## Controllable double tunneling induced transparency and solitons formation in a quantum dot molecule |

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

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. 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] .

*ω*and half Rabi frequency Ω

_{p}*=*

_{p}*p*

_{12}

*E*/2

_{p}*h̄*) and a control field (with center angular frequencies

*ω*and half Rabi frequency Ω

_{c}*=*

_{c}*p*

_{34}

*E*/2

_{c}*h̄*) couple the transitions |1〉 ↔ |2〉 and |3〉 ↔ |4〉, respectively. Here

*p*

_{12(34)}=

*p⃗*

_{12(34)}·

*e⃗*are the dipole moments of transitions |1〉 ↔ |2〉 (|3〉 ↔ |4〉) with

_{l}*e⃗*being the polarization unit vector of the laser fields. The electric-field vector for the probe and control field can be written as

_{l}*E⃗*(

*r*,

*t*) = ∑

_{l}_{=}

*[*

_{p,c}e⃗_{l}ε_{l}exp*i*(

*k⃗*) ·

_{l}*z⃗*−

*ω*] +

_{l}t*c.c.*, where

*k⃗*,

_{l}*ω*,

_{l}*e⃗*and

_{l}*c.c.*are wave vector, envelope amplitudes, polarization direction of the probe and control fields, and complex conjugate, respectively.

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] .

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] .

*H.c.*represents the Hamiltonian complex conjugate, and

*Te*is interdot tunneling coupling strength. Detunings are defined by Δ

_{2}=

*ω*− (

_{p}*E*

_{2}−

*E*

_{1})/

*h̄*, Δ

_{4}=

*ω*− (

_{p}*E*

_{2}−

*E*

_{1})/

*h̄*+ (

*E*

_{4}−

*E*

_{2})/

*h̄*, and Δ

_{3}=

*ω*− (

_{p}*E*

_{2}−

*E*

_{1})/

*h̄*− (

*E*

_{4}−

*E*

_{3})/

*h̄*+ (

*E*

_{4}−

*E*

_{2})/

*h̄*, where

*E*(

_{j}*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

*A*of the four states Here

_{j}*d*= Δ

_{j}*+*

_{j}*iγ*with

_{j}*γ*=

_{j}*γ*+ Γ

_{ij}*, where*

_{ij}*γ*and Γ

_{ij}*represent lifetime broadening and dephasing broadening linewidths, respectively. Here Γ*

_{ij}*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, Γ*

_{ij}*depends on temperature of the system, and the range can be changed from several*

_{ij}*μ*eV 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] .

*can be obtained by the Maxwell equation with*

_{p}*N*and

_{a}*ε*

_{0}are the atomic concentration and dielectric coefficient in vacuum, respectively. Under a slowly varying envelope approximation, Eq. (3) is turned into where

*κ*

_{12}= (

*N*|

_{a}ω_{p}*p*

_{12}|

^{2})/2

*ε*

_{0}

*h̄c*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 Ω

*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 (*

_{c}*x*and

*y*) directions.

*A*

_{1}≈ 1). Taking

*A*

_{2},

*A*

_{3},

*A*

_{4}and Ω

*to be proportional to exp[*

_{p}*i*(

*Kz*−

*ωt*)], and making use of the Maxwell-Schrödinger Eqs. (2) and (4), the linear dispersion relation of the system is with

*D*= |Ω

_{c}*|*

_{c}^{2}− (

*ω*+

*d*

_{3})(

*ω*+

*d*

_{4}) and

*D*= |Ω

*|*

_{c}^{2}(

*ω*+

*d*

_{2}) + |

*Te*|

^{2}(

*ω*+

*d*

_{3}) − (

*ω*+

*d*

_{3})(

*ω*+

*d*

_{4})(

*ω*+

*d*

_{2}). 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*) =

*K*

_{0}+

*K*

_{1}

*ω*+

*K*

_{2}

*ω*

^{2}/2 +⋯. Here

*K*= (

_{j}*∂*)|

^{j}K/∂ω^{j}

_{ω}_{=0}where

*K*

_{0}=

*φ*+

*iα*/2 describes the phase shift

*φ*per unit length and the linear absorption coefficient

*α*of the probe field,

*V*= Re[1/

_{g}*K*

_{1}(

*ω*)] gives the group velocity, and

*K*

_{2}determines the group-velocity dispersion that contributes to the probe fields shape change and additional loss of the probe field intensity.

