Directional phase-sensitive amplifier between microwave and optical photons

We study the directional phase-sensitive amplification between microwave and optical photons in a triple-cavity optomechanical system consisting of three cavities and one mechanical resonator. When the microwave cavity is pumped on both red and blue sidebands but the directly coupled optical cavities are only pumped on the red sidebands, directional phase-sensitive amplification can be realized by tuning the effective optomechanical couplings and the coupling strength between the optical cavities. We show that both gain and bandwidth of the amplifier can be enhanced by increasing the optomechanical cooperativities, indicating an unlimited gain-bandwidth product. Meanwhile, the proposed phase-sensitive amplifier can approach the quantum limit of zero added noise in the large cooperativity limit.

Figure 1

(a) Schematic diagram of the triple-cavity optomechanical system. Cavity modes $a_1$, $a_2$, and $a_3$ are respectively coupled to the common mechanical resonator via radiation pressure. Cavity $a_1$ is pumped on both red and blue sidebands of the mechanical resonator by two-tone driving fields at frequencies ${\omega }_{d,1}$ and ${\omega }_{d,1}^{\prime }$, but cavities $a_2$ and $a_3$ are only pumped on their respective red sidebands by driving fields at the frequency ${\omega }_{d,2}$ and ${\omega }_{d,3}$. Moreover, cavity $a_2$ and cavity $a_3$ are coupled directly via hopping interaction $J$. (b) Schematic panel of the frequencies of the cavity modes and driving fields.

Figure 2

Scattering probabilities $|S_{U_1\to U_2}{|}^2$ and $|S_{V_2\to V_1}{|}^2$ as functions of the probe detuning $\omega$ for (a) $\varphi =-\pi /2$ and (b) $\varphi =\pi /2$, respectively. Other parameters are ${\kappa }_1/2\pi =2$ MHz, ${\kappa }_2/2\pi =3$ MHz, ${\kappa }_3/2\pi =3$ MHz, ${\gamma }_m={\kappa }_2/100$, $C_2=10$, $C_1=C_2^2$, $J=\sqrt{{\kappa }_2{\kappa }_3}/2$, and $G_3=G_2{\kappa }_3/\left(2J\right)$.

Figure 3

Scattering probabilities $|S_{U_1\to U_2}{|}^2$ and $|S_{V_2\to V_1}{|}^2$ as functions of (a) phase difference $\varphi$ and probe detuning $\omega$ and (b) phase difference $\varphi$ with $\omega =0$. The other parameters are the same as those in Fig. 2.

Figure 4

Scattering probability $|S_{U_1\to U_2}{|}^2$ vs. (a) the probe detuning $\omega$ for different values of cooperativity $C_2$ and (b) cooperativity $C_2$ with $\omega =0$. Red dashed-dotted-dotted line in Fig. 4 plots the bandwidth as a function of cooperativity $C_2$. Other parameters are the same those in Fig. 2 except $\varphi =-\pi /2$.

Figure 5

Added noise ${\mathcal{N}}_{2,\mathrm{add}}$ emerging from cavity mode $a_2$ as functions of (a) the probe detuning $\omega$ for different values of cooperativity $C_2$ and (b) cooperativity $C_2$ with $\omega =0$. Other parameters are the same as those in Fig. 4 except $n_1=n_2=n_3=0,n_m=50$.

Figure 6

Scattering probabilities $|S_{i\to j}{|}^2(i,j=U_1,V_1,U_2,V_2,U_3,V_3)$ as functions of the probe detuning $\omega$ when $\varphi =-\pi /2$ (a–c) and $\varphi =\pi /2$ (d–f), respectively. Other parameters are the same as those in Fig. 2.