The novel dynamical and spectral features of (single photon) spontaneous emission from an atomic transition near a photonic band edge are investigated by frustrating these features by spontaneous emission through another atomic transition far from the photonic band edge and deep in the photon continuum, the two atomic transitions forming the two dipole-allowed transitions of a three-level atom in a Λ configuration. Though spontaneous emission through the atomic transition far from the photonic band edge (the “far” transition) is not directly affected by the photonic band gap (PBG), it is indirectly affected by the PBG through its coupling to the atomic transition near the photonic band edge (the “near” transition) via the Λ configuration. As a result of this coupling, spontaneous emission through the “far” transition can be used as a probe to measure the strength of the effects of the PBG on spontaneous emission through the “near” transition. We have shown that the effects of the PBG on spontaneous emission via the “near” transition are strongly affected (and, therefore, can be controlled by) not only by the detuning of the “near” transition from photonic band edge but also by the vacuum decay rate of spontaneous emission through the “far” transition which is coupled to the “near” transition through the Λ configuration. In particular, we have shown that the oscillatory behavior of spontaneous emission near a photonic band edge as well as the Autler-Townes doublet of the spontaneous emission spectrum (due to the splitting of the atomic level near the photonic band edge) are strongly dependent on the decay rate of spontaneous emission through the probe transition which is far from the band edge.
The novel dynamical and spectral features of (single photon) spontaneous emission near the edge of a photonic band gap (PBG) are well known and are extensively discussed in the literature 1, 2, 3, 4, 5, 6. Chief among these novel feature of spontaneous emission near the edge of a PBG are its oscillatory behavior (as opposed to simple exponential decay in vacuum), and non-zero steady-state populations on excited levels (as opposed to zero in the case of vacuum) 6, 7, 8, 9, leading to many actualized and potential applications such as enhancing the light extracting efficiency of LED’s 10, 11, 12, 13, improving photocatalysis 14, 15, optical memory on atomic scale 8, qubits for quantum computation 9, super-radiance 16 as well as photon hopping conduction 17, 18.
In this paper we apply a nearly exact operator formalism (the only approximations being electric dipole and rotating wave approximations) to investigate the dynamical and spectral features of (single photon) spontaneous emission from a three-level atom in a Λ configuration embedded in a PBG. Of the two dipole- allowed transitions of the three-level atom in a Λ configuration, one transition (the “near” transition) is close to the edge of the photonic band gap, whereas the other transition (the “far” transition) is far from the photonic band gap and deep in the photon continuum. As a result, spontaneous emission through the “near” transition is strongly affected by the photonic band gap, whereas that through the “far” transition is not directly affected by the it. However, even though the “far” transition is sufficiently far from the photonic band gap to be directly affected by it, it is indirectly affected by the gap because of its coupling to the “near” transition via the Λ configuration. As a result of this coupling, spontaneous emission through the “far” transition can be used as a probe to measure the strength of the effects of the photonic band gap on spontaneous emission through the “near” transition.
The atomic transition near the photonic band edge (the “near” transition) is characterized by its detuning from the band edge, whereas the atomic transition far from the photonic band gap and deep in the photon continuum (the “far” transition) is characterized by its vacuum decay rate. Our emphasis is on the combined effects of the detuning of the “near” transition from the photonic band edge, and the vacuum decay rate of the “far” transition on the dynamics of spontaneous emission from the “near” transition. We have shown that the effects of the photonic band gap on spontaneous emission via the “near” transition are strongly affected (and, therefore, can be controlled by) not only by the detuning of the “near” transition from photonic band edge but also by the vacuum decay rate of the “far” transition which is coupled to the “near” transition through the Λ configuration. In particular, we have shown that the oscillatory behavior of spontaneous emission near a photonic band edge as well as the Autler-Townes doublet 19 of the spontaneous emission spectrum (due to the splitting of the atomic level near the photonic band edge as a result of the strong non-Markovian 16 interaction between the atom and its own localized radiation) are strongly dependent of the decay rate of the probe transition which is far from the band edge.
An arrangement of a Λ system in a PBG similar to ours has been briefly investigated by John and Quang 7. However, their investigation was based on an isotropic dispersion model for the PBG which is but a non-realistic instructive toy model for studying the effects of a PBG qualitatively. Our detailed investigation in this paper is based on a realistic anisotropic dispersion model for the PBG which introduces several important quantitative corrections over and above the isotropic model 6.
This paper is organized as follows. In the next section (section 2) we describe our model system and write the Hamiltonian of the system under the electric dipole and rotating wave approximations 21. In section 3, we write the general state vector of our model system in terms of the basis vectors of the Hamiltonian of the system as well as on time-dependent population amplitudes, and then derive the general solutions for these time-dependent amplitudes via the time-dependent Schödinger equation. In section 4, we introduce the memory kernels (delay Green’s functions) describing the highly non-Markovian atom-photon interactions in PBG materials, for both isotropic and anisotropic dispersion models of a photonic band gap. Section 5 deals with detailed investigation of the dynamical and spectral behavior of spontaneous emission from our model Λ system under three subsections. Section 6 summarizes our conclusions. Finally, appendices A and B are used to derive the formulas employed in our investigation.
