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Fluorescence Spectroscopy as a Tool to Investigate the Interaction and Geometry

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Abstract

This paper studies a system that contains various atoms coupled by dipole–dipole interaction. The fluorescence spectrum and the dressed energy levels are calculated by a new method. It first solves the problem for two different spatial configurations of three coupled two-level emitters. Then, it is shown that the coupling among dressed levels and consequently energies and number of sidebands in the fluorescence spectrum differs in the two configurations. Accordingly, the fluorescence spectrum can serve as a tool to give information about the interaction and configuration of atoms. In addition, to study the fluorescence spectrum, the Bogoliubov-Born-Green-Kirkwood-Yvon approximation is used, and the accuracy of this approximation is analyzed for the different sizes of the system. It is proven that this approximation is suitable for studying the fluorescence spectrum for a weak dipole–dipole interaction, which occurs when the distance between the emitters is comparable to or larger than the optical wavelength.

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References

  1. Cohen-Tannoudji, C.; Reynaud, S.: Modification of resonance Raman scattering in very intense laser fields. J. Phys. B 10, 365 (1977)

    Article  Google Scholar 

  2. Reynaud, S.: La fluorescence de résonance : etude par la méthode de l’atome habillé. Ann. Phys. 8, 315 (1983)

    Article  Google Scholar 

  3. Carmichael, H.J.; Walls, D.F.: A quantum-mechanical master equation treatment of the dynamical Stark effect. J. Phys. B 9, 1199 (1976)

  4. Scully, M.O.; Zubairy, M.S.: Quantum optics. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  5. Cohen-Tannoudji, C.; Dupont-Roc, J.; Grynberg, G.: Atom-photon interactions. Wiley (1998)

    Book  Google Scholar 

  6. Senitzky, I.R.: Sidebands in strong-field resonance fluorescence. Phys. Rev. Lett. 40, 1334 (1978)

    Article  Google Scholar 

  7. Agarwal, G.S.; Saxena, R.; Narducci, L.M.; Feng, D.H.; Gilmore, R.: Analytical solution for the spectrum of resonance fluorescence of a cooperative system of two atoms and the existence of additional sidebands. Phys. Rev. A 21, 257 (1980)

    Article  Google Scholar 

  8. Ben-Aryeh, Y.; Bowden, C.: Resonance fluorescence spectra of two driven two-level atoms. IEEE J. Quantum Electron. 24, 1376 (1988)

    Article  Google Scholar 

  9. Pucci, L.; Roy, A.; Espirito Santo, T.S.; Kaiser, R.; Kastner, M.; Bachelard, R.: Quantum effects in the cooperative scattering of light by atomic clouds. Phys. Rev. A 95, 053625 (2017)

    Article  Google Scholar 

  10. Peng, Z.A.; Yang, G.Q.; Wu, Q.L.; Li, G.X.: Filtered strong quantum correlation of resonance fluorescence from a two-atom radiating system with interatomic coherence. Phys. Rev. A 99, 033819 (2019)

    Article  Google Scholar 

  11. Darsheshdar, E.; Hugbart, M.; Bachelard, R.; Villas-Boas, C.J.: Photon-photon correlations from a pair of strongly coupled two-level emitters, arXiv e-prints, arXiv:2012.03735 [quant-ph], (2020)

  12. Agarwal, G.S.: Springer tracts in modern physics. Springer, Heidelberg (1974)

  13. Das, S.; Agarwal, G.S.; Scully, M.O.: Quantum interferences in cooperative Dicke emission from spatial variation of the laser phase. Phys. Rev. Lett. 101, 153601 (2008)

    Article  Google Scholar 

  14. Gardiner, C.; Zoller, P.: The quantum world of ultra-cold atoms and light book I: foundations of quantum optics. Imperial College Press (2014)

    Book  Google Scholar 

  15. Gisin, N.: Time correlations and Heisenberg picture in the quantum state diffusion model of open systems. J. Mod. Opt. 40, 2313 (1993)

    Article  Google Scholar 

  16. Brun, T.A.; Gisin, N.: Quantum state diffusion and time correlation functions. J. Mod. Opt. 43, 2289 (1996)

    Article  Google Scholar 

  17. Breuer, H.-P.; Kappler, B.; Petruccione, F.: Stochastic wave-function approach to the calculation of multi-time correlation functions of open quantum systems. Phys. Rev. A 56, 2334 (1997)

