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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 17392–17403
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Controllable double tunneling induced transparency and solitons formation in a quantum dot molecule

Yanchao She, Xuejun Zheng, Denglong Wang, and Weixi Zhang  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 17392-17403 (2013)
http://dx.doi.org/10.1364/OE.21.017392


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Abstract

We consider the coupling effect between interdot tunneling coupling and external optical control field to study the linear optical property and the formation of temporal optical solitons in a quantum dot molecules system, analytically. The results show that the double tunneling induced transparency (TIT) windows are appeared in the absorption curve of probe field because of the formation of dynamic Stark splitting and quantum destructive interference effect from the two upper levels. Interestingly, the width of the TIT window becomes wider with the increasing intensity of the optical control field. We also find that the Kerr nonlinear effect of the probe field can be modulated effectively through coherent control both the control field and the interdot tunneling coupling in this system. Meanwhile, we demonstrate that the formation of dark or bright solitons can be practical regulated by varying the intensity of the optical control field.

© 2013 OSA

1. Introduction

2. The QD molecules model and linear properties

Recently, the self-assembled lateral QD molecules system were produced in GaAs (001) substrates by a unique combination of molecules beam epitaxy [18

18. R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett. 82(17), 2892(1–3) (2003) [CrossRef] .

21

21. B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B 72(8), 085339(1–12) (2005) [CrossRef] .

]. Based on the recent experiment condition, we consider an asymmetrical QD molecules system consisting of two dots with different band structures coupled by interdot tunneling, as shown in Fig. 1. The tunnel barrier in the QD molecules can be controlled by placing a gate electrode between the two QDs. The energy levels |1〉 (|3〉) and |2〉 (|4〉) are the crystal ground state and excited state of them. A probe field (with center angular frequencies ωp and half Rabi frequency Ωp = p12Ep/2) and a control field (with center angular frequencies ωc and half Rabi frequency Ωc = p34Ec/2) couple the transitions |1〉 ↔ |2〉 and |3〉 ↔ |4〉, respectively. Here p12(34) = p⃗12(34) · e⃗l are the dipole moments of transitions |1〉 ↔ |2〉 (|3〉 ↔ |4〉) with e⃗l being the polarization unit vector of the laser fields. The electric-field vector for the probe and control field can be written as E⃗(r, t) = ∑l=p,ce⃗lεlexp[i(k⃗l) · z⃗ωlt] + c.c., where k⃗l, ωl, e⃗l and c.c. are wave vector, envelope amplitudes, polarization direction of the probe and control fields, and complex conjugate, respectively.

Fig. 1 Energy-level diagram and excitation scheme of QD molecules system interacting with a strong control field (with half Rabi frequency Ωc) and a weak probe optical field (with half Rabi frequency Ωp). A bias voltage, which is added between two quantum dots, induces electron appearing tunneling effect, and the corresponding tunneling strength is Te. The detail description on the diagram sees the contexts of the paper.

Because the effect of the electron-electron interactions are expected to be rather weak in the present analysis, many body effects arising from electron-electron interactions are not included in our study [22

22. X. Hao, J. Wu, and Y Wang, “Steady-state absorptionCdispersion properties and four-wave mixing process in a quantum dot nanostructure,” J. Opt. Soc. Am. B 29(3), 420–428 (2012) [CrossRef] .

]. In the interaction picture, by using the rotating wave approximation [21

21. B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B 72(8), 085339(1–12) (2005) [CrossRef] .

], the semiclassical Hamiltonian of the system is given by
HI/h¯=j=24Δj|jj|(Ωp|21|+Ωc|43|Te|42|+H.c.),
(1)
where H.c. represents the Hamiltonian complex conjugate, and Te is interdot tunneling coupling strength. Detunings are defined by Δ2 = ωp − (E2E1)/, Δ4 = ωp − (E2E1)/ + (E4E2)/, and Δ3 = ωp − (E2E1)/ − (E4E3)/ + (E4E2)/, where Ej(j = 1 – 4) are eigenenergy of the state |j〉. Under the rotating wave and electro-dipole approximation, we obtain the evolution equations for the slowly varying probability amplitudes Aj of the four states
(it+d2)A2+ΩpA1TeA4=0,
(2a)
(it+d3)A3+Ωc*A4=0,
(2b)
(it+d4)A4+Ωc*A3TeA2=0,
(2c)
|A1|2+|A2|2+|A3|2+|A4|2=1.
(2d)
Here dj = Δj + j with γj = γij + Γij, where γij and Γij represent lifetime broadening and dephasing broadening linewidths, respectively. Here Γij is the dominant mechanism in semiconductor solid-state systems. It originates from not only electron-electron and electron-phonon scattering but also inhomogeneous broadening due to scattering on interface roughness. Also, Γij depends on temperature of the system, and the range can be changed from several μeV to several meV with the increasing temperature [23

