Phonon induced phase grating in quantum dot system Guang-Ling Cheng,1 Wen-Xue Zhong,1 and Ai-Xi Chen1,2,∗ 1 Department 2 Institute
of Applied Physics, East China Jiaotong University, Nanchang, 330013, China for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada ∗ [email protected]
Abstract: Electromagnetically induced phase grating is theoretically investigated in the driven two-level quantum dot exciton system at the presence of the exciton-phonon interactions. Due to the phonon-induced coherent population oscillation, the dispersion and absorption spectra are sharply changed and the phase modulation is enhanced via the high refractive index with nearly-vanishing absorption, which could effectively diffract a weak probe light into the first-order direction with the help of a standing-wave control field. Moreover, the diffraction efficiency of the grating can be easily manipulated by controlling the Huang-Rhys factor representing the exciton-phonon coupling, the intensity and detuning of the control field, and the detuning of the probe field. The scheme we present has potential applications in the photon devices for optical-switching and optical-imaging in the micro-nano solid-state system. © 2015 Optical Society of America OCIS codes: (270.1670) Coherent optical effects; (050.1950) Diffraction gratings; (230.3990) Micro-optical devices.
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1.
Introduction
Quantum coherence and interference effect has been one of important research focuses in quantum optics and laser physics [1]. In particular, the laser-induced coherence and interference is found to be an effective and feasible method for achieving many interesting quantum phenomena [2–10]. Recently, electromagnetically induced grating (EIG), which is first proposed by Ling et al. [11], has attracted considerable interest due to its potential applications in optical switching and routing [12], light storage [13], self-imaging [14], and so on. EIG refers to the phenomenon that the probe beam is diffracted into high-order direction by the medium acting as an absorption or phase grating at the presence of the standing-wave driving field, and has been experimentally observed in hot and cold atomic systems [12, 15–17]. Due to the tunability and easy implementation of EIG, many schemes have been proposed for the preparation of EIG [18–26]. The grating with the enhanced intensity of higher-order diffraction was proposed in a driven three-level ladder system [18]. The phase grating with the improved efficiency was shown via microwave modulation in a double-dark-state atomic system [19]. The EIG scheme based on nonlinear modulation has been proposed in the electromagneticallyinduced-transparency atomic medium [20]. Later, the spontaneously generated coherence [21] of the atomic sample and Fano-interference in asymmetric quantum well [22, 23] were applied to produce the quantum grating. The photoinduced diffraction grating has been proposed via exciton induced transparency in the coupled metal nanoparticle-quantum dot system [24]. The latest researches indicate that the gain-phase grating could be obtained in the active Raman gain cold atoms [25] and the two-dimensional electromagnetically induced cross-grating could be existent in a four-level tripod-type atomic system [26]. On the other hand, semiconductor quantum dot nanostructure has been a promising candidate for quantum information processing in recent years due to its striking inherent advantages [27]. For example, the quantum dots behave in many ways as the simple stationary atomic system with the discrete energy levels. And the strong coupling could be existent in the present solidstate system and high-efficiency quantum gate and state operations could be implemented. Furthermore, such a system could be artificially structured with the flexible designs and are easily controlled for the experimental parameters, which are helpful for implementing the information processing via combining the other quantum system to develop the hybrid quantum system. Due to quantum confinement effects, the quantum dot system exhibits the strong coherence and it could be formed the exciton model more easily. And the discrete levels are existent and can be modulated via changing its size. As a consequence, many interesting quantum optical phenomena have already been studied in this system, such as coherent optical spectroscopy [28], electromagnetically induced transparency [29, 30], coherent population trapping [31], vacuum Rabi splitting [32], and so on. In the quantum dot system, the exciton in quantum dot acts as a two-level system [33] and generally the strong exciton-phonon interactions are present in the two-level quantum dot exciton system [30, 34–36]. Furthermore, the effect of exciton-phonon couplings can play an important role in such a system and make natural difference from that in the usual two-level atomic systems. It is worth mentioning that the above scheme for the diffraction gating in the quantum dot is based on electromagnetically induced transparency and the metal nanoparticle is required. In this paper, we focus on the electromagnetically induced phase grating via phonon induced #234070 - $15.00 USD (C) 2015 OSA
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coherent population oscillation in the driven two-level quantum dot system. In our scheme, due to the exciton-phonon interactions, the enhanced refractive index could be obtainable, which is accompanied by the nearly-vanishing absorption and is useful for generating the phase grating and improving the diffraction efficiency. With the help of the standing-wave control field, the refractive index of the probe field experiences a periodic variation and the phase grating is formed in the quantum dot system, which can diffract the probe light beam in the first-order and the higher-order directions. The diffraction efficiency could be manipulated via changing the Huang-Rhys factor corresponding to the exciton-phonon coupling, the detunings of the control and probe fields and the interaction length. Our scheme is based on the phonon-induced coherent population oscillation and is drastically different from the scheme of photoinduced diffraction grating proposed by Xiao et al., which is based on electromagnetically induced transparency and is required for the external metal nanoparticle. This paper is organized as follows. In Sec. II, we describe the physical model under consideration and derive the required equations. In Sec. III, we discuss the phase grating properties in quantum dot system. And a conclusion is given in the last section. 2.
