Nonlinear effects in modulated quantum optomechanics

The nonlinear quantum regime is crucial for implementing interesting quantum effects, which have wide applications in modern quantum science. Here we propose an effective method to reach the nonlinear quantum regime in a modulated optomechanical system (OMS), which is originally in the weak-coupling regime. The mechanical spring constant and optomechanical interaction are modulated periodically. This leads to the result that the resonant optomechanical interaction can be effectively enhanced into the single-photon strong-coupling regime by the modulation-induced mechanical parametric amplification. Moreover, the amplified phonon noise can be suppressed completely by introducing a squeezed vacuum reservoir, which ultimately leads to the realization of photon blockade in a weakly coupled OMS. The reached nonlinear quantum regime also allows us to engineer the nonclassical states (e.g., Schrödinger cat states) of the cavity field, which are robust against the phonon noise. This work offers an alternative approach to enhance the quantum nonlinearity of an OMS, which should expand the applications of cavity optomechanics in the quantum realm.

Figure 1

(a) Schematic illustration of an OMS consisting of an optical cavity mode $a$ coupled to the mechanical mode $b$ with the modulated coupling strength $g\left(t\right)=g_{0}cos\left({\omega }_{d}t\right).$ In addition, the mechanical spring constant is modulated with frequency $2{\omega }_d$ and phase ${\mathrm{\Phi }}_d$, i.e., $k\left(t\right)=k_0-k_{r}cos(2{\omega }_{d}t-{\mathrm{\Phi }}_d)$. A weak probe field with frequency ${\omega }_p$ and amplitude ${\varepsilon }_p$ is applied to the cavity to demonstrate the nonlinear quantum regime. (b) The phase-matching condition ${\mathrm{\Phi }}_e-{\mathrm{\Phi }}_d=\pm n\pi (n=1,3,5,...)$ for suppressing the noise of $\tilde{b}$ induced by mechanical parametric amplification.

Figure 2

(a) The effective single-photon optomechanical coupling strength $\tilde{g}/\kappa$ versus the squeezing parameter $r_d$. The squeezing parameter $r_d$ versus (b) driving amplitude $\lambda /\kappa$ and (c) frequency detuning ${\mathrm{\Delta }}_m/\kappa$. The system parameters are scaled by the decay rate $\kappa$, i.e., (a) $g_0=0.5\kappa$, (b) ${\mathrm{\Delta }}_m=4000\kappa$, and (c) $\lambda =4000\kappa$.

Figure 3

The steady-state equal-time second-order correlation function $g_{\mathrm{ss}}^2\left(0\right)$ versus the scaled coupling (a) $g_0/\kappa$ and (b) $g_0/{\omega }_m$ in the single-photon-resonance case ${\mathrm{\Delta }}_c={\tilde{g}}^2/{\tilde{\omega }}_m$ with $H_{\mathrm{OMS}}^{\prime }$ for the ideal parameter matching. The parameters we take are ${\mathrm{\Delta }}_m=4000\kappa$, ${\omega }_d=30\kappa$, $\delta =0.02\kappa$, $\gamma =0.01\kappa$, ${\varepsilon }_p=0.1\kappa$, and $\mathrm{\Phi }=\pi$. Here the symbol $\delta$ denotes the difference between the parameters ${\mathrm{\Delta }}_m$ and $\lambda$ as $\delta ={\mathrm{\Delta }}_m-\lambda$. The relevant squeezing parameters are $r_e=r_d\approx 3.22$.

Figure 4

The equal-time second-order correlation function $g^2\left(0\right)$ versus the scaled time $\kappa t$ in the single-photon-resonance case ${\mathrm{\Delta }}_c={\tilde{g}}^2/{\tilde{\omega }}_m$. The cavity and mechanical modes are initially in a thermal state and vacuum state, respectively. All the curves are obtained by numerically calculating Eq. (4) with $H_{\mathrm{tot}}^{\prime }$ or $H_{\mathrm{OMS}}^{\prime }$. The parameters are the same as that in Fig. 3 except for $g_0=0.5\kappa$. Here the green dotted line (i.e., $r_e=0$) actually corresponds to the case in which the mechanical oscillator is initially in a vacuum bath and an amplified thermal noise $\tilde{N}={sinh}^2\left(r_d\right)$ is induced for the considered squeezed mode $\tilde{b}$.

Figure 5

The steady-state equal-time second-order correlation function $g_{\mathrm{ss}}^2\left(0\right)$ versus (a) $g_0/{\omega }_m$ with different $\mathrm{\Phi }$, (b) $\mathrm{\Phi }$ for the case of single-photon resonance ${\mathrm{\Delta }}_c={\tilde{g}}^2/{\tilde{\omega }}_m$. The system parameters are the same as in Fig. 3 except for (b) $g_0=0.5\kappa$.

Figure 6

Wigner function of the cavity field at time $t=2\pi /{\tilde{\omega }}_m$ for different $\tilde{g}/{\tilde{\omega }}_m$ when (a)–(c) $r_e=r_d$ and (d)–(f) $r_e=0$. The case of $r_e=0$ is the same as for the green dotted line in Fig. 4. The quadrature variables are $x=(a+a^{†})/2$, $y=-i(a-a^{†})/2$. The cavity field and the mechanical mirror are initially in coherent states $|\alpha \rangle ,|\beta \rangle$ with the amplitudes $\alpha =\beta =2$. Other parameters are the same as that in Fig. 3 except for (a), (d) $\kappa /{\omega }_m=3.16\times {10}^{-5}$, $\gamma /\kappa ={10}^{-2}$, $g_0/{\omega }_m=1.26\times {10}^{-4}$, (b) (e) $g_0/{\omega }_m=1.03\times {10}^{-4}$, and (c), (f) $g_0/{\omega }_m=0.89\times {10}^{-4}$.

Figure 7

Wigner function of the cavity field at time $t=2\pi /{\tilde{\omega }}_m$ for different $\tilde{g}/{\tilde{\omega }}_m$. The system parameters are the same as in Fig. 6 except for $r_e=0$ and (a)–(c) $\kappa /{\omega }_m=3.16\times {10}^{-5}$, (d)–(f) $\kappa /{\omega }_m=1.58\times {10}^{-4}$, (h)–(j) $\kappa /{\omega }_m=3.16\times {10}^{-4}$, (a), (d), (h) $g_0/{\omega }_m=1.26\times {10}^{-4}$, (b), (e), (i) $g_0/{\omega }_m=1.03\times {10}^{-4}$, and (c), (f), (j) $g_0/{\omega }_m=0.89\times {10}^{-4}$.