Coupling to a microdisk cavity containing a three-level quantum-dot with two orthogonal modes

Optics Communications 284 (2011) 2937–2942

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Coupling to a microdisk cavity containing a three-level quantum-dot with two orthogonal modes X.W. Mi ⁎, J.X. Bai, D.J. Li, H.P. Zhao College of Physics Science and Information Engineering, Jishou and University, Jishou 416000, Hunan, PR China

a r t i c l e

i n f o

Article history: Received 2 December 2010 Received in revised form 21 January 2011 Accepted 11 February 2011 Available online 9 March 2011 Keywords: Mode coupling Whispering gallery mode Quantum dot

a b s t r a c t The dynamics of a composite system containing two orthogonal degenerate whispering-gallery cavity modes coupling to a quantum dot (QD) is presented by a full quantum approach. The energy levels of the quantum dot are modeled as a V-type three-level system, which consist of the ground state, right- and left-polarized excitons. The counterclockwise mode a and the clockwise mode b are coupled with the transitions corresponding to the right- and left-polarized excitons with coupling rates gR and gL, respectively. An exact solution is proposed in a real-space approach. We majorly discuss the effects of the backscattering rate β on the spectra of the transmission and reflection in a strong coupling regime. A new insight is that one can overcome the excitons' fine structure splitting of a real QD with appropriate backscattering rate β by fine designing the cavity, which would be possible for applications to produce the degenerate entangled photon pairs in a real QD system. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The study of coherent quantum interactions between the electromagnetic field and matter inside a resonator is known as cavity quantum electrodynamics (CQED)[1], which has been extensively studied both in experiments [2–8] and theory [9–15] due to their potential applications for optical quantum computer and all-optical quantum information processing [16]. When an atom is strongly coupled to a cavity mode, it is possible to realize important quantum information processing tasks, such as controlled coherent coupling and entanglement of distinguishable quantum systems. Realizing these tasks in the solid state is desirable for developing an all-solid-state quantum device. The pioneering work of Weisbuch et al. [17] opens a new door of solid state quantum device research. Coupling semiconductor self-assembled quantum dots (QDs) to monolithic optical cavities is promising to realize quantum information processing in solid-state materials [3,18,19]. Recently, the system of a waveguide coupled to a cavity embedded with an atom or a QD was proposed to study single-photon transport properties [5,6,11,15,20–22] . Srinivasan and Painter have carried out a comprehensive study of this system based on the standard quantum master equation model [22]. In addition, the property of single-photon transmission spectra in a single-mode waveguide coupling to a whispering-gallery resonator containing a two-level atom has been investigated by Jung-Tsung Shen and Shanhui Fan using a full quantum-mechanical and deterministic approach [15]. In general, an atom or a QD embedded in a cavity is modeled as a simple two-level system for brevity. But an atom or a QD always has ⁎ Corresponding author. E-mail address: [email protected] (X.W. Mi). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.02.024

more than two energy levels, and it should be necessary for us to consider the more complicated system such as three- or four-level system. Furthermore, entangled photon pairs can be produced through a multi-level quantum state, such as the biexciton decay of a QD strongly coupled to the modes of a photonic crystal [23–25]. However, even though QDs are often thought of as artificial atoms, they deviate from the ideal scheme because of the coupling between the exciton states with total angular momentum +1 and −1. The anisotropic electronhole interaction splits the exciton states into two modes linearly polarized along the crystallographic axis of the crystals H (horizontal) and V (vertical) [23] and produces fine structure splitting (FSS) in the exciton states [26,27]. The FSS of cross-polarized excitons prevents entangled photon generation. Several significant efforts have been made to overcome this problem, for example, by spectrally filtering indistinguishable photon pairs [28], by applying external fields to make the exciton states degenerate [29,30], by thermal annealing of QDs [31], by carefully selecting QDs with smaller anisotropic energy difference, [32,33] and by using temporal gates [34]. Recently, a new type of entangled-photon generation from a V-type system in a microcavity has been proposed [35]. In contrast to the entangled-photon generation in context, all four Bell states can be freely generated from an identical cavity system by simply selecting applied-field polarizations and frequencies due to the excitation of dressed states in CQED. In this paper, we present the dynamics of a composite system containing two orthogonal degenerate whispering-gallery cavity modes coupling to a V-type three-level QD by a full quantum approach. The energy levels of the QD are modeled as a three-level system consisting of the ground state, right- and left-polarized excitons states. The counterclockwise whispering-gallery mode (WGM) and the clockwise WGM are coupled with the transitions corresponding to the right- and

