## Semiclassical analysis of two-level collective population inversion using photonic crystals in three-dimensional systems |

Optics Express, Vol. 20, Issue 15, pp. 17201-17213 (2012)

http://dx.doi.org/10.1364/OE.20.017201

Acrobat PDF (1817 KB)

### 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

1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**2486–2489 (1987). [CrossRef] [PubMed]

2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**2059–2062 (1987). [CrossRef] [PubMed]

5. M. Lindberg and C. M. Savage, “Steady-state two-level atomic population inversion via a quantized cavity field,” Phys. Rev. A **38**5182–5192 (1988). [CrossRef] [PubMed]

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]

*ω*, populations at ground and excited states oscillate with the Rabi frequency 2Ω. Then, there appear three frequency components

_{L}*ω*and

_{L}*ω*± 2Ω (Mollow triplet) in radiative electromagnetic waves [9

_{L}9. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. **188**1969–1975 (1969). [CrossRef]

*ω*=

*ω*± 2Ω is large, population inversion can be achieved even in two-level atoms [6

_{L}6. S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. **78**1888–1891 (1997). [CrossRef]

10. M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A **64**033801 (2001). [CrossRef]

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

6. S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. **78**1888–1891 (1997). [CrossRef]

10. M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A **64**033801 (2001). [CrossRef]

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

13. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. **93**99–110 (1954). [CrossRef]

16. R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: Superfluorescence,” Phys. Rev. A **11**1507–1521 (1975). [CrossRef]

17. H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A **83**053811 (2011). [CrossRef]

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

17. H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A **83**053811 (2011). [CrossRef]

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]

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]

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]

_{2}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. **94**061108 (2009). [CrossRef]

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. 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]

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. 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]

## 2. First-principle Maxwell-Bloch theory

*H*=

*h*̄

*ω*

_{A}σ_{22}−

**d**

_{0}

*·*

**E**(

*t*)(

*σ*

_{12}+

*σ*

_{21}), where

*σ*= |

_{ij}*i*〉〈

*j*| is the atomic operator,

*ω*is the atomic frequency defined by the difference of ground and excited energies,

_{A}**d**

_{0}=

*d*

_{0}

**u**

*is the transition dipole moment, and*

_{d}**E**(

*t*) is the electric field. I define the operators

*σ*

_{1}(

*t*) =

*σ*

_{12}(

*t*)+

*σ*

_{21}(

*t*),

*σ*

_{2}(

*t*) =

*i*[

*σ*

_{12}(

*t*) −

*σ*

_{21}(

*t*)] and

*σ*

_{3}(

*t*) =

*σ*

_{22}(

*t*) −

*σ*

_{11}(

*t*). Taking averages of the operators

*ρ*is the atomic density operator, 〈

*σ*

_{11}(

*t*)〉 and 〈

*σ*

_{22}(

*t*)〉 indicate the populations at the ground and excited states, respectively [〈

*σ*

_{11}(

*t*)〉 + 〈

*σ*

_{22}(

*t*)〉 = 1]. Applying

*σ*and

_{i}*H*to the Heisenberg equation of motion, I obtain Equation (1) is called the optical Bloch equation [28]. In the semiclassical theory, 〈

*σ*

_{1}(

*t*)〉

^{2}+ 〈

*σ*

_{2}(

*t*)〉

^{2}+ 〈

*σ*

_{3}(

*t*)〉

^{2}= 1 (probability conservation law) is satisfied, since

*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 **83**053811 (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*)〉 =

**d**

_{0}〈

*σ*

_{1}(

*t*)〉. Especially, 〈

*σ*

_{3}(

*t*)〉 becomes the criterion of population inversion. When population inversion is achieved, 〈

*σ*

_{3}(

*t*)〉 becomes positive.

*ε*

_{0}(= 8.854 × 10

^{−12}F/m) and

*μ*

_{0}(= 4

*π*× 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**)

**d**

_{0}〈

*σ*

_{1}(

*t*)〉 is the electric polarization and

*N*(

**r**) is the number of atoms per volume. In 3D systems,

*N*(

**r**) =

*N*(

_{p}δ**r**−

**r**′), where

*N*is the point density or the number of two-level atoms at

_{p}**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 〈

*σ*

_{1}(

*t*)〉. Therefore, solving the optical Bloch and Maxwell equations self-consistently, I obtain temporal behaviors of 〈

*σ*

_{3}(

*t*)〉. In terms of two-level atoms, I set up three factors,

**d**

_{0},

*ω*and

_{A}*N*. I choose |

_{p}**d**

_{0}| = 1.0×10

^{−28}C m, assuming semiconductor quantum dots with large transition dipole moments.