23. P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett. 91(26), 267401(1–4) (2003) [CrossRef] .

].

The motion equation of the system for Ωp can be obtained by the Maxwell equation
2E1c22t2E=1ε0c22t2P,
(3)
with P=Na(p12A1*A2+p34A3*A4+c.c.) being the electric polarization intensity. Here Na and ε0 are the atomic concentration and dielectric coefficient in vacuum, respectively. Under a slowly varying envelope approximation, Eq. (3) is turned into
i(z+1ct)Ωp+c2ωp(2x2+2y2)Ωp+κ12A1*A2=0,
(4)
where κ12 = (Naωp|p12|2)/2ε0h̄c is the propagation coefficient. c is light speed in vacuum. For simplicity, we assume that the control field is strong enough and unchanged so that Ωc can be considered as a constant. The second term in Eq. (4) depicts the transverse diffraction effect of the system. We also assume that the probe field is homogeneous in the transverse (x and y) directions.

We first examine the linear properties of the system, which are the major contributors to pulse spreading and attenuation. For this purpose, we assume the pulsed probe field is much smaller than the control field and the electrons initially populated the energy level |1〉(i.e., A1 ≈ 1). Taking A2, A3, A4 and Ωp to be proportional to exp[i(Kzωt)], and making use of the Maxwell-Schrödinger Eqs. (2) and (4), the linear dispersion relation of the system is
K(ω)=ωcκ12DcD,
(5)
with Dc = |Ωc|2 − (ω + d3)(ω + d4) and D = |Ωc|2(ω + d2) + |Te|2(ω + d3) − (ω + d3)(ω + d4)(ω + d2). In the process of obtaining Eq. (5), the transverse diffraction effect to leading order, which is usually very small, has been neglected. In most cases, K(ω) can be Taylor expanded around the center frequency of the probe field, which corresponds to ω = 0, i.e., K(ω) = K0 + K1ω + K2ω2/2 +⋯. Here Kj = (jK/∂ωj)|ω=0 where K0 = φ + /2 describes the phase shift φ per unit length and the linear absorption coefficient α of the probe field, Vg = Re[1/K1(ω)] gives the group velocity, and K2 determines the group-velocity dispersion that contributes to the probe fields shape change and additional loss of the probe field intensity.

In Fig. 2, we plot the linear absorption and the linear dispersion curves of the probe field Ωp, which are characterized by ImK(ω) and ReK(ω), respectively, as a function of the frequencies ω with the different tunneling strength Te or/and control field Ωc. We here consider the realistic parameters for a typical QD molecules [17

17. H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B 85(11), 115425 (2012) [CrossRef] .

, 24

24. R. Espiau de Lamaëstre and H. Bernas1, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B 73(12), 125317(1–18) (2006) [CrossRef] .

