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We theoretically present a highly efficient optical method to obtain slow and fast light in a coupled system consisting of a nanomechanical resonator and quantum dots in terms of mechanically induced coherent population oscillation (MICPO). Turning on or turning off the specific detuning of pump field from exciton resonance, this coupling system can provide us a direct optical way to obtain the slow or fast group velocity without absorption. Our coupling scheme proposed here works as a fast-and slow-light knob and may have potential applications in various domains such as optical communication and biology sensor.
©2009 Optical Society of America
1. Introduction
The development of slow- and fast-light techniques has recently received a lot of interest in view of photonic applications, where the realization of all-optical communication networks relies on the ability to store, switch, and delay optical pulses [1, 2]. Chu et al.[3] first observed the fast light propagation in a resonant system, where the laser pulses propagated without appreciable shape distortion but experienced very strong resonant absorption. To reduce absorption, most of the recent work on slow and fast light propagation has used electromagnetically induced transparency (EIT) [4] or coherent population oscillation (CPO) [5] to render the material system highly transparent to resonant laser radiation. The first published results of slow light made use of electromagnetically induced transparency (EIT) to minimize signal absorption while retaining the large contribution to the group index associated with working close to an atomic absorption frequency [6, 7]. However, EIT based slow light in general has limitations in potential applications of ultrahigh speed information processing due to its narrow transparency spectrum. Coherent population oscillation (CPO) was introduced as a robust physical mechanism to overcome the defect of EIT and had less limitations to achieve ultraslow group velocity of light in solids [8, 9]. Superluminal and slow light at room temperature originated by CPO has been experimentally observed in solid state crystals[9, 10], erbium doped fibers (EDFs) [11], photo-refractive materials [12], and biological thin films [13].
On the other hand, bestriding the realms of classical and quantum mechanics, nanomechanical resonator offers greater promise for a huge variety of applications and fundamental researches [14]. Because of the limited environment and small size, an exciting possibility is to physically couple nanomechanical resonator to a condensed matter system such as a semiconductor quantum dot. Such a coupled quantum system can be used to study fundamental quantum effects, and also it has possible applications in high precision measurement [15], zeptogram-scale mass sensing [16] and laser cooling of a nanomechanical resonator mode to its quantum ground state [17].
In the present paper, we theoretically demonstrate the possibility to propagate superluminal and slow light by a nano-knob with quantum dots (QDs) embedded in a nanomechanical resonator (NR), whose potentialities for this purpose have remained unexplored to date. We obtain superluminal and slow light in this coupled scheme with different detuning of pump field from exciton frequency by two independent optical lasers in terms of mechanically induced coherent population oscillation (MICPO). The slow or fast light can be switched obviously when we turn on or turn off the detuning between pump laser and exciton resonance from the frequency of nanomechanical resonator. In addition to the optical method, this coupling scheme is a trend to mechanically control superluminal and slow light optical device. Moreover, this technique is expect to paves the way of optical buffer, nonlinear optics, optical communication and variable true-time delay lines.
2. Theory
In the following, we consider a semiconductor quantum dot embedded in a nanomechanical resonator in the simultaneous presence of a strong pump field and a weak signal field. The coherent optical spectroscopy of a strongly driven quantum dot without coupling to a nanomechanical resonator has been investigated experimentally by Xu et al. [18]. At low temperature the quantum dot can generally be regarded as a two-level system which consists of the ground state |g > and the single exciton state |ex>. The quantum dot via exciton interacts with a strong pump field (ωp) and a weak signal field (ωs). As usual the two-level exciton can be characterized by the pseudospin-1/2 operators S ± and Sz. We also assume that the nanomechanical resonator vibrates in its fundamental mode, and we treat it as a quantum resonator with Hamiltonian Hn=h̄ωna + a, where ωn is the frequency of the mode and a +(a) is the creation (annihilation) operator. The interaction Hamiltonian of the resonator mode and the quantum dot is derived by Wilson-Rae et al [17]. Then the total Hamiltonian of this optical knob including quantum dot, nanomechanical resonator and two optical fields in a rotating frame at the pump field frequency ωp reads as follows [19]:
where Δp=ωex-ωp. And β is coupling strength of nanomechanical resonator and quantum dot; Ω=µEp/h̄ is the Rabi frequency of the pump field; Ep and Es are the slowly varying envelope of the pump field and signal field; µ is the electric dipole moment of the exciton, assumed to be real; δ=ωs-ωp is the detuning of the signal and the pump field.