*ω*corresponds to the wavelength of approximately 1.5

_{A}*μ*m. 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]

*x*,

*y*and

*z*directions are placed at

**r**=

**r**′ (

*N*= 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 SiO

_{p}_{2}[29

29. I. Kang and F. W. Wise, “Electronic structure and optical properties of PbS and PbSe quantum dots,” J. Opt. Soc. Am. B **14**1632–1646 (1997). [CrossRef]

31. M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, “Spectroscopy of 1.55 *μ*m PbS quantum dots on Si photonic crystal cavities with a fiber taper waveguide,” Appl. Phys. Lett. **96**161108 (2010). [CrossRef]

*x*/

*a*= Δ

*y*/

*a*= Δ

*z*/

*a*= 1/10, and

*c*Δ

_{l}*t*/

*a*= 1/20, respectively, where

*a*is the lattice constant of PC’s and

## 3. Numerical results and discussion

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

_{2}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. **94**061108 (2009). [CrossRef]

_{2}are

*ε*= 11.9 and

_{Si}*ε*

_{SiO2}= 2.25, respectively. These materials are chosen for large contrasts of dielectric constants. Thicknesses of Si and SiO

_{2}layers are

*t*/

_{Si}*a*= 0.4 and

*t*

_{SiO2}

*/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*/2

*πc*=

_{l}*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 SiO

_{2}region, and this structure is symmetric at the origin.

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

33. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E **58**3896–3908 (1998). [CrossRef]

*a*× 5

*a*×

*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 SiO

_{2}layers at (

*x*/

*a*,

*y*/

*a*,

*z*/

*a*) = (0, 0, ±0.5) (Si region) and (0, 0, 0) (SiO

_{2}region). The

**u**-polarized EM LDOS is defined by where

*ω*and

_{λ}**E**

*(*

_{λ}**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. **4**499–516 (2010). [CrossRef]

_{2}layers, and the total length is 20

*a*+

*t*

_{SiO2}= 10.64

*μ*m in the

*z*direction. A gray line indicates the EM LDOS in free space and is proportional to

*ω*

^{2}. (The EM LDOS in free space is proportional to

*ω*

^{d−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]. 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.

*ωa*/2

*πc*≃ 0.1. As a result, the EM LDOS’s increase exponentially (linearly in the semi log scale) in the lower frequency region.

*ωa*/2

*πc*= 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 SiO

_{l}_{2}region with lower dielectric constants, on the other hand, the sharp peak near

*ωa*/2

*πc*= 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

_{l}**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) (SiO

_{2}region) near

*ωa*/2

*πc*= 0.3442.

_{l}### 3.3. Collective spontaneous emission

*x*/

*a*,

*y*/

*a*,

*z*/

*a*) = (0, 0, 0) (SiO

_{2}region). The spontaneous emission rate

*γ*is proportional to the atomic frequency and the EM LDOS [

*γ*∝

*ω*

_{A}ρ_{u,u}(

*ω*)] [34

_{A}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. **4**499–516 (2010). [CrossRef]

*x*/

*a*,

*y*/

*a*,

*z*/

*a*) = (0, 0, 0) for 0.32 ≤

*ωa*/2

*πc*≤ 0.38. I focus on

_{l}*ω*/2

_{A}a*πc*= 0.3300, 0.3442, 0.3479 and 0.3525.

_{l}*γ*at

*ωa*/2

*πc*= 0.3525 is normalized to unity. Figure 3(b) shows the temporal behavior of 〈

_{l}*σ*

_{3}(

*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,

**d**

_{0}

*·*

**E**(

*t*) =

*d*

_{0}

*E*(

_{x}**r**′,

*t*), where

**r**′

**/**

*a*= (0, 0, 0). As an initial state, I set up 〈

*σ*

_{1}(0)〉 = [1 − 〈

*σ*

_{3}(0)〉

^{2}]

^{1/2}cos

*θ*, 〈

*σ*

_{2}(0)〉 = [1 − 〈

*σ*

_{3}(0)〉

^{2}]

