An efficient optical knob from slow light to fast light in a coupled nanomechanical resonator-quantum dot system

An efficient optical knob from slow lightto fast light in a coupled nanomechanical

resonator-quantum dot system

Jin-Jin Li and Ka-Di ZhuDepartment of Physics, Shanghai Jiao Tong University, 800 DongChuan Road,

Shanghai 200240, P.R.China

corresponding author: [email protected]

Abstract: We theoretically present a highly efficient optical method toobtain slow and fast light in a coupled system consisting of a nanomechani-cal resonator and quantum dots in terms of mechanically induced coherentpopulation oscillation (MICPO). Turning on or turning off the specificdetuning of pump field from exciton resonance, this coupling system canprovide us a direct optical way to obtain the slow or fast group velocitywithout absorption. Our coupling scheme proposed here works as a fast-and slow-light knob and may have potential applications in various domainssuch as optical communication and biology sensor.

© 2009 Optical Society of America

OCIS codes: (230.1150) All-optical devices; (270.1670) Coherent optical effects.

References and links1. S. Residori, U. Bortolozzo, and J. P. Huignard, “ Slow and fast light in liquid crystal light valves” Phys. Rev.

Lett. 100, 203603(2008).2. R. W. Boyd and D. J. Gauthier, Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam), Vol. 43, pp. 497-530

(2002).3. S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738(1982).4. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an

ultracold atomic gas,” Nature 397, 594-598(1999).5. P. -C. Ku, R. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S. -W. Chang, and S. -L. Chuang,

“Slow light in semiconductor quantum wells,” Opt. Lett. 29, 2291-2293(2004).6. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coher-

ent media,” Rev. Mod. Phys. 77, 633-673(2005).7. M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry and

M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomicgas,” Phys. Rev. Lett. 82, 5229 (1999).

8. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystalat room temperature,” Phys. Rev. Lett. 90, 113903 (2003).

9. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature Solid ,” Science 301, 200-202 (2003).

10. E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson and V, Rouget, “Ultraslow light propagation in an inhomo-geneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).

11. S. Melle, O. G. Calderon, C. E. Caro, E. Cabrera-Granado, M. A. Anton and F. Carreno, “ Modulation-frequency-controlled change from sub- to superluminal regime in highly doped erbium fibers ,” Opt. Lett. 33, 827-829(2008).

12. A. Shumelyuk, K. Shcherbin, S. Odulov, B. Sturman, E. Podivilov, K. Buse, “Slowing down of light in photore-fractive crystals with beam intensity coupling reduced to zero,” Phys. Rev. Lett. 93, 243604 (2004).

13. P. Wu and D. V. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys. Rev. Lett.95, 253601 (2005).

14. K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58, 36-42 (2005).

(C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19874#113676 - $15.00 USD Received 6 Jul 2009; revised 23 Aug 2009; accepted 15 Sep 2009; published 19 Oct 2009

15. Y.J. Wang, M. Eardley, S. Knappe, J. Moreland, L. Hollberg, and J. Kitching, “Magnetic resonance in an atomicvapor excited by a mechanical resonator,” Phys. Rev. Lett. 97, 227602 (2006).

16. Y.T. Yang, C.Callegari, X.L. Feng, K.L.Ekinci, and M.L.Roukes, “Zeptogram-scale nanomechanical mass sens-ing,” Nano. Lett. 6, 583-586 (2006).

17. I.Wilson-Rae, P. Zoller, and A. Imamoglu, “Laser cooling of a nanomechanical resonator mode to its quantumground state,” Phys. Rev. Lett. 92, 075507 (2004).

18. X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Coherent optical spec-troscopy of a strongly driven quantum dot ,” Science 317, 929-932(2007).

19. R. W. Boyd, Nonlinear Optics (San Diego, CA: Academic) (1992).20. G. S. Agarwal, “Electromagnetic-field-induced transparency in high-density exciton systems,” Phys. Rev. A 51,

R2711-R2714 (1995).21. J. F. Lam, S. R. Forrest and G. L. Tangonan, “Optical nonlinearities in crystalline organic multiple quantum

wells,” Phys. Rev. Lett. 66, 1614 (1991).22. R. S. Bennink, R. W. Boyd, C. R. Stroud and V. Wong, “Enhanced self-action effects by electromagnetically

induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001).23. S. E. Harris, J. E. Field and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,”

Phys. Rev. A 46, R29-R32 (1992).24. V.Preisler, T.Grange, R.Ferreira, L.A.de Vaulchier, Y.Guldner, F.J.Teran, M.Potemski, and A.Lemaitre, ”Evi-

dence for excitonic polarons in InAs/GaAs quantum dots”, Phys. Rev. B 73, 075320(2006).25. K.D.Zhu and W.S.Li, ”Electromagnetically induced transparency due to exciton-phonon interaction in an organic

quantum well,” J.Phys.B: At. Mol. Opt. Phys.34, L679-L686(2001).26. Y.W.Jiang, K.D.Zhu, Z.J.Wu, X.Z.Yuan and M.Yao,”Electromagnetically induced transparency in quantum dots”,

J.Phys.B: At. Mol. Opt. Phys.39, 2621-2632(2006).

