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

Hideaki Taniyama; Hisashi Sumikura; Masaya Notomi

doi:10.1364/OE.19.023067

1. Introduction

Optical microcavities are attracting considerable research interest, and great progress has been made on the quality (Q) factor of ultrasmall optical resonators, such as photonic crystal (PhC) cavities [1, 2, 3, 4] and whispering-gallery cavities [5]. In particular, the planar PhC structure is well matched with other device such as photo detectors and optical lasers. Furthermore, photonic crystal cavities are attracting growing interest due to their potential for confining light in extremely small volumes and their high-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, 7, 8, 9], whose principal signature is vacuum-Rabi oscillations. Strong coupling between matter and cavity modes has been shown to create quantum nonlinearity. The cQED regime has been intensively studied in atomic physics for several decades [10]. cQED in the strong coupling regime has proved to be an invaluable tool in investigating and understanding quantum phenomena. One of the principal applications of cQED techniques has been in the field of quantum information.

Our main interest lies in a two-level system embedded in an optical cavity, which is an idealized model of a single exciton that interacts with the cavity mode field. In the past few years, there have been several experimental reports on these systems [11, 12, 13]. There have also been theoretical reports based on a master equation for an idealized structure [14, 15, 16]. However, a numerical simulation using the finite-difference time-domain (FDTD) method is a suitable approach for precisely understanding the physical phenomena in realistic device structures. The FDTD method has become a powerful tool in computational electrodynamics [17]. A spontaneous emission from a two-level system in a cavity is simulated using the FDTD method to estimate the spontaneous emission coupling factor[18]. In this study, they calculated the coupled component fraction of a radiated electromagnetic field from the oscillating dipole current, but their simulation ignored the re-absorption of the electromagnetic field that had been emitted by the two-level system. When precisely modeling the optical emission process in a strong coupling regime, the re-absorption of the electromagnetic field by two-level system is important and cannot be ignored. FDTD simulations of the nonlinear active material in optical cavities were studied based on the Maxwell-Bloch equations [19]. Homogeneously broadened active material was assumed in the study. On the other hand, we will concentrate on a point dipole embedded in an optical cavity, which is a model of single atom interacting with the electromagnetic field. It is more appropriate as the model of cQED.

In this study, we report the simple numerical modeling of a light-atom (two-level system) interaction. Light-atom interactions are modeled using the harmonic polarization response to a local electric field 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

cQED is used to study the interaction between atoms and the quantized electromagnetic modes inside a cavity, which is shown schematically in Fig. 1. The key parameters describing a cQED system are the cavity resonance frequency, the atomic transition frequency ω_p, and the strength of the atom-photon coupling g appearing in the Jaynes-Cummings Hamiltonian [20],

H = \bar{h} ω_{p} (a^{†} a + \frac{1}{2}) + \frac{\bar{h} Ω}{2} σ^{z} + \bar{h} g (a^{†} σ^{-} + σ^{+} a) + H_{κ} + H_{r} .

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

\frac{d^{2}}{{dt}^{2}} J (t) + 2 δ \frac{d}{dt} J (t) + ω_{p}^{2} J (t) = ɛ_{0} Δ ɛ ω_{p}^{2} E (t),

where J(t) is the time dependent polarization current, δ is its damping factor, ω_p is the polarization frequency, and Δɛ 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, Δɛ 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 Δɛ and the atom-photon coupling strength g.

Fig. 1 Standard representation of a cavity quantum electrodynamic system comprising a single mode of the electromagnetic field in a cavity with decay rate κ coupled with a coupling strength g to a two-level system with spontaneous decay rate γ. Schematic representing a two-level system in an optical cavity. The QD mode is coupled to a cavity mode with coupling strength g. The excited state of the dot decays incoherently with the decay rate γ.

Download Full Size | PDF

The amount of splitting in a spectrum under a resonant condition is closely related to the strength of coupling g between a two-level system and a cavity field. Although Eq. (2) does not include g explicitly, it can be expressed by the parameters used in Eq. (2) and values that characterize the optical cavity. The parameters used in Eq. (2) are associated with optical susceptibility as follows [17]

χ (ω) = \frac{Δ ɛ_{p} ω_{p}^{2}}{ω_{p}^{2} + 2 i ω δ - ω^{2}} .