^{1/2}sin

*θ*(

*θ*= 0) and 〈

*σ*

_{3}(0)〉 = 0.9 [〈

*σ*

_{1}(0)〉

^{2}+ 〈

*σ*

_{2}(0)〉

^{2}+ 〈

*σ*

_{3}(0)〉

^{2}= 1]. It depends not only on

*N*but also on 〈

_{p}*σ*

_{3}(0)〉 whether collective spontaneous emission can be recaptured by the semiclassical theory. In spite of

*N*≫ 1, as 〈

_{p}*σ*

_{3}(0)〉 is close to unity, electromagnetic fields behave quantum mechanically, because of larger quantum fluctuation. However, when

*N*> 100 and 〈

_{p}*σ*

_{3}(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]

*ω*/2

_{A}a*πc*= 0.3442 〈

_{l}*σ*

_{3}(

*t*)〉 rapidly decays (Purcell effect) [3], at

*ω*/2

_{A}a*πc*= 0.3300 it slowly decays. This is since

_{l}*ω*/2

_{A}a*πc*= 0.3442 and 0.3300 are at the photonic band edge and inside the PBG, respectively. In fact, orders of fast collective spontaneous emission (

_{l}*ω*/2

_{A}a*πc*= 0.3442, 0.3525, 0.3479 and 0.3300) coincide with magnitudes of

_{l}*γ*. 〈

*σ*

_{3}(

*t*)〉 at

*ω*/2

_{A}a*πc*= 0.3442, 0.3479 and 0.3525 decay to zero (〈

_{l}*σ*

_{3}(

*t*)〉 ≤ −0.9945) near

*c*/

_{l}t*a*= 1.48 × 10

^{4}(

*t*= 25.47 ps), 1.81×10

^{5}(

*t*= 311.50 ps) and 7.0 × 10

^{4}(

*t*= 120.47 ps), respectively (Multiplying

*c*/

_{l}t*a*by

*a/c*= 1.721 × 10

_{l}^{−3}ps gives real time). Ratios of the inverse of the relaxation time at

*ω*/2

_{A}a*πc*= 0.3442, 0.3479 and 0.3525 coincide with those of

_{l}*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

**83**053811 (2011). [CrossRef]

**85**, 023837 (2012). [CrossRef]

*σ*

_{3}(

*t*)〉 decays monotonically with time.

### 3.4. Collective population inversion driven by an external laser

**E**(

*t*) =

*A*

_{0}

**u**

*[exp(−*

_{d}*iω*) + exp(

_{L}t*iω*)] in the absence of spontaneous emission, the Hamiltonian including the coupling of electrons and input electric fields is Using the unitary transformation

_{L}t*U*(

*t*) = exp(−

*iω*

_{L}tσ_{22}),

*=*

_{AL}*ω*−

_{A}*ω*is the detuning. Higher frequency terms exp(±2

_{L}*iω*) are neglected. In the bare-state representation, the interaction term −

_{L}t*d*

_{0}

*A*

_{0}(

*σ*

_{12}+

*σ*

_{21}) remains. Redefining the ground and excited states (dressed state), the interaction term disappears in where is the Rabi frequency. While the dressed-state atomic operator is

*R*= |

_{ij}*ĩ*〉〈

*j*̃|, the redefined ground and excited states are |1̃〉 =

*c*|1〉+

*s*|2〉 and |2̃〉 = −

*s*|1〉+

*c*|2〉, respectively. (

*R*and

_{ij}*H*̃ to the Heisenberg equation of motion, I obtain 〈

*R*

_{12}(

*t*)〉 = 〈

*R*

_{12}(0)〉 exp(−2

*i*Ω

*t*), 〈

*R*

_{21}(

*t*)〉 = 〈

*R*

_{21}(0)〉 exp(2

*i*Ω

*t*) and 〈

*R*

_{3}(

*t*)〉 = 〈

*R*

_{3}(0)〉, where 〈

*R*

_{3}(

*t*)〉 = 〈

*R*

_{22}(

*t*)〉 − 〈

*R*

_{11}(

*t*)〉. Finally, where 〈

*R*

_{12}(0)〉 = |〈

*R*

_{12}(0)〉| exp(−

*iθ*

_{0}). In other words, 〈

*σ*

_{3}(

*t*)〉 keeps oscillating with the Rabi frequency (well-known Rabi oscillation) in the absence of spontaneous emission.

