Zeng et al.
Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B
2253
Spontaneous emission spectrum of a V-type three-level
atom in a Fabry–Perot cavity containing
left-handed materials
Xiaodong Zeng,2,* Mingzhang Yu,2 Dawei Wang,3 Jingping Xu,1,2 and Yaping Yang1,2,3
1
Key Laboratory of Advanced Micro-Structure Materials, Ministry of Education,
Tongji University, Shanghai 200092, China
2
Department of Physics, Tongji University, Shanghai 200092, China
3
Beijing Computational Science Research Center, Beijing 100084, China
*Corresponding author: zengxdgood@163.com
Received March 21, 2011; revised June 11, 2011; accepted July 22, 2011;
posted July 25, 2011 (Doc. ID 144507); published August 26, 2011
The spontaneous emission of a V-type Zeeman atom in a Fabry–Perot cavity containing left-handed materials
(LHMs) is investigated. Because of the strong indirect quantum interference induced by the refocusing and phase
compensation of LHMs, the population evolution and the emission spectrum are much different from that in
isotropic environments at different initial conditions. For the degenerate cases, by preparing different initial states,
the population decays much faster or slower than that in free vacuum, while the spontaneous emission spectra are
narrowed or broadened, respectively. For large detuning cases, the population exchange between two upper levels
could be weakened, and the Fano minimum appears in the emission spectrum. In addition, the influence of the
dipole orientation on the spectrum is discussed. © 2011 Optical Society of America
OCIS codes: 160.3918, 270.1670.
1. INTRODUCTION
Left-handed materials (LHMs) were first investigated by
Veselago in 1948 [1]. Since both the effective permittivity and
permeability of LHMs are negative in a certain frequency
range, the refractive index is also negative according to
Maxwell’s equations. Therefore, the electric field, magnetic
field, and wave vector form a left-handed system, and the
energy flux of the electromagnetic field is opposite to the
wave vector. After the fabrication of LHMs in experiment [2],
LHMs have attracted a great deal of attention during the past
decade [3–8]. Potential applications have been widely proposed, such as in highly efficient low reflectance surfaces [9],
superlenses [10,11], and broadband ground-plane cloaks [12].
Furthermore, atomic spontaneous decay in the presence of an
LHM was recently investigated [13].
On the other hand, quantum interference among different
decay channels of multilevel atoms is one of the basic topics
in quantum optics, because it leads to many interesting
effects, such as electromagnetically induced transparency
[14], ultranarrow spectral lines [15], coherence trapping of
populations [16], and lasing or gain without inversion [17].
Most of the previous studies about quantum interference
assumed that the two degenerate transition dipole moments
were nearly parallel or antiparallel. There are few systems that
can satisfy the above condition. The Zeeman atom can have
two nearly degenerate sublevels, but the two corresponding
transition dipoles are orthogonal to each other. In 2000,
Agarwal suggested that quantum interference between the
two orthogonal dipoles can be achieved by placing the atom
in an anisotropic vacuum [18]. Recently, Yang and co-workers
pointed out that a cavity containing LHMs could provide a
large anisotropic space and the Zeeman atom in it could have
0740-3224/11/092253-07$15.00/0
strong quantum interference [19,20]. Since such quantum
interference originates from the coupling of one dipole with
the reflected field emitted from the other dipole, they defined
such interference as “indirect quantum interference.” Because
of such indirect interference, the photon can exchange between the two upper levels, which induces the spontaneous
evolution and the spectrum to be much different from those
in free vacuum. By preparing different initial conditions, the
decay could be canceled or accelerated, and the corresponding emission spectrum could be narrowed or broadened,
respectively. In addition, we can change the atomic dipole
directions to make the two degenerate dipoles have different
decay rates and still have strong quantum interference. In this
paper, we will investigate the population evolution and emission spectrum of a V-type Zeeman atom in such a cavity based
on indirect quantum interference.
This paper is organized as follows. In Section 2, we introduce the model and derive the atomic evolution equations
and the formula of emission spectrum under proper approximations. In Section 3, we discuss atomic spontaneous emission in company with indirect quantum interference, and
compare it with that in free vacuum. In last section, we draw
a conclusion.
2. MODEL AND THE QUANTIZATION
SCHEME
We consider a V-type Zeeman atom placed in the middle of a
Fabry–Perot cavity containing LHMs, as is shown in Fig. 1.
