Design of electro-optic switching via a photonic crystal cavity coupled to a quantum-dot molecule and waveguides

Physics Letters A 374 (2010) 3762–3767

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Design of electro-optic switching via a photonic crystal cavity coupled to a quantum-dot molecule and waveguides Jiahua Li a,∗ , Rong Yu b , Xiaoxue Yang a a b

Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China School of Science, Hubei Province Key Laboratory of Intelligent Robot, Wuhan Institute of Technology, Wuhan 430073, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 17 May 2010 Received in revised form 8 July 2010 Accepted 9 July 2010 Available online 15 July 2010 Communicated by R. Wu

a b s t r a c t We demonstrate the possibility to realize an electro-optic switching operation in a weak-excitation limit, where one can control the path of an optical field propagating in a waveguide by suitably varying the applied bias voltage to tune the electron tunnel coupling. © 2010 Elsevier B.V. All rights reserved.

Keywords: Quantum-dot molecule (QDM) Photonic crystal (PC) cavity Electro-optic switching Dipole-induced transparency (DIT)

1. Introduction During the past decade considerable research efforts have been exerted towards the design and the realization of solid-state optical devices with micron-scale footprint for the next generation of optical networks [1–4] and future optical quantum information processing [5–12]. In these optical networks and quantum computing architectures, a key and important element for sending and routing information is switching light in an effective way. One of the methods for achieving optical switching is based on quantum coherence and interference in the light-matter interaction. In conventional media, the main drawback of this technique is the weakness of optical nonlinearities and the high power consumption of the switching operation. In order to enhance nonlinear coupling and overcome the limitation of power consumption, the possibility of combining both semiconductor quantum dot (QD) and photonic crystal (PC) cavity is very appealing, because, on the one hand, semiconductor QD and QD molecule (QDM) are similar to artificial atoms and molecules exhibiting a high density of states confined in a PC cavity [13–17]; on the other hand, PC structures are very suitable for confining light in line and point defects in a very small volume [18–21]. Recently, Waks and Vuckovic [22] theoretically investigated the effect of dipole-induced-transparency (DIT) in a PC cavity-wave-

*

Corresponding author. Tel.: +86 2787557477; fax: +86 2787557477. E-mail address: [email protected] (J. Li).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.07.027

guide coupling system: an optical field would be normally transmitted from one waveguide to another waveguide through the resonant coupling to the PC cavity, in other words, the waveguide is normally opaque at the cavity resonance. However, when a dipole (atom, QD, etc.) is placed in the PC cavity, the resonant coupling between the dipole and the cavity makes the waveguide highly transparent even in the bad cavity regime. Starting from these results, we propose a method to realize an electro-optic switching operation via suitably varying the applied bias voltage to tune the electron tunnel coupling in a coupled QDM-cavity-waveguide system (see Fig. 1) in the low- Q regime. It should be pointed out that tunneling control by bias voltage is a phenomenon exploited in devices such as resonant tunneling diodes. But here, it is used for an efficient and convenient electronic control of the transmission of a beam of light. Previous investigations [23–27] on the manipulation of the light transmission in the PC cavity-waveguide system usually required a second electromagnetic field (it is a time-varying perturbation, nevertheless the tunnel coupling here is a stationary ingredient of a coupled QD nanostructure) and therefore are substantially different from our proposed scheme. This Letter is organized as follows. In Section 2, we establish the physical model under study and present the coupled Heisenberg equations of motion for the PC cavity field and the QDM. In terms of perturbation expansion, we further derive analytical expressions for the transmission coefficients of two waveguides. In Section 3, we investigate the possibility to realize controllable electro-optic switching operation and then give

