Geometric discord in a dissipative double-cavity optomechanical system

Abstract

In this paper, we propose a theoretical scheme to study the dynamics of the geometric measure of quantum discord (GMQD) between two non-interacting qubits in a dissipative optomechanical system composed of two Fabry–Pérot cavities. In this system, each cavity contains a single-mode quantized radiation field which, in the rotating wave approximation, interacts with both mechanical resonator and two-level atom. In addition, the effects of dissipation are taken into account by considering cavity decay, losses of cavity mirror, and spontaneous emission from the atom. We start with the master equation approach and under some circumstances, we find a non-Hermitian Hamiltonian. Adopting this procedure yields the problem with an acceptable analytical solution. This means that all enough information in the study of the considered open quantum system is exactly obtained. Thereupon, we study the quantum correlations between the atoms with the help of the GMQD. It can be seen that the GMQD between two atoms can be controlled by the optomechanical coupling coefficient, the atom-field coupling strength and the dissipation parameters. Moreover, decreasing the coefficient of optomechanical coupling as well as reducing the strength of atom-field coupling improves the GMQD between two atoms. It is also mentioned that, taking the dissipation effects into account, we see that, as the time proceeds, the GMQD approaches to a stable value. In addition, eliminating the effect of spontaneous emission leads to a diminution in the amount of GMQD. Consequently, the quantum correlations between two atoms can be enhanced by considering the effect of spontaneous emission.

Proof of Eq. (3)

Here, we want to demonstrate how the effective Hamiltonian of the whole system, which is introduced in Eq. (3), can be obtained. To this end, first suppose that an interaction Hamiltonian could be written as follows

$$\begin{aligned} \hat{\mathscr {H}}_{\mathrm{I}}(t) = \sum _{\mathrm{n}=0}^{\mathrm{N}} \hat{h}_{n} \mathrm{e}^{-i \omega _{n} t} +\hat{h}^\dagger _{n} {e}^{i \omega _{n} t}, \end{aligned}$$

(17)

where N is the total number of different harmonic terms, which make up the interaction Hamiltonian, with oscillating frequency $\omega _{n} > 0$. Then, the effective Hamiltonian reduces to

$$\begin{aligned} \hat{{{\mathbb {H}}}}_{\mathrm{eff}}(t) = \sum _{{m,n}=0}^{N} \frac{1}{\hbar \bar{\omega }_{ mn}} [\hat{h}^\dagger _{m},\hat{h}_{n}] {e}^{i(\omega _{m}-\omega _{n})t}, \end{aligned}$$

(18)

in which $\bar{\omega }_{mn}$ is the harmonic average of $\omega _{m}$ and $\omega _{n}$, defined as $\frac{1}{\bar{\omega }_{mn}} = \frac{1}{2}(\frac{1}{\omega _{m}}+\frac{1}{\omega _{n}})$ [80]. Thus, by repeating the mentioned-above procedure for the present system, it is required to write the Hamiltonian in Eq. (1) in the interaction picture as

$$\begin{aligned} \hat{\mathcal {H}}_{I}= & {} \sum _{k=1}^{2} \lambda _{k} \left( \hat{a}_{k} \hat{\sigma }_{+}^{(k)} e^{-i\varDelta _{k}t}+ \hat{a}^\dagger _{k} \hat{\sigma }_{-}^{(k)}e^{i\varDelta _{k}t} \right) \nonumber \\&-\,\sum _{k=1}^{2} G_{k} \hat{a}^\dagger _{k} \hat{a}_{k} \left( \hat{b}_{k}e^{-i\omega _{km}t}+\hat{b}^\dagger _{k}e^{i\omega _{km}t} \right) , \end{aligned}$$

(19)

with $\varDelta _{k}=\varOmega _{k}-\omega _{k}$ being the detuning parameter between the cavity field and the atom. Under the condition $\varDelta _{k}=\omega _{km}$ [28, 31, 61], Eq. (19) gets the form

$$\begin{aligned} \hat{H}_{I}= & {} \sum _{k=1}^{2} \left( \lambda _{k} \hat{a}_{k} \hat{\sigma }_{+}^{(k)} e^{-i\omega _{km}t}+H.c. \right) \nonumber \\&-\,\sum _{k=1}^{2} \left( G_{k} \hat{a}^\dagger _{k} \hat{a}_{k} \hat{b}_{k}e^{-i\omega _{km}t}+H.c. \right) . \end{aligned}$$

(20)

Comparing Eq. (17) with that of (20), one would be able to find the operators $\hat{h}_{n}$, associated with the Hamiltonian (20), as $\hat{h}_{1} = \lambda _{1} \hat{a}_{1} \hat{\sigma }_{+}^{(1)}$, $\hat{h}_{2} = \lambda _{2} \hat{a}_{2} \hat{\sigma }_{+}^{(2)}$, $\hat{h}_{3} = - G_{1} \hat{a}^\dagger _{1} \hat{a}_{1} \hat{b}_{1}$, and $\hat{h}_{4} = - G_{2} \hat{a}^\dagger _{2} \hat{a}_{2} \hat{b}_{2}$ with $\omega _{n} = \omega _{km}, i=1,2,3,4$. Applying Eq. (18) along with doing some manipulations, we arrive at the effective Hamiltonian of the whole system introduced in Eq. (3).