] under coherent laser excitation, such as k12 = 3.66 × 104cm−1meV, Δ2 = Δ4 = 0.1meV, Δ3 = 0.2meV, γ2 = 1.0meV, γ4 = 0.1meV and γ3 = 1.0 × 10−3meV. Figs. 2(a)–2(c) give the linear optical properties of the system for the case Ωc = 0. From Fig. 2(a), we observed that, in the absence of tunneling (Te = 0), the absorption profile has only a Lorentz line-shape peak, hence the probe field with central angular frequency ωp (corresponding to ω = 0) is largely absorbed. However, for a weak tunneling strength (Te = 1.5meV), the absorption profile [see the red dash curve in Fig. 2(b)] splits into separate peaks, and meanwhile a deep dip in the optical absorption line can be produced [i.e., in this case, the TIT transparency window (TW) have formed]. The suppression of the probe field absorption is caused by the quantum destructive interference effect, which drives the interdot tunneling coupling and then renders the population in level |2〉 and |4〉 into dark states. When the strength Te increases to 2.5meV [see the red dash curve in Fig. 2(c)], the positions of two absorption peaks are away from the central angular frequency (i.e., ω = 0). It illustrates that the width of the TW becomes wider than that of Te = 1.5meV. When the control field is present, the linear absorption and linear dispersion curves of the probe field is shown in the Figs. 2(d) and 2(e). For a suitable control field (Ωc = 1.5meV), compared with Fig. 2 (a), two new peaks appear at the region ω = −2.26 × 1012s−1 and ω = 2.01 × 1012s−1 in the absorption profile [see the red dash curve in Fig. 2(d)]. Meanwhile, there is an absorption peak around central angular frequency (i.e., ω = 0). Therefore, two deep dips in the optical absorption line can be produced [i.e., the case of forming the double TIT TW]. It is from that both the interdot tunneling and the control field interact with the QDs molecules, the levels |3〉 and |4〉 have formed dynamic Stark splitting, which induce the corresponding levels splitting to two dark dressed levels. This leads two TIT configurations so the double TIT TWs appear in the system. With the control field ωc increases to 2.5meV [see the red dash curve in Fig. 2(e)], both of the width of the double TWs become wider than the width of Ωc = 1.5meV. From above discussion, we can conclude that the width of both the signal TIT TW and double TIT TWs can be modulated by the strength of the interdot tunneling and the coupling intensity of the control field. This result may provide new directions for current laser experiments.

Fig. 2 The linear absorption ImK(ω) (red dash curve) and linear dispersion Re(K) (black solid curve) of the probe field Ωp as a function of the frequencies ω under the consideration of (a)Te = Ωc = 0, (b)Te = 1.5meV, Ωc = 0, (c)Te = 2.5meV, Ωc = 0, (d)Te = Ωc = 1.5meV, and (e)Te = 1.5meV, Ωc = 2.5meV. The other parameters used are given in the text.

It is well known that the dispersion curves can be divided into normal and anomalous dispersion regimes. For the normal dispersion regimes (i.e. [ReK(ω)]/∂ω < 0), the group velocity Vg < c so the probe field is a slow light; in the anomalous dispersion regimes (i.e. [ReK(ω)]/∂ω > 0), ones find Vg > c and thus the probe field is superluminal. From ReK(ω) in Fig. 2, it is obviously that the normal (anomalous) dispersion regimes correspond to the TIT TW (the absorption peaks). It means that in the normal (anomalous) dispersion regimes the probe field has a negligible (strong) absorption. As a consequence, the probe field is nearly transparent (opaque) for the slow-light (superluminal) propagation when working in the normal (anomalous) dispersion regimes. Meanwhile, we find that the sign of [ReK(ω)]/∂ω has changed from negative to positive value with the enhancement of the interdot tunneling coupling strength Te. This result demonstrates that it is possible to obtain double switching, in which switching from the anomalous dispersion regime to the normal dispersion regime occurs.

Furthermore, as shown in Fig. 3, we depict the linear absorption and the linear dispersion of the probe field Ωp by taking into account both the influence of the interdot tunneling coupling and the external control optical field Ωc with different level detuning Δ (Δ = (E4E2)/) existing between states |2〉 and |4〉. The system parameters used are Te = Ωc = 2.0meV, k12 = 3.66×104cm−1meV, Δ2 = 0.1meV, Δ3 = 0.2meV, γ2 = 1.0meV, γ4 = 0.1meV, and γ3 = 1.0× 10−3meV. One sees that, when the detuning Δ increases, both the linear absorption and the dispersion curves standing in both side of the central angular frequencies ωp shift slight toward left, although the curves around the central angular frequencies have hardly changed. It is also obvious that with changing the detuning Δ from −1.0meV to 1.0meV, the width of the left-hand side TW increases accompanied by decreasing that of the right-hand one (just as shown in Fig. 3 (a)). Simultaneously, the dispersion curves which correspond to the left-hand side TW become gentle and that of the right-hand one turn into abrupt (as shown in Fig. 3 (b)). It means that the group velocity of the probe field in the left-hand side being in normal dispersion regimes decreases following increasing that in the right-hand one. In some words, both the transparency window in the linear absorption curve and the group velocity of the probe field can be modulated by the level detuning Δ.