11. M. Florescu and S. John, “Resonance fluorescence in photonic band gap waveguide architectures: Engineering the vacuum for all-optical switching,” Phys. Rev. A **69**053810 (2004). [CrossRef]

*< 0 is assumed. |*

_{AL}*i*,

*m*〉 indicates the |

*i*〉 state with the photon number

*m*of the laser frequency

*ω*. For example, the energy of |

_{L}*i*,

*m*〉 + 1〉 is larger by

*h*̄

*ω*than that of |

_{L}*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

*A*

_{0}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 (10

^{6}− 10

^{7}

*V/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

*ω*(

_{L}*ω*=

_{out}*ω*). However, in the transitions |2̃,

_{L}*n*〉

*→*|1̃,

*n*−1〉 (

*ω*=

_{out}*ω*+ 2Ω) and |1̃,

_{L}*n*〉 → |2̃,

*n*−1〉 (

*ω*=

_{out}*ω*− 2Ω), the output frequencies are different from

_{L}*ω*. This leads to three frequency components

_{L}*ω*=

*ω*and

_{L}*ω*± 2Ω in radiative electromagnetic waves (Mollow triplet) [9

_{L}9. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. **188**1969–1975 (1969). [CrossRef]

*ω*=

*ω*+ 2Ω,

_{L}*ω*− 2Ω and

_{L}*ω*are referred to as

_{L}*γ*=

*γ*

_{+},

*γ*

_{−}and

*γ*

_{0}, 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〉 (〈

*R*

_{11}〉 > 〈

*R*

_{22}〉). This means that in the bare state population inversion (〈

*σ*

_{22}〉 > 〈

*σ*

_{11}〉) 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

*A*

_{0}is large, the higher Mollow sideband (

*ω*=

*ω*+ 2Ω) is pushed into the propagation region, and then,

_{L}*γ*

_{+}>>

*γ*

_{−}is satisfied. In practice, however, probabilities of the four transitions are different, and those from |

*j*̃,

*n*〉 to |

*ĩ*,

*n*− 1〉 are represented by |〈

*ĩ*|

*σ*

_{12}|

*j*̃〉|

^{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 **69**053810 (2004). [CrossRef]

*σ*

_{12}= −

*scR*

_{3}+

*c*

^{2}

*R*

_{12}−

*s*

^{2}

*R*

_{21}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

*s*

^{2}

*c*

^{2},

*s*

^{2}

*c*

^{2},

*c*

^{4}and

*s*

^{4}, respectively (The total is

*c*

^{4}+

*s*

^{4}+2

*s*

^{2}

*c*

^{2}= (

*c*

^{2}+

*s*

^{2})

^{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

*c*

^{4}

*γ*

_{+}and

*s*

^{4}

*γ*

_{−}, respectively, the precise condition of population inversion is

*c*

^{4}

*γ*

_{+}>

*s*

^{4}

*γ*

_{−}. 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 **69**053810 (2004). [CrossRef]

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

**85**, 023837 (2012). [CrossRef]

*s*

^{2}= 1 and

*c*

^{2}= 0 at

*A*

_{0}= 0, and

*s*

^{2}(

*c*

^{2}) decreases (increases) with larger

*A*

_{0}(

*s*

^{2}=

*c*

^{2}= 1/2 for

*A*

_{0}→ ∞). When

*A*

_{0}exceeds the threshold value,

*c*

^{4}

*γ*

_{+}>

*s*

^{4}

*γ*

_{−}is achieved. In conventional optical materials with

*γ*

_{−}≃

*γ*

_{+}, this condition is never achieved, because of

*s*

^{2}≥

*c*

^{2}for Δ

*< 0. That is the reason why two-level population inversion is generally considered impossible.*

_{AL}*x*/

*a*,

*y*/

*a*,

*z*/

*a*) = (0, 0, 0) (SiO

_{2}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*/2

*πc*

_{l}*≤*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

*ω*/2

_{A}a*πc*= 0.3405 and

_{l}*ω*/2

_{L}a*πc*= 0.3415. In Fig. 1, the

_{l}*x*-polarized electric-field plane wave

*E*(

_{x}*t*) =

*E*

_{0}sin(

*ω*) is input in the

_{L}t*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

*E*

_{0}, and then, it is easy to estimate 2Ω. The frequency positions of the lower and higher Mollow sidebands described by two arrows are (

*ω*− 2Ω)

