## Finite-Difference Time-Domain Analysis of Photonic Crystal Slab Cavities with Two-Level Systems |

Optics Express, Vol. 19, Issue 23, pp. 23067-23077 (2011)

http://dx.doi.org/10.1364/OE.19.023067

Acrobat PDF (1517 KB)

### Abstract

In this paper, we report the numerical simulation of an atom-cavity interaction within photonic crystal nano-cavities. The numerical model is based on a damping oscillator description of a dipole current and it is implemented with a finite-difference time-domain method. Using the method, we successfully simulate the atom-cavity mode field interactions of a two-level system embedded in a photonic crystal cavity under several coupling strength conditions. We show that enhancement and suppression of optical emission rate from a two-level system are also shown by this model.

© 2011 OSA

## 1. Introduction

*Q*) factor of ultrasmall optical resonators, such as photonic crystal (PhC) cavities [1

1. Y. Akahane, T. Asano, B. Song, and S. Noda, “High-*Q* photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003). [CrossRef]

2. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. **4**, 207–210 (2005). [CrossRef]

3. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, and T. Tanabe, “Ultrahigh-*Q* photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006). [CrossRef]

4. E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. Roh, and M. Notomi, “Ultrahigh-*Q* one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO_{2} claddings and on air claddings,” Opt. Express **18**, 15859–15869 (2010). [CrossRef] [PubMed]

5. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-*Q* toroid microcavity on a chip,” Nature (London) **421**, 925–928 (2003). [CrossRef]

*Q*factor. This suggests a large enhancement of the optical nonlinear effect and light-matter interaction. If the electric field per photon inside the cavity is sufficiently large, then the single-photon dipole coupling strength

*g*can exceed the decoherence rates in the system owing to cavity losses

*κ*and dipole dephasing

*γ*. This corresponds to the strong-coupling regime of cavity quantum electrodynamics (cQED) [6

6. Kerry J. Vahala, “Optical microcavities,” Nature (London) **424**, 839–846 (2003). [CrossRef]

7. M. Yamaguchi, T. Asano, K. Kojima, and S. Noda, “Quantum electrodynamics of a nanocavity coupled with exciton complexes in a quantum dot,” Phys. Rev. B **80**, 155326 (2009). [CrossRef]

9. T. Tawara, H. Kamada, T. Tanabe, T. Sogawa, H. Okamoto, P. Yao, P. K. Pathak, and S. Hughes, “Cavity-QED assisted *attraction* between a cavity mode and an exciton mode in a planar photonic-crystal cavity,” Opt. Express **18**, 2719–2728 (2010). [CrossRef] [PubMed]

10. J. M. Raimond, M. Brune, and S. Haroche, “, “Manipulating quantum entanglement atoms and photons in a cavity,” Rev. Mod. Phys. **73**, 565–582 (2001). [CrossRef]

11. T. Yoshie, Scherer, J. Hendrickson, G. Khitroya, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) **432**, 200–203 (2004). [CrossRef]

12. G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nature Physics **2**, 81–90 (2006). [CrossRef]

13. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature (London) **445**, 896–899 (2007). [CrossRef]

14. G. S. Agarwal and R. R. Puri, “Exact quantum-electrodynamics results for scattering emission, and absorption from a Rydberg atom in a cavity with arbitrary Q,” Phys. Rev. A **33**, 1757–1764 (1986). [CrossRef] [PubMed]

15. H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A **40**,, 5516–5519 (1989). [CrossRef] [PubMed]

16. F. P. Laussy, E. del Valle, and C. Tejedor, “Strong Coupling of Quantum Dots in Microcavities,” Phys. Rev. Lett. **101**, 083601 (2008). [CrossRef] [PubMed]

18. J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, “Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,” IEEE J. Quant. Electron. **35**, 1168–1175 (1999). [CrossRef]

19. G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, IEEE J. Select. Top. Quantum Electron. **10**, 1052–1062 (2004). [CrossRef]

*E*and they are incorporated in an FDTD method. The equation has the same shape as that of Lorentz dispersive media [17]. We adopt the equation to express the dynamics of a two-level system and the interaction with light at a single grid point. The parameters of the differenccial equation are different with those used in cQED theory. When we use it for the analysis of cQED, the meaning of the parameters is not clear. We will make it clear through the simulation. To test the validity of the model, we study the light emission from an initially excited atom in a PhC cavity. The result shows the Rabi splitting of the resonant frequency and the time-dependent energy exchange between the two-level system and the cavity mode. The qualitative characteristics of the spectrum agree with both the experiments and the theoretical prediction. The enhancement and suppression of the radiation rate dependent on the relation of

*κ*and

*γ*are also successfully described using this model.

