Macroscopic Quantum Superposition in Cavity Optomechanics

Quantum superposition in mechanical systems is not only key evidence for macroscopic quantum coherence, but can also be utilized in modern quantum technology. Here we propose an efficient approach for creating macroscopically distinct mechanical superposition states in a two-mode optomechanical system. Photon hopping between the two cavity modes is modulated sinusoidally. The modulated photon tunneling enables an ultrastrong radiation-pressure force acting on the mechanical resonator, and hence significantly increases the mechanical displacement induced by a single photon. We study systematically the generation of the Yurke-Stoler-like states in the presence of system dissipations. We also discuss the experimental implementation of this scheme.

Figure 1

(a) The maximum amplitude $|\beta {|}_{\max }$ vs $\delta /g_0$ at $\xi =1.5271$ and 4.9847, which correspond to the peak values of the Bessel function $J_2(2\xi )$, as shown in the inset. (b) Time dependence of $\sin (\delta t/2)$ and $\tan [\mu (t)/2]$ near the positions that give the large oscillation amplitude and the equal probability superposition in states $|{\phi }_{L(R)}(t){⟩}_M$, where ${\omega }_M/\delta =80$, $n_0=1$, and $\xi =1.527$.

Figure 2

(a) The probability $P_L(t)$ and (b) the fidelity $F_L(t)$ vs $\delta t$ at selected values of the cavity-field decay rate ${\gamma }_c$. Insets: The probability $P_{L(R)}(t_d)$ and the fidelity $F_{L(R)}(t_d)$ at time $t_d$ versus ${\gamma }_c/g_0$. (c) The Wigner function $W_L(\eta )$ (with $\eta ={\eta }_r+i{\eta }_i$) and (d) the probability distribution of the rotated quadrature operator $P_L[X({\theta }_0)]$ for the state ${\widehat{\rho }}_M^{(L)}(t_d)$. Other parameters are ${\omega }_M/g_0=20$, $n_0=1$, $\xi =1.5271$, $\delta =g$, ${\gamma }_M/g_0=0.0001$, and $n_{\mathrm{th}}=4$.

Figure 3

The fidelity $F_L(t)$ vs $\delta t$ at selected values of (a) the mechanical decay rate ${\gamma }_M/g_0$ and (b) the thermal phonon occupation number $n_{\mathrm{th}}$. Insets: The fidelity $F_{L(R)}(t_d)$ at time $t_d$ vs ${\gamma }_M/g_0$ and $n_{\mathrm{th}}$. The probability distribution of the rotated quadrature operator $P_L[X({\theta }_0)]$ of the state ${\widehat{\rho }}_M^{(L)}(t_d)$ at selected values of (c) ${\gamma }_M/g_0$ and (d) $n_{\mathrm{th}}$. Other parameters are ${\omega }_M/g_0=20$, $n_0=1$, $\xi =1.5271$, $\delta =g$, and ${\gamma }_c/g_0=0.2$.