Optics and Photonics Journal, 2013, 3, 293-297
doi:10.4236/opj.2013.32B069 Published Online June 2013 (http://www.scirp.org/journal/opj)
Intracavity Tunneling Intr oduced Transpar ency in
Ultrastrong-coupling Regime
Tao Wang, Rui Zhang, Chunxiao Zhou, Xuemei Su
Physics College, Jilin University Changchun 130012, People’s Republic of China
Email: suxm@jlu.edu.cn
Received 2013
ABSTRACT
Intracavity tunneling induced transparency in asymmetric double-quantum wells embedded in a microcavity in the ul-
trastrong-coupling regime is investigated by the input-output theory developed by Ciuti and Carusotto. In this system a
narrow spectra can be realized under anti-resonant terms of the external dissipation. Fano-interference asymmetric line
profile is found in the absorption spectr a .
Keywords: Ultrastrong Coupling; Spectra Narrowing; Anti-resonant Terms; Fano-interferences
1. Introduction
Intersubband electronic transitions in doped semicon-
ductor quantum wells play an important role in many re-
markable devices, such as quantum cascade lasers, quan-
tum-well infrared photodetectors and ultrafast optical
modulators [1]. If semiconductor quantum wells in a mi-
crocavity were explored by using the intersubband tran-
sitions, it can achieve an ultrastrong light-matter cou-
pling regime where vacuum Rabi frequency becomes
comparable to the intersubband electronic transitions [2].
In the ultrastrong coupling regime, the usual rotating-
wave approximation is not exact enough to explain the
experimental data and the antiresonant terms should be
considered. The contribution of antiresonant terms can
create many new effects such as back-reaction, photon
blocked and two-mode squeezed vacuum [3].
Quantum coherence is a fundamental problem in
quantum mechanics and has been investigated for a long
time in atom physics and optics physics. Strong cou pling
is necessary for quantum coherence, but we still do not
understand what will happen when the coupling strength
reaches the ultrastrong-coupling region. The intersubband
electronic transition in doped semiconductor quantum
wells embedded in a microcavity provide a chance to
discuss and test the problem.
A full quantum descriptio n of the ultrastrong-coupling
regime in doped semiconductor quantum wells microcavity
has been developed by Ciuti and Carusotto [4] which can
explicitly include the coupling to external dissipation
baths including probe photons and injection electrons.
The dissipation baths are not only responsible for damping
rates but also provide a way of exciting and observing
the cavity dynamics. The photonic mode is coupled to the
external world mostly because of the finite reflectivity of
the cavity mirrors, while the intersubband transition is
coupled to other excitations in the semiconductor
material-e.g., acoustic and optical phonons and free
carriers in levels other than the ones involved in the
considered transition. In particular, the coupling to this
electronic bath allows one to electrically excite the
intersubband transitions and induce electroluminescence.
In the theory cavity field and electronic transition are
treated as two oscillators . The vacuum Rabi splitting of a
collection of oscillators in a single-mode cavity is
proportional to the square root of their number. So the
control of polariton coupling can be realized through the
variatio n o f c arrier dens ity.
In this paper we investigate quantum coherence in a
multiple asymmetric double-well structure embedded in
macrocavities. The cavity mode and the electronic excita-
tions in each qu antum well interact strongly in the u ltras-
trong-coupling regime and the in tersubb and transitio ns of
each quantum well are coupled by quantum tunneling which
can be tuned using bias voltage. Asymmetric semicon-
ductor double quantum well structure has been studied
for many interesting phenomena such as lasers without
population inversion, tunneling induced transparency
(TIT) [5] and opt i ca l swi tc h i ng [6].
TIT is a typical interesting quantum coherence pheno-
mena similar as those of the idea of Electromagnetically
induced transparency (EIT) [7]. Recent years intracavity
EIT has attracted much attentions and has been studied
experimentally in the multiatoms-cavity systems [8,9]. It
has been shown that intracavity EIT results in an ultra-
narrow spectral linewidth which may be used for
Copyright © 2013 SciRes. OPJ
T. WANG ET AL.
294
frequency stabilization and high-resolution spectroscopic
measurements. When the intracavity dark state is induced
by a coupling laser in the cavity and -type atomic system ,
the cavity transmission spectrum exhibits three peaks:
two side-bands associated with the multiatom vacuum
Rabi splitting and a narrow central peak representing the
cavity dark-state resonance of the two-photon Raman
transition.
Ulike atomic systems, semiconductor quantum well
embedded in a microcavity is easily use to realize the
ultrastrong coupling due to very large subband transition
matrix elements [6]. In the asymmetric quantum wells
microcavity, the transmission spectrum is similar with
the multilevels atom-cavity system but is not symmetric
and the narrow peak is not in the central position in the
ultrastrong-coupling regime. Intuitively the antiresonant
terms reduce the simultaneous creation and annihilation
of the cavity photon and the electronic transition so the
dark state must be destroyed. However the zero-absorp-
tion at resonance is not changed by the aniresonant terms
in the ideal condition and can be used to reduce the ab-
sorption induced by the aniresonant terms to zero. The
results mean that the intracavity TIT still hold even in the
ultrastrong-coupling regime.
2. Model
Figure 1 is the energy level diagram of the double qu an-
tum wells separated by a thin tunneling barrier. The deep
quantum well populates two-dimensional electron gas
density (2DEG) in the first subband 1 at low tempera-
tures with gas density. The energy sp litting of the energy
subband 2 and 3 is caused by resonant tunneling.
12
, 13
are corresponding energies of the two in-
tersubband transition energy from 1 to 2 and 3.
If we denote with z the growth direction of the multiple
quantum well structure, then the dipole moment of the
transition is aligned along z, i.e., 12 12
ˆ
, 13 13
ˆ
ˆ
ddzz
ˆ
dd
.
The in-plane wave vector is a conserved quantity.
k
3
2
1
2DE G
continuum
Figure 1. Subband energy level diagram for double quan-
tum wells separated by a thin tunneling barrier. The deep
quantum well contains a two-dimensional electron gas in
the lowest subband.
The photons mode is chosen to be resonant to the fun-
damental cavity photon mode, whose frequency disper-
sion is given by 22
,z
cav k
ckk
, where
is
the dielectric constant of the cavity spacer and
z
k s the
quantized photon wav e vector along the growth direction,
which depends on the boundary conditions imposed by
the specific mirror structures. In the simplest case of me-
llic mirrors, z
cav
π
kL
, with he cavity thickness.
cav
L
The Hamiltonian of the present system is



