Temporal nonreciprocity in gently modulated three-mode optomechanical systems

We investigate temporal dynamics of nonreciprocal photon transmission between two optical modes in a three-mode optomechanical system, where the optical modes are simultaneously driven by two independent periodically amplitude-modulated lasers. We reveal that the nonreciprocal optical transmission oscillating with time can be achieved in the modulated three-mode optomechanical system. By elaborately tuning the strength of the modulation lasers, we can significantly improve the nonreciprocal optical transmission and shorten the convergence time to reach an asymptotic oscillating regime. In particular, the dependence of the temporal photon transmission on the phase difference between two independent amplitude-modulated lasers provides a flexible controllability of optical communication based on optomechanical systems.

Figure 1

(a) The three-mode optomechanical system includes common mechanical resonator $\widehat{b}$ and two optical cavities ${\widehat{a}}_1$ and ${\widehat{a}}_2$, which are directly coupled with a coherent coupling strength $J$. (b) Equivalent schematic diagram of the three-mode optomechanical system, consisting of a dielectric membrane as a mechanical resonator, separating an optical cavity into two cavities ${\widehat{a}}_1$ and ${\widehat{a}}_2$. The two optical cavities are coupled with the dielectric membrane $\widehat{b}$ via the radiation pressure difference, and tunneling of photons can go through the membrane.

Figure 2

(a)–(d) Time evolution of the mean values of the cavity fields $n_1\left(t\right)=\langle {\widehat{a}}_1^{†}\left(t\right){\widehat{a}}_1\left(t\right)\rangle$ (red solid lines) and $n_2\left(t\right)=\langle {\widehat{a}}_2^{†}\left(t\right){\widehat{a}}_2\left(t\right)\rangle$ (blue solid lines) for different phases $\phi$: (a) $\phi =0$, (b) $\phi =0.4\pi$, (c) $\phi =\pi$, and (d) $\phi =-0.4\pi$. All the lines are based on the numerical solutions with Eq. (8) and the markers correspond to the analytical solutions Eqs. (11) and (12). Other parameters are $J/\kappa =1, P_{1,0}=P_{2,0}=10\mathrm{mW}, P_{1,\pm 1}=P_{2,\pm 1}=2.2\mathrm{mW}$, and ${\kappa }_1={\kappa }_2=\kappa$.

Figure 3

The temporal nonreciprocal transmission coefficients $T_{1\to 2}$ (red solid curves) and $T_{2\to 1}$ (blue solid curves) for different power $P_1$ of the modulation driving laser: (a) $P_{1,\pm 1}=P_{2,\pm 1}=2.2\mathrm{mW}$, (b) $P_{1,\pm 1}=P_{2,\pm 1}=2.4\mathrm{mW}$, and (c) $P_{1,\pm 1}=P_{2,\pm 1}=2.6\mathrm{mW}$. The insets show the corresponding temporal nonreciprocal transmission in the long-time limit $t/\tau \in [0,200]$. Other parameters are the same as in Fig. 2 except for $\phi =0.4\pi , P_{1,0}=P_{2,0}=10\mathrm{mW}$, and $J/\kappa =1$.

Figure 4

The temporal transmission coefficients $T_{1\to 2}$ (red solid curves) and $T_{2\to 1}$ (blue solid curves) at a fixed first-order modulated laser $P_{1,\pm 1}=P_{2,\pm 1}=2.4\mathrm{mW}$ with different powers of the zeroth-order modulated laser: (a) $P_{1,0}=P_{2,0}=8\mathrm{mW}$, (b) $P_{1,0}=P_{2,0}=9\mathrm{mW}$, and (c) $P_{1,0}=P_{2,0}=10\mathrm{mW}$. Other parameters are the same as in Fig. 2 except $\phi =0.4\pi$.