On the other hand, optical susceptibility can also be expressed as

χ (ω) = \frac{2 N ɛ}{ω_{p}} \frac{V_{eff}}{V} \frac{g^{2}}{ω_{p} - ω + i δ},

where V_eff is the effective mode volume of the cavity

V_{eff} = \frac{\int ɛ (r) {| E (r) |}^{2} d^{3} r}{max [ɛ (r) {| E (r) |}^{2}]},

and V is the volume of a dipole. We assume a point dipole in this study and V corresponds to the volume of the Yee cell in FDTD. Because these two expressions are equivalent, we can deduce the following expression for the coupling strength g:

2 g = ω_{p} \sqrt{\frac{Δ ɛ_{p} V}{ɛ V_{eff}}} .

This equation must be verified through calculations.

The motion of electromagnetic fields is described by Maxwell’s equation with dipole current term,

\nabla \times H (t) = ɛ \frac{d}{dt} E (t) + J (t)

\nabla \times E (t) = - μ \frac{d}{dt} H (t)

The numerical technique adopted to solve this equation of motion in the FDTD method is similar to that of the auxiliary differential equation method [17]. The finite difference expression of the dipole current is

J^{n + 1} = α J^{n} + ξ J^{n - 1} + η [\frac{E^{n + 1} - E^{n - 1}}{2 Δ t}],

where

α = \frac{2 - ω_{p}^{2} Δ t^{2}}{1 + δ Δ t}, ξ = \frac{δ Δ t - 1}{δ Δ t + 1}, η = \frac{ɛ_{0} Δ ɛ ω_{p}^{2} Δ t^{2}}{1 + δ Δ t} .

Using Eq. (7), the time evolution of an electric field is calculated by

E^{n + 1} = E_{0}^{n + 1} + Δ E^{n + 1},

where

E_{0}^{n + 1}

is an unperturbed electric field and ΔEⁿ⁺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

Δ E^{n + 1} = - C_{1} (E_{0}^{n + 1} - E^{n + 1}) - \frac{1}{2} C_{2} \frac{Δ t}{ɛ_{0} ɛ_{\infty}} [(1 + α) J^{n} + ξ J^{n - 1}] .

where C₁ and C₂ are defined as

C_{1} = \frac{\frac{1}{2} η}{2 ɛ_{0} ɛ_{\infty} + \frac{1}{2} η}, C_{2} = \frac{2 ɛ_{0} ɛ_{\infty}}{2 ɛ_{0} ɛ_{\infty} + \frac{1}{2} η} .

3. Results

Figure 2 (a) shows the silicon PhC slab cavity structure considered in the simulation. The thickness of the photonic crystal slab is 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_y 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-Q) cavity has a Q factor of 8.3×10⁴ (κ = 14GHz) and the other (low-Q) has a Q factor of 1.1 × 10⁴ (κ = 109GHz). The effective mode volume of the PhC cavity is V_eff = 0.071μm³.

Fig. 2 (a) Silicon photonic crystal cavity structure with three missing holes (L3) and (b) H_y field profile of fundamental mode. The two-level system is located in the middle of the cavity, where the horizontal component of the electric field of the fundamental mode has its maximum values.

Download Full Size | PDF

As the first step in the calculation, we need to check the validity of Eq. (6). We simulate the radiation process from a two-level system embedded in a PhC cavity and calculate the spectrum of the electromagnetic field in the cavity. The two-level atom is located exactly on a single grid point in the center of the optical cavity. As an initial condition of the calculation, we set the dipole current at J(0) = (J_x, 0, 0), and we assume that there is initially no electromagnetic field in the system. δ(= γ) = 3.0 × 10²GHz and Δɛ = 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.

Fig. 3 Calculated g-factor as a function of Δɛ. Theoretical results are obtained using Eq. (6).