The cavity wall is made of a LHM slab, with indices εA and μA ,
and thickness dA , mounted on a mirror. The distance between
the two LHM slabs is 2dA . Interfaces are in the x–y plane, and
the atom is located at the origin. The upper levels jai and jbi
© 2011 Optical Society of America
2254
J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011
A
LHM
A
A
LHM
x
atom
0
dA
dA
Perfect mirror
Perfect mirror
eΑ mΑ
Zeng et al.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^jN ðr; ωÞ ¼ ω ℏε0 Imεðr; ωÞ^f e ðr; ωÞ
π
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℏ
þ∇× −
Im½1=μðr; ωÞ^f m ðr; ωÞ;
πμ0
a
b
z
where ε0 and μ0 are the permittivity and permeability in
vacuum, respectively. Gðr; r0 ; ωÞ is the Green’s tensor in the
environment of Fig. 1 satisfying the following equation:
c
dA
Fig. 1. Zeeman atom with two upper levels jai, jbi and a ground level
jci is placed in the middle of two LHM slabs attached on two perfect
mirrors. The thickness of the LHM slab is dA , and the distance
between the slabs is 2dA .
are coupled by the same vacuum modes to the lower level jci
with transition frequencies ωa and ωb , respectively. The
corresponding transition dipoles of jai → jci and jbi → jci
are orthogonal to each other, and the transition process jai →
jbi is forbidden. The atomic dipole moment is given by
d ¼ dðAac e1 þ Abc e2 Þ þ H:c:;
ð1Þ
∇×
↔
1
ω2
∇ × − 2 εðr; ωÞ Gðr; r0 ; ωÞ ¼ I δðr − r0 Þ:
μðr; ωÞ
c
jψðtÞi ¼ C a ðtÞe−iωa t jf0gijai
þ C b ðtÞe−iωb t jf0gijbi þ
Z
×
pffiffiffi
¼ ðez iex Þ= 2;
ð2Þ
Aij ¼ jiihjjði; j ∈ fa; b; cgÞ are the atomic transition ði ≠ jÞ and
population ði ¼ jÞ operators, ek ðk ¼ x; y; zÞ are the normalized
Cartesian basis vectors, and d is the atomic dipole strength,
chosen to be real.
Under the rotating-wave approximation, the Hamiltonian of
the total system can be derived as [20,21]
Z ∞
X Z
^ ¼
^
H
d3 r
dωℏω^f þ
λ ðr; ωÞf λ ðr; ωÞ
λ¼e;m
0
þ ℏωa jaihaj þ ℏωb jbihbj
Z ∞
^ ðþÞ ðrA ; ωÞ
− ½jaihcjda ·
dωE
0
Z ∞
^ ðþÞ ðrA ; ωÞ þ H:c::
dωE
þ jbihcjdb ·
¼ δλλ0 δij δðr −
½^f λ;i ðr; ωÞ; ^f λ0 ;j ðr0 ; ω0 Þ ¼ 0;
−
ω0 Þ;
∞
d3 r
λ¼e;m
dωe−iωt Cλc ðr; ω; tÞ · j1λ ðr; ωÞijci;
ð9Þ
where Cλc ðr; ω; 0Þ ¼ 0 and j1λ ðr; ωÞi ¼ ^f þ
λ ðr; ωÞjf0gi. By sub_
stituting Eqs. (3) and (9) into Schrödinger equation iℏjψðtÞi
¼
^
HjψðtÞi,
we obtain the following differential equations:
Z ∞
1
dωe−iðω−ωa Þt
C_ a ðtÞ ¼ − pffiffiffiffiffiffiffiffiffiffi
ℏπε0 0
Z
ω
ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d3 r · da ·
Imεðr; ωÞ
×
c
c
⃖ r · Cmc ðr; ω; tÞ ;
× ½GðrA ; r; ωÞ × ∇
ð3Þ
The first term on the right side of Eq. (3) represents the
Hamiltonian of the LHM-assisted field, the second and the
third terms are the Hamiltonian of the atom, and the last
one is the interaction Hamiltonian. Here, ^f e ðr; ωÞ and ^f m ðr; ωÞ
are basic bosonic vector operators, corresponding to the
annihilation operators of electric and magnetic excitons [21].
They satisfy the following commutation relations:
r0 Þδðω
0
X Z
× GðrA ; r; ωÞ · Cec ðr; ω; tÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ −Im½1=μðr; ωÞ
0
0
0
½^f λ;i ðr; ωÞ; ^f þ
λ0 ;j ðr ; ω Þ
ð8Þ
We assume the atom is in the superposition state between
the upper levels initially, i.e., jC a ð0Þj2 þ jC b ð0Þj2 ¼ 1.The state
function of the system at any time t can be written as
where
e1;2
ð7Þ
ð4Þ
ð5Þ
where λ; λ0 ¼ e; m and i, j run over all three spatial coordinates. The positive frequency part of the electric field in
frequency space is
Z
^ ðþÞ ðr; ωÞ ¼ iωμ0 d3 r0 Gðr; r0 ; ωÞ · ^jN ðr0 ; ωÞ:
E
ð6Þ
The noise current operator ^jN ðr0 ; ωÞ is the function of bosonic
vector operators [21]:
ð10aÞ
Z ∞
1
dωe−iðω−ωb Þt
C_ b ðtÞ ¼ − pffiffiffiffiffiffiffiffiffiffi
ℏπε0 0
Z
ω
ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d3 r · db ·
Imεðr; ωÞ
×
c
c
× GðrA ; r; ωÞ · Cec ðr; ω; tÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ −Im½1=μðr; ωÞ
⃖ r · Cmc ðr; ω; tÞg;
× ½GðrA ; r; ωÞ × ∇
ð10bÞ
1 ω2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C_ ec ðr; ω; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi 2 Imεðr; ωÞ
ℏπε0 c
× G ðr; rA ; ωÞ · ½da C a ðtÞe−iðωa −ωÞt
þ db C b ðtÞe−iðωb −ωÞt ;
ð10cÞ
1 ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−Im½1=μðr; ωÞ
C_ mc ðr; ω; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi
ℏπε0 c
~ × G ðr; r ; ωÞ · ½d C ðtÞe−iðωa −ωÞt
× ½∇
a a
r
A
þ db C b ðtÞe−iðωb −ωÞt :
ð10dÞ
Zeng et al.
Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B
0
Here, ½Gðr; r0 ; ωÞ × ∇⃖ r0 ij ≡ εjkl ∂rk Gil ðr; r0 ; ωÞ, and the indices
i, j, k, and l run over all three spatial components. By integrating Eqs. (10c) and (10d) and substituting them into Eqs. (10a)
and (10b), we get
γ
κ
C_ a ðtÞ ¼ − a C a ðtÞ − a C b ðtÞeiωab t ;
2
2
ð11Þ
emission is canceled. For the dipole component normal to the
interface, the reflected field has no phase difference as the
emitted field, so the emission is accelerated. Because of
the above reasons, p ≠ 0 and strong quantum interference
appears [13,19,20]. Eqs. (11) and (12) have analytical solutions as
C a ðtÞ ¼ C 1 es1 t þ C 2 es2 t ;
γ
κ
C_ b ðtÞ ¼ − b C b ðtÞ − b C a ðtÞeiωba t ;
2
2
ð12Þ
where ωij ¼ ωi − ωj ði; j ¼ a; b; i ≠ jÞ. Here we omit the frequency shifts induced by the environment. γ i ði ¼ a; bÞ are
the decay rates from the corresponding upper jii level to
jci, which are
γi ¼
2ω2i d · ImGðrA ; rA ; ωi Þ · di ;
ℏε0 c2 i
ð13aÞ
and κ i are the quantum interference between two transitions:
κi ¼
2ω2j
ℏε0 c2
di · ImGðrA ; rA ; ωj Þ · dj :
ð13bÞ
In order to measure the degree of quantum interference, we
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
adopt the p ¼ κa κ b =ðγ a γ b Þ in Refs. [18–20]. When the frequency difference between two upper levels is small, it is reasonable to get γ a ≈ γ b ≈ ðΓ⊥ þ Γ∥ Þ=2, κa ¼ κ b ≈ ðΓ⊥ − Γj∥ Þ=2,
and p ¼ ðΓ⊥ − Γ∥ Þ=ðΓ⊥ þ Γ∥ Þ, where Γ⊥ (Γ∥ ) are the spontaneous decay rates of the dipole moment d perpendicular (parallel) to the interface having the following expressions:
3
Γ∥ ¼ Γ0 Reμ0 n0
4
Z ∞
2iβ0 dA
2iβ0 dA
dk∥ k∥
2r TE r TE e4iβ0 dA þ r TE
þ r TE
L e
R e
1þ L R
×
TE 4iβ0 dA
k0 β 0
1 − r TE
0
L rR e
2iβ0 dA − r TM e2iβ0 dA
β2
2r TM r TM e4iβ0 dA − r TM
L e
R
þ 20 1 þ L R
; ð14Þ
TM 4iβ0 dA
1 − r TM
k0
L rR e
Z ∞
dk∥ k3∥
3
Γ⊥ ¼ Γ0 Reμ0 n0
2
k30 β0
0
TM 4iβ0 dA þ r TM e2iβ0 dA þ r TM e2iβ0 dA
2r TM
L rR e
L
R
;
× 1þ
TM 4iβ0 dA
1 − r TM
L rR e
2255
C b ðtÞ ¼ −
2
γ ab
s1 þ
ð16aÞ
γa
γ
C 1 es1 t þ s2 þ a C 2 es2 t e−iωab t ;
2
2
ð16bÞ
where
γ
γ
s1;2 ¼ − a þ b þ iωab D
2;
2
2
ffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
γa γb
− þ iωab þ κa κb ;
D¼
2 2
γa
κ
D:
C 1;2 ¼ ð∓Þ
þ s2;1 C a ð0Þ þ a C b ð0Þ
2
2
The spontaneous emission spectrum Sðr; ωÞ observed at a
position r is proportional to the Fourier transform of the field
correlation function:
^ ð−Þ ðr; ωÞE
^ ðþÞ ðr; ωÞjψðtÞi
Sðr; ωÞ ¼ limhψðtÞjE
t→∞
ω4 μ2 d − ð2s1 þ γ a Þdb =κa
C1
¼ 2 0 ImGðr; rA ωÞ · a
ω − ωa − is1
π
2
d − ð2s2 þ γ a Þdb =κ a
þ a
C 2 :
ω − ω − is
a
ð17Þ
2
Because of the indirect quantum interference of the
Zeeman atom in the structure of Fig. 1, the properties of the
spontaneous emission spectrum would be much different
from those of two orthogonal dipoles in an isotropic environment, and also different from those of two parallel dipoles in
previous studies [22].