J. Li et al. / Physics Letters A 374 (2010) 3762–3767

3763

Fig. 1. (Color online.) (a) Schematic diagram of proposed electro-optic switching system, which is composed of one single mode photonic-crystal (PC) cavity, one threesubband QDM and two identical parallel waveguides. The QDM is localized in the PC cavity. For other symbols, see the text for details. (b) Band diagram of a QDM, which consists of two dots (the left dot (LD) and the right dot (RD)) with different band structures coupled by tunneling. With an external voltage applied to a gate electrode, the conduction-band levels get closer to resonance, greatly increasing their coupling, while the valence-band levels get more off-resonance, resulting in effective decoupling of those levels. (c) Schematic of the energy level arrangement under study. The PC cavity with central frequency ωc and coupling strength gc excites one electron from the valence to the conduction band in the LD, which can in turn tunnel to the RD. δ is the detuning of the electron-hole ground state transition |1 ↔ |0 of the QDM from the cavity resonance. The ground state |0 is the system without excitations, the (direct) exciton state |1 is a pair of electron and hole bound in the LD, and the indirect exciton state |2 is one hole in the LD with one electron in the RD.

the corresponding physical explanations. Alternatively, we also discuss the influences of the dissipation of the system on the transmission. Finally, the main results are summarized in Section 4. 2. Model description and transmission coefficient derivation of QDM-cavity-waveguide system The system of interest in this Letter is schematically shown in Fig. 1(a): a single mode PC cavity containing an embedded threesubband QDM is symmetrically coupled to two identical parallel waveguides a and b, with a geometry similar to the one reported in Ref. [22]. The band structure of the considered QDM [28], which consists of two dots (the left dot (LD) and the right dot (RD)) with different band structures coupled by electron tunneling, is shown in Fig. 1(b). At nanoscale interdot separation, the hole states are localized in the QDs and the electron states are rather delocalized. With the effective coupling of a single mode PC cavity, an electron is excited from the valence band to the conduction band of one of the QDs. This electron can be transferred by the tunneling to the other QD. The tunnel barrier in a double QD can be controlled by placing a gate electrode between the two QDs. In the absence of the gate voltage, the conduction-band electron energy levels are out of resonance and the electron tunneling between two QDs is very weak. Contrarily, in the presence of a gate voltage the conduction-band electron levels come close to resonance

and the electron tunneling between the two QDs is significantly enhanced. In addition, in the latter case the valence-band energy levels become more off-resonant and thus the hole tunneling can be neglected. Fig. 1(c) describes energy-level diagram of an asymmetric double QD, i.e., QDM. The ground state |0 is the system without excitations, and the exciton state |1 is a pair of electron and hole bound in the LD, and the indirect exciton state |2 is one hole in the LD with an electron in the RD. To proceed further, we firstly address briefly the basic idea of our proposed protocol. Making use of the two-dimensional (2D) PC, we assume that an input signal is guided by the waveguide b in the crystal plane (see Fig. 1(a)). A weak monochromatic laser field through the PC cavity coherently drives the |0 ↔ |1 transition of the LD in Fig. 1(b). Under the resonant electron tunneling condition (ω12 = 0), the waveguide b becomes opaque and the PC cavity becomes transparent. The input signal transmits from the original waveguide b into the other waveguide a via the PC cavity. At this time, the signal field propagates in the waveguide a. However, when the applied bias voltage tilts the energy levels of the structure such that the levels of the different dots move out of resonance (ω12 = 0, i.e., the nonresonant tunneling), a strong coupling between the PC cavity and the LD makes the waveguide b transparent and the cavity opaque. In this case, the signal field is switched to the waveguide b. This can work as a kind of optical switching where a signal is fully controlled by the external bias voltage.

3764

J. Li et al. / Physics Letters A 374 (2010) 3762–3767

Following the method developed in Refs. [29–32], in the rotating-wave approximation (RWA) the whole Hamiltonian for the QDM-cavity-waveguide system under study is given by

ˆ = h¯ ω10 σˆ 11 + h¯ ω20 σˆ 22 H

    + h¯ ωc cˆ † cˆ + h¯ gc cˆ † σˆ 01 + cˆ σˆ 10 + T e σˆ 12 + T e∗ σˆ 21 ωa

ωb h¯ ωaˆ (ω)ˆa(ω) dω + †

+ −ωa

 + ih¯ 

h¯ ωbˆ † (ω)bˆ (ω) dω

η 2π

η + ih¯ 2π





−ωa





bˆ † (ω)ˆc − cˆ † bˆ (ω) dω,

(1)