Model and Equation
We consider a single two-level semiconductor quantum dot exciton system with the ground state |g and the first excited state |e at the presence of exciton-phonon interactions [30-36]. As shown in Fig. 1(a), the single quantum dot is driven by a strong standing-wave field with frequency ωc and probed by a weak traveling-wave field with frequency ω p , and simultaneously is coupled with the longitudinal optical phonons with small momenta [37]. Here we consider the quantum dot with the large transition energies and then the interaction with longitudinal acoustic phonons could be neglected [37]. With the rotating-wave approximation and in the rotating frame, the Hamiltonian of the whole system reads as [30, 38–40] H = h¯ ω ph ∑b†q bq + h¯ ∑λq (bq + b†q )Sz + h¯ Δc Sz − h¯ Ωc (x) (S+ + S− ) − h¯ Ω p (S+ e−iδ t + S− eiδ t ), q
q
(1) where Sz = (|e e| − |g g|) /2, S+ = |e g| and S− = |ge| are the pseudospin-1/2 operators characterizing the single quantum dot exciton, and b†q bq is the creating (annihilation) operator of the phonon with momentum q and frequency ω ph . For the present system, we focus on the interaction between longitudinal optical phonons and the exciton and so we may consider the longitudinal optical phonons with the monochrmatic frequency. Δc = ωeg − ωc is the detuning between quantum dot and the control field and δ = ω p − ωc is the detuning of the probe field with respect to the control field. λq is the phonon-exciton coupling constant. Ωn = μeg ·en En /¯h are the Rabi frequencies of the applied fields with the amplitudes En , where μeg is the transition dipole matrix element, anden are the polarization vectors of the electromagnetically fields. As shown in Fig. 1(b), a sketch of prototype experimental setup is presented. Two counterpropagating components of the control fields form a standing wave field along the x direction, which lead to the space-dependent atom-field interaction. The probe field passes through the standing-wave region in the z axis and then exhibits the periodical phase modulation. Thus the phase-diffraction grating is achieved. The Rabi frequency of the standing-wave field can be expressed as Ωc (x) = Ωc sin (π x/Λ), where the parameter Ωc is the amplitude of the control field and Λ is the space period of the standing wave. The angle θ1 denotes the first-order diffraction angle of the probe beam with respect to the z direction. As shown in the result section, the first-order diffraction occurs at θ1 = arcsin 12 . It is obvious that the motion equations of the operators with respect to exciton and phonon are derived in terms of the Heisenberg equation of motion. In order to obtain the steady-state response results, we consider the equations of motion of the mean values of the physical oper#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9873
Δc
Ωc(x) phonons
|e²
Δp
Ωc Ωp
Ωp
Mq
θ1 θ1
Dot
+1 0 -1
x
z
Ωc
y
|g²
(a)
(b)
Fig. 1. (a) The sketch consisting of many longitudinal optical phonons with small momenta and a single two-level quantum dot driven by a control field and probed by a weak field. (b) The schematic plot of the control and probe fields propagating through the medium.
ators and utilize the semiclassical approach via ignoring the quantum correlation properties of the operators [41, 42]. For simplicity, the average value A is characterized as A, and so the final equations of motions are
dS− 1 − −iδ t Sz , dt = −( T2 + i(Δc + N))S − 2i Ωc (x) + Ω p e dSz 1 1 z −iδ t S+ − i Ω∗ (x) + Ω∗ eiδ t S− , c p dt = − T1 (S + 2 ) + i Ωc (x) + Ω p e d2N dt 2
2 3 z + γn dN dt + ω ph N = −2ω ph λ0 S ,
(2) (3) (4)
2 , which represents the exciton-phonon coupling where N = ∑λq bq + b†q and λ0 = ∑λq2 /ω pn q
q
and is named to the Huang-Rhys factor [43]. T1 is the exciton spontaneous lifetime and T2 is its dephasing time, which are included into the above equations phenomenologically. γ ph is the phonon decay rate. Now we make the following assumptions [44] S− (t) Sz (t) N(t)
= S0 + S+1 e−iδ t + S−1 eiδ t , z z = S0z + S+1 e−iδ t + S−1 eiδ t ,
(5) (6)
= N0 + N+1 e−iδ t + N−1 eiδ t .
(7)
We then substitute Eqs. (5-7) into Eqs. (2-4) and ignore the second-order and the high-term small terms. After doing the above steps, we obtain the steady-state solution as follows S+1 = T2 Ω p
2 −8λ0 ω0 η T1 /T2 Ω2 0 α k0 Π(1+iΔ0 −4iλ0 ω0 k0 )
+
2iT1 /T2 Ω2 0 α k0 Π
+ ik0
1 + iΔ0 − 4iλ0 ω0 k0 − iδ0
,
(8)
where ω0 = ω ph T2 , Ω0 = Ω0 sin (π x/Λ) , Ω0 = Ωc T2 , Δ0 = Δc T2 , δ0 = δ T2 , γ0 = γ ph T2 , and η =
ω02 ,α ω02 −iγ0 δ0 −δ02
=
1 1+iΔ0 −4iλ0 ω0 k0 −iδ0
+
1 1−iΔ0 +4iλ0 ω0 k0 , β
=
4iλ0 ω0 k0 η Ω2 0 +Ω2 0 1+iΔ0 −4iλ0 ω0 k0 1+iΔ0 −4iλ0 ω0 k0 −iδ0
+
−4iλ0 ω0 k0 η Ω2 0 +Ω2 0 1−iΔ0 +4iλ0 ω0 k0 1−iΔ0 +4iλ0 ω0 k0 −iδ0
, and Π = TT12 iδ0 − 1 − 2 TT21 β . The parameter k0 represents the population inversion of the exciton with k0 = 2S0z , and it is satisfied as (9) (k0 + 1) (Δ0 − 4λ0 ω0 k0 )2 + 1 + 4T1 /T2 Ω2 0 k0 = 0. #234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9874
The above cubic equation has either a single root or three real roots. The latter case just corresponds to the intrinsic optical bistability which arises from the interactions of the exciton and phonon [41]. However, only a single real root is existent in the present system. For the case of weak probe field, the effective linear susceptibility is described by 2 2 Nq μeg S+1 Nq T2 μeg χe f f (ω p ) = = χ (ω p ) h¯ ε0 h¯ ε0 Ωp with
χ (ω p ) =
2 −8λ0 ω0 η T1 /T2 Ω2 0 α k0 Π(1+iΔ0 −4iλ0 ω0 k0 )
+
2iT1 /T2 Ω2 0 α k0 Π
+ ik0
1 + iΔ0 − 4iλ0 ω0 k0 − iδ0
(10)
.
