# From Hamiltonians to complex symplectic transformations

## TL;DR

This paper introduces a new complex symplectic transformation framework for Gaussian unitaries, unifying and extending existing representations, and reveals that the final displacement depends on both quadratic and linear Hamiltonian components.

## Contribution

It develops a comprehensive complex symplectic transformation approach that unifies Bogoliubov and real symplectic transformations, and shows the displacement depends on quadratic terms.

## Key findings

- Final displacement depends on quadratic and linear Hamiltonian parts.
- Combining squeezing and rotation achieves arbitrary displacement.
- New representation simplifies analysis of Gaussian unitaries.

## Abstract

Gaussian unitaries are specified by a second order polynomial in the bosonic operators, that is, by a quadratic polynomial and a linear term. From the Hamiltonian other equivalent representations of the Gaussian unitaries are obtained, such as Bogoliubov and real symplectic transformations. The paper develops an alternative representation, called complex symplectic transformation, which is more compact and is comprehensive of both Bogoliubov and real symplectic transformations. Moreover, it has other advantages. One of the main results of the theory, not available in the literature, is that the final displacement is not simply given by the linear part of the Hamiltonian, but depends also on the quadratic part. In particular, it is shown that by combining squeezing and rotation, it is possible to achieve a final displacement with an arbitrary amount.