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Atom-Atom Entanglement in a Hybrid Fiber-Atom-Optomechanical System

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Abstract

In this paper, we consider a hybrid fiber-atom-optomechanical system consisting of two optomechanical cavities where they are connected together with an optical fiber. Moreover, each cavity interacts separately with a two-level atom. Obtaining the effective Hamiltonian of the whole system, we find the state vector of the system numerically. Afterward, we study the degree of entanglement (DEM) between two atoms with the benefit of concurrence. We show that the DEM between the atoms can be appropriately controlled by adopting cavity-fiber and optomechanical couplings. Furthermore, we observe from the numerical results that the so-called phenomena of entanglement sudden death and birth (ESD and ESB) happen in the considered system.

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Authors and Affiliations

  1. Department of Physics, Faculty of Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

    Mohammad Ali Fathi, Hamid Reza Baghshahi, Mohammad Khanzadeh & Sayyed Yahya Mirafzali

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  1. Mohammad Ali Fathi
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  2. Hamid Reza Baghshahi
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  3. Mohammad Khanzadeh
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  4. Sayyed Yahya Mirafzali
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Correspondence to Hamid Reza Baghshahi.

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Appendix

Appendix

Here, we want to demonstrate how the effective Hamiltonian of the whole system, which is introduced in (7), can be obtained. To this end, first suppose that an interaction Hamiltonian could be written as follows

$$ \begin{array}{@{}rcl@{}} \hat{H}_{\mathrm{I}}(t) = \sum\limits_{{ n}=0}^{{ N}} \hat{h}_{n} { e}^{-i \omega^{\prime}_{n} t} +\hat{h}^{\dagger}_{n} \mathrm{e}^{i \omega^{\prime}_{n} t}, \end{array} $$
(15)

where N is the total number of different harmonic terms,which make up the interaction Hamiltonian, with oscillating frequency ωn > 0. So the effective Hamiltonian reduces to:

$$ \begin{array}{@{}rcl@{}} \hat{H}_{\text{eff}}(t) = \sum\limits_{{n,m}=0}^{N} \frac{1}{\hbar \bar{\omega}^{\prime}_{mn}} [\hat{h}^{\dagger}_{m},\hat{h}_{n}] {e}^{i(\omega^{\prime}_{m}-\omega^{\prime}_{n})t} \end{array} $$
(16)

where \(\bar {\omega }^{\prime }_{mn}\) is the harmonic average of \({\omega }^{\prime }_{m}\) and \({\omega }^{\prime }_{n}\), defines as \(\frac {1}{\bar {\omega }^{\prime }_{mn}} = \frac {1}{2}(\frac {1}{{\omega }^{\prime }_{m}}+\frac {1}{{\omega }^{\prime }_{n}})\) [41].

Now, we use the above equations and calculate the effective Hamiltonian for our considered model. In our system, the interaction Hamiltonian reads as (\(\hbar =1\)):

$$ \begin{array}{@{}rcl@{}} \hat{H}_{\mathrm{I}}&=& \sum\limits_{{j}=1}^{2} \lambda_{j} (\hat{a}_{j} \hat{\sigma}_{j+} {e}^{-i ({\varOmega}_{j} - \omega_{j}) t} + \hat{a}^{\dagger}_{j} \hat{\sigma}_{j-} {e}^{i ({\varOmega}_{j} - \omega_{j} ) t} ) \\ &+& \sum\limits_{{j}=1}^{2} {J}(\hat{a}_{j}\hat{b}^{\dagger} {e}^{-i ({\varOmega}_{j} - {\varOmega}_{f}) t}+\hat{a}^{\dagger}_{j}\hat{b} {e}^{i ({\varOmega}_{j} - {\varOmega}_{f}) t}) \\ &-& \sum\limits_{{j}=1}^{2} {G}_{j}\hat{a}^{\dagger}_{j} \hat{a}_{j} (\hat{c}_{j} { e}^{-i\omega_{jm} t}+\hat{c}^{\dagger}_{j} {e}^{i \omega_{jm} t}). \end{array} $$
(17)

