Effects of quadratic coupling and squeezed vacuum injection in an optomechanical cavity assisted with a Bose-Einstein condensate

We investigate theoretically a hybrid system consisting of a Bose-Einstein condensate (BEC) trapped inside a laser-driven membrane-in-the-middle optomechanical cavity assisted with squeezed vacuum injection whose moving membrane interacts both linearly and quadratically with the radiation pressure of the cavity. It is shown that such a hybrid system is very suitable for generating strong quadrature squeezing in the mechanical mode of the membrane and the Bogoliubov mode of the BEC in the unresolved sideband regime. More interestingly, by choosing a suitable sign for the quadratic optomechanical coupling (QOC), one can achieve a very high degree of squeezing in the mechanical mode and a strong entanglement between the mechanical and atomic modes without the necessity of using squeezed light injection. Furthermore, the QOC changes the effective oscillation frequencies of both the mechanical and the atomic modes and affects their relaxation times. It can also make the system switch from optical bistability to tristability.

Figure 1

A BEC trapped in a membrane-in-the-middle optomechanical cavity interacting with a single-cavity mode. The cavity which decays at rate $\kappa$ is driven through the left mirror by a laser with frequency ${\omega }_p$ and also is injected by a squeezed vacuum field. The membrane is free to oscillate at mechanical frequency ${\omega }_m$.

Figure 2

The mean values of (a) the $q$ quadrature of the mechanical mode, (b) the $Q$ quadrature of the Bogoliubov mode of the BEC, and (c) the mean number of cavity photons versus the normalized time ${\gamma }_{m}t$ when the cavity is pumped at rate $\eta =100\kappa$ and the effective detuning has been fixed at ${\delta }_c=50\kappa$ for ${\xi }_2=0$ (the line indicated by ${\xi }_0$), ${\xi }_2=-0.003{\xi }_1$ (the line indicated by ${\xi }_{-}$), and ${\xi }_2=+0.003{\xi }_1$ (the line indicated by ${\xi }_{+}$). The cavity contains $N={10}^5$ Rb atoms and has a length of $L=178\mu \mathrm{m}$. The other parameters are $\kappa =2\pi \times 1.3\mathrm{MHz}, \lambda =780\mathrm{nm}, {\xi }_1=0.05\kappa , {\gamma }_m=2\pi \times 100\mathrm{Hz}, {\gamma }_c=0.001\kappa$, and ${\omega }_{sw}=0.5{\omega }_R$. Panels (d)–(f) show the temporal behaviors of the same mean fields in a shorter time interval.

Figure 3

The mean number of cavity photons versus the normalized effective detuning ${\delta }_c/\kappa$ for ${\xi }_2=-0.002{\xi }_1,0$ [(a) and (b)], ${\xi }_2=\pm 0.003{\xi }_1$ [(c) and (d)], and ${\xi }_2=\pm 0.005{\xi }_1$ [(e) and (f)] when the cavity is driven at rate $\eta =100\kappa$. Each numbered curve corresponds to the stable or unstable solution for the steady-state mean value of the optical field (see the text for details). The other parameters are the same as those of Fig. 2.

Figure 4

The quadrature squeezing in the dB unit [(a), (c), and (e)] of the mechanical mode and the Bogoliubov mode, and the BEC-membrane entanglement [(b), (d), and (f)] vs normalized effective detuning ${\delta }_c/\kappa$ for three different values of ${\xi }_2/{\xi }_1\le 0$. The lines indicated by $q\left(qs\right),Q\left(Qs\right)$, and $E_N\left(E_{Ns}\right)$ correspond, respectively, to the quadrature squeezing of the mechanical mode, the Bogoliubov mode, and the BEC-membrane entanglement in the absence of the squeezed vacuum injection (presence of the squeezed vacuum injection with $N_s=10,\varphi =\pi$). In (a), (c), and (e), the thick horizontal line corresponds to the 3-dB limit. The other parameters are the same as those in Fig. 2.

Figure 5

The quadrature squeezing in the dB unit [(a), (c), and (e)] of the mechanical mode and the Bogoliubov mode, and the BEC-membrane entanglement [(b), (d), and (f)] vs normalized effective detuning ${\delta }_c/\kappa$ for three different values of ${\xi }_2/{\xi }_1>0$. The lines indicated by $q\left(qs\right),Q\left(Qs\right)$, and $E_N\left(E_{Ns}\right)$ correspond, respectively, to the quadrature squeezing of the mechanical mode, the Bogoliubov mode, and the BEC-membrane entanglement in the absence of the squeezed vacuum injection (presence of the squeezed vacuum injection with $N_s=10,\varphi =\pi$). In (a), (c), and (e), the thick horizontal line corresponds to the 3-dB limit. The other parameters are the same as those in Fig. 2.