(11)
Here the parameter Nq is the number density of quantum dot and χ (ω p ) is dimensionless linear susceptibility. Next, we focus on the diffraction pattern of the probe field through the quantum dot. According to the Maxwell’s theories, the propagation equation of the probe field in the slowly-varying amplitude approximation and the paraxial approximation is given by [1]
∂ E p (x, z,t) 1 ∂ E p (x, z,t) iπ χe f f (ω p ) + = E p (x, z,t), ∂z c ∂t λp
(12)
where λ p is the wavelength of the probe field. On replacing z by z = z/z0 with z0 = 2 h¯ ε0 λ p /π Nq T2 μeg and considering the steady-state response, we have the spatial propagation equation as ∂ Ep = iχ E p , (13) ∂ z where z can be made dimensionless when the parameter z0 is taken as the unit of z. The transmission function of the quantum dot at z = L can be derived via solving the above equation as (14) T (x) = eiχ L/z0 = e−Im[χ ]L/z0 eiRe[χ ]L/z0 . Here the term e−Im[χ ]L/z0 corresponds to the absorption modulation and the term eiRe[χ ]L/z0 describes the phase modulation. Based on the standing-wave coupling and Fraunhofer diffraction theory, we have the Fraunhofer-diffraction intensity [11] I p (θ ) = |F (θ )|2
sin2 (M π Λx sin θ /λ p ) , M 2 sin2 (π Λx sin θ /λ p )
(15)
where F (θ ) = 01 T (x) exp (−i2π xΛx sin θ /λ p ) dx is the Fourier transform of T (x), which represents the Fraunhofer diffraction of single space period. θ is the diffraction angle with respect to the z-direction and M is the number of space period in x-axis. The parameter Λx is spatial period of the control standing-wave field. It is noted that the n-order diffraction intensity is determined by Eq. (15) with sin θ = nλ p /Λx along x-axis. In particular, the first diffraction intensity is given by 1 2 Ip (θ1 ) = |F (θ1 )| = T (x) exp (−i2π x) dx . 0 2
3.
(16)
Results and Discussions
In this section, we will discuss the Fraunhofer diffraction intensity of the probe field through a single driven two-level quantum dot exciton system in terms of the diffraction theories and #234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9875
Fig. 2. The dispersion (Blue solid line) and absorption (Red dash line) spectra vs the probe detuning Δ p0 . The corresponding parameters are given by T2 = 2T1 , ω0 = 15, γ0 = 0.01, Δ0 = 0, and λ0 = 0, Ω0 = 0.2 (a), λ0 = 0, Ω0 = 0.5 (b), λ0 = 0, Ω0 = 2 (c), λ0 = 0.01, Ω0 = 0.2 (d), λ0 = 0.01, Ω0 = 0.5 (e), λ0 = 0.01, Ω0 = 2 (f).
equations of the the above section. Here we choose the realistic parameters in the InAs quantum dots. As shown in Refs. [45–47], the interband dipole moment is about 2.1enm [45] and the dephasing times range between 300 fs at room temperature and 630 ps at low temperature [46]. The interband transition energy is 1.1 eV and the LO-phonon energy in InAs quantum dot is about 32.3 meV [47]. Meanwhile, the Huang–Rhys factor λ0 is about 0.015 in self-organized and between 0.1 and 1 for nearly spherical II–VI quantum dots [35]. From these realistic parameters of quantum dots, we will analyze the diffraction properties of the probe light beam. In order to clearly clarify the physical mechanism of phase diffraction, we first focus on the influence of exciton-phonon couplings on the dispersion and absorption of the quantum dot system. As shown in Fig. 2, the dispersion and absorption spectra are plotted with the invariable Ωc (x). The corresponding parameters are given by T2 = 2T1 , ω0 = 15, γ0 = 0.01, Δ0 = 0, and λ0 = 0, Ω0 = 0.2 (a), λ0 = 0, Ω0 = 0.5 (b), λ0 = 0, Ω0 = 2 (c), λ0 = 0.01, Ω0 = 0.2 (d), λ0 = 0.01, Ω0 = 0.5 (e), λ0 = 0.01, Ω0 = 2 (f). At the absence of phonon-exciton couplings (Fig. 2(a), 2(b) and 2(c)), the two-level quantum dot exciton system exhibits the similar behavior as
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Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9876
the natural atomic system driven by a monochromatic field. When the driven field is weak, the maximal absorption with anomalous dispersion appears at the central probe region. For the increasing intensity of the control field, the simultaneous presence of absorption and gain can be clearly observed. And the high refractive index is obtainable with Reχ (ω p ) ≈ 0.053 and Imχ (ω p ) ≈ 0, nearly at Δ p0 = 3.84 with Δ p0 = (ω p − ωeg ) T2 . When the phonon-exciton coupling is involved, as shown in Fig. 2(d), 2(e) and 2(f), the phonon-induced coherent population oscillation occurs, which dramatically changes the absorption and dispersion properties of the quantum dot system. Compared with the case of no-coupling, the narrow absorption and gain profiles are present at about probe detuning Δ p0 = ±15. At the same time, the greatly-enhanced refractive index can be obtained with respect to the nearly-vanishing absorption. Especially, for the case of Fig. 2(e) with parameters λ0 = 0.01, Ω0 = 0.5, we could obtain Reχ (ω p ) ≈ 0.16 and Imχ (ω p ) ≈ 0 approximately at the detuning Δ p0 = −14.9709. Furthermore, from the insert maps in Fig. 2(d), 2(e) and 2(f) we see that the value of dispersion at Δ p0 = −14.97 is strongly dependent on the Rabi frequency of the control field. It increases to a maximal value and then decreases with the increasing of Ω0 . All in all, the phonon-exciton couplings lead to the enhancement of refractive index, which is useful for preparing phase grating and improving the diffraction efficiency.
Fig. 3. The Fraunhofer diffractions of phase modulation exp (iΦ) (Φ = Re [χ ] L/z0 ) (a), absorption modulation |T (x)| (|T (x)| = exp (−Im [χ ] L/z0 )) (b), and the diffraction intensity I p (θ ) (c) as functions of sin θ . The parameters are chosen as Ω0 = 0.5, Δ p0 = −14.97, L = 40z0 , M = 5, Λx /λ p = 4, λ0 = 0 (Red dash line) and λ0 = 0.01 (Blue solid line). The other parameters are as same as those in Fig. 2.