*, which are characterized by Im*

_{p}*K*(

*ω*) and Re

*K*(

*ω*), respectively, as a function of the frequencies

*ω*with the different tunneling strength

*Te*or/and control field Ω

*. We here consider the realistic parameters for a typical QD molecules [17*

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

*k*

_{12}= 3.66 × 10

^{4}cm

^{−1}meV, Δ

_{2}= Δ

_{4}= 0.1meV, Δ

_{3}= 0.2meV,

*γ*

_{2}= 1.0meV,

*γ*

_{4}= 0.1meV and

*γ*

_{3}= 1.0 × 10

^{−3}meV. Figs. 2(a)–2(c) give the linear optical properties of the system for the case Ω

*= 0. From Fig. 2(a), we observed that, in the absence of tunneling (*

_{c}*Te*= 0), the absorption profile has only a Lorentz line-shape peak, hence the probe field with central angular frequency

*ω*(corresponding to

_{p}*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 (Ω

*= 1.5meV), compared with Fig. 2 (a), two new peaks appear at the region*

_{c}*ω*= −2.26 × 10

^{12}

*s*

^{−1}and

*ω*= 2.01 × 10

^{12}

*s*

^{−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

*ω*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 Ω

_{c}*= 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.*

_{c}*∂*[Re

*K*(

*ω*)]/

*∂ω*< 0), the group velocity

*V*<

_{g}*c*so the probe field is a slow light; in the anomalous dispersion regimes (i.e.

*∂*[Re

*K*(

*ω*)]/

*∂ω*> 0), ones find

*V*>

_{g}*c*and thus the probe field is superluminal. From Re

*K*(

*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

*∂*[Re

*K*(

*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.

*by taking into account both the influence of the interdot tunneling coupling and the external control optical field Ω*

_{p}*with different level detuning Δ (Δ = (*

_{c}*E*

_{4}−

*E*

_{2})/

*h̄*) existing between states |2〉 and |4〉. The system parameters used are

*Te*= Ω

*= 2.0meV,*

_{c}*k*

_{12}= 3.66×10

^{4}cm

^{−1}meV, Δ

_{2}= 0.1meV, Δ

_{3}= 0.2meV,

*γ*

_{2}= 1.0meV,

*γ*

_{4}= 0.1meV, and

*γ*

_{3}= 1.0× 10

^{−3}meV. 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

*ω*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.

_{p}## 3. Asymptotic expansion and the slow-light solitons

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. 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] .

*l*= 1–4) and

*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

*x*

_{1}=

*εx*,

*y*

_{1}=

*εx*,

*z*=

_{j}*ε*(

^{j}z*j*= 0–2), and

*t*=

_{j}*ε*(

^{j}t*j*= 0, 1). Substituting them into Eqs. (2) and (4), we have Here, the explicit expressions of

*α*

^{(j)},

*β*

^{(j)},

*γ*

^{(j)}and

*ρ*

^{(j)}are omitted here to save space.

*j*= 1), we obtain where

*F*is a yet to be determined envelope function depending on the slow variable

*t*

_{1},

*z*

_{1}and

*z*

_{2}. For the second order (

*j*= 2), the solvability condition for

*F*travels with the group velocity

*V*.