Fig. 3 The linear absorption ImK(ω) (Fig.3 (a)) and linear dispersion Re(K) (Fig.3(b)) of the probe field Ωp as a function of the frequencies ω under the different level detuning Δ. The black solid, red dash, and blue dotted curves correspond to the detuning Δ = −1.0meV, 0.0meV, 1.0meV, respectively. Here, Te = Ωc = 2.0meV, Δ2 = 0.1meV, Δ4 = Δ + Δ2. The other parameters used are the same as those in Fig. 2.

3. Asymptotic expansion and the slow-light solitons

In this section, we consider weak nonlinear excitations of the system. The aim is to get a nonlinear probe pulse that allows shape-preserving propagation (i.e., the case of forming optical solitons), which has applications to optical transmission and information processing. For this aim, we use multiple-scale method [25

25. C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18(3), 2952–2966 (2010) [CrossRef] [PubMed] .

27

27. Y. She, D. Wang, W. Zhang, Z. He, and J. Ding, “Formation and interaction characteristics of two-component spatial weak-light soliton in a four-level double- type system,” J. Opt. Soc. Am. B 27(2), 208–214 (2010) [CrossRef] .

] to analyze the Maxwell-Schrödinger Eqs. (2) and (4). We introduce the asymptotic expansions Al=j=1εjAl(j) (l = 1–4) and Ωp=j=1εjΩp(j) with A1(0)=1, A2(0)=A3(0)=A4(0)=0. Here ε is a small parameter characterizing the small depletion of the ground state. In order to obtain divergence-free expansions, all quantities on the right-hand sides of the asymptotic expansions are considered as functions of multi-scale variables x1 = εx, y1 = εx, zj = εjz (j = 0–2), and tj = εjt (j = 0, 1). Substituting them into Eqs. (2) and (4), we have
it0A2(j)+d2A2(j)TeA4(j)+Ωp(j)=α(j),
(6a)
it0A3(j)+d3A3(j)+ΩcA4(j)=β(j),
(6b)
it0A4(j)+d4A4(j)+ΩcA3(j)TeA2(j)=γ(j),
(6c)
iz0Ωp(j)+i1ct0Ωp(j)+κ12A2(j)=ρ(j).
(6d)
Here, the explicit expressions of α(j), β(j), γ(j) and ρ(j) are omitted here to save space.

For the first order (j = 1), we obtain
Ωp(1)=Feiθ=Fei(Kz0wt0),
(7a)
A2(1)=1κ12(Kωc)Feiθ,
(7b)
A3(1)=1Ωc[Te1κ12(Kωc)1κ12Te(ω+d4)(ω+d2)(Kωc)1Te(ω+d4)]Feiθ,
(7c)
A4(1)=1Te[1κ12(ω+d2)(Kωc)+1]Feiθ,
(7d)
where F is a yet to be determined envelope function depending on the slow variable t1, z1 and z2. For the second order (j = 2), the solvability condition for Al(2) and Ωp(2) requires Fz1+1VgFt1=0 and the envelope F travels with the group velocity Vg.