Here we focus on the phase grating based on the large refractive index enhanced by the coherent population oscillation, which is induced by the phonon-exciton couplings in the quantum dot exciton system. In Fig. 3, the Fraunhofer diffractions of phase modulation exp (iΦ) (Φ = Re [χ ] L/z0 ) (a), absorption modulation |T (x)| (|T (x)| = exp (−Im [χ ] L/z0 )) (b), and the diffraction intensity Ip (θ ) (c) are displayed as functions of sin θ . The parame-
#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9877
ters are chosen as T2 = 2T1 , ω0 = 15, γ0 = 0.01, Δ0 = 0, Ω0 = 0.5, Δ p0 = −14.97, L = 40z0 , M = 5, Λx /λ p = 4, λ0 = 0 (Red dash line) and λ0 = 0.01 (Blue solid line). It is obvious that the phonon-exciton couplings play a crucial role in the phase and absorption modulations. At the absence of phonon-exciton couplings (λ0 = 0), the diffraction efficiency of first-order diffraction is very low and most intensity are limited to the zero-order direction. The root is that the small index of refraction accompanied by the nearly-vanishing absorption is existent at the large probe-detuning in the weak monochromatic-field-driving two-level physical system, as shown in Fig. 2(b). As a consequence, the phase modulation is very small although the absorption modulation |T (x)| is nearly equal to 1 and high transmissivity is achievable. At the presence of phonon-exciton interactions (λ0 = 0.01), however, the large phase modulation with high transmissivity is clearly obtained, which transfers energy from zero-order to high-order diffractions. Even though the driving field is quite weak in two-level system, phonon-exciton couplings induce the coherent population oscillation phenomenon and change the dispersion and absorption properties. At about the probe detuning Δ p0 = −14.97, the great enhancement of refractive index with nearly-vanishing absorption is present, which are shown in Fig. 2(e). Therefore, the phase modulation is dominant in the Fraunhofer diffraction, which could improve the first-order diffraction efficiency.
Fig. 4. The first-order diffraction intensity I p (θ1 ) as a function of Rabi frequency Ω0 (a), interaction length L (b), and coupling constant λ0 (c). Other parameters are the same as in Fig. 3.
Finally, we are concentrated on the first-order diffraction intensity Ip (θ1 ). In order to do so, we plot the variance of Ip (θ1 ) as functions of Ω0 , L and λ0 in Fig. 4, where the other parameters are same as those in Fig. 3. It is seen from Fig. 4(a) that the first-order diffraction intensity gradually becomes large as the increasing of Ω0 , then it oscillates with the Rabi frequency and finally it decreases drastically. For the very small Ω0 , only the small dispersion could be achievable nearby the probe detuning Δ p0 = −14.97, which limit the diffraction efficiency.
#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9878
About at the region of Ω0 = 0.5, the dispersion is maximal and that leads to the enhancement of phase modulation. Hence, the high diffraction efficiency is existent in the first-order diffraction direction in the present system. In addition, for the case of large Ω0 , the absorption is dominant and the first-order diffraction energy is restrained and the efficiency of diffraction is highly decreased. Similarly, we can see from Fig. 4(b) that the value of Ip (θ1 ) gradually becomes large as the increasing of L. At the length L ≈ 40z0 , the intensity value is maximal and then it becomes small. From Fig. 4(c) we see that the value of Ip (θ1 ) could be effectively modulated via changing the Huang-Rhys factor λ0 . Under the condition of λ0 0.03, the value could be close to 0.3 owing to the optimal phase modulation with high transmissivity. All in all, the high-efficiency first-order diffraction could be obtained via properly controlling the HuangRhys factor λ0 , the Rabi frequency Ω0 and the interaction length L.
|e,N+² |e,N−²
|e²
ωeg−ωph
ωeg |g,N+² |g,N−²
|g²
(a)
ωeg+ωph |g,N+² |g,N−²
|g,N+² |g,N−²
|g,N+² |g,N−²
(b)
|e,N+² |e,N−²
|e,N+² |e,N−²
|e,N+² |e,N−²
(c)
(d)
Fig. 5. (a) The dressed states of the exciton at the presence of phonon. (b) The transition process with respect to the emission of a photon for the three-photon resonance. (c) The case for the stimulated Rayleigh resonance. (d) The corresponding transition for the absorption resonance as modified by the ac Stark effect.
Before conclusion, we provide the origin of phonon-induced coherent features via using the dressed-state picture at the presence of phonon couplings. In order to do so, here we consider the system consisting of a single quantum dot and a phonon mode with momentum q and express the quantum dot-phonon Hamiltonian as Hd−p = h¯ ωeg Sz + h¯ ω ph b†q bq + h¯ λq (bq + b†q )Sz .
(17)
In terms of the Hamiltonian-diagonalization method [48], the energy eigenstates are derived as |±, N± = |± ⊗ e
∓(λq /ω ph )(bq +b†q )
|Nq ,
(18)
with the corresponding eigenvalues E± = ±¯hωeg /2 + h¯ ω ph (Nq − λ /2). In the above equation, the states |± are the eigenstates of the operator Sz with |+ = |e and |− = |g, Nq is the number operator of phonon, and |N± is the position-displaced Fock states. As shown in Fig. 5, the dressed states (Fig. 5(a)) and the corresponding transition processes are provided. Fig. 5(b) indicates the electromagnetically induced three-photon resonance. The control field excites the electron into the excited state and then makes a transition from |e, N− to |g, N+ , which is accompanied with the emission of photon with frequency ωeg − ω ph . This process corresponds to an amplified light at the probe detuning Δ p0 = −ω ph0 = −15, as indicated by the region of negative absorption in Fig. 2(d), 2(e) and 2(f). Fig. 5(c) refers to the process of electromagnetically induced stimulated Rayleigh resonance, in which the transition frequency is centred #234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9879
on the frequency of the control field. And Fig. 5(d) represents the electromagnetically induced absorption resonance as modified by the ac Stark effect. Here we focus on the amplification process indicated by Fig. 5(b) and apply it to prepare the phase grating. 4.
Conclusion
In conclusion, the phase grating is theoretically investigated in the two-level quantum dot exciton system under the condition of phonon-exciton couplings. The results show that the phononexciton interactions play a crucial role in preparing the phase grating and controlling the diffraction efficiency. At the absence of phonon-exciton couplings, the very low first-order diffraction intensity could be obtained for the probe beam. At the moment, the probe light is difficultly diffracted due to the ineffective phase modulation. However as the phonon-exciton interactions are existent, the phase modulation, accompanied with the high transmissivity, is greatly enhanced and the first-order diffraction becomes increasing with the reduced center light energy. Furthermore, the diffraction intensity could be effectively improved via appropriately adjusting the physical parameters including Huang-Rhys factor, Rabi frequency of the control field, the detunings of external fields and the interaction length. The phase grating in the quantum dot system has potential application in all-optical switching and routing, developing new photon devices and measuring the solid materials. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant Nos. 11165008 and 11365009), the Foundation of Young Scientist of Jiangxi Province, China (Grant No. 20142BCB23011), and China Scholarship Council (Grant No. 201408360040).