Comparing (15) with that of (17), one would be able to find the operators hi(i = 1, 2,...6) and frequencies \({\omega }^{\prime }_{i} (i=1,2,...6)\), associated with the Hamiltonian (17), as the following form:

$$ \begin{array}{@{}rcl@{}} \hat{h}_{1} &=& \lambda_{1} \hat{a}_{1} \hat{\sigma}_{1+}, \quad \quad \quad \quad \hat{h}_{2} = \lambda_{2} \hat{a}_{2} \hat{\sigma}_{2+},\\ \hat{h}_{3} &=& J \hat{a}_{1}\hat{b}^{\dagger}, \quad \quad \quad \quad \quad \hat{h}_{4} = J \hat{a}_{2}\hat{b}^{\dagger},\\ \hat{h}_{5} &=& -G_{1} \hat{a}^{\dagger}_{1} \hat{a}_{1} \hat{c}_{1}, \quad \quad \quad \hat{h}_{6} = -G_{2} \hat{a}^{\dagger}_{2} \hat{a}_{2} \hat{c}_{2}, \\ {\omega}^{\prime}_{1} &=& {\varOmega}_{1} - \omega_{1}, \quad \quad \quad \quad {\omega}^{\prime}_{2} = {\varOmega}_{2} - \omega_{2}, \\ \omega_{3}^{\prime} &=& {\varOmega}_{1}-{\varOmega}_{f}, \quad \quad \quad \quad \omega_{4}^{\prime} = {\varOmega}_{2}-{\varOmega}_{f}, \\ \omega_{5}^{\prime}&=& \omega_{5m}, \quad \quad \quad \quad \quad \quad \omega_{6}^{\prime} = \omega_{6m}. \end{array} $$
(18)

Now, using the (16), we get to the following formula:

$$ \begin{array}{@{}rcl@{}} \hat{H}_{\text{eff}} &=& \frac{1}{\bar{\omega}^{\prime}_{11}} [\hat{h}^{\dagger}_{1},\hat{h}_{1}]+\frac{1}{\bar{\omega}^{\prime}_{12}} [\hat{h}^{\dagger}_{1},\hat{h}_{2}]e^{i(\omega_{1}^{\prime} - \omega_{2}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{13}} [\hat{h}^{\dagger}_{1},\hat{h}_{3}]e^{i(\omega_{1}^{\prime} - \omega_{3}^{\prime})t} \\ &+&\frac{1}{\bar{\omega}^{\prime}_{14}} [\hat{h}^{\dagger}_{1},\hat{h}_{4}]e^{i(\omega_{1}^{\prime} - \omega_{4}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{15}} [\hat{h}^{\dagger}_{1},\hat{h}_{5}]e^{i(\omega_{1}^{\prime} - \omega_{5}^{\prime})t} \\ &+&\frac{1}{\bar{\omega}^{\prime}_{16}} [\hat{h}^{\dagger}_{1},\hat{h}_{6}]e^{i(\omega_{1}^{\prime} - \omega_{6}^{\prime})t}+ \frac{1}{\bar{\omega}^{\prime}_{21}} [\hat{h}^{\dagger}_{2},\hat{h}_{1}]e^{i(\omega_{2}^{\prime} - \omega_{1}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{22}} [\hat{h}^{\dagger}_{2},\hat{h}_{2}]\\ &+&\frac{1}{\bar{\omega}^{\prime}_{23}} [\hat{h}^{\dagger}_{2},\hat{h}_{3}]e^{i(\omega_{2}^{\prime} - \omega_{3}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{24}}[\hat{h}^{\dagger}_{2},\hat{h}_{4}]e^{i(\omega_{2}^{\prime} - \omega_{4}^{\prime})t} \\ &+&\frac{1}{\bar{\omega}^{\prime}_{25}} [\hat{h}^{\dagger}_{2},\hat{h}_{5}]e^{i(\omega_{2}^{\prime} - \omega_{5}^{\prime})t}+ \frac{1}{\bar{\omega}^{\prime}_{26}} [\hat{h}^{\dagger}_{2},\hat{h}_{6}]e^{i(\omega_{2}^{\prime} - \omega_{6}^{\prime})t}\\ &+&\frac{1}{\bar{\omega}^{\prime}_{31}} [\hat{h}^{\dagger}_{3},\hat{h}_{1}]e^{i(\omega_{3}^{\prime} - \omega_{1}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{32}} [\hat{h}^{\dagger}_{3},\hat{h}_{2}]e^{i(\omega_{3}^{\prime} - \omega_{2}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{33}} [\hat{h}^{\dagger}_{3},\hat{h}_{3}] \\ &+&\frac{1}{\bar{\omega}^{\prime}_{34}} [\hat{h}^{\dagger}_{3},\hat{h}_{4}]e^{i(\omega_{3}^{\prime} - \omega_{4}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{35}} [\hat{h}^{\dagger}_{3},\hat{h}_{5}]e^{i(\omega_{3}^{\prime} - \omega_{5}^{\prime})t}\\ &+&\frac{1}{\bar{\omega}^{\prime}_{36}} [\hat{h}^{\dagger}_{3},\hat{h}_{6}]e^{i(\omega_{3}^{\prime} - \omega_{6}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{41}} [\hat{h}^{\dagger}_{4},\hat{h}_{1}]e^{i(\omega_{4}^{\prime} - \omega_{1}^{\prime})t}\\ &+&\frac{1}{\bar{\omega}_{42}} [\hat{h}^{\dagger}_{4},\hat{h}_{2}]e^{i(\omega_{4}^{\prime} - \omega_{2}^{\prime})t} +\frac{1}{\bar{\omega}_{43}} [\hat{h}^{\dagger}_{4},\hat{h}_{3}]e^{i(\omega_{4}^{\prime} - \omega_{3}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{44}} [\hat{h}^{\dagger}_{4},\hat{h}_{4}]\\ &+&\frac{1}{\bar{\omega}^{\prime}_{45}} [\hat{h}^{\dagger}_{4},\hat{h}_{5}]e^{i(\omega_{4}^{\prime} - \omega_{5}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{46}} [\hat{h}^{\dagger}_{4},\hat{h}_{6}]e^{i(\omega_{4}^{\prime} - \omega_{6}^{\prime})t}\\ &+&\frac{1}{\bar{\omega}^{\prime}_{51}} [\hat{h}^{\dagger}_{5},\hat{h}_{1}]e^{i(\omega_{5}^{\prime} - \omega_{1}^{\prime})t} +\frac{1}{\bar{\omega}^{\prime}_{52}} [\hat{h}^{\dagger}_{5},\hat{h}_{2}]e^{i(\omega_{5}^{\prime} - \omega_{2}^{\prime})t}\\ &+&\frac{1}{\bar{\omega}^{\prime}_{53}} [\hat{h}^{\dagger}_{5},\hat{h}_{3}]e^{i(\omega_{5}^{\prime} - \omega_{3}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{54}} [\hat{h}^{\dagger}_{5},\hat{h}_{4}]e^{i(\omega_{5}^{\prime} - \omega_{4}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{55}} [\hat{h}^{\dagger}_{5},\hat{h}_{5}] \\ &+&\frac{1}{\bar{\omega}^{\prime}_{56}} [\hat{h}^{\dagger}_{5},\hat{h}_{6}]e^{i(\omega_{5}^{\prime} - \omega_{6}^{\prime})t} + \frac{1}{\bar{\omega}^{\prime}_{61}} [\hat{h}^{\dagger}_{6},\hat{h}_{1}]e^{i(\omega_{6}^{\prime} - \omega_{1}^{\prime})t} \\ &+&\frac{1}{\bar{\omega}^{\prime}_{62}} [\hat{h}^{\dagger}_{6},\hat{h}_{2}]e^{i(\omega_{6}^{\prime} - \omega_{2}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{63}} [\hat{h}^{\dagger}_{6},\hat{h}_{3}]e^{i(\omega_{6}^{\prime} - \omega_{3}^{\prime})t} \\ &+&\frac{1}{\bar{\omega}^{\prime}_{64}} [\hat{h}^{\dagger}_{6},\hat{h}_{4}]e^{i(\omega_{6}^{\prime} - \omega_{4}^{\prime})t}+\frac{1}{\bar{\omega}^{\prime}_{65}} [\hat{h}^{\dagger}_{6},\hat{h}_{5}]e^{i(\omega_{6}^{\prime} - \omega_{5}^{\prime})t} +\frac{1}{\bar{\omega}^{\prime}_{66}} [\hat{h}^{\dagger}_{6},\hat{h}_{6}]. \end{array} $$
(19)