For the third order (j = 3), the solvability condition yields the nonlinear Schrödinger (NLS) equation
iFz2+c2ωp(2x12+2y12)FK222Ft12W|F|2Feα˜z2=0,
(8)
where W=2κ12DcD{|DcD|2+|DDc(ω+d2)DTe|2+|Dc(ω+d4)(ω+d2)D(ω+d4)Dc|Te|2DΩcTe|2} and α̃ = ε−2α. Combining above results, we obtain
i(z+α˜2)U+c2ωp(2x2+2y2)UK222Uτ2W|U|2U=0,
(9)
after returning to the original variables, with τ = tz/Vg, U = εFeα̃z/2. The last three terms on the left-hand side of the Eq. (9) characterize diffraction, group-velocity dispersion and Kerr nonlinearity effect, respectively. In Fig. 4(a), we plot that the Kerr nonlinear coefficient |W| changes with the control field Ωc under the different interdot tunneling coupling strength Te. For Te = 1.5meV, one can see that with the increasing Ωc the Kerr nonlinear effect first tardily decreases with a minimum |W| existing at Ωc = 1.86meV. It is also found that, for the stronger control field (i.e., Ωc > 1.86meV), the Kerr nonlinear effect can be enhanced markedly. The similar variation tendency of Kerr nonlinear effect are observed with larger tunneling coupling strength (e.g., Te = 2.0meV and Te = 2.4meV). Moreover, the minimum of |W| increases when the tunneling coupling strength Te becomes stronger [for clarity, see in the inset in Fig. 4(a)], while the position of the minimum corresponds to the same control field (i.e. Ωc = 1.86meV).Therefore, we can conclude that the Kerr nonlinear effect of the system can be modulated by the control field. We further show in Fig. 4 (b) the Kerr nonlinear coefficient |W| as a function of the control field Ωc with different one-photon detuning Δ4. From Fig. 4 (b), one finds the Kerr nonlinear coefficient first decreases slowly and then enhances markedly with the increasing control field Ωc, which is similar with the phenomenon in Fig. 4(a). Notice that the Kerr nonlinear coefficient |W| reaches its minimum when the control optical field Ωc increases. Simultaneously, there is an enhancement of the control field Ωc (corresponding to the minimum |W|) with the increasing detuning Δ4. It illustrates that the detuning Δ4 may determine the Ωc where the |W| reaches the minimum.

Fig. 4 The Kerr nonlinear coefficient |W| as a function of the control field Ωc with different tunneling strength (Fig.4(a)) and one-photon detuning Δ4 (Fig.4(b)). In the Fig.4(a), the black solid, red dash, and blue dotted curves correspond to the tunneling strength Te = 1.5meV, 2.0meV and 2.5meV, respectively. In the Fig.4(b), the black solid, red dash, and blue dotted curves correspond to the detuning Δ4 = 3.0meV, 3.5meV, 4.0meV. Here Δ2 = −3.0meV, Δ3 = 1.0meV. The other parameters used are the same as those in Fig. 2.

Firstly, we assume K2r > 0 and Wr > 0, then we rewrite Eq. (9) in the dimensionless form with z = −2LDs as follow,
ius+2uσ2+2|u|2u=0,
(10)
which is a standard NLS equation [36

36. G. P. Agrawal, Nonlinear Fiber Optics4th ed. (Academic, 2009).

] having the bright solitons solutions including multi-solitons solutions.

Secondly, if K2r < 0 and Wr < 0, an equation similar to Eq. (10) can be obtained from Eq. (9) by introducing a dimensionless variable z = 2LDs.

In the third case, if the conditions K2r > 0 and Wr < 0 are satisfied, Eq. (9) can be reduced to
ius2uσ2+2|u|2u=0,
(11)
with z = 2LDs. The Eq. (11) is a dimensionless equation which admits solutions describing dark solitons.

In the last case, when K2r < 0 and Wr > 0, a dimensionless equation analogous to Eq. (11) can be get by using new variables z = −2LDs.

From the discussed above, it is obvious to find that the solutions of Eq. (9) are bright or dark solitons can be determined by the sign of product K2rWr. Under the condition K2rWr > 0, a simple bright soliton solution of Eq. (10) is u = sechσeis, or in terms of field
Ωp=1τ0K2rWrsech[1τ0(tzVg)]exp[iKz+iz2LD].
(12)
It describes a fundamental bright soliton traveling with the propagating velocity Vg and amplitude (K2r/Wr)1/2/τ0. To cross-check the validity of our analytical results, we plot the ratios WrWi and K2rK2i versus the tunneling coupling strength Te with Ωc = 2.5meV in Fig. 5(a). It is clearly seen that the coefficients of Kerr nonlinearity effect and group-velocity dispersion effect satisfy WrWi and K2rK2i, as shown in Fig. 5(a). Thus, we can safely neglect the imaginary part of the coefficients of Kerr nonlinearity effect and group-velocity dispersion effect. In addition, we find that when K2rWr > 0, the bright solitons can be formed in the parameter regime of the Fig. 5(a).

