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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 17201–17213
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Semiclassical analysis of two-level collective population inversion using photonic crystals in three-dimensional systems

Hiroyuki Takeda  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 17201-17213 (2012)
http://dx.doi.org/10.1364/OE.20.017201


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Abstract

I theoretically demonstrate the population inversion of collective two-level atoms using photonic crystals in three-dimensional (3D) systems by self-consistent solution of the semiclassical Maxwell-Bloch equations. In the semiclassical theory, while electrons are quantized to ground and excited states, electromagnetic fields are treated classically. For control of spontaneous emission and steady-state population inversion of two-level atoms driven by an external laser which is generally considered impossible, large contrasts of electromagnetic local densities of states (EM LDOS’s) are necessary. When a large number of two-level atoms are coherently excited (Dicke model), the above properties can be recaptured by the Maxwell-Bloch equations based on the first-principle calculation. In this paper, I focus on the realistic 1D PC’s with finite structures perpendicular to periodic directions in 3D systems. In such structures, there appear pseudo photonic band gaps (PBG’s) in which light leaks into air regions, unlike complete PBG’s. Nevertheless, these pseudo PBG’s provide large contrasts of EM LDOS’s in the vicinity of the upper photonic band edges. I show that the realistic 1D PC’s in 3D systems enable the control of spontaneous emission and population inversion of collective two-level atoms driven by an external laser. This finding facilitates experimental fabrication and realization.

© 2012 OSA

1. Introduction

In photonic crystals (PC’s) composed of periodic dielectric materials [1

1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 582486–2489 (1987). [CrossRef] [PubMed]

,2

2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 582059–2062 (1987). [CrossRef] [PubMed]

], electromagnetic local densities of states (EM LDOS’s) are strongly modulated due to photonic band gaps (PBG’s) in which light in a certain frequency region cannot propagate. In uniform media, EM LDOS’s are constant or monotonically increase with frequencies. However, while EM LDOS’s of PC’s become very small inside PBG’s, they are greatly enhanced near photonic band edges. In spontaneous emission from two-level atoms, electromagnetic waves with the atomic frequency (energy difference of ground and excited states) are emitted. Since spontaneous emission rates are proportional to EM LDOS’s, PC’s enable the control of spontaneous emission. For example, spontaneous emission is greatly inhibited and greatly enhanced (Purcell effect) [3

3. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69681 (1946).

] inside PBG’s and near photonic band edges, respectively.

In the atom-laser interaction, on the other hand, it is generally considered that steady-state population inversion is impossible in two-level atoms, since both spontaneous and stimulated emissions occur in the excitation of two-level atoms by stimulated absorption, and therefore, three-level or four-level atoms are necessary for population inversion [4

4. P. W. Milloni and J. H. Eberly, Laser Physics (Wiley, 2009).

]. However, if it were possible to achieve population inversion in two-level atoms, it would provide new functionality for optical materials. From this scientific interest, possibility of two-level population inversion has been discussed for a long time [5

5. M. Lindberg and C. M. Savage, “Steady-state two-level atomic population inversion via a quantized cavity field,” Phys. Rev. A 385182–5192 (1988). [CrossRef] [PubMed]

8

8. S. Hughes and H. J. Carmichael, “Stationary inversion of a two level system coupled to an off-resonant cavity with strong dissipation,” Phys. Rev. Lett. 107, 193601 (2011). [CrossRef] [PubMed]

]. In fundamental physics, in other words, this is recognized as one of important topics. PC’s are considered as powerful candidates to achieve two-level population inversion driven by an external laser. When inputting electric fields with the laser frequency ωL, populations at ground and excited states oscillate with the Rabi frequency 2Ω. Then, there appear three frequency components ωL and ωL ± 2Ω (Mollow triplet) in radiative electromagnetic waves [9

9. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 1881969–1975 (1969). [CrossRef]

]. When a contrast of EM LDOS’s at ω = ωL ± 2Ω is large, population inversion can be achieved even in two-level atoms [6

6. S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. 781888–1891 (1997). [CrossRef]

, 10

10. M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A 64033801 (2001). [CrossRef]

12

12. S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. 3S103–S120 (2001). [CrossRef]

].

In Ref. [17

17. H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A 83053811 (2011). [CrossRef]

], for numerical simplicity, idealized one-dimensional (1D) and two-dimensional (2D) PC’s with infinite structures perpendicular to periodic directions were considered. In idealized 1D and 2D PC’s, two-level atoms at specific points correspond to the area and line densities, respectively. In idealized 1D and 2D PC’s, for example, two-level atoms are uniformly distributed in the infinite 2D perpendicular plane and 1D perpendicular line, respectively. Therefore, leakage of light perpendicular to periodic directions was neglected, and then, spontaneous emission depends on EM LDOS’s for the propagation in periodic directions. In 3D systems with finite structures in any direction, however, such leakage of light cannot be neglected, since two-level atoms at a specific point correspond to the point density. Although certainly idealized 1D and 2D models are useful for easily illustrating and understanding optical phenomena, they do not always show correct results, because of hypothetical infinite structures.

