Solution of Cross-Kerr Interaction Combined with Parametric Amplification

Sina Khorasani

doi:10.1038/s41598-018-38377-7

Abstract

We present a full operator approach to treatment of the cross-Kerr interaction combined with parametric amplification. It is shown that this problem can be exactly integrated using the method of higher-order operators. While the initial basis is infinite-dimensional, an orthogonal transformation can reduce the problem exactly into a six-dimensional basis which can be integrated conveniently.

Highlights

We demonstrate that the cross-Kerr interaction with parametric amplification could be exactly solvable using the method of higher-order operators[11,12,13,14,15,16], which has evolved out of the rich domain of quadratic optomechanics[17,18,19,20,21,22,23]
Ω and Ω respectively refer to the pump and probe frequencies with annihilators denoted by aand b, g represents the cross-Kerr nonlinear interaction rate, and f is the parametric amplification rate
Such type of cross-Kerr mixing can happen in a non-linear cavity where the strength of nonlinear interaction is proportional to the energies in each of the two fields

Summary

Introduction

We demonstrate that the cross-Kerr interaction with parametric amplification could be exactly solvable using the method of higher-order operators[11,12,13,14,15,16], which has evolved out of the rich domain of quadratic optomechanics[17,18,19,20,21,22,23]. Ω and Ω respectively refer to the pump and probe frequencies with annihilators denoted by aand b, g represents the cross-Kerr nonlinear interaction rate, and f is the parametric amplification rate. The contributing part of the multiplicative operators which operate on the white Gaussian noise processes ain and bin as shown in § S2 of Supplementary Information are the silent or noiseless parts of these operators, which can be found by solving the corresponding Langevin equations with all zero-mean stochastic processes dropped and only keeping the drive terms.

Results

Conclusion