#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9880
of Applied Physics, East China Jiaotong University, Nanchang, 330013, China for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada ∗ [email protected]
Abstract: Electromagnetically induced phase grating is theoretically investigated in the driven two-level quantum dot exciton system at the presence of the exciton-phonon interactions. Due to the phonon-induced coherent population oscillation, the dispersion and absorption spectra are sharply changed and the phase modulation is enhanced via the high refractive index with nearly-vanishing absorption, which could effectively diffract a weak probe light into the first-order direction with the help of a standing-wave control field. Moreover, the diffraction efficiency of the grating can be easily manipulated by controlling the Huang-Rhys factor representing the exciton-phonon coupling, the intensity and detuning of the control field, and the detuning of the probe field. The scheme we present has potential applications in the photon devices for optical-switching and optical-imaging in the micro-nano solid-state system. © 2015 Optical Society of America OCIS codes: (270.1670) Coherent optical effects; (050.1950) Diffraction gratings; (230.3990) Micro-optical devices.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
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#234070 - $15.00 USD (C) 2015 OSA
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1.
Introduction
Quantum coherence and interference effect has been one of important research focuses in quantum optics and laser physics [1]. In particular, the laser-induced coherence and interference is found to be an effective and feasible method for achieving many interesting quantum phenomena [2–10]. Recently, electromagnetically induced grating (EIG), which is first proposed by Ling et al. [11], has attracted considerable interest due to its potential applications in optical switching and routing [12], light storage [13], self-imaging [14], and so on. EIG refers to the phenomenon that the probe beam is diffracted into high-order direction by the medium acting as an absorption or phase grating at the presence of the standing-wave driving field, and has been experimentally observed in hot and cold atomic systems [12, 15–17]. Due to the tunability and easy implementation of EIG, many schemes have been proposed for the preparation of EIG [18–26]. The grating with the enhanced intensity of higher-order diffraction was proposed in a driven three-level ladder system [18]. The phase grating with the improved efficiency was shown via microwave modulation in a double-dark-state atomic system [19]. The EIG scheme based on nonlinear modulation has been proposed in the electromagneticallyinduced-transparency atomic medium [20]. Later, the spontaneously generated coherence [21] of the atomic sample and Fano-interference in asymmetric quantum well [22, 23] were applied to produce the quantum grating. The photoinduced diffraction grating has been proposed via exciton induced transparency in the coupled metal nanoparticle-quantum dot system [24]. The latest researches indicate that the gain-phase grating could be obtained in the active Raman gain cold atoms [25] and the two-dimensional electromagnetically induced cross-grating could be existent in a four-level tripod-type atomic system [26]. On the other hand, semiconductor quantum dot nanostructure has been a promising candidate for quantum information processing in recent years due to its striking inherent advantages [27]. For example, the quantum dots behave in many ways as the simple stationary atomic system with the discrete energy levels. And the strong coupling could be existent in the present solidstate system and high-efficiency quantum gate and state operations could be implemented. Furthermore, such a system could be artificially structured with the flexible designs and are easily controlled for the experimental parameters, which are helpful for implementing the information processing via combining the other quantum system to develop the hybrid quantum system. Due to quantum confinement effects, the quantum dot system exhibits the strong coherence and it could be formed the exciton model more easily. And the discrete levels are existent and can be modulated via changing its size. As a consequence, many interesting quantum optical phenomena have already been studied in this system, such as coherent optical spectroscopy [28], electromagnetically induced transparency [29, 30], coherent population trapping [31], vacuum Rabi splitting [32], and so on. In the quantum dot system, the exciton in quantum dot acts as a two-level system [33] and generally the strong exciton-phonon interactions are present in the two-level quantum dot exciton system [30, 34–36]. Furthermore, the effect of exciton-phonon couplings can play an important role in such a system and make natural difference from that in the usual two-level atomic systems. It is worth mentioning that the above scheme for the diffraction gating in the quantum dot is based on electromagnetically induced transparency and the metal nanoparticle is required. In this paper, we focus on the electromagnetically induced phase grating via phonon induced #234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9872
coherent population oscillation in the driven two-level quantum dot system. In our scheme, due to the exciton-phonon interactions, the enhanced refractive index could be obtainable, which is accompanied by the nearly-vanishing absorption and is useful for generating the phase grating and improving the diffraction efficiency. With the help of the standing-wave control field, the refractive index of the probe field experiences a periodic variation and the phase grating is formed in the quantum dot system, which can diffract the probe light beam in the first-order and the higher-order directions. The diffraction efficiency could be manipulated via changing the Huang-Rhys factor corresponding to the exciton-phonon coupling, the detunings of the control and probe fields and the interaction length. Our scheme is based on the phonon-induced coherent population oscillation and is drastically different from the scheme of photoinduced diffraction grating proposed by Xiao et al., which is based on electromagnetically induced transparency and is required for the external metal nanoparticle. This paper is organized as follows. In Sec. II, we describe the physical model under consideration and derive the required equations. In Sec. III, we discuss the phase grating properties in quantum dot system. And a conclusion is given in the last section. 2.
