Effect of laser phase noise on the fidelity of optomechanical quantum memory

Optomechanical and electromechanical cavities have been widely used in quantum memories and quantum transducers. We theoretically investigate the robustness of optomechanical and electromechanical quantum memories against the noise of the control laser. By solving the Langevin equations and using the covariance matrix formalism in the presence of laser noise, the storing fidelity of Gaussian states is obtained. It is shown that the destructive effect of phase noise is more significant in higher values of coupling laser amplitude and optomechanical coupling strength $G$. However, by further increasing the coupling coefficient, the interaction time between photons and phonons decreases below the coherence time of laser frequency noise and the destructive effect of laser phase noise on the storing fidelity drops as well.

Figure 1

Schematics of the following systems: (a) An optomechanical Fabry-Perot cavity consisting one vibrating mirror. The cavity is driven by a laser from the left mirror. (b) A microwave cavity.

Figure 2

A schematic outline of coupling $\frac{\pi }{2}$ pulse for quantum memory. The first pulse applies for cooling the mirror near the ground state. This pulse will change the state of the initial optical mode (vacuum state) with the thermal mechanical mode. After a while, the intracavity thermal optical mode transfers to the reservoir and the input Gaussian state enters the cavity at the beginning of the write pulse. After the storage time $\tau$, the reading $\frac{\pi }{2}$ pulse with an opposite sign of the writing pulse enters the cavity.

Figure 3

Variation of (a) storing fidelity, (b) heating parameter ${\overline{n}}_h$, and (c) damping parameter $\lambda$ vs mechanical quality factor $Q_m=\frac{{\omega }_m}{\gamma }$ $({\omega }_m=2\pi \times 10.69$ MHz), for ${\Gamma }_L=0$ (solid red line), ${\Gamma }_L=1$ kHz, ${\gamma }_c=100$ kHz (dash-dotted green line), ${\Gamma }_L=1$ kHz, ${\gamma }_c=0.5$ MHz (dashed blue line), and ${\Gamma }_L=1$ kHz, ${\gamma }_c=10$ MHz $({\gamma }_c\to \infty$ or white noise; purple line with marker $\circ )$ [white noise has not been shown in (b) and (c)]. The input state is a coherent state $|\alpha \rangle$ with $\alpha =1$, the thermal phonon number is $N_m=3$, and the storage time is set to $\tau =\frac{64}{{\omega }_m}$.

Figure 4

(a) Variation of fidelity as a function of control linewidth ${\Gamma }_L$. (b) The variation of fidelity as a function of the control cutoff frequency ${\gamma }_c$ for ${\Gamma }_L=1$ kHz (solid blue line), ${\Gamma }_L=5$ kHz (dash-dotted green line), and ${\Gamma }_L=10$ kHz (dashed red line). The input state is a coherent state $|\alpha \rangle$ with $\alpha =1$. $Q_m=360000$.

Figure 5

Storing fidelity as a function of the storing time $\tau$ for different values of reservoir temperature, laser linewidth ${\Gamma }_L$, and cutoff frequency ${\gamma }_c$ : $T=1.7$ mK, ${\Gamma }_L=0$ (solid red line), $T=1.7$ mK, ${\Gamma }_L=1$ kHz, ${\gamma }_c=300$ kHz (dotted blue line), $T=0.01$ mK, ${\Gamma }_L=0$ (dashed green line), and $T=0.01$ mK, ${\Gamma }_L=1$ kHz, ${\gamma }_c=300$ kHz (dash-dotted black line). The input state is a coherent state $|\alpha \rangle$ with $\alpha =1$.

Figure 6

(a),(b) Storing fidelity as a function of $\frac{G}{\kappa }$ for two ranges of variation of $G$ in the absence and in the presence of laser phase noise effect: $\Gamma =0$ (solid red line), $\Gamma =1$ kHz, ${\gamma }_c=100$ kHz (dashed green line), $\Gamma =1$ kHz, ${\gamma }_c=200$ kHz (dash-dotted blue line), and $\Gamma =1$ kHz, ${\gamma }_c=300$ kHz (dotted black line). (c) Schematic description of cascading a low pass filter with cutoff frequency ${\gamma }_c$ and a band pass filter with center frequency determined by the coupling coefficient $G$ for two values of coupling coefficient $G_1\propto {\alpha }_{s1}$ and $G_2\propto {\alpha }_{s2}$. ${\alpha }_{s2}>{\alpha }_{s1}$. The storage time is $\tau =\frac{64}{{\omega }_m}$.

Figure 7

(a) Storing fidelity as a function of mechanical quality factor $Q_m$ with ${\Gamma }_L=1$ kHz, ${\gamma }_c=10$ kHz for different values of squeezing parameter $r$: $r=0$ (solid red line), $r=0.2$ (dash-dotted green line), $r=0.5$ (dashed blue line), and $r=0.8$ (dotted black line). (b) Variation of fidelity as a function of squeezing parameter $r$ for fixed value of $Q_m=360000$. The storage time is set to $\tau =0.95\mu \mathrm{s}$.

Figure 8

Storing fidelity as a function of mechanical quality factor $Q_m$, for ${\Gamma }_L=0$ (solid red line), ${\Gamma }_L=1$ kHz, ${\gamma }_c=0.5$ kHz (dashed green line), and ${\Gamma }_L=1$ kHz, ${\gamma }_c=1$ kHz (dash-dotted blue line). Storage time and thermal phonon number are $\tau =0.95\mu \mathrm{s}$ and $N_m=3$, respectively.