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left-polarized excitons with coupling rates gR and gL, respectively. An exact solution is proposed in a real-space approach. The effects of the backscattering rate β on the spectra of the transmission and reflection in a strong coupling regime are majorly discussed. Moreover, an insight is that one can overcome the excitons' fine structure splitting (FSS) of the QD with appropriate backscattering rate β by fine designing the cavity. Thus one can get the degenerate exciton states, which are important to produce the degenerate entangled photon pairs in a biexciton QD [23]. 2. The configuration of the system and theory A sketch of the composite system under consideration is depicted in Fig. 1. Two waveguide couples to a whispering-gallery-type microcavity embedded a QD, and the degenerate left- and right-polarized modes propagate in the top and bottom waveguides, respectively. The QD locates in the symmetric position in the cavity, and it is modeled as a V-type three-level system. The relevant states of the quantum dot include the ground state |G〉 and the exciton states |R〉 and |L〉 with frequencies ωR and ωL, respectively. The fine structure splitting (FSS) Δ = ωR − ωL. The counterclockwise WGM a and the clockwise WGM b are coupled with the transitions corresponding to the right-and left-polarized excitons, respectively. Considering the reservoir [36] and assuming the energy of the ground state is zero with the rotating wave approximation [37], the effective Hamiltonian of the composite system S ⊕ R is given by [15,22,38]

 ∂ ∂ H =
h = ∫dxc†Ra ðxÞ ω0 −iυg cRa ðxÞ + ∫dxc†La ðxÞ ω0 + iυg c ðxÞ ∂x ∂x La

 ∂ ∂ † † cRb ðxÞ + ∫dxcLb ðxÞ ω0 + iυg c ðxÞ + ∫dxc ðxÞ ω0 −iυg Rb ∂x ∂x Lb †

†

†

creation operator of the exciton state of the QD with left (right) polarization. The interactions with the reservoirs give rise to intrinsic dissipation, and γc, γL and γR are the intrinsic dissipation rates of the cavity WGMs and the quantum-dot, respectively. The fourth and fifth lines in Eq. (1) describe the interactions between the waveguides and the cavity. Va/b is the coupling rate between the cavity and the † − † waveguide with respect to mode a/b. σ+ R = aRag(σR = agaR) and † + − † σL = aLag(σL = agaL) are the raising (lowering) ladder operator of the QD corresponding to the transition of |R〉 − |G〉, and |L〉 − |G〉, respectively. The last item describes the coupling between the clockwise and counterclockwise whispering-gallery modes with the backscattering rate β caused by surface roughness of the microcavity. For the microcavity, this Hamiltonian is effective that the detail can be seen in Fan's paper [36]. The exact solutions are derived by a full quantum method [15]. The temporal evolution of an arbitrary single photon state |Φ(t)〉 is described by the Schröinger equation
ih

∂ jΦðt Þ〉 = H jΦðt Þ〉; ∂t

where H is the Hamiltonian of Eq. (1). The system is considered that it is optically pumped in such a way as the QD has an initial ground state with no cavity photons, and the waveguides are empty. It is in a weak excitation limit where a limited Fock space can be employed, thus the state of the system at time t can be written as ˜ Rb ðx; t Þc†Rb ðxÞj∅〉 jΦðt Þ〉 = ∫dx½˜ φRa ðx; t Þc†Ra ðxÞ j∅〉 + φ ˜ La ðx; t Þc†La ðxÞj∅〉 + φ ˜ Lb ðx; t Þc†Lb ðxÞj∅〉 + e˜a ðt Þa† j∅〉 +φ + e˜b ðt Þb† jj∅〉 + e˜L ðt ÞσLþ j∅〉 + e˜R ðt ÞσRþ jj∅〉;

