Skip to main content

The Correlated Two-Photon Transport in a One-Dimensional Waveguide Coupling to a Hybrid Atom-Optomechanical System

Abstract

We investigate the two-photon transport properties inside one-dimensional waveguide side coupled to an atom-optomechanical system, aiming to control the two-photon transport by using the nonlinearity. By generalizing the scheme of Phys. Rev. A 90, 033832, we show that Kerr nonlinearity induced by the four-level atoms is remarkable and can make the photons antibunching, while the nonlinear interaction of optomechanical coupling participates in both the single photon and the two photon processes so that it can make the two photons exhibiting bunching and antibunching.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. Shen, J.T., Fan, S.: Phys. Rev. Lett. 98, 153003 (2007)

    ADS  Article  Google Scholar 

  2. Shen, J.T., Fan, S.: Phys. Rev. A 76, 062709 (2007)

    ADS  Article  Google Scholar 

  3. Liao, J.Q., Law, C.K.: Phys. Rev. A 82, 053836 (2010)

    ADS  Article  Google Scholar 

  4. Yan, W.B., Fan, Q.B., Zhou, L.: Phys. Rev. A 85, 015803 (2012)

    ADS  Article  Google Scholar 

  5. Yan, W.B., Fan, H.: Phys. Rev. A 90, 053807 (2014)

    ADS  Article  Google Scholar 

  6. Xu, X.W., Li, Y.: Phys. Rev. A 90, 033832 (2014)

    ADS  Article  Google Scholar 

  7. Jia, W.Z., Wang, Z.D.: Phys. Rev. A 88, 063821 (2013)

    ADS  Article  Google Scholar 

  8. Song, J., Xia, Y., Sun, X.-D., Song, H.-S.: Quantum Inf. Comput. 12, 601 (2012)

    Google Scholar 

  9. Zhou, L., Yang, S., Liu, Yu-xi, Sun, C.P., Nori, F.: Phys. Rev. A 80, 062109 (2009)

    ADS  Article  Google Scholar 

  10. Zhang, W.Z., Cheng, J., Liu, J.Y., Zhou, L.: Phys. Rev. A 91, 063836 (2015)

    ADS  Article  Google Scholar 

  11. Liao, J.-Q., Nori, F.: Phys. Rev. A 88, 023853 (2013)

    ADS  Article  Google Scholar 

  12. Buchmann, L.F., Wright, E.M., Meystre, P.: Phys. Rev. A 88, 041801 (2013)

    ADS  Article  Google Scholar 

  13. Zhou, L., Han, Y., Jing, J.T., Zhang, W.P.: Phys. Rev. A 83, 052117 (2011)

    ADS  Article  Google Scholar 

  14. Ma, Y.H., Zhang, X.F.: Appl. Phys. B 112, 485 (2013)

    ADS  Article  Google Scholar 

  15. Mi, X., Bai, J., Song, K.H.: Eur. Phys. J. D 67, 115 (2013)

    ADS  Article  Google Scholar 

  16. Xiao, R.J., Pan, G.X., Zhou, L.: Eur. Phys. J. D 67, 113 (2013)

    ADS  Article  Google Scholar 

  17. Zhang, W.Z., Cheng, J., Zhou, L.: J. Phys. B 48, 015502 (2015)

    ADS  Article  Google Scholar 

  18. Liao, J.Q., Law, C.: Phys. Rev. A 87, 043809 (2013)

    ADS  Article  Google Scholar 

  19. Gong, Z.R., Ian, H., Liu, Y., Sun, C.P., Nori, F.: Phys. Rev. A 80, 065801 (2009)

    ADS  Article  Google Scholar 

  20. Ludwig, M., Safavi-Naeini, A.H., Painter, O., Marquardt, F.: Phys. Rev. Lett 109, 063601 (2012)

    ADS  Article  Google Scholar 

  21. Rabl, P.: Phys. Rev. Lett. 107, 063601 (2011)

    ADS  Article  Google Scholar 

  22. Zhou, L., Cheng, J., Han, Y., Zhang, W.P.: Phys. Rev. A 88, 063854 (2013)

    ADS  Article  Google Scholar 

  23. Imamoḡlu, A., Schmidt, H., Woods, G., Deutsch, M.: Phys. Rev. Lett. 79, 1467 (1997)

    ADS  Article  Google Scholar 

  24. Werner, M.J., Imamoḡlu, A.: Phys. Rev. A 61, 011801 (1999)

    ADS  Article  Google Scholar 

  25. Hartmann, M.J., Plenio, M.B.: Phys. Rev. Lett. 99, 103601 (2007)

    ADS  Article  Google Scholar 

  26. Zhou, L., Liu, Z.J., Yan, W.B., Mu, Q.X.: Chinese Phys. B 20, 074205 (2011)

    ADS  Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the NSF of China under Grant No. 11474044 and No.11505023.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ling Zhou.