The physical system we consider consists of a single three-level atom embedded in a PBG material. We let |0> denote the ground level of the atom; and |1> and |2> the two excited levels, Figure 1. The three level atom is assumed to be in so called Λ configuration where the uppermost atomic level |2> is dipole coupled to the lower levels |1> and |0> by radiation modes (photon reservoir) in a three dimensional PBG, whereas the transition |1) → |0> is dipole forbidden.
We designate the energy of an atomic level 
by 
and the frequency separation between level 
and 
by
 | (2.1) |
Transitions between atomic levels are described by atomic operators 
with the property 
from which the commutation relation
 | (2.2) |
readily follows. We let 
denote the atomic dipole moment vector and express it in terms of the atomic operators 
as
 | (2.3) |
where 
is the magnitude of 
, and 
is the unit vector in the direction of 
.
We assume that the atom is initially excited to the uppermost level 
by a resonant laser pulse of frequency 
. We then consider the scattering of this pulse via the transitions 
and 
into Rayleigh and Stokes components with frequencies 
and 
. Accordingly, we divide the photon reservoir into two parts, a Raleigh part (identified by the subscript 
) and a Stokes part (identified by the subscript 
). With such a division, the Hamiltonian 
describing our model atom-field system can be written as 6, 20.
 | (2.4) |
where
 | (2.5a) |
 | (2.5b) |
 | (2.5c) |
 | (2.5d) |
 | (2.5e) |
Here 
represents the Hamiltonian of the bare atom, whereas 
and 
represent, respectively, the Hamiltonians of the Rayleigh and Stokes components of the photon reservoir, 
and 
being the wavevector and polarization index of a photon of mode 
. Associated with the Rayleigh and Stokes components are the annihilation (
) and creation (
) operators satisfying the commutation rule
 | (2.6) |
whereby we assumed that the Rayleigh and Stokes components are well separated in frequency that their corresponding field operators commute. We also assume that atomic operators 
commute with the field operators 
and 
for the quantized Stokes and Rayleigh modes.
 | (2.7) |
The Hamiltonians 
and 
given by Eqs. 2.5d and 2.5e are interaction Hamiltonians; 
describes interaction between the atom and the Rayleigh component of the photon reservoir, whereas 
describes the same between the atom and the Stokes component of the photon reservoir. These interaction Hamiltonians are written in the electric dipole approximation whereby the spatial variation of the photon field over the atom is ignored. They are also written in the rotating wave approximation whereby virtual processes of excitation (de-excitation) of the atom with simultaneous creation (annihilation) of a photon (terms of the form 
and 
) are neglected 21, 22, 23, 24, 25. In these interaction Hamiltonians, the factors 
represent the frequency dependent coupling constants between the atomic transitions 
and the modes 
of the radiation field. These coupling constants are derived in 6 and are given by
 | (2.8) |
where 
is the atomic transition frequency given by Eq.2.1, 
and 
are the magnitude and unit vector of the atomic dipole moment 
given by 2.3, 
is the quantization volume of the radiation field, and 
is the Coulomb constant. The vectors 
are the two transverse (polarization) unit vectors satisfying
 | (2.9) |
where 
is the unit vector in the direction of the wavevector 
. The condition 
expresses transversality of the photon field, whereas the condition 
shows that unit vectors 
form a right-handed triad.
For our model atom-field system, we assume that initially the atom is resonantly excited from the ground level 
to the uppermost level 
by a laser pulse of frequency 
whereas both the Rayleigh and Stokes components of the photon reservoir are in the vacuum state. As a result, the initial state of our model system can be written as the direct product of the atomic state 
and the radiation state 
, since the atomic operators are assumed to commute with the radiation field operators (Eq. 2.7).
 | (3.1) |
The notation 
is a short way of writing the direct product state 
which means the atom is in state 
whereas both the Rayleigh and Stokes fields are in the vacuum state (that is, no Rayleigh or Stokes photons) 26.
As a result of the perturbation by the interaction Hamiltonians 
and 
applied at initial time 
, the initial state 
of Eq. 3.1 evolves in time according to the time-dependent Schrödinger equation
 | (3.2) |
where 
is the total Hamiltonian of the system given by Eq. 2.4. At any time 
, the state vector of our model system can be written as a linear combination of the eigenstates of the non-interaction Hamiltonians 
, 
and 
as
 | (3.3) |
where time dependencies due to 
, 
and 
are explicitly factored out in the form of exponentials. The state vector 
means atom in state 
and a single Rayleigh photon of mode 
. Similarly, the state vector 
means atom in state 
and a single Stokes photon of mode 
From Eqs. (3.3) and (3.1), we obtain
 | (3.4) |
as the initial values for the amplitudes 
, 
and 
corresponding to the initial state (3.1).
Using Eq. 2.4 for the total Hamiltonian, and Eq. 3.3 for the general state vector in the Schrödinger equation (Eq. 3.2), and projecting the result onto the eigenstates 
, 
and 
of 
, 
and 
, respectively, we obtain the following (infinite) set of coupled equations for the amplitudes 
, 
and 
 | (3.5a) |
 | (3.5b) |
 | (3.5c) |
Here a dot over an amplitude signifies total time derivative, whereas
 | (3.6) |
represent the detunings of the radiation mode frequencies 
and 
from the atomic transition frequencies 
and 
. Eqs. 3.5b and 3.5c can be integrated (in time), using the initial conditions (3.4), to give
 | (3.7a) |
 | (3.7b) |
Substituting these expressions for 
and 
in Eq. 3.5a and exchanging the order of summation over 
and integration over time, we obtain the following integro-differential equation for 
.
 | (3.8) |
Here
 | (3.9a) |
 | (3.9b) |
are the delay Green's functions associated with the Rayleigh and Stokes components of the photon reservoir, and depend very strongly on the photon density of states of the relevant photon reservoir. In essence, these Green's functions measure the photon reservoir's memory of its previous state at a later time and, therefore, are alternately known as memory kernels 6.
