Time-evolution of nonlinear optomechanical systems: interplay of mechanical squeezing and non-Gaussianity

Sofia Qvarfort,Alessio Serafini,Dennis Rätzel,Daniel Braun,David Edward Bruschi,André Xuereb

doi:10.1088/1751-8121/ab64d5

Abstract

We solve the time evolution of a nonlinear optomechanical Hamiltonian with arbitrary time-dependent mechanical displacement, mechanical single-mode squeezing and a time-dependent optomechanical coupling up to the solution of two second-order differential equations. The solution is based on identifying a minimal and finite Lie algebra that generates the time-evolution of the system. This reduces the problem to considering a finite set of coupled ordinary differential equations of real functions. To demonstrate the applicability of our method, we compute the degree of non-Gaussianity of the time-evolved state of the system by means of a measure based on the relative entropy of the non-Gaussian state and its closest Gaussian reference state. We find that the addition of a constant mechanical squeezing term to the standard optomechanical Hamiltonian generally decreases the overall non-Gaussian character of the state. For sinusoidally modulated squeezing, the two second-order differential equations mentioned above take the form of the Mathieu equation. We derive perturbative solutions for a small squeezing amplitude at parametric resonance and show that they correspond to the rotating-wave approximation at times larger than the scale set by the mechanical frequency. We find that the non-Gaussianity of the state increases with both time and the squeezing parameter in this specific regime.

Highlights

The mathematical understanding of optomechanical systems operating in the nonlinear quantum regime is a major topic of current interest
We solved the time-evolution of a nonlinear optomechanical system with a timedependent mechanical displacement term and a time-dependent mechanical single-mode squeezing term
We found analytic expressions for all first and second moments of the quadratures of the nonlinear system and used them to compute the amount of non-Gaussianity of the state

Summary

Introduction

The mathematical understanding of optomechanical systems operating in the nonlinear quantum regime is a major topic of current interest. While most experiments effectively undergo linear dynamics, governed by quadratic Hamiltonians that emerge following a ‘linearisation’ procedure [1,2,3], many experiments operate in the fully nonlinear regime [4,5,6] where this procedure fails. It is highly desirable to provide a complete and analytic characterisation of the fully nonlinear system dynamics. The inherently nonlinear interaction between the optical field and the mechanical element in an optomechanical system allows for the generation of non-Gaussian states. As such, investigating the non-Gaussianity of optomechanical states can only be performed once the time-evolution in the nonlinear regime has been solved, which is the primary aim of this work.

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Conclusion