−ωb

where ω10 and ω20 are the subband transition frequencies, related to the subbands |1 ↔ |0 and |2 ↔ |0, respectively. σˆ mn = |mn| (m, n = 0, 1, 2) for m = n, are the electronic transition operators between the states |m and |n and σˆ mm = |mm| (m = 1, 2) represent the population operators for the QDM. cˆ and cˆ † are the bosonic annihilation and creation operators of the cavity mode with the central frequency ωc . T e is the tunneling coupling constant, depicted on Fig. 1(b). aˆ (ω) and aˆ † (ω) are the annihilation and creation operators of one waveguide mode and bˆ (ω) and bˆ † (ω) are the annihilation and creation operators of another waveguide mode, with the commutation relations [ˆa(ω), aˆ † (ω )] = δ(ω − ω ) and [bˆ (ω), bˆ † (ω )] = δ(ω − ω ), respectively. In the above Hamiltonian operator (1), the first and second terms are unperturbed parts, which represent the energies of the states |1 and |2 of the QDM, where the energy of the ground state |0 has been taken as the energy origin. The third term is the energy of the cavity mode. The forth term describes the situation that the cavity couples the ground state |0 to the excited state |1 with a coupling strength g c and a detuning δ (δ = ω10 − ωc ). The fifth term presents the electron tunnel-induced coupling between the LD and the RD. The sixth and seventh terms stand for the energy of the optical modes of two waveguides a and b with the finite bandwidths [ω w g − ωa , ω w g + ωa ] and [ω w g − ωb , ω w g + ωb ], respectively. The two waveguides are identical, so we have taken the same center frequency, denoted by ω w g , for both of them. Within this bandwidth, the eighth and ninth terms describe the coupling rate between the cavity and each waveguide. The cavitywaveguide coupling rate can be taken as a constant within the first Markov approximation [29–31] and is expressed by the symbol η . The Heisenberg equations of motion for the PC cavity field and the QDM can be derived from the Hamiltonian operator (1) and ˆ

ˆ , Hˆ ], yielding i h¯ ddtA = [ A dcˆ dt

dσˆ 00 dt dσˆ 11 dt dσˆ 22 dt dσˆ 01 dt dσˆ 02 dt

√

2

(2)

= γ01 σˆ 11 + γ02 σˆ 22 + igc cˆ σˆ 10 − igc cˆ σˆ 01 ,

(3)

= −γ01 σˆ 11 + igc cˆ † σˆ 01 − igc cˆ σˆ 10 + iT e∗ σˆ 21 − iT e σˆ 12 ,

(4)

†

∗

= −γ02 σˆ 22 + iT e σˆ 12 − iT e σˆ 21 ,

(5)

= −(i ω10 + Γ01 )σˆ 01 − igc cˆ (σˆ 00 − σˆ 11 ) − iT e σˆ 02 ,

(6)

= −(i ω20 + Γ02 )σˆ 02 − iT e∗ σˆ 01 + igc cˆ σˆ 12 ,

(7)

(9) (10)

which are so-called standard “input–output relation” [29–31]. Specifically, as shown in Fig. 1(a), the operators aˆ in and bˆ in are field operators for the flux of the two input ports of the waveguides, while aˆ out and bˆ out are corresponding output field operators for the two output ports of the waveguides. We outline the solution of Eqs. (2)–(8) in the weak-excitation approximation [22,23], where the intensity of the PC cavity field is sufficiently weak. In the weak-excitation limit, the perturbation approach can be applied to the electronic part of QDM, which is introduced in terms of perturbation expansion, ( 0) (1 ) (2 ) σˆ mn = σˆ mn + λσˆ mn + λ2 σˆ mn + · · · ,