    Article  Google Scholar 

  18. Compagno, G.; Passante, R.; Persico, F.: Atom-field interactions and dressed atoms. Cambridge University Press (1995)

  19. Pucci, L.; Roy, A.; Kastner, M.: Simulation of quantum spin dynamics by phase space sampling of Bogoliubov-Born-Green-Kirkwood-Yvon trajectories. Phys. Rev. B 93, 174302 (2016)

    Article  Google Scholar 

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Acknowledgements

E. D. benefited from Grant from São Paulo Research Foundation (FAPESP, Grant Number 2018/10813-2).

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Authors and Affiliations

  1. Basic Sciences Group, Golpayegan College of Engineering, Isfahan University of Technology, Golpayegan, 8771767498, Iran

    Seyed Mostafa Moniri

  2. Laser and Plasma Research Institute, Shahid Beheshti University, Tehran, 1983969411, Iran

    Marjan Fani

  3. Departamento de Física, Universidade Federal de São Carlos, P.O. Box 676, São Carlos, São Paulo, 13565-905, Brazil

    Elnaz Darsheshdar

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  1. Seyed Mostafa Moniri
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  2. Marjan Fani
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  3. Elnaz Darsheshdar
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Correspondence to Seyed Mostafa Moniri.

Appendix A: Quantum Regression Theorem in BBGKY Hierarchical Equations

Appendix A: Quantum Regression Theorem in BBGKY Hierarchical Equations

The (A1–A8) hierarchical equations are needed to be solved together with Eq. (24) in order to obtain \(g^{(1)}(t,\tau )\). Using quantum regression theorem into Eqs. (18) and (19), we can obtain

$$\begin{aligned} \frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _j(t+\tau )\rangle&=\left( -i\Delta _0 -\frac{\Gamma }{2}\right) \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _j(t+\tau )\rangle \nonumber \\&\quad - \frac{i\Omega _0}{2} e^{-i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\rangle \nonumber \\&\quad + \frac{\Gamma }{2} \sum _{k\ne j}^N G^*_{jk} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau ) \sigma ^\dagger _{k}(t+\tau ) \rangle , \end{aligned}$$
(A1)
$$\begin{aligned} \frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\rangle&= i\Omega _0 \left[ e^{-i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau )\rangle \nonumber \right. \\&\left. \quad - e^{i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\rangle \right] \nonumber \\&\quad - \Gamma \left[ \langle \sigma ^\dagger _{m}(t)\rangle + \langle \sigma ^\dagger _{m}(t) \sigma ^z_j(t+\tau )\rangle \right] \nonumber \\&\quad - \Gamma \sum _{k\ne j}^N \left[ G_{jk}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau ) \sigma _{k}(t+\tau ) \rangle \nonumber \right. \\&\left. \quad + G^*_{jk}\langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau ) \sigma ^\dagger _{k}(t+\tau )\rangle \right] . \end{aligned}$$
(A2)

Using the same method, following equations can be obtained from Eqs. (20–23)