## 2. Numerical method

*ω*, and the strength of the atom-photon coupling

_{p}*g*appearing in the Jaynes-Cummings Hamiltonian [20], The above equation describes both the emission and absorption of an electromagnetic field through the transition between the fundamental and excited states of a two-level system. In a semiclassical approximation, this model can be simply expressed by an electromagnetic field interacting with an oscillating dipole current. The equation of the motion for a dipole current is as follows where

*J*(

*t*) is the time dependent polarization current,

*δ*is its damping factor,

*ω*is the polarization frequency, and Δ

_{p}*ɛ*is the change in relative permittivity [17]. This semiclassical model correctly describes vacuum Rabi oscillation[21]. The differential equation for a dipole current, Eq. (2), is mathematically equivalent to that of a Lorentz dispersive medium, where dispersive material is assumed to occupy a certain finite volume. We adopt this equation as a model of a single dipole moment to simulate a two-level atom embedded in an optical cavity. In the equation, the damping factor

*δ*corresponds to parameter

*γ*in Fig. 1. The radiative reaction force[22] and pure dephasing effect of a two-level system is phenomenologically included in the equation. As we can recognize from the equation, Δ

*ɛ*expresses the strength of acceleration and deceleration of a dipole current caused by the electric field. When employing this equation, it is necessary to clarify the relation between Δ

*ɛ*and the atom-photon coupling strength

*g*.

*E*

^{n}^{+1}is the correction of the electric field required because of the emission and absorption of an electromagnetic field by a two-level system. It is explicitly written as where

*C*

_{1}and

*C*

_{2}are defined as

## 3. Results

*t*= 210 nm, the lattice constant is

*a*= 420 nm, the radius of the air hole is

*r*= 115.5 nm, and the refractive index of the slab material is assumed to be 3.475. The cavity is a three-hole missing cavity (L3). Prior to simulating the PhC cavity with a two-level system, we calculate the fundamental cavity mode profile of the PhC cavity without a dipole. Figure 2 (b) shows the field profile of the

*H*component of the fundamental mode. Because it is a TE-like mode, the electric field is symmetrical in terms of vertical direction and has its maximum horizontal amplitude in the middle of the cavity. We consider two cavity structures with the same parameters but with different numbers of rows of air holes surrounding the cavity area. One (high-

_{y}*Q*) cavity has a

*Q*factor of 8.3×10

^{4}(

*κ*= 14GHz) and the other (low-

*Q*) has a

*Q*factor of 1.1 × 10

^{4}(

*κ*= 109GHz). The effective mode volume of the PhC cavity is

*V*= 0.071

_{eff}*μm*

^{3}.

*J*(0) = (

*J*, 0, 0), and we assume that there is initially no electromagnetic field in the system.

_{x}*δ*(=

*γ*) = 3.0 × 10

^{2}GHz and Δ

*ɛ*= 0.5 is assumed. Figure 3 is the calculated coupling strength

*g*for a low-

*Q*PhC cavity. Theoretical values obtained with Eq. (6) are also shown in the figure. As seen in the figure, the results agree well, and the modeling of a two-level system using the present formulation is physically consistent. As a result, the validity of Eq. (6) can be reasonably inferred.

*Q*cavity, we calculated the radiation process from a two-level system for several polarization frequencies

*ω*with Δ

_{p}*ɛ*= 0.5, which corresponds to

*g*= 3.5THz according to Eq. (6). The assumed

*g*value is very large and unrealistic, although we adopt it to reduce the calculation time. The spectra of the emitted electric field are obtained by Fourier transforming the electromagnetic field in the cavity. The calculated results are shown in Fig. 4 as a function of a polarization frequency. The result shows the characteristics of the anti-crossing behavior, which are typical characteristics of strong coupling [11

11. T. Yoshie, Scherer, J. Hendrickson, G. Khitroya, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) **432**, 200–203 (2004). [CrossRef]

*J*

^{2}(

*t*) in a cavity divided by the damping rate in free space. We study the emission from a dipole located in a low-

*Q*cavity. The enhancement ratio as a function of Δ

*ɛ*is shown in Fig. 6. The calculation is performed with a damping factor of

*δ*= 0GHz to exclude the effects of an unknown pure dephasing term and

*ω*with a resonant condition. In this calculation, the damping term of Eq. (2) is omitted. Therefore the damping of the dipole current is caused only by the interaction with the electromagnetic field. The emission rate is greatly enhanced for a small Δ

_{p}*ɛ*.