Our goal is to solve Eq. 3.8 for the amplitude 
for different kinds of photon reservoirs which determine the nature of the Green's functions 
and 
. The integral on the right hand side of Eq. 3.8 is a convolution integral which suggests solution by Laplace transformation 27. Upon taking the Laplace transform of both sides of Eq. 3.8, and using the initial condition for 
(Eq. 3.4), we obtain
 | (3.10) |
where 
, 
and 
are, respectively, the Laplace transforms of 
, 
, and 
. For given dispersion relations 
and 
, we evaluate 
and 
from Eqs. 3.9 which, in turn, are used to evaluate 
and 
. These expressions for 
and 
are then used in Eq. 3.10 to find 
. Finally, the amplitude 
is obtained by inverting 
. Once 
is calculated, the excited state population 
on level 
, and the steady-state value 
of this excited state population are obtained from 
by
 | (3.11) |
The memory kernels (delay Green's functions) of Eqs. 3.9 for Rayleigh and Stokes photon reservoirs are highly dependent on the the dispersion relation for the relevant photon reservoir. In the case of vacuum, the dispersion relation is linear and is given by 
, resulting in a density of propagating photon modes 
which varies with frequency 
continuously like 
, Figure 2. For such a linear dispersion relation, the Green's functions of Eq. 3.9 are given by 6
 | (4.1) |
where
 | (4.2) |
is one-half of the spontaneous emission rate 
for the transition 
and 
is the Dirac delta function.
In free space, the memory kernels 
and 
are proportional to the delta function (Eq. 4.1). This is because free space is an infinitely broad photon reservoir (flat spectrum), and, therefore, its response should be instantaneous. Interactions governed by such delta function dependent memory kernels are said to be Markovian 22, 23. From Eq. 4.1, we obtain
 | (4.3) |
for the Laplace transforms of the Green's functions in the case of vacuum.
In a PBG, unlike in vacuum, there is a gap (or gaps) in which the density of propagating photon modes 
is absolutely zero 28, meaning photons with frequencies in the gap cannot propagate in the PBG, Figure 2. A simple model dispersion relation which exhibits a gap in the photon density of states is the so called isotropic effective mass dispersion relation given by 4, 5
 | (4.4) |
Here 
is the upper band edge frequency (Figure 2), 
is the modulus of the wave vector 
, and 
is a constant characteristic of the PBG. The factor 
is a constant which measures the curvature of the dispersion curve 
at 
and is given by
 | (4.5) |
The dispersion relation of Eq. 4.4 is valid only for frequencies close to the upper photonic band edge 
depicted in Figure 2. However, if the width 
of photonic band gap is large enough, and if the relevant atomic transitions are near enough the upper photonic band edge 
, the effects of the lower photonic band would be negligible, making Eq. 4.4 a good approximation.
The dispersion relation of Eq. 4.4 is isotropic because it depends only on the magnitude 
of the wave vector 
. Such a dispersion relation associates the band edge wave vector with the entire sphere 
in 
space (spherical Brillouin zone) and, therefore, artificially increases the true phase space available for photon propagation near the band edge. This results in a photonic density of states 
which, near the band edge 
, behaves as 
for 
, the square-root singularity being characteristic of a one-dimensional phase space 4, 5. While there is no physical PBG material with isotropic gap, the isotropic dispersion relation of Eq. 4.4 gives qualitatively correct results. Realistic anisotropic dispersion relations lead only to quantitative corrections 6.
For the isotropic dispersion relation of 4.4, the Green's functions of Eq. 3.9 are given by 6
 | (4.6a) |
 | (4.6b) |
where
 | (4.7) |
represent the detunings of the atomic transition frequencies 
and 
from the upper band edge frequency 
. At optical frequencies 
and 
so that 
. Thus, if the band-edge 
is also in the optical regime, we have 
.
Eq. 4.6 shows that, in the case of an isotropic PBG, the memory kernels 
decay in time like 
, unlike in the case free space where the 
have delta function time dependence (Eq. 4.1). As a result, unlike in the case of vacuum where atom-photon interaction is Markovian 22, 23, in the case of PBG, atom-photon interactions are highly non-Markovian 16. From Eqs.4.6, we obtain
 | (4.8) |
for the Laplace transforms of the green's functions in the case of isotropic PBG.
In a real 3D PBG, the gap is highly anisotropic and the band edge is associated with a finite collection of symmetry related points in 
space, rather than with the entire sphere 
. For such a realistic PBG, the effective mass approximation of the photon dispersion relation near the upper band edge takes the vector form 4, 5
 | (4.9) |
In this case 
, is approximately given by 
where 
is a dimensionless factor for scaling the different slopes the dispersion curve exhibits in different directions. The anisotropic dispersion relation of Eq. 4.9 leads to a photonic density of states which behaves as 
for 
characteristic of a three-dimensional phase space 4, 5.