Model and Equation
We consider a single two-level semiconductor quantum dot exciton system with the ground state |g and the first excited state |e at the presence of exciton-phonon interactions [30-36]. As shown in Fig. 1(a), the single quantum dot is driven by a strong standing-wave field with frequency ωc and probed by a weak traveling-wave field with frequency ω p , and simultaneously is coupled with the longitudinal optical phonons with small momenta [37]. Here we consider the quantum dot with the large transition energies and then the interaction with longitudinal acoustic phonons could be neglected [37]. With the rotating-wave approximation and in the rotating frame, the Hamiltonian of the whole system reads as [30, 38–40] H = h¯ ω ph ∑b†q bq + h¯ ∑λq (bq + b†q )Sz + h¯ Δc Sz − h¯ Ωc (x) (S+ + S− ) − h¯ Ω p (S+ e−iδ t + S− eiδ t ), q
q
(1) where Sz = (|e e| − |g g|) /2, S+ = |e g| and S− = |ge| are the pseudospin-1/2 operators characterizing the single quantum dot exciton, and b†q bq is the creating (annihilation) operator of the phonon with momentum q and frequency ω ph . For the present system, we focus on the interaction between longitudinal optical phonons and the exciton and so we may consider the longitudinal optical phonons with the monochrmatic frequency. Δc = ωeg − ωc is the detuning between quantum dot and the control field and δ = ω p − ωc is the detuning of the probe field with respect to the control field. λq is the phonon-exciton coupling constant. Ωn = μeg ·en En /¯h are the Rabi frequencies of the applied fields with the amplitudes En , where μeg is the transition dipole matrix element, anden are the polarization vectors of the electromagnetically fields. As shown in Fig. 1(b), a sketch of prototype experimental setup is presented. Two counterpropagating components of the control fields form a standing wave field along the x direction, which lead to the space-dependent atom-field interaction. The probe field passes through the standing-wave region in the z axis and then exhibits the periodical phase modulation. Thus the phase-diffraction grating is achieved. The Rabi frequency of the standing-wave field can be expressed as Ωc (x) = Ωc sin (π x/Λ), where the parameter Ωc is the amplitude of the control field and Λ is the space period of the standing wave. The angle θ1 denotes the first-order diffraction angle of the probe beam with respect to the z direction. As shown in the result section, the first-order diffraction occurs at θ1 = arcsin 12 . It is obvious that the motion equations of the operators with respect to exciton and phonon are derived in terms of the Heisenberg equation of motion. In order to obtain the steady-state response results, we consider the equations of motion of the mean values of the physical oper#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9873
Δc
Ωc(x) phonons
|e²
Δp
Ωc Ωp
Ωp
Mq
θ1 θ1
Dot
+1 0 -1
x
z
Ωc
y
|g²
(a)
(b)
Fig. 1. (a) The sketch consisting of many longitudinal optical phonons with small momenta and a single two-level quantum dot driven by a control field and probed by a weak field. (b) The schematic plot of the control and probe fields propagating through the medium.
ators and utilize the semiclassical approach via ignoring the quantum correlation properties of the operators [41, 42]. For simplicity, the average value A is characterized as A, and so the final equations of motions are
dS− 1 − −iδ t Sz , dt = −( T2 + i(Δc + N))S − 2i Ωc (x) + Ω p e dSz 1 1 z −iδ t S+ − i Ω∗ (x) + Ω∗ eiδ t S− , c p dt = − T1 (S + 2 ) + i Ωc (x) + Ω p e d2N dt 2
2 3 z + γn dN dt + ω ph N = −2ω ph λ0 S ,
(2) (3) (4)
2 , which represents the exciton-phonon coupling where N = ∑λq bq + b†q and λ0 = ∑λq2 /ω pn q
q
and is named to the Huang-Rhys factor [43]. T1 is the exciton spontaneous lifetime and T2 is its dephasing time, which are included into the above equations phenomenologically. γ ph is the phonon decay rate. Now we make the following assumptions [44] S− (t) Sz (t) N(t)
= S0 + S+1 e−iδ t + S−1 eiδ t , z z = S0z + S+1 e−iδ t + S−1 eiδ t ,
(5) (6)
= N0 + N+1 e−iδ t + N−1 eiδ t .
(7)
We then substitute Eqs. (5-7) into Eqs. (2-4) and ignore the second-order and the high-term small terms. After doing the above steps, we obtain the steady-state solution as follows S+1 = T2 Ω p
2 −8λ0 ω0 η T1 /T2 Ω2 0 α k0 Π(1+iΔ0 −4iλ0 ω0 k0 )
+
2iT1 /T2 Ω2 0 α k0 Π
+ ik0
1 + iΔ0 − 4iλ0 ω0 k0 − iδ0
,
(8)
where ω0 = ω ph T2 , Ω0 = Ω0 sin (π x/Λ) , Ω0 = Ωc T2 , Δ0 = Δc T2 , δ0 = δ T2 , γ0 = γ ph T2 , and η =
ω02 ,α ω02 −iγ0 δ0 −δ02
=
1 1+iΔ0 −4iλ0 ω0 k0 −iδ0
+
1 1−iΔ0 +4iλ0 ω0 k0 , β
=
4iλ0 ω0 k0 η Ω2 0 +Ω2 0 1+iΔ0 −4iλ0 ω0 k0 1+iΔ0 −4iλ0 ω0 k0 −iδ0
+
−4iλ0 ω0 k0 η Ω2 0 +Ω2 0 1−iΔ0 +4iλ0 ω0 k0 1−iΔ0 +4iλ0 ω0 k0 −iδ0
, and Π = TT12 iδ0 − 1 − 2 TT21 β . The parameter k0 represents the population inversion of the exciton with k0 = 2S0z , and it is satisfied as (9) (k0 + 1) (Δ0 − 4λ0 ω0 k0 )2 + 1 + 4T1 /T2 Ω2 0 k0 = 0. #234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9874
The above cubic equation has either a single root or three real roots. The latter case just corresponds to the intrinsic optical bistability which arises from the interactions of the exciton and phonon [41]. However, only a single real root is existent in the present system. For the case of weak probe field, the effective linear susceptibility is described by 2 2 Nq μeg S+1 Nq T2 μeg χe f f (ω p ) = = χ (ω p ) h¯ ε0 h¯ ε0 Ωp with
χ (ω p ) =
2 −8λ0 ω0 η T1 /T2 Ω2 0 α k0 Π(1+iΔ0 −4iλ0 ω0 k0 )
+
2iT1 /T2 Ω2 0 α k0 Π
+ ik0
1 + iΔ0 − 4iλ0 ω0 k0 − iδ0
(10)
.
(11)
Here the parameter Nq is the number density of quantum dot and χ (ω p ) is dimensionless linear susceptibility. Next, we focus on the diffraction pattern of the probe field through the quantum dot. According to the Maxwell’s theories, the propagation equation of the probe field in the slowly-varying amplitude approximation and the paraxial approximation is given by [1]
∂ E p (x, z,t) 1 ∂ E p (x, z,t) iπ χe f f (ω p ) + = E p (x, z,t), ∂z c ∂t λp
(12)
where λ p is the wavelength of the probe field. On replacing z by z = z/z0 with z0 = 2 h¯ ε0 λ p /π Nq T2 μeg and considering the steady-state response, we have the spatial propagation equation as ∂ Ep = iχ E p , (13) ∂ z where z can be made dimensionless when the parameter z0 is taken as the unit of z. The transmission function of the quantum dot at z = L can be derived via solving the above equation as (14) T (x) = eiχ L/z0 = e−Im[χ ]L/z0 eiRe[χ ]L/z0 . Here the term e−Im[χ ]L/z0 corresponds to the absorption modulation and the term eiRe[χ ]L/z0 describes the phase modulation. Based on the standing-wave coupling and Fraunhofer diffraction theory, we have the Fraunhofer-diffraction intensity [11] I p (θ ) = |F (θ )|2
sin2 (M π Λx sin θ /λ p ) , M 2 sin2 (π Λx sin θ /λ p )
(15)
where F (θ ) = 01 T (x) exp (−i2π xΛx sin θ /λ p ) dx is the Fourier transform of T (x), which represents the Fraunhofer diffraction of single space period. θ is the diffraction angle with respect to the z-direction and M is the number of space period in x-axis. The parameter Λx is spatial period of the control standing-wave field. It is noted that the n-order diffraction intensity is determined by Eq. (15) with sin θ = nλ p /Λx along x-axis. In particular, the first diffraction intensity is given by 1 2 Ip (θ1 ) = |F (θ1 )| = T (x) exp (−i2π x) dx . 0 2
3.