12
,
††
13 12,
†† ††
13,
††
12,
13,
cavkkk kk
kk
kkk kkkk
kk
kkk kkkkkkk
kkk kk
k
kkk k
Haabb
ddiab ab
iadadD aaaa
iabab
iadad





 
 
 

 

 

  
 
 



††
kkkkkk
Daa aa



(1)
The first three terms describe the energy of the cavity
mode and the two intersunbband electronic transitions.
The fouth term describe the resonant interactions be-
tween the cavity photons and the electronic transitions
and the fifth terms is the antiresonant interactions be-
tween them.
k
a
is the creation operators of the cavity
with in-plane wave vector and energy ,cav k
k
and
k
b
,
k
d
are respectively the creation operatrs of the
electronic excitation mode of wave vector o
k
and en-
ergy 12
, 13
. 12,k
, 13,k are the vacuum Rabi
frequency of the cavity photon and the two electronic
excitation which quantifies the strength of the light-matter
dipole coupling and it can become a significant fraction
of the intersubband transition. The explicit expression is
 
12
221
12 3
12 3,0
2sin
el QW
eff
k
cav
eNf
mL





(2)
where QW is the number of quantum wells present in
the cavity,
N
el el
NS
is the electron density per unit
area in the deep quantum wells, el is the number of
the electron in deep well and S is the quantization area.
is the effective length of the cavity mode,
N
eff
cav
L
is
the cavity dielectric constant, and 1
is the intracavity
probe photon propagation angle.
H
are bilinear in the field operators and can be di-
agonalized through a Bogoliubov transformation. Three
intersubband polariton annihilation operators can be in-
troduced as following
,, , ,,
††
,,
j
kjkkjkkjkkjk
jk kjkk
cwaxbydza
pb qd
k




 
(3)
Copyright © 2013 SciRes. OPJ
T. WANG ET AL. 295
where . The Hamiltonian of the system can
be written as
{1,2,3}j
3
,,,
1
G
j
kjkjk
jk
H
Ec

 
c (4)
where the constant term G is the vacuum energy and
not considered here. The vectors
E

,,,,,,,
,,,,, T
jkjk jkjkjkjkjk
vwxyzpq

(5)
Inserting expression (5) into (1) and notes the Bose
commutation rule
,
,, ,
,jj
j
kjk kk
cc
 


 
(6)
imposes the normalization condition
****
,,,,,, ,,
** ,
,, ,,
j
kjkjkjkjkjk jkjk
jj
jkjkjkj k
wwxxyyzz
pp qq
 



 
 
(7)
We found five excitations shou ld be taken into accoun t
which are related with the cavity mode and the two elec-
tronic transitions and treated as ensembles of quantum
oscillators. In the double-sided cavity the cavity mode is
coupled to two external electromagnetic field reservoirs
through the front and back mirrors which are described
by ph
H
and ph
H
. where ,
ph
qk
(,
ph
qk
) is the frequency
of an extra-cavity photon with in-plane wave vector k
and wave vector q in the orthogonal direction and
,qk
(
,qk
) is the corresponding creation operator, obeying
the commutation rule

,, ,
,-
qkq kkk
qq
 
 


 
(

,, ,
,-
qkq kkk
qq
 
 

 

 
).
The coupling between the cavity and extracavity radia-
tion fields is quantified by the tunneling matrix element
,
h
qk
(,
p
h
qk
) through the cavity mirror. The damping and
decoherence of the electronic transition is due to the in-
terplay of different processes, such as the interaction with
crystal phonons (optical and acoustical) and the scatter-
ing with impurities and with free carriers in levels other
than the ones involved in the considered electronic tran-
sition. In the TIT media there is a shared continuum res-
ervoir so the two electronic excitations are coupled by
three baths which are 2
el
H
, 3
el
H
and el
H
. Here, the
bath operators () is only coupled to elec-
tronic transition from
2, ,qk
3, ,
qk
1 to 2 (3) and ,qk
is
coupled to both. The operators satisfy the harmonic oscill-
ator commutation rule and has frequency 2, ,
el
qk
(3, ,
el
qk
).
are the matrix elements quantifying the coupling
2, ,
el
qk
to the electronic polarization.