_{L}*a*/2

*πc*≤ 0.3405 and (

_{l}*ω*+ 2Ω)

_{L}*a*/2

*πc*≥ 0.3425, respectively. Then, the contrast

_{l}*γ*

_{+}/

*γ*

_{−}increases with larger

*E*

_{0}. The Mollow triplet spectrum can be calculated by the Fourier transform of the output wave on the right side with respect to time [17

**83**053811 (2011). [CrossRef]

*ω*=

*ω*± 2Ω. Figure 5(b) shows the temporal behavior of 〈

_{L}*σ*

_{3}(

*t*)〉 as a function of time at

*E*

_{0}= 2.0 × 10

^{6}V/m and 8.0 × 10

^{6}V/m, using the first-principle Maxwell-Bloch theory (Sec. 2). I take 〈

*σ*

_{1}(0)〉 = 0.0, 〈

*σ*

_{2}(0)〉 = 0.0 and 〈

*σ*

_{3}(0)〉 = −1.0 at

*t*= 0. In Eq. (1), then,

**d**

_{0}

*·*

**E**(

*t*) =

*d*

_{0}

*E*(

_{x}**r**′,

*t*), where

**r**′/

*a*= (0, 0, 0). In the presence of spontaneous emission, 〈

*σ*

_{3}(

*t*)〉 converges to a certain value with Rabi oscillations. While at

*E*

_{0}= 2.0 × 10

^{6}V/m 〈

*σ*

_{3}(

*t*)〉 converges to a negative value near

*c*/

_{l}t*a*= 8.0 × 10

^{5}(

*t*= 1.377 ns) with diminishing Rabi oscillations, at

*E*

_{0}= 8.0 × 10

^{6}V/m it converges to a positive value near

*c*/

_{l}t*a*= 7.0 × 10

^{5}(

*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

**83**053811 (2011). [CrossRef]

*σ*

_{3}〉

*(steady-state value of 〈*

^{st}*σ*

_{3}(

*t*)〉) as a function of

*E*

_{0}for various atomic frequencies, using the first-principle Maxwell-Bloch theory (Sec. 2). The laser frequency is

*ω*/2

_{L}a*πc*= 0.3415. At

_{l}*ω*/2

_{A}a*πc*= 0.3405 (Δ

_{l}*< 0), a threshold occurs between*

_{AL}*E*

_{0}= 5.0 × 10

^{6}V/m and 5.5 × 10

^{6}V/m. 〈

*σ*

_{3}〉

*(circle) discontinuously changes near the threshold and becomes positive. This discontinuous change of 〈*

^{st}*σ*

_{3}〉

*is the characteristic behavior of collective population inversion [6*

^{st}6. S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. **78**1888–1891 (1997). [CrossRef]

**3**S103–S120 (2001). [CrossRef]

**85**, 023837 (2012). [CrossRef]

**78**1888–1891 (1997). [CrossRef]

**3**S103–S120 (2001). [CrossRef]

*N*= 5000 are taken to obtain the discontinuous change because of large dipole dephasing, in this paper no dipole dephasing is considered for simplicity.) At

_{p}*ω*/2

_{A}a*πc*= 0.3415 (Δ

_{l}*= 0), 〈*

_{AL}*σ*

_{3}〉

*(square) becomes zero with small intensities (*

^{st}*E*

_{0}> 2.0 × 10

^{5}V/m). At

*ω*/2

_{A}a*πc*= 0.3425 (Δ

_{l}*> 0), 〈*

_{AL}*σ*

_{3}〉

*(triangle) always remains negative, although it increases with larger*

^{st}*E*

_{0}. Therefore, when

*γ*

_{+}>

*γ*

_{−}, Δ

*< 0 is necessary for collective population inversion. Moreover, 〈*

_{AL}*σ*

_{3}〉

*derived from the quantum theory is represented by where*

^{st}*η*= (

*c*

^{4}

*γ*

_{+})/(

*s*

^{4}

*γ*

_{−}) [6

**78**1888–1891 (1997). [CrossRef]

**3**S103–S120 (2001). [CrossRef]

*σ*

_{3}〉

*at*

^{st}*N*= 1 (single atom) and 125 (collective atoms), respectively, derived from Eq. (10). (I have used the gray line in Fig. 6(b) for