So far, no one has illustrated the two-level collective population inversion in concrete 3D geometries of PC’s, based on the first-principle calculation. In order to realize the two-level collective population inversion, it is crucial to explore concrete 3D geometries of PC’s, positions of two-level atoms and intensities of input electric fields. Otherwise, this population inversion might be misunderstood as an impractical proposition. In this paper, therefore, I theoretically demonstrate the collective population inversion in the absence of phonon dephasing, using the 3D Maxwell-Bloch theory (based on the first-principle calculation). Intuitively, 3D PC’s such as woodpiles with large complete PBG’s, in which light cannot exist in any direction, are necessary for large contrasts of EM LDOS’s in 3D systems. Actually, however, theoretical and experimental results of transmittance do not agree very well, although frequency regions of PBG’s are very close in both results [20

20. I. Staude, M. Thiel, S. Essig, C. Wolff, K. Busch, G. von Freymann, and M. Wegener, “Fabrication and characterization of silicon woodpile photonic crystals with a complete bandgap at telecom wavelengths,” Opt. Lett. 35, 1094–1096 (2010). [CrossRef] [PubMed]

]. Certainly, experimental techniques have been improved so far. For example, the inhibition of spontaneous emission has been observed in a woodpile [21

21. M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited spontaneous emission of quantum dots observed in a 3D photonic band gap,” Phys. Rev. Lett. 107, 193903 (2011). [CrossRef] [PubMed]

]. In the context of magnetic resonance at radio frequencies, moreover, both suppression of spontaneous emission and non-Markovian dynamics have been demonstrated in 3D PC’s [22

22. U. Hoeppe, C. Wolff, J. Kuchenmeister, J. Niegemann, M. Drescher, H. Benner, and Kurt Busch, “Direct observation of non-markovian radiation dynamics in 3D bulk photonic crystals”, Phys. Rev. Lett. 108, 043603 (2012). [CrossRef] [PubMed]

]. However, it is still a challenging issue to accurately fabricate these structures in experiments. In 3D systems, therefore, I consider the realistic 1D PC’s composed of stacked layers of Si and SiO2 with finite structures perpendicular to periodic directions. The realistic 1D PC’s correspond to the pillars composed of two kinds of different stacked layers in the z direction on the substrate parallel to the 2D xy plane. Such pillars have already been fabricated experimentally [23

23. S. Reitzenstein, N. Gregersen, C. Kistner, M. Strauss, C. Schneider, L. Pan, T. R. Nielsen, S. Hofling, J. Mork, and A. Forchel, “Oscillatory variations in the Q factors of high quality micropillar cavities,” Appl. Phys. Lett. 94061108 (2009). [CrossRef]

]. At first glance, these structures cannot be considered to have large contrasts of EM LDOS’s in 3D systems, because of the leakage of light perpendicular to periodic directions. In the realistic 1D PC’s, nevertheless, the pseudo PBG’s, in which light leaks into air regions, provide large contrasts of EM LDOS’s in the vicinity of the upper photonic band edges. In other words, the realistic 1D PC’s are the simplest structures to obtain large contrasts of EM LDOS’s in 3D systems. I show that the realistic 1D PC’s in 3D systems enable the control of spontaneous emission and population inversion of collective two-level atoms driven by an external laser. This finding facilitates experimental fabrication and realization.

In the strong atom-photon coupling, for example, the vacuum Rabi splitting, in which the frequency spectrum of emitted electromagnetic fields is split to two peaks, has attracted much attention. While 2D PC-slab cavities are used for strong coupling [24

24. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, 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 432, 200–203 (2004). [CrossRef] [PubMed]

, 25

25. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

], the vacuum Rabi splitting has also been reported in 1D micropillar cavities [26

26. J. P. Reithmaier, G. Se.k, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity,” Nature 432, 197–200 (2004). [CrossRef] [PubMed]

, 27

27. M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Hofling, J. Mork, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012). [CrossRef] [PubMed]

]. In 3D systems, in other words, the realization in simple structures provides the novelty, significant impact and new applications.

This paper is organized as follows. In Sec. 2, I explain the first-principle Maxwell-Bloch equations. In Sec. 3, I present my numerical results. Section 4 contains my overall conclusions.

2. First-principle Maxwell-Bloch theory

In a two-level atom with the ground and excited states |1〉 and |2〉, respectively, the Hamiltonian is represented by H = h̄ωAσ22d0 · E(t)(σ12 + σ21), where σij = |i〉〈j| is the atomic operator, ωA is the atomic frequency defined by the difference of ground and excited energies, d0 = d0ud is the transition dipole moment, and E(t) is the electric field. I define the operators σ1(t) = σ12(t)+σ21(t), σ2(t) = i[σ12(t) − σ21(t)] and σ3(t) = σ22(t) − σ11(t). Taking averages of the operators σi(t)=tr[ρσi(t)]=n=12n|ρσi(t)|n, where ρ is the atomic density operator, 〈σ11(t)〉 and 〈σ22(t)〉 indicate the populations at the ground and excited states, respectively [〈σ11(t)〉 + 〈σ22(t)〉 = 1]. Applying σi and H to the Heisenberg equation of motion, I obtain
ddt[σ1(t)σ2(t)σ3(t)]=[0ωA0ωA02d0E(t)/h¯02d0E(t)/h¯0][σ1(t)σ2(t)σ3(t)],
(1)
Equation (1) is called the optical Bloch equation [28

28. L. Allen and J. H. Eberly, Optical Resonance and Two-level Atoms (Dover Publications, Inc., New York, 1987).

]. In the semiclassical theory, 〈σ1(t)〉2 + 〈σ2(t)〉2 + 〈σ3(t)〉2 = 1 (probability conservation law) is satisfied, since d[σ1(t)2+σ2(t)2+σ3(t)2]dt=2[σ1(t)dσ1(t)dt+σ2(t)dσ2(t)dt+σ3(t)dσ3(t)dt]=0, using Eq. (1). This equation is valid for low temperature such as T = 0 K. At higher temperature, however, influences of phonon dephasing cannot be neglected, and they hinder effective population inversion [17

17. H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A 83053811 (2011). [CrossRef]

]. E(t) includes both input and radiative electric fields. The radiative electric fields are generated from two-level atoms. The electric dipole per two-level atom is defined by 〈d(t)〉 = −〈∂H/E(t)〉 = d0σ1(t)〉. Especially, 〈σ3(t)〉 becomes the criterion of population inversion. When population inversion is achieved, 〈σ3(t)〉 becomes positive.