By substituting the defined parameters hi and \({\omega }^{\prime }_{i}\) from (18) into (19) and evaluating the commutators, we arrive at the effective Hamiltonian as follows:

$$ \begin{array}{@{}rcl@{}} \hat{H}_{\text{eff}}&=& \sum\limits_{{j}=1}^{2} \frac{\lambda_{j}{G}_{j}}{2} \left( \frac{1}{{\varOmega}_{j} - \omega_{j}} + \frac{1}{\omega_{jm}}\right) (\hat{a}^{\dagger}_{j} \hat{c}_{j} \hat{\sigma}_{j-} {e}^{i({\varOmega}_{j} - \omega_{j} -\omega_{jm}t )} + H.C) \\ &+& \sum\limits_{{j}=1}^{2} \frac{{J} {G}_{j}}{2} \left( \frac{1}{\omega_{jm}} + \frac{1}{{\varOmega}_{j} - {\varOmega}_{f}}\right) (\hat{a}^{\dagger}_{j} \hat{c}_{j} \hat{b} e^{i({\varOmega}_{j}-{\varOmega}_{f}-\omega_{jm}) t} + H.C ) \end{array} $$
$$ \begin{array}{@{}rcl@{}} &-& \sum\limits_{{j}=1}^{2} \frac{\lambda_{j} {J}}{2}\left( \frac{1}{{\varOmega}_{j} - \omega_{j}} +\frac{1}{{\varOmega}_{j} - {\varOmega}_{f}}\right) (\hat{b}^{\dagger} \hat{\sigma}_{j-} {e}^{-i (\omega_{j} - {\varOmega}_{f}) t} + H.C ) \\ &+& \frac{J^{2}}{2} \left( \frac{1}{{\varOmega}_{1} - \omega_{f}} + \frac{1}{{\varOmega}_{2} - \omega_{f}}\right) (\hat{a}^{\dagger}_{1} \hat{a}_{2} {e}^{i({\varOmega}_{1} - {\varOmega}_{2} )t} + H.C) \\ &-& \sum\limits_{{j}=1}^{2} \frac{{\lambda_{j}^{2}}}{{\varOmega}_{j} - \omega_{j}} (\hat{a}^{\dagger}_{j} \hat{a}_{j} \hat{\sigma}_{jz} + \hat{\sigma}_{j+} \hat{\sigma}_{j-}) \\ &-& \sum\limits_{{j}=1}^{2} \frac{{G}^{2}_{j}}{\omega_{jm}} (\hat{a}^{\dagger}_{j} \hat{a}_{j})^{2}\\ &+& \sum\limits_{{j}=1}^{2} \frac{{J}^{2}}{{\varOmega}_{j} - {\varOmega}_{f}}(\hat{a}^{\dagger}_{j}\hat{a}_{j} - \hat{b}^{\dagger}\hat{b}), \end{array} $$
(20)

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Fathi, M.A., Baghshahi, H.R., Khanzadeh, M. et al. Atom-Atom Entanglement in a Hybrid Fiber-Atom-Optomechanical System. Int J Theor Phys 61, 62 (2022). https://doi.org/10.1007/s10773-022-05056-3

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