Abstract
We theoretically investigate the spectral features of tunnelinginduced transparency (TIT) and AutlerTownes (AT) doublet and triplet in a triplequantumdot system. By analyzing the eigenenergy spectrum of the system Hamiltonian, we can discriminate TIT and double TIT from AT doublet and triplet, respectively. For the resonant case, the presence of the TIT does not exhibit distinguishable anticrossing in the eigenenergy spectrum in the weaktunneling regime, while the occurrence of double anticrossings in the strongtunneling regime shows that the TIT evolves to the AT doublet. For the offresonance case, the appearance of a new detuningdependent dip in the absorption spectrum leads to double TIT behavior in the weaktunneling regime due to no distinguished anticrossing occurring in the eigenenergy spectrum. However, in the strongtunneling regime, a new detuningdependent dip in the absorption spectrum results in AT triplet owing to the presence of triple anticrossings in the eigenenergy spectrum. Our results can be applied to quantum measurement and quantumoptics devices in solid systems.
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Introduction
Quantum coherence and interference effects can lead to considerably interesting phenomena of quantum optics such as lasing without inversion^{1,2}, coherent population trapping^{3}, correlated spontaneous emission^{4}, and electromagnetically induced transparency (EIT)^{5,6,7,8,9}. As a phenomenon closely related to EIT, AutlerTownes (AT) splitting^{10,11} is indicated by a level anticrossing in the eigenenergy spectrum and a transparency window owing to the AT doublet rather than the quantum interference. This phenomenon has been utilized to measure the state of the electromagnetic field^{12,13,14}, as well as the AT triplet and multiplet spectroscopy^{15}. Both EIT and AT splitting have been investigated theoretically and experimentally in different quantum systems, including atomic and molecular systems^{8,16,17}, solidstate and metamaterials systems^{18,19}, superconducting quantum circuits^{20,21,22,23,24,25,26} and whisperinggallerymode optical resonators^{27,28,29}. It is also interesting to investigate EIT and AT splitting in semiconductor nanostructures because the trapped carriers behave like atoms and can be conveniently manipulated via external fields.
In the semiconductor quantum dot (QD), excitons form bound states and play an important role in the optical properties of these systems^{10,30,31,32,33,34,35}. Moreover, tunnelinginduced transparency (TIT)^{36,37,38} can occur for the excitonic states, which is similar to EIT in a threelevel atomic system, but no pump field is needed to apply to the excitonic system. As shown by Borges et al.^{37}, there is an evidence regarding the coexistence of both TIT and AT doublet in the intermediate regime, when the tunneling coupling is slightly above a threshold in a doubleQD system. However, a tripleQD system can offer new possibilities to study intriguing phenomena that are not observed in single and doubleQD systems^{39,40,41}. In the present paper, we show that when the electron is resonanttunneling in such a tripleQD system, which is in contrast to the case considered in a doubleQD system, the coexistence of both TIT and AT doublet regime does not occur in this system and the threshold of the tunneling coupling just corresponds to a transition point. More specifically, we find that, by analyzing the eigenenergy spectrum of the system Hamiltonian, there exist degenerate points in the eigenenergy spectrum for the resonant tunneling case. Therein the TIT presents in the weaktunneling regime without displaying wellresolved anticrossing in eigenenergy spectrum. However, in the strongtunneling regime, the double anticrossings in the eigenenergy spectrum illustrate the emergence of the AT doublet. For the offresonance case, i.e., only the right dot is not resonant with the central dot, we show that the degenerate points are absent in the eigenenergy spectrum. The double TIT can be realized in the weaktunneling regime along with the undistinguishable anticrossing in the eigenenergy spectrum and a new detuningdependent transparency dip in the absorption spectrum. However, in the strongtunneling regime, the presence of triple anticrossings in the eigenenergy spectrum reveals the realization of AT triplet in the absorption spectrum, where a new redshifted (blueshifted) transparency dip in the absorption spectrum is due to the presence of the blue (red) detuning from the right dot.