Next, I consider the Maxwell equations with electric polarizations.
×E(r,t)=μ0H(r,t)t
(2)
×H(r,t)=P(r,t)t+ε0ε(r)E(r,t)t,
(3)
where ε0(= 8.854 × 10−12 F/m) and μ0(= 4π × 10−7 H/m) are the permittivity and permeability, respectively, in vacuum. ε(r) is the dielectric constant of PC’s, P(r,t) = N(r)〈d(t)〉 = N(r)d0σ1(t)〉 is the electric polarization and N(r) is the number of atoms per volume. In 3D systems, N(r) = Npδ(rr′), where Np is the point density or the number of two-level atoms at r = r′. In other words, r′ is the position of two-level atoms, and P(r,t) is set up only at r = r′. While Eq. (1) includes E(t) = E(r′,t), Eq. (3) includes 〈σ1(t)〉. Therefore, solving the optical Bloch and Maxwell equations self-consistently, I obtain temporal behaviors of 〈σ3(t)〉. In terms of two-level atoms, I set up three factors, d0, ωA and Np. I choose |d0| = 1.0×10−28 C m, assuming semiconductor quantum dots with large transition dipole moments. ωA corresponds to the wavelength of approximately 1.5 μm. For rapid saturation of calculations [18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

], I take a large number of atoms. 5×5×5 two-level atoms in the x, y and z directions are placed at r = r′ (Np = 53 = 125), and these two-level atoms are assumed to be coherently excited by an external laser. As two-level atoms, for example, PbS quantum dots are preferable for experimental fabrications in Si and SiO2 [29

29. I. Kang and F. W. Wise, “Electronic structure and optical properties of PbS and PbSe quantum dots,” J. Opt. Soc. Am. B 141632–1646 (1997). [CrossRef]

31

31. M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, “Spectroscopy of 1.55 μm PbS quantum dots on Si photonic crystal cavities with a fiber taper waveguide,” Appl. Phys. Lett. 96161108 (2010). [CrossRef]

]. In the case of a single quantum dot with a width of approximately 5 nm, 5 × 5 × 5 quantum dots have a volume of 25 nm ×25 nm ×25 nm. Since this volume is small enough for dielectric layers with widths of approximately 250–500 nm, the above point density approximation of collective two-level atoms is valid.

I use the finite-difference time-domain (FDTD) method [32

32. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

] to solve the first-principle Maxwell-Bloch equations, and then, discretizations of space and time are Δx/a = Δy/a = Δz/a = 1/10, and clΔt/a = 1/20, respectively, where a is the lattice constant of PC’s and cl=1/ε0μ0 is the speed of light in vacuum. In the FDTD method, the second-order Higdon’s absorbing boundary condition is used for electromagnetic waves not to reflect at edges of computational regions [32

32. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

].

3. Numerical results and discussion

3.1. Schematic model of a realistic 1D PC in 3D systems

Figure 1 shows the structure of a realistic 1D PC in 3D systems. The realistic 1D PC composed of stacked layers of Si and SiO2 is periodic in the z direction, and finite in the x and y directions. (Idealized 1D PC’s are periodic in the z direction, but infinite in the x and y directions, and then, the photonic band structure and EM LDOS for the propagation in the z direction are considered.) In practice, this structure corresponds to the pillar composed of two kinds of different stacked layers [23

23. S. Reitzenstein, N. Gregersen, C. Kistner, M. Strauss, C. Schneider, L. Pan, T. R. Nielsen, S. Hofling, J. Mork, and A. Forchel, “Oscillatory variations in the Q factors of high quality micropillar cavities,” Appl. Phys. Lett. 94061108 (2009). [CrossRef]

]. Dielectric constants of Si and SiO2 are εSi = 11.9 and εSiO2 = 2.25, respectively. These materials are chosen for large contrasts of dielectric constants. Thicknesses of Si and SiO2 layers are tSi/a = 0.4 and tSiO2 /a = 0.6, respectively, where a is the lattice constant of 1D PC’s. Widths in the x and y directions are a. Although in idealized 1D PC’s effective PBG’s can easily be obtained by two kinds of different stacked layers, in realistic 1D PC’s they are not always obtained, because of surface modes and light cones [Fig. 2(a)]. Therefore, I have searched for thickness and width parameters to obtain effective PBG’s. In this paper, these parameters are used. I assume that ωa/2πcl = a/λ = 0.3442 corresponds to λ = 1.5 μm, obtained by choosing a = 516.3 nm. This frequency corresponds to the second photonic band edge [Fig. 3(a)]. The wavelength widely used in optical communication is approximately 1.5 μm. I choose this frequency as a standard, since frequencies in the vicinity of the second photonic band edge are focused, as mentioned later. There is the origin (x, y, z) = (0, 0, 0) in the SiO2 region, and this structure is symmetric at the origin.

