Cite this Article
Ba Nuo, Fei Jin-You, Li Dong-Fei, Zhong Xin, Wang Dan, Wang Lei, Wang Hai-Hua, Bao Qian-Qian. Dynamic manipulation of probe pulse and coherent generation of beating signals based on tunneling-induced inference in triangular quantum dot molecules. Chinese Physics B, 2020, 29(3): 034204  
Permissions
Dynamic manipulation of probe pulse and coherent generation of beating signals based on tunneling-induced inference in triangular quantum dot molecules
Ba Nuo1, †, Fei Jin-You1, Li Dong-Fei1, Zhong Xin1, Wang Dan1, Wang Lei2, ‡, Wang Hai-Hua2, Bao Qian-Qian3
National Demonstration Center for Experimental Physics Education, College of Physics, Jilin Normal University, Siping 136000, China
College of Physics, Jilin University, Changchun 130012, China
College of Physics, Liaoning University, Shenyang 110036, China

 

† Corresponding author. E-mail: banuo2008@163.com wang_lei@jlu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11704151 and 11247201), the Twelfth Five-Year Program for Science and Technology of Education Department of Jilin Province, China (Grant No. 20150215), and the Innovation and Entrepreneurship Program of Liaoning University (Grant Nos. x201710140122 and x201710140127).

Abstract

We investigate the dynamic propagation of a probe field via the tunneling-induced interference effect in a triple model of quantum dot molecules. By theoretical analysis and numerical simulation, we find that the number of transparency window relate to the energy splitting and the group velocity of probe field can be effectively controlled by the tunneling coupling intensity. In addition, in the process of light storage and retrieval, when the excited states have no energy splitting in the storage stage but opposite values of the energy splitting in the retrieval stage, the beating signals can be generated.

PACS: ;42.50.Gy;;42.50.Wk;;37.10.Vz;
Keyword:;coherent optical effects;;electromagnetically induced transparency;;triple quantum dots;
1. Introduction

During the past decade, the techniques of controlling light propagation have been paid much attention for its scientific merits. One effective method for achieving the velocity manipulation of light pulse is based on electromagnetically induced transparency (EIT), which can eliminate the absorption of a weak probe field at the resonant frequency via inducing atomic coherence by a strong coupling field.[1–3] Using the EIT technique, one can obtain the slowdown of light by changing the intensity of the coupling field.[4–6] Typically, people have termed “dark-state polaritons” (DSP) defined as form-stable coupled excitations of field and atom, which can explain the propagation of quantum field in EIT medium.[7,8] Subsequently, researchers have demonstrated experimentally that the storage and read-out processes are observed in cold sodium atoms[9] and the solid-state materials.[5,10] Recently, Wang et al. experimentally obtained the storage and retrieval of 2D Airy wavepackets in a doped solid driven by EIT.[11] In addition, it is worth noting that after storage procedure, the beating signals are generated based on the quantum interference effect between two weak probe fields,[12] a weak probe field and a control field.[13] Especially, Bao et al. obtained the beating signals by modulating the detunings of two coupling fields in the retrieval stage,[14] and by controlling a microwave field in the retrieval stage.[15]

A quantum dot (QD) is a semiconductor nanostructure that restrains the behavior of the conduction band electrons and valence band holes in the three-dimensional space.[16,17] The particle size of QD is generally between 1 nm and 10 nm. So, the electrons and holes in QD occupy the discrete energy level states due to the three-dimensional quantum confinement. By comparing with atomic system, QD has many merits, such as large electric-dipole moments, high nonlinear optical coefficients, controllable energy levels spacing, and ease of integration. While, quantum dot molecules (QDMs) can be formed by two or more closely spaced dots coupled via tunneling effect using the self-assembled dot growth method.[18] Double quantum dots (DQDs) have been deeply investigated in both theories and experiments, because the induced quantum interference[19–21] can be achieved by the tunneling effect of electrons between the dots with an external electric field.[22–24] Some investigators have exploited the DQDs to obtain EIT,[25,26] coherent population transfer,[27–29] optical bistability,[30–32] narrowing of fluorescence spectrum,[33] and the enhancement of Kerr nonlinearity.[34] Recently, triple quantum dots (TQDs) have also been paid much attention, because TQDs have been achieved in some experimental studies[35–38] and they have possessed multi-level structure and more tunable extra parameters. Subsequently, some theoretical works have utilized the TQDs to realize multiple transparency windows,[39,40] the enhanced Kerr nonlinearity,[41,42] and tunneling-assisted coherent population transfer and creation of coherent superposition states,[43,44] as well as controlling the Goos–Hänchen shift.[45]