1. Introduction

The development of slow- and fast-light techniques has recently received a lot of interest in viewof photonic applications, where the realization of all-optical communication networks relies onthe ability to store, switch, and delay optical pulses [1, 2]. Chu et al.[3] first observed the fastlight propagation in a resonant system, where the laser pulses propagated without appreciableshape distortion but experienced very strong resonant absorption. To reduce absorption, mostof the recent work on slow and fast light propagation has used electromagnetically inducedtransparency (EIT) [4] or coherent population oscillation (CPO) [5] to render the material sys-tem highly transparent to resonant laser radiation. The first published results of slow light madeuse of electromagnetically induced transparency (EIT) to minimize signal absorption while re-taining the large contribution to the group index associated with working close to an atomicabsorption frequency [6, 7]. However, EIT based slow light in general has limitations in po-tential applications of ultrahigh speed information processing due to its narrow transparencyspectrum. Coherent population oscillation (CPO) was introduced as a robust physical mecha-nism to overcome the defect of EIT and had less limitations to achieve ultraslow group velocityof light in solids [8, 9]. Superluminal and slow light at room temperature originated by CPO hasbeen 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, nanomechan-ical resonator offers greater promise for a huge variety of applications and fundamental re-searches [14]. Because of the limited environment and small size, an exciting possibility is tophysically couple nanomechanical resonator to a condensed matter system such as a semicon-ductor quantum dot. Such a coupled quantum system can be used to study fundamental quantumeffects, 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 quantumground state [17].

In the present paper, we theoretically demonstrate the possibility to propagate superluminaland slow light by a nano-knob with quantum dots (QDs) embedded in a nanomechanical res-onator (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 fromexciton frequency by two independent optical lasers in terms of mechanically induced coherentpopulation oscillation (MICPO). The slow or fast light can be switched obviously when we turnon or turn off the detuning between pump laser and exciton resonance from the frequency ofnanomechanical resonator. In addition to the optical method, this coupling scheme is a trendto mechanically control superluminal and slow light optical device. Moreover, this technique isexpect to paves the way of optical buffer, nonlinear optics, optical communication and variabletrue-time delay lines.

2. Theory

In the following, we consider a semiconductor quantum dot embedded in a nanomechanical res-onator in the simultaneous presence of a strong pump field and a weak signal field. The coherentoptical spectroscopy of a strongly driven quantum dot without coupling to a nanomechanicalresonator has been investigated experimentally by Xu et al. [18]. At low temperature the quan-tum 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 pumpfield (ωp) and a weak signal field (ωs). As usual the two-level exciton can be characterized bythe pseudospin−1/2 operators S± and Sz. We also assume that the nanomechanical resonatorvibrates in its fundamental mode, and we treat it as a quantum resonator with HamiltonianHn = hωna+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 byWilson-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 frequencyωp reads as follows [19]:

H = hΔpSz + hωna+a+ hSzωnβ (a+ +a)− h(ΩS+ +Ω∗S−)− μh

(S+Ese−iδ t +S−E∗

s eiδ t), (1)

where Δp = ωex −ωp . And β is coupling strength of nanomechanical resonator and quantumdot; Ω = μEp/h is the Rabi frequency of the pump field; Ep and Es are the slowly varyingenvelope 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 ih dO/dt = [O,H] and the commutationrelation [Sz,S±] = ±S±, [S+,S−] = 2Sz, [a,a+] = 1. The temporal evolutions of the exciton andnanomechanical resonator system are given by setting N = a+ +a. In what follows, we ignorethe 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. Theresults of calculations in the limits appropriated to this work are read as follows:

dS−

dt= [−Γ2 − i(Δp +ωnβN)]S−−2iΩSz −2i

μh

Ese−iδ tSz, (2)

dSz

dt= −(Sz +

12)Γ1 + iΩ(S+ −S−)+ i

μh

(S+Ese−iδ t −S−E∗

s eiδ t), (3)

d2Ndt2 + γn

dNdt

+ω2n N = −2ω2

n βSz, (4)

where Γ1 is the exciton relaxation rate, Γ2 is the exciton dephasing rate, γn is the decay rate ofthe nanomechanical resonator due to the coupling to a reservoir of ”background” modes and theother intrinsic processes[17]. In order to solve Eqs. (2)− (4), we make the ansatz[19] S−(t) =S0+S+e−iδ t +S−eiδ t , Sz(t) = Sz