*δ*= 3.0GHz and (a) Δ

*ɛ*= 0.005(

*g*= 356GHz) and (b) Δ

*ɛ*= 0.0005(

*g*= 113GHz). In both cases, the enhancement ratio has resonant characteristics under a zero detuning frequency condition. However, the resonance of (b) is narrower than that of (a), where their widths are approximately proportional to 2

*g*. The enhancement ratio for a strong coupling condition is smaller than that with weak coupling. This is because, with a strong coupling regime, pure emission is enhanced by coupling with the cavity mode but the emitted photons will be re-absorbed by the two-level system. This process will partially cancel out the emission enhancement.

*Q*cavity. As experimentally demonstrated by several researchers [1

1. Y. Akahane, T. Asano, B. Song, and S. Noda, “High-*Q* photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003). [CrossRef]

3. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, and T. Tanabe, “Ultrahigh-*Q* photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006). [CrossRef]

4. E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. Roh, and M. Notomi, “Ultrahigh-*Q* one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO_{2} claddings and on air claddings,” Opt. Express **18**, 15859–15869 (2010). [CrossRef] [PubMed]

*Q*factors. Considering this finding, the fact that the radiation from an optical cavity is much slower than the pure dephasing of a two-level system is an interesting and realistic case. Figure 8 shows the inverse damping time of the dipole current and cavity field. For this study, we consider a high-

*Q*PhC cavity. The damping term is

*δ*= 3.0 × 10

^{2}GHz, and the coupling strength is Δ

*ɛ*= 0.5(

*g*= 3.56THz). For large frequency detuning, a two-level system decays quickly mainly as a result of damping term

*γ*, while for a resonant condition, a two-level system couples with the cavity mode. This means that a two-level system emits the cavity-mode field before damping. The decay rate of the cavity-mode field is determined by the leakage from the cavity and also by coupling with the two-level system. In this case, the cavity’s leakage rate is slow (

*κ*= 14GHz) and the field is re-absorbed by the two-level system during its stay in the cavity before leaking out. This means that the energy is stored as the cavity mode field for a long time with this strong coupling. This alternating emission and re-absorption phenomenon leads to the suppression of the decay rate of a two-level system under resonant conditions.

## References and links

1. | Y. Akahane, T. Asano, B. Song, and S. Noda, “High- |

2. | B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. |

3. | E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, and T. Tanabe, “Ultrahigh- |

4. | E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. Roh, and M. Notomi, “Ultrahigh- |

5. | D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high- |

6. | Kerry J. Vahala, “Optical microcavities,” Nature (London) |

7. | M. Yamaguchi, T. Asano, K. Kojima, and S. Noda, “Quantum electrodynamics of a nanocavity coupled with exciton complexes in a quantum dot,” Phys. Rev. B |

8. | Y. Ota, S. Iwamoto, N. Kumagai, and Y. Arakawa, “Impact of electron-phonon interactions on quantum-dot cavity quantum electrodynamics,” arXive:0908.0788v1 [cond-mat.mes-hall]. |

9. | T. Tawara, H. Kamada, T. Tanabe, T. Sogawa, H. Okamoto, P. Yao, P. K. Pathak, and S. Hughes, “Cavity-QED assisted |

10. | J. M. Raimond, M. Brune, and S. Haroche, “, “Manipulating quantum entanglement atoms and photons in a cavity,” Rev. Mod. Phys. |

11. | T. Yoshie, Scherer, J. Hendrickson, G. Khitroya, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) |

12. | G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nature Physics |

13. | K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature (London) |

14. | G. S. Agarwal and R. R. Puri, “Exact quantum-electrodynamics results for scattering emission, and absorption from a Rydberg atom in a cavity with arbitrary Q,” Phys. Rev. A |

15. | H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A |

16. | F. P. Laussy, E. del Valle, and C. Tejedor, “Strong Coupling of Quantum Dots in Microcavities,” Phys. Rev. Lett. |