3. SPONTANEOUS EMISSION
ð15Þ
where Γ0 ¼ d2 ω30 =ð3πε0 ℏc3 Þ is the decay rate of dipole moment d with frequency ω0 in free space. r TE;TM
(r TE;TM
) are
L
R
the reflection coefficients of the left (right) slabs for the TE
and TM polarization fields. Here, r TE;TM
¼ r TE;TM
.
L
R
If the quantum interference between the two transition
channels vanishes, i.e., κi ¼ 0, p ¼ 0, then Eqs. (11) and (12)
would be two independent differential equations, which
means the population evolutions of the upper levels would
not interfere with each other. In the present model (see Fig. 1),
the phase shift for the cycle path is zero due to the negative
phase shift in the LHM layer, and the LHM focuses the field to
the atom and keeps the amplitude the same as that of the
emitted field. For the dipole component parallel to the interface, the reflected field has a π phase shift at the mirror, so the
The spontaneous decay of the Zeeman atom is greatly influenced by the indirect quantum interference. In this section,
we discuss this influence by examining the population in
the upper levels and the emission spectrum. According to
Eqs. (16) and (17), it is clear that the properties of the atomic
population evolution and emission spectrum are dependent
on the atomic initial states and the detuning between the
two upper levels. In the following discussion, we adopt the
parameters in Refs. [19,20] and set εA ¼ μA ¼ nA ¼ −0:999 þ
i0:003 for the LHMs, r TE ¼ −0:99 and r TM ¼ 0:99 for the reflection coefficients of the mirrors for TE and TM waves, and
dA ¼ λ0 ¼ 2πc=ωa for the width of the LHM slabs.
A. Degenerate Case with Small ωab
First, we choose a very small frequency difference between
two upper levels ωab ≪ Γa0 , where Γao is the decay rate from
jai to jci in free vacuum. In this case, it is a proper approximation that Gðωa Þ ≈ Gðωb Þ, γ a ≈ γ b , and κ a ¼ κb . Using
Eqs. (14) and (15), we can get Γ⊥ ¼ 17:4Γa0 , Γ∥ ¼ 0:17Γa0 ,
2256
J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011
Zeng et al.
1.0
1.0
0.8
|Ca(t)|
2
|Ca(t)|
2
|Cb(t)|
2
0.8
2
0.6
2
0.6
P
P
|Ca(t)| +|Cb(t)|
0.4
0.4
0.2
0.2
0.0
0
1
2
3
4
5
6
Γaot
Fig. 2. Evolution of jC a ðtÞj2 (dotted curve), jC b ðtÞj2 (dashed curve),
and jC a ðtÞj2 þ jC b ðtÞj2 (dashed–dotted curve) of a Zeeman atom in the
LHM cavity and jC a ðtÞj2 (solid curve) in free vacuum with C a ð0Þ ¼ 1,
C b ð0Þ ¼ 0.
and p ¼ 0:98, which is different from the case of two (anti-)
parallel dipoles (p ¼ 1) and the case of two orthogonal dipoles
in an isotropic environment (p ¼ 0).
By preparing the atom initially in the state C a ð0Þ ¼ 1,
C b ð0Þ ¼ 0, we plot the population evolution of levels jai
and jbi in Fig. 2. We can see that the population jC a ðtÞj2 in
the upper level jai can decay to the lower level jci and then
jump in part to the other upper level jbi. For the case in free
vacuum, there is no population in level jbi because the quantum interference between the two transitions disappears.
During the starting period, the population jC b ðtÞj2 in level jbi
increases with time, and goes to a maximum value (about
0.22) at time t ≈ 0:5=Γa0 . Then jC a ðtÞj2 and jC b ðtÞj2 decay to
zero with the same decay rate. It is obvious that, at the
beginning, jC a ðtÞj2 or the total population in upper levels
(jC a ðtÞj2 þ jC b ðtÞj2 ) decreases faster than that in free vacuum
(the solid curve). The appearance of such a phenomenon is
because the decay rate γ a of the atom in our structure is much
larger than that in free vacuum. But after jC b ðtÞj2 reaches the
maximum value, the indirect quantum interference between
the two transition channels leads to the population exchanging between the two upper levels. The jC a ðtÞj2 is the same
as jC b ðtÞj2 . Because the phase difference between C a ðtÞ and
C b ðtÞ approximates to π, the decay is inhibited and is much
slower than that in free vacuum. Compared with the complete
quantum interference, the population cannot be trapped on
the upper levels forever, because the two decay channels
cannot cancel each other absolutely for p ≠ 1.