In this paper, we investigate the pulse propagation dynamics in triangular quantum dot molecules which exhibit the electron tunneling coupling between three QDs in the presence of the externally applied voltages. We first deduced theoretically the expression of probe field susceptibility and further obtained the expression of probe group velocity. Then, by the numerical simulation, we found that the number of transparency window relates to the energy slitting and we utilized the dressed state representation to explain the result. Using the Bloch–Maxwell equations, it is found that the tunneling intensity of coupling field has important effects on the probe group velocity in the absence of the energy splitting. On this basis, a method to generate the beating signals (a series of maxima and minima in intensity) is designed. In detail, the probe field could be transformed into the spin coherence by turning off the tunneling coupling intensity without the energy splitting and then we retrieve it after a short storage time by turning on the tunneling coupling intensity with the opposite energy splitting. The retrieved probe field possesses two different optical components with time-dependent phase and leads to the beating signals occurrence owing to the alternate constructive and destructive interferences.

2. Atomic model and relevant equations

We consider the setup of the TQDs as composed of three QDs with different band structures and a triangular arrangement as shown in Fig. 1(a). The QDs are formed with a thin barrier of GaAs/AlGaAs, so the tunneling effect between the dots can be created and controlled by the gate voltage. Figure 1(b) shows the level configuration of the TQDs. In this system, the ground level |1〉 has no excitation, the direct exciton level |4〉 has an electron–hole pair in QD1, and the two indirect exciton levels |2〉 and |3〉 have the hole in QD1, as well as the electron in QDi (i = 2,3), respectively. When the gate electrode is applying, the tunneling effect between the QDs can take place. The ground level |1〉 is coupled with the direct exciton level |4〉 by a weak probe field in the direction of . The weak probe field is a time- and space-dependent electric field and can be described by

where , ωp, kp, V, and f(z,t) correspond to the polarization vector, the carrier frequency, the wave number, the quantization volume, and the dimensionless pulse envelope, respectively. Using the rotating-wave and electric-dipole approximation, with the assumption of ħ = 1, the interaction Hamiltonian can be obtained

where Ωp = gpEp is Rabi frequency of the probe and is the coupling constant of the probe as well as is the dipole moment for the transition |1〉 ↔ |4〈. T2 and T3 denote the intensities of tunneling coupling, which relate to the barrier properties and the external electric field. The probe detuning is defined as ωp = ω41 − ωp, where ω41 is the transition frequency from |4〉 to |1〉. In addition, ω42 and ω43 express the energy splitting of the excited states, which depend on the effective confinement potential controlled by the external electric field.

Fig. 1. (a) The diagram of a setup of the triangular TQDs. The probe field with central frequency ωp transmits the QD1. V denotes a bias voltage. (b) The scheme of the level configuration for the TQDs. (c) The dressed state representation for the two tunneling couplings T2 and T3.

To examine the dynamical evolution of the probe field, the following density matrix equations can be derived:

constrained by ρ11 + ρ22 + ρ33 + ρ44 = 1 and , (i = 2,3,4) denote the types of effective decay rate, Γi1 is the radiative decay rate from level |i〉 to |1〉, and is the pure dephasing decay rates.

In the limit of a weak field, the steady-state solutions of ρ14 can be obtained, furthermore the probe susceptibility can be derived as

where N is the quantum dot density. In order to investigate the dynamic evolution of a probe field in these TQDs, the wave propagation equation in the slowly varying envelope approximation can be given

with κ2 = cγ4α and being the propagation constant and c being light velocity in vacuum. Then, in the local retarded frame forτ = t − z/c and ξ = z, equation (5) can be rewritten as

