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Coherent transport of nanowire surface
plasmons coupled to quantum dots
Wei Chen, Guang-Yin Chen, and Yueh-Nan Chen∗
Department of Physics and National Center for Theoretical Sciences, National Cheng-Kung
University, Tainan 701, Taiwan
*[email protected]
Abstract:
The coherent transport of surface plasmons with nonlinear
dispersion relations on a metal nanowire coupled to two-level emitters
is investigated theoretically. Real-space Hamiltonians are used to obtain
the transmission and reflection spectra of the surface plasmons. For the
single-dot case, we find that the scattering spectra can show completely
different features due to the non-linear quadratic dispersion relation. For
the double-dot case, we obtain the interference behavior in transmission
and reflection spectra, similar to that in resonant tunneling through a
double-barrier potential. Moreover, Fano-like line shape of the transmission
spectrum is obtained due to the quadratic dispersion relation. All these
peculiar behaviors indicate that the dot-nanowire system provides a onedimensional platform to demonstrate the bandgap feature widely observed
in photonic crystals.
© 2010 Optical Society of America
OCIS codes: (230.4320) Nonlinear optical devices; (230.5298) Photonic crystals; (240.6680)
Surface plasmons; (270.1670) Coherent optical effects.
References and links
1. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008).
2. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.
Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature (London) 431, 162–167 (2004).
3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an
optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
4. K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk–
quantum dot system,” Nature (London) 450, 862–865 (2007).
5. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A Photon Turnstile Dynamically
Regulated by One Atom,” Science 319, 1062–1065 (2008).
6. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in
waveguide quantum circuits,” Nature Photon. 3, 346–350 (2009).
7. M. Rosenblit, P. Horak, S. Helsby, and R. Folman, “Single-atom detection using whispering-gallery modes of
microdisk resonators,” Phys. Rev. A 70, 053808 (2004).
8. P. Bermel, A. Rodriguez, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Single-photon all-optical switching using waveguide-cavity quantum electrodynamics,” Phys. Rev. A 74, 043818 (2006).
9. J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,”
Opt. Lett. 30, 2001 (2005).
10. J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity
containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
11. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable Scattering of a Single Photon inside a
One-Dimensional Resonator Waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
12. L. Zhou, H. Dong, Y. X. Liu, C. P. Sun, and F. Nori, “Quantum supercavity with atomic mirrors,” Phys. Rev. A
78, 063827 (2008).
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(C) 2010 OSA
Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10360
13. J. Q. Liao, Z. R. Gong, L. Zhou, Y. X. Liu, C. P. Sun, and F. Nori, “Controlling the transport of single photons
by tuning the frequency of either one or two cavities in an array of coupled cavities,” Phys. Rev. A 81, 042304
(2010).
14. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin,
“Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450,
402–406 (2007).
15. Y. Fedutik, V. V. Temnov, O. Schops, U. Woggon, and M. V. Artemyev, “Exciton-Plasmon-Photon Conversion in
Plasmonic Nanostructures,” Phys. Rev. Lett. 99, 136802 (2007).
16. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum Optics with Surface Plasmons,” Phys.
Rev. Lett. 97, 053002 (2006).
17. G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons
in a nanowire,” Opt. Lett. 33, 2212–2214 (2008).
18. Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, “Quantum-dot exciton dynamics with a surface plasmon:
Band-edge quantum optics,” Phys. Rev. A 79, 033815 (2009).
19. Y. N. Chen, G. Y. Chen, Y. Y. Liao, N. Lambert, and F. Nori, “Detecting non-Markovian plasmonic band gaps in
quantum dots using electron transport,” Phys. Rev. B 79, 245312 (2009).
20. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface
plasmons,” Nature Phys. 3, 807–812 (2007).
21. A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. D. Snapp, A. V. Akimov, M. H. Jo, M. D. Lukin, and H.
Park, “Near-field electrical detection of optical plasmons and single-plasmon sources ,” Nature Phys. 5, 475–479
(2009).
22. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel, ev, and F. Nori, “Unusual resonators: Plasmonics, metamaterials,
and random media,” Rev. Mod. Phys. 80, 1201 (2008).
23. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58,
2059 (1987).
24. S. John, “Localization of Light,” Phys. Today 44, 32–40 (1991).
25. S. Savel, ev, A. L. Rakhmanov, and F. Nori, “Using Josephson Vortex Lattices to Control Terahertz Radiation:
Tunable Transparency and Terahertz Photonic Crystals,” Phys. Rev. Lett. 94, 157004 (2005).
