Macroscopic quantum coherence and mechanical squeezing of a graphene sheet

We theoretically investigate the macroscopic quantum coherence and the mechanical squeezing of a mechanical oscillator in a hybrid optomechanical system consisting of a suspended graphene sheet and an ultracold atomic ensemble trapped inside a Fabry-Pérot cavity. In the study the vacuum is used to mediate an effective optomechanical coupling between the graphene oscillator and the cavity field driven by an external laser beam. We find that in the presence of this coupling, the macroscopic quantum coherence and the mechanical squeezing of the graphene sheet can be attained in a certain range of driving power. In particular, the quantum coherence in the optomechanical system can be transferred from the optical field to the mechanical oscillator. We also investigate in detail the spectrum and the squeezing of the output field and the attained results may be used to study the mechanical squeezing of a graphene sheet.

Figure 1

Plot of the setup studied in the paper. An ensemble of $N$ two-level ultracold atoms is trapped inside an optical cavity, which interacts with a driven cavity mode ${\omega }_c$ and a suspended graphene sheet. Vibration of the graphene sheet changes the internal state of the atoms due to the vacuum fluctuation. The cavity mode is driven by a classical laser with frequency ${\omega }_p$ and amplitude $\eta$.

Figure 2

The quantum coherence of the graphene sheet as a function of the input laser power $P$ with different distances. The other parameter values are ${\lambda }_p\simeq 780$ nm, $\kappa =2\pi \times 2\times {10}^5$ Hz, $m_a=1.42\times {10}^{-25}$ kg, $N={10}^4$, $U_{1s}=2\pi$ Hz, ${\mathrm{\Delta }}_{as}=2\pi \times 3\times {10}^8$ Hz, ${\mathrm{\Gamma }}_0=2\pi \times 6.1\times {10}^6$ Hz, ${\omega }_0\simeq {\omega }_p$, ${\gamma }_a=2\pi \times 1000$ Hz, $M=2.81\times {10}^{-18}$ kg, ${\omega }_m=2\pi \times 5\times {10}^6$ Hz, ${\gamma }_m=2\pi \times 30$ Hz, $T=10$ mK, $\mu =0.8\hslash {\omega }_0$, ${\gamma }_g={\omega }_0$, and ${\mathrm{\Delta }}_{\text{eff}}=-{\omega }_m$.

Figure 3

The quantum coherence of the optical field as a function of the input laser power $P$ with different distances. Other parameter values we select are the same as in Fig. 2.

Figure 4

The quantum coherence of the graphene sheet as a function of the input laser power $P$ with different optical lattice potentials. ${\omega }_{0}d/c=0.15$, and other parameter values are the same as in Fig. 2.

Figure 5

The quantum coherence of the optical field as a function of the input laser power $P$ with different optical lattice potentials. Other parameter values are the same as in Fig. 4.

Figure 6

The mean-square fluctuation $\langle \delta p_y{\left(t\right)}^2\rangle$ as a function of the input laser power $P$ with different distances. Other parameter values we select are the same as in Fig. 2.

Figure 7

The mean-square fluctuation $\langle \delta p_y{\left(t\right)}^2\rangle$ as a function of the input laser power $P$ with different optical lattice potential. The other parameter values we select are the same as in Fig. 2.

Figure 8

The spectrum $S_{Z\text{out}}\left(\omega \right)$ of the phase fluctuation of the output field is plotted as a function of the normalized frequency $\omega /{\omega }_m$ when $\varphi =\pi /2$ and $P=1$ mW in the absence and presence of the vacuum-induced coupling. Other parameter values are the same as in Fig. 2.

Figure 9

The contour spectrum $S_{Z\text{out}}\left(\omega \right)$ of the quadrature fluctuation of the output field versus the normalized frequency $\omega /{\omega }_m$ and the phase $\varphi /\pi$ in the absence and presence of the vacuum-induced optomechanical coupling. Other parameter values are the same as in Fig. 8.