Electromagnetically induced transparency with quantized fields in a dissipative optomechanical system

We study electromagnetically induced transparency (EIT) using quantized fields in a dissipative optomechanical system with two mechanical oscillators coupled to each other via the Coulomb interaction. The weak probe field is a finite bandwidth squeezed vacuum field. In the absence of the Coulomb coupling, we show that an EIT dip is observable in the homodyne spectrum of the output field even for the squeezed vacuum field at the single-photon level. We find that the thermal environment has a negative impact on the EIT behavior. In the presence of the Coulomb coupling, we show that double EIT dips appear in the homodyne spectrum of the output field. The separation between the two EIT dips can be used to measure the Coulomb coupling strength. Compared to the case of the purely dissipative coupling, the combined dispersive and dissipative coupling can make the EIT dip broader and the minimum value of the EIT dip smaller.

Figure 1

Sketch of the Michelson-Sagnac interferometer formed by three fixed perfect reflecting mirrors (${\mathrm{M}}_1, {\mathrm{M}}_2$, and ${\mathrm{M}}_3$) and a fixed beam splitter (BS). The mechanical oscillator ${\mathrm{MO}}_1$ is placed at the middle point between two mirrors ${\mathrm{M}}_2$ and ${\mathrm{M}}_3$. The two mechanical oscillators ${\mathrm{MO}}_1$ and ${\mathrm{MO}}_2$ are coupled via the Coulomb interaction, and the equilibrium distance between them is $r_0$. A strong-coupling field with amplitude ${\varepsilon }_c$ and a quantized field $c_{\mathrm{in}}$ are sent into the cavity. The output field from the cavity is $c_{\mathrm{out}}$. Here $q_1$ and $q_2$ are the displacements of the two mechanical oscillators ${\mathrm{MO}}_1$ and ${\mathrm{MO}}_2$ from their respective equilibrium positions.

Figure 2

Sketch of the homodyne measurement of the output field. The output field ${\tilde{c}}_{\mathrm{out}}\left(t\right)$ and a strong local oscillator $c_{LO}\left(t\right)$ are combined through a lossless 50:50 beam splitter, where ${\tilde{c}}_{\mathrm{out}}\left(t\right)$ is defined as the sum of the output field $c_{\mathrm{out}}\left(t\right)$ from the cavity and the incident quantized field $c_{\mathrm{in}}\left(t\right)$. Here PD denotes photodetector and SA spectrum analyzer.

Figure 3

Homodyne spectrum $X\left(\omega \right)$ as a function of the normalized frequency $\omega /{\omega }_m$ in the absence (dotted curve) and presence of the coupling field with different powers when $\lambda =0, N=5, M=\sqrt{N(N+1)}$ and 0, and $T=1\mathrm{mK}$. The long-dashed curve, solid curve, medium-dashed curve, dash-dotted curve, short-dashed curve, and solid curve with one circle marker are for $M=\sqrt{N(N+1)}$ and $\wp =5$, 10, 15, 20, 25, and $30\mu \mathrm{W}$, respectively, and the short-dashed curve with two circles is for $M=0$ and $\wp =20\mu \mathrm{W}$.

Figure 4

Homodyne spectrum $X\left(\omega \right)$ as a function of the normalized frequency $\omega /{\omega }_m$ for different squeezing phases $\varphi$ of the squeezed vacuum field when $\lambda =0, N=5, M=\sqrt{N(N+1)}{e}^{i\varphi }, \wp =10\mu \mathrm{W}$, and $T=1\mathrm{mK}$. The curves from top to bottom (dotted curve, dash-dotted curve, short-dashed curve, medium-dashed curve, long-dashed curve, solid curve with one circle, solid curve with two circles, solid curve with three circles, and solid curve) correspond to $\varphi =0, \pi /6, \pi /4, \pi /3, \pi /2, 2\pi /3, 3\pi /4, 5\pi /6$, and $\pi$, respectively.