Results
The model
We study a tripleQD artificial molecule consisting of three aligned QD separated by two barriers [see Fig. 1(a)]. For the purpose of inhibiting the hole tunneling between valence bands, we here consider that the central dot is identical to the left dot but structural asymmetry with the right dot. This structural asymmetry is similar to the case shown in the double quantum dots system^{36,37,42,43,44}. With a gate (bias) voltage being applied along the growth direction, as shown in Fig. 1(b), the conductionbandlevel in the left and right dot are onresonant with the central dot^{38,42,43}. In doing so, the excitonic states are composed of mostly delocalized electron states in the tripleQD accompanied with entirely localized hole states. This tripleQD system can be achieved using, e.g., a selfassembled (In, Ga) As tripleQD fabricated on a GaAs (001) substrate by molecule beam epitaxy and insitu atomic layer precise etching, corresponding to a homogeneous tripleQD along the [1\(\bar{1}\)0] direction^{39}. The system is driven by a weak probe laser field with frequency ω_{ p } and the Rabi frequency Ω_{ p } corresponds to the driving strength for generating the direct excitonic state in the central dot (More specifically, the electron and hole are both in the central dot). As shown in Fig. 1(c), we assume that an electron can be excited from the valence band to the conduction band via a pulsed laser field to form a direct excitonic state (denoted as 4〉) only in the central dot, where 1〉 denotes vacuum state without excitons due to the absence of optical excitation in the tripleQD system. The gate (bias) voltage just only allow the electron to tunnel from the central dot to either left or right dot, yielding an indirect excitonic state 2〉 or 3〉 in the interdot (i.e., the electron is in the central dot but the hole is in left or right dot at this moment). The Hamiltonian for this tripleQD system reads (we set ħ = 1)
where σ_{ mn } ≡ m〉 〈n, and \({{\rm{\Omega }}}_{p}={\mu }_{14}{ {\mathcal E} }_{p}\mathrm{/2}\hslash \) is the Rabi frequency related to the probe laser field, with μ_{14} being the electricdipole transition matrix element between 1〉 and 4〉, and \({ {\mathcal E} }_{p}\) the electricfield amplitude of the probe field. T_{e1} (T_{e2}) denotes the tunneling coupling between the central dot and the left (right) dot, which can be controlled by regulating the width of the barrier and the applied gate voltage between the central dot and the left (right) dot.
Using a unitary transformation \(U(t)=\exp [i{\omega }_{p}({\sum }_{m\mathrm{=2}}^{4}\,{\sigma }_{mm})t]\) to remove the timedependent oscillatory terms^{24}, we can write the Hamiltonian in the interaction picture as
with Δ_{ p } = ω_{ p } − ω_{41}, Δ_{2} = Δ_{ p } + ω_{42}, and Δ_{3} = Δ_{ p } + ω_{43}. Here Δ_{ p } denotes the detuning of the probe field from ω_{41}, and ω_{4n}(n = 2, 3) is the energy difference between 4〉 and n〉. For the tripleQD system described by Hamiltonian (2), when Δ_{ p } = 0, the system has the degenerate dark states ψ_{1}〉_{dark} and ψ_{2}〉_{dark} as follows
where
Note that when T_{e2} = 0, ψ_{2}〉_{dark} is reduced to the dark state in a Λtype threelevel QD system, ψ〉_{dark} = sin Θ1〉 − cos Θ2〉 with tan Θ = T_{e1}/Ω_{ p }.
The dynamics of the system can be described by a Lindblad master equation:
where \({\mathscr{D}}[\hat{{\mathscr{O}}}]\rho =2\hat{{\mathscr{O}}}\rho {\hat{{\mathscr{O}}}}^{\dagger }{\hat{{\mathscr{O}}}}^{\dagger }\hat{{\mathscr{O}}}\rho \rho {\hat{{\mathscr{O}}}}^{\dagger }\hat{{\mathscr{O}}}\), Γ_{m1} are the relaxation rates between m〉 and 1〉, and \({\gamma }_{m}^{\varphi }\) describe the pure dephasing rates of the states m〉 (m = 2, 3, 4). The decoherence of the excitonic states is induced by both spontaneous radiation and pure dephasing processes. For the explicit expression of the master equation, see Eq. (12) in Methods.