(11)

where m, n = {0, 1, 2} and λ is a continuously varying parameter (0) ranging from zero to unity. Here σˆ mn is of the zeroth order in g cˆ , (1) (2) σˆ mn is of the first order in g cˆ , σˆ mn is of the second order in g cˆ , and so on. Our analysis works in the weak excitation limit, where the electron in the QDM is predominantly populated in the ground (0) (0) state |0. In this limit, we have σˆ 00 = 1 while others σˆ mn = 0 [22, 23] for the zeroth-order electronic transition operators. By substituting Eq. (11) into Eqs. (3)–(8) and keeping the terms up to the first order in the cavity field amplitude, after some mathematical manipulation the equations for the first order electronic transition (1) operators σˆ mn read (1 )

dσˆ 01

(1 )

dt (1 )

dσˆ 02 dt

(1 )

dσˆ 21 dt

(1 )

= −(i ω10 + Γ01 )σˆ 01 − igc cˆ − iT e σˆ 02 ,

(12)

(1 ) (1 ) = −(i ω20 + Γ02 )σˆ 02 − iT e∗ σˆ 01 ,

(13)

 (1 ) (1 ) (1 )  . = −(i ω12 + Γ21 )σˆ 21 − iT e σˆ 22 − σˆ 11

(14)

ˆ (ω) = √1 By taking the Fourier transformations A

2π

i ωt

+∞

ˆ −∞ A (t ) ×

e dt on the operators in Eqs. (2), (12) and (13), we have the results



i (ωc − ω) + η +

 κ √ = − i ωc + η + cˆ − igc σˆ 01 − η(ˆain + bˆ in ),

(8)

where the cavity intrinsic decay rate (κ ) and the QDM dissipation rates (γ01 , γ02 , Γ01 , Γ02 , and Γ21 ) have been added. The parameter ω12 related to interdot tunneling can be tuned with the applied bias voltage [28,33,34]. The input optical fields aˆ in and bˆ in in Eq. (2) are respectively associated with the output optical fields aˆ out and bˆ out by

ηcˆ , ˆbout − bˆ in = √ηcˆ ,

aˆ † (ω)ˆc − cˆ † aˆ (ω) dω

ωb

= −(i ω12 + Γ21 )σˆ 21 − iT e (σˆ 22 − σˆ 11 ) − igc cˆ σˆ 20 ,

dt

aˆ out − aˆ in =

−ωb

ωa

dσˆ 21

 

+

√ 

κ 2



(1 )

cˆ (ω) + igc σˆ 01 (ω)



η aˆ in (ω) + bˆ in (ω) = 0, 

(1 )



(1 )

(15) (1 )

i (ω10 − ω) + Γ01 σˆ 01 (ω) + igc cˆ (ω) + iT e σˆ 02 (ω) = 0,

(16)

i (ω20 − ω) + Γ02 σˆ 02 (ω) + iT e∗ σˆ 01 (ω) = 0.

(17)

(1 )

The solution to Eqs. (15), (16) and (17) can be found as

cˆ (ω) =

√ − η(ˆain (ω) + bˆ in (ω)) i (ωc − ω) + η + κ2 +

gc2

.

(18)

| T e |2

i (ω10 −ω)+Γ01 + i (ω −ω)+Γ 20 02

Substituting Eq. (18) into Eqs. (9) and (10), we then obtain directly the relations between the input optical field mode and the output optical field mode of two waveguides a and b as follows

J. Li et al. / Physics Letters A 374 (2010) 3762–3767

3765

aˆ out (ω)

−ηbˆ in (ω) + [i ω + κ2 + =

i ω + η + κ2 +

gc2

| T |2

i (ω+δ)+Γ01 + i (ω+δ−eω )+Γ 12 02 gc2

]ˆain (ω) ,

| T e |2

i (ω+δ)+Γ01 + i (ω+δ−ω )+Γ 12 02

(19) bˆ out (ω)