Fig. 1 Structure of a realistic 1D PC composed of stacked layers of Si and SiO2. Dielectric constants of Si and SiO2 are εSi = 11.9 and εSiO2 = 2.25, respectively. Thicknesses of Si and SiO2 layers are tSi/a = 0.4 and tSiO2 /a = 0.6, respectively, where a = 516.3 nm is the lattice constant of 1D PC’s. Widths in the x and y directions are a. There is the origin (0, 0, 0) in the SiO2 region, and this structure is symmetric at the origin.
Fig. 2 (a) Photonic band structure of the realistic 1D PC. A shaded region indicates the light cone in which light leaks into air regions. Below the light cone, solid and dashed lines indicate the doubly-degenerate and single modes, respectively. (b) x-polarized EM LDOS’s of the realistic 1D PC composed of 21 SiO2 layers at (x/a, y/a, z/a) = (0, 0, ±0.5) (Si region) and (0, 0, 0) (SiO2 region). A gray line indicates the EM LDOS in free space and is proportional to ω2.
Fig. 3 (a) Collective spontaneous emission rate at (x/a, y/a, z/a) = (0, 0, 0) for 0.32 ≤ ωa/2πcl ≤ 0.38. ωAa/2πcl = 0.3300, 0.3442, 0.3479 and 0.3525 are focused. (b) Temporal behavior of 〈σ3(t)〉 for various atomic frequencies, using the first-principle Maxwell-Bloch theory (Sec. 2). Multiplying clt/a by a/cl = 1.721 × 10−3 ps gives real time.

3.2. Photonic band structure and EM LDOS in the realistic 1D PC

Figure 2(a) shows the photonic band structure of the realistic 1D PC calculated by the plane wave expansion method [33

33. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 583896–3908 (1998). [CrossRef]

] with 6727 plane waves in a supercell of 5a × 5a × a. A shaded region indicates the light cone in which light leaks into air regions. Below the light cone, solid and dashed lines indicate the doubly-degenerate and single modes, respectively. The doubly-degenerate modes result from identical structures in the x and y directions. On the other hand, the single modes are localized at surfaces. In Fig. 2(a), there appear pseudo PBG’s. These pseudo PBG’s are different from conventional ones generally discussed in idealized 1D PC’s, because of the presence of the light cone. In Fig. 2(b), I show the x-polarized EM LDOS of the realistic 1D PC composed of 21 SiO2 layers at (x/a, y/a, z/a) = (0, 0, ±0.5) (Si region) and (0, 0, 0) (SiO2 region). The u-polarized EM LDOS is defined by
ρu,u(ω;r)=1Vλ|uEλ(r)|2δ(ωωλ),
(4)
where ωλ and Eλ (r′) are the eigenfrequency and eigen electric field, respectively, V is the volume and u is the unit vector. The computational method of EM LDOS’s is described in Ref. [34

34. P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photonics Rev. 4499–516 (2010). [CrossRef]

]. The left and right edges of the realistic 1D PC are SiO2 layers, and the total length is 20a +tSiO2 = 10.64 μm in the z direction. A gray line indicates the EM LDOS in free space and is proportional to ω2. (The EM LDOS in free space is proportional to ωd−1, where d is the dimension.) All EM LDOS’s increase as ω approaches zero, because of the second-order Higdon’s absorbing boundary condition in the FDTD method [32

32. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

]. The EM LDOS’s in Fig. 2(b) strongly depend on the doubly degenerate modes (solid lines) in Fig. 2(a), and do not depend on the single modes localized at surfaces (dashed line). This is since the excitation point r′ is inside the PC.

In the lower frequency region, EM LDOS’s are smaller than that of free space (gray line). Since 1D waveguides with finite widths have cutoff frequencies, EM LDOS’s are very small below the cutoff frequencies, because of the absence of photonic bands. In Fig. 2(a), the first photonic band is truncated by the light cone, and then, the cutoff frequency is ωa/2πc ≃ 0.1. As a result, the EM LDOS’s increase exponentially (linearly in the semi log scale) in the lower frequency region.

Next, I focus on the pseudo PBG between the first and second solid lines in Fig. 2(a). In the Si region with higher dielectric constants, the sharp peak near ωa/2πcl = 0.3080 corresponds to the lower photonic band edge. However, the EM LDOS inside the PBG is higher than that of free space (gray line). Therefore, a large contrast of EM LDOS’s cannot be obtained in the vicinity of the lower photonic band edge. In the SiO2 region with lower dielectric constants, on the other hand, the sharp peak near ωa/2πcl = 0.3442 corresponds to the upper photonic band edge. Unlike in the Si region, the EM LDOS inside the PBG is lower than that of free space (gray line). Moreover, the EM LDOS is higher than that of free space (gray line) in the frequency region of the second solid line of Fig. 2(a). In other words, a large contrast of EM LDOS’s can be obtained in the vicinity of the upper photonic band edge, even in the presence of the light cone. In 3D PC’s with complete PBG’s, EM LDOS’s inside the PBG’s become very small, regardless of internal higher or lower dielectric constant regions. Even in realistic 1D PC’s, however, there appear very small EM LDOS’s at specific points. (Unfortunately, this physical mechanism is unclear.) The pseudo PBG depends on the polarized direction u in Eq. (4). With increasing the width of the realistic 1D PC, effective pseudo PBG’s cannot be obtained, because of propagation modes in the direction oblique to the z direction. In what follows, I focus on the x-polarized EM LDOS at (x/a, y/a, z/a) = (0, 0, 0) (SiO2 region) near ωa/2πcl = 0.3442.