Fig. 5 The ratios of the imaginary parts and the corresponding real parts of the Kerr nonlinearity coefficient W (red solid curve) and group-velocity dispersion coefficient K2 (blue dashed curve) versus the tunneling coupling strength Te with different control fields Ωc. The Figs. 4(a) and 4(b) correspond respectively to the case of bright solitons and dark solitons due to K2rWr > 0 and K2rWr < 0, in the case of Ωc = 2.5meV and 3.5meV. The other parameters used are the same as those in Fig. 4.

Likewise, for the condition of K2rWr < 0, we can get a simple dark solitons solution of Eq. (10) reads u = tanhσeis, which is rewritten in terms of field
Ωp=1τ0K2rWrtanh[1τ0(tzVg)]exp[iKz+iz2LD].
(13)
We should cross-check the validity of our analytical results too. Using the same parameters as Fig. 5(a) except for Ωc = 3.5meV, we show the ratios WrWi and K2rK2i versus the tunneling coupling strength Te, in Fig. 5(b). Similarly, the values of WrWi and K2rK2i are far less than one. However, we find K2rWr < 0 in the parameter regime of the Fig. 5(b). It means that the parameter regime admits the existence of dark solitons. As shown in Figs. 5(a) and 5(b), the phenomenon of the optical solitons exists over quite wide parameter regimes. It is a quite robust phenomenon that the bright or dark solitons generated can be modulated only by changed the control field in this QD molecules system.

4. Discussion and conclusion

Acknowledgments

This work was supported by PCSIRT under Grant No. IRT1080, NNSF of China under Grant Nos. 51272158, Changjiang Scholar Incentive Program under Grant No. [2009]17, Shanghai Nano Special Foundation under Grant No. 11nm0502600, Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, Scientific Research Fund of Hunan Provincial Education Department of China under Grant No. 12A140, the Science and Technology Foundation of Guizhou Province of China under Grant Nos. J20112219 and J20122314, and Hunan Provincial Innovation Foundation for Postgraduate under Grant No. CX2012B248.

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K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett. 108(19), 197402(1–4) (2012).

16.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B 80(20), 205304 (2009) [CrossRef] .

17.

H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B 85(11), 115425 (2012) [CrossRef] .

18.

R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett. 82(17), 2892(1–3) (2003) [CrossRef] .

19.

C. Yuan and K. Zhu, “Voltage-controlled slow light in asymmetry double quantum dots,” Appl. Phys. Lett. 89(5), 052115 (1–3) (2006) [CrossRef] .

20.

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett. 96(13), 137401(1–4) (2006) [CrossRef] .

21.

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B 72(8), 085339(1–12) (2005) [CrossRef] .

22.

X. Hao, J. Wu, and Y Wang, “Steady-state absorptionCdispersion properties and four-wave mixing process in a quantum dot nanostructure,” J. Opt. Soc. Am. B 29(3), 420–428 (2012) [CrossRef] .

23.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett. 91(26), 267401(1–4) (2003) [CrossRef] .

24.

R. Espiau de Lamaëstre and H. Bernas1, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B 73(12), 125317(1–18) (2006) [CrossRef] .

25.

C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18(3), 2952–2966 (2010) [CrossRef] [PubMed] .

26.

C. Zhu and G. Huang, “Giant Kerr nonlinearity, controlled entangled photons and polarization phase gates in coupled quantum-well structures,” Opt. Express 19(23), 23364–23376 (2011) [CrossRef] [PubMed] .

27.

Y. She, D. Wang, W. Zhang, Z. He, and J. Ding, “Formation and interaction characteristics of two-component spatial weak-light soliton in a four-level double- type system,” J. Opt. Soc. Am. B 27(2), 208–214 (2010) [CrossRef] .

28.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

29.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003) [CrossRef] .

30.

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers,” Phys. Rev. B 80(5), 052302(1–4) (2009).

31.

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A 80(4), 045803(1–4) (2009) [CrossRef] .