0 +Sz+e−iδ t +Sz

−eiδ t , N(t) = N0+N+e−iδ t +N−eiδ t . Upon work-ing 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)e f f (ωs) = ρμS+/Es = ρμ2χ(1)(ωs)/hΓ2, and the

dimensionless susceptibility is given by

χ(1)(ωs) =2Bw0(Ω2

R +C)−Ew0

AE −2B(Ω2R +C)(B−δ0)

, (5)

where we have assumed that many quantum dots couple to a single mechanical resonator, sohere ρ is the number density of quantum dots. Γ1 = 2Γ2,ωn0 = ωn/Γ2,γn0 = γn/Γ2,ΩR =Ω/Γ2,δ0 = δ/Γ2,Δp0 = Δp/Γ2, A = Δp0 −ωn0β 2w0 − i− δ0, B = Δp0 −ωn0β 2w0 + i + δ0,C = Ω2

Rωn0β 2ηw0/(Δp0−ωn0β 2w0− i), D = Ω2Rωn0β 2ηw0/(Δp0−ωn0β 2w0 + i), E = (2Ω2

R +2D−2iB−Bδ0)(B−δ0), and η = ω2

n0/(ω2n0 − iδ0γn0 −δ 2

0 ) is the auxiliary function.The population inversion of the exciton w0 is determined by the following equation

(w0 +1)[(Δp0 −β 2ωn0w0)2 +1]+2Ω2Rw0 = 0. (6)

In terms of this model, we can determine the light group velocity as [22, 23]

vg =c

n+ωs(dn/dωs), (7)

where n ≈ 1+2πχ(1)e f f , and then

cvg

= 1+2πReχ(1)e f f (ωs)ωs=ωex +2πωsRe(

dχ(1)e f f

dωs)ωs=ωex . (8)

It is clear from this expression for vg that when Reχ(ωs)ωs=ωex is zero and the dispersion issteeply positive or negative, the group velocity is significantly reduced or increased, and then

cvg

−1 =2πωexρμ2

hΓ2Re(

dχ(1)(ωs)dωs

)ωs=ωex = Γ2ΣRe(dχ(1)(ωs)

dωs)ωs=ωex , (9)

where Σ = 2πωexρμ2/hΓ22.

3. Numerical results and discussion

For illustration of the numerical results, we choose the realistic parameters of our optical knobconsisting of InAs quantum dots and a GaAs nanomechanical resonator. In experiment InAsquantum dots have a typical area density of about 4× 1010cm−2 [24]. The relevant couplingstrength β = 0.06, Γ1 = 0.3 GHz, ωn = 1.2 GHz and Q = 3× 104 [17]. The dephasing rateof exciton is Γ2 = Γ1/2 = 0.15 GHz and the decay rate of the nanomechanical resonator isγn = 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-excitondetuning Δs for Δp = 1.2GHz, ωn = 1.2GHz, Ω2 = 0.15(GHz)2. We can see clearly that atΔs = 0, there is a steep positive slope (the dash curve) related to zero absorption (the solidcurve). This large dispersive characteristics can lead to the possibility of implementation ofslow light.

Figure 2 illustrates the group velocity index ng (in units of Σ) as a function of the Rabifrequency Ω2. It is obvious that near Ω2 = 0.001(GHz)2, the most slow-light index can beproduced in such optical knob as 1800 for Γ2 = 0.15GHz. That is, the output pulse will be1.8× 103 times slower than the input light. The physical origin of this result is due to the socalled mechanically induced coherent population oscillation (MICPO) which makes quantum


Fig. 1. The imaginary part and real part of the linear optical susceptibility as a function ofthe signal detuning form exciton resonance Δs with parameters Ω2 = 0.15(GHz)2, ωn =1.2GHz, Δp = 1.2GHz, γn = 4×10−5GHz, and β = 0.06.

Fig. 2. The group velocity index ng(= c/vg) of slow light (in units of Σ) as a functionof the detuning Δs with parameters ωn = 1.2GHz, Δp = 1.2GHz, γn = 4×10−5GHz, andβ = 0.06.