In Fig. 3, we plot theptotal
ffiffiffi population evolution for
pffiffiffiinitial
states C a ð0Þ ¼ C b ð0Þ ¼ 2=2 and C a ð0Þ ¼ −C b ð0Þ ¼ 2=2. It
shows that the total population of the two upper levels
pffiffiffi
pffiffiffi
with C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (C a ð0Þ ¼ −C b ð0Þ ¼ 2=2) decays
much faster (slower) than that in free vacuum. If the atomic
pffiffiffi
initial state is C a ð0Þ ¼ C b ð0Þ ¼ 2=2, as the indirect quantum
interference inhibits the emission of x components of the
dipoles and enhances the emission of z components of
the dipoles, jC a ðtÞj2 and jC b ðtÞj2 decrease in exponential
form expð−Γ⊥ tÞ. Here, Γ⊥ is equal to 17:4Γa0 , which is much
larger than Γa0 , so the atom decays much faster than that
in free vacuum. In contrast, if the atomic initial state is
0.0
0
1
2
3
4
5
6
Γa0t
pffiffiffi
2
Fig. 3. Evolution of jC a ðtÞj2 þ jC bpðtÞj
ffiffiffi with C a ð0Þ ¼ C b ð0Þ ¼ 2=2
(dashed curve), C a ð0Þ ¼ −C b ð0Þ ¼ 2=2 (dottedpcurve)
of a Zeeman
ffiffiffi
atom in the LHM cavity and C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (solid curve) in
free vacuum.
pffiffiffi
C a ð0Þ ¼ −C b ð0Þ ¼ 2=2, the π phase difference between
C a ð0Þ and C b ð0Þ induces the indirect quantum interference
to cancel the emission of z components of the dipoles and accelerates the emission of x components of the dipoles. jC a ðtÞj2
and jC b ðtÞj2 decrease in exponential form expð−Γ∥ tÞ, while Γ∥
is equal to 0:17Γa0 and much smaller than Γa0 , so the atom
decays much slower than that in free vacuum. Because of
the indirect quantum interference, the population evolution
of the upper levels displays different behavior compared with
that in free vacuum. The decay time is prolonged or reduced,
which is determined by the atomic initial state.
The indirect quantum interference and the initial atomic
state also influence the spontaneous emission spectrum. In
the following discussion, the spectra at the position r ¼
ð−λ0 ; 0; 0Þ are considered. By preparing the atom initially in
state C a ð0Þ ¼ 1, C b ð0Þ ¼ 0, we get s1 ¼ −Γ∥ =2, s2 ¼ −Γ⊥ =2,
C 1 ¼ 1=2, and C 2 ¼ 1=2. From Eq. (17), the spontaneous emission spectrum of Zeeman atom can be rewritten as
Sðr; ωÞ ¼
ez
ω4 μ20 d2 ImGðr;
r
;
ωÞ
·
A
2
ω − ωa þ i Γ2⊥
2π
iex
2
þ
Γ∥ :
ω − ωa þ i 2
ð18Þ
This means the spectrum can be divided into two parts:
the spectrum of the x-polarized field with width Γ∥ and the
z-polarized field with width Γ⊥ . In Fig. 4, we plot the
spontaneous emission spectrum as function of ω − δ ¼
ω − ðωa þ ωb Þ=2. Because of Γ∥ (Γ⊥ ) much smaller (larger)
than the spontaneous decay rate in free vacuum, the emission
spectrum of the x (z)-polarized field is narrower (flatter) compared with that in free vacuum. The total spectrum is sharp
on the top part near ω − δ ¼ 0 and broad in other regions.
Compared with the population decay process, the faster
population decay during the starting time period contributes
to the flat part of the spontaneous emission spectrum, and
the slower decay after a certain time leads to one peak in
the top spectrum. The behavior of the spontaneous emission
spectrum is quite different from that in free space. If the atom
Zeng et al.
Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B
12
2257
18
16
10
14
12
S(r,ω)
S(r,ω)
8
6
10
8
6
4
4
2
2
-10
-5
0
5
10
ω -δ
Fig. 4. Spontaneous emission spectrum with ωab ¼ 0 and C a ð0Þ ¼ 1,
C b ð0Þ ¼ 0.
is in an isotropic environment, there is no quantum interference, the spontaneous emission from the Zeeman atom (initially in one upper level) is the same as that from a two-level
atom, and the corresponding spectrum is Lorentzian, so
the ultranarrow spectrum comes from the indirect quantum
interference.
pffiffiffi
If the atom is in the state C a ð0Þ ¼ C b ð0Þ ¼ 2=2 and
pffiffiffi
C a ð0Þ ¼ −C b ð0Þ ¼ 2=2 initially, we can get s1 ¼ −Γ∥ =2,
pffiffiffi
s2 ¼ −Γ⊥ =2 and C 1 ¼ 0, C 2 ¼ 2=2 for C a ð0Þ ¼ C b ð0Þ ¼
pffiffiffi
pffiffiffi
pffiffiffi
2=2 (or C 1 ¼ 2=2, C 2 ¼ 0 for C a ð0Þ ¼ −C b ð0Þ ¼ 2=2).
The spontaneous emission spectrum corresponding to different atomic initial states can be rewritten as
2
ez
ω4 μ20 d2 ImGðr; r ; ωÞ ·
A
Γ
⊥ 2
ω
−
ω
þ
i
π
a
2
pffiffiffi
¼ C b ð0Þ ¼ 2=2;
Sðr; ωÞ ¼
2
ex
ω4 μ20 d2 ImGðr; r ; ωÞ ·
A
Γj∥ 2
ω − ωa þ i 2 π
pffiffiffi
¼ −C b ð0Þ ¼ 2=2:
Sðr; ωÞ ¼
0
-15
15
for C a ð0Þ
ð19aÞ
for C a ð0Þ
ð19bÞ
This shows that the emission spectrum stems frompffiffithe
ffi
z-polarized field with width Γ⊥ for C a ð0Þ ¼ C b ð0Þ ¼ 2=2
(or
pffiffiffi the x-polarized field with width Γ∥ for C a ð0Þ ¼ −C b ð0Þ ¼
2=2). The two transition dipoles from upper p
levels
jai and
ffiffiffi
jbi to lower
level
jci
are
d
¼
dðe
þ
ie
Þ=
2
and
a
z
x
pffiffiffi
pffiffiffi db ¼
dðez − iex Þ= 2. In the case of C a ð0Þ ¼ C b ð0Þ ¼ 2=2, the
phase difference between two x-polarized components of
the field emitted from two transition dipoles is π and the
amplitude of the x-polarized field is canceled to zero. Corresponding to the z-polarized component of the field, the phase
difference is zero and the amplitude of the z-polarized field is
doubled. As a result, only the z-polarized component remains
in the emitted field.
if the atomic initial state is
pffiffiConversely,
ffi
C a ð0Þ ¼ −C b ð0Þ ¼ 2=2, there exists an additional phase
difference π between the two levels. The phases of two
x-polarized components of the field are the same, and the
amplitude of the x-polarized field is enhanced. The phase difference between the two z-polarized components of the field
is π and the amplitude of the z-polarized field is reduced to
-10
-5
0
10
5
15
ω -δ
Fig. 5. p
Spontaneous
emission spectrum with ωab ¼ 0 and
ffiffiffi
pffiffiffi C a ð0Þ ¼
C b ð0Þ ¼ 2=2 (dashed curve) and C a ð0Þ ¼ −C b ð0Þ ¼ 2=2 (solid
curve).
zero, so only the x-polarized component occurs in the emitted
field (see Fig. 5).
From Figs. 4 and 5, it is seen that the indirect quantum
interference could affect the spontaneous emission spectrum,
and the spectral narrowing or broadening is determined by the
atomic initial state.
B. Case of Large ωab
Let us consider the case of a large detuning between two
upper levels, i.e., ωab ≥ Γa0 . In Fig. 6, we plot the population
evolution for different detuning ωab . By preparing the atom in
upper level jai initially, the decay rate of the population in
level jai increases as the detuning ωab increases. Because of
indirect quantum interference, the population in the lower
level jci can jump in part to the other upper level jbi. As the
detuning between the reflected field and the resonant frequency ωb increases, the population coming back to the upper
level jbi decreases. If the parameters of the system, γ a , γ b , κa ,
κb , and ωab , satisfy the relations γ a − γ b ≠ 0 and ωab ≠ 0 or
γ a − γ b ¼ 0 and ω2ab ≥ κ a κb , the indirect quantum interference
leads to the population exchanging between the two upper
levels, and the population evolution displays an oscillatory
behavior during the decay process (see Fig. 7).
1.0
2
|Ca(t)| (ωab=2Γa0 )
0.8
2
|Cb(t)| (ωab=2Γa0 )
2
|Ca(t)| (ωab=8Γa0 )
0.6
2
|Cb(t)| (ωab=8Γa0 )
P
0
-15
0.4
0.2
0.0
0
1
2
3
4
5
6
Γ0 t
Fig. 6. Evolution of jC a ðtÞj2 (solid curve), jC b ðtÞj2 (dotted curve)
with ωab ¼ 2Γa0 , and jC a ðtÞj2 (dashed curve), jC b ðtÞj2 (dashed–dotted
curve) with ωab ¼ 8Γa0 .