†

+ ðωc −iγc Þa a + ðωc −iγc Þb b + ðωR −iγR ÞaR aR + ðωL −iγL ÞaL aL ð1Þ h i h i †  † †  † + ∫dxδðxÞ Va cRa ðxÞa + Va cRa ðxÞa + ∫dxδðxÞ Vb cLa ðxÞb + Vb cLa ðxÞb h i h i †  † †  † + ∫dxδðxÞ Vb cRb ðxÞb + Vb cRb ðxÞb + ∫dxδðxÞ Va cLb ðxÞa + Va cLb ðxÞa     þ þ † 4 † + gR aσR + gL bσL + h:c + βb a + β ba :

This Hamiltonian includes the waveguide, the cavity, and the QD, as well as the interaction between the waveguide and the cavity, and the interaction between QD and the cavity. The first two lines in Eq. (1) † describe the propagating photon modes in the waveguides. cRa/b(x) † (cLa/b(x)) is a bosonic operator creating a right-moving (left-moving) photon at x with respect to mode a/b. ω0 is a reference frequency, around which the waveguide dispersion relation is linearized [36]. And υg is the group velocity of the photons in the waveguide. The third line in Eq. (1) describes the modes in the cavity and the QD states considering the effects of the reservoir. a†(a) is the bosonic creation (annihilation) operator for the counterclockwise WGM, and b†(b) is the bosonic creation (annihilation) operator for the clockwise WGM. The modes a and b have the same frequency ωc and couple with the transitions † † corresponding to |R〉 − |G〉 and |L〉 − |G〉, respectively. aL(aR) is the

ð2Þ

where | ∅ 〉 represents the vacuum state with zero photons in the cavity. ˜ R = L ðx; t Þ is the single-photon wave function in the Right/Left mode. φ e˜a = b ðt Þ is the excitation amplitude of the whispering-gallery mode, and e˜L ðt Þ; e˜R ðt Þ are the excitation amplitudes of the QD with respect to the transitions |L〉 − |G〉 and |R〉 − |G〉, respectively. Substituting Eq. (3) to the Schröinger equation [Eq. (2)], we get the following set of equations of motion: −iνg

∂ ∂ ˜ Ra ðx; t Þ = i φ ˜ ðx; t Þ + δðxÞVa e˜a ðt Þ + ω0φ ˜ ðx; t Þ; φ ∂x Ra ∂t Ra

ð4aÞ

−iνg

∂ ∂ ˜ Rb ðx; t Þ = i φ ˜ ðx; t Þ + δðxÞVb e˜b ðt Þ + ω0φ ˜ ðx; t Þ; φ ∂x Rb ∂t Rb

ð4bÞ

iνg

∂ ∂ ˜ La ðx; t Þ = i φ ˜ ðx; t Þ + δðxÞVb e˜b ðt Þ + ω0φ ˜ ðx; t Þ; φ ∂x La ∂t La

ð4cÞ

iνg

∂ ∂ ˜ Lb ðx; t Þ = i φ ˜ ðx; t Þ + δðxÞVa e˜a ðt Þ + ω0φ ˜ ðx; t Þ; φ ∂x Lb ∂t Lb

ð4dÞ

4 ˜ Ra ð0; t Þ + Vaφ ˜ Lb ð0; t Þ + gR4 e˜R ðt Þ + β4 e˜b ðt Þ = i ðωc −iγc Þ˜ea ðt Þ + Va φ

4

4

4

˜ Rb ð0; t Þ + Vb φ ˜ La ð0; t Þ + gL e˜L ðt Þ + β˜ ðωc −iγc Þ˜ eb ðt Þ + Vb φ e a ðt Þ = i

∂ e˜ ðt Þ; ∂t R ∂ ðωL −iγLG Þ˜ eL ðt Þ + gL e˜b ðt Þ = i e˜L ðt Þ: ∂t ðωR −iγRG Þ˜ eR ðt Þ + gR e˜a ðt Þ = i

Fig. 1. Left panel: sketch of the system under consideration. Two optical fiber taper waveguides couple with a quantum dot-microcavity system. Cavity-waveguide coupling rate is Γ. Ta, Tb and Ra, Rb are the transmitted and reflected powers, respectively. The two degenerate whispering-gallery modes (WGMs) are described by counterclockwise mode a and clockwise mode b. β is the backscattering rate. Right panel: schematic representation of V-type three-level quantum dot. gR and gL are the right and left coupling rate between cavity and quantum dot, respectively. Δ is the fine structure splitting (FSS) of the QD.