3. Numerical calculation and discussion

We have investigated the steady optical response of TQDs and given the imaginary and real parts of the probe susceptibility χp as a function of the probe detuning ωp as shown in Fig. 2. It is clear that the numbers of transparency windows relate to the energy splitting ω42 and ω43. In the case of ω42 ≠ ω43, a pair of EIT windows appears between three absorption peaks and they are accompanying by steep normal dispersions at two different frequencies. However, for ω42 = ω43 the only one EIT window arises in the probe absorption spectrum. The above phenomenon can be well explained in the dressed-state picture of T2 and T3. When ω42 = −ω43 = ω and T2 = T3 = T, the three dressed states can be written as

with the eigenvalues of the dressed levels being , λ0 = ω, . From Eqs. (7)–(9), the three dressed states are all comprised of the bare states |2〉, |3〉, and |4〉, and the three absorptive peaks are generated by the transitions from levels | + 〉, |0〉, | − 〉 to the level |1〉. Therefore, the quantum destructive interference among the three dipole-allowed transition leads to form two EIT windows.

Fig. 2. The real part (black line) and imaginary part (red line) of probe susceptibilities for ω42 = −ω43 = 0.5γ in panel (a) and for ω42 = ω43 = 0 in panel (b). Relevant parameters are γ4 = γ = 10 μeV, γ2 = γ3 = 10−3γ, T2 = T3 = 0.7γ.

While, when ω42 = ω43 = ω and T2 = T3 = T, the expressions of the dressed states are

The corresponding eigenvalues of the dressed levels are , λ0 = 0, . From Eq. (11) it is shown that the dressed state |0〉 is the coherent superposition state of the levels |2〉 and |3〉, and it is independent of the excitation level |4〉. According to the selection rule, the transitions |2〉 ↔ |1〉 and |3〉 ↔ |1〉 are dipole-forbidden, so the electric dipole moment of the dressed state |0〉 and level |1〉 is zero. Hence, the two absorptive peaks correspond to the dressed-state transition pathways | + 〉 ↔ |1〉 and | − 〉 ↔ |1〉, which lead to the only one transparency window appearing due to the quantum destructive interference in the two transition pathways (see Fig. 2(b)).

In the transparency window region, there are steep varieties of the dispersion as shown in Fig. 2, which will make the light slow. The group velocity of the probe pulse can be defined as

which is further derived as

in the case of ωp = ω42 = ω43 = 0, meanwhile the time delay for a medium of length ξ is given by

From Eq. (14), it can be found that υg depends on the tunneling coupling intensity. Figure 3 shows the magnitude square of the probe pulse at different penetration lengths as a function of time delay. The system parameters can be taken as γ = 10 μeV, d14 = 10−16 esu⋅ cm, N = 1018 cm−3, and ωp = ω41 at resonance, with these parameters we can obtain α = 6.7 × 107 cm−1.[46–48] As we can see that the group velocity of probe pulse could significantly reduce by changing the tunneling coupling intensities. It can be found that the group velocity of probe field υg = 27.32 m/s with the time delay Δτ = 16γ−1 for T2 = T3 = 0.7γ, and υg = 97.20 m/s with the time delay Δτ = 4.4γ−1 for T2 = T3 = 1.35γ if αξ = 16. As we know that the width of transparency window becomes wider and the dispersions slope gently when the tunneling coupling intensities become strong. In the following, we design a scheme to realized stored light and dynamically control the beating signals based on the tunneling-induced quantum interference in the three quantum dots. In order to explain the scheme, we used the definition of DSP (the dark state polariton) as a coherent mixture of the quantum field and spin coherence. According to Refs. [7,8], equations (3) and (6) can be transformed into

which indicates a shape preserving propagation of the two-mode DSP

with β = d14/0127, tanθ = κ/T′, tanϕ = T3/T2, . In the first stage, we turn on the two tunneling couplings T2 and T3 to guide the probe field Ep(z,t) into the TQDs with slow speed without energy splitting. In the next stage, we turn off the two tunneling couplings T2 and T3 to convert the probe field Ep(z,t) into a pair of spin coherences cosερ12 and sinερ13, so the intensity of probe field gets zero. In the final stage, we turn on the two tunneling couplings T2 and T3 with the energy splitting ω42 ≠ ω43, so the two spin coherences become into slowly propagating DSP described by Eqs. (18) and (19). It is worth to note that the field component of Ψa(z,t) and Ψb(z,t) gains time-dependent phases ω42t and ω43t, which lead to produce a series of beating signals due to the alternative constructive and destructive interferences. As shown in Fig. 4, a quantum field E(z,t) evolves more slowly in the TQDs and is turned into spin coherence at the sample center when the tunneling effect is absent (T2 = T3 = 0) at t = 37γ−1. During a short storage time with Δt = 20γ−1, we apply the tunneling couplings T2 and T3 to retrieve the quantum field under different energy splittings ω42 and ω43, i.e., ω42 = −ω43 = 0 in panels (a) and (b), ω42 = −ω43 = 0.2γ in panels (c) and (d), ω42 = −ω43 = 0.4γ in panels (e) and (f), ω42 = −ω43 = 0.6γ in panels (g) and (h). We have noticed that the retrieved probe field slowly propagates with a series of maxima and minima (beating signals) in the TQDs when the opposite energy splitting is present.

