Tunable optical response in a hybrid quadratic optomechanical system coupled with single semiconductor quantum well

Abstract

Quantum optomechanical system serves as an interface for coupling between photons and phonons via radiation pressure. We theoretically investigate the optical response of a hybrid optomechanical system that contains a single undoped semiconductor quantum well inside a cavity as well as a thin dielectric movable membrane in the middle, quadratically coupled to the cavity photons. We find that in the presence of both quadratic optomechanical coupling and exciton–cavity field coupling, two additional absorption dips appear in the output field spectrum of the probe field as compared to a standard quadratic optomechanical system which gives only two-phonon optomechanical induced transparency and optomechanical induced absorption phenomena with probe field detuning. This is due to the formation of the dressed state mediated by the single-photon state and the exciton mode. Furthermore, we have shown that the optical transmission of the probe field at these two absorption dips can be controlled by a number of parameters present in the system like exciton–cavity field coupling strength, decay rate of exciton as well as the mean number of thermal phonons- in the environment. We also explore the possibility of slow light in this absorption regime due to exciton–photon coupling. Our study shall provide a method to control the propagation of light in quadratic hybrid optomechanical system containing semiconductor nanostructures.

Appendices

A Simplification for Equation (6):

$$\begin{aligned} i\varOmega c_{+}= & {} -\left( \kappa _{c}/2 +i\varDelta ^{\prime }_{1}\right) c_{+} - ig_{om}c_{s}Q_{+}-iga_{+}, \end{aligned}$$

(A-1)

$$\begin{aligned} i\varOmega c_{-}= & {} \left( \kappa _{c}/2 +i\varDelta ^{\prime }_{1}\right) c_{-} + ig_{om}c_{s}Q_{-} + iga_{-}-\varepsilon _{P}, \end{aligned}$$

(A-2)

$$\begin{aligned} i\varOmega a_{+}= & {} -\left( \kappa _{a}/2 +i\varDelta _{a}\right) a_{+}- igc_{+}, \end{aligned}$$

(A-3)

$$\begin{aligned} a_{+}= & {} \dfrac{-igc_{+}}{\left[ \left( \kappa _{a}/2 +i\varDelta _{a}\right) +i\varOmega \right] }, \end{aligned}$$

(A-4)

$$\begin{aligned} i\varOmega a_{-}= & {} \left( \kappa _{a}/2 +i\varDelta _{a}\right) a_{-} + igc_{-}, \end{aligned}$$

(A-5)

$$\begin{aligned} a_{-}= & {} \dfrac{igc_{-}}{\left[ i\varOmega -\left( \kappa _{a}/2 +i\varDelta _{a}\right) \right] }, \end{aligned}$$

(A-6)

$$\begin{aligned} i\varOmega P_{+}= & {} -\left( \omega _{m} + 2 g_{om}\vert c_{s}\vert ^{2}\right) X_{+} - 2\gamma _{m}P_{+}, \end{aligned}$$

(A-7)

$$\begin{aligned} i\varOmega P_{-}= & {} \left( \omega _{m} + 2 g_{om}\vert c_{s}\vert ^{2}\right) X_{-} + 2\gamma _{m}P_{-}, \end{aligned}$$

(A-8)

$$\begin{aligned} i\varOmega Q_{+}= & {} \omega _{m}X_{+}, \end{aligned}$$

(A-9)

$$\begin{aligned} i\varOmega Q_{-}= & {} -\omega _{m}X_{-}, \end{aligned}$$

(A-10)

$$\begin{aligned} i\varOmega X_{+}= & {} 2\omega _{m}P_{+}-2\left( \omega _{m} + 2g_{om}\vert c_{s}\vert ^{2}\right) Q_{+}-4 g_{om}Q_{s}c_{+}\left( c^{\star }_{s} + c_{s}\right) -\gamma _{m}X_{+}, \end{aligned}$$

(A-11)

$$\begin{aligned} i\varOmega X_{-}= & {} -2\omega _{m}P_{-} + 2\left( \omega _{m} + 2g_{om}\vert c_{s}\vert ^{2}\right) Q_{-} + 4g_{om}Q_{s}c_{-}\left( c^{\star }_{s} + c_{s}\right) + \gamma _{m}X_{-}.\nonumber \\ \end{aligned}$$

(A-12)

B Hamiltonian in terms of dressed state operators:

Our system Hamiltonian given in Eq. (1) can be also expressed in terms of dressed states operators as follows:

$$\begin{aligned} {\hat{H}}= & {} \hbar \varDelta _{+}{\hat{d}}_{+}^{\dagger }{\hat{d}}_{+} + \hbar \varDelta _{-}{\hat{d}}_{-}^{\dagger }{\hat{d}}_{-} + \dfrac{\hbar \omega _m}{2}({\hat{p}}^2+{\hat{q}}^2)+\hbar \dfrac{g_{om}}{2}({\hat{d}}_{+}^{\dagger }{\hat{d}}_{+} + {\hat{d}}_{-}^{\dagger }{\hat{d}}_{-}){\hat{q}}^2 \nonumber \\&+ \hbar \dfrac{g_{om}}{2}({\hat{d}}_{+}^{\dagger }{\hat{d}}_{-}+ {\hat{d}}_{+}{\hat{d}}^{\dagger }_{-}){\hat{q}}^2 + i\hbar \dfrac{\varepsilon _{L}}{\sqrt{2}}\left( {\hat{d}}_{+}^{\dagger }-{\hat{d}}_{+}\right) + i\hbar \dfrac{\varepsilon _{L}}{\sqrt{2}}\left( {\hat{d}}_{-}^{\dagger }-{\hat{d}}_{-}\right) \nonumber \\&+ i\hbar \dfrac{\varepsilon _{p}}{\sqrt{2}}\left( {\hat{d}}_{+}^{\dagger }e^{-i\varOmega t}-{\hat{d}}_{+}e^{i\varOmega t}\right) + i\hbar \dfrac{\varepsilon _{p}}{\sqrt{2}}\left( {\hat{d}}_{-}^{\dagger }e^{-i\varOmega t}-{\hat{d}}_{-}e^{i\varOmega t}\right) , \end{aligned}$$

(B-1)

where $\varDelta _{\pm } = \varDelta _{c} \pm g = \varDelta _{a} \pm g$. The first two terms in the first line of Eq. (B1) represent the free evolution Hamiltonian of the two dressed states formed by the single-photon state and the exciton mode, i.e., ${\hat{d}}_{\pm } = {\hat{c}} \pm {\hat{a}}$ and the third term is the free evolution Hamiltonian for the coupled optomechanical membrane. These two dressed states interact quadratically with the membrane through the fourth terms given in first line which is responsible for the OMIT (OMIA) phenomena. This coupled membrane can also scatter photons in between these two dressed states ${\hat{d}}_{+}$ and ${\hat{d}}_{-}$ which is given by last or fifth term in the first line of Eq. (B1). All the terms in second line of Eq.(B1) represent driven terms for these two dressed states ${\hat{d}}_{+}$ and ${\hat{d}}_{-}$, with driving amplitudes $\dfrac{\varepsilon _{L}}{\sqrt{2}}$ and $\dfrac{\varepsilon _{p}}{\sqrt{2}}$, respectively.