SCIENCE CHINA
Physics, Mechanics & Astronomy
• Article •
March 2013 Vol.56 No.3: 524–529
doi: 10.1007/s11433-012-4884-5
The spontaneous emission of an excited atom embedded in photonic
crystals with two atomic position-dependent bands
HUANG XianShan*, LIU HaiLian, WANG Dong, MO XuTao & LIU HouTong
School of Mathematics and Physics, Anhui University of Technology, Maanshan 243002, China
Received March 27, 2012; accepted May 8, 2012; published online September 10, 2012
The spontaneous emission of an excited atom embedded in photonic crystals with two atomic position-dependent bands is investigated. The distribution of the density of states between two bands depends on the atomic position in a unit cell of the photonic crystal and is described with an atomic position-dependent parameter. The result shows that the emitted field and the time
evolution of the upper-level population are affected by the atomic position and the gap width. The spontaneous emission spectrum in free space can be shifted and narrowed with the photonic reservoir and the gap width.
spontaneous emission, photonic crystals, two atomic position-dependent bands
PACS number(s): 42.70.Qs, 42.50.Gy
Citation:
Huang X S, Liu H L, Wang D, et al. The spontaneous emission of an excited atom embedded in photonic crystals with two atomic position-dependent bands. Sci China-Phys Mech Astron, 2013, 56: 524529, doi: 10.1007/s11433-012-4884-5
1 Introduction
Spontaneous emission (SE) is one of the basic problems in
quantum optics. Purcell was the first to predict that the SE
rate for atomic transitions resonant with cavity frequencies
could be enhanced [1]. Later, Purcell’s concept was intensively pursued with different metallic cavities in experimental and theoretical studies [2,3]. Until 1987, John and
Yablonovitch [4,5] respectively discovered photonic crystals (PCs). As a new kind of artificial optical material with
periodic dielectric structures, PCs, through redistributing the
density of states (DOS), create a photonic band gap (PBG)
to forbid those electromagnetic waves whose frequencies
are located in the PBG from propagating in PCs. Now, the
ability to control SE with PCs has many potential applications, such as light emitting diodes [6,7], miniature lasers
[8], high-Q optical microcavities [9,10], photocatalysis materials [11], solar cells [12,13] and so on.
*Corresponding author (email: [email protected])
© Science China Press and Springer-Verlag Berlin Heidelberg 2012
As the photonic DOS near the PBG is significantly different from that of free space vacuum, the SE properties of
an excited atom within PCs are modified, and lots of novel
phenomena in quantum optics have been studied, such as
photon-atom bound states [14,15], spectral splitting [16],
reservoir-induced transparency [17], giant Lamb shift [18],
non-Markovian effects [19,20], etc. However, most previous studies of SE in PC have used a one-band approximation [14–16], which is valid for the wide gap case and
atomic transitions near a PBG. For a real electromagnetic
structure, the width of the band gap is often limited. The
photon-atom coupling in such structures should come from
more than one band. Many of the earlier studies began to
consider two-band approximations, but double bands usually were regarded as two independent parts [21–24], which
were assumed to equally couple to atoms. In fact, according
to Fermi’s golden rule, the field-atom interaction relates to
the atomic position. Li and co-workers [25] investigated the
local density of states (LDOS) in PCs, and they found that
the decay rate of SE was proportional to the LDOS, which
depended on the atomic position inside a unit cell of the PC.
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Huang X S, et al.
Sci China-Phys Mech Astron
Nikolaev et al. [26] observed experimentally that the SE
from quantum dots was controlled by the LDOS in a PC.
Thus, in the narrow gap case, two bands should be considered as a whole reservoir, which depends on the atomic position. Cheng et al. [27] discussed the SE spectrum, absorption and dispersion properties of an atom in the PC with two
bands linked together through the atomic position and they
showed that the SE properties were affected by different
positions of the atom. However, they analyzed the spectrum
of an atomic transition in free space and the absorption and
dispersion properties of the atomic system they obtained
were consistent with ref. [22].
In this paper, we also consider an excited  -typed atom
embedded within a PC with two atomic position-dependent
bands. There are two atomic transitions simultaneously associated with the excited level. One transition is near a
two-band photonic reservoir, while the other is in free space.
Near two band edges, the atomic transition coupling to the
photonic reservoir should be stronger than that in free space.
Due to the quantum interference between two transitions,
we can use one transition channel to control the other transition. With the variation of the relative position of the atom,
we discuss the change of the radiation modes from the
atomic transition modified in PCs. The result further shows
that the dynamic and radiative properties of an excited atom
are affected by the gap width, the atomic transition frequencies, and the distribution of the photonic DOS between two
bands, varying with the relative position of an embedded
atom in a unit cell of the crystal. The SE light from the excited atom can simultaneously travel out via two associated
transitions. The result also shows that the spectrum of the
SE light in free space can be shifted and narrowed by the
distribution of the photonic DOS between two bands. This
means that we may be able to use this property to control
the SE light in free space.
The paper is organized as follows. In sect. 2 we derive
the basic theory from the model we use. In sect. 3 we analyze the dynamic and radiative properties of the atom influenced by the PBG reservoir. Finally, we summarize our
results in sect. 4.
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March (2013) Vol. 56 No. 3