3.3. Collective spontaneous emission

I consider the collective spontaneous emission at (x/a, y/a, z/a) = (0, 0, 0) (SiO2 region). The spontaneous emission rate γ is proportional to the atomic frequency and the EM LDOS [γωAρu,u(ωA)] [34

34. P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photonics Rev. 4499–516 (2010). [CrossRef]

]. In Fig. 3(a), I show the spontaneous emission rate at (x/a, y/a, z/a) = (0, 0, 0) for 0.32 ≤ ωa/2πcl ≤ 0.38. I focus on ωAa/2πcl = 0.3300, 0.3442, 0.3479 and 0.3525. γ at ωa/2πcl = 0.3525 is normalized to unity. Figure 3(b) shows the temporal behavior of 〈σ3(t)〉 for various atomic frequencies, using the first-principle Maxwell-Bloch theory (Sec. 2). I assume that the transition dipole moment is polarized in the x direction. In Eq. (1), then, d0 · E(t) = d0Ex(r′, t), where r/a = (0, 0, 0). As an initial state, I set up 〈σ1(0)〉 = [1 − 〈σ3(0)〉2]1/2 cosθ, 〈σ2(0)〉 = [1 − 〈σ3(0)〉2]1/2 sinθ (θ = 0) and 〈σ3(0)〉 = 0.9 [〈σ1(0)〉2 + 〈σ2(0)〉2 + 〈σ3(0)〉2 = 1]. It depends not only on Np but also on 〈σ3(0)〉 whether collective spontaneous emission can be recaptured by the semiclassical theory. In spite of Np ≫ 1, as 〈σ3(0)〉 is close to unity, electromagnetic fields behave quantum mechanically, because of larger quantum fluctuation. However, when Np > 100 and 〈σ3(0)〉 = 0.9, the semiclassical theory is still valid, since quantum fluctuation can be neglected [18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

]. While at ωAa/2πcl = 0.3442 〈σ3(t)〉 rapidly decays (Purcell effect) [3

3. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69681 (1946).

], at ωAa/2πcl = 0.3300 it slowly decays. This is since ωAa/2πcl = 0.3442 and 0.3300 are at the photonic band edge and inside the PBG, respectively. In fact, orders of fast collective spontaneous emission (ωAa/2πcl = 0.3442, 0.3525, 0.3479 and 0.3300) coincide with magnitudes of γ. 〈σ3(t)〉 at ωAa/2πcl = 0.3442, 0.3479 and 0.3525 decay to zero (〈σ3(t)〉 ≤ −0.9945) near clt/a = 1.48 × 104 (t = 25.47 ps), 1.81×105 (t = 311.50 ps) and 7.0 × 104 (t = 120.47 ps), respectively (Multiplying clt/a by a/cl = 1.721 × 10−3 ps gives real time). Ratios of the inverse of the relaxation time at ωAa/2πcl = 0.3442, 0.3479 and 0.3525 coincide with those of γ (4.730 : 0.3867 : 1). Even in the presence of the leakage of light, the pseudo PBG enables the effective inhibition of spontaneous emission. Near the photonic band edge, collective spontaneous emission strongly depends on frequencies. This is the evidence of large contrasts of spontaneous emission rates. The above results verify the validity of the 3D Maxwell-Bloch theory (based on the first-principle calculation) for recapturing collective spontaneous emission. In the semiclassical theory, collective spontaneous emission corresponds to the classical dipole radiation [17

17. H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A 83053811 (2011). [CrossRef]

, 18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

]. The dipole radiation emits electromagnetic waves from collective atoms, and then, the atoms lose the excitation energies. Therefore, 〈σ3(t)〉 decays monotonically with time.

3.4. Collective population inversion driven by an external laser

I consider the atom-laser interaction in the absence of spontaneous emission, prior to two-level collective population inversion driven by an external laser. When inputting E(t) = A0ud[exp(−Lt) + exp(Lt)] in the absence of spontaneous emission, the Hamiltonian including the coupling of electrons and input electric fields is
H=h¯ωAσ22d0E(t)(σ12+σ21)=h¯ωAσ22d0A0[exp(iωLt)σ12+exp(iωLt)σ21+exp(iωLt)σ12+exp(iωLt)σ21]
(5)
Using the unitary transformation U(t) = exp(−L22),
H˜=U(t)HU(t)ih¯dU(t)dtU(t)=h¯ΔALσ22d0A0[σ12+σ21+exp(2iωLt)σ12+exp(2iωLt)σ21]h¯ΔALσ22d0A0(σ12+σ21),
(6)
where ΔAL = ωAωL is the detuning. Higher frequency terms exp(±2Lt) are neglected. In the bare-state representation, the interaction term −d0A0(σ12 + σ21) remains. Redefining the ground and excited states (dressed state), the interaction term disappears in
H˜=2h¯ΩR22,
(7)
where
2Ω=ΔAL2+(2d0A0/h¯)2|ΔAL|
(8)
is the Rabi frequency. While the dressed-state atomic operator is Rij = |ĩ〉〈j̃|, the redefined ground and excited states are |1̃〉 = c|1〉+s|2〉 and |2̃〉 = −s|1〉+c|2〉, respectively. ( c2=12[1+ΔAL2Ω] and s2=12[1ΔAL2Ω]). Applying Rij and H̃ to the Heisenberg equation of motion, I obtain 〈R12(t)〉 = 〈R12(0)〉 exp(−2iΩt), 〈R21(t)〉 = 〈R21(0)〉 exp(2iΩt) and 〈R3(t)〉 = 〈R3(0)〉, where 〈R3(t)〉 = 〈R22(t)〉 − 〈R11(t)〉. Finally,
σ3(t)=(c2s2)R3(t)+2sc[R12(t)+R21(t)]=(c2s2)R3(0)+4sc|R12(0)|cos(2Ωt+θ0),
(9)
where 〈R12(0)〉 = |〈R12(0)〉| exp(−0). In other words, 〈σ3(t)〉 keeps oscillating with the Rabi frequency (well-known Rabi oscillation) in the absence of spontaneous emission.

