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Journal of Physics A Mathematical and Theoretical

Four-wave mixing in all degrees of freedom

Filippus Stefanus Roux

Published Sep 25, 2023• 0 citations
Publisher

Abstract

Abstract A Wigner functional approach is used to derive an evolution equation for a photonic state propagating through a Kerr medium. The resulting evolution equation incorporates all the spatiotemporal degrees of freedom together with the photon-number degrees of freedom and thus allows thorough analyses of the effects of experimental parameters in physical quantum information systems. We then use the evolution equation to consider four-wave mixing as a spontaneous process and finally we impose some approximations to obtain an expression for the optical field due to self-phase modulation.

DOI
10.1088/1751-8121/acfcf5

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Abstract A Wigner functional approach is used to derive an evolution equation for a photonic state propagating through a Kerr medium. The resulting evolution equation incorporates all the spatiotemporal degrees of freedom together with the photon-number degrees of freedom and thus allows thorough analyses of the effects of experimental parameters in physical quantum information systems. We then use the evolution equation to consider four-wave mixing as a spontaneous process and finally we impose some approximations to obtain an expression for the optical field due to self-phase modulation.
The aim of this study is to analyze the quantum correlation of entangled photons for four-wavemixing process within Kerr nonlinear susceptibility χ(3) of an optical fiber ring resonator. The main system Hamiltonian composes of two types of coupling photon modes, including pumping and parametric-down conversion. Wigner representation is applied to the Heisenberg equation of motion derived from the system Hamiltonian in order to obtain the corresponding stochastic equations of motion in a c-number. Quantum trajectories obtained from the stochastic equation of motion of photon states are then derived. Finally, the entanglement inseparability criteria for a pair of entangled photonfrom a numerical approach is also satisfied.
Evolution equations for the moments of a photonic quantum state propagating through atmospheric turbulence are derived. These evolution equations are obtain from an evolution equation for the characteristic functional of the state, incorporating all spatiotemporal degrees of freedom. The measured quantities, such as the intensity or photon distribution, of the evolving state can be expressed in terms of such moments without having to know the exact final state. The case of an initial coherent state is considered as an example.
As four-wave-mixing-based photon-pair sources mature, accurate modeling of the photon-pair properties becomes important. Unlike spontaneous parametric down-conversion, four-wave mixing is accompanied by a number of parasitic effects such as nonlinear self- and cross-phase modulation. Currently, most modelings of photon-pair states are analytic in nature, which limits the number and type of effects that can be taken into account. In this work, we derive a complete, dual-pump evolution equation for the joint amplitude of photon pairs, wherein any desired effects can be included. We describe how to efficiently obtain numerical solution to this equation using a split-step approach. Lastly, we cover a few analytical solutions and compare two schemes for pure-photon generation under three different parasitic effects. We show how one scheme is highly sensitive to parasitic effects, while the other is very robust.
Using the quadrature bases that incorporate the spatiotemporal degrees of freedom, we develop a Wigner functional theory for quantum optics, as an extension of the Moyal formalism. Since the spatiotemporal quadrature bases span the complete Hilbert space of all quantum optical states, it does not require factorization as a tensor product of discrete Hilbert spaces. The Wigner functions associated with such a space become functionals and operations are expressed by functional integrals -- the functional version of the star product. The resulting formalism enables tractable calculations for scenarios where both spatiotemporal degrees of freedom and particle-number degrees of freedom are relevant. To demonstrate the approach, we compute examples of Wigner functionals for a few well-known states and operators.
Using a perturbative approach, we investigate the parametric down-conversion process without the semi-classical approximation. A Wigner functional formalism, which incorporates both the spatiotemproal degrees of freedom and the particle-number degrees of freedom is used to perform the analysis. First, we derive an evolution equation for the down-conversion process in the nonlinear medium. Then we use the perturbative approach to solve the equation. The leading order contribution is equivalent to the semi-classical solution. The next-to-leading order contribution provides a solution that includes the evolution of the pump field as a quantum field.