,,,
,,, ,
,,,
,,,,
22,
1
2
1
2
ph ph
qkqk qk
k
ph ph
qkqk kqkqk k
k
ph ph
qkqkqk
k
ph ph
qkqk kqkqkk
k
el
Hdq
idqa a
Hdq
idqa a
Hdq

 

 














 

 


,2,,2,,
††
2, ,2, ,2, ,2, ,
33, ,3, ,3, ,
††
3,, 3,,3,, 3,,
1
2
1
2
el
qkqk qk
k
el el
qkqk kqkqk k
k
el el
qkqk qk
k
el el
qkqk kqkqk k
k
idqb b
Hdq
idqdd

 

 










 




,,,
,, ,,,,
††
,, ,,,,
1
2
el el
qkqk qk
k
el el
bqk qkkbqk qkk
k
el el
dqkqk kdqkqk k
k
Hdq
idqb b
idqd d

 
 








 
(8)
3. Results
In the multiatoms-cavity system when the intracavity
dark state is induced b y a couplin g laser in the cavity and
multiple
-type atomic system, the cavity transmission
spectrum exhibits three peaks shown in (a): two side-
bands associated with the multiatom vacuum Rabi split-
ting and an ultranarrow central peak representing the
cavity dark-state resonance of the two-photon Raman
transition. In the quantum wells microcavity the ul-
tranarrow polariton is a dark state which results from the
coherent superposition of the cavity photons and the
electronic transition. This is quantum interference phe-
nomenon induced by multiple energy transfer pathways.
In order to show intracavity TIT characteristics we use
the input-output theory to solve equation (1) and obtain
the spectra of reflectivity (a), transmission (b) and ab-
sorption (c) spectra as functions of the incident photon
energy E with (solid line) and without (dashed line) the
antiresonant terms of the syste m and the external dissipa-
tion bath. The calculations have been performed for a
symmetric double-sided cav ity with doub le quantu m well
structure in Ref. [6]. The parameters are 12 90
meV,
1312 17.6
meV, 25.6
meV, 3 meV .
The parameters related with the cavity mode are chosen
as 12,k
7.0
meV and 6
2
meV where is the loss of
the cavity mode and
and are decay rates of the
3
h
sitions. Ottwo intersubband traner pare de- arameters
Copyright © 2013 SciRes. OPJ
T. WANG ET AL.
296
cided by the relations,

212 13
12,
2
kk
D
 


12 13
,2
cav kk
D


 . 2
Ac s 2(a) and (b), a narrow dip or
pe cording to Figure
ak really demonstrate intracavity TIT in the reflectivity
Figure 2. Transmission (a), absorption (b) and absorption (c)
cavity with parameters in ref.[6].
spectra as a function of the incident photon energy Ewith
(solid line) and without (dashed line) the antiresonant terms
of the system and the external dissipation bath. The calcu-
lations have been performed for a symmetric double-sided
and transmission spectra which shows that an asymmetric
polariton can be produced at the transparent window at
middle-infrared wavelength. The spectra are similar to
those in three-levels atoms-cavity system which has two
bandsides and a much narrow symmetric middle polariton
[9]. The linewidth of the middle dip or peak is one-sixth
of that in the bare cavity.
Secondly, the asymmetric degree between two band-
sides got larger when th e anti-reson ant terms ar e consider ed
than that without considering them. However the posi-
tions of the two Rabi splitting peaks are nearly the same.
The anti-reson ant ter ms results in the trans mission enhance-
ment of the left polariton and suppression of the right
polariton. This is caused by asymmetric effect of quantum
interference of two transition path due to anti-resonant
interaction of the external dissipation bath. In the trans-
mission spectra the transmission ratio of the two
polaritions enhances about 72 %.
Thirdly the absorption spectra shows that there is a
zero absorption near TIT window caused by Fano inter-
ference of two transition path through two level 2 and
3 into the same continuum [10]. The anti-resonant
terms leads to suppression of the resonant peak in an
asymmetric absorption line profile at zero absorption
position.
4. Conclusions
rove intracavity TIT in an asymmetric
owledgements
m National Natural Science
[1] C. Gmachl, F.nd A. Y. Cho, Re-
ports on Prol. 64, No. 11,
In conclusion , we p
double-quantum well microcavity can be realized in the
ultrastrong-coupling regime. The quantum wells micro-
cavity may be useful for the precision measurement and
the control of the photons in the ultrastrong-coupling
regime.
5. Ackn
We acknowledge suppor ts fro
Foundat i on of C h i na ( Grant N o.1117 4109).
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