$$\begin{aligned}&\frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma _{k}(t+\tau ) \rangle \nonumber \\&\quad = \left( i\Delta _0 - \frac{3\Gamma }{2} \right) \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma _{k}(t+\tau ) \rangle \nonumber \\&\qquad - \Gamma \langle \sigma ^\dagger _{m}(t)\sigma _{k}(t+\tau ) \rangle \nonumber \\&\qquad + i\Omega _0 \Big [ e^{-i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau )\sigma _{k}(t+\tau )\rangle \nonumber \\&\qquad - e^{i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\sigma _{k}(t+\tau )\rangle \nonumber \\&\qquad + \frac{1}{2}e^{i\mathbf {k}_0.\mathbf {r}_k} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^z_k(t+\tau )\rangle \Big ] \nonumber \\&\qquad - \Gamma \sum _{l\ne j,k}^N \left[ G_{jl}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau ) \sigma _{k}(t+\tau )\sigma _{l}(t+\tau ) \rangle \nonumber \right. \\&\left. \qquad + G^*_{jl}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{l}(t+\tau ) \sigma _{k}(t+\tau ) \sigma _{j}(t+\tau )\rangle \right] \nonumber \\&\qquad + \frac{\Gamma }{2} \sum _{l\ne j,k}^N G_{lk}\langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau ) \sigma ^z_j(t+\tau )\sigma _{l}(t+\tau ) \rangle \nonumber \\&\qquad - 2\Gamma \Gamma _{jk} \langle \sigma ^\dagger _m(t) \sigma _j(t+\tau )\sigma ^z_k(t+\tau )\rangle \nonumber \\&\qquad - \Gamma G^*_{jk}\langle \sigma ^\dagger _m(t)\sigma _j(t+\tau )\rangle , \end{aligned}$$
(A3)
$$\begin{aligned}&\frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^\dagger _{k}(t+\tau ) \rangle \nonumber \\&\quad = \left( -i\Delta _0 - \frac{3\Gamma }{2} \right) \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^\dagger _{k}(t+\tau ) \rangle \nonumber \\&\quad \quad - \Gamma \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{k}(t+\tau ) \rangle \nonumber \\&\qquad - i\Omega _0 \Big [ e^{i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\sigma ^\dagger _{k}(t+\tau )\rangle \nonumber \\&\qquad - e^{-i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau )\sigma ^\dagger _{k}(t+\tau )\rangle \nonumber \\&\qquad + \frac{1}{2}e^{-i\mathbf {k}_0.\mathbf {r}_k} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^z_k(t+\tau )\rangle \Big ] \nonumber \\&\qquad - \Gamma \sum _{l\ne j,k}^N \left[ G^*_{jl}\langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau ) \sigma ^\dagger _{k}(t+\tau )\sigma ^\dagger _{l}(t+\tau ) \rangle \nonumber \right. \\&\left. \qquad + G_{jl}\langle \sigma ^\dagger _{m}(t)\sigma _{l}(t+\tau ) \sigma ^\dagger _{k}(t+\tau ) \sigma ^\dagger _{j}(t+\tau )\rangle \right] \nonumber \\&\qquad + \frac{\Gamma }{2} \sum _{l\ne j,k}^N G^*_{lk}\langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau ) \sigma ^z_j(t+\tau )\sigma ^\dagger _{l}(t+\tau ) \rangle \nonumber \\&\qquad - 2\Gamma \Gamma _{jk} \langle \sigma ^\dagger _m(t) \sigma ^\dagger _j(t+\tau )\sigma ^z_k(t+\tau )\rangle \nonumber \\&\qquad - \Gamma G^*_{jk}\langle \sigma ^\dagger _m(t)\sigma ^\dagger _j(t+\tau )\rangle , \end{aligned}$$
(A4)
$$\begin{aligned}&\frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\sigma _{k}(t+\tau ) \rangle \nonumber \\&\quad = - \Gamma \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\sigma _{k}(t+\tau ) \rangle \nonumber \\&\qquad - i\frac{\Omega _0}{2} \left[ e^{-i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma _{k}(t+\tau )\rangle \nonumber \right. \\&\left. \qquad - e^{i\mathbf {k}_0.\mathbf {r}_k} \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\sigma ^z_k(t+\tau )\rangle \right] \nonumber \\&\qquad + \frac{\Gamma }{2} \sum _{l\ne j,k}^N \left[ G^*_{jl}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{l}(t+\tau ) \sigma _{k}(t+\tau )\sigma ^z_j(t+\tau ) \rangle \nonumber \right. \\&\left. \qquad + G_{lk}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau ) \sigma _{l}(t+\tau ) \sigma ^z_k(t+\tau )\rangle \right] \nonumber \\&\qquad + \frac{\Gamma }{4} \left[ G_{jk}\langle \sigma ^\dagger _m(t)\sigma ^z_k(t+\tau )\rangle + G^*_{jk}\langle \sigma ^\dagger _m(t)\sigma ^z_j(t+\tau )\rangle \right] \nonumber \\&\qquad + \frac{\Gamma }{2}\Gamma _{jk}\langle \sigma ^\dagger _m(t)\sigma ^z_j(t+\tau )\sigma ^z_k(t+\tau )\rangle , \end{aligned}$$