According to the Heisenberg equation of motion ih̄ dO/dt=[O,H] and the commutation relation [Sz,S ±]=±S ±, [S +,S-]=2Sz, [a,a +]=1. The temporal evolutions of the exciton and nanomechanical resonator system are given by setting N=a ++a. In what follows, we ignore the quantum properties of S-, Sz and N [20, 21]. For calculations of the absorption spectrum, we can use the semiclassical approach where the optical fields are taken to be classical. The results of calculations in the limits appropriated to this work are read as follows:
where Γ1 is the exciton relaxation rate, Γ2 is the exciton dephasing rate, γn is the decay rate of the nanomechanical resonator due to the coupling to a reservoir of “background” modes and the other intrinsic processes[17]. In order to solve Eqs. (2)–(4), we make the ansatz[19] S -(t)=S 0+S+e-iδt+S-eiδt, Sz(t)=Sz 0 +Sz+e-iδ t+Sz-eiδt, N(t)=N 0+N+e-iδ t+N-ei δt. Upon working to the lowest order in Es, but to all orders in Ep, we can obtain S +, which corresponds to the linear optical susceptibility as follows: χ (1) eff(ωs)=ρµS + /Es=ρµ 2 χ (1)(ωs)/h̄Γ2, and the dimensionless susceptibility is given by
where we have assumed that many quantum dots couple to a single mechanical resonator, so here ρ is the number density of quantum dots. Γ1=2Γ2,ωn0=ωn/Γ2,γn0=γn/Γ2,Ω R=Ω/Γ2,δ 0=δ/Γ2,Δp0=Δp/Γ2, A=Δp0-ω n0 β 2 w0-i-δ 0, B=Δp0-ω n0 β 2 w 0+i+δ 0, C=Ω2 R ω n0 β 2 ηw 0/(Δp0-ω n0 β 2 w0-i), D=Ω2 R ω n0 β 2 ηw 0/(Δp0-ω n0 β 2 w 0+i), E=(2Ω2 R+2D-2iB-Bδ 0)(B-δ 0), and η=ω 2 n 0/(ω 2 n 0-iδ 0 γn0-δ 2 0) is the auxiliary function.
The population inversion of the exciton w 0 is determined by the following equation
In terms of this model, we can determine the light group velocity as [22, 23]
where n≈1+2πχ (1) eff, and then
It is clear from this expression for vg that when Reχ(ωs)ωs=ωex is zero and the dispersion is steeply positive or negative, the group velocity is significantly reduced or increased, and then
where ∑=2πωex ρµ 2/h̄Γ2 2.
3. Numerical results and discussion
For illustration of the numerical results, we choose the realistic parameters of our optical knob consisting of InAs quantum dots and a GaAs nanomechanical resonator. In experiment InAs quantum dots have a typical area density of about 4×1010 cm -2 [24]. The relevant coupling strength β=0.06, Γ1=0.3 GHz, ωn=1.2 GHz and Q=3×104 [17]. The dephasing rate of exciton is Γ2=Γ1/2=0.15 GHz and the decay rate of the nanomechanical resonator is γn=ωn/Q=4.0×10-5 GHz. Figure 1 shows the behavior of the imaginary part (Imχ (1)) and the real part (Reχ (1)) of the linear optical susceptibility as a function of the signal-exciton detuning Δs for Δp=1.2GHz, ωn=1.2GHz, Ω2=0.15(GHz)2. We can see clearly that at Δs=0, there is a steep positive slope (the dash curve) related to zero absorption (the solid curve). This large dispersive characteristics can lead to the possibility of implementation of slow light.
Figure 2 illustrates the group velocity index ng (in units of ∑) as a function of the Rabi frequency Ω2. It is obvious that near Ω2=0.001(GHz)2, the most slow-light index can be produced in such optical knob as 1800 for Γ2=0.15GHz. That is, the output pulse will be 1.8×103 times slower than the input light. The physical origin of this result is due to the so called mechanically induced coherent population oscillation (MICPO) which makes quantum interference between the resonator and the two optical fields via the quantum dot as the pump-signal detuning δ is equal to the resonator frequency ωn. In the picture of the dressed states (see Fig.3(b) below), the condition Δp=ωn=1.2GHz just corresponds to that the pump field couples to the optical transition via the Stokes process and the system becomes fully transparent to the signal beam. In this case, the system is similar to the conventional three-level systems in EIT [6]. Here coupling to a mechanical resonator seems to provide the exciton with additional energy levels to realize EIT phenomena. Therefore in our structure one can obtain the slow output light without absorption only by simply adjusting the pump-exciton detuning to the frequency of nanomechanical resonator.
Fig. 1. The imaginary part and real part of the linear optical susceptibility as a function of the signal detuning form exciton resonance Δs with parameters Ω2=0.15(GHz)2, ωn=1.2GHz, Δp=1.2GHz, γn=4×10-5 GHz, and β=0.06.
Fig. 2. The group velocity index ng(=c/vg) of slow light (in units of ∑) as a function of the detuning Δs with parameters ωn=1.2GHz, Δp=1.2GHz, γn=4×10-5 GHz, and β=0.06.