I explain the physical meaning of the bare and dressed states. The left and right sketches of Fig. 4(a) show the ground and excited states for the bare and dressed states, respectively [11

11. M. Florescu and S. John, “Resonance fluorescence in photonic band gap waveguide architectures: Engineering the vacuum for all-optical switching,” Phys. Rev. A 69053810 (2004). [CrossRef]

]. ΔAL < 0 is assumed. |i, m〉 indicates the |i〉 state with the photon number m of the laser frequency ωL. For example, the energy of |i, m〉 + 1〉 is larger by h̄ωL than that of |i, m〉. In the left sketch, the bare-state Hamiltonian describes that |2, m〉 and |1, m + 1〉 interact, as described by gray circles with arrows. In the right sketch, on the other hand, the interaction disappears. Instead, the frequency gap of |1̃, m〉 and |2̃, m〉 becomes 2Ω. Since in the quantum theory A0 is proportional to the square root of the photon number, 2Ω originally depends on the photon number. However, since the photon number considered here is very large (m ≫ 1) for strong laser intensity (106 − 107V/m), the photon number-dependence of 2Ω can be neglected even if increasing or decreasing by one photon. I focus on the following four transitions at a certain n determined by the laser intensity. In the transitions |1̃, n〉 → |1̃, n − 1〉 and |2̃, n〉 → |2̃, n −1〉, the output frequency is the same as ωL (ωout = ωL). However, in the transitions |2̃, n |1̃, n−1〉 (ωout = ωL + 2Ω) and |1̃, n〉 → |2̃, n−1〉 (ωout = ωL − 2Ω), the output frequencies are different from ωL. This leads to three frequency components ω = ωL and ωL ± 2Ω in radiative electromagnetic waves (Mollow triplet) [9

9. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 1881969–1975 (1969). [CrossRef]

]. The spontaneous emission rates at ω = ωL + 2Ω, ωL − 2Ω and ωL are referred to as γ = γ+, γ and γ0, respectively. If γ+ >> γ is assumed, the transition |2̃, n〉 → |1̃, n − 1〉 is dominant rather than |1̃, n〉 → |2̃, n − 1〉, and then, electrons tend to stay at |1̃〉 (〈R11〉 > 〈R22〉). This means that in the bare state population inversion (〈σ22〉 > 〈σ11〉) can be achieved. This is since |2, m − 1〉 and |1, m〉 in the bare state are modified to |1̃, m − 1〉 and |2̃, m − 1〉 in the dressed state, respectively, as connected by dashed lines. In other words, |2̃〉 and |1̃〉 correspond to |1〉 and |2〉, respectively. Figure 4(b) shows the schematic model of EM LDOS’s in the vicinity of the upper photonic band edge. Lower and higher gray regions indicate the PBG and the propagation region, respectively. I set up the atomic and laser frequencies, as shown in Fig. 4(b). When A0 is large, the higher Mollow sideband (ω = ωL + 2Ω) is pushed into the propagation region, and then, γ+ >> γ is satisfied. In practice, however, probabilities of the four transitions are different, and those from |j̃, n〉 to |ĩ, n − 1〉 are represented by |〈ĩ|σ12|j̃〉|2 [11

11. M. Florescu and S. John, “Resonance fluorescence in photonic band gap waveguide architectures: Engineering the vacuum for all-optical switching,” Phys. Rev. A 69053810 (2004). [CrossRef]

], where the operator σ12 = −scR3 + c2R12s2R21 describes the transition from |2〉 to |1〉 in the bare state. The probabilities of the transitions from |1̃, n〉 to |1̃, n − 1〉, from |2̃, n〉 to |2̃, n − 1〉, from |2̃, n〉 to |1̃, n − 1〉 and from |1̃, n〉 to |2̃, n − 1〉 are s2c2, s2c2, c4 and s4, respectively (The total is c4 +s4 +2s2c2 = (c2 +s2)2 = 1). In other words, since the effective spontaneous emission rates from |2̃, n〉 to |1̃, n − 1〉 and from |1̃, n〉 to |2̃, n − 1〉 are c4γ+ and s4γ, respectively, the precise condition of population inversion is c4γ+ > s4γ. This condition coincides with that derived from the quantum theory [11

11. M. Florescu and S. John, “Resonance fluorescence in photonic band gap waveguide architectures: Engineering the vacuum for all-optical switching,” Phys. Rev. A 69053810 (2004). [CrossRef]

, 12

12. S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. 3S103–S120 (2001). [CrossRef]

, 18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

]. s2 = 1 and c2 = 0 at A0 = 0, and s2 (c2) decreases (increases) with larger A0 (s2 = c2 = 1/2 for A0 → ∞). When A0 exceeds the threshold value, c4γ+ > s4γ is achieved. In conventional optical materials with γγ+, this condition is never achieved, because of s2c2 for ΔAL < 0. That is the reason why two-level population inversion is generally considered impossible.

Fig. 4 (a) Ground and excited states for bare and dressed states in the left and right sketches, respectively. ΔAL < 0 is assumed. |i, m〉 indicates the |i〉 state with the photon number m of the frequency ωL. (b) Schematic model of EM LDOS’s near the upper photonic band edge. Lower and higher gray regions indicate the PBG and the propagation region, respectively.