ð3Þ

∂ e˜ ðt Þ; ∂t a

ð4eÞ

∂ e˜ ðt Þ; ∂t b

ð4fÞ

ð4gÞ ð4hÞ

One can obtain the dynamics of the system directly by integrating this set of equations [Eqs. (4a–4h)] for any given initial state |Φ(t = 0)〉. In this way, one could study the time-dependent transport of an arbitrary single-photon wave packet. In the following, we only consider the steady-state properties. When |Φ(t)〉 is an eigenstate of frequency ,

X.W. Mi et al. / Optics Communications 284 (2011) 2937–2942

i.e. |Φ(t)〉 = e− it|+〉 Eq. (2) yields the time-independent eigen equation þ

þ

H j 〉 =
h j 〉;

ð5Þ

where -h is the total energy of the system. For an input state of onephoton Fock state, the most general time-independent interacting eigenstate can be given as þ

†

†

ω + iγR − ωR, k3 = ω + iγL − ωL, k12 = k1k2,k123 = k1k2k3, we arrive the following exact solutions by solving Eqs. (8a)–(8h)     2 2 2 2 2 k23 jβj + 2Γiβ4 −k1 −2Γik1 + jgL j k12 −jgR j + jgR j k3 ðk1 + 2ΓiÞ  2    ; ta = 2 2 2 2 k23 jβj + 4Γ −k1 −4Γik1 + jgL j k12 + 2Γik2 −jgR j + jgR j2 k3 ðk1 + 2ΓiÞ

tb =

†

j  〉 = ∫dx½φRa ðxÞcRa ðxÞj∅〉 + φRb ðxÞcRb ðxÞj∅〉 + φLa ðxÞcLa ðxÞj∅〉 +

† φLb ðxÞcLb ðxÞj∅〉

+

þ eR σR j∅〉

†

†

+ ea a j∅〉 + eb b jj∅〉 +

þ eL σL j∅〉

ra =

ð6Þ

where we denote the time-independent amplitudes by the corresponding untilded symbols, e.g. e˜a ðt Þ = ea e−it , etc. The connection between the interacting eigenstate and a scattering experiment is described by the Lippmann–Schwinger formalism [15,39,40]. According to the time-independent Schröinger equation of Eq. (5), the state |+〉 in Eq. (6) yields the following equations of motion −iυg −iυg

∂ φ ðxÞ + δðxÞVa ea = ð−ω0 ÞφRa ðxÞ; ∂x Ra ∂ φ ðxÞ + δðxÞVb eb = ð−ω0 ÞφRa ðxÞ; ∂x Rb

eR =

∂ φ ðxÞ + δðxÞVb eb = ð−ω0 ÞφLa ðxÞ; ∂x La

ð7cÞ

iυg

∂ φ ðxÞ + δðxÞVa ea = ð−ω0 ÞφLb ðxÞ; ∂x Lb

ð7dÞ

4

4

ð7eÞ

ðωc −iγc Þeb + Vb φRb ð0Þ + Vb φLa ð0Þ + gL4 eL + βea = eb ;

ð7fÞ

ðωR −iγR ÞeR + gR ea = eR ;

ð7gÞ

ðωL −iγL ÞeL + gL eb = eL ;

ð7hÞ

with  = ω, and ω = ω0 + υgk. We take φR(x) = eikx[θ(− x) + t(x)], and ω−ω0 , t is the transmission ampliφL(x) = re− ikxθ(− x), where k = υg tude, r is the reflection amplitude. The set of equations of motion, Eqs. (7a)–(7h) now reads −iυg ðta −1Þ + Va ea = 0

ð8aÞ

−iυg ðtb −1Þ + Vb eb = 0

ð8bÞ

−iυg ra + Vb eb = 0;

ð8cÞ

−iυg rb + Va ea = 0;

ð8dÞ 

1 + ta + rb + gR4 e˜R + β4 eb = 0; 2

ð8eÞ



1 + tb + ra + gL4 eL + βea = 0 2

ð8fÞ

ðωc −iγc −ωÞea + Va

ðωc −iγc −ωÞeb + Vb

ðωR −iγR −ωÞeR + gR ea = 0;

ð8gÞ

ðωL −iγL −ωÞeL + gL eb = 0;

