## 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 Optical Society of America

## 1. Introduction

In photonic crystals (PC’s) composed of periodic dielectric materials [1,2], 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] 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]. 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 [5–8]. 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 *ω _{L}*, populations at ground and excited states oscillate with the Rabi frequency 2Ω. Then, there appear three frequency components

*ω*and

_{L}*ω*± 2Ω (Mollow triplet) in radiative electromagnetic waves [9]. When a contrast of EM LDOS’s at

_{L}*ω*=

*ω*± 2Ω is large, population inversion can be achieved even in two-level atoms [6, 10–12].

_{L}In previous studies, the above discussions were mainly conducted by the quantum theory in which both electrons and photons are second quantized [6, 10–12]. However, when a large number of two-level atoms are coherently excited, electromagnetic fields emitted from them can be treated classically (Dicke model), since quantum fluctuation of emitted electromagnetic fields can be neglected [13–16]. For example, it is well known that the superradiance, in which electromagnetic fields emitted from collective atoms are greatly enhanced before decaying, can be recaptured by the semiclassical theory. In the semiclassical theory, while electrons are quantized to ground and excited states, electromagnetic fields are treated classically. Therefore, I consider the population inversion of collective two-level atoms driven by an external laser, using the self-consistent Maxwell-Bloch equations based on the semiclassical theory [17]. The validity of the semiclassical theory for two-level collective population inversion is discussed in Ref. [18]. Optical Bloch and Maxwell equations describe electronic states of atoms and electromagnetic fields, respectively. Unlike the quantum theory, the self-consistent Maxwell-Bloch equations enable me to estimate the above properties in concrete geometries of PC’s, based on the first-principle calculation without using empirical parameters such as decay terms. In the first-principle calculation, only structural parameters such as dielectric constants are used. Although the Maxwell-Bloch equations are widely used in laser physics, phenomenological decay terms are generally attached by hand for spontaneous emission and dipole dephasing [19], and moreover, population inversion by pumping is initially set up for active media. What I investigate in this paper is the steady-state population inversion from the ground state driven by an external laser. Therefore, such phenomenological Maxwell-Bloch equations are not valid for this analysis.

In Ref. [17], 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]. Certainly, experimental techniques have been improved so far. For example, the inhibition of spontaneous emission has been observed in a woodpile [21]. 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]. 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 SiO_{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]. 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, 25], the vacuum Rabi splitting has also been reported in 1D micropillar cavities [26, 27]. 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* = *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 $\u3008{\sigma}_{i}(t)\u3009=\mathit{tr}\left[\rho {\sigma}_{i}(t)\right]={\sum}_{n=1}^{2}\u3008n\left|\rho {\sigma}_{i}(t)\right|n\u3009$, where

*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

*σ*

_{1}(

*t*)〉

^{2}+ 〈

*σ*

_{2}(

*t*)〉

^{2}+ 〈

*σ*

_{3}(

*t*)〉

^{2}= 1 (probability conservation law) is satisfied, since $\frac{d\left[{\u3008{\sigma}_{1}(t)\u3009}^{2}+{\u3008{\sigma}_{2}(t)\u3009}^{2}+{\u3008{\sigma}_{3}(t)\u3009}^{2}\right]}{dt}=2\left[\u3008{\sigma}_{1}(t)\u3009\frac{d\u3008{\sigma}_{1}(t)\u3009}{dt}+\u3008{\sigma}_{2}(t)\u3009\frac{d\u3008{\sigma}_{2}(t)\u3009}{dt}+\u3008{\sigma}_{3}(t)\u3009\frac{d\u3008{\sigma}_{3}(t)\u3009}{dt}\right]=0$, 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].

**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.

Next, I consider the Maxwell equations with electric polarizations.

*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], 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**′ (

*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}[29–31]. 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] to solve the first-principle Maxwell-Bloch equations, and
then, discretizations of space and time are
Δ*x*/*a* =
Δ*y*/*a* =
Δ*z*/*a* = 1/10, and
*c _{l}*Δ

*t*/

*a*= 1/20, respectively, where

*a*is the lattice constant of PC’s and ${c}_{l}=1/\sqrt{{\epsilon}_{0}{\mu}_{0}}$ 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].

## 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 SiO_{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]. Dielectric constants of Si and SiO_{2} are *ε _{Si}* = 11.9 and

*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

Figure 2(a) shows the photonic band structure of the realistic 1D PC calculated by the plane wave expansion method [33] with 6727 plane waves in a supercell of 5*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

*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]. The left and right edges of the realistic 1D PC are SiO

_{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.

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*/2*πc* ≃ 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*/2*πc _{l}* = 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

_{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

I consider the collective spontaneous emission at (*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]. In Fig. 3(a), I show the spontaneous emission rate at (

_{A}*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]. While at

*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, 18]. The dipole radiation emits electromagnetic waves from collective atoms, and then, the atoms lose the excitation energies. Therefore, 〈

*σ*

_{3}(

*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*) = *A*_{0}**u*** _{d}*[exp(−

*iω*) + exp(

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

_{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

*R*= |

_{ij}*ĩ*〉〈

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

*c*|1〉+

*s*|2〉 and |2̃〉 = −

*s*|1〉+

*c*|2〉, respectively. ( ${c}^{2}=\frac{1}{2}\left[1+\frac{{\mathrm{\Delta}}_{AL}}{2\mathrm{\Omega}}\right]$ and ${s}^{2}=\frac{1}{2}\left[1-\frac{{\mathrm{\Delta}}_{AL}}{2\mathrm{\Omega}}\right]$). Applying

*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,

*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.

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]. Δ* _{AL}* < 0 is assumed. |

*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]. The spontaneous emission rates at

_{L}*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], where the operator

*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, 12, 18].

*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}I assume that 125 two-level atoms placed at (*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], and then, the spectrum shows broad peaks at

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

Figure 6(a) shows the behavior of 〈*σ*_{3}〉* ^{st}* (steady-state value of 〈

*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, 12, 18]. (Although in Refs. [6, 12] a very large number of two-level atoms*

^{st}*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*

^{st}*η*= (

*c*

^{4}

*γ*

_{+})/(

*s*

^{4}

*γ*

_{−}) [6, 12]. Gray dashed and solid lines correspond to the behaviors of 〈

*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, 12], when

*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]. In fact, while saturation time is very slow near the threshold value because of |

*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]. However,

*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]. 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.

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]. Moreover, when referring to positive and negative 〈*σ*_{3}〉* ^{st}* 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 SiO_{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.

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.

In the atom-laser interaction, moreover, there appear three frequency components of the laser frequency, lower and higher Mollow sidebands in radiative electromagnetic waves. When the spontaneous emission rates at the lower and higher Mollow sidebands have a large contrast of EM LDOS’s, population inversion of collective two-level atoms can be achieved, even in the presence of leakage of light perpendicular to periodic directions. Then, intensities of input electric fields must exceed the threshold value. The condition of collective population inversion coincides with that predicted by the previous study.

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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