−ηaˆ in (ω) + [i ω + κ2 + =

i ω + η + κ2 +

gc2

| T |2

i (ω+δ)+Γ01 + i (ω+δ−eω )+Γ 12 02 gc2

]bˆ in (ω) ,

| T |2

i (ω+δ)+Γ01 + i (ω+δ−eω )+Γ 12 02

(20) where ω = ωc − ω is the cavity detuning from the input signal field. Initially, the PC cavity is in the vacuum state and the input field in the waveguides (see Fig. 1(a)) is bˆ in (ω), then the waveguide outputs in Eqs. (19) and (20) can be reduced into the following form

aˆ out (ω) =

bˆ out (ω) =

−ηbˆ in (ω) i ω + η + κ2 +

[i ω + κ2 +

,

gc2

(21)

| T |2 i (ω+δ)+Γ01 + i (ω+δ−eω )+Γ 12 02

gc2

| T |2

i (ω+δ)+Γ01 + i (ω+δ−eω )+Γ 12 02

i ω + η + κ2 +

gc2

]bˆ in (ω) . (22)

| T |2 i (ω+δ)+Γ01 + i (ω+δ−eω )+Γ 12 02

The intensity transmission coefficients of two waveguides a and b are readily written as

T a (ω) ≡

2

†

ˆaout (ω)ˆaout (ω)

η(d1 d2 + | T e |2 )

, =

2 † 2 (d3 + η)(d1 d2 + | T e | ) + d2 gc bˆ (ω)bˆ in (ω) in

(23)

2

† bˆ out (ω)bˆ out (ω)

d3 (d1 d2 + | T e |2 ) + d2 gc2

= , T b (ω) ≡ † (d3 + η)(d1 d2 + | T e |2 ) + d2 gc2 bˆ (ω)bˆ in (ω) in

(24) with d1 = i (ω + δ) + Γ01 , d2 = i (ω + δ − ω12 ) + Γ02 , and d3 = i ω + κ2 , respectively. Here, T a describes the transmission

of the field from bˆ in to aˆ out and T b describes the transmission of the field from bˆ in to bˆ out . Alternatively, assuming that the initial field at frequency ω begins in mode aˆ in , it is easy to check from Eqs. (19) and (20) that the intensity transmission coefficients of the PC cavity-waveguide system has the same form as Eqs. (23) and (24) except the exchange a ⇔ b in T a and T b due to the symmetry of two waveguides a and b in Fig. 1(a). 3. Discussion of results We adopt the following system parameters for calculation: η = 1 THz, g c = 0.33 THz, and κ = 0.1 THz based on Waks and Vuckovic’s papers [22,23]. Typical values of Γ01 and Γ02 for realistic QDM materials can be found in Refs. [35–37]: Γ01 = 1.6 GHz and Γ02 = 10−3 Γ01 . In Fig. 2, the intensity transmission T a and T b of PC cavitywaveguide system as a function of the voltage-controlled detuning of the levels |1 and |2 (ω12 ) is plotted based on Eqs. (23) and (24), when keeping all other parameters fixed. From this figure, we can find interesting and useful phenomena: (i) When the levels of the different dots are in resonance ω12 = 0 (resonant tunneling), the input signal field bˆ in transmits into aˆ out . In this case,

Fig. 2. (Color online.) Intensity transmission T a and T b as a function of voltagecontrolled detuning ω12 (GHz). The other system parameters are chosen as T e = 5 GHz, ω = 0, δ = 0, η = 1 THz, gc = 0.33 THz, κ = 0.1 THz, Γ01 = 1.6 GHz, and Γ02 = 10−3 Γ01 , respectively.

