Fully quantum approach to optomechanical entanglement

The radiation-pressure-induced coupling between an optical cavity field and a mechanical oscillator can create entanglement between them. In previous works this entanglement was treated as that of the quantum fluctuations of the cavity and mechanical modes around their classical mean values. Here we provide a fully quantum approach to optomechanical entanglement, which goes beyond the approximation of classical mean motion plus quantum fluctuation and applies to arbitrary cavity drive. We illustrate the real-time evolution of optomechanical entanglement under drive of arbitrary detuning to show the existence of high, robust, and stable entanglement in the blue-detuned regime and highlight the quantum noise effects that can cause entanglement sudden death and revival.

Figure 1

Setup for coupling the cavity field $\widehat{a}\left(t\right)$ with the quantum-mechanical oscillator. The cavity field built up by the drive $E\left(t\right)$ is entangled with the mechanical oscillator, which is initially in thermal equilibrium with its environment via their effective coupling described by the Hamiltonian $-\sqrt{2}g{\widehat{x}}_m{\widehat{a}}^{†}\widehat{a}$.

Figure 2

Evolution of entanglement for blue-detuned cw drives. (a) is obtained with the drive intensity $E/\kappa =3\times {10}^5$, while (b) is for a drive of $E/\kappa =2\times {10}^6$. The long-dashed red curve is for ${\Delta }_0=-0.5{\omega }_m$, the short-dashed blue curve is for ${\Delta }_0=-{\omega }_m$, the thin solid black curve is for ${\Delta }_0=-1.5{\omega }_m$, and the thick solid purple curve is for ${\Delta }_0=-2{\omega }_m$. Here $g/\kappa ={10}^{-6}$, ${\omega }_m/\kappa =2.5$, ${\omega }_m/{\gamma }_m={10}^7$, and $T=0$. The entanglement measure by $E_{\mathcal{N}}$ for these blue-detuned drives tends to a stable value with time. The maximum entanglement is reached at the SQ resonant point ${\Delta }_0=-{\omega }_m$. For the stronger drive, the entanglement at the smaller detuning ${\Delta }_0=-0.5{\omega }_m$ dies a sudden death after a finite time period.

Figure 3

Evolution of entanglement for red-detuned cw drives. (a) is obtained with the same system parameters as in Fig. 2, while (b) is found with the same parameters as in Fig. 2. The thin solid blue curve is for ${\Delta }_0=0.5{\omega }_m$, the long-dashed purple curve is for ${\Delta }_0={\omega }_m$, the thick solid red curve is for ${\Delta }_0=1.5{\omega }_m$, and the short-dashed black curve is for ${\Delta }_0=2{\omega }_m$. The entanglement dies sooner for a detuning closer to the BS resonant point ${\Delta }_0={\omega }_m$ or the stronger drive. Given the stronger drive in (b), the entanglement at ${\Delta }_0=1.5{\omega }_m$ exhibits sudden death and revival.

Figure 4

(a) and (b) Entanglement vs detuning. (a) is under the same conditions as in Figs. 2 and 3, and (b) corresponds to the situation in Figs. 2 and 3. The long-dashed blue curves show the entanglement obtained at zero temperature without the noise-drive terms in (5), while the thick solid red curves include the effect of the noise-drive terms at zero temperature. In (a) the zero-temperature entanglement is eliminated around ${\Delta }_0={\omega }_m$. The thin solid purple curves in (a) and (b) give the exact degree of entanglement at the temperature corresponding to $n_m={10}^4$. (c) and (d) Entanglement vs cavity-drive intensity. In (c), the long-dashed blue curve stands for ${\Delta }_0=-{\omega }_m$, the short-dashed red curve is for ${\Delta }_0=-0.5{\omega }_m$, and the solid purple curve is for ${\Delta }_0=-2{\omega }_m$. In (d), the long-dashed green curve stands for ${\Delta }_0=0$, the short-dashed purple curve is for ${\Delta }_0={\omega }_m$, and the solid blue curve is for ${\Delta }_0=0.5{\omega }_m$. Except for the thin solid purple curves about finite-temperature entanglement in (a) and (b), all plots are found for the initial temperature $T=0$ at the moment $\kappa t=15$, when the concerned entanglement has stabilized.