17. | A. Taflove and S. C. Hagness, “Computational Electronics: The Finite-Difference Time-Domain Method,” 2nd ed (Artech House, Norwood2000). |

18. | J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, “Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,” IEEE J. Quant. Electron. |

19. | G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, IEEE J. Select. Top. Quantum Electron. |

20. | D. Walls and G. Milburn, “Quantum Optics” (Springer-Verlag, Berlin, 1994). |

21. | J. J. Childs, K. An, R. R. Dasari, and M. S. Feld, “Single Atom Emission in an Optical Resonator,” in |

22. | J. D. Jackson, “Classical Electrodynamics,” 3rd ed, (Wiley, NY1999). |

23. | E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5580) Quantum optics : Quantum electrodynamics

**ToC Category:**

Quantum Optics

**Citation**

Hideaki Taniyama, Hisashi Sumikura, and Masaya Notomi, "Finite-Difference Time-Domain Analysis of Photonic Crystal Slab Cavities with Two-Level Systems," Opt. Express **19**, 23067-23077 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23067

Sort: Year | Journal | Reset

### References

- Y. Akahane, T. Asano, B. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London)425, 944–947 (2003). [CrossRef]
- B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater.4, 207–210 (2005). [CrossRef]
- E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, and T. Tanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006). [CrossRef]
- E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on air claddings,” Opt. Express18, 15859–15869 (2010). [CrossRef] [PubMed]
- D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London)421, 925–928 (2003). [CrossRef]
- Kerry J. Vahala, “Optical microcavities,” Nature (London)424, 839–846 (2003). [CrossRef]
- M. Yamaguchi, T. Asano, K. Kojima, and S. Noda, “Quantum electrodynamics of a nanocavity coupled with exciton complexes in a quantum dot,” Phys. Rev. B80, 155326 (2009). [CrossRef]
- Y. Ota, S. Iwamoto, N. Kumagai, and Y. Arakawa, “Impact of electron-phonon interactions on quantum-dot cavity quantum electrodynamics,” arXive:0908.0788v1 [cond-mat.mes-hall].
- T. Tawara, H. Kamada, T. Tanabe, T. Sogawa, H. Okamoto, P. Yao, P. K. Pathak, and S. Hughes, “Cavity-QED assisted attraction between a cavity mode and an exciton mode in a planar photonic-crystal cavity,” Opt. Express18, 2719–2728 (2010). [CrossRef] [PubMed]
- J. M. Raimond, M. Brune, and S. Haroche, “, “Manipulating quantum entanglement atoms and photons in a cavity,” Rev. Mod. Phys.73, 565–582 (2001). [CrossRef]
- T. Yoshie, Scherer, J. Hendrickson, G. Khitroya, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London)432, 200–203 (2004). [CrossRef]
- G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nature Physics2, 81–90 (2006). [CrossRef]
- K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature (London)445, 896–899 (2007). [CrossRef]
- G. S. Agarwal and R. R. Puri, “Exact quantum-electrodynamics results for scattering emission, and absorption from a Rydberg atom in a cavity with arbitrary Q,” Phys. Rev. A33, 1757–1764 (1986). [CrossRef] [PubMed]
- H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A40,, 5516–5519 (1989). [CrossRef] [PubMed]
- F. P. Laussy, E. del Valle, and C. Tejedor, “Strong Coupling of Quantum Dots in Microcavities,” Phys. Rev. Lett.101, 083601 (2008). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, “Computational Electronics: The Finite-Difference Time-Domain Method,” 2nd ed (Artech House, Norwood2000).
- J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, “Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,” IEEE J. Quant. Electron.35, 1168–1175 (1999). [CrossRef]
- G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, IEEE J. Select. Top. Quantum Electron.10, 1052–1062 (2004). [CrossRef]
- D. Walls and G. Milburn, “Quantum Optics” (Springer-Verlag, Berlin, 1994).
- J. J. Childs, K. An, R. R. Dasari, and M. S. Feld, “Single Atom Emission in an Optical Resonator,” in Cavity Quantum Electrodynamics, P. R. Berman, Editor, Academic Press, San Diego (1994).
- J. D. Jackson, “Classical Electrodynamics,” 3rd ed, (Wiley, NY1999).
- E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev.69, 681 (1946).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.