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 fieldcouples to the optical transition via the Stokes process and the system becomes fully transparentto the signal beam. In this case, the system is similar to the conventional three-level systems inEIT [6]. Here coupling to a mechanical resonator seems to provide the exciton with additionalenergy levels to realize EIT phenomena. Therefore in our structure one can obtain the slowoutput light without absorption only by simply adjusting the pump-exciton detuning to thefrequency of nanomechanical resonator.

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 thepump-exciton detuning Δp0 = 2. In the middle of figure, we see that three features appear in thesignal absorption spectrum as those in atomic two-level systems[19]. However, the new featureswhich are different from those in atomic systems without nanomechanical resonator also appearin the both sides of the spectrum. Fig.3(b) demonstrates the origin of these novel features. The


ex

1 2 3 4

p np

np

(a)

(b)

Fig. 3. (a) The absorption spectrum of a signal field in the presence of a strong pumpfield for the case Ω2

R = 6, ωn0 = 8, Δp0 = 2, γn0 = 3.0×10−4, and β = 0.06. (b) The newfeatures in the spectrum shown in (a) are identified by the corresponding transition betweenthe dressed states of exciton.

leftmost (1) of Fig.3(b) shows the dressed states of exciton (|n > denotes the number states ofthe 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 photonsand emission of a photon at ωp −ωn. This process can amplify a wave at δ0 = −ωn0 = −8,as indicated by the region of negative absorption in Fig.3(a). The part (3) in Fig.3(b) showsthe origin of mechanically induced stimulated Rayleigh resonance. The Rayleigh resonancecorresponds 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 Starkeffect.

Alternatively, this optical knob can also implement the superluminal light without absorptionas we modulate the pump laser detuning Δp = 0. In order to show this more clearly, we plotFig.4 and Fig.5 with the same experimental data as Fig.2. In Figure 4, we report the theoreticalvariation of Reχ(1) and Imχ(1) as a function of detuning Δs for detuning Δp = 0. From thisfigure, we can also obtain a large dispersive characteristics. In contrary to Figure 1, this largedispersion relates to a very steep negative slope (the dash curve) which results in fast groupvelocity without absorption.

Figure 5 plots the group velocity index ng (in units of Σ) of fast light as a function of theRabi frequency Ω2. This picture tells us the output pulse can be about 10 times faster than inputpulse in vacuum simply via coupling to a nanomechanical resonator. Up to now, this opticalknob provides us an efficient and convenient way to achieve slow- and fastlight in terms ofmechanically induced coherent population oscillation (MICPO), and it performs by interferinga weak signal beam together with a higher intensity frequency-adjustable pump laser. FromFig.2 and Fig.5, one can first fix the signal beam with frequency ωs = ωex, and then scan thepump frequency across the exciton frequency ωex, one can efficiently obtain the signal light


Fig. 4. The dimensionless imaginary part and real part of the linear optical suscepti-bility as a function of the signal detuning from exciton resonance Δs with parametersΩ2 = 0.15(GHz)2, ωn = 1.2GHz, Δp = 0, γn = 4×10−5GHz, and β = 0.06.

Fig. 5. The group velocity index ng(= c/vg) of superluminal light (in units of Σ) as afunction of the detuning Δs with parameters ωn = 1.2GHz, Δp = 0, γn = 4× 10−5GHz,and β = 0.06.

from slow to fast as the pump detuning Δp is equal to the nanoresonator frequency or zero,respectively.

Figure 6 illustrates the the imaginary part of χ(1) as a function of Δs for three differentdecay rates of γn. The inset of Figure 6 is the amplification of the most remarkable region oftransparency. From this figure, we can demonstrate that the width of the MICPO resonanceand the absorption increase as the decay rate γn increases. Therefore the longer the lifetimeof resonator, the more obvious the transparency effect. When the decay rate of the resonatoris 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 quantumsystems such as quantum wells and quantum dots, the lifetime of phonons in the dot (well) isvery short as compared with that of nanomechanical resonator, so such an effect is weak and thetransparency 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 toobserve 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 asignal field and exciton with three different decay rates of resonator. The other parametersused are Ω2 = 0.1(GHz)2, ωn = 1.2GHz, Δp = 1.2GHz, and β = 0.06. The inset is theamplification of the most remarkable region of transparency.

4. Conclusion

In conclusion, we have presented superluminal and slow light in our optical knob consisting ofquantum dots and a nanomechanical resonator due to mechanically induced coherent popula-tion oscillation (MICPO). The greatest advantage of our optical knob is that we can efficientlyachieve fast and slow light only by using a frequency-adjustable laser device. Through turningon or turning off the pump laser detuning from exciton resonance, a large dispersion can beobtained, 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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An efficient optical knob from slow light to fast light in a coupled nanomechanical resonator-quantum dot system