Interference-modulated photon statistics in whispering-gallery-mode microresonator optomechanics

Whispering-gallery-mode (WGM) microresonator optomechanical systems that can attain high quality factors, exhibit small optical mode volume, and can be excited through their evanescent field are versatile platforms for both theoretical and experimental studies in quantum and nonlinear optics. Investigating photon statistical properties in WGM microresonator optomechanical systems is an important avenue to understand their inner interaction mechanism. Here, the interference-modulated photon statistics in a three-mode coupling, i.e., a pair of counterpropagating optical cavity modes and a mechanical mode, in WGM microresonator optomechanical systems is studied. In the case that one optical mode is driven by an external field, strong antibunching photon statistics can be observed in the presence of mode coupling. When the two cavity modes are driven simultaneously, it is found that the photon statistical properties can be well steered by modulating the interference between different transition paths with the help of the amplitudes of the two input fields and their relative phase. Especially, we show in detail that the antibunching photon statistics can be optimized within the weak optomechanical coupling regime by properly adjusting the relative phase. We also find that it is necessary to prepare the mechanical resonator near to the ground state to eliminate the detrimental effect of thermal phonon number on the photon statistical properties. This investigation can deepen our understanding of the interaction between clockwise or counterclockwise light and mechanical motion as well as be useful for the construction of integrated on-chip single-photon sources.

Figure 1

Schematic diagram of the WGM microresonator optomechanical system which is coherently driven by two input fields through an optical tapered fiber waveguide on both the left- and right-hand ports. The WGM microresonator contains a mechanical radial breathing mode $b$ with a resonance frequency ${\omega }_m$ and two counterpropagating modes (one CW mode $a_{\text{CW}}$ and one CCW mode $a_{\text{CCW}}$), which are coupled to each other by scattering of light at a rate $J$ because of internal defect centers or surface roughness. The left-hand input field ${\varepsilon }_{\text{in}}^L$ is driven by a coherent light ${\varepsilon }_L$ of frequency ${\omega }_L$, whereas the right-hand input field ${\varepsilon }_{\text{in}}^R$ is driven by a coherent light ${\varepsilon }_R$ of frequency ${\omega }_R$ with a phase $\theta$ with respect to the left-hand input field. The left- and right-hand output fields are described by ${\varepsilon }_{\text{out}}^L$ and ${\varepsilon }_{\text{out}}^R$, respectively. ${\kappa }_{\text{ex}}$ denotes the cavity-waveguide coupling rate and ${\kappa }_i$ is the intrinsic cavity decay rate of the two cavity modes.

Figure 2

Calculation results of the normalized power for the forward transmission spectra $P_F$ (blue solid line) and the backward reflection spectra $P_B$ (red dashed line) varying with $\mathrm{\Delta }/\kappa$ under the four different values of optical mode-coupling strength $J$: (a) $J=0.3\kappa$, (b) $J=\kappa$, (c) $J=3\kappa$, and (d) $J=6\kappa$.

Figure 3

Contour plot of the photon statistics spectrum in logarithmic scale ${log}_{10}\left[g_{a_{\text{CCW}}}^{\left(2\right)}\left(0\right)\right]$ of the cavity mode $a_{\text{CCW}}$ plotted as a function of $g/\kappa$ and $\mathrm{\Delta }/\kappa$ when only the cavity mode $a_{\text{CCW}}$ is driven by the input field ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$. All other system parameters are given as ${\varepsilon }_{\text{in}}^R=0$, ${\overline{n}}_{\text{th}}=0$, $\kappa =0.1{\omega }_m$, ${\gamma }_m/{\omega }_m={10}^{-4}$, $J=3\kappa$, and ${\omega }_m/2\pi =10$ MHz.

Figure 4

Energy-level and transition path diagram of the coupled three-mode WGM microresonator optomechanical system. The quantum interference between different transition pathways for the two-photon state of the cavity mode $a_{\text{CCW}}$ leads to the counterintuitive photon antibunching effect.