ð8hÞ

From the equations above, ta, tb, ra, rb, ea, eb, eR and eL can be solved straightforwardly. Defining the parameters k1 = ω + iγc − ωc, k2 =

ð9aÞ

    k23 jβj2 + 2Γiβ−k21 −2Γik1 + jgR j2 k13 −jgL j2 + jgL j2 k2 ðk1 + 2ΓiÞ  2    ; ð9bÞ k23 jβj + 4Γ2 −k21 −4Γik1 + jgL j2 k12 + 2Γik2 −jgR j2 + jgR j2 k3 ðk1 + 2ΓiÞ   ijVb j2 k3 k12 + 2Γik2 −jgR j2 + iVb Va4 k23 β  2     ; ð9cÞ υg k23 jβj + 4Γ2 −k21 −4Γik1 + jgL j2 k12 + 2Γik2 −jgR j2 + jgR j2 k3 ðk1 + 2ΓiÞ 

  ijVa j2 k2 k13 + 2Γik3 −jgL j2 + iVa k23 β      2  ; ð9dÞ υg k23 jβj + 4Γ2 −k21 −4Γik1 + jgL j2 k12 + 2Γik2 −jgR j2 + jgR j2 k3 ðk1 + 2ΓiÞ

    2 −k2 Va4 k13 + 2Γik3 −jgL j + k3 β4 Vb4     ea = ; ð9eÞ k23 jβj2 + 4Γ2 −k21 −4Γik1 + jgL j2 k12 + 2Γik2 −jgR j2 + jgR j2 k3 ðk1 + 2ΓiÞ

ð7aÞ

iυg

4 4 ðωc −iγc Þea + Va φRa ð0Þ + Va φLb ð0Þ + gR4 e˜R + β4 eb = ea ;

rb =

eb =

ð7bÞ

2939

k23



    −k3 Vb4 k12 + 2Γik2 −jgR j2 + k2 βVa4    ; ð9fÞ jβj2 + 4Γ2 −k21 −4Γik1 + jgL j2 k12 + 2Γik2 −jgR j2 + jgR j2 k3 ðk1 + 2ΓiÞ

    2 −gR Va4 k13 + 2Γik3 −jgL j + k3 β4 Vb4  2    ; ð9gÞ k23 jβj + 4Γ2 −k21 −4Γik1 + jgL j2 k12 + 2Γik2 −jgR j2 + jgR j2 k3 ðk1 + 2ΓiÞ

    2 −gL Vb4 k12 + 2Γik2 −jgR j + k2 βVa4     : ð9hÞ eL = k23 jβj2 + 4Γ2 −k21 −4Γik1 + jgL j2 k12 + 2Γik2 −jgR j2 + jgR j2 k3 ðk1 + 2ΓiÞ

where ta, tb are the transmission amplitudes and ra, rb are the reflection amplitudes. ea, eb and eR, eL are the amplitudes with respect to cavity modes V2 , which and excitation modes, respectively. We set jVa j = jVb j≡V; Γ = 2υg are real number. Γ is the external linewidth of the WGMs due to waveguide–cavity coupling. Analytic expressions of the amplitudes shown in Eqs. (9a)–(9h) provide a complete description on the single-photon transport properties of the composite system. 3. Results and analysis We next show the effects of the detuning of the frequencies, the FSS Δ and the backscattering rate β on the spectra. The FSS of the QD exhibits frequency shifts and splittings and the frequency detuning only exhibits frequency shifts. The two cavity modes couple with each other due to backscattering and form standing wave modes, which enhance the frequency splitting as the cavity-modes strongly couple with the QD and exhibit the frequency shift [22]. From above similar effects on the spectra, an insight is that one can overcome the FSS by varying β and then achieve energy degenerate exciton–photon (polariton) modes which are valuable to recover the ideal biexciton decay picture for generating entangled photon pair [23]. A semiconductor composite system is considered, consisting a GaAs microdisk cavity and an InAs QD. The parameters are taken from current experiment [6]. The cavity quality factor Q≃105, and the disk diameter D=2.5μm. The QD in the whispering-gallery cavity is modeled as a V-type three-level system [35]. First, considering an ideal cavity (the backscattering rate β is zero), and (gR =gL)/2π=6.0 GHz,ωR =ωL =ωc, we get Fig. 2 by properly choosing the orientation of the quantum dot dipole polarization, and the azimuthal origin. The spectra of transmission are expressed as Ta ≡|ta|2, Tb ≡|tb|2; the reflected signals are expressed as Ra ≡|ra|2, Rb ≡|rb|2. In Fig. 2(a) the two excited levels of the QD are degenerate (ωR =ωL), and two pair overlapped symmetric resonances are shown, which is consistent with two modes being equally spatially coupled to the QD. In this case, the cross-polarized photon pairs can be extracted by spectral filtering due to vacuum Rabi splitting [35]. While in Fig. 2(b), ΔcR =(ωc −ωR)/2π=−9.6 GHz, ωc =ωL, the line of Ta gets asymmetrical, and a polarization splitting S about 3.0 GHz at lower