I assume that 125 two-level atoms placed at (x/a, y/a, z/a) = (0, 0, 0) (SiO2 region) are coherently excited by an external laser. Figure 5(a) shows the spontaneous emission rate at (x/a, y/a, z/a) = (0, 0, 0) for 0.335 ≤ ωa/2πcl 0.350. The structure is the same as that discussed in Fig. 2 (b). To satisfy the condition of Fig. 4(b), I set up ωAa/2πcl = 0.3405 and ωLa/2πcl = 0.3415. In Fig. 1, the x-polarized electric-field plane wave Ex(t) = E0 sin(ωLt) is input in the z direction from the left side. When there are no atoms, the amplitude of the x-polarized electric field at (x/a, y/a, z/a) = (0, 0, 0) converges to a certain value with oscillations. When choosing this laser frequency, the steady-state electric field at (x/a, y/a, z/a) = (0, 0, 0) is the same as E0, and then, it is easy to estimate 2Ω. The frequency positions of the lower and higher Mollow sidebands described by two arrows are (ωL − 2Ω)a/2πcl ≤ 0.3405 and (ωL + 2Ω)a/2πcl ≥ 0.3425, respectively. Then, the contrast γ+/γ increases with larger E0. The Mollow triplet spectrum can be calculated by the Fourier transform of the output wave on the right side with respect to time [17

17. H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A 83053811 (2011). [CrossRef]

], and then, the spectrum shows broad peaks at ω = ωL ± 2Ω. Figure 5(b) shows the temporal behavior of 〈σ3(t)〉 as a function of time at E0 = 2.0 × 106 V/m and 8.0 × 106 V/m, using the first-principle Maxwell-Bloch theory (Sec. 2). I take 〈σ1(0)〉 = 0.0, 〈σ2(0)〉 = 0.0 and 〈σ3(0)〉 = −1.0 at t = 0. In Eq. (1), then, d0 · E(t) = d0Ex(r′, t), where r′/a = (0, 0, 0). In the presence of spontaneous emission, 〈σ3(t)〉 converges to a certain value with Rabi oscillations. While at E0 = 2.0 × 106 V/m 〈σ3(t)〉 converges to a negative value near clt/a = 8.0 × 105 (t = 1.377 ns) with diminishing Rabi oscillations, at E0 = 8.0 × 106 V/m it converges to a positive value near clt/a = 7.0 × 105 (t = 1.205 ns) after increasing Rabi oscillations. In other words, when input electric fields exceed a certain value, collective population inversion can be achieved at steady states, even in the presence of leakage of light perpendicular to periodic directions. Such leakage of light was neglected in idealized 1D and 2D PC’s [17

17. H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A 83053811 (2011). [CrossRef]

]. In this respect, collective population inversion in 3D systems is relatively robust for leakage of light.

Fig. 5 (a) Spontaneous emission rate at (x/a, y/a, z/a) = (0, 0, 0) for 0.335 ≤ ωa/2πcl ≤ 0.350. I set up ωAa/2πcl = 0.3405 and ωLa/2πcl = 0.3415. (b) Temporal behavior of 〈σ3(t)〉 as a function of time at E0 = 2.0 × 106 V/m and 8.0 × 106 V/m, using the first-principle Maxwell-Bloch theory (Sec. 2). Multiplying clt/a by a/cl = 1.721 × 10−3 ps gives real time.

Figure 6(a) shows the behavior of 〈σ3st (steady-state value of 〈σ3(t)〉) as a function of E0 for various atomic frequencies, using the first-principle Maxwell-Bloch theory (Sec. 2). The laser frequency is ωLa/2πcl = 0.3415. At ωAa/2πcl = 0.3405 (ΔAL < 0), a threshold occurs between E0 = 5.0 × 106 V/m and 5.5 × 106 V/m. 〈σ3st (circle) discontinuously changes near the threshold and becomes positive. This discontinuous change of 〈σ3st is the characteristic behavior of collective population inversion [6

6. S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. 781888–1891 (1997). [CrossRef]

, 12

12. S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. 3S103–S120 (2001). [CrossRef]

, 18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

]. (Although in Refs. [6

6. S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. 781888–1891 (1997). [CrossRef]

, 12

12. S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. 3S103–S120 (2001). [CrossRef]

] a very large number of two-level atoms Np = 5000 are taken to obtain the discontinuous change because of large dipole dephasing, in this paper no dipole dephasing is considered for simplicity.) At ωAa/2πcl = 0.3415 (ΔAL = 0), 〈σ3st (square) becomes zero with small intensities (E0 > 2.0 × 105 V/m). At ωAa/2πcl = 0.3425 (ΔAL > 0), 〈σ3st (triangle) always remains negative, although it increases with larger E0. Therefore, when γ+ > γ, ΔAL < 0 is necessary for collective population inversion. Moreover, 〈σ3st derived from the quantum theory is represented by
σ3st=ΔAL2Ω[2(1+1Np)11ηNp+12Np11η1],
(10)
where η = (c4γ+)/(s4γ) [6

6. S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. 781888–1891 (1997). [CrossRef]

, 12

12. S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. 3S103–S120 (2001). [CrossRef]

]. Gray dashed and solid lines correspond to the behaviors of 〈σ3st at Np = 1 (single atom) and 125 (collective atoms), respectively, derived from Eq. (10). (I have used the gray line in Fig. 6(b) for η.) While the gray dashed line increases continuously with larger E0, the gray solid line changes discontinuously near the threshold value. With increasing the number of two-level atoms, 〈σ3st becomes discontinuous near the threshold value. At Np = 125, especially, behaviors of gray solid lines and circles are very similar. This means that the quantum and semiclassical theories coincide in the case that a large number of two-level atoms are coherently excited (Dicke model). Figure 6(b) shows the behavior of (c4γ+)/(s4γ) as a function of E0 at ωAa/2πcl = 0.3405 (ΔAL < 0). γ± can be estimated from ω = ωL ± 2Ω and Fig. 5(a). I have confirmed that 2Ω calculated by 2A0 = E0 in Eq. (8) coincides with the resonant frequency obtained from the fast Fourier transform of 〈σ3(t)〉 with respect to time. While circles indicate the numerical data, a gray line is the spline interpolation of them. As predicted by Refs. [11

