Microwave and optical photons entanglement in a hybrid electro-optomechanical system: effect of a mechanical plasmonic waveguide at high temperatures

Abstract

In this paper, we describe a hybrid electro-optomechanical system consisting of a microwave resonator and an optical cavity which are coupled together by a plasmonic waveguide with a fixed side and another mechanically oscillating one. We show that a considerable entanglement can be obtained between optical and microwave photons at temperatures between $T=10K$ and $T=15K$ through our proposed system. Also, we show that it is possible to suppress effect of the temperature on entanglement between optical and microwave photons by a direct current voltage applied on both sides of the plasmonic waveguide. Furthermore, we show importance of the presence of optomechanical interaction in the system to obtain a significant entanglement between optical and microwave photons. Overall we show our system is an appropriate candidate to perform as a detector in detection of received microwave signals from an object with low reflectivity which is embedded in a bright thermal environment.

Appendix: Details on the Coulomb’s interaction

In this appendix we present details of calculation of the Coulomb’s interaction between the two sides of MOPW and how it is related to the applied DC voltage between them. We consider for each side a corresponding capacitance $C_\mathrm{i}$ and an electric potential $V_\mathrm{i}$ ($i=1$, 2). Therefore, the Coulomb’s interaction between the two sides of MOPW can be written as [22, 59, 60]

$$\begin{aligned} H_\mathrm{C}= & {} \frac{-C_\mathrm{1}V_\mathrm{1}C_\mathrm{2}V_\mathrm{2}}{4\pi \epsilon _\mathrm{0}|d_\mathrm{0}+x|}, \end{aligned}$$

(39)

where $d_0$ is the free distance between the two sides of MOPW and x accounts for the displacement of the mechanically oscillating side. If we consider that x is much smaller than $d_0$, i.e., $d_0\gg x$, Tailor series of $H_\mathrm{C}$ in Eq. 39 gives

$$\begin{aligned} H_\mathrm{C}= & {} \frac{-C_\mathrm{1}V_\mathrm{1}C_\mathrm{2}V_\mathrm{2}}{4\pi \epsilon _\mathrm{0}d_\mathrm{0}}\left[ 1-\frac{x}{d_\mathrm{0}}+(\frac{x}{d_\mathrm{0}})^2+\dots \right] . \end{aligned}$$

(40)

One can rewrite Eq. 40 as the following form

$$\begin{aligned}&H_\mathrm{C}\approx \frac{-C_\mathrm{1}V_\mathrm{1}C_\mathrm{2}V_\mathrm{2}}{4\pi \epsilon _\mathrm{0}d_\mathrm{0}}\left[ \left( \frac{x}{d_\mathrm{0}}-\frac{1}{2}\right) ^2+\frac{3}{4}\right] . \end{aligned}$$

(41)

In this paper we have $x=\sqrt{\frac{\hbar }{m \omega _\mathrm{m}}}Q$. Therefore, after omitting constant terms from Eq. 41, one can obtain

$$\begin{aligned}&H_\mathrm{C}\approx \frac{-\hbar C_\mathrm{1}V_\mathrm{1}C_\mathrm{2}V_\mathrm{2}}{4\pi \epsilon _\mathrm{0}m\omega _\mathrm{m}d_\mathrm{0}^3}\left[ \left( Q-\frac{d_0}{2}\sqrt{\frac{m\omega _\mathrm{m}}{\hbar }}\right) ^2\right] . \end{aligned}$$

(42)

If we consider applied DC potential between the two sides of MOPW as $V_\mathrm{DC}$ and a corresponding capacitance of C (indeed the stationary capacitance between two sides of MOPW), and change the reference of the potential energy of the system, one can obtain the following form of Eq. 42

$$\begin{aligned} H_\mathrm{C}= & {} \hbar \Omega \left( Q-\frac{d_0}{2}\sqrt{\frac{m\omega _\mathrm{m}}{\hbar }}\right) ^2, \end{aligned}$$

(43)

where $\Omega =\frac{C^2V^2}{4\pi \epsilon _\mathrm{0}m\omega _\mathrm{m}d_0^3}$ is defined as the Coulomb’s interaction strength. Without losing generality, we change the zero point of measuring Q (and hence x) and we consider $(Q-\frac{d_0}{2}\sqrt{\frac{m\omega _\mathrm{m}}{\hbar }})\rightarrow Q$. Therefore, from Eq. 43 one can obtain

$$\begin{aligned} H_\mathrm{C}=\hbar \Omega Q^2. \end{aligned}$$

(44)

In this paper, chosen parameters for Coulomb’s interaction are as $C=1$pF and $d_0=1\mu$m.