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X.W. Mi et al. / Optics Communications 284 (2011) 2937–2942

1.0

a

0.8

transmission/reflection

0.6 0.4 0.2 0.0 1.0

b

0.8 0.6 0.4 0.2 0.0 -20

-15

-10

-5

0

5

10

15

20

(w-wc)/2π (GHz) Fig. 2. Normalized optical transmission spectra (top curves) and reflection spectra (bottom curves) of the coupled waveguide–microcavity–quantum dot: ωc = ωL = ωR (top panel), (ωc − ωR)/2π = − 9.6 GHz GHz (bottom panel). Cavity and QD parameters are {gR,gL, γc,γR, γL, Γ}/2π = {6, 6,0.16, 0.16, 0.44} GHz, respectively.

frequency and about 6.5 GHz at higher frequency. Moreover, both of the splitting S in lower frequency and in higher frequency are less than the FSS which is Δ=((ωR −ωL)/2π=9.6 GHz). The results show the FSS of the polarization quantum dot can be suppressed in this compound system by strong coupling. It is also shown that the line moves to the range of the higher frequency due to the detuning ΔcR; and the frequency splitting gets larger, which can be seen from the black arrow in Fig. 2(b) comparing with the red arrow in Fig. 2(a). To have an insight into the effects of the backscattering rate β on the FSS, we studied the normalized optical transmission and reflection spectra of the coupled system with different β, which is shown in Fig. 3. In this case the coupling rate (gR = gL)/2π = 6.0 GHz is invariable and ωR = ωL = ωc. When β is negative (corresponding to the phase of β which is π), the normalized optical transmission and reflection spectra are the mirror image of the spectra of positive β. The resonance peaks shift from the left (lower frequency) to right (higher frequency); and the frequency splitting gets enhanced with the increasing of the |β|, which can be seen from the blue arrow at the bottom comparing with the red arrow in the center in Fig. 3. The backscattering effects on the spectra are similar to the effects of the detuning ΔcR/L and the FSS of the QD on the spectra. For a better understanding of this phenomenon, we take gR = gL = 0 with β = 0 and β/2π = 9.6 GHz for a comparison in the plot

