Non-Hermiticity in quantum nonlinear optics through symplectic transformations

TL;DR

This paper introduces a quantum optical scheme using symplectic transformations to simulate non-Hermitian dynamics, overcoming quantum noise limitations and enabling efficient, scalable quantum simulations of non-unitary processes.

Contribution

It establishes a novel connection between PT-symmetry and symplectic Bogoliubov transformations, enabling non-Hermitian quantum simulation without post-selection.

Findings

Success probability is independent of system size or simulation time.

The scheme can be efficiently Trotterised like unitary transformations.

It allows simulation of arbitrary non-unitary processes in quantum optics.

Abstract

Over the past decade classical optical systems with gain or loss, modelled by non-Hermitian parity-time symmetric Hamiltonians, have been deeply investigated. Yet, their applicability to the quantum domain with number-resolved photonic states is fundamentally voided by quantum-limited amplifier noise. Here, we show that second-quantised Hermitian Hamiltonians on the Fock space give rise to non-Hermitian effective Hamiltonians that generate the dynamics of corresponding creation and annihilation operators. Using this equivalence between $\mathcal{PT}$-symmetry and symplectic Bogoliubov transformations, we create a quantum optical scheme comprising squeezing, phase-shifters, and beam-splitters for simulating arbitrary non-unitary processes by way of singular value decomposition. In contrast to the post-selection scheme for non-Hermitian quantum simulation, the success probability in this approach is independent of the system size or simulation time, and can be efficiently Trotterised similar to a unitary transformation.