Figure 5

Evolution of entanglement under the pulsed drive $E\left(t\right)=E{e}^{-\Delta {\omega }^2{t}^2}$ with $\Delta \omega ={\omega }_m$. (a) Red-detuned central frequencies. The long-dashed purple curve is for ${\Delta }_0={\omega }_m$ of the central frequency, the solid red curve is for ${\Delta }_0=1.5{\omega }_m$, and the short-dashed black curve is for ${\Delta }_0=2{\omega }_m$. (b) Blue-detuned and resonant central frequencies. The long-dashed red curve is for ${\Delta }_0=0$, the short-dashed blue curve is for ${\Delta }_0=-{\omega }_m$, the thin solid black curve is for ${\Delta }_0=-1.5{\omega }_m$, and the thick solid purple curve is for ${\Delta }_0=-2{\omega }_m$. The system parameters are $g/\kappa ={10}^{-6}$, $E/\kappa =2\times {10}^6$, ${\omega }_m/\kappa =2.5$, ${\omega }_m/{\gamma }_m={10}^7$, and $T=0$. The entanglement measured by $E_{\mathcal{N}}$ goes down at almost the same pace for the drives of different central frequency detuning.

Figure 6

The workable regime for the standard fluctuation expansion approach. The scale represents the value of $s_1$ defined in (7). The second condition, $s_2>0$, is always satisfied in the concerned regime. The fluctuation expansion approach works in the regime where the scale takes the positive value, which is separated from the other part of the whole parameter space by the boundary of the dashed line. On the other hand, the approach presented in this paper is valid at any point of the parameter space.

Figure 7

Comparison of entanglement obtained in our approach (solid curve) and in the fluctuation expansion approach of [11] (dashed curve). (a) is obtained with the drive intensity $E/\kappa =3\times {10}^5$, while (b) is for a drive of $E/\kappa =2\times {10}^6$. The system parameters are $g/\kappa ={10}^{-6}$, ${\omega }_m/\kappa =2.5$, ${\omega }_m/{\gamma }_m={10}^7$, and $T=0$. The entanglement represented by the solid curves is obtained at $\kappa t=15$.

Figure 8

Comparison of the cavity fluctuation $\langle \delta {\widehat{a}}^{†}\delta \widehat{a}\rangle =\langle ({\widehat{a}}^{†}-\langle {\widehat{a}}^{†}\rangle )(\widehat{a}-\langle \widehat{a}\rangle )\rangle =\langle {\widehat{a}}^{†}\widehat{a}\rangle -\langle {\widehat{a}}^{†}\rangle \langle \widehat{a}\rangle$ at zero temperature obtained in the approach of the present work (solid curves) and in the fluctuation expansion approach of Ref. [11] (dashed curves). The plots for our approach are obtained at the time $\kappa t=15$. The thick solid red and long-dashed curves compare the fluctuation at the detuning ${\Delta }_0={\omega }_m$, while the thin solid black and short-dashed curves compare that at ${\Delta }_0=0.5{\omega }_m$.

Figure 9

Entanglement evolution without quantum noise effect. (a) Blue-detuned regime. The long-dashed red curve is for ${\Delta }_0=-0.5{\omega }_m$, the short-dashed blue curve is for ${\Delta }_0=-{\omega }_m$, the thin solid black curve is for ${\Delta }_0=-1.5{\omega }_m$, and the thick solid purple curve is for ${\Delta }_0=-2{\omega }_m$. (b) Red-detuned regime. The thin solid blue curve is for ${\Delta }_0=0.5{\omega }_m$, the long-dashed purple curve is for ${\Delta }_0={\omega }_m$, the thick solid red curve is for ${\Delta }_0=1.5{\omega }_m$, and the short-dashed black curve is for ${\Delta }_0=2{\omega }_m$. The system parameters are $g/\kappa ={10}^{-6}$, $E/\kappa =2\times {10}^6$, ${\omega }_m/\kappa =2.5$, ${\omega }_m/{\gamma }_m={10}^7$, and $T=0$. The plots in (a) and (b) correspond to those in Figs. 1 and 2 of the main text, respectively. Here the entanglement without the effect of quantum noises becomes steady with time, while that under the quantum noise effect shown in the main text could be destroyed after a finite period of time.