Absorption, Transmission and Amplification in a Double-Cavity Optomechanical System with Coulomb-Interaction

Abstract

We explore three interesting phenomena in a double-cavity optomechanical system: coherent perfect absorption, coherent perfect transmission and output signal amplification, and find that these phenomena can be realized and controlled by the coulomb-interaction between the dissipative oscillator locates in the cavity and the gain oscillator locates outside. They originate from the efficient hybrid coupling of optical and mechanical modes, and can be used for realizing novel photonic devices in quantum information networks.

Appendices

Appendix A: Instructions on Simplifying H _c

Equation (3) can be rewritten as follow,

$$\begin{array}{@{}rcl@{}} {H_{c}} &=& \frac{{ - {C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi {\varepsilon_{0}}{r_{0}}}}\left[ {1 - \frac{{{q_{1}} - {q_{2}}}}{{{r_{0}}}} + {{\left( {\frac{{{q_{1}} - {q_{2}}}}{{{r_{0}}}}} \right)}^{2}}} \right] \\ &=& \frac{{ - {C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi {\varepsilon_{0}}{r_{0}}}} + \frac{{{C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi {\varepsilon_{0}}{r_{0}}^{2}}}\left( {{q_{1}} \!- {q_{2}}} \right) - \frac{{{C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi {\varepsilon_{0}}{r_{0}}^{3}}}\left( {{q_{1}^{2}} + {q_{2}^{2}}} \right) \!+ \frac{{{C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{2\pi {\varepsilon_{0}}{r_{0}}^{3}}}{q_{1}}{q_{2}} \\ &=& \frac{{ - {C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi {\varepsilon_{0}}{r_{0}}}} + \hbar \xi \left( {{q_{1}} - {q_{2}}} \right) - \hbar {\Theta} \left( {{q_{1}^{2}} + {q_{2}^{2}}} \right) + \hbar \lambda {q_{1}}{q_{2}}, \end{array} $$

(22)

where $\xi = \frac {{{C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi \hbar {\varepsilon _{0}}{r_{0}}^{2}}}$, ${\Theta } = \frac {{{C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi \hbar {\varepsilon _{0}}{r_{0}}^{3}}}$, $\lambda = \frac {{{C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{2\pi \hbar {\varepsilon _{0}}{r_{0}}^{3}}}$. The constant term $\frac {{ - {C_{1}}{U_{1}}{C_{2}}{U_{2}}}}{{4\pi {\varepsilon _{0}}{r_{0}}}}$ can be omitted. The quadratic term includes a renormalization of the oscillation frequencies for both NR₁ and NR₂. There are quadratic terms ${\frac {1}{2}{m_{1}}\omega _{m1}^{2}{q_{1}^{2}}}$ and ${\frac {1}{2}{m_{2}}\omega _{m2}^{2}{q_{2}^{2}}}$ in (5), then the calculation process is,

$$\begin{array}{@{}rcl@{}} \frac{1}{2}{m_{1}}\omega_{m1}^{2}{q_{1}^{2}} - \hbar {\Theta} {q_{1}^{2}} &=& \frac{1}{2}{m_{1}}\left( {\omega_{m1}^{2} - \frac{{2\hbar {\Theta} }}{{{m_{1}}}}} \right){q_{1}^{2}} \equiv \frac{1}{2}{m_{1}}{\omega_{1}^{2}}{q_{1}^{2}}, \\ \frac{1}{2}{m_{2}}\omega_{m2}^{2}{q_{2}^{2}} - \hbar {\Theta} {q_{2}^{2}} &=& \frac{1}{2}{m_{2}}\left( {\omega_{m2}^{2} - \frac{{2\hbar {\Theta} }}{{{m_{2}}}}} \right){q_{2}^{2}} \equiv \frac{1}{2}{m_{2}}{\omega_{2}^{2}}{q_{2}^{2}}, \end{array} $$

(23)

where ${\omega _{1}^{2}} = \omega _{m1}^{2} - \frac {{2\hbar {\Theta } }}{{{m_{1}}}},{\omega _{2}^{2}} = \omega _{m2}^{2} - \frac {{2\hbar {\Theta } }}{{{m_{2}}}}$. Thus, H _c can be reduced as follow,

$$\begin{array}{@{}rcl@{}} {H_{c}} = \hbar \xi \left( {{q_{1}} - {q_{2}}} \right) + \hbar \lambda {q_{1}}{q_{2}}, \end{array} $$

(24)

where the linear term may be absorbed into the definition of the equilibrium positions (This can be seen from the (9)). For the sake of simplicity, the coordinates and momentum operators of the harmonic oscillators are written in the form of the phonon creation operator (b^‡) and the annihilation operator (b), i.e. ${q_{\alpha } } = \sqrt {\frac {\hbar }{{2{m_{\alpha } }{\omega _{\alpha } }}}} \left ({{b_{\alpha } } + b_{\alpha }^{\dag } } \right )$, ${p_{\alpha } } = i\sqrt {\frac {{\hbar {m_{\alpha } }{\omega _{\alpha } }}}{2}} \left ({b_{\alpha }^{\dag } - {b_{\alpha } }} \right )$, with α = 1, 2.

