components.

Figure 2

Detailed scheme of the architecture obtained with the BM reduction: an $N$-mode Gaussian unitary decomposed into a rotation $R\left(\psi \right)$, a parallel set of $N$ single-mode real squeezers, a rotation $R\left(\beta \right)$, and a parallel set of $N$ single-mode displacements.

Figure 3

Implementation of a two-mode rotation operator and of its adjoint.

Figure 4

Implementation of the complex beam splitters $T_{12}{(c,\beta )}_3$ and $T_{13}{(c,\beta )}_3$.

Figure 5

Implementation of the rotation matrix $U=e^{i\psi }$ for $N=4$ by phase shifters and beam splitters. The schemes with primitive components for the complex BSs are given in Fig. 4, and for the unitary matrix $U_2$ the scheme is given in Fig. 3.

Figure 6

Implementation by primitive components of the four-mode rotation $R\left(\beta \right)$.