11. M. Florescu and S. John, “Resonance fluorescence in photonic band gap waveguide architectures: Engineering the vacuum for all-optical switching,” Phys. Rev. A 69053810 (2004). [CrossRef]

, 12

12. S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. 3S103–S120 (2001). [CrossRef]

], when c4γ+ > s4γ (η > 1) is satisfied, collective population inversion is achieved. Saturation time becomes faster with increasing |c4γ+s4γ| [18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

]. In fact, while saturation time is very slow near the threshold value because of |c4γ+s4γ| ≃ 0 (η ≃ 1), at E0 = 1.0 × 107 V/m it is very fast [clt/a = 2.5 × 105 (t = 430.25 ps)]. Behaviors in Figs. 5(b) and 6(a) coincide with those derived from the quantum theory based on the mean-field approximation [Figs. 3(a) and 3(b) in Ref. [18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

]. However, γ > γ+ is assumed, because of an opposite condition ΔAL > 0]. According to the quantum theory, the contrast of γ± larger than 10 is preferable for large population inversion and low threshold values [18

18. H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A 85, 023837 (2012). [CrossRef]

]. Such a large contrast is difficult to achieve in simple cavities, since resonant peaks are relatively broad and EM LDOS’s are not very low except at resonant frequencies, and therefore, PBG’s are necessary. In Figs. 5(b) and 6(a), no phenomenological decay terms γ± are attached to Eq. (1). Nevertheless, the 3D Maxwell-Bloch theory (based on the first-principle calculation) can recapture the collective population inversion of two-level atoms. Even simple structures such as realistic 1D PC’s enable the two-level collective population inversion which is generally considered impossible. This fact is crucial for experimental realization.

Fig. 6 (a) Behavior of 〈σ3st as a function of E0 for various frequencies, using the first-principle Maxwell-Bloch theory (Sec. 2). ωAa/2πcl = 0.3405, 0.3415 and 0.3425 are considered. The laser frequency is ωLa/2πcl = 0.3415. Gray dashed and solid lines correspond to the behaviors of 〈σ3st at Np = 1 (single atom) and 125 (collective atoms), respectively, derived from Eq. (10). (b) Behavior of (c4γ+)/(s4γ) as a function of E0 at ωAa/2πcl = 0.3405 (ΔAL < 0). While circles indicate the numerical data, a gray line is the spline interpolation of them.

Finally, I introduce the application of population inversion of two-level atoms driven by an external laser. When population inversion is achieved by an external laser, input signal light is amplified. In other words, transmittance of signal light greatly changes with an external laser, which corresponds to all-optical transistors in which light (signal light) can be controlled by light (external laser) [12

12. S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. 3S103–S120 (2001). [CrossRef]

]. Moreover, when referring to positive and negative 〈σ3st as 1 and 0, respectively, logic gates controlled by light can be realized. This is useful for optical information processing. Since widths of the structure demonstrated in this paper are narrow (Fig. 1), these applications could be realized in compact devices.

4. Conclusions

Using the 3D semiclassical Maxwell-Bloch theory (based on the first-principle calculation), I have theoretically demonstrated the two-level collective population inversion in realistic 1D PC’s composed of stacked layers of Si and SiO2 with finite structures perpendicular to periodic directions. The semiclassical theory is valid for the case that a large number of two-level atoms are coherently excited (Dicke model). Even in the presence of light cones, a large contrast of EM LDOS’s in the vicinity of the upper photonic band edge can be obtained in the SiO2 region with lower dielectric constants. Such simple structures facilitate experimental fabrication and realization.

Orders of fast collective spontaneous emission coincide with magnitudes of EM LDOS’s. Especially, collective spontaneous emission is very slow and greatly enhanced inside the PBG and at the photonic band edge, respectively.

This verifies that the self-consistent Maxwell-Bloch theory based on the semiclassical theory is an effective tool for the analysis of spontaneous emission and population inversion of collective two-level atoms in 3D systems.

Acknowledgments

This work was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology.

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T. M. Stace, A. C. Doherty, and S. D. Barrett, “Population inversion of a driven two-level system in a structureless bath,” Phys. Rev. Lett. 95106801 (2005). [CrossRef] [PubMed]

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S. Hughes and H. J. Carmichael, “Stationary inversion of a two level system coupled to an off-resonant cavity with strong dissipation,” Phys. Rev. Lett. 107, 193601 (2011). [CrossRef] [PubMed]

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OCIS Codes
(270.0270) Quantum optics : Quantum optics
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Quantum Optics

History
Original Manuscript: May 29, 2012
Revised Manuscript: June 15, 2012
Manuscript Accepted: June 25, 2012
Published: July 12, 2012

Citation
Hiroyuki Takeda, "Semiclassical analysis of two-level collective population inversion using photonic crystals in three-dimensional systems," Opt. Express 20, 17201-17213 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17201


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References

  1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.582486–2489 (1987). [CrossRef] [PubMed]
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.582059–2062 (1987). [CrossRef] [PubMed]
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