Fig. 3. Pulse dynamics of the probe field at different medium depths ξ as a function of time for T2 = T3 = 0.7γ in panel (a), and for T2 = T3 = 1.35γ in panel (b), with ρ11(0) = 1, ρ22(0) = ρ33(0) = ρ44(0) = 0, ω42 = ω43 = 0. Other parameters are the same as those in Fig. 2.
Fig. 4. Pulse dynamics of the probe field inside the TQDs [panels (a), (c), (e), (g)] and the probe pulse at the sample exit [panels (b), (d), (f), (h)] as a function of time for different energy splittings: (a) and (b) ω42 = −ω43 = 0; (c) and (d) ω42 = −ω43 = 0.2γ; (e) and (f) ω42 = −ω43 = 0.4γ; (g) and (h) ω42 = −ω43 = 0.6γ; T2 = T3 = 1.35γ. The black solid line is the probe field at the entrance of TQDs and the blue solid line is the probe field at the exit, and the dotted line is the time sequence of the tunneling coupling T2 and T3. Other parameters are the same as those in Fig. 2.
4. Conclusion

We have theoretically investigated the dynamic propagation of a probe field in triple quantum dot molecules with the tunneling-induced interference effect. Our analytical and numerical results show that when the tunneling detunings are zero, a narrow transparency window with steep normal dispersion appears, which leads the probe field to propagating in the TQD at a reduced group velocity. In addition, we have obtained the dynamic generation of beating signals in an asymmetric procedure of light storage and retrieval. It is notable that the quantum probe field, incident on the TQD, is transformed into the spin coherence in the storage stage and formed the beating signals exhibiting a series of maxima and minima in intensity for opposite energy splittings during the retrieval stage.