Figure 5

(a) The photon statistics spectrum in logarithmic scale ${log}_{10}\left[g_{a_{\text{CCW}}}^{\left(2\right)}\left(0\right)\right]$ of the cavity mode $a_{\text{CCW}}$ varying with the detuning $\mathrm{\Delta }/\kappa$ for the case that the two cavity modes are simultaneously driven. Blue starred line, ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$, ${\varepsilon }_R=0$, and $\theta /\pi =0$; red dashed line, ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$, ${\varepsilon }_R=0.1{\varepsilon }_L$, and $\theta =0$; black dotted line, ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$, ${\varepsilon }_R=0.3{\varepsilon }_L$, and $\theta =0$; green dot-dashed line, ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$, ${\varepsilon }_R=0.5{\varepsilon }_L$, and $\theta =0$; black solid line, ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$, ${\varepsilon }_R=0.3{\varepsilon }_L$, and $\theta /\pi =0.6$. (b) Partially enlarged view of the photon statistics spectrum in (a). All other system parameters are given as ${\overline{n}}_{\text{th}}=0$, ${\gamma }_m/{\omega }_m={10}^{-4}$, $\kappa =0.1{\omega }_m$, $J=3\kappa$, $g=0.6\kappa$, and ${\omega }_m/2\pi =10$ MHz.

Figure 6

(a) Contour plot of the photon statistics spectrum in logarithmic scale ${log}_{10}\left[g_{a_{\text{CCW}}}^{\left(2\right)}\left(0\right)\right]$ of the cavity mode $a_{\text{CCW}}$ plotted as a function of the relative phase $\theta /\pi$ and the detuning $\mathrm{\Delta }/\kappa$ under the weak optomechanical coupling regime. (b) The photon statistics spectrum in logarithmic scale ${log}_{10}\left[g_{a_{\text{CCW}}}^{\left(2\right)}\left(0\right)\right]$ of the cavity mode $a_{\text{CCW}}$ varying with the detuning $\mathrm{\Delta }/\kappa$ under different phases $\theta$: $\theta /\pi =0.1$ (blue dotted line), $\theta /\pi =0.2$ (red dashed line), $\theta /\pi =0.6$ (black solid line), and $\theta /\pi =0.8$ (green dot-dashed line). The WGM microresonator optomechanical system is driven by the two input fields ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$ and ${\varepsilon }_R=0.3{\varepsilon }_L$ at the same time. All other system parameters are given as ${\overline{n}}_{\text{th}}=0$, ${\gamma }_m/{\omega }_m={10}^{-4}$, $\kappa =0.1{\omega }_m$, $J=3\kappa$, $g=0.6\kappa$, and ${\omega }_m/2\pi =10$ MHz.

Figure 7

The photon statistics spectrum in logarithmic scale ${log}_{10}\left[g_{a_{\text{CCW}}}^{\left(2\right)}\left(0\right)\right]$ of the cavity mode $a_{\text{CCW}}$ as a function of the detuning $\mathrm{\Delta }/\kappa$ under different mean thermal phonon number ${\overline{n}}_{\text{th}}$ (blue dotted curve for ${\overline{n}}_{\text{th}}=0.1$, red dashed curve for ${\overline{n}}_{\text{th}}=0.01$, and black solid curve for ${\overline{n}}_{\text{th}}=0.001$). (a) $\theta /\pi =0.2$. (b) $\theta /\pi =0.6$. All other system parameters are given as ${\varepsilon }_L=3\times {10}^{-3}{\omega }_m$, ${\varepsilon }_R=0.3{\varepsilon }_L$, ${\gamma }_m/{\omega }_m={10}^{-4}$, $\kappa =0.1{\omega }_m$, $J=3\kappa$, $g=0.6\kappa$, and ${\omega }_m/2\pi =10$ MHz.