Pulsed entanglement and quantum steering in a three-mode electro-optomechanical system

Abstract

We investigate bipartite entanglement and quantum steering in a three-mode, hybrid electro-optomechanical system consisting of a Fabry–Perot optical cavity, a nanomechanical oscillator and a lumped-element microwave cavity. The nanomechanical resonator is directly coupled to the optical cavity on the one side via the moving mirror of the Fabry–Perot cavity, and on the other side, it is capacitively coupled to the microwave cavity. There is no direct coupling between the cavity modes. The optical cavity is blue-detuned through excitation by a laser source emitting short pulses, while the microwave cavity is red-detuned through excitation by a voltage pulse generator. The presence of bipartite entanglement between different modes is verified based on the asymmetric entanglement criteria. Specifically, the system is shown to behave effectively as a two-mode system in which a perfect bipartite EPR state can be formed by the optical cavity and the mirror. The possibility of generating genuine tripartite entanglement is also investigated by examining the necessary conditions for multiparty multimode entanglement. Simultaneous coupling of the three modes is shown to be possible in accordance with the genuine tripartite entanglement criteria. Furthermore, we establish the existence of quantum steering between the modes using the steering parameter formality and examine the monogamy relations to quantify the amount of bipartite steering shared between different modes. Notably, steering is found to be present only between the mechanical and the optical cavity modes.

Appendix A

Taking the input noise correlations into account, the expressions for the covariance matrix elements are given herein as a function of the normalized interaction time parameter r, parameters $\alpha$ and $\beta$ (defined in the main text), the scaled damping parameter $\gamma$, the mean number of thermal phonons, $n_{m}$, and the equilibrium number of photons, $n_{a}$ and $n_{b}$, in the optical and microwave cavities, respectively. Expressly, the elements of the covariance matrix can be written in the explicit form as

$$\begin{aligned} V_{11}= & {} V_{22}=(-\beta ^2e^{r}+\alpha ^2)^2(n_{a}+1/2) +4\alpha ^2\beta ^2 sinh^2r(n_{b}+1/2)\nonumber \\&+\beta ^2(-\sqrt{e^{2r}-1}+2\gamma sinhr)^2(n_{m}+1/2), \end{aligned}$$

(A.1)

$$\begin{aligned} V_{33}= & {} V_{44}=4\alpha ^2\beta ^2 sinh^2r(n_{a}+1/2) +(-\beta ^2+\alpha ^2e^{r})^2(n_{b}+1/2)\nonumber \\&+\alpha ^2(-\sqrt{e^{2r}-1}+2\gamma sinhr)^2(n_{m}+1/2), \end{aligned}$$

(A.2)

$$\begin{aligned} V_{55}= & {} V_{66}=\beta ^2(e^{2r}-1)(n_{a}+1/2) +\alpha ^2(e^{2r}-1)(n_{b}+1/2)\nonumber \\&+(e^{2r}-\gamma \sqrt{e^{2r}-1})(n_{m}+1/2), \end{aligned}$$

(A.3)

$$\begin{aligned} V_{12}= & {} -V_{21}=((-\beta ^2e^{r}+\alpha ^2)^2-4\alpha ^2\beta ^2 sinh^2r\nonumber \\&-\beta ^2(-\sqrt{e^{2r}-1}+2\gamma sinhr)^2)i/2, \end{aligned}$$

(A.4)

$$\begin{aligned} V_{13}= & {} V_{31}=2(-\beta ^2e^{r}+\alpha ^2)\alpha \beta sinhr(n_{a}+1/2)\nonumber \\&-2\alpha \beta sinhr(-\beta ^2+\alpha ^2e^{r})(n_{b}+1/2)\nonumber \\&-\alpha \beta (-\sqrt{e^{2r}-1}+2\gamma sinhr)^2(n_{m}+1/2), \end{aligned}$$

(A.5)

$$\begin{aligned} V_{14}= & {} -V_{41}=(-2(-\beta ^2e^{r}+\alpha ^2)\alpha \beta sinhr -2\alpha \beta sinhr(-\beta ^2+\alpha ^2e^{r})\nonumber \\&-\alpha \beta (-\sqrt{e^{2r}-1}+2\gamma sinhr)^2)i/2, \end{aligned}$$

(A.6)

$$\begin{aligned} V_{15}= & {} -V_{51}=((-\beta ^2e^{r}+\alpha ^2)\beta \sqrt{e^{2r}-1}\nonumber \\&+2 \alpha ^2 \beta sinhr\sqrt{e^{2r}-1} - \beta (-\sqrt{e^{2r}-1}\nonumber \\&+2\gamma sinhr)(e^r-\gamma \sqrt{e^{2r}-1}))i/2, \end{aligned}$$

(A.7)

$$\begin{aligned} V_{16}= & {} V_{61}=(-\beta ^2e^{r}+\alpha ^2)\beta \sqrt{e^{2r}-1}(n_{a}+1/2)\nonumber \\&-2\alpha ^2 \beta sinhr \sqrt{e^{2r}-1}(n_{b}+1/2)\nonumber \\&+\beta (-\sqrt{e^{2r}-1}+2\gamma sinhr)(e^r-\gamma \sqrt{e^{2r}-1})(n_{m}+1/2), \end{aligned}$$

(A.8)

$$\begin{aligned} V_{23}= & {} -V_{32}=V_{14}, \end{aligned}$$

(A.9)

$$\begin{aligned} V_{24}= & {} V_{42}=-V_{13}, \end{aligned}$$

(A.10)

$$\begin{aligned} V_{25}= & {} V_{52}=V_{16}, \end{aligned}$$

(A.11)

$$\begin{aligned} V_{26}= & {} -V_{62}=-V_{15}, \end{aligned}$$

(A.12)

$$\begin{aligned} V_{34}= & {} -V_{43}=(-4\alpha ^2\beta ^2 sinh^2r+(-\beta ^2+\alpha ^2e^{r})^2\nonumber \\&+\alpha ^2(-\sqrt{e^{2r}-1}+2\gamma sinhr)^2)i/2, \end{aligned}$$

(A.13)

$$\begin{aligned} V_{35}= & {} -V_{53}=(2\alpha \beta ^2 sinhr\sqrt{e^{2r}-1} -(-\beta ^2+\alpha ^2e^{r})\alpha \sqrt{e^{2r}-1}\nonumber \\&+\alpha (-\sqrt{e^{2r}-1}+2\gamma sinhr)(e^r-\gamma \sqrt{e^{2r}-1}))i/2, \end{aligned}$$

(A.14)

$$\begin{aligned} V_{36}= & {} V_{63}=2\alpha \beta ^2 sinhr\sqrt{e^{2r}-1}(n_{a}+1/2)\nonumber \\&+(-\beta ^2+\alpha ^2e^{r})\alpha \sqrt{e^{2r}-1}(n_{b}+1/2)\nonumber \\&-\alpha (-\sqrt{e^{2r}-1}+2\gamma sinhr)(e^r-\gamma \sqrt{e^{2r}-1})(n_{m}+1/2), \end{aligned}$$

(A.15)

$$\begin{aligned} V_{45}= & {} V_{54}=-V_{36}, \end{aligned}$$

(A.16)

$$\begin{aligned} V_{46}= & {} -V_{64}=V_{35} \end{aligned}$$

(A.17)

and

$$\begin{aligned} V_{56}=-V_{65}=(-\beta ^2(e^{2r}-1)+\alpha ^2(e^{2r}-1) +(e^r-\gamma \sqrt{e^{2r}-1})^2)i/2. \end{aligned}$$

(A.18)