2258
J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011
Zeng et al.
0.07
6
ωab=1Γa0
0.06
ωab=3Γa0
5
ωab=6Γa0
0.05
4
P
S(r,ω)
0.04
0.03
3
0.02
2
0.01
1
0.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Γa0t
Fig. 7. Population evolution of jC b ðtÞj2 with C a ð0Þ ¼ 1, C b ð0Þ ¼ 0,
and ωab ¼ 20Γa0 .
In Fig. 8, we plot the spontaneous emission spectrum with
initial state C a ð0Þ ¼ 1, C b ð0Þ ¼ 0 and different ωab . The emission spectrum takes a Fano-type form [23] characterized by a
Fano minimum corresponding to the transition frequency ωb .
It is different from the case of quantum interference between
two parallel dipoles where the Fano minimum is replaced by
a completely dark point [22,24]. The Fano minimum cannot
be completely dark, mainly because the indirect quantum
interference here is incomplete for the dissipation of LHMs.
This Fano minimum is lower as the frequency difference ωab
increases. The spectrum at the peak close to ωa is narrow,
whereas the spectrum at the other peak is flat.
In Fig. 9, we p
plot
ffiffiffi the spectrum of the atom initially in
C a ð0Þ ¼ C b ð0Þ ¼ 2=2 with different ωab . If the atom is in
isotropic surroundings, there will be no indirect quantum
interference between the two orthogonal dipoles, and the
spectrum will be a combination of emission from two twolevel atoms. Consequently, the two peaks corresponding to
the two transition dipoles in the emission spectrum could
not separate until the detuning of the two upper levels is larger
than maxðγ a =2; γ b =2Þ. In the present structure, there is indirect
quantum interference between the two transitions with orthogonal dipoles, and the emission spectrum is not a simple
0
-15
-5
-10
0
ω-δ
5
10
15
pffiffiffi
Fig. 9. Spontaneous emission spectrum with C a ð0Þ ¼ C b ð0Þ ¼ 2=2
and ωab ¼ 1Γa0 (solid curve), 3Γa0 (dashed–dotted curve), and 6Γa0
(dotted curve).
combination of p
emission
from two two-level atoms. For
ffiffiffi
C a ð0Þ ¼ C b ð0Þ ¼ 2=2, the spectra has two peaks and there
is a darklike line at the transition frequency from the middle
point of two upper levels to the lower level, which is due to a
destructive interference (See Fig. 9). Even if ωab (such as Γa0 ,
3Γa0 ) is much smaller than γ a;b =2 (γ a ≈ γ b ≈ 8:7Γa0 ), the two
peaks in the emission spectrum will also separate, and the
spontaneous spectrum near two
pffiffiffi peaks could be extremely
narrow. For C a ð0Þ ¼ −C b ð0Þ ¼ 2=2, the destructive interference will merge the two peaks to one.
In the above discussion, the angle θ between the dipole da
and the z axis is π=4. If we change the angle θ from 0 to π=2,
the strength of quantum interference increases from zero to a
maximal value in the region 0 ≤ θ ≤ π=4, and then decreases to
zero in the region π=4 ≤ θ ≤ π=2 (See Fig. 10). In Fig. 11, we
plot the spontaneous emission spectrum with different directions of dipole da . The ratio of the decay rate γ a =γ b decreases
with θ. For example, if θ ¼ π=6; π=4; π=3, we can obtain
γ a =γ b ¼ 2:92; 1:0; 0:34, respectively, and the strengths of the
quantum interference are nearly the same (See Fig. 10). A
darklike line also occurs at the frequency ωb . The Fano
minimum becomes smaller as γ a =γ b increases, and the frequency of the right peak is farther away from ωa for bigger θ.
80
70
60
1.0
0.5
0.4
ωab=2Γa0
0.3
ωab=4Γa0
0.2
40
0.0
-6
0.8
ωab=8Γa0
0.1
-5
-4
-3
-2
-1
0.7
0
0.6
p
S(r,ω)
50
0.9
30
0.5
0.4
20
0.3
10
0.2
0
-10
0.1
-8
-6
-4
-2
0
ω -δ
2
4
6
8
10
Fig. 8. Spontaneous emission spectrum with C a ð0Þ ¼ 1, C b ð0Þ ¼ 0,
and ωab ¼ 2Γa0 (dotted curve), 4Γa0 (dashed–dotted curve), 8Γa0
(solid curve).
0.0
0.0
0.1
0.2
θ(π)
0.3
0.4
0.5
Fig. 10. Strength of quantum interference of a Zeeman atom in the
LHM cavity varying with θ.
Zeng et al.
Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B
120
2.
1.5
100
S(r,ω)
80
1.0
θ=π/6
θ=π/4
θ=π/3
0.5
0.0
-4
-3
-2
-1
3.
0
4.
60
40
5.