For the anisotropic dispersion relation of 4.9, the Green's functions of Eq. 3.9 are given by 6
 | (4.10a) |
 | (4.10b) |
where 
and 
are detuning frequencies given by Eq. 4.7, whereas 
(given by Eq. 4.2) is one-half of the spontaneous emission rate 
for the transition 
. At optical frequencies, 
and 
so that 
. For example, when 
, we obtain 
so that 
, for 
is in the optical regime.
Eqs. 4.10 show that, for the anisotropic dispersion relation of Eq. 4.9, the memory kernels decay in time faster like 
as opposed to the slower 
time decay for the isotropic dispersion relation of Eq. 4.4. The enhanced memory effect for the isotropic model is an artifact of artificially increasing the phase space available for propagating photon modes near the upper band edge by associating the band edge with an entire sphere 
in 
space rather than with a finite collection of symmetry-related points. From Eqs. 4.10, we obtain
 | (4.11) |
for the Laplace transforms of the green's functions in the case of anisotropic PBG.
When the three-level atom in the 
configuration is in vacuum, we use Eq. 4.3 in Eq. 3.10 to obtain
 | (5.1) |
which can be easily inverted to give
 | (5.2) |
showing that, in the case of vacuum, spontaneous emission from level 
is purely exponential, and that, in the stead-state (long-time) limit, the population of the excited level 
is zero. That spontaneous emission is purely exponential, and that all populations on excited levels eventually decay to the ground level is a general result valid not only for vacuum but for any broadband smoothly varying electromagnetic density of states in which the Wigner-Weisskopf approximation is valid 25.
When the electromagnetic density of states changes abruptly in the vicinity of an atomic transition (such as near a photonic band edge), spontaneous emission through the transition displays behaviors dramatically different from those in vacuum. In particular, spontaneous emission near a photonic band edge is shown to be oscillatory (as opposed to simple exponential decay in vacuum), and there may be non-zero (fractionalized) steady-state population in excited states unlike in the case of vacuum in which all excited state populations eventually decay to the ground level 6, 7, 8, 9.
The case of spontaneous emission from a three-level atom in the 
configuration embedded in a PBG material modeled by the isotropic dispersion relation of Eq. 4.4 is briefly discussed in 7. In this work, we discuss the same case in much more detail, with the PBG modeled by the more realistic anisotropic dispersion relation of Eq. 4.9 which gives quantitative corrections over and above the isotropic model.
In our analysis of the three-level atom in the 
configuration, we make two important assumptions. The first assumption is that the Stokes frequency 
lies in the photon continuum far outside the gap so that atom-photon interaction for the transition 
is described by the vacuum memory kernel of Eq. 4.1 for which the Laplace transform is given by 
(Eq. 4.3). The second assumption is that the Rayleigh frequency 
lies near the photonic band edge 
so that atom-photon interaction for the transition 
is described by the anisotropic PBG memory kernel of Eq. 4.10b for which the Laplace transform is given by Eq. 4.11. Applying these two assumptions in Eq. 3.10, we obtain
 | (5.3) |
where
 | (5.4) |
Substituting 
in the equation for 
, we obtain
 | (5.5) |
where
 | (5.6) |
is a quadratic equation whose roots (separated into real and imaginary parts) are given by
 | (5.7) |
where
 | (5.8) |
In terms of the roots of 
, Eq. 5.4 for 
is expressed as
 | (5.9) |
Using Eq. 5.9 in Eq. 5.3, we obtain
 | (5.10) |
The amplitude 
is obtained from the inverse Laplace transforms of 
given by Eq. 5.10 via the complex inversion formula 27
 | (5.11) |
where the real number 
is chosen so that 
lies to the right of all the singularities (poles and branch points) of the function 
. Once 
is calculated, the excited state population 
on level 
as well as the steady-state value 
of this excited state population are obtained from 
via Eq. 3.11.
Our objective in this work is to evaluate 
for a PBG described by the anisotropic dispersion relation of Eq. 4.9, and then investigate the nature and behavior of the excited state population 
as well as its steady-state value 
for such a dispersion relation. To this end, we consider the case in which 
separately from the case in which 
in the following subsections.
Our system of three-level atom in the 
configuration depicted in Figure 1 is initially assumed to be in the upper most level 
(Eq. 3.4). If we further assume that 
, the decay channel 
for the initial population on level 
is completely closed, so that the three-level atom essentially acts as a two-level atom consisting of atomic levels 
and 
. This case of spontaneous emission from a two-level atom near the edge of a photonic band gap has been discussed in detail in 7, albeit under the non-realistic isotropic dispersion of Eq. 4.4. In this subsection, we briefly discuss this two-level atom case under the realistic anisotropic dispersion relation of Eq. 4.9. This 
case involving only two of the three atomic levels, besides being an interesting case in its own right, will be used as a reference for the 
case, whereby all three levels of the three-level atom have to be taken into consideration.
For 
, Appendix B shows that the amplitude 
is given by
 | (5.12) |
where
 | (5.13a) |
 | (5.13b) |
the upper signs of 
applying for 
and 
. The term 
is given by
 | (5.14) |
and represents the branch-cut contribution arising from the deformation of the contour of integration around a branch point in the complex inversion of Eq. 5.10 via Eq. 5.11.
