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Temporal Dynamics and Nonclassical Photon Statistics of Quadratically Coupled Optomechanical Systems

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Abstract

Quantum optomechanical system serves as an interface for coupling between photons and phonons due to mechanical oscillations. We used the Heisenberg-Langevin approach under Markovian white noise approximation to study a quadratically coupled optomechanical system which contains a thin dielectric membrane quadratically coupled to the cavity field. A decorrelation method is employed to solve for a larger number of coupled equations. Transient mean numbers of cavity photons and phonons that provide dynamical behaviour are computed for different coupling regime. We have also obtained the two-boson second-order correlation functions for the cavity field, membrane oscillator and their cross correlations that provide nonclassical properties governed by quadratic optomechanical system.

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Authors and Affiliations

  1. Institute of Nuclear Sciences, Hacettepe University, 06800, Ankara, Turkey

    Shailendra Kumar Singh

  2. Center of Theoretical and Computational Physics, Department of Physics, University of Malaya, 50603, Kuala Lumpur, Malaysia

    S. V. Muniandy

Authors
  1. Shailendra Kumar Singh
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Correspondence to Shailendra Kumar Singh.

Appendix A: Decorrelated and Closed set of Coupled Equations

Appendix A: Decorrelated and Closed set of Coupled Equations

$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}\right\rangle &=&-i{\Delta}_{c}\left\langle \hat{a}\right\rangle -i{\Omega} -\frac{1}{2}{\Gamma}_{a}\left\langle \hat{a} \right\rangle -ig_{opt}\left[ \left\{ \left\langle \hat{a}\right\rangle \left\langle \hat{b}^{\dagger 2}\right\rangle +2\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{\dagger} \right\rangle \right\} \right. \\ &&\left. +\left\{ \left\langle \hat{a}\right\rangle \left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{a}\hat{b}\right\rangle \left\langle \hat{b}\right\rangle \right\} +2\left\{ \left\langle \hat{a}\right\rangle \left\langle \hat{b}^{\dagger} \hat{b}\right\rangle +\left\langle \hat{a} \hat{b}^{\dagger} \right\rangle \left\langle \hat{b}\right\rangle +\left\langle \hat{a}\hat{b}\right\rangle \left\langle \hat{b}^{\dagger} \right\rangle \right\} +\left\langle \hat{a}\right\rangle \right] . \end{array} $$
(36)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}^{\dagger} \right\rangle &=&i{\Delta}_{c}\left\langle \hat{a}^{\dagger} \right\rangle +i{\Omega} -\frac{1}{2}{\Gamma}_{a}\left\langle \hat{a}^{\dagger} \right\rangle +ig_{opt}\left[ \left\{ \left\langle \hat{a}^{\dagger} \right\rangle \left\langle \hat{b}^{\dagger 2}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{\dagger} \right\rangle \right\} \right. \\ &&\left. +\left\{ \left\langle \hat{a}^{\dagger} \right\rangle \left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b} \right\rangle \left\langle \hat{b}\right\rangle \right\} +2\left\{ \left\langle \hat{a}^{\dagger} \right\rangle \left\langle \hat{b}^{\dagger} \hat{b}\right\rangle +\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{b}\right\rangle +\left\langle \hat{a}^{\dagger }\hat{b}\right\rangle \left\langle \hat{b}^{\dagger} \right\rangle \right\} +\left\langle \hat{a}^{\dagger }\right\rangle \right] . \end{array} $$
(37)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{b}\right\rangle &=&-i\omega_{M}\left\langle \hat{b}\right\rangle -\frac{1}{2}{\Gamma}_{b}\left\langle \hat{b} \right\rangle -2ig_{opt}\left[ \left\langle \hat{a}^{\dagger} \right\rangle \left\{ \left\langle \hat{a}\hat{b}\right\rangle +\left\langle \hat{a}\hat{b} ^{\dagger} \right\rangle \right\} \right. \\ &&\left. +\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\{ \left\langle \hat{b}\right\rangle +\left\langle \hat{b}^{\dagger} \right\rangle \right\} +\left\langle \hat{a}\right\rangle \left\{ \left\langle \hat{a}^{\dagger} \hat{b}\right\rangle +\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \right\} \right] . \end{array} $$
(38)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{b}^{\dagger} \right\rangle &=&i\omega_{M}\left\langle \hat{b}^{\dagger} \right\rangle -\frac{1}{2}{\Gamma}_{b}\left\langle \hat{b}^{\dagger} \right\rangle +2ig_{opt}\left[ \left\langle \hat{a}^{\dagger} \right\rangle \left\{ \left\langle \hat{a} \hat{b}\right\rangle +\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \right\} \right. \\ &&\left. +\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\{ \left\langle \hat{b}\right\rangle +\left\langle \hat{b}^{\dagger} \right\rangle \right\} +\left\langle \hat{a}\right\rangle \left\{ \left\langle \hat{a}^{\dagger} \hat{b}\right\rangle +\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \right\} \right] . \end{array} $$
