Non-Hermiticity in Quantum Nonlinear Optics Through Symplectic Transformations

Over the past decade, classical optical systems with gain or loss, modeled 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-quantized 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 P T-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.