From the population amplitude 
given by Eq. 5.12, the excited state population 
on level 
is obtained by 
(according to Eq. 3.11. This excited state population on level 
is depicted Figure 3 as a function of the scaled time 
for various values of the detuning parameter 
. From this figure we see that, unlike in the case of vacuum where spontaneous emission decay is purely exponential, spontaneous emission near the edge of a photonic band gap exhibits pronounced oscillatory behavior 7. This oscillatory behavior of spontaneous emission near a photonic band edge was first predicted by John and Quang 7 using the isotropic dispersion model of Eq. 4.4. Our result depicted in Figure 3 confirms their result for the more realistic anisotropic dispersion model of Eq. 4.9.
Besides its oscillatory behavior, the other novel feature of spontaneous emission near a photonic band edge is the presence of non-zero (fractionalized) steady-state population on the excited level 
, as opposed to zero in the case of vacuum. This feature of spontaneous emission was first predicted again by John and Qaung 7, and again for the case of the isotropic dispersion relation of Eq. 4.4. We hereby have shown that their prediction is also true for the more realistic anisotropic dispersion relation of Eq. 4.9, albeit with one major difference. For the case of the isotropic dispersion relation, non-zero steady-state population on an excited level was predicted even when the excited level lies slightly outside the gap but not far from it 7. However, for the case of anisotropic dispersion relation, non-zero steady-state population exists on an excited level only when the level is strictly inside the gap so that its detuning 
from the band edge 
is negative. This discrepancy between the two dispersion relations is due to the enhanced memory effect of the isotropic one which varies as 
(according to Eq. 4.6) as opposed to that of the anisotropic one which varies as 
(according to Eq. 4.10).
For the 
case, Appendix B shows that the steady-state value of the excited state population on level 
is given by
 | (5.15) |
where 
are given by Eq. 5.13a. Therefore, even when the excited level 
is right on the band edge so that 
there is no excited population on level 
in the steady-state limit. The variation of this steady-state population of level 
according to Eq. 5.15 is depicted in Figure 4.
For the 
case, Appendix A shows that 
is given by
 | (5.16) |
where 
are given by Eq. 5.7, and
 | (5.17) |
represents the branch-cut contribution of the complex inversion of Eq. 5.10.
Figure 1 depicts the variations of the excited state population 
of level 
for 
given by Eq. 5.16 as a function of the scaled time 
for various values of the detuning parameter 
. From this figure we see that, like in the 
case, in the 
case the spontaneous decay of the excited state population displays strong oscillatory behavior. However, unlike in the 
case, in the 
case all excited state population eventually decays to the ground level in the long-time (steady-state) limit, even when the excited level lies deep inside the photonic band gap. This total decay in the excited population of level 
occurs through decay channel 
which is assumed to be far outside the gap and deep in the photon continuum.
We can easily see why the steady-state population on the exited level 
is always zero when 
, by investigating the three terms on the RHS of Eq. 5.16. Since 
and 
at optical frequencies, from Eq. 5.7 we see that the real and imaginary parts of root 
are both positive whereas those of root 
are both negative. Therefore, the roots 
and 
can be expressed as
 | (5.18) |
where 
and 
are positive real numbers given by
 | (5.19a) |
 | (5.19b) |
 | (5.19c) |
As a result, the exponentials on the RHS of Eq. 5.16 involving the roots 
and 
can be cast as
 | (5.20a) |
 | (5.20b) |
showing that the first two terms on the RHS of Eq. 5.16 are both oscillatory due to the factors 
and 
but both decay to zero in the steady-state limit (
) due to the factors 
and 
. Moreover, the integrand on the RHS Eq. 5.16 decays to zero as 
. It follows that all three terms on RHS of Eq. 5.16 decay to zero as 
, so that the amplitude 
and, therefore, the population 
decay to zero in the steady-state limit.
The oscillatory behavior of the excited state population 
on level 
(which is assumed to be near the photonic band edge 
) depends on the strength of the decay channel 
which is specified by the decay parameter 
. As shown in Figure 6, the smaller the value of 
, the slower the decay of 
and the longer the oscillation life-time of 
, though the oscillation frequency of 
depends only on the value of 
. The ability to control the oscillatory behavior of spontaneous emission near the edge of a photonic band gap using 
as depicted in Figure 6 is an important conclusion of our investigation.
With 
given by Eq. 5.10, the spectrum of the spontaneous emission through the decay channel 
is given by 7
 | (5.21) |
Figure 7 depicts the spectrum 
for 
and for different values of the detuning 
of the atomic transition frequency 
from the band edge frequency 
. From this figure we see that for small 
(that is, when level 
is close to the band edge) the spectrum 
splits into a doublet. This splitting is analogous to the Autler-Townes splitting (dynamic Stark splitting) of an excited level into two dressed states by a resonant (or nearly resonant) external driving field. In the PBG case, however, there is no external driving field, and the splitting is solely due to the strong non-Markovian interaction of the atom with its localized radiation field 1, 2, 14. This splitting was first predicted by John and Quang 7, albeit for a PBG described by the isotropic dispersion relation of Eq. 4.4 which artificially enhances the effects of the PBG by artificially increasing the phase space available for band edge photons. The splitting depicted in Figure 7 is for a PBG describe by the more realistic anisotropic relation of Eq. 4.9 and, therefore, is not as pronounced as that in 7.