_{p}*η*.) While the gray dashed line increases continuously with larger

*E*

_{0}, the gray solid line changes discontinuously near the threshold value. With increasing the number of two-level atoms, 〈

*σ*

_{3}〉

*becomes discontinuous near the threshold value. At*

^{st}*N*= 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 (

_{p}*c*

^{4}

*γ*

_{+})/(

*s*

^{4}

*γ*

_{−}) as a function of

*E*

_{0}at

*ω*/2

_{A}a*πc*= 0.3405 (Δ

_{l}*< 0).*

_{AL}*γ*

_{±}can be estimated from

*ω*=

*ω*± 2Ω and Fig. 5(a). I have confirmed that 2Ω calculated by 2

_{L}*A*

_{0}=

*E*

_{0}in Eq. (8) coincides with the resonant frequency obtained from the fast Fourier transform of 〈

*σ*

_{3}(

*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

**69**053810 (2004). [CrossRef]

**3**S103–S120 (2001). [CrossRef]

*c*

^{4}

*γ*

_{+}>

*s*

^{4}

*γ*

_{−}(

*η*> 1) is satisfied, collective population inversion is achieved. Saturation time becomes faster with increasing |

*c*

^{4}

*γ*

_{+}−

*s*

^{4}

*γ*

_{−}| [18

**85**, 023837 (2012). [CrossRef]

*c*

^{4}

*γ*

_{+}−

*s*

^{4}

*γ*

_{−}| ≃ 0 (

*η*≃ 1), at

*E*

_{0}= 1.0 × 10

^{7}V/m it is very fast [

*c*/

_{l}t*a*= 2.5 × 10

^{5}(

*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

**85**, 023837 (2012). [CrossRef]

*γ*

_{−}>

*γ*

_{+}is assumed, because of an opposite condition Δ

*> 0]. According to the quantum theory, the contrast of*

_{AL}*γ*

_{±}larger than 10 is preferable for large population inversion and low threshold values [18

**85**, 023837 (2012). [CrossRef]

*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.

**3**S103–S120 (2001). [CrossRef]

*σ*

_{3}〉

*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.*

^{st}## 4. Conclusions

_{2}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 SiO

_{2}region with lower dielectric constants. Such simple structures facilitate experimental fabrication and realization.

## Acknowledgments

## References and links

1. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

2. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

3. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

4. | P. W. Milloni and J. H. Eberly, |

5. | M. Lindberg and C. M. Savage, “Steady-state two-level atomic population inversion via a quantized cavity field,” Phys. Rev. A |

6. | S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. |

7. | 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. |

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. |

9. | B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. |

10. | M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A |

11. | M. Florescu and S. John, “Resonance fluorescence in photonic band gap waveguide architectures: Engineering the vacuum for all-optical switching,” Phys. Rev. A |

12. | S. John and M. Florescu, “Photonic bandgap materials: towards an all-optical micro-transistor,” J. Opt. A: Pure Appl. Opt. |

13. | R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. |

14. | F. T. Arecchi and E. Courtens, “Cooperative phenomena in resonant electromagnetic propagation,” Phys. Rev. A |

15. | R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A |

16. | R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: Superfluorescence,” Phys. Rev. A |

17. | H. Takeda and S. John, “Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals,” Phys. Rev. A |

18. | H. Takeda, “Collective population evolution of two-level atoms based on mean-field theory,” Phys. Rev. A |

19. | A. A. Belyanin, V. V. Kocharovsky, Vl. V. Kocharovsky, and D. S. Pestov, “Novel schemes and prospects of superradiant lasing in heterostructures,” Laser Phys. |

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. |

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. |

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. |

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. |

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 |

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 |

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 |

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. |

28. | L. Allen and J. H. Eberly, |

29. | I. Kang and F. W. Wise, “Electronic structure and optical properties of PbS and PbSe quantum dots,” J. Opt. Soc. Am. B |

30. | P. M. Naves, T. N. Gonzaga, A. F. G. Monte, and N. O. Dantas, “Band gap energy of PbS quantum dots in oxide glasses as a function of concentration,” J. Non-Crystalline Solids |

31. | M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, “Spectroscopy of 1.55 |

32. | A. Taflove and S. C. Hagness, |

33. | K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E |

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. |

**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

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- 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]
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- 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]
- 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,” Nature432, 200–203 (2004). [CrossRef] [PubMed]
- 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,” Nature445, 896–899 (2007). [CrossRef] [PubMed]
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