Appendix A: Two-Photon Scattering Eigenstate

Appendix A: Two-Photon Scattering Eigenstate

In this Appendix, we provide a step to derive the scattering states for two incident photons in the right-going mode. Decomposing the right-going mode to the even and odd modes by \(a_{R}^{\dag } (x)=[a_{e}^{\dag } (x)+a_{o}^{\dag } (x)]/\sqrt {2}\), the incoming state [(8)] can be written as

$$\begin{array}{@{}rcl@{}} |\psi_{i}\rangle &=&\sum\limits_{n}[\frac{1}{2}\int dx_{1}dx_{2}\phi_{k}(x_{1},x_{2})\frac{1}{\sqrt{2}}a_{e}^{\dag} (x_{1})a_{e}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b} \\ &&+\frac{1}{2}\int dx_{1}dx_{2}\phi_{k}(x_{1},x_{2})\frac{1}{\sqrt{2}} a_{o}^{\dag} (x_{1})a_{o}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b} \\ &&+\frac{1}{\sqrt{2}}\int dx_{1}dx_{2}\phi_{k}(x_{1},x_{2})\frac{1}{\sqrt{2} }a_{e}^{\dag} (x_{1})a_{o}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b}] \end{array} $$
(A.1)

where |ϕ〉 is no photon in the waveguide. The general two-photon scattering state of the system in the spaces with even and odd modes can be written as (10) with

$$\begin{array}{@{}rcl@{}} \psi_{ee} &=& \sum\limits_{n}\int\int dx_{1} dx_{2}\phi_{ee}(x_{1},x_{2},n)\frac{1}{ \sqrt{2}}a^{\dag}_{e}(x_{1})a^{\dag}_{e}(x_{2})|0\rangle_{c}|\phi\rangle|n \rangle_{b}\\ &&+\sum\limits_{n}\int dx\phi_{ae}(x,n)a^{\dag}_{e}(x)a^{\dag}|0\rangle_{c}|\phi\rangle|\widetilde{n} (1)\rangle_{b} \\ &&+\sum\limits_{n}\phi_{aa}\frac{1}{\sqrt{2}} a^{\dag}a^{\dag}|0\rangle_{c}|\phi \rangle|\widetilde{n}(2)\rangle_{b} \end{array} $$
(A.2)
$$\begin{array}{@{}rcl@{}} \phi_{oe} &=& \sum\limits_{n}\int\int dx_{1} dx_{2}\phi_{oe}(x_{1},x_{2},n)a^{\dag}_{o}(x_{1})a^{\dag}_{e}(x_{2})|0 \rangle_{c}|\phi\rangle|n\rangle_{b}\\ && +\sum\limits_{n}\int dx\phi_{oa}(x,n)a^{\dag}_{o}(x)a^{\dag}|0\rangle_{c}|\phi\rangle|\widetilde{n} (1)\rangle_{b} \end{array} $$
(A.3)
$$\begin{array}{@{}rcl@{}} \phi_{eo} &=& \sum\limits_{n}\int\int dx_{1} dx_{2}\phi_{eo}(x_{1},x_{2},n)a^{\dag}_{o}(x_{1})a^{\dag}_{e}(x_{2})|0 \rangle_{c}|\phi\rangle|n\rangle_{b}\\ && +\sum\limits_{n}\int dx\phi_{ao}(x,n)a^{\dag}_{o}(x)a^{\dag}|0\rangle_{c}|\phi\rangle|\widetilde{n} (1)\rangle_{b} \end{array} $$
(A.4)
$$\begin{array}{@{}rcl@{}} \phi_{oo}&=&{\sum}_{n}\int \int dx_{1}dx_{2}\phi_{oo}(x_{1},x_{2},n)\frac{1}{ \sqrt{2}}a_{o}^{\dag} (x_{1})a_{o}^{\dag} (x_{2})|0\rangle_{c}|\phi \rangle |n\rangle_{b} \end{array} $$
(A.5)