of Fig. 4. But there is only a frequency shift that is equal to 9.6 GHz, and no frequency splitting, which is different from the results of Fig. 3. It means that the major effects of backscattering rate are exhibiting frequency shifts, while enhancing frequency splitting happens when there is coupling between the QD and the cavity modes. According to Eqs. (9e)–(9h), the spectra Ea = |ea|2, Er = |er|2, Eb = |eb|2, El = |el|2 are shown in Fig. 5. We take β = 0 in the top panel and β/2π = − 9.6 GHz in the bottom panel, ωR = ωL, (ωc − ωL)/2π = − 9.6 GHz. The two cavity modes couple with the transitions corresponding to the right- and left-polarized excitons, respectively. The same character of curves in the top panel a1–a2 and in the bottom panel b1–b2 is obtained as the two exciton levels are degenerate (ωR = ωL). From Fig. 5, we can see that Ea and Er have the same resonance frequency as well as Eb and El, and it is in agreement with which the modes a and b couple to the transitions corresponding to |R〉 − |G〉 and |L〉 − |G〉, respectively. For a larger backscattering rate (β/2π = − 9.6 GHz) in the bottom panel, the distance of two resonance peaks in the bottom panel is larger than that in the top panel. It indicates that the frequency splitting has been greatly enhanced with the increasing of the modulus of the β. It is also shown in the top panel that the detuning ΔcR/L = ωc − ωR/L makes the spectra line shift to higher frequency. While the frequency shift is compensated due to backscattering as shown in the bottom panel in Fig. 5. From the above discussions, the backscattering effects on the spectra are similar to the effects of the detuning ΔcR/L and the FSS of the QD on the spectra. A new insight is that one can overcome the excitons' fine structure splitting of the real QD with appropriate backscattering rate β by fine designing the cavity. We next show that it is indeed possible. We take β/2π = 9.6 GHz, (ωc − ωL)/2π = − 6.6 GHz, the FSS Δ = (ωR − ωL)/ 2π = 10 GHz which is about 40 μeV, and the transmission and reflection spectra of the system are shown in Fig. 6(a). The resonance peaks of a mode and b mode are nearly overlapped, which indicate that the exciton states are in degenerate states. The interpretation of these results is: The two WGMs are strongly coupled to each other due to backscattering except their coupling to the cavity and QD, which has formed standing wave modes through a superposition of the initial traveling wave modes; when the phase of the β is 0, only the higher frequency mode of the doublet has any spatial overlap with the QD, while the lower frequency mode of the doublet has any spatial overlap with the QD in the case that the phase of the β is π; as a result, the backscattering enhances the frequency splitting as the cavity-modes strongly couple with the QD and exhibit frequency shift [22]. The compositive effects both of the coupling of the QD and the backscattering result in implementation of energy degenerate exciton–photon (polariton)

β/2π=−9.6 ω =ω =ω

β/2π=−4.8

β=0

β/2π=4.8

β/2π=9.6

)/2π (

)

Fig. 3. Normalized optical transmission spectra and reflection spectra of the coupled waveguide–microcavity–quantum dot system for different β. In all case, ωc = ωR = ωL. Cavity and QD parameters for these simulations are {gR, gL, γc, γR, γL,Γ}/2π = {6, 6,0.76, 0.16,0.16, 0.44} GHz.

( Fig. 4. Normalized optical transmission spectra and reflection spectra of the coupled waveguide–microcavity system without coupling to QD (gR =gL =0), and with β=0 and β/2π=−9.6 GHz for compare. Other parameters of cavity and QD are {γc,γR,γL,Γ}/2π= {0.76,0.16,0.16,0.44}GHz, respectively.

X.W. Mi et al. / Optics Communications 284 (2011) 2937–2942

2941

ω −ω =0 (ω ω )/2π=−9.6

β=0

β=0

β/2π=−9.6

β/2π=−9.6

ω−ω )/2π (

)

ω−ω )/2π (

)

Fig. 5. Spectra Ea = |ea|2, Er = |er|2 (left panel); Eb = |eb|2, El = |el|2 (right panel) of the coupled system: ωR = ωL, (ωc − ωR)/2π = − 9.6 GHz; β = 0 in the top panel and β/2π = − 9.6 GHz in the bottom panel. Other parameters of cavity and QD are {gR, gL, γc, γR, γL, Γ}/2π = {6, 6, 0.76, 0.16, 0.16, 0.44} GHz, respectively.

modes. It means that one can overcome the QD FSS in the coupled system by choosing the backscattering rate β and then recover the ideal biexciton decay picture for generating entangled photon pair [23]. When β = 0, the transmission and reflection spectra of the system are shown in Fig. 6(b). The resonance peaks of Ta and Tb are separated each other. The separate gap is about 1.7 GHz at lower frequency and about 8.3 GHz at higher frequency, and both of them are less than the FSS Δ = 10 GHz. That means the polarized quantum dot FSS has been compensated. Furthermore, comparing Fig. 6(a) to (b), one can find there are more resonance peaks in Fig. 6(a). The reason is that the two

(ω −ω )/2π=10 (ω −ω )/2π=−6.6 (

WGMs are strongly coupling with the QD when the two WGMs are coupling with each other with β/2π = 9.6 GHz rather than β = 0. To make a better understanding of the results in Fig. 6(a), we plot the spectra Ea = |ea|2, Eb = |eb|2, Er = |er|2, El = |el|2 in Fig. 7 with the same parameters as in Fig. 6(a). The resonance peaks are also almost overlapped. It is also in agreement with which the counterclockwise mode a and the clockwise mode b are coupled with the transitions corresponding to |R〉 − |G〉 and |L〉 − |G〉, respectively. Finally, the transmission and reflection spectra of the system depending on the dephasing rate produced by radiation decay are investigated, shown in Fig. 8. We set γR/2π=γL/2π=2.56 GHz and