20
0
-4
6.
-3
-2
-1
0
1
2
3
4
ω -δ
Fig. 11. Spontaneous emission spectrum with C a ð0Þ ¼ 1, C b ð0Þ ¼ 0,
ωab ¼ 2Γa0 , and θ ¼ π=6 (dotted curve), π=4 (solid curve), and π=3
(dashed–dotted curve).
4. SUMMARY
In this paper, we consider a V-type Zeeman atom with two
orthogonal transition dipoles that is located in the focal plane
of LHM lenses. The population evolution and spontaneous
emission spectrum are investigated in detail. Because of the
properties of refocusing and phase compensation of LHMs,
there exists strong indirect quantum interference between
the two orthogonal dipoles. If the atom is in state C a ð0Þ ¼ 1,
C b ð0Þ ¼ 0 initially, the total population decay could be faster
during the starting period and slower after a certain period.
The population exchange between the two upper levels
occurs due to existence of the indirect quantum interference
and can display an oscillatory behavior for large frequency
difference ωab . There is a sharp peak close to ωa or the Fano
minimum at ωb in the spontaneous spectrum. When the atom
is initially in the coherent
pffiffiffi superposition of the two pupper
ffiffiffi
levels C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (or C a ð0Þ ¼ −C b ð0Þ ¼ 2=2),
the population decay becomes faster (or slower) than that
in free vacuum, the spontaneous emission spectra are narrowed (or broadened), and there could be a darklike line. The
properties of the spontaneous emission spectrum are different
from the cases of two parallel dipoles or two orthogonal
dipoles in an isotropic environment.
ACKNOWLEDGMENTS
This work is supported in part by the National Natural Science
Foundation of China (NSFC) (Nos. 91021012, 10904113), the
Foundation of the Ministry of Science and Technology
(Nos. 2007CB13201 and 2011CB922203).
REFERENCES
1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov.
Phys. Usp. 10, 509–514 (1968).
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
2259
D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and
S. Schultz, “Composite medium with simultaneously negative
permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187
(2000).
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental
verification of a negative index of refraction,” Science 292,
77–79 (2001).
R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handedmedia simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys.
Rev. Lett. 89, 183901–183904 (2002).
A. Grbic and G. V. Eleftheriades, “Experimental verification
of backward-wave radiation from a negative refractive index
metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and
M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90,
107401–107405 (2003).
L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,”
Phys. Rev. B 66, 155411–155427 (2002).
M. L. Povinelli, S. G. Johnson, J. D. Joannopoulos, and J. B.
Pendry, “Toward photonic-crystal metamaterials: creating
magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82,
1069–1071 (2003).
D. R. Smith and N. Kroll, “Negative refractive index in lefthanded materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
J. B. Pendry, “Negative refraction makes a perfect lens,” Phys.
Rev. Lett. 85, 3966–3969 (2000).
D. R. Smith and D. Schurig, “Electromagnetic wave propagation
in media with indefinite permittivity and permeability tensors,”
Phys. Rev. Lett. 90, 077405–077408 (2003).
R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith,
“Broadband ground-plane cloak,” Science 323, 366–369 (2009).
J. Kästel and M. Fleischhauer, “Suppression of spontaneous
emission and superradiance over macroscopic distance in media with negative refraction,” Phys. Rev. A 71, 011804–011807
(2005).
S. E. Harris, J. E. Field, and A. Imamoglu, “Nolinear optical
processes using electromagnetically induced transparency,”
Phys. Rev. Lett. 64, 1107–1110 (1990).
P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum
interference,” Phys. Rev. Lett. 77, 3995–3998 (1996).
P. M. Radmore and P. L. Knight, “Population trapping and
dispersion in a three-level system,” J. Phys. B 15, 561–573
(1982).
M. O. Scully and S. Y. Zhu, “Degenerate quantum-beat laser:
lasing without inversion and inversion without lasing,” Phys.
Rev. Lett. 62, 2813–2816 (1989).
G. S. Agarwal, “Anisotropic vacuum-induced interference in
decay channels,” Phys. Rev. Lett. 84, 5500–5503 (2000).
Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett.
100, 043601–043604 (2008).
J. P. Xu and Y. P. Yang, “Quantum interference of V-type threelevel atom in structures made of left-handed materials and
mirrors,” Phys. Rev. A 81, 013816–013823 (2010).
H. T. Dung, S. Y. Buhmann, L. Knöll, and D. G. Welsch,
“Electromagnetic-field quantization and spontaneous decay in
left-handed media,” Phys. Rev. A 68, 043816–043830 (2003).
S. Y. Zhu, R. F. Chan, and C. P. Lee, “Spontaneous emission from
a three-level atom,” Phys. Rev. A 52, 710–716 (1995).
U. Fano, “Effects of configuration interaction on intensities and
phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys.
Rev. Lett. 76, 388–391 (1996).