where ϕ i j (i,j = e,o,a) is the amplitude of the two photons with one photon in mode i and the other in mode j, subscript e(o) stands for the even (odd) mode and subscript a stands for the cavity mode. In order to satisfy the statistical property of photons, the amplitudes satisfy the relations ϕ e e (x 1, x 2) = ϕ e e (x 2, x 1), ϕ o o (x 1, x 2) = ϕ o o (x 2, x 1),ϕ e o (x 1, x 2) = ϕ o e (x 2, x 1), and ϕ o a (x) = ϕ a o (x). Under the two-photon transport of the frequency E = v g (k 1 + k 2) + n 0 ω m , by solving Schr\(\ddot {o}\)dinger equation H|Ψ〉 = E|Ψ〉, we obtain the amplitudes for the two-photon scattering state

$$\begin{array}{@{}rcl@{}} \phi_{ee}(x_{1},x_{2},n) \!&\,=\,&\!\frac{1}{\sqrt{2}}[\phi_{e,k_{1}}(x_{1})\phi_{e,k_{2}}(x_{2})+\phi_{e,k_{1}}(x_{2})\phi_{e,k_{2}}(x_{1}) \\ &&+\{\theta (x_{2}-x_{1})\theta (x_{1})Be^{iE\frac{x_{c}}{v_{g}}} e^{\{i[E-2(\omega_{eff}-n\omega_{m}-\delta )]-(\kappa -{\Gamma} )\}\frac{x}{ 2v_{g}}}\,+\,(x_{2}\longleftrightarrow x_{1})\}\\ && \end{array} $$
(A.6)
$$\begin{array}{@{}rcl@{}} \phi_{oe}(x_{1},x_{2},n)=\frac{1}{\sqrt{2}}[\phi_{o,k_{1}}(x_{1})\phi _{e,k_{2}}(x_{2})+\phi_{e,k_{1}}(x_{2})\phi_{o,k_{2}}(x_{1})] \end{array} $$
(A.7)
$$\begin{array}{@{}rcl@{}} \phi_{oo}(x_{1},x_{2})=\frac{1}{\sqrt{2}}[\phi_{o,k_{1}}(x_{1})\phi _{o,k_{2}}(x_{2})+\phi_{o,k_{1}}(x_{2})\phi_{o,k_{2}}(x_{1})] \end{array} $$
(A.8)
$$\begin{array}{@{}rcl@{}} \phi_{oa}(x_{i},m)=\rho_{k_{1}}e^{ik_{1}x_{i}}+\rho_{k_{1}}e^{ik_{2}x_{i}} \end{array} $$
(A.9)
$$\begin{array}{@{}rcl@{}} \phi_{ae}(x_{i},m) &=&\theta (-x_{i})(\mu_{k_{1}}e^{ik_{1}x_{i}}+\mu_{k_{2}}e^{ik_{2}x_{i}})+ \\ &&\theta (x_{i})\{\eta_{k_{1}}e^{i[k_{1}+(n_{0}-m)\frac{\omega_{m}}{v_{g}} ]}x_{i}+\eta_{k_{2}}e^{i[k_{2}+(n_{0}-m)\frac{\omega_{m}}{v_{g}} ]})x_{i}+\xi (m)e^{\lambda (m)x_{i}}\}\\ \end{array} $$
(A.10)
$$\begin{array}{@{}rcl@{}} \phi_{aa}(m^{\prime} ) &=&-\sum\limits_{n}U_{n_{1}m_{2}^{\prime} }^{*}\frac{\sqrt{2} \overline{V}\phi_{ae}(0^{-},n)}{2U+2\omega_{eff}-i(\kappa+{\Gamma})+m^{\prime} \omega_{m}-E-4\delta} \end{array} $$
(A.11)

with

$$\begin{array}{@{}rcl@{}} \phi_{e,k_{i}}(x_{j},n) &=& \frac{1}{\sqrt{2\pi}}[\delta_{nn_{0}} \theta(-x_{j})+t_{k_{i}}\theta(x_{j})]e^{i[k_{i}+(n_{0}-n)\frac{\omega_{m}}{ v_{g}}]x_{j}} \end{array} $$
(A.12)
$$\begin{array}{@{}rcl@{}} \phi_{o,k_{i}}(x_{j},n) &=& \frac{\delta_{nn_{0}}}{\sqrt{2\pi}} e^{i[k_{i}+(n_{0}-n)\frac{\omega_{m}}{v_{g}}]x_{j}} \end{array} $$
(A.13)