β/2π=9.6

)

β=0

)/2π (

)

Fig. 6. Transmission spectra and reflection spectra of the coupled system for β/2π=9.6 GHz (top panel) and β=0 GHz (bottom panel). The FSS is (ωR −ωL)/2π=10 GHz. Other parameters of cavity and QD are {gR,gL,γc,γR,γL,Γ}/2π={6, 6, 0.76,0.16,0.16,0.44}GHz, respectively.

)/2π Fig. 7. Spectra Ea =|ea|2, Eb =|eb|2, Er =|er|2, El =|el|2 of the coupled system: β/2π=9.6 GHz. The QD FSS Δ is (ωR −ωL)/2π=10 GHz. Other parameters of cavity and QD are {gR, gL,γc,γR,γL,Γ}/2π={6,6,0.76,0.16,0.16,0.44}GHz, respectively.

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X.W. Mi et al. / Optics Communications 284 (2011) 2937–2942

Acknowledgments We acknowledge informative discussions with K.Srinivasan in emails. This work is supported by the National Science Foundation of China (Grant no. 10647132) and the Scientific Research Fund of Hunan Provincial Education Department (Grant nos. 09C825 and 10A100).

γ /2π=2.4

References

γ /2π=9.4

)/2π (

)

Fig. 8. Transmission spectra (top panel) and reflection spectra (bottom panel) of the coupled system: γR/2π = γL/2π = 2.56 GHz (top panel) and γR/2π = γL/2π = 9.56 GHz (bottom panel). QD FSS is (ωR − ωL)/2π = 10 GHz. Other parameters of cavity and QD are {gR, gL, γc, γR, γL, Γ}/2π={6, 6, 0.76, 0.16, 0.16, 0.44} GHz, respectively.

γR/2π=γL/2π=9.56 GHz. The relation between the parameters used in this paper and Ref. [22] is Γ=ke, γc =ki =kT −ke, γR =γ‖R +γpR. From Fig. 8, we can see that the resonance peaks decrease with the increasing of the dephasing rates. When γR/2π=γL/2π=9.56 GHz, the radiation decay rates are greater than the coupling rate g/2π=6 GHz between QD and cavity. The system is in weak coupling situation, and there is only one resonance peak in the transmission and reflection spectra, shown in Fig. 8(b). It indicates that the influence of dephasing rate on the dynamics of the coupling between a QD and a cavity is very important. How to overcome the influence of dephasing rate on the system is still an interesting topic [41,42]. From the discussions above, the FSS can be overcome in the cavityassisted composite system. The real GaAs and In(Ga)As QDs are good candidates to implement entangled-photon pairs. The main parameters, such as backscattering rate β, can be adjusted by experimental methods. In recent experiments so far, the backscattering rate is not precisely controlled, since it is due to fabrication imperfections. There is some general correspondence between the size of the microdisk and the backscattering rate, in that smaller disks tend to have higher rates because of the larger field overlap with the surface imperfections. In principle, one could engineer a backscattering rate by deliberately introducing features in the structure (i.e., fabricating a circular grating around its periphery). Such a device, including two waveguides coupled to the microdisk, is within reach of current technology [6].

4. Summary In conclusion, we extend a full quantum mechanical approach for a coupled waveguide-ring resonator interacting with a V-type three-level QD, and derive the analytic solutions for the single-photon transport. The real-space approach is outlined in this paper. The Rabi splitting of the results shows a strong coupling of the composite system. One can overcome the excitons' fine structure splitting of the real QD with appropriate backscattering rate β by fine designing the cavity. This analytic results will be likely valuable to interpret experimental spectra of a single V-type three-level QD strongly coupled to the WGMs which are coupled to two waveguides in the same time. We have shown that our configuration allows transforming the splitting QD exciton levels into degenerate polariton levels and it will likely be valuable for potential device applications generating entangled photon pairs.

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