Figure 5

Homodyne spectrum $X\left(\omega \right)$ as a function of the normalized frequency $\omega /{\omega }_m$ for different values of the parameter $N$ in the absence (dotted curves) and presence (solid and dashed curves) of the coupling field with the power $\wp =20\mu \mathrm{W}$ when $\lambda =0, M=\sqrt{N(N+1)}$ and 0, and (a) $T=10\mathrm{mK}$ and (b) $T=50\mathrm{mK}$. In each plot the upper three curves are for $N=5$ and the lower three curves are for $N=1$; the solid curves are for $M=\sqrt{N(N+1)}$ and the dashed curves are for $M=0$.

Figure 6

Homodyne spectrum $X\left(\omega \right)$ as a function of the normalized frequency $\omega /{\omega }_m$ for different Coulomb coupling strengths $\lambda$ in the presence of the coupling field when $N=5, M=\sqrt{N(N+1)}$ and 0, $T=1\mathrm{mK}$, and (a) $\wp =10\mu \mathrm{W}$ and (b) $\wp =20\mu \mathrm{W}$. In each plot the upper three curves are for $M=\sqrt{N(N+1)}$ and the lower three curves are for $M=0$; the dash-dotted, solid, and dashed curves represent $\lambda =0, 0.5\kappa$, and $\kappa$, respectively.

Figure 7

Separation $D$ between the two EIT dips as a function of the Coulomb coupling strength $\lambda /\kappa$ when $N=5, M=\sqrt{N(N+1)}, \wp =20\mu \mathrm{W}$, and $T=1\mathrm{mK}$.

Figure 8

(a) Energy-level diagram of the optomechanical system. Here $|n+1,n_1,n_2\rangle \leftrightarrow |n,n_1,n_2\rangle$ is the excitation at the cavity frequency ${\omega }_0, |n,n_1,n_2\rangle \leftrightarrow |n,n_1+1,n_2\rangle$ is the mechanical excitation at frequency ${\omega }_1$, the coupling field drives the transition $|n+1,n_1,n_2\rangle \leftrightarrow |n,n_1+1,n_2\rangle$, and the Coulomb coupling drives the transition $|n,n_1+1,n_2\rangle \leftrightarrow |n,n_1,n_2+1\rangle$, where $n, n_1$, and $n_2$ are the intracavity photon number, the phonon number of the ${\mathrm{MO}}_1$, and the phonon number of the ${\mathrm{MO}}_2$, respectively. (b) Energy-level diagram of the optomechanical system in the dressed-state picture. Here $|n,\pm \rangle$ are the two dressed states generated by the Coulomb coupling, whose frequency difference is $\lambda$.

Figure 9

Homodyne spectrum $X\left(\omega \right)$ as a function of the normalized frequency $\omega /{\omega }_m$ for different values of the parameter $N$ in the presence of the coupling field when $\lambda =0.5\kappa , M=\sqrt{N(N+1)}$ and 0, $\wp =20\mu \mathrm{W}$, and (a) $T=10\mathrm{mK}$ and (b) $T=20\mathrm{mK}$. In each plot the upper two curves are for $N=5$ and the lower two curves are for $N=1$; the solid curves are for $M=\sqrt{N(N+1)}$ and the dashed curves are for $M=0$.

Figure 10

Homodyne spectrum $X\left(\omega \right)$ as a function of the normalized frequency $\omega /{\omega }_m$ in the absence (dotted curve) and presence of the coupling field with different powers when $\lambda =0, N=5, M=\sqrt{N(N+1)}$ and 0, and $T=1\mathrm{mK}$. The long-dashed, solid, medium-dashed, and dash-dotted curves are for $M=\sqrt{N(N+1)}$ and $\wp =5$, 10, 15, and $20\mu \mathrm{W}$, respectively, and the short-dashed curve is for $M=0$ and $\wp =20\mu \mathrm{W}$.

Figure 11

Homodyne spectrum $X\left(\omega \right)$ as a function of the normalized frequency $\omega /{\omega }_m$ for different Coulomb coupling strengths $\lambda$ in the presence of the coupling field when $N=5, M=\sqrt{N(N+1)}$ and 0, $\wp =10\mu \mathrm{W}$, and $T=1\mathrm{mK}$. The upper three curves are for $M=\sqrt{N(N+1)}$ and the lower three curves are for $M=0$. The dash-dotted, solid, and dashed curves represent $\lambda =0, 0.5\kappa$, and $\kappa$, respectively.