the intensity transmission T a in the waveguide a reaches to 0.9 while the intensity transmission T b in the waveguide b reaches to 0; (ii) When ω12 = 0 (nonresonant tunneling), namely when the applied bias voltage tilts the energy levels of the structure such that levels |1 and |2 move out of resonance, the intensity transmission T a in the waveguide a quickly decreases to a zero steady-state value (T a = 0). On the other hand, the intensity transmission T b in the waveguide b rapidly increases to a saturation value (T b = 0.97) and is independent of the voltage-controlled detuning ω12 at the output of the waveguide. From what has been analyzed above, we can see that the voltage-controlled detuning ω12 = 0 suggests one channel bˆ in ⇒ aˆ out is switched on whereas another channel bˆ in ⇒ bˆ out is switched off. In contrast, the voltagecontrolled detuning ω12 = 0 (e.g., ω12 = 10 GHz below) indicates one channel bˆ in ⇒ aˆ out is switched off whereas another channel bˆ in ⇒ bˆ out is switched on. It is clearly shown that one can well control the path of an optical field propagating in a waveguide with the voltage-controlled detuning. As mentioned above, the tunnel barrier in such a driven QDM can be directly controlled by placing a gate electrode between the two QDs. As a result, by suitably varying the applied bias voltage, we can realize a type of electro-optic switching operation for our considered system. Now, we give a quantitative explanation for the above observed switching operation. Consider the situation where the LD is resonant with the PC cavity, so that δ = 0. In the ideal case, the bare cavity decay rate κ is very small and can be set to zero (this assumption is reasonable when η κ , as is normally achieved in experimental situations). In this limit, when the input signal field is resonant with the PC cavity ω = 0 and the voltage-controlled detuning is tuned to ω12 = 0, from Eqs. (21) and (22) we have aˆ out ≈ −bˆ in and bˆ out ≈ 0 for the experimentally available parameters used in the present Letter. Consequently, the field propagates in the waveguide a. In the opposite regime, when the voltage-controlled detuning is adjusted to an appropriate nonzero value ω12 > T e2 /Γ01 , from Eqs. (21) and (22) we have

aˆ out ≈ 0 and bˆ out ≈ bˆ in , so that the field is switched to the original waveguide b. That is to say, the input field can efficiently transmit from the waveguide a into the waveguide b through the PC cavity by appropriately tuning the voltage-controlled detuning ω12 , and in this process the switching manipulation is completed.

3766

J. Li et al. / Physics Letters A 374 (2010) 3762–3767

Fig. 3. (Color online.) Intensity transmission T a and T b versus tunneling constant T e (GHz). The other system parameters are chosen as ω = 0, δ = 0, η = 1 THz, gc = 0.33 THz, κ = 0.1 THz, Γ01 = 1.6 GHz, and Γ02 = 10−3 Γ01 , respectively.

Fig. 4. (Color online.) Intensity transmission T a and T b versus cavity intrinsic decay rate κ (THz). The other system parameters are chosen as T e = 5 GHz, ω = 0, δ = 0, η = 1 THz, gc = 0.33 THz, Γ01 = 1.6 GHz, and Γ02 = 10−3 Γ01 , respectively.

In order to gain a deeper insight into the dependence of the intensity transmission spectrum of PC cavity-waveguide switching system on the tunneling strength, Fig. 3 plots the intensity transmission T a and T b versus the tunneling strength T e for two different values of the voltage-controlled detuning (ω12 = 0 and ω12 = 10 GHz), when keeping all other parameters fixed. It is found from Fig. 3 that, when the voltage-controlled detuning is zero, i.e., ω12 = 0, the intensity transmission T a increases rapidly with the increase of the tunneling strength T e firstly, then the amount of the intensity transmission reaches to the saturation value (T a = 0.9) and is independent of the tunneling strength T e . On the other hand, the intensity transmission T b decreases quickly with increasing T e , up to a zero steady-state value (T b = 0). However, when the voltage-controlled detuning is tuned to the constant value ω12 = 10 GHz, the intensity transmission curves T a and T b are almost insensitive to T e . From these analysis, we can conclude that a change of T e in higher value ranges has little influence on the intensity transmission of PC cavity-waveguide switching system. Finally, we analyse how the intensity transmission is related to the dissipations of the optical system for two cases that ω12 = 0

Fig. 5. (Color online.) Intensity transmission T a and T b versus cavity-waveguide coupling rate η (THz). The other system parameters are chosen as T e = 5 GHz, ω = 0, δ = 0, gc = 0.33 THz, κ = 0.1 THz, Γ01 = 1.6 GHz, and Γ02 = 10−3 Γ01 , respectively.