_{g}*j*= 3), the solvability condition yields the nonlinear Schrödinger (NLS) equation where

*α̃*=

*ε*

^{−2}

*α*. Combining above results, we obtain after returning to the original variables, with

*τ*=

*t*−

*z/V*,

_{g}*U*=

*εFe*

^{−α̃z/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 Ω

*under the different interdot tunneling coupling strength*

_{c}*Te*. For

*Te*= 1.5meV, one can see that with the increasing Ω

*the Kerr nonlinear effect first tardily decreases with a minimum |*

_{c}*W*| existing at Ω

*= 1.86meV. It is also found that, for the stronger control field (i.e., Ω*

_{c}*> 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.,*

_{c}*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. Ω

*= 1.86*

_{c}*meV*).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 Ω

*with different one-photon detuning Δ*

_{c}_{4}. From Fig. 4 (b), one finds the Kerr nonlinear coefficient first decreases slowly and then enhances markedly with the increasing control field Ω

*, 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 Ω*

_{c}*increases. Simultaneously, there is an enhancement of the control field Ω*

_{c}*(corresponding to the minimum |*

_{c}*W*|) with the increasing detuning Δ

_{4}. It illustrates that the detuning Δ

_{4}may determine the Ω

*where the |*

_{c}*W*| reaches the minimum.

*K*

_{2r}> 0 and

*W*> 0, then we rewrite Eq. (9) in the dimensionless form with

_{r}*z*= −2

*L*as follow, which is a standard NLS equation [36] having the bright solitons solutions including multi-solitons solutions.

_{D}s*K*

_{2r}< 0 and

*W*< 0, an equation similar to Eq. (10) can be obtained from Eq. (9) by introducing a dimensionless variable

_{r}*z*= 2

*L*.

_{D}s*K*

_{2r}> 0 and

*W*< 0 are satisfied, Eq. (9) can be reduced to with

_{r}*z*= 2

*L*. The Eq. (11) is a dimensionless equation which admits solutions describing dark solitons.

_{D}s*K*

_{2r}< 0 and

*W*> 0, a dimensionless equation analogous to Eq. (11) can be get by using new variables

_{r}*z*= −2

*L*.

_{D}s*K*

_{2r}

*W*. Under the condition

_{r}*K*

_{2r}

*W*> 0, a simple bright soliton solution of Eq. (10) is

_{r}*u*= sec

*hσe*, or in terms of field It describes a fundamental bright soliton traveling with the propagating velocity

^{is}*V*and amplitude (

_{g}*K*

_{2r}/

*W*)

_{r}^{1/2}/

*τ*

_{0}. To cross-check the validity of our analytical results, we plot the ratios

*W*≫

_{r}*W*and

_{i}*K*

_{2r}≫

*K*

_{2i}versus the tunneling coupling strength Te with Ω

*= 2.5meV in Fig. 5(a). It is clearly seen that the coefficients of Kerr nonlinearity effect and group-velocity dispersion effect satisfy*

_{c}*W*≫

_{r}*W*and

_{i}*K*

_{2r}≫

*K*

_{2i}, 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

*K*

_{2r}

*W*> 0, the bright solitons can be formed in the parameter regime of the Fig. 5(a).

_{r}*K*

_{2r}

*W*< 0, we can get a simple dark solitons solution of Eq. (10) reads

_{r}*u*= tan

*hσe*, which is rewritten in terms of field We should cross-check the validity of our analytical results too. Using the same parameters as Fig. 5(a) except for Ω

^{is}*= 3.5meV, we show the ratios*

_{c}*W*≫

_{r}*W*and

_{i}*K*

_{2r}≫

*K*

_{2i}versus the tunneling coupling strength

*Te*, in Fig. 5(b). Similarly, the values of

*W*≫

_{r}*W*and

_{i}*K*

_{2r}≫

*K*

_{2i}are far less than one. However, we find

*K*

_{2r}

*W*< 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.

_{r}## 4. Discussion and conclusion

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] .

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] .

4. W. Yang and R. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express **16**(22), 17161–17170 (2008) [CrossRef] [PubMed] .

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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 |

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 |

34. | H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dual-wavelength domain wall solitons in a fiber ring laser,” Opt. Express |

35. | H. Zhang, D. Tang, L. Zhao, and R. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express |

36. | G. P. Agrawal, |

37. | Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. |

38. | M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. |

**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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