H  i  g  b 2 0   gu (r0 )bu 2 1
u


  gl (r0 )bl 2 1   H.c..
l

Here g  , gu (r0 ) and gl (r0 ) characterize the coupling
strength of the atom with free space, the upper band and the
lower band, respectively, with
gu (l ) (r0 )  21 D21 1 2  0u (l )V d 21  Eu (l ) (r0 ).
Eu (l ) (r0 ) embodies the emitted field from the atom inter-
acting with the upper (lower) band. For simplicity, the
emitted fields can be expressed as Eu (r0 )  Ek sin  (r0 )
and El (r0 )  Ek cos  (r0 ) . The parameter  (r0 ) depends
on the atomic position r0 in a unit cell of the PC. This
means that the two bands correspondingly vary with the
atomic position. Thus, gu (r0 ) and gl (r0 ) can be respectively
written
as
gu (r0 )  g k sin  (r0 )
and
gl (r0 ) 
g k cos  (r0 ) with g k  21 D21 1 2  0kV (d 21  Ek ) . mn
is the atomic transition frequency between the two states
m and n . k is the frequency of the photonic eigen-mode k. D21 ( d 21 ) is the atomic dipole moment (a unit
vector of the atomic dipole). b , bu and bl are the emitted field annihilation operations for the , u and lth reservoir modes with the frequencies  , u and l , respectively. For a two-band isotropic PBG reservoir, the dispersion relation near a gap can be expressed as k 
c1  C1 (k  k0 )2 for k  c1 and k  c 2  C2 (k  k0 )2
for k  c 2 . C1 and C2 are two model-related constants, and c1 ( c 2 ) is the upper (lower) band edge.
We assume the atom is initially in the excited state 2 .
The state vector of the system at arbitrary time t can be
written in terms of the state vectors as:
 (t )  A(t )e  i2 t 2,{0}   B1u (t )e  i(1 u )t 1,{1u }
u
  B1l (t )e
2 Basic theory
 i(1  l ) t
1,{1l }
l
  B0  (t )e  i(0  )t 0,{1 } .
the atom in the states 2,{0}
The atomic transition 2  1 is assumed to be near the
spectively describe the atom in state 2
2  0
is far from the gap.
Under the rotating-wave and electronic dipole approximations, the interaction Hamiltonian for the system can be
written as:
(2)

The model is an excited -type atom embedded in a
PC with two atomic position-dependent bands. The upper
level is 2 , while 1 and 0 are the two lower levels.
gap, but the transition
(1)
Here A(t ) and B0  (t ) are the probability amplitudes of
and 0,{1 } , which rewith no photon
and in the state 0 with a single th mode photon. B1u (t )
( B1l (t ) ) is the probability amplitude of the atom in the state
1,{1u }
with a single uth mode photon in the upper band
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Sci China-Phys Mech Astron
March (2013) Vol. 56 No. 3
(in the state 1,{1l } with a single lth mode photon in the
Here 1  21  c1 ,
lower band).
Substituting eqs. (1) and (2) into the time-dependent
Schrödinger equation, we can obtain the following differential equations for the amplitudes A(t ) , B0  (t ) , B1u (t )

sume C1  C2  C and 1   2   in the paper.
and B1l (t ) .
3 Atomic SE properties in the PBG reservoir
i
dA(t )
  g  B0  (t )e  i( 20 )t
dt

 i(l 21 ) t
(3a)
,
l
dB (t )
i 0   g  A(t )e i( 20 )t ,
dt
(3b)
i
dB1u (t )
 gu (r0 ) A(t )e i(u 21 )t ,
dt
(3c)
i
dB1l (t )
 gl (r0 ) A(t )e i(l 21 )t .
dt
(3d)
is assumed to be a constant  ), from eq. (3), we can obtain
j

x( 2 )  t
x( 3)  t
e j
ej
ej


(1)
(2)
G ( x j ) j F ( x j ) j K ( x(3)
j )