(A5)
$$\begin{aligned}&\frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau )\sigma _{k}(t+\tau ) \rangle \nonumber \\&\quad = \left( 2i\Delta _0 - \Gamma \right) \langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau )\sigma _{k}(t+\tau ) \rangle \nonumber \\&\qquad + i\frac{\Omega _0}{2} \left[ e^{i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma _{k}(t+\tau )\rangle \nonumber \right. \\&\left. \qquad + e^{i\mathbf {k}_0.\mathbf {r}_k} \langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau )\sigma _{j}(t+\tau )\rangle \right] \nonumber \\&\qquad + \frac{\Gamma }{2} \sum _{l\ne j,k}^N \left[ G_{jl}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau ) \sigma _{k}(t+\tau )\sigma _{l}(t+\tau ) \rangle \nonumber \right. \\&\left. \qquad + G_{kl}\langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau ) \sigma _{j}(t+\tau ) \sigma _{l}(t+\tau )\rangle \right] , \end{aligned}$$
(A6)
$$\begin{aligned}&\frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\sigma ^\dagger _{k}(t+\tau ) \rangle \nonumber \\&\quad = \left( -2i\Delta _0 - \Gamma \right) \langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau )\sigma ^\dagger _{k}(t+\tau ) \rangle \nonumber \\&\qquad - i\frac{\Omega _0}{2} \left[ e^{-i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^\dagger _{k}(t+\tau )\rangle \nonumber \right. \\&\left. \qquad + e^{-i\mathbf {k}_0.\mathbf {r}_k} \langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau )\sigma ^\dagger _{j}(t+\tau )\rangle \right] \nonumber \\&\qquad + \frac{\Gamma }{2} \sum _{l\ne j,k}^N \left[ G^*_{jl}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau ) \sigma ^\dagger _{k}(t+\tau )\sigma ^\dagger _{l}(t+\tau ) \rangle \nonumber \right. \\&\left. \qquad + G^*_{kl}\langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau ) \sigma ^\dagger _{j}(t+\tau ) \sigma ^\dagger _{l}(t+\tau )\rangle \right] , \end{aligned}$$
(A7)
$$\begin{aligned}&\frac{d}{dt}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^z_k(t+\tau ) \rangle \nonumber \\&\quad = - \Gamma \left[ \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau ) + \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau ) \nonumber \right. \\&\left. \qquad + 2 \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^z_k(t+\tau )\rangle \right] \nonumber \\&\qquad + i{\Omega _0} \Big [ e^{-i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau )\sigma _{j}(t+\tau )\rangle \nonumber \\&\qquad + e^{-i\mathbf {k}_0.\mathbf {r}_k} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma _{k}(t+\tau )\rangle \nonumber \\&\qquad - e^{i\mathbf {k}_0.\mathbf {r}_j} \langle \sigma ^\dagger _{m}(t)\sigma ^z_k(t+\tau )\sigma ^\dagger _{j}(t+\tau )\rangle \nonumber \\&\qquad - e^{i\mathbf {k}_0.\mathbf {r}_k} \langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau )\sigma ^\dagger _{k}(t+\tau )\rangle \Big ] \nonumber \\&\qquad -{\Gamma } \sum _{l\ne j,k}^N \Big [ G_{jl}\langle \sigma ^\dagger _{m}(t)\sigma ^\dagger _{j}(t+\tau ) \sigma ^z_k(t+\tau )\sigma _{l}(t+\tau ) \rangle \nonumber \\&\qquad + G_{lk}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau ) \sigma ^\dagger _{k}(t+\tau ) \sigma _{l}(t+\tau )\rangle \nonumber \\&\qquad + G^*_{jl}\langle \sigma ^\dagger _{m}(t)\sigma _{j}(t+\tau ) \sigma ^z_k(t+\tau )\sigma ^\dagger _{l}(t+\tau ) \rangle \nonumber \\&\qquad + G^*_{lk}\langle \sigma ^\dagger _{m}(t)\sigma ^z_j(t+\tau ) \sigma _{k}(t+\tau ) \sigma ^\dagger _{l}(t+\tau )\rangle \Big ] \nonumber \\&\qquad + 2\Gamma \Gamma _{jk} \left[ \langle \sigma ^\dagger _m(t)\sigma ^\dagger _{j}(t+\tau )\sigma _{k}(t+\tau )\nonumber \right. \\&\left. \qquad + \sigma ^\dagger _m(t)\sigma _{j}(t+\tau )\sigma ^\dagger _{k}(t+\tau )\rangle \right] . \end{aligned}$$
(A8)

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Moniri, S.M., Fani, M. & Darsheshdar, E. Fluorescence Spectroscopy as a Tool to Investigate the Interaction and Geometry. Arab J Sci Eng 48, 8011–8020 (2023). https://doi.org/10.1007/s13369-022-07386-0

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