In order to understand the mechanism of this MICPO more clearly, we plot Figure 3. Fig.3(a) shows the absorption spectrum of a signal field as a function of pump-signal detuning for the pump-exciton detuning Δp0=2. In the middle of figure, we see that three features appear in the signal absorption spectrum as those in atomic two-level systems[19]. However, the new features which are different from those in atomic systems without nanomechanical resonator also appear in the both sides of the spectrum. Fig.3(b) demonstrates the origin of these novel features. The leftmost (1) of Fig.3(b) shows the dressed states of exciton (|n > denotes the number states of the nanomechanical resonator). The part (2) shows the origin of mechanically induced three-photon resonance. Here the electron makes a transition from the lowest dressed level |g,n> to the highest dressed level |ex,n+1> by the simultaneous absorption of two pump photons and emission of a photon at ωp-ωn. This process can amplify a wave at δ 0=-ω n0=-8, as indicated by the region of negative absorption in Fig.3(a). The part (3) in Fig.3(b) shows the origin of mechanically induced stimulated Rayleigh resonance. The Rayleigh resonance corresponds to a transition from the lowest dressed level |g,n> to the dressed level |ex,n>. Each of these transitions is centered on the frequency of the pump laser. The rightmost part (4) corresponds to the mechanically induced absorption resonance as modified by the ac Stark effect.
Fig. 3. (a) The absorption spectrum of a signal field in the presence of a strong pump field for the case Ω2 R=6, ω n0=8, Δp0=2, γn0=3.0×10-4, and β=0.06. (b) The new features in the spectrum shown in (a) are identified by the corresponding transition between the dressed states of exciton.
Alternatively, this optical knob can also implement the superluminal light without absorption as we modulate the pump laser detuning Δp=0. In order to show this more clearly, we plot Fig.4 and Fig.5 with the same experimental data as Fig.2. In Figure 4, we report the theoretical variation of Reχ (1) and Imχ (1) as a function of detuning Δs for detuning Δp=0. From this figure, we can also obtain a large dispersive characteristics. In contrary to Figure 1, this large dispersion relates to a very steep negative slope (the dash curve) which results in fast group velocity without absorption.
Figure 5 plots the group velocity index ng (in units of ∑) of fast light as a function of the Rabi frequency Ω2. This picture tells us the output pulse can be about 10 times faster than input pulse in vacuum simply via coupling to a nanomechanical resonator. Up to now, this optical knob provides us an efficient and convenient way to achieve slow- and fast- light in terms of mechanically induced coherent population oscillation (MICPO), and it performs by interfering a weak signal beam together with a higher intensity frequency-adjustable pump laser. From Fig.2 and Fig.5, one can first fix the signal beam with frequency ωs=ωex, and then scan the pump frequency across the exciton frequency ωex, one can efficiently obtain the signal light from slow to fast as the pump detuning Δp is equal to the nanoresonator frequency or zero, respectively.
Fig. 4. The dimensionless imaginary part and real part of the linear optical susceptibility as a function of the signal detuning from exciton resonance Δs with parameters Ω2=0.15(GHz)2, ωn=1.2GHz, Δp=0, γn=4×10-5 GHz, and β=0.06.
Fig. 5. The group velocity index ng(=c/vg) of superluminal light (in units of ∑) as a function of the detuning Δs with parameters ωn=1.2GHz, Δp=0, γn=4×10-5 GHz, and β=0.06.
Figure 6 illustrates the the imaginary part of χ (1) as a function of Δs for three different decay rates of γn. The inset of Figure 6 is the amplification of the most remarkable region of transparency. From this figure, we can demonstrate that the width of the MICPO resonance and the absorption increase as the decay rate γn increases. Therefore the longer the lifetime of resonator, the more obvious the transparency effect. When the decay rate of the resonator is 0.04 GHz, the hole width in the spectrum becomes flat as shown in the inset. As a result, the resonator with small decay rate is beneficial to the transparency window. In most quantum systems such as quantum wells and quantum dots, the lifetime of phonons in the dot (well) is very short as compared with that of nanomechanical resonator, so such an effect is weak and the transparency due to lattice vibrations is not obvious in quantum well and quantum dot systems [25, 26]. Hence, quantum dots coupled to a nanomechanical resonator have the advantage to observe this transparency effect and the slow/fast light in experiment.
Fig. 6. The absorption spectrum of a signal field as a function of the detuning Δs between a signal field and exciton with three different decay rates of resonator. The other parameters used are Ω2=0.1(GHz)2, ωn=1.2GHz, Δp=1.2GHz, and β=0.06. The inset is the amplification of the most remarkable region of transparency.
4. Conclusion
In conclusion, we have presented superluminal and slow light in our optical knob consisting of quantum dots and a nanomechanical resonator due to mechanically induced coherent population oscillation (MICPO). The greatest advantage of our optical knob is that we can efficiently achieve fast and slow light only by using a frequency-adjustable laser device. Through turning on or turning off the pump laser detuning from exciton resonance, a large dispersion can be obtained, and thus make the output pulse light slower or faster without absorption.
Acknowledgements
The part of this work was supported by National Natural Science Foundation of China (No.10774101) and the National Ministry of Education Program for Ph.D.
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