E (r , t )  
k  i(k t  k r )
e
b1k (t )ek .
2 0V
From eqs. (3c), (3d), and (4), we find that 21  Im( x (jn) )
( n  1, 2,3 ) is the frequency of the emitted field E (r , t )
[23]. Obviously, due to c 2  21  Im( x (1)
j )  c1 in the
band gap, the root x (1)
reflects the emitted field E (r , t )
j
with localized characteristic. As the associated transition
2  0 couples to free space, x (1)
is not a purely imj
time. However, for the roots x (2)
and x (3)
j
j , because of
ix   2
bands, the corresponding fields E (r , t ) are the propagating
modes which can travel away from the atom.
According to the number and the value of the roots, we
can divide the roots into six regions. The emitted fields depend on the transition frequency and the width of the PBG,
root; in
as shown in Figure 1. In region I there is one x (2)
j
 ix  1  i cos2 
(with Re( x)  0 or 1  Im( x)   2 );
ix  1  i cos2  i ix   2
(with Re( x)  0 and Im( x)   2 );
K ( x)  x   / 2  sin 2 
directly related to studying the main components of the
emitted fields. For radiation modes very close to the band
edge, the emitted field E (r , t ) at a particular space point r
can be written as [28]:

In eq. (4) these functions are defined as:
F ( x)  x   / 2  i sin 2 
and Im( x (3)
=0 with Re( x (3)
j )  0
j )   1 . The roots are
aginary root, but a complex one. This means that the intencan decay with the
sity of E (r , t ) relating to the root x (1)
j
1  i(21 c 2 )t  x t
e
e J1 ( x)  e i(21 c1 )t e  x t J 2 ( x) dx.
i  0
(4)
G( x)  x   / 2  i sin 2 
k ) . For simplicity, we as-
(2)
is the root of K ( x)
Re( x (2)
x (3)
j )  0 and Im( x j )   2 ;
j
k
Using the Laplace transform, residue theorem and Weisskopf-Wigner approximation to free space modes  (i.e.,
the transition 2  0 is in free space, and the decay rate
x(1)  t
 (21 D21 ) (6  0 C
  c1  c 2 and
3
3
1(2) 0
x (2)
is the root of F ( x)  0 with
or 1  Im( x (1)
j )  2;
j
u
A(t )  
 2  21  c 2 ,
2
In eq. (4), x (1)
is the root of G( x)  0 with Re( x (1)
j )  0
j
  gu (r0 ) B1u (t )e  i(u 21 )t
  gl (r0 ) B1l (t )e
3/ 2
1(2)
(3)
21  Im( x(2)
j )  c 2 and 21  Im( x j )  c1 both in the
root and one x (2)
root; in reregion II there is one x (1)
j
j
gion III there is one x (1)
root; in region IV there is one
j
ix  1  i cos2 
ix   2
(with Re( x)  0 and Im( x)  1 ); G ( x), F ( x) and
K ( x) are the derivative functions of G( x), F(x) and K(x).
J1 ( x)  cos2  ix [ixM 2 ( x)  cos4  ],
J 2 ( x)  i sin 2  ix [ixN 2 ( x)  sin 4  ],
M ( x)  x  i 2   / 2  i sin 2 
 ix  
N ( x)  x  i1   / 2  i cos2 
ix   .
and
x (1)
root and one x (3)
root; in region V there is one x (3)
j
j
j
root; in region VI there are three roots in all: one x (1)
root,
j
root and one x (3)
root. In regions I and V the
one x (2)
j
j
atomic transition frequency deeply locates in the bands
where the photon DOS is relatively small, and so there is
one propagating mode in each region. In region III the transition deeply lies in the gap where the DOS is zero, and
there is one localized mode in this region. In regions II and
IV the transition is respectively near the lower and upper
band edges where the DOS is very large, which leads to the
Huang X S, et al.
Sci China-Phys Mech Astron
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March (2013) Vol. 56 No. 3
Figure 1 The region distribution for the roots with the atomic transition
frequency 21/ and the PBG width /. The decay rate of the transition
2  0 is assumed to be /. The atomic position-dependent parameter
(r0)=/6 (solid line) and (r0)=/3 (dot line).
atomic energy level being split, and there exists one localized mode and one propagating mode in each region. In
region VI, for a narrower gap, the transition strongly couples to double bands, so there are two propagating modes
and one localized mode in the region. We also consider the
influence of different positions of the atom on the radiation
properties in Figure 1. The boundaries of the different regions, i.e., radiation properties, can be manipulated with the
atomic position (the parameter  (r0 ) ). As  (r0 ) varies
from /6 to /3, the area of region II increases, but that of
region IV decreases. During this process, the photonic DOS
near the upper band edge is enhanced, while that of the
lower band edge is reduced. The center of the photonic reservoir moves towards the upper band edge. Consequently,
the coupling between the atom and reservoir with the upper
(lower) band is increased (decreased).
The dynamic properties of the atom can be investigated
by examining the time evolution of the population in the