Fig. 6. (Color online.) Intensity transmission T a and T b versus QDM dissipation rate Γ01 (GHz). The other system parameters are chosen as T e = 5 GHz, ω = 0, δ = 0, η = 1 THz, gc = 0.33 THz, κ = 0.1 THz, and Γ02 = 10−3 Γ01 , respectively.

and ω12 = 10 GHz, as shown in Figs. 4–6. Fig. 4 displays the influence of the cavity intrinsic decay rate κ on the intensity transmission T a and T b of PC cavity-waveguide switching system. With the value of ω12 = 0, an increase of κ significantly decreases the transmission T a of the signal field from bˆ in to aˆ out through the PC cavity but it has weak influence on the transmission T b from bˆ in to bˆ out . While with the constant value of ω12 = 10 GHz, the intensity transmission T a and T b are independent of κ . Therefore, with the higher intrinsic quality factor Q of the PC cavity (here Q is defined as Q = ωc /κ ), the larger intensity transmission for bˆ in ⇒ aˆ out can be achieved for the case of resonant tunneling. Fig. 5 plots the intensity transmission T a and T b of PC cavity-waveguide switching system as a function of the cavity-waveguide coupling rate η . It can be seen from Fig. 5 that, when ω12 = 0, the intensity transmission T a is enhanced continuously with the increase of η , then the amount of the intensity transmission reaches to a larger value (T a = 0.96) at η = 3 THz. At the same time, the intensity transmission T b is depressed continuously with increasing η , finally it arrives at a zero steady-state value (T b = 0) at the output of the waveguide. While when ω12 = 10 GHz, the intensity

J. Li et al. / Physics Letters A 374 (2010) 3762–3767

transmission T a is zero all the time. The intensity transmission T b is affected weakly by η and is depressed slightly with increasing T e . In Fig. 6, the intensity transmission T a and T b versus the QDM dissipation rate Γ01 is plotted. For the case that ω12 = 0, both of T a and T b are independent of Γ01 . For the case that ω12 = 10 GHz, the intensity transmission T a is almost kept unchanged with the increase of Γ01 , but the intensity transmission T b obviously decreases as Γ01 increases. So with the longer dipole emitter lifetime τ (here τ = 1/Γ01 ), the higher intensity transmission for bˆ in ⇒ bˆ out would be attained for the case of nonresonant tunneling. 4. Conclusion In summary, the transmission spectra of a QDM-cavity-waveguide coupling system have been studied theoretically in the low- Q regime, where the coupling strength between the QDM and the cavity is smaller than the cavity decay rate. Using realistic system parameters, we show that, by suitably varying the applied bias voltage to tune the electron tunnel coupling, one can well control the path of an optical field propagating in a waveguide and electro-optic switching operation is possible for such a QDM-cavity-waveguide system. We would like to point out here that most optical switching schemes are proposed by means of the strong coupling of the trapped atoms with the high- Q cavity in the general case of the non-linear system. In contrast with previous work, the weak excitation through the low- Q PC cavity in our scheme is employed to realize the switching operation. This work may be useful for the implementation of the optical information processing and optical network for chip to chip interconnects in photonic crystal nanostructures. Acknowledgements We would like to thank Professor Ying Wu for helpful discussion and his encouragement. The project is partially supported by the National Natural Science Foundation of Chinas under Grant Nos. 10975054, 10634060 and 10874050, by the Doctoral Foundation of the Ministry of Education of China under Grant No. 200804870051 as well as by National Basic Research Program of China under Contract No. 2005CB724508.