(39)
$$ \frac{d}{dt}\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle =-i{\Omega} \left(\left\langle \hat{a}^{\dagger} \right\rangle -\left\langle \hat{a} \right\rangle \right) -{\Gamma}_{a}\left\langle \hat{a}^{\dagger} \hat{a} \right\rangle +{\Gamma}_{a}\bar{n}_{th}^{a}. $$
(40)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{b}^{\dagger} \hat{b}\right\rangle &=&-2ig_{opt} \left[ \left\{ \left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\langle \hat{b}^{\dagger 2}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \right\} -\left\{ \left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left\langle \hat{a}\hat{b} \right\rangle \right\} \right] \\ &&-{\Gamma}_{b}\left\langle \hat{b}^{\dagger} \hat{b}\right\rangle +{\Gamma}_{b}\bar{n}_{th}^{b}. \end{array} $$
(41)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle &=&i\left(\omega_{M}-{\Delta}_{c}\right) \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle -i{\Omega} \left\langle \hat{b}^{\dagger} \right\rangle -\frac{1} {2}\left({\Gamma}_{a}+{\Gamma}_{b}\right) \left\langle \hat{a}\hat{b} ^{\dagger} \right\rangle \\ &&+ig_{opt}\left[ 2\left\{ 2\left\langle \hat{a}^{\dagger} \hat{a} \right\rangle \left\langle \hat{a}\hat{b}\right\rangle +\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left\langle \hat{a}^{2}\right\rangle \right\} -\left\{ \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{a}\hat{b} \right\rangle \left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right\} \right. \\ &&+2\left\{ \left(2\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle +\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{a}^{2}\right\rangle \right) -\left(2\left\langle \hat{b}^{\dagger} \hat{b} \right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle +\left\langle \hat{a}\hat{b}\right\rangle \left\langle \hat{b}^{\dagger 2}\right\rangle \right) \right\} \\ &&\left. -\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \left(3\left\langle \hat{b}^{\dagger 2}\right\rangle +1\right) \right] . \end{array} $$
(42)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle &=&i\left({\Delta}_{c}-\omega_{M}\right) \left\langle \hat{a}^{\dagger} \hat{b} \right\rangle +i{\Omega} \left\langle \hat{b}\right\rangle -\frac{1}{2}\left({\Gamma}_{a}+{\Gamma}_{b}\right) \left\langle \hat{a}^{\dagger} \hat{b} \right\rangle \\ &&+ig_{opt}\left[ \left\{ \left\langle \hat{a}^{\dagger} \hat{b} \right\rangle \left\langle \hat{b}^{\dagger 2}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right\} -2\left\{ \left\langle \hat{a}^{\dagger 2}\right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle +2\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \right\} \right. \\ &&+2\left\{ \left(\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left\langle \hat{b}^{\dagger} \hat{b} \right\rangle \right) -\left(\left\langle \hat{a}^{\dagger 2}\right\rangle \left\langle \hat{a}\hat{b}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \right) \right\} \\ &&\left. +\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left(3\left\langle \hat{b}^{2}\right\rangle +1\right) \right] . \end{array} $$
(43)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}\hat{b}\right\rangle &=&-i\left({\Delta}_{c}+\omega_{M}\right) \left\langle \hat{a}\hat{b}\right\rangle -i{\Omega} \left\langle \hat{b}\right\rangle -\frac{1}{2}\left({\Gamma}_{a}+{\Gamma}_{b}\right) \left\langle \hat{a}\hat{b}\right\rangle \\ &&-ig_{opt}\left[ 2\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle +3\left\langle \hat{a}\hat{b}\right\rangle \left(\left\langle \hat{b}^{2}\right\rangle +1\right) \right] -ig_{opt}\left[ \left\langle \hat{a}\hat{ b}\right\rangle \left\langle \hat{b}^{\dagger 2}\right\rangle +2\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{\dagger} \hat{b} \right\rangle \right] \\ &&-2ig_{opt}\left[ \left(2\left\langle \hat{a}^{\dagger} \hat{a} \right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle +\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{a}^{2}\right\rangle \right) +\left(\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{a}\hat{b}\right\rangle \left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right) \right. \\ &&\left. +\left(2\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\langle \hat{a}\hat{b}\right\rangle +\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left\langle \hat{a}^{2}\right\rangle \right) \right] . \end{array} $$