(16)
Results and Discussions
In this section, we will discuss the Fraunhofer diffraction intensity of the probe field through a single driven two-level quantum dot exciton system in terms of the diffraction theories and #234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9875
Fig. 2. The dispersion (Blue solid line) and absorption (Red dash line) spectra vs the probe detuning Δ p0 . The corresponding parameters are given by T2 = 2T1 , ω0 = 15, γ0 = 0.01, Δ0 = 0, and λ0 = 0, Ω0 = 0.2 (a), λ0 = 0, Ω0 = 0.5 (b), λ0 = 0, Ω0 = 2 (c), λ0 = 0.01, Ω0 = 0.2 (d), λ0 = 0.01, Ω0 = 0.5 (e), λ0 = 0.01, Ω0 = 2 (f).
equations of the the above section. Here we choose the realistic parameters in the InAs quantum dots. As shown in Refs. [45–47], the interband dipole moment is about 2.1enm [45] and the dephasing times range between 300 fs at room temperature and 630 ps at low temperature [46]. The interband transition energy is 1.1 eV and the LO-phonon energy in InAs quantum dot is about 32.3 meV [47]. Meanwhile, the Huang–Rhys factor λ0 is about 0.015 in self-organized and between 0.1 and 1 for nearly spherical II–VI quantum dots [35]. From these realistic parameters of quantum dots, we will analyze the diffraction properties of the probe light beam. In order to clearly clarify the physical mechanism of phase diffraction, we first focus on the influence of exciton-phonon couplings on the dispersion and absorption of the quantum dot system. As shown in Fig. 2, the dispersion and absorption spectra are plotted with the invariable Ωc (x). The corresponding parameters are given by T2 = 2T1 , ω0 = 15, γ0 = 0.01, Δ0 = 0, and λ0 = 0, Ω0 = 0.2 (a), λ0 = 0, Ω0 = 0.5 (b), λ0 = 0, Ω0 = 2 (c), λ0 = 0.01, Ω0 = 0.2 (d), λ0 = 0.01, Ω0 = 0.5 (e), λ0 = 0.01, Ω0 = 2 (f). At the absence of phonon-exciton couplings (Fig. 2(a), 2(b) and 2(c)), the two-level quantum dot exciton system exhibits the similar behavior as
#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9876
the natural atomic system driven by a monochromatic field. When the driven field is weak, the maximal absorption with anomalous dispersion appears at the central probe region. For the increasing intensity of the control field, the simultaneous presence of absorption and gain can be clearly observed. And the high refractive index is obtainable with Reχ (ω p ) ≈ 0.053 and Imχ (ω p ) ≈ 0, nearly at Δ p0 = 3.84 with Δ p0 = (ω p − ωeg ) T2 . When the phonon-exciton coupling is involved, as shown in Fig. 2(d), 2(e) and 2(f), the phonon-induced coherent population oscillation occurs, which dramatically changes the absorption and dispersion properties of the quantum dot system. Compared with the case of no-coupling, the narrow absorption and gain profiles are present at about probe detuning Δ p0 = ±15. At the same time, the greatly-enhanced refractive index can be obtained with respect to the nearly-vanishing absorption. Especially, for the case of Fig. 2(e) with parameters λ0 = 0.01, Ω0 = 0.5, we could obtain Reχ (ω p ) ≈ 0.16 and Imχ (ω p ) ≈ 0 approximately at the detuning Δ p0 = −14.9709. Furthermore, from the insert maps in Fig. 2(d), 2(e) and 2(f) we see that the value of dispersion at Δ p0 = −14.97 is strongly dependent on the Rabi frequency of the control field. It increases to a maximal value and then decreases with the increasing of Ω0 . All in all, the phonon-exciton couplings lead to the enhancement of refractive index, which is useful for preparing phase grating and improving the diffraction efficiency.
Fig. 3. The Fraunhofer diffractions of phase modulation exp (iΦ) (Φ = Re [χ ] L/z0 ) (a), absorption modulation |T (x)| (|T (x)| = exp (−Im [χ ] L/z0 )) (b), and the diffraction intensity I p (θ ) (c) as functions of sin θ . The parameters are chosen as Ω0 = 0.5, Δ p0 = −14.97, L = 40z0 , M = 5, Λx /λ p = 4, λ0 = 0 (Red dash line) and λ0 = 0.01 (Blue solid line). The other parameters are as same as those in Fig. 2.
Here we focus on the phase grating based on the large refractive index enhanced by the coherent population oscillation, which is induced by the phonon-exciton couplings in the quantum dot exciton system. In Fig. 3, the Fraunhofer diffractions of phase modulation exp (iΦ) (Φ = Re [χ ] L/z0 ) (a), absorption modulation |T (x)| (|T (x)| = exp (−Im [χ ] L/z0 )) (b), and the diffraction intensity Ip (θ ) (c) are displayed as functions of sin θ . The parame-
#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9877
ters are chosen as T2 = 2T1 , ω0 = 15, γ0 = 0.01, Δ0 = 0, Ω0 = 0.5, Δ p0 = −14.97, L = 40z0 , M = 5, Λx /λ p = 4, λ0 = 0 (Red dash line) and λ0 = 0.01 (Blue solid line). It is obvious that the phonon-exciton couplings play a crucial role in the phase and absorption modulations. At the absence of phonon-exciton couplings (λ0 = 0), the diffraction efficiency of first-order diffraction is very low and most intensity are limited to the zero-order direction. The root is that the small index of refraction accompanied by the nearly-vanishing absorption is existent at the large probe-detuning in the weak monochromatic-field-driving two-level physical system, as shown in Fig. 2(b). As a consequence, the phase modulation is very small although the absorption modulation |T (x)| is nearly equal to 1 and high transmissivity is achievable. At the presence of phonon-exciton interactions (λ0 = 0.01), however, the large phase modulation with high transmissivity is clearly obtained, which transfers energy from zero-order to high-order diffractions. Even though the driving field is quite weak in two-level system, phonon-exciton couplings induce the coherent population oscillation phenomenon and change the dispersion and absorption properties. At about the probe detuning Δ p0 = −14.97, the great enhancement of refractive index with nearly-vanishing absorption is present, which are shown in Fig. 2(e). Therefore, the phase modulation is dominant in the Fraunhofer diffraction, which could improve the first-order diffraction efficiency.
