Tunable Optomechanically Induced Absorption in a Hybrid Optomechanical System

Abstract

We study the tunable optomechanically induced absorption (OMIA) with the quantized field in the system, which consists of a driven cavity and a mechanical resonator with a super-conducting charge qubit via Jaynes-Cummings interaction. Such a OMIA can be achieved by controlling the strength of the Jaynes-Cummings interaction. Moreover, our work shows this OMIA for the quantized fields can be robust against cavity decay in somehow. With the combination of optomechanically induced transparency (OMIT), our proposal may have paved a new avenue towards quantum photon router.

Appendix A

A set of linear quantum Langevin equations for the fluctuation operators can be written as

$$\begin{array}{@{}rcl@{}} \mathbf{A}\mathbf{X}=\mathbf{B} \end{array} $$

(A1)

where

$$\begin{array}{@{}rcl@{}} \mathbf{A}=\left( \begin{array}{ccccccc} A_{1}+A_{2} & 0 & -igc_{s} & -igc_{s}^{*} & 0 & 0 & 0 \\ 0 & A_{1}-A_{2} & igc_{s} & 0 & 0 & 0 \\ -igc_{s} & -igc_{s}^{*} & A_{3}+i\omega_{b} & 0 & i\lambda & 0 & 0 \\ igc_{s}^{*} & igc_{s} & 0 & A_{3}-i\omega_{b} & 0 & -i\lambda & 0 \\ 0 & 0 & -i\lambda\sigma_{zs} & 0 & A_{4}+i\omega_{q} & 0 & -igb_{s} \\ 0 & 0 & 0 & i\lambda\sigma_{zs} & 0 & A_{4}-i\omega_{q} & i\lambda b_{s}^{*} \\ 0 & 0 & 2i\lambda(\sigma_{-s})^{*} & -2i\lambda\sigma_{-s} & -2i\lambda b_{s} & 2i\lambda b_{s}^{*} & A_{5} \end{array} \right) \end{array} $$

(A2)

and

$$\begin{array}{@{}rcl@{}} \mathbf{B}=(\sqrt{2\gamma_{c}}c_{in}(\omega),\sqrt{2\gamma_{c}}c_{in}^{\dagger}(-\omega), \sqrt{2\gamma_{b}}b_{in}(\omega),\sqrt{2\gamma_{b}}b_{in}^{\dagger}(-\omega),\sqrt{\gamma_{q}}{\Gamma}_{-}(\omega), \sqrt{\gamma_{q}}{\Gamma}_{+}(-\omega), \sqrt{\gamma_{q}}{\Gamma}_{z}(\omega))^{T}\\ \end{array} $$

(A3)

with A ₁ = γ _c −i ω, A ₂ = iΔ _c −2i g R e[b _s ], A ₃ = γ _b −i ω, $A_{4}=\frac {\gamma _{q}}{2}-i\omega$ and A ₅ = γ _q −i ω, where R e[Z] is define as the real part of Z.