26. U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124, 1866–1878
(1961).
1.
Introduction
In the realization of quantum network [1], coherent single-photon transport is of central issue,
in which the photons conveying information (signal field) are controlled by other photons (gate
field). Much experimental [2–6] and theoretical [7–13] work has been focused on the photon
transport properties. With the development of technologies, a coupled system comprising of a
single metal nanowire with a quantum dot (QD) has been fabricated successfully [14, 15]. This
leads to the possibilities of investigating cavity quantum electrodynamics [16–19] and coherent
single surface plasmon (SP) transport [20–22] within such a device.
Inspired by the work mentioned above, we consider here the scattering properties of nanowire
SP coupled to QDs. Based on the findings that the dispersion relations of metal nanowire SPs
are parabola-like for higher excitation modes [17, 18], we investigate the transmission and reflection properties of SPs by approximating the dispersion relations with a quadratic form.
With reasonable assumption of strong coupling between QDs and SPs, we consider the coherent transport of surface plasmons, i.e. the incoherent scattering can be neglected [9, 10]. As
will be seen, the transmission (or reflection) spectrum for the single-dot case is found to have
double peaks due to the quadratic form of the dispersion relation. For the double-dot case, the
interference curves, similar to resonant tunneling phenomenon, are obtained in the scattering
spectra. In addition, the transmission spectrum can reveal a Fano-like line shape because of the
non-linear dispersion relation.
2.
Model and formulas
Let us consider now two QDs placed close to a metal nanowire, such that the SPs are evanescently coupled to the dots as shown in Fig. 1(a). The Hamiltonian of the SPs and two two-level
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Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
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QDs, separated by a distance d, is written as
H
=
∑ h̄ωk a†k ak + ∑
k
+ ∑ h̄
k
h̄Ωσe j ,e j
j=1,2
2π
Ω D · ek [(ak σe1 ,g1 + ak eikd σe2 ,g2 ) + h.c.],
h̄ωkV
(1)
where ωk is the frequency (dispersion relation) of the SPs with wave vector k, a†k (ak ) is the
creation (annihilation) operator
plasmon
ofthe
mode, Ω is the two-level energy spacing of the
QDs, and σe j ,e j (σe j ,g j ) = e j e j (e j g j ) is the diagonal (off-diagonal) element of the QD
operator. Here, Vk ≡ h̄ω2πV ΩD · ek describes the coupling strength between SPs and the QD,
k
where D is the dipole moment of the QD, V is the quantization volume, and ek is the polarization
vector of the SPs. For Simplicity, we first assume that these two QDs are equal in distance to the
nanowire, which means that their couplings to the nanowire SPs are the same. In addition, we
also assume the couplings between QDs and SPs are strong, such that the incoherent scattering
can be neglected. The validity of this assumption will be clarified later.
SP mode
Fig. 1. (a) Schematic view of the model: a silver nanowire coupled to two QDs. (b) Dispersion relations of the nanowire surface plasmons for the modes n = 0 to n = 3 [17, 18].
The unit for the axes are Ω = ω /ω p , and K = kc/ω p . This figure is for the case of R = 0.1,
where R ≡ ω p a/c is the effective radius of the QD, which is roughly equal to 53.8 nm.
Here, the separation between QDs and nanowire is 10.76 nm.
The dispersion relations of nanowire SP in Fig. 1(b) are quoted from Refs. [17, 18]. As seen,
the dispersion relations for n = 0 modes show highly non-linear behaviors. Let us now focus
on the n = 1 mode (red line). One immediately observes that there is a local minimum around
the value of the wavevector K(= kc/ω p )∼ 15, where h̄ω p = 3.76 eV is the plasma energy of
bulk silver. The dispersion relation around this local minimum can be analogous to the case
when a two-level is put in a photonic crystal. The density of states becomes singular near the
bandedge and the dispersion relation can be approximated as a parabolic curve. This parabolic
dispersion curve results from the strong interaction between the two-level atom and its own
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localized radiation [18, 19, 23–25]. In our model, the interactions between QDs and SPs can
be also very strong, resulting from a similar feature of local extremum in the dispersion curve.
We thus approximate the dispersion relation ωk around this bottom with a quadratic form:
ωc + A(k − k0 )2 , where k0 is the wavevector for the local minimum frequency ωc . Adopting the
units in Refs [16, 17], the value of A is about 0.0001, which is very
small. Therefore, one can
−1/2
0)
in Vk to the lowest-order term in k, such that Vk ≈ h̄ω2πcV ΩD · ek [1 − A(k−k
expand ωk
2ωc ].