Reference
[1] Harris S E 1997 Phys. Today 50 36
[2] Fleischhauer M Imamoglu A Marangos J P 2005 Rev. Mod. Phys. 77 633
[3] Jiao Y C Han X X Yang Z W Zhao J M Jia S T 2016 Chin. Phys. Lett. 33 123201
[4] Hau L V Harris S E Dutton Z Behroozi C H 1999 Nature 397 594
[5] Turukhin A V Sudarshanam V S Shahriar M S Musser J A Ham B S Hemmer P R 2001 Phys. Rev. Lett. 88 023602
[6] Cui C L Jia J K Gao J W Xue Y Wang G Wu J H 2007 Phys. Rev. 76 033815
[7] Fleischhauer M Lukin M D 2000 Phys. Rev. Lett. 84 5094
[8] Raczyński A Rzepecka M Zaremba J Zielinńska Kaniasty S 2006 Opt. Commun. 260 73
[9] Liu C Dutton Z Behroozi C H Hau L V 2001 Nature 409 490
[10] Wang H H Kang Z H Jiang Y Du D M Wei X G Wu J H Gao J Y 2008 Appl. Phys. Lett. 92 011105
[11] Wang L Sun Y H Wang R Zhang X J Chen Y Kang Z H Wang H H Gao J Y 2019 Opt. Express 27 6370
[12] Karpa L Vewinger F Weitz M 2008 Phys. Rev. Lett. 101 170406
[13] Mair A Hager J Phillips D F Walsworth R L Lukin M D 2002 Phys. Rev. 65 031802(R)
[14] Bao Q Q Gao J W Cui C L Wang G Xue Y Wu J H 2011 Opt. Express 19 11832
[15] Bao Q Q Yang L Ba N Cui C L Wu J H 2013 J. Opt. Soc. Am. 30 1532
[16] Goldstein A N Echer C M Alivisatos A P 1992 Science 256 1425
[17] Alivisatos A P 1992 Science 271 933
[18] Wang L J Rastelli A Kiravittaya S Benyoucef M Schmidt O G 2009 Adv. Mater. 21 2601
[19] Boyer de la Giroday A Sköld N Stevenson R M Farrer I Ritchie D A Shields A J 2011 Phys. Rev. Lett. 106 216802
[20] Müller K Bechtold A Ruppert C Zecherle M Reithmaier G Bichler M Krenner H J Abstreiter G Holleitner A W Villas-Boas J M Betz M Finley J J 2012 Phys. Rev. Lett. 108 197402
[21] Weiss K M Elzerman J M Delley Y L Miguel-Sanchez J Imamoǧlu A 2012 Phys. Rev. Lett. 109 107401
[22] Villas-Bôas J M Govorov A O Ulloa Sergio E 2004 Phys. Rev. 69 125342
[23] Borges H S Sanz L Villas-Bôas J M Alcade A M 2010 Phys. Rev. 81 075322
[24] Sköld N Boyer de la Giroday A Bennett A J Farrer I Ritchie D A Shields A J 2013 Phys. Rev. Lett. 110 016804
[25] Yuan C H Zhu K D 2006 Appl. Phys. Lett. 89 052115
[26] Borges H S Sanz L Villas-Boas J M Alcalda A M 2013 Appl. Phys. Lett. 103 222101
[27] Paspalakis E Kalini A Terzis A F 2006 Phys. Rev. 73 073305
[28] Fountoulakis A Paspalakis E 2013 J. Appl. Phys. 113 174301
[29] Voutsinas E Terzis A F Paspalakis E 2014 Phys. Lett. 378 219
[30] Li J Yu R Liu J Huang P Yang X 2008 Physica 41 70
[31] Wang Z Zhen S Wu X Zhu J Cao Z Yu B 2013 Opt. Commun. 307 7
[32] Wang Z P Yu B L 2014 J. Opt. Soc. Am. 30 2915
[33] Tian S C Tong C Z Wang C L Wang L J Wu H Xing E B Ning Y Q Wang L J 2014 Opt. Commun. 321 296
[34] Peng Y D Yang A H Li D H Zhang H Niu Y P Gong S Q 2014 Laser Phys. Lett. 11 065201
[35] Hsieh C Y Shim Y P Korkusinski M Hawrylak P 2012 Rep. Prog. Phys. 75 114501
[36] Xie Q H Madhukar A Chen P Kobayashi N P 1995 Phys. Rev. Lett. 75 2542
[37] Rainò G Salhi A Tasco V De Vittorio M Passaseo A Cingolani R De Giorgi M Luna E Trampert A 2008 J. Appl. Phys. 103 096107
[38] Songmuang R Kiravittaya S Schmidt O G 2003 Appl. Phys. Lett. 82 2892
[39] Tian S C Wan R G Tong C Z Fu X H Cao J S Ning Y Q 2015 Laser Phys. Lett. 12 125203
[40] Tian S C Wan R G Li L H Tong C Z Ning Y Q 2015 Opt. Commun. 334 94
[41] Tian S C Wan R G Tong C Z Ning Y Q Qin L Liu Y 2014 J. Opt. Soc. Am. 31 1436
[42] Tian S C Wan R G Tong C Z Ning Y Q 2015 AIP Adv. 5 027111
[43] Tian S C Wan R G Wang L J Shu S L Tong C Z Wang L J 2016 Laser Phys. Lett. 13 125203
[44] Tian S C Wan R G Wang C L Shu S L Wang L J Tong C Z 2016 Nanoscale Research Letters 11 219
[45] Nasehi R Asadpour S H Rahimpour Soleimani H Mahmoudi M 2016 Chin. Phys. Lett. 33 014204
[46] Paillard M Marie X Renucci P Amand T Jbeli A Gérard J M 2001 Phys. Rev. Lett. 86 1634
[47] Borri P Langbein W Schneider S Woggon U Sellin R L Ouyang D Bimberg D 2001 Phys. Rev. Lett. 87 157401
[48] Kim J Chuang S L Ku P C Chang-Hasnain C J 2004 J. Phys.: Condens. Matter 16 s3727