So H _c can be transformed finally as follow,

$$\begin{array}{@{}rcl@{}} {H_{c}} &=& \hbar \xi \sqrt {\frac{\hbar }{{2{m_{1}}{\omega_{1}}}}} \left( {{b_{1}} + b_{1}^{\dag} } \right) - \hbar \xi \sqrt {\frac{\hbar }{{2{m_{2}}{\omega_{2}}}}} \left( {{b_{2}} + b_{2}^{\dag} } \right) \\ &+& \hbar \lambda \sqrt {\frac{\hbar }{{2{m_{1}}{\omega_{1}}}}} \sqrt {\frac{\hbar }{{2{m_{2}}{\omega_{2}}}}} \left( {{b_{1}} + b_{1}^{\dag} } \right)\left( {{b_{2}} + b_{2}^{\dag} } \right) \\ &=& \hbar {\xi_{1}}\left( {{b_{1}} + b_{1}^{\dag} } \right) - \hbar {\xi_{2}}\left( {{b_{2}} + b_{2}^{\dag} } \right) + \hbar \eta \left( {{b_{1}} + b_{1}^{\dag} } \right)\left( {{b_{2}} + b_{2}^{\dag} } \right), \end{array} $$

(25)

where ${\xi _{1}} = \xi \sqrt {\frac {\hbar }{{2{m_{1}}{\omega _{1}}}}} ,{\xi _{2}} = \xi \sqrt {\frac {\hbar }{{2{m_{2}}{\omega _{2}}}}} ,\eta = \lambda \sqrt {\frac {\hbar }{{2{m_{1}}{\omega _{1}}}}} \sqrt {\frac {\hbar }{{2{m_{2}}{\omega _{2}}}}}$.

Appendix B: Instruction for Three Situations of Coherent Perfect Transmission

In the discussion of coherent perfect transmission, there are three cases of the solutions of (20). In order to better illustrate this, we write (20) as follow,

$$\begin{array}{@{}rcl@{}} n &=& 1, \\ {\gamma_{1}} &\to& 0, \\ {\gamma_{2}} &\to& 0, \\ {x_ \pm } &=& \pm \sqrt {{y_{1, 2}}}, \\ {y_{1, 2}} &=& \frac{{ - {\kappa^{2}} + {\eta^{2}} + 2{G^{2}} \pm \sqrt {{{\left( { - {\kappa^{2}} + {\eta^{2}} + 2{G^{2}}} \right)}^{2}} + 4{\kappa^{2}}{\eta^{2}}} }}{2}. \end{array} $$

(26)

The value of y_{1, 2} must be greater than or equal to zero, because ${x_ \pm } = \pm \sqrt {{y_{1, 2}}}$. When − κ² + η² + 2G² < 0, we must take + in the expression of y_{1, 2} to ensure that y_{1, 2} is greater than zero. Thus, there are two solutions in this case as follow,

$$\begin{array}{@{}rcl@{}} {x_ \pm } &=& \pm \sqrt {{y_{1, 2}}} , \\ {y_{1, 2}} &=& \frac{{ - {\kappa^{2}} + {\eta^{2}} + 2{G^{2}} + \sqrt {{{\left( { - {\kappa^{2}} + {\eta^{2}} + 2{G^{2}}} \right)}^{2}} + 4{\kappa^{2}}{\eta^{2}}} }}{2}. \end{array} $$

(27)

When − κ² + η² + 2G² = 0, we can take ± in the expression of y_{1, 2}. Then there are two solutions in this case as follow,

$$\begin{array}{@{}rcl@{}} {x_ \pm } &=& \pm \sqrt{{y_{1, 2}}} , \\ {y_{1, 2}} &=& \kappa \eta. \end{array} $$

(28)

When − κ² + η² + 2G² > 0, we can take + in the expression of y_{1, 2} to ensure that y_{1, 2} is greater than zero. Thus, there are two solutions in this case as follow,

$$\begin{array}{@{}rcl@{}} {x_ \pm } &=& \pm \sqrt{{y_{1, 2}}} , \\ {y_{1, 2}} &=& \frac{{ - {\kappa^{2}} + {\eta^{2}} + 2{G^{2}} + \sqrt {{{\left( { - {\kappa^{2}} + {\eta^{2}} + 2{G^{2}}} \right)}^{2}} + 4{\kappa^{2}}{\eta^{2}}} }}{2}. \end{array} $$

(29)

According to the above analysis, we obtain three situations of the solutions of (20) solution and draw Figs. 5, 7 and 8.

Fig. 7

(Color online) The second situation (− κ² + η² + 2G² = 0): a Normalized output probe field energy ${E_{Lt}} = {\left | {{\varepsilon _{outL + }}/{\varepsilon _{L}}} \right |^{2}}$ versus normalized input probe field detuning x/κ for η = 0.2κ (red-dotted), η = 0.5κ (black-solid), and η = 0.8κ (blue-dashed) with $G = \sqrt {\frac {{{\kappa ^{2}} - {\eta ^{2}}}}{2}}$. b Normalized output probe field energy ${E_{Rt}} = {\left | {{\varepsilon _{outR + }}/{\varepsilon _{L}}} \right |^{2}}$ versus normalized input probe field detuning x/κ for η = 0.2κ (red-dotted), η = 0.5κ (black-solid), and η = 0.8κ (blue-dashed) with $G = \sqrt {\frac {{{\kappa ^{2}} - {\eta ^{2}}}}{2}}$. Other parameters have been given at the beginning of Section 3

Fig. 8

(Color online) The third situation (${\eta ^{2}} > {\kappa ^{2}} - {G^{2}}\left ({{n^{2}} + 1} \right )$): a Normalized output probe field energy ${E_{Lt}} = {\left | {{\varepsilon _{outL + }}/{\varepsilon _{L}}} \right |^{2}}$ versus normalized input probe field detuning x/κ for η = 0.8κ (red-dotted), η = κ (black-solid), and η = 1.2κ (blue-dashed) with G = 0.6κ. b Normalized output probe field energy ${E_{Rt}} = {\left | {{\varepsilon _{outR + }}/{\varepsilon _{L}}} \right |^{2}}$ versus normalized input probe field detuning x/κ for η = 0.8κ (red-dotted), η = κ (black-solid), and η = 1.2κ (blue-dashed) with G = 0.6κ. Other parameters have been given at the beginning of Section 3