2
upper level 2 , which can be written as P2 (t )  A(t ) . As
shown in Figure 2(a), when the gap width is very narrow,
such as   0.2  , the atom strongly couples to two bands
and the emitted field can be easily transferred out via PCs,
and thus the population of the level 2 shows rapid decay.
With increasing gap width, the influence on the atom from
two band edges gradually decreases, and the atom can emit
and reabsorb photons back and forth more times. So, the
frequency of the population oscillation increases. In Figure
2(b), we consider the population evolution influenced by the
atomic position in a unit cell of the PC. With an increase of
the parameter  (r0 ) , the coupling to the upper band is enhanced. As the atom transition 2  1 is resonant with
the upper band edge, the population shows slow decay in
the beginning. For long times, the decay rate depends on the
Figure 2
The time evolution of the population in the upper level 2
with (a) the PBG width  and (b) the atomic position parameter . The
parameters (in a unit of ) are (a) 21=99.5 in the gap, = /4 and =0.1;
(b) 21/1c=100, 2c=99, =1 and =0.1.
associated transition 2  0
coupling to the Markovian
reservoir (free space vacuum).
In Figure 3 the SE spectrum of the atomic transition
2  0 in free space vacuum is plotted. From eq. (3), by
using the Laplace transform and final-value theorem, we
can obtain S (  )   | A ( i  ) |2 with      20 , where
A ( s) is the Laplace transform of the probability amplitude
of A(t ) . As plotted in Figure 3, we can see that the SE
spectrum is split into three peaks that are separated by two
dark lines at the band edges. The middle peak can be shifted
with the atomic position [27]. With the increase of the
parameter  (r0 ) from /6 to /3, the increase of the
photon-atom coupling in the upper band can push the
dressed level towards the lower band, as shown in Figure
3(a). In the model, the excited level 2 relates with the
two atomic transitions 2  1
and 2  0 , which
can interfere with each other. In Figure 3(b), we consider
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March (2013) Vol. 56 No. 3
model, one atomic transition couples to the structured reservoir, and the other one interacts with free space vacuum.
For the atomic transition | 2 | 1 near the band edges,
the narrow gap leads to the strong coupling between the
transition and two bands. The relationship between two
bands, originating from the relative position of the embedded atom, results in the different distribution of the photonic
DOS near the two band edges. The characteristic of the
emitted fields modified in PCs, the time evolution of the
population in the upper level 2 , and the SE spectrum of
the transition 2  0
in free space have been discussed,
respectively. The results show that the SE properties are
modified by the photonic reservoir, varying with the atomic
position in a unit cell of the crystal. The quantum interference between two transitions is further illustrated by analyzing the SE spectrum of the transition 2  0 . The
spectrum of SE light in free space can be narrowed with the
relative position of the atom.
The work was supported by the Natural Science College Key Projects of
Anhui Province (Grant Nos. KJ2010A335 and KJ2012Z023), the National
Natural Science Foundation of China (Grant Nos. 41075027 and 61205115)
and the Key Project of Chinese Ministry of Education (Grant No. 212076).
1
2
3
Figure 3 The SE spectrum (arbitrary unit) modified with (a) the atomic
position (i.e., the parameter (r0)) and (b) the gap width . The parameters
(in a unit of ) are 21=99.5 in the gap, =1, (a) =1 and (b) (r0)=/3.
4
5
the SE spectrum modified with the gap width. When the
atomic transition 2  1 lies in the gap, for a large band
6
gap, the spectrum keeps a Lorentzian shape because the
transition is inhibited in the gap and the coupling from two
photonic bands almost can be neglected. With the decrease
of the gap width, the middle peak of the spectrum is significantly narrowed, while those at the two sides are enhanced.
In physics, the transition 2  1 coupling to two bands
7
becomes stronger and more emitted energy can travel out
from the PC. In Figure 1, when the PBG is narrow, we find
that there are more radiation modes from the transitions
modified in the PC.
4 Conclusion
The dynamic and radiative properties of an excited -type
atom embedded in a two-band reservoir, depending on the
atomic position, have been investigated in the paper. In the
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