Figure 8 depicts the spectrum 
for 
, and for different values of the decay rate 
of the spontaneous emission through the transition 
which is assumed to be sufficiently far from the gap to be directly affected by the gap. From this figure, we see that the smaller the value of 
the more pronounced the splitting effect on 
, though the positions of the peaks is determined only by 
. The ability to control the spectrum of spontaneous emission near the edge of a photonic band gap using 
as depicted in Figure 8 is another important conclusion of our investigation.
We investigated spontaneous emission from a three-level atom in a 
configuration embedded in a photonic band gap (PBG) material. Of the two dipole-allowed transitions of the atom, one is near the photonic band edge (the “near'' transition), whereas the other one is sufficiently far from the photonic band gap not to be directly affected by the gap (the “far” transition). Though the “far” transition is not directly affected by the photonic band gap, it is indirectly affected by the gap because of its coupling to the “near” transition via the 
configuration. As a result of this coupling, spontaneous emission through the “far” transition can be used as a probe to measure the strength of the effects of the photonic band gap on spontaneous emission through the “near” transition. In this paper, we showed that the oscillatory behavior as well as the spectrum of the spontaneous emission through the “near” transition are strongly dependent (and, therefore, can be controlled by) the vacuum decay rate of the “far” transition which lies far outside the gap and deep in the photon continuum. In short, we have shown that spontaneous emission through an atomic transition near the edge of a photonic band gap can be investigated by frustrating it by another atomic transition far from the photonic band gap but coupled to the near transition via a 
configuration.
The atomic amplitude 
is obtained from the inverse Laplace transform of 
given by Eq. 5.10 following similar Laplace inversion by Woldeyohannes and John 6. This inverse Laplace transform is evaluated via the complex inversion formula 27
 | (A.1) |
where the real number 
is chosen so that 
lies to the right of all the singularities (poles and branch points) of the function 
. It is apparent from Eq. (5.10) that 
is a branch point of 
. In order to evaluate (A1), we consider the contour C shown in Figure 9 where the branch cut of the integrand is chosen to lie along the negative real axis. According to the residue theorem,
 | (A.2) |
where 
is the sum of the residues of the integrand at the poles enclosed by the contour 
. Omitting the integrands, we obtain
 | (A.3) |
In the limit 
and 
(so that 
), the integrals over the curves BDE, HJL and LNA on the RHS of Eq. (A1) approach zero and, according to Eq. (5.10), the integral over the line AB gives 
Therefore,
 | (A.4) |
Along EH, 
; and using this in Eq. 5.10 we obtain
 | (A.5) |
Similarly, along KL, 
; and using this in Eq. 5.10 we obtain
 | (A.6) |
Using Eqs. A.5 and A.6 in Eq.A4 we obtain
 | (A.7) |
Next we evaluate the total residue 
. From Eq. (5.10), we obtain
 | (A.8) |
Clearly, the function 
has simple poles at 
and 
. The residues at 
and 
are given by
 | (A.9) |
 | (A.10) |
For the complex roots 
and 
given by Eq. 5.7, we choose the positive branch of the square root function and set 
, 
. The residues at 
and 
are then given by
 | (A.11) |
so that the total residue is
 | (A.12) |
Using Eq. A.12 in Eq. A.7, we finally obtain Eq. 5.16.
With the initial condition of Eq. 3.4 (that is, atom initially on the upper most level 
), and with the assumption that 
(so that the decay channel 
for the population of level 
is completely closed), the three-level atom of Figure 1 essentially acts as a two-level atom consisting of levels 
and 
. In this two-level atom case, the expression for the amplitude 
is obtained as follows 6.
For 
, Eq. 5.16 reduces to
 | (B.1) |
and its roots are given by
 | (B.2) |
where the upper sign of 
applies for 
. Just like in the 
case, in the 
case, the amplitude 
is obtained via the complex inversion formula of Eq. A.1 using the contour of Figure 9. The inversion gives different results depending on the value of the detuning 
of the atomic transition frequency 
from the band edge frequency 
. We consider three different cases, the 
case (case I), the 
case (case II), and the 
case (case III).
1. Case I: δ < 0
For 
, the roots of Eq. B.1 are given by
 | (B.3) |
In this case, 
is a positive real number and, therefore, lies inside the contour of integration so that it has a non-zero residue. On the other hand, 
is a negative real number and, therefore, lies outside the contour of integration so that its residue is zero. As a result, when 
, the amplitude 
is given by
 | (B.4) |
where
 | (B.5) |
and 
are given by B.3.
2. Case II: 
In this case
 |
As a result, both roots of Eq. B.1 are negative real numbers given by
 | (B.6) |
and, therefore, lie outside the contour of integration so that their residues are zero. The only contribution to 
is the branch-cut contribution. The amplitude 
is then given by
 | (B.7) |
where 
is given by Eq. B.5.
Case III: 
In this case,
 |
As a result, the roots of Eq. B.1 are complex conjugates of each other and can be expressed in terms of positive real numbers 
and 
as
 | (B.8) |
so that
 | (B.9) |
 | (B.10) |
The residue corresponding to root 
increases exponentially in time due to the factor 
and, therefore, is unphysical. As a result, when 
, the amplitude 
is given by
 | (B.11) |
where 
are given by Eq. B.8 and 
is given by Eq. B.5. Summarizing our results for the 
case, we finally obtain
 |
where
 | (B.13a) |
 | (B.13b) |
the upper signs of 
applying for 
and 
, and 
is given by Eq. B.5.