Since A again is very small compared to ωc , one can further drop the second term in the square
bracket. The Hamiltonian, H = HSP + Hint + HQD , can now be written as
2
HSP
=
Hint
=
∑ h̄[ωc + A(k − k0 )2 ]a†k ak
k
∑ h̄g[(ak σe1 ,g1 + a†k σg1 ,e1 ) + (ak eikd σe2 ,g2 + a†k e−ikd σg2 ,e2 )]
k
=
HQD
where g =
∑
h̄Ωσe j ,e j ,
(2)
j=1,2
2π
h̄ωcV ΩD · ek .
Transforming the Hamiltonian into real-space [9, 10], it reads
SP
H
=
int
H
∂
∂
CR (x) + iCL† (x) CL (x)]
∂x
∂x
∂
∂
∂
∂
+h̄A[ CR† (x) CR (x) + CL† (x) CL (x)]}
∂x
∂x
∂x
∂x
√
=
dx{ 2π g ∑ δ (x − ( j − 1)d)[CR† (x)σg j ,e j
dx{(h̄ωc + h̄Ak02 )[CR† (x)CR (x) +CL† (x)CL (x)]
−2h̄k0 [−iCR† (x)
j=1,2
+CR (x)σe j ,g j +CL† (x)σg j ,e j +CL (x)σe j ,g j ]}
QD
H
=
∑ (Ee σe j ,e j + Eg σg j ,g j ),
(3)
j=1,2
where CR† (x) [CL† (x)] is a bosonic operator which creates a right-going (left-going) photon at
real-space position x. Ee (Eg ) is the energy of the QD’s excited (ground) state, and Ee − Eg =
h̄Ω.
The stationary state |Ek of the system with energy Ek can be described as
|Ek =
+
†
†
dx[φk,R
(x)CR† (x) + φk,L
(x)CL† (x)]|0, g1 , g2 ∑
ek j a†e j ag j |0, g1 , g2 (4)
j=1,2
where |0, g1 , g2 means that the QDs are both in the ground state with no SP present, and ek j
is the probability amplitude of the j-th QD in the excited state. For a SP incident from the left,
†
†
φk,R
(x) ≡ eikx [θ (−x) + a θ (x)θ (d − x) +t θ (x − d)] and φk,L
(x) ≡ e−ikx [r θ (−x) + b θ (x)θ (d −
x)], where t and r are the transmission and reflection amplitudes respectively. Here, a and b
are the probability amplitudes of the SP between x = 0 and d, and θ (x) is the step function.
By solving the eigenvalue equation H |Ek = Ek |Ek , one can obtain the exact forms of the
transmission and reflection coefficients:
T ≡ |t|2 =
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[F(k)]4
D(k)
(5)
Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10363
Fig. 2. Transmission (solid
black) and reflection (dashed red) spectra of the single-QD
case for g = 1 (in unit of 10−4 ω p c/4π ) for detunings (in unit of 10−4 ω p ) (a) δ = 0, (b)
δ = 0.1, (c) δ = −0.1, and (d) δ = −0.2. Here, the detuning is defined as: δ ≡ ωc − Ω.
and
2
16g4 F(k) cos (kd) − 2g2 sin (kd)
,
R ≡ |r| =
D(k)
2
(6)
Here, we have defined the function F(k) ≡ h̄A(k + 2k0 )[δ + A (k − k0 )2 ] and D(k) ≡
{8g4 [F(k)]2 + [F(k)]4 − 32g6 F(k) sin (2kd) − 8g4 (4g4 − [F(k)]2 ) cos (2kd) + 32g8 , where δ =
ωc − Ω is the detuning between ωc and the two-level energy spacing.
Fig. 3. Transmission (solid black) and reflection (dashed red) spectra of the double-QD
case with g = 1 and δ = 0 for different inter-dot distance (a) d = 0, (b) d= 3, (c) d = 6,
and (d) d = 12. In plotting this figure, δ is in unit of 10−4 ω p , g is in unit of
and d is in unit of c/ω p .
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10−4 ω p c/4π ,
Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10364
3.
Results and discussions
Let us first study how the quadratic dispersion relation affects the transmission/reflection. For
simplicity, the inter-dot distance d in Fig. 2 is assumed to be zero, i.e. the single-dot case. As
seen, the main feature in Fig. 2 is the double peak when the detuning, δ = ωc − Ω, is negative.
