Analytic Solutions To Fully Quantized Parametric Amplification

Abstract

We derive analytic expressions for observable dynamics under the fully quantized trilinear Hamiltonian, governing quantum three-wave mixing and parametric amplification. Starting from a coherent pump and vacuum signal and idler, the calculation reorganizes the Heisenberg expansion into resummed depletion and return profiles, yielding directly evaluable expressions for all 1st-, 2nd- and 3rd-order correlations. The construction uses no parametric approximation or fitted parameters. Representative numerical benchmarks show errors of order $10^{-12}$ throughout the entire pump depletion, and below $10^{-3}$ across dozens of depletion-and-back-conversion cycles, up to the first total collapse of the oscillation. The first-depletion treatment additionally provides 4th-order correlations, accommodates bright squeezed-pump inputs and reveals phenomena important to future optical quantum technologies. The framework can be extended to higher moments.

The paper addresses observable dynamics in fully quantized three-wave mixing, governed by $H=i\hbar g(a^\dagger b^\dagger c-abc^\dagger)$, beyond the undepleted classical-pump approximation. It primarily considers a coherent pump of amplitude $\alpha$ with vacuum signal and idler, including strong depletion and repeated back-conversion. The method reorganizes the Heisenberg expansion in $\epsilon=\alpha^{-2}$ into successive depletion and return passages. Borel resummation gives integral profiles, with systematic finite-pump corrections. Five pump moments determine all correlations of total operator degree up to three through symmetries, conservation laws, and derivatives. Interference between passages and averaging over photon-number-dependent evolution rates describe repeated oscillations and their collapse envelope without propagating large Fock-space sectors. For the benchmark $\alpha=25$, corrections through $O(\epsilon^{50})$ give relative pump-population errors of order $10^{-13}$ during the first depletion and below $10^{-3}$ through $\chi=\alpha gt=200$. Quantum-noise predictions require subleading corrections even when leading mean-field profiles are accurate. For a bright coherent pump, the first signal–idler EPR-variance minimum scales as $V_{\min}\simeq1/(2\alpha)$ at $\chi_{\min}\simeq\tfrac12\log(4\alpha)$, rather than decreasing indefinitely as in the classical-pump model. Fourth-order correlations and bright displaced squeezed-pump inputs are also treated, including changes in third-order non-Gaussian correlations. These extensions are restricted to the first-depletion window, $0\leq\chi\leq2\log(4\alpha)$. The quoted numerical precision is benchmark-specific; losses, multimode interactions, and more general inputs are identified as future extensions.