Weak-Field Expansion: A Time-Closed Solution of Quantum Three-Wave Mixing

TL;DR

This paper introduces a perturbative method for solving quantum three-wave mixing dynamics that yields time-closed expressions and converges faster than traditional approaches, enabling higher-order corrections for quantum optical applications.

Contribution

A new perturbative expansion method for quantum three-wave mixing that provides time-closed solutions and improves convergence over standard BCH expansion.

Findings

First-order solution reproduces known quantum parametric amplification results.

Second-order correction acts as an effective detector of time-energy entanglement.

Third-order correction sets limits on quantum state transfer fidelity.

Abstract

We present a systematic derivation of the Heisenberg evolution of a trilinear bosonic Hamiltonian system in presence of a strong drive beyond the standard approximation of a classical, undepleted driving field. We employ a perturbative expansion of the Hamiltonian propagator in orders of the input field amplitudes, as opposed to the standard Baker-Campbell-Hausdorff (BCH) expansion of the propagator in orders of time. Our method automatically provides time-closed expressions; and converges considerably faster than BCH, especially in the regime of high parametric gain because the small parameter it uses is natural to the problem. We obtain the well-known quantum solution for optical parametric amplification of down-conversion simply as the first order of the expansion, and present the rigorous procedure to derive higher order corrections one by one. To demonstrate the utility of higher corrections, we discuss the 2nd order correction to the pump field as an ideal detector of time-energy entanglement in parametric down-conversion. We also use the 3rd order correction to calculate the limits on the fidelity of quantum state-transfer from one optical mode to another using sum/difference frequency generation, due to the quantum properties of the strong driving field.