To explain this, we note in the limit of d → 0, T and R can be reduced to
T=
[F(k)]2
16π 2 g4 + [F(k)]2
(7)
R=
16π 2 g4
.
16π 2 g4 + [F(k)]2
(8)
and
From Eqs. (7) and (8) one realizes that, for negative δ , it yields two zeros (unities) in T (R),
corresponding to the two dips (peaks) in the curves. For the case of δ > 0, the incident SP
is affected slightly for positive detunings. To understand this clearly, one recalls that the approximation of quadratic dispersion relation with a bandgap is commonly used for photons in
photonic crystals [19, 23–25]. The propagation of photons is determined by whether the energy
of photons is above the bandgap. The effects in Eqs. (7) and (8) are analogous to the transport
of a photon in a photonic crystal with a bandgap. This means the QD-SP system provides a
one-dimensional platform to demonstrate the bandgap feature.
Figure 3 displays the transmission and reflection spectra, T and R, for different inter-dot
distance d with g = 1 and δ = 0. We can see that, for d = 0, the results return to the ordinary single dot scattering spectrum. When increasing d, the jiggling behavior becomes more
obvious. Mathematically, the zero-reflection (R=0) occurs when the two functions
X = h̄A(k + 2k0 )[A (k − k0 )2 + δ ] cos (kd)
(9)
Y = 2g2 sin (kd)
(10)
and
coincide with each other. Figure 4(a) displays the transmission and reflection spectra for the
conditions of d = 6, g = 1, δ = 0, and k0 . Figure 4(c) demonstrates the two functions X and
Y numerically. One can identify that the intersections of X and Y are the zeros of R. However,
for the case of K = k0 , R is not zero, but unity. This is because for this particular choice, the
denominator of R is also zero. Therefore, one can not determine R only from the numerator.
The limit of R at k = k0 is identified to be unity by applying the L’Hopital’s rule. One could
interpret this as the phenomenon of resonant tunneling of an electron through double barriers
in quantum mechanics.
Figure 5 shows how the coupling g affects R and T under the conditions of d = 6 and δ = 0.
When increasing the coupling, the region (light green area) for forbidden propagation of SPs
becomes wider. The coupling strength g can be controlled by the dot-wire separation and the
polarization of the QD exciton. In reality, it is difficult to alter the dot-wire separation. However,
one can still vary the coupling g by applying an external field to orient the direction of the dipole
moment of the QD exciton.
Keeping SP-QD1 coupling (g1 ) fixed, Fig. 6 shows the transmission and reflection spectra
for different SP-QD2 couplings (g2 ). When decreasing g2 , the feature of Fano-like resonance
becomes more evident as seen in Fig. 6(d). Fano resonance [26] is a general property whenever there is interference between localized and delocalized states (channels). The interference
between these two channels leads to an asymmetric line shape universally found in many physical systems. To illustrate the feature here is related to Fano resonance, we further deduce the
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Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10365
Fig. 4. (a) Transmission (solid black) and reflection (dashed red) spectra with d = 6, g = 1,
δ = 0 and k0 = 15. (b) Magnification of the region with probability ranging from 0 to
4 × 10−4 . (c) The intersections of functions X (green dotted curve) and Y (blue solid curve)
represent the zeros of R. In plotting this figure, δ is in unit of 10−4 ω p , g is in unit of
10−4 ω p c/4π , k0 is in unit of ω p /c, and d is in unit of c/ω p .
explicit forms of the transmission and reflection coefficients for δ1 = 0, d = 0, and g1 = 1:
T=
A4 (k − k0 )4 (k + 2k0 )2 Δ22
(11)
4A2(k − k0 )4 g42 + 8A (k − k0 )2 g22 Δ2
+ 4 + A4 (k − k0 )4 (k + 2k0 )2 Δ22
and
R=
2
4 A (k − k0 )2 1 + g22 + δ2
4A2(k − k0 )4 g42 + 8A (k − k0 )2 g22 Δ2
,
(12)
+ 4 + A4 (k − k0 )4 (k + 2k0 )2 Δ22
where Δ2 ≡ δ2 + A (k − k0 )2 . The zeros of the reflection coefficient R (totally transmitted) are
− g22 δ1 + g21 δ2
.