Figure 5

The temporal transmission coefficients $T_{1\to 2}$ (red solid curves) and $T_{2\to 1}$ (blue solid curves) at a fixed ratio $P_{1,\pm 1}/P_{1,0}=P_{2,\pm 1}/P_{2,0}=0.24$ with different power of driving laser: (a) $P_{1,0}=P_{2,0}=8\mathrm{mW}$, (b) $P_{1,0}=P_{2,0}=9\mathrm{mW}$, and (c) $P_{1,0}=P_{2,0}=10\mathrm{mW}$. Other parameters are the same as in Fig. 2 except for $\phi =0.4\pi$.

Figure 6

The temporal transmission coefficients $T_{1\to 2}$ (red solid curves) and $T_{2\to 1}$ (blue solid curves) for different powers of the zeroth-order driving lasers: $P_{1,0}=9.2,9.5$, and $9.8\mathrm{mW}$ in (a)–(c) with $P_{1,\pm 1}=P_{2,\pm 1}=2.4\mathrm{mW}$ and $P_{2,0}=10\mathrm{mW}$, respectively. The temporal transmission coefficients for different powers of the modulated lasers are $P_{1,\pm 1}=2.5,2.6$,and $2.8\mathrm{mW}$ in (d)–(f) with $P_{2,\pm 1}=2.4\mathrm{mW}$ and $P_{1,0}=P_{2,0}=10\mathrm{mW}$, respectively. Other parameters are the same as those in Fig. 2 except for $\phi =0.4\pi$.

Figure 7

The temporal transmission coefficients $T_{1\to 2}$ (red solid curves) and $T_{2\to 1}$ (blue solid curves) for the different phase difference $\phi$: (a) $\phi =0.14\pi$, (b) $\phi =0.28\pi$, (c) $\phi =0.42\pi$, (d) $\phi =(2-0.14)\pi$, (e) $\phi =(2-0.28)\pi$, and (f) $\phi =(2-0.42)\pi$. Other parameters are the same as in Fig. 2 except for $P_{1,\pm 1}=P_{2,\pm 1}=2.4\mathrm{mW}$ and $P_{1,0}=P_{2,0}=10\mathrm{mW}$.

Figure 8

The temporal transmission coefficients $T_{1\to 2}$ and $T_{2\to 1}$ as functions of the phase $\phi /\pi$ and the time $t/\tau$. Other parameters are the same as those in Fig. 2.

Figure 9

The time evolution of the temporal transmission coefficients $T_{1\to 2}$ (red solid curves) and $T_{2\to 1}$ (blue solid curves) for $J/\kappa =2, \phi =\pi$. Other parameters are the same as those in Fig. 2.

Figure 10

The scaled isolation parameter $\mathfrak{R}$ as a function of time. Periodic modulation (red curve) can lead to a more significant isolation rate compared to the absence of periodic modulation with $\mathrm{\Omega }=0$ (blue curve). Other parameters are the same as in Fig. 2 in the main text except for $\phi =0.42\pi , P_{1,0}=P_{2,0}=10\mathrm{mW}$, and $P_{1,\pm 1}=P_{2,\pm 1}=2.4\mathrm{mW}$.

Figure 11

The time evolutional isolation rate $\mathfrak{R}$ for different modulation frequencies of the driving lasers: (a) ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=0.96$, (b) ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=0.98$, and (c) ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=1$. Other parameters are the same as in Fig. 2 except for $\phi =0.42\pi , P_{1,0}=P_{2,0}=10\mathrm{mW}, P_{1,\pm 1}=P_{2,\pm 1}=2.4\mathrm{mW}$, and ${\mathrm{\Omega }}_1=\mathrm{\Omega }$.

Figure 12

Temporal evolution and steady value of the transmission ratio $T_{1\to 2}/T_{2\to 1}$. The green (purple) solid lines indicate the analytical steady solutions (numerical simulations) for the phase differences $\varphi =0.5\pi$. Other parameters are the same as those in Fig. 2 except for $P_{1,\pm 1}=P_{2,\pm 1}=2.4\mathrm{mW}, P_{1,0}=P_{2,0}=10\mathrm{mW}$, and $\overline{\delta }/{\omega }_m=0$.