3767

References [1] A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, M. Paniccia, Nature 427 (2004) 615. [2] Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, M. Lipson, Opt. Express 15 (2007) 430. [3] Y. Vlasov, W.M.J. Green, F. Xia, Nat. Photonics 2 (2008) 242. [4] G. Manzacca, K. Hingerl, G. Cincotti, IEEE J. Quantum Electron. 44 (2008) 872. [5] J.L. O’Brien, Science 318 (2007) 1567. [6] D. Englund, A. Faraon, B. Zhang, Y. Yamamoto, J. Vuckovic, Opt. Express 15 (2007) 5550. [7] P. Yao, S. Hughes, Opt. Express 17 (2009) 11505. [8] A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, J.L. O’Brien, Science 320 (2008) 646. [9] E. Paspalakis, A. Kalini, A.F. Terzis, Phys. Rev. B 73 (2006) 073305. [10] E. Paspalakis, Phys. Rev. B 67 (2003) 233306. [11] Y.F. Xiao, J. Gao, X. Yang, R. Bose, G.C. Guo, C.W. Wong, Appl. Phys. Lett. 91 (2007) 151105. [12] A.S. Sørensen, K. Mølmer, Phys. Rev. Lett. 90 (2003) 127903. [13] H.J. Kimble, Nature 453 (2008) 1023. [14] A. Imamoglu, H. Schmidt, G. Woods, M. Deutsch, Phys. Rev. Lett. 79 (1997) 1467. ´ Nature 450 [15] D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, J. Vuˇckovic, (2007) 857. ´ Science 320 [16] I. Fushman, D. Englund, A. Faraon, N. Stoltz, P. Petroff, J. Vuˇckovic, (2008) 769. ´ [17] D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, J. Vuˇckovic, Phys. Rev. Lett. 104 (2010) 073904. [18] S. Noda, M. Fujita, T. Asano, Nat. Photonics 1 (2007) 449. [19] Y. Wu, Phys. Rev. A 61 (2000) 033803. [20] Y. Wu, P.T. Leung, Phys. Rev. A 60 (1999) 630. [21] Y. Qian, J. Qian, Y.Z. Wang, Chin. Phys. Lett. 26 (2009) 084203. [22] E. Waks, J. Vuckovic, Phys. Rev. Lett. 96 (2006) 153601. [23] E. Waks, J. Vuckovic, Phys. Rev. A 73 (2006) 041803(R). [24] W.X. Lai, H.C. Li, R.C. Yang, Opt. Commun. 281 (2008) 4048. [25] K. Eftekhari, K. Abbasian, A. Rostami, Opt. Commun. 283 (2010) 1817. [26] G. Manzaccaa, G. Cincotti, K. Hingerl, Appl. Phys. Lett. 91 (2007) 231920. [27] G. Manzacca, H. Habibian, K. Hingerl, G. Cincotti, Photonics Nanostruct. Fundam. Appl. 7 (2009) 39. [28] J.M. Villas-Bôas, A.O. Govorov, S.E. Ulloa, Phys. Rev. B 69 (2004) 125342. [29] C.W. Gardiner, M.J. Collett, Phys. Rev. A 31 (1985) 3761. [30] D. Walls, G. Milburm, Quantum Optics, Springer, Berlin, 1994. [31] C.W. Gardiner, P. Zoller, Noise Quantum, 3rd ed., Springer, Berlin, 2004. [32] L.M. Duan, A. Kuzmich, H.J. Kimble, Phys. Rev. A 67 (2003) 032305. [33] G. Shinkai, T. Hayashi, Y. Hirayama, T. Fujisawa, Appl. Phys. Lett. 90 (2007) 103116. [34] W.Q. Ma, X.J. Yang, M. Chong, T. Yang, L.H. Chen, J. Shao, X. Lü, W. Lu, C.Y. Song, H.C. Liu, Appl. Phys. Lett. 93 (2008) 013502. [35] C.J. Chang-Hasnain, P.C. Ku, J. Kim, S.L. Chuang, Proc. IEEE 91 (2003) 1884. ´ D. Fattal, C. Santori, G.S. Solomon, Y. Yamamoto, Appl. Phys. Lett. 82 [36] J. Vuˇckovic, (2003) 3596. [37] C.H. Yuan, K.D. Zhu, Appl. Phys. Lett. 89 (2006) 052115.