(13)
k = k0 ±
A g21 + g22
From Eq. (12) and (13), one knows the condition for the presence of the double dips in R is
δ2 < 0. This means the appearance of Fano-like resonance in our case is due to the quadratic
dispersion relation, which produces a photonic band-gap similar to those in photonic crystals
[23, 24]. The effect of QD1 (QD2) is just like a delocalized (localized) channel for the SP
passing through it. One notes that the Fano-like line shape due to the non-linear dispersion
was also found in Refs. [11–13]. However, the origins of the non-linear dispersion relations
are different. In Refs. [11–13], the non-linear dispersion relation comes form the tight-binding
consideration of the of the cavity array. In our case, the non-linear dispersion is solely due to the
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Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10366
Fig. 5. Transmission (solid black) and reflection (dashed red) spectra of the double-QD
case with d = 6 and δ = 0 for different couplings (a) g = 0.7, (b) g =1.5, (c) g = 7 and (d)
g = 15. In plotting this figure, δ is in unit of 10−4 ω p , g is in unit of
is in unit of c/ω p .
10−4 ω p c/4π , and d
geometric nature of the cylindrical wire, i.e. a single wire is enough to produce the non-linearity.
Furthermore, a single dot coupled to the wire can not reveal the Fano-like line shape. Two dots
with proper energy differences to the bandgap (ωc ) are needed to show this phenomenon. Our
work clearly demonstrating the interference between the localized and de-localized channels
due to the bandgap.
A few remarks on the validity of our assumption for real parameters and experimental readout
should be given here. From the numerical calculations for CdSe QD coupled to Ag nanowire in
Refs. [16] and [17], one realizes the corresponding energy scale for the case of g = 1 is about
0.376meV. If the QD is 10 nm away from the metal nanowire, the corresponding coupling
strength in this special case is: g ≈ 1.6. Compared with the dissipated energy scale of the QD
exciton (h̄/lifetime≈ 1μ eV), the QD-SP coupling is two to three orders of magnitude larger.
This validates our assumption of strong coupling. In real experiments, surface plasmons inevitably experience dissipations like Ohmic losses during propagation. To preserve coherence,
we propose a setup to carry out the experiments: Instead of using an infinite long nanowire, we
consider two separate wires with finite length evanescently coupled to a phase-matched dielectric waveguide [5, 20]. The two QDs are coupled to these two wires as shown in Fig. 7. In this
case, one can have both the advantages of strong coupling from SP and long-distance transport
by the dielectric waveguide. With this setup, one can measure the transmission and reflection
spectra of SP by scanning the wave vector spectrum of the incident SP. In addition, one can also
apply the detecting technique developed in Ref. [21], such that the surface plasmons in metal
wire can be converted into electron-hole pairs in semiconductor wire. This would allow one to
detect the surface plasmons with an electrical way.
4.
Summary
In summary, real-space Hamiltonians with nonlinear quadratic dispersion relation are used to
obtain the transport properties of SPs propagating on the surface of a silver nanowire coupled
to two QDs. For the single-QD case, the incident SP is affected slightly for positive detunings,
while it shows two peaks (dips) in reflection (transmission) spectra for negative detunings.
For the double-QD case, the region for forbidden propagation of SPs becomes wider when
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Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10367
Fig. 6. Transmission (solid black) and reflection (dashed red) spectra of the double-QD
case with δ1 = 0, δ2 = −0.05, d = 0, and g1 = 1 for different coupling (a) g2 = 1, (b)
g2 = 0.5, and (c) g2 = 0.1 between SP and QD2; (d) is for g2 = 0.1 in a small
scale of K.
In plotting this figure, δ1 and δ2 are both in unit of 10−4 ω p , g is in unit of
and d is in unit of c/ω p .
|e1 >
10−4 ω p c/4π ,
|e2 >
|g 1 >
|g 2 >
incident
photons
dielectric waveguide
Fig. 7. Schematic diagram of the two quantum dots coupled to two separate wires with
finite length.
increasing the SP-QD coupling g. In addition, we also found the Fano-like resonance can appear
if the SP-QD coupling for each dot is different (g1 = g2 ). The reasons of these phenomena are
attributed to the quadratic dispersion relation and related to the band-edge effect in photonic
crystals.
Acknowledgements
This work is supported partially by the National Science Council, Taiwan under the grant number NSC 98-2112-M-006-002-MY3.
#125617 - $15.00 USD
(C) 2010 OSA
Received 17 Mar 2010; revised 22 Apr 2010; accepted 27 Apr 2010; published 4 May 2010
10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10368