Fig. 4. The first-order diffraction intensity I p (θ1 ) as a function of Rabi frequency Ω0 (a), interaction length L (b), and coupling constant λ0 (c). Other parameters are the same as in Fig. 3.
Finally, we are concentrated on the first-order diffraction intensity Ip (θ1 ). In order to do so, we plot the variance of Ip (θ1 ) as functions of Ω0 , L and λ0 in Fig. 4, where the other parameters are same as those in Fig. 3. It is seen from Fig. 4(a) that the first-order diffraction intensity gradually becomes large as the increasing of Ω0 , then it oscillates with the Rabi frequency and finally it decreases drastically. For the very small Ω0 , only the small dispersion could be achievable nearby the probe detuning Δ p0 = −14.97, which limit the diffraction efficiency.
#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9878
About at the region of Ω0 = 0.5, the dispersion is maximal and that leads to the enhancement of phase modulation. Hence, the high diffraction efficiency is existent in the first-order diffraction direction in the present system. In addition, for the case of large Ω0 , the absorption is dominant and the first-order diffraction energy is restrained and the efficiency of diffraction is highly decreased. Similarly, we can see from Fig. 4(b) that the value of Ip (θ1 ) gradually becomes large as the increasing of L. At the length L ≈ 40z0 , the intensity value is maximal and then it becomes small. From Fig. 4(c) we see that the value of Ip (θ1 ) could be effectively modulated via changing the Huang-Rhys factor λ0 . Under the condition of λ0 0.03, the value could be close to 0.3 owing to the optimal phase modulation with high transmissivity. All in all, the high-efficiency first-order diffraction could be obtained via properly controlling the HuangRhys factor λ0 , the Rabi frequency Ω0 and the interaction length L.
|e,N+² |e,N−²
|e²
ωeg−ωph
ωeg |g,N+² |g,N−²
|g²
(a)
ωeg+ωph |g,N+² |g,N−²
|g,N+² |g,N−²
|g,N+² |g,N−²
(b)
|e,N+² |e,N−²
|e,N+² |e,N−²
|e,N+² |e,N−²
(c)
(d)
Fig. 5. (a) The dressed states of the exciton at the presence of phonon. (b) The transition process with respect to the emission of a photon for the three-photon resonance. (c) The case for the stimulated Rayleigh resonance. (d) The corresponding transition for the absorption resonance as modified by the ac Stark effect.
Before conclusion, we provide the origin of phonon-induced coherent features via using the dressed-state picture at the presence of phonon couplings. In order to do so, here we consider the system consisting of a single quantum dot and a phonon mode with momentum q and express the quantum dot-phonon Hamiltonian as Hd−p = h¯ ωeg Sz + h¯ ω ph b†q bq + h¯ λq (bq + b†q )Sz .
(17)
In terms of the Hamiltonian-diagonalization method [48], the energy eigenstates are derived as |±, N± = |± ⊗ e
∓(λq /ω ph )(bq +b†q )
|Nq ,
(18)
with the corresponding eigenvalues E± = ±¯hωeg /2 + h¯ ω ph (Nq − λ /2). In the above equation, the states |± are the eigenstates of the operator Sz with |+ = |e and |− = |g, Nq is the number operator of phonon, and |N± is the position-displaced Fock states. As shown in Fig. 5, the dressed states (Fig. 5(a)) and the corresponding transition processes are provided. Fig. 5(b) indicates the electromagnetically induced three-photon resonance. The control field excites the electron into the excited state and then makes a transition from |e, N− to |g, N+ , which is accompanied with the emission of photon with frequency ωeg − ω ph . This process corresponds to an amplified light at the probe detuning Δ p0 = −ω ph0 = −15, as indicated by the region of negative absorption in Fig. 2(d), 2(e) and 2(f). Fig. 5(c) refers to the process of electromagnetically induced stimulated Rayleigh resonance, in which the transition frequency is centred #234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9879
on the frequency of the control field. And Fig. 5(d) represents the electromagnetically induced absorption resonance as modified by the ac Stark effect. Here we focus on the amplification process indicated by Fig. 5(b) and apply it to prepare the phase grating. 4.
Conclusion
In conclusion, the phase grating is theoretically investigated in the two-level quantum dot exciton system under the condition of phonon-exciton couplings. The results show that the phononexciton interactions play a crucial role in preparing the phase grating and controlling the diffraction efficiency. At the absence of phonon-exciton couplings, the very low first-order diffraction intensity could be obtained for the probe beam. At the moment, the probe light is difficultly diffracted due to the ineffective phase modulation. However as the phonon-exciton interactions are existent, the phase modulation, accompanied with the high transmissivity, is greatly enhanced and the first-order diffraction becomes increasing with the reduced center light energy. Furthermore, the diffraction intensity could be effectively improved via appropriately adjusting the physical parameters including Huang-Rhys factor, Rabi frequency of the control field, the detunings of external fields and the interaction length. The phase grating in the quantum dot system has potential application in all-optical switching and routing, developing new photon devices and measuring the solid materials. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant Nos. 11165008 and 11365009), the Foundation of Young Scientist of Jiangxi Province, China (Grant No. 20142BCB23011), and China Scholarship Council (Grant No. 201408360040).
#234070 - $15.00 USD (C) 2015 OSA
Received 18 Feb 2015; revised 27 Mar 2015; accepted 27 Mar 2015; published 9 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.009870 | OPTICS EXPRESS 9880