(44)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle &=&i\left({\Delta}_{c}+\omega_{M}\right) \left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle +i{\Omega} \left\langle \hat{b}^{\dagger} \right\rangle -\frac{1}{2}\left({\Gamma}_{a}+{\Gamma}_{b}\right) \left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \\ &&+ig_{opt}\left[ 2\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle +3\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left(\left\langle \hat{b}^{\dagger 2}\right\rangle +1\right) \right] +ig_{opt} \left[ \left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b} \right\rangle \left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right] \\ &&+2ig_{opt}\left[ \left(\left\langle \hat{a}^{\dagger 2}\right\rangle \left\langle \hat{a}\hat{b}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \right) +\left(\left\langle \hat{b}^{\dagger 2}\right\rangle \left\langle \hat{a}^{\dagger} \hat{b}\right\rangle +2\left\langle \hat{a}^{\dagger} \hat{ b}^{\dagger} \right\rangle \left\langle \hat{b}^{\dagger} \hat{b} \right\rangle \right) \right. \\ &&\left. +\left(2\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle +\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \left\langle \hat{a}^{\dagger 2}\right\rangle \right) \right] . \end{array} $$
(45)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}^{2}\right\rangle &=&-2i{\Delta}_{c}\left\langle \hat{a}^{2}\right\rangle -2i{\Omega} \left\langle \hat{a} \right\rangle -{\Gamma}_{a}\left\langle \hat{a}^{2}\right\rangle -2ig_{opt}\left\langle \hat{a}^{2}\right\rangle \\ &&-2ig_{opt}\left[ \left\langle \hat{a}^{2}\right\rangle \left(\left\langle \hat{b}^{\dagger 2}\right\rangle +\left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right) +2\left(\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle^{2}+\left\langle \hat{a} \hat{b}\right\rangle^{2}+2\left\langle \hat{a}\hat{b}\right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \right) \right] . \end{array} $$
(46)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{a}^{\dagger 2}\right\rangle &=&2i{\Delta}_{c}\left\langle \hat{a}^{\dagger 2}\right\rangle +2i{\Omega} \left\langle \hat{a}^{\dagger} \right\rangle -{\Gamma}_{a}\left\langle \hat{a}^{\dagger 2}\right\rangle +2ig_{opt}\left\langle \hat{a}^{\dagger 2}\right\rangle \\ &&+2ig_{opt}\left[ \left\langle \hat{a}^{\dagger 2}\right\rangle \left(\left\langle \hat{b}^{\dagger 2}\right\rangle +\left\langle \hat{b}^{2}\right\rangle +2\left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right) +2\left(\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle^{2}+\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle ^{2}+2\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle \left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \right) \right] . \end{array} $$
(47)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{b}^{2}\right\rangle &=&-2i\omega_{M}\left\langle \hat{b}^{2}\right\rangle -{\Gamma}_{b}\left\langle \hat{b}^{2}\right\rangle -2ig_{opt}\left\langle \hat{a}^{\dagger} \hat{a} \right\rangle \\ &&-4ig_{opt}\left[ \left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left(\left\langle \hat{b}^{2}\right\rangle +\left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right) +\left\langle \hat{a}\hat{b}\right\rangle \left(\left\langle \hat{a}^{\dagger} \hat{b}^{\dagger} \right\rangle +2\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \right) +\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \right] . \end{array} $$
(48)
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}\left\langle \hat{b}^{\dagger 2}\right\rangle &=&2i\omega_{M}\left\langle \hat{b}^{\dagger 2}\right\rangle -{\Gamma}_{b}\left\langle \hat{b}^{\dagger 2}\right\rangle +2ig_{opt}\left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \\ &&+4ig_{opt}\left[ \left\langle \hat{a}^{\dagger} \hat{a}\right\rangle \left(\left\langle \hat{b}^{\dagger 2}\right\rangle +\left\langle \hat{b}^{\dagger} \hat{b}\right\rangle \right) +\left\langle \hat{a}^{\dagger} \hat{ b}^{\dagger} \right\rangle \left(\left\langle \hat{a}\hat{b}\right\rangle +2\left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \right) +\left\langle \hat{a}^{\dagger} \hat{b}\right\rangle \left\langle \hat{a}\hat{b}^{\dagger} \right\rangle \right] . \end{array} $$
(49)

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Singh, S.K., Muniandy, S.V. Temporal Dynamics and Nonclassical Photon Statistics of Quadratically Coupled Optomechanical Systems. Int J Theor Phys 55, 287–301 (2016). https://doi.org/10.1007/s10773-015-2661-8

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