where

$$\begin{array}{@{}rcl@{}} \mu_{k_{1}}(m) &=& \frac{\overline{V}U_{n_{0}m_{1}}^{*}}{2\pi}[\frac{1}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{2} +(n_{0}-m)\omega_{m}+\delta} ] \end{array} $$
(A.14)
$$\begin{array}{@{}rcl@{}} \mu_{k_{2}}(m) &=& \frac{\overline{V}U_{n_{0}m_{1}}^{*}}{2\pi}[\frac{1}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{1} +(n_{0}-m)\omega_{m}+\delta }] \end{array} $$
(A.15)
$$\begin{array}{@{}rcl@{}} t_{k_{1}}(n) &=& \sum\limits_{m}U_{nm_{1}}U_{n_{0}m_{1}}^{*}[\frac{-\omega_{eff}+i \frac{\kappa-{\Gamma}}{2}+v_{g}k_{2} +(n_{0}-m)\omega_{m}+\delta}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{2} +(n_{0}-m)\omega_{m}+\delta} ] \end{array} $$
(A.16)
$$\begin{array}{@{}rcl@{}} t_{k_{2}}(n) &=& \sum\limits_{m}U_{nm_{1}}U_{n_{0}m_{1}}^{*}[\frac{-\omega_{eff}+i \frac{\kappa-{\Gamma}}{2}+v_{g}k_{1} +(n_{0}-m)\omega_{m}+\delta}{ -\omega_{eff}+i\frac{\kappa+{\Gamma}}{2}+v_{g}k_{1} +(n_{0}-m)\omega_{m}+\delta} ] \end{array} $$
(A.17)
$$\begin{array}{@{}rcl@{}} \eta_{k_{1}}(m^{\prime} ) &=& \sum\limits_{n}U_{nm_{1}^{\prime} }^{*}\frac{ \overline{V}}{2\pi}\frac{t_{k_{1}}} {-\omega_{eff}+i\frac{\kappa-{\Gamma}}{2} +v_{g}k_{2}+(n-m^{\prime} )\omega_{m}+\delta} \end{array} $$
(A.18)
$$\begin{array}{@{}rcl@{}} \eta_{k_{2}}(m^{\prime} ) &=& \sum\limits_{n}U_{nm_{1}^{\prime} }^{*}\frac{ \overline{V}}{2\pi}\frac{t_{k_{2}}} {-\omega_{eff}+i\frac{\kappa-{\Gamma}}{2} +v_{g}k_{1}+(n-m^{\prime} )\omega_{m}+\delta} \end{array} $$
(A.19)
$$\begin{array}{@{}rcl@{}} \xi(n) &=& \sum\limits_{m}U_{nm_{1}}\frac{\sqrt{2}\overline{V}}{iv_{g}} \phi_{aa}(m)+\mu_{k_{1}}(n)+\mu_{k_{2}}(n)-\eta_{k_{1}}(n)-\eta_{k_{2}}(n) \end{array} $$
(A.20)
$$\begin{array}{@{}rcl@{}} B(n) &=& \sum\limits_{m^{\prime} }U_{nm_{1}^{\prime} }\frac{\overline{V}}{\sqrt{2} iv_{g}}\xi \end{array} $$
(A.21)
$$\begin{array}{@{}rcl@{}} \lambda(m^{\prime} ) &=& \frac{1}{iv_{g}}(\omega_{eff}-i\frac{\kappa+{\Gamma}}{ 2}+m^{\prime} \omega_{m}-E-\delta) \end{array} $$
(A.22)

and \(\rho _{k_{i}}=\mu _{k_{i}}/\sqrt {2}\)

$$\begin{array}{@{}rcl@{}} U_{nm_{1}} &=& \sum\limits_{l}e^{\frac{1}{2}(\frac{g_{0}}{\omega_{m}})^{2}}(-\frac{ g_{0}}{\omega_{m}})^{l+m-n}(-\frac{g_{0}}{\omega_{m}})^{l} \frac{(l+m)!}{ (l+m-n)!l!\sqrt{n!m!}} \end{array} $$
(A.23)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Zhang, W., Li, X. et al. The Correlated Two-Photon Transport in a One-Dimensional Waveguide Coupling to a Hybrid Atom-Optomechanical System. Int J Theor Phys 55, 4620–4630 (2016). https://doi.org/10.1007/s10773-016-3083-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-016-3083-y

Keywords

  • Hybrid cavity optomechanical system
  • Control of two-photon transport
  • Bunching
  • Antibunching