4. Steady-state population on level 
when 
In Eq. B.12a, the exponential 
is a non-decaying oscillatory factor because root 
(which is given by Eq. B.13a) is a positive real number. On the other hand, in Eq. B.12c the exponential 
decays in time like 
as shown by Eq. B.10. The term 
in Eqs. B.12a-c also decays in time and tends to zero as 
. As a result, in Eqs. B.12a-c for 
, only the first term in Eq. B.12a remains in the long-time limit. Therefore, in the long-time limit, the amplitude 
is given by
 | (B.14) |
where 
are both real and are given by Eq. 13a, the upper sign of 
applying for 
. The steady-state population of level 
is then given by
 | (B.15) |
| [1] | S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett., 53, 2169 (1984). | ||
| In article | View Article | ||
| [2] | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett., 58, 2486 (1987). | ||
| In article | View Article PubMed | ||
| [3] | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett., 58, 2059 (1987). | ||
| In article | View Article PubMed | ||
| [4] | S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett., 64, 2418 (1990). | ||
| In article | View Article PubMed | ||
| [5] | S. John and J. Wang, “Quantum optics of localized light,” Phys. Rev. B, 43, 12772 (1991). | ||
| In article | View Article PubMed | ||
| [6] | M. Woldeyohannes and S. John, “Coherent control of spontaneous emission near a photonic band edge: Ph.D Tutorial”, J. Opt. B, 5, R43-R82 (2003) | ||
| In article | View Article | ||
| [7] | S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A, 50, 1764 (1994). | ||
| In article | View Article PubMed | ||
| [8] | T. Quang, M. Woldeyohannes, S. John, and G. S. Agarwal, “Coherent control of spontaneous emission near a photonic band edge: a single-atom optical memory device,” Phys. Rev. Lett., 79, 5238 (1997). | ||
| In article | View Article | ||
| [9] | M. Woldeyohannes and S. John, “Coherent control of spontaneous emission near a photonic band edge: a qubit for quantum computation,” Phys. Rev. A, 60, 5046 (1999). | ||
| In article | View Article | ||
| [10] | M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol., 17, 2096 (1999). | ||
| In article | View Article | ||
| [11] | T. Baba, K. Inoshita, H. Tanaka, J. Yoenkura, M. Ariga, A. Matsutani, T. Miyamoto, F. Koyama, and K. Iga, “Strong enhancement of light extraction efficiency in GaInAsP 2-D-arranged micro-columns,” J. Lightwave Technol., 17, 2113 (1999). | ||
| In article | View Article | ||
| [12] | W. L. Barnes, “Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices,” J. Lightwave Technol., 17, 2170 (1999). | ||
| In article | View Article | ||
| [13] | S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Ref. Lett., 78, 3294 (1997). | ||
| In article | View Article | ||
| [14] | C. M. Soukoulis, Ed., “Photonic band gaps and localization,” in Proc. NATO Advanced Research Workshop on Localization and Propagation of Classical Waves in Random and Periodic Media, Heraklion, Crete, Greece, May 26-30, 1992; see also, Nato Advanced Studies Institute Series B: Physics. New York: Plenum, 1993, Vol. 308. | ||
| In article | |||
| [15] | N. M. Lawandy and G. Kweon, “Molecular and free electron spontaneous emission in periodic three- dimensional dielectric structures,” in Ref [14]. | ||
| In article | |||
| [16] | N. Vats and S. John, “Non-Markovian quantum fluctuations and superradiance near a photonic band edge”, Phys. Rev. A, 58, 4168 (1998). | ||
| In article | View Article | ||
| [17] | S. John and T. Quang, “Photon-hopping conduction and collectively induced transparency in a photonic band gap”, Phys. Rev. A, 55, 4083 (1995). | ||
| In article | View Article PubMed | ||
| [18] | M. Woldeyohannes, I. Idehenre, and T. Hardin, “Coherent control of cooperative spontaneous emission from two identical three-level atoms in a photonic crystal”, J. Opt,, 17, 085105 (2015). | ||
| In article | View Article | ||
| [19] | H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev., 100, 703 (1955). | ||
| In article | View Article | ||
| [20] | M. Woldeyohannes, S. John, and V. I. Rupasov, “Resonance Raman scattering in photonic band gap mate- rials,” Phys. Rev. A, 63, 013814 (2001). | ||
| In article | View Article | ||
| [21] | L. Allen and J. H. Eberly, Optical Resonance and Two-level atoms. New York: Wiley, 1975. | ||
| In article | |||
| [22] | P. Meystre and M. Sargent, Elements of Quantum Optics. New York: Springer-Verlag, 1991. | ||
| In article | View Article | ||
| [23] | M.O Scully and S. Zubairy, Quantum Optics. London: Cambridge University Press, 1997. | ||
| In article | View Article | ||
| [24] | D. F. Walls and G. J. Milburn, Quantum Optics. New York: Springer-Verlag, 1995. | ||
| In article | View Article | ||
| [25] | W. H. Louisell, Quantum Statistical Properties of Radiation. New York: Wiley, 1973. | ||
| In article | |||
| [26] | C. Cohen-Tannoudji, B. Diu and F. Lalo¨e, Quantum Mechanics, Vol.1. Toronto: Wiley, 1977. | ||
| In article | |||
| [27] | M. R. Spiegel, Schaum’s Outline of Theory and Problems of Laplace Transforms. New York: McGraw-Hill, 1965. | ||