32.

H. Sun, S. Fan, X. Feng, C. Wu, S. Gong, G. Huang, and C. H. Oh, “Strongly interacting photons in asymmetric quantum well via resonant tunneling,” Opt. Express 20(8), 8485–8495 (2012) [CrossRef] [PubMed] .

33.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Spatial vector solitons in a four-level tripod-type atomic system,” Phys. Rev. A 84(2), 023814(1–6) (2011) [CrossRef]

34.

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dual-wavelength domain wall solitons in a fiber ring laser,” Opt. Express 19(4), 3525–3530 (2011) [CrossRef] [PubMed] .

35.

H. Zhang, D. Tang, L. Zhao, and R. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express 18(5), 4428–4433 (2010) [CrossRef] [PubMed] .

36.

G. P. Agrawal, Nonlinear Fiber Optics4th ed. (Academic, 2009).

37.

Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29(17), 2064–2066 (2004) [CrossRef] [PubMed] .

38.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005) [CrossRef] .

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(270.5585) Quantum optics : Quantum information and processing
(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 22, 2013
Revised Manuscript: July 3, 2013
Manuscript Accepted: July 4, 2013
Published: July 12, 2013

Citation
Yanchao She, Xuejun Zheng, Denglong Wang, and Weixi Zhang, "Controllable double tunneling induced transparency and solitons formation in a quantum dot molecule," Opt. Express 21, 17392-17403 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17392


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References

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  16. V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009). [CrossRef]
  17. H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B85(11), 115425 (2012). [CrossRef]
  18. R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett.82(17), 2892(1–3) (2003). [CrossRef]
  19. C. Yuan and K. Zhu, “Voltage-controlled slow light in asymmetry double quantum dots,” Appl. Phys. Lett.89(5), 052115 (1–3) (2006). [CrossRef]
  20. G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006). [CrossRef]
  21. B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005). [CrossRef]
  22. X. Hao, J. Wu, and Y Wang, “Steady-state absorptionCdispersion properties and four-wave mixing process in a quantum dot nanostructure,” J. Opt. Soc. Am. B29(3), 420–428 (2012). [CrossRef]
  23. P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003). [CrossRef]
  24. R. Espiau de Lamaëstre and H. Bernas1, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B73(12), 125317(1–18) (2006). [CrossRef]
  25. C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express18(3), 2952–2966 (2010). [CrossRef] [PubMed]
  26. C. Zhu and G. Huang, “Giant Kerr nonlinearity, controlled entangled photons and polarization phase gates in coupled quantum-well structures,” Opt. Express19(23), 23364–23376 (2011). [CrossRef] [PubMed]
  27. Y. She, D. Wang, W. Zhang, Z. He, and J. Ding, “Formation and interaction characteristics of two-component spatial weak-light soliton in a four-level double- type system,” J. Opt. Soc. Am. B27(2), 208–214 (2010). [CrossRef]
  28. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  29. A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003). [CrossRef]
  30. H. Zhang, D. Tang, L. Zhao, and X. Wu, “Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers,” Phys. Rev. B80(5), 052302(1–4) (2009).
  31. H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A80(4), 045803(1–4) (2009). [CrossRef]
  32. H. Sun, S. Fan, X. Feng, C. Wu, S. Gong, G. Huang, and C. H. Oh, “Strongly interacting photons in asymmetric quantum well via resonant tunneling,” Opt. Express20(8), 8485–8495 (2012). [CrossRef] [PubMed]
  33. Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Spatial vector solitons in a four-level tripod-type atomic system,” Phys. Rev. A84(2), 023814(1–6) (2011) [CrossRef]
  34. H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dual-wavelength domain wall solitons in a fiber ring laser,” Opt. Express19(4), 3525–3530 (2011). [CrossRef] [PubMed]
  35. H. Zhang, D. Tang, L. Zhao, and R. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express18(5), 4428–4433 (2010). [CrossRef] [PubMed]
  36. G. P. Agrawal, Nonlinear Fiber Optics4th ed. (Academic, 2009).
  37. Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett.29(17), 2064–2066 (2004). [CrossRef] [PubMed]
  38. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005). [CrossRef]

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