$$\begin{array}{@{}rcl@{}} E(\omega)&=&\frac{\sqrt{2\gamma_{c}}[2ig^{2}|c_{s}|^{2}C(\omega)+(A_{1}-A_{2})D(\omega)]}{d(\omega)} \\ F(\omega)&=&\frac{2\sqrt{2\gamma_{c}}ig^{2}|a_{s}|^{2}C(\omega)}{d(\omega)} \\ G(\omega)&=&\sqrt{2\gamma_{b}}(A_{1}-A_{2})gc_{s}\{4A_{4}\lambda^{2}|b_{s}|^{2}(iA_{3}+\omega_{b}) +4\lambda^{3}b_{s}Re[\sigma_{-s}](A_{4}+i\omega_{q}) \\ &&+A_{5}(A_{4}+i\omega_{q})[i(A_{3}A_{4}+\lambda^{2}\sigma_{zs}) +\omega_{b}(A_{4}-i\omega_{q})+A_{3}\omega_{q}]\}/d(\omega) \\ H(\omega)&=&i\sqrt{2\gamma_{b}}(A_{1}-A_{2})gc_{s}^{*}\{4A_{4}\lambda^{2}|b_{s}|^{2}(A_{3}+i\omega_{b}) +4\lambda^{3}b_{s}Re[\sigma_{-s}](iA_{4}+\omega_{q}) \\ &&+A_{5}(A_{4}-i\omega_{q})[(A_{3}A_{4}-\lambda^{2}\sigma_{zs}) +i\omega_{b}(A_{4}+i\omega_{q})+iA_{3}\omega_{q}]\}/d(\omega) \\ I(\omega)&=&\{(A_{1}-A_{2})\lambda gc_{s}\sqrt{\gamma_{q}}[-2i\lambda^{2}Re[b_{s}](2\lambda Re[\sigma_{-s}] +2b_{s}\omega_{b})\\ &&+A_{5}(A_{3}A_{4}+\lambda^{2}\sigma_{zs}-iA_{4}\omega_{b}-iA_{3}\omega_{q}-\omega_{b}\omega_{q})]\}/d(\omega) \\ J(\omega)&=&\{-(A_{1}-A_{2})\lambda gc_{s}^{*}\sqrt{\gamma_{q}}[2i\lambda^{2}Re[b_{s}](2\lambda Re[\sigma_{-s}] +2b_{s}^{*}\omega_{b})\\ &&+A_{5}(A_{3}A_{4}+\lambda^{2}\sigma_{zs}+iA_{4}\omega_{b}+iA_{3}\omega_{q}-\omega_{b}\omega_{q})]\}/d(\omega) \\ K(\omega)&=&\{(A_{1}-A_{2})\lambda gc_{s}\sqrt{\gamma_{q}}[A_{5}(A_{3}A_{4}+\lambda^{2}\sigma_{zs}-iA_{4}\omega_{b}-iA_{3}\omega_{q}-\omega_{b}\omega_{q}) \\ &&+2\lambda b_{s}(-A_{3}A_{4}+2\lambda^{2}Re[\sigma_{-s}]+\omega_{b}\omega_{q}+2i\lambda b_{s}\omega_{b}-2i\lambda^{2}Re[b_{s}])]\}/d(\omega)\\ \end{array} $$

(A4)

with

$$\begin{array}{@{}rcl@{}} d(\omega)&=&4iA_{2}g^{2}|c_{s}|^{2}C(\omega)+({A_{1}^{2}}-{A_{2}^{2}})D(\omega) \end{array} $$

(A5)

where $C(\omega )=iA_{4}A_{5}\lambda ^{2}\sigma _{zs}+4A_{4}\lambda ^{3}Re[b_{s}Re[\sigma _{-s}] +4A_{4}\lambda ^{2}|b_{s}|^{2}\omega _{b}+A_{5}\omega _{b}({A_{4}^{2}}+{\omega _{q}^{2}})$ and $D(\omega )=\{4A_{4}\lambda ^{2}|b_{s}|^{2}({A_{3}^{2}}+{\omega _{b}^{2}})+A_{5}(A_{3}-i\omega _{b})(A_{4}-i\omega _{q})[A_{3}A_{4}-\lambda ^{2}|\sigma _{zs}|^{2} +i\omega _{b}(A_{4}+i\omega _{q})+iA_{3}\omega _{q}] +2\lambda ^{3}Re[b_{s}][2A_{3}A_{4}Re[\sigma _{-s}] +2A_{3}\omega _{q}Re[\sigma _{-s}]+2\omega _{b}(A_{4}Re[\sigma _{-s}]-\omega _{q}Re[\sigma _{-s}])]\}$. It is clear that the real and imaginary parts, with R e[E(ω)] and I m[E(ω)], describe the absorption and dispersion of the optomechanical system, respectively.