Macroscopic Entangled Cat State in Cavity Optomechanics

Abstract

We propose a double-opto-mechanical set-up capable of generating a macroscopic entangled cat state of mechanical oscillators. Bridged by the ultra-strong coupling of light and atom, a modulation of opto-mechanical can amplify the observable displacement of the mechanical oscillators. We calculate the logarithmic negativity and joint Wigner functions to demonstrate the entanglement and quantum coherence effects between two mechanical oscillators.

Appendix

Since the unitary evolution operator U(t) is obtained by the equation of motion i∂U(t) = H(t)U(t), with the 9-Lie algebras S,

$$ \begin{array}{@{}rcl@{}} S&=& \{|-\rangle\langle-|b,|-\rangle\langle-|b,|+\rangle\langle+|b^{+},|-\rangle\langle-|b^{+},\\ &&|+\rangle\langle+|,|-\rangle\langle-|,b,b^{+}, \textit{I}\}, \end{array} $$

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U(t) can be written as

$$ \begin{array}{@{}rcl@{}} U(t)&=&e^{f_{1}|-\rangle\langle-|b}e^{f_{2}|-\rangle\langle-|b}e^{f_{3}|+\rangle\langle+|b^{+}}e^{f_{4}|-\rangle\langle-|b^{+}}\\ &&e^{f_{5}|+\rangle\langle+|}e^{f_{6}|-\rangle\langle-|}e^{f_{7}b}e^{f_{8}b^{+}}e^{f_{9}}, \end{array} $$

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where

$$ \begin{array}{@{}rcl@{}} f_{1}&=& \frac{g\alpha_{+}}{2\delta}(e^{-i\delta t}-1)\\f_{2}&=& \frac{g\alpha_{-}}{2\delta}(e^{-i\delta t}-1)\\ f_{3}&=& -\frac{g\alpha_{+}}{2\delta}(e^{i\delta t}-1)\\ f_{4}&=& -\frac{g\alpha_{-}}{2\delta}(e^{i\delta t}-1)\\ f_{5}&=& i\frac{g^{2}\alpha_{+}\xi t}{4\delta}+\frac{g^{2}\alpha_{+}\xi}{4\delta^{2}}(e^{-i\delta t}-1) \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&+i\frac{g^{2}\alpha_{+}t}{4\delta}(\xi+\alpha_{+})-\frac{g^{2}\alpha_{+}t}{4\delta^{2}}(e^{i\delta t}-1)\\ f_{6}&=& i\frac{g^{2}\alpha_{-}\xi t}{4\delta}+\frac{g^{2}\alpha_{-}\xi}{4\delta^{2}}(e^{-i\delta t}-1)\\ &&+i\frac{g^{2}\alpha_{-}t}{4\delta}(\xi+\alpha_{-})-\frac{g^{2}\alpha_{-}t}{4\delta^{2}}(e^{i\delta t}-1)\\ f_{7}&=& \frac{g\xi}{2\delta}(e^{-i\delta t}-1)\\ f_{8}&=& -\frac{g\xi}{2\delta}(e^{i\delta t}-1)\\ f_{9}&=& 0. \end{array} $$

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