| In article | |||
| [28] | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals. Molding the Flow of Light. Princeton NJ: Princeton University Press, 1995. | ||
| In article | |||
Published with license by Science and Education Publishing, Copyright © 2023 Mesfin Woldeyohannes
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
https://creativecommons.org/licenses/by/4.0/
| [1] | S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett., 53, 2169 (1984). | ||
| In article | View Article | ||
| [2] | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett., 58, 2486 (1987). | ||
| In article | View Article PubMed | ||
| [3] | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett., 58, 2059 (1987). | ||
| In article | View Article PubMed | ||
| [4] | S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett., 64, 2418 (1990). | ||
| In article | View Article PubMed | ||
| [5] | S. John and J. Wang, “Quantum optics of localized light,” Phys. Rev. B, 43, 12772 (1991). | ||
| In article | View Article PubMed | ||
| [6] | M. Woldeyohannes and S. John, “Coherent control of spontaneous emission near a photonic band edge: Ph.D Tutorial”, J. Opt. B, 5, R43-R82 (2003) | ||
| In article | View Article | ||
| [7] | S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A, 50, 1764 (1994). | ||
| In article | View Article PubMed | ||
| [8] | T. Quang, M. Woldeyohannes, S. John, and G. S. Agarwal, “Coherent control of spontaneous emission near a photonic band edge: a single-atom optical memory device,” Phys. Rev. Lett., 79, 5238 (1997). | ||
| In article | View Article | ||
| [9] | M. Woldeyohannes and S. John, “Coherent control of spontaneous emission near a photonic band edge: a qubit for quantum computation,” Phys. Rev. A, 60, 5046 (1999). | ||
| In article | View Article | ||
| [10] | M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol., 17, 2096 (1999). | ||
| In article | View Article | ||
| [11] | T. Baba, K. Inoshita, H. Tanaka, J. Yoenkura, M. Ariga, A. Matsutani, T. Miyamoto, F. Koyama, and K. Iga, “Strong enhancement of light extraction efficiency in GaInAsP 2-D-arranged micro-columns,” J. Lightwave Technol., 17, 2113 (1999). | ||
| In article | View Article | ||
| [12] | W. L. Barnes, “Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices,” J. Lightwave Technol., 17, 2170 (1999). | ||
| In article | View Article | ||
| [13] | S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Ref. Lett., 78, 3294 (1997). | ||
| In article | View Article | ||
| [14] | C. M. Soukoulis, Ed., “Photonic band gaps and localization,” in Proc. NATO Advanced Research Workshop on Localization and Propagation of Classical Waves in Random and Periodic Media, Heraklion, Crete, Greece, May 26-30, 1992; see also, Nato Advanced Studies Institute Series B: Physics. New York: Plenum, 1993, Vol. 308. | ||
| In article | |||
| [15] | N. M. Lawandy and G. Kweon, “Molecular and free electron spontaneous emission in periodic three- dimensional dielectric structures,” in Ref [14]. | ||
| In article | |||
| [16] | N. Vats and S. John, “Non-Markovian quantum fluctuations and superradiance near a photonic band edge”, Phys. Rev. A, 58, 4168 (1998). | ||
| In article | View Article | ||
| [17] | S. John and T. Quang, “Photon-hopping conduction and collectively induced transparency in a photonic band gap”, Phys. Rev. A, 55, 4083 (1995). | ||
| In article | View Article PubMed | ||
| [18] | M. Woldeyohannes, I. Idehenre, and T. Hardin, “Coherent control of cooperative spontaneous emission from two identical three-level atoms in a photonic crystal”, J. Opt,, 17, 085105 (2015). | ||
| In article | View Article | ||
| [19] | H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev., 100, 703 (1955). | ||
| In article | View Article | ||
| [20] | M. Woldeyohannes, S. John, and V. I. Rupasov, “Resonance Raman scattering in photonic band gap mate- rials,” Phys. Rev. A, 63, 013814 (2001). | ||
| In article | View Article | ||
| [21] | L. Allen and J. H. Eberly, Optical Resonance and Two-level atoms. New York: Wiley, 1975. | ||
| In article | |||
| [22] | P. Meystre and M. Sargent, Elements of Quantum Optics. New York: Springer-Verlag, 1991. | ||
| In article | View Article | ||
| [23] | M.O Scully and S. Zubairy, Quantum Optics. London: Cambridge University Press, 1997. | ||
| In article | View Article | ||
| [24] | D. F. Walls and G. J. Milburn, Quantum Optics. New York: Springer-Verlag, 1995. | ||
| In article | View Article | ||
| [25] | W. H. Louisell, Quantum Statistical Properties of Radiation. New York: Wiley, 1973. | ||
| In article | |||
| [26] | C. Cohen-Tannoudji, B. Diu and F. Lalo¨e, Quantum Mechanics, Vol.1. Toronto: Wiley, 1977. | ||
| In article | |||
| [27] | M. R. Spiegel, Schaum’s Outline of Theory and Problems of Laplace Transforms. New York: McGraw-Hill, 1965. | ||
| In article | |||
| [28] | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals. Molding the Flow of Light. Princeton NJ: Princeton University Press, 1995. | ||
| In article | |||