Download Full Size | PDF

The coupling between a cavity mode and a two-level system can be clearly shown by the spectra as a function of detuning the eigen frequency. For a high-Q cavity, we calculated the radiation process from a two-level system for several polarization frequencies ω_p with Δɛ = 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]. In this regime, the energy is exchanging back and force between the two-level system and the cavity mode. This dynamic energy exchange is clearly shown by the time-dependent dipole current and the electromagnetic field in the cavity. The time dependent energy exchange is shown in Fig. 5 for several coupling strengths. For a small coupling strength, the amplitude of the dipole current decreases monotonically. The amplitude of the dipole current decreases mainly through the damping term (the second term of Eq. (2)). With a strong coupling strength, the amplitude of the dipole current decreases and exhibits vacuum Rabi oscillation. The dipole current radiates the cavity mode field. Some fraction of it gradually leaks out of the cavity and the rest is re-absorbed by the dipole. The energy exchange between the dipole and the cavity mode is clearly recognized by comparing the amplitude of the dipole current and the field.

Fig. 4 Calculated cavity field spectra for a two-level system located in a photonic crystal nanocavity. The Q-factor is 8.3 × 10⁴ and Δɛ = 0.5(g = 3.56GHz).

Download Full Size | PDF

Fig. 5 Time evolution of J_x and H_y. The two-level system is located in the center of the cavity(Q = 8.3 × 10⁴). The coupling parameter is (a) and (b) Δɛ = 5 × 10⁻⁶(g = 11GHz), (c) and (d) Δɛ = 5 × 10⁻³(g = 360GHz), (e) and (f) Δɛ = 5 × 10⁻²(g = 1.1THz)

Download Full Size | PDF

A two-level system in an excited state makes a transition to a fundamental state by emitting a photon with a certain probability. The probability of this transition is dependent on its environment, and the photon density of states. This is called the Purcell effect [23] and it occurs in relatively weak coupling regimes. To check the applicability of the FDTD method to a weak coupling regime, we simulate the time evolution of a two-level system initially set in an excited state. As an estimation of the change in emission rate, we define the enhancement ratio as the damping rate of J²(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 Δɛ 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_p 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 Δɛ.

Fig. 6 Enhancement ratio of radiation from a two-level system is calculated as the dacay ratio versus that in a vacuum. The Q-factor of the cavity is 1.1 × 10⁴ (low-Q) and the damping factor is δ = 0.0GHz.

Download Full Size | PDF

Figure 7 shows the enhancement ratio as a function of dipole frequency. For this calculation, we assume δ = 3.0GHz and (a) Δɛ = 0.005(g = 356GHz) and (b) Δɛ = 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 2g. 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.

Fig. 7 Enhancement ratio of radiation from a two-level system located in a photonic crystal cavity. The Q-factor is 1.1×10⁴ (κ = 109GHz), the damping factor is δ = 3.0GHz, and the coupling strength are (a) Δɛ = 0.005 (g = 356GHz) and (b) Δɛ = 0.0005 (g = 113GHz).

Download Full Size | PDF

Let us consider another situation with a high-Q cavity. As experimentally demonstrated by several researchers [1, 3, 4], a photonic crystal cavity achieves extremely high-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 × 10²GHz, and the coupling strength is Δɛ = 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.

Fig. 8 Inverse damping time of J²(t) and E². The Q-factor is 8.3 × 10⁴(κ = 14GHz), the damping factor is δ = 3.0×10²GHz, and Δɛ = 0.5(g = 3.56THz). Under the zero detuning condition, the dipole current decay becomes slow and the field mode decay becomes fast and their decay rates coincide.

Download Full Size | PDF

In summary, we have presented a numerical technique that allows us to analyze basic optical phenomena for a two-level system embedded in a photonic crystal cavity. The method is based on a simple numerical technique, namely the FDTD method. However, it reasonably describes the radiation process from a two-level system embedded in an optical cavity. The model is purely classical, although it helps us to better understand the phenomenon.

Finally, we believe that the numerical method proposed here would be useful for a better understanding of the physical phenomena displayed by a two-level system embedded in an optical cavity. We thank Y. Tokura for his encouragement throughout this work. This work was supported by JST-CREST.

References and links

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 SiO₂ 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]

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]

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 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]

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. 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]

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 Cavity Quantum Electrodynamics, P. R. Berman, Editor, Academic Press, San Diego (1994).

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

23. E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev. 69, 681 (1946).

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

Abstract

1. Introduction

2. Numerical method

3. Results

References and links

Cited By

Figures (8)

Equations (13)

Optics Express