Also, as shown in Methods, the density matrix element ρ_{14} in the steady state is given by
where d_{2(3)} = Δ_{2(3)} + iΓ_{2(3)}, and d_{4} = Δ_{ p } + iΓ_{4}. It can be seen that when T_{e2} = 0, i.e., in the absence of the rightside electron tunneling in Fig. 1, Eq. (6) is reduced to the result for the linear response of a Λtype threelevel QD system^{37}.
Next we decompose the density matrix element ρ_{14} into two components. In this way, ρ_{14} in Eq. (6) is decomposed as
where
The two terms R_{I} and R_{II} represent the first (“I”) and second (“II”) resonances, respectively. The two resonances can be directly used to analyze the characters of the probefield absorption in our scheme^{37,45}.
Tunnelinginduced transparency and AutlerTownes doublet
We first consider the case of an electron resonantly tunneling in the tripleQD system, i.e., ω_{42} = ω_{43} = 0. For simplicity, let Γ_{2} = Γ_{3}. Then, d_{2} = d_{3}, and one can analytically solve α = 0 in Eq. (8b) (see refs^{25,29,37} and^{45}). It is shown that a transition point turns out at the threshold coupling strength \({T}_{t}\cong {{\rm{\Gamma }}}_{4}\mathrm{/2}\). Note that in the intermediate coupling regime \(({T}_{t} < \sqrt{{T}_{e1}^{2}+{T}_{e2}^{2}} < {{\rm{\Gamma }}}_{4})\), in contrast to the case shown in the threelevel Λ system^{37}, the firstresonance R_{I} and the secondresonance R_{II}, which have the same sign in absorption profile are apparently separated, indicating that there is no interference in this case. In other words, the crossover regime does not occur in this tripleQD system. Thus, there are only two regimes in such a system, i.e., the threshold T_{ t } separates TIT in the weaktunneling regime \((0 < \sqrt{{T}_{e1}^{2}+{T}_{e2}^{2}} < {T}_{t})\) from AT doublet in the strongtunneling regime \((\sqrt{{T}_{e1}^{2}+{T}_{e2}^{2}} > {T}_{t})\).
(1) The weaktunneling regime
In the weaktunneling regime with \(0 < \sqrt{{T}_{e1}^{2}+{T}_{e2}^{2}} < {T}_{t}\), α is a pure imaginary number. It gives rise to a pure real number \({R}_{\pm }=\mathrm{1/2}\mp {\varepsilon }_{1}/\alpha \), with ε_{1} = (Γ_{4} − Γ_{2})/2, and a pure imaginary number \({{\rm{\Delta }}}_{\pm }=i({\varepsilon }_{2}\pm \alpha \mathrm{/2)}\), with ε_{2} = (Γ_{4} + Γ_{2})/2. Thus, the imaginary part of ρ_{14} is given by
where
In the weaktunneling case, 0 < α/2 < ε_{1}, ε_{2}, so that both the parameters C_{I} and C_{II} are positive. Then, the firstresonance term Im(R_{I})_{TIT} is a wide, positive Lorentz line profile, while the secondresonance term Im(R_{II})_{TIT} is a narrow, negative Lorentz line profile. The different signs of the first and second resonances profile indicate the realization of the TIT, as due to the destructive interference.
As shown in Fig. 2(a), when the tunneling couplings T_{e1} and T_{e2} are both weak (i.e., T_{e1} = T_{e2} = Γ_{4}/5), it is interesting to see that the eigenenergies of this system Hamiltonian do not display an obvious anticrossing, but show degenerate points when the electron is resonant tunneling among the tripleQD system (ω_{42} = ω_{43} = 0). However, there is an evidence of a double anticrossing in the eigenenergy spectrum of the system Hamiltonian [see the red dashed loops in Fig. 2(d)], when the relative tunneling coupling (\({T}_{re}=\sqrt{{T}_{e1}^{2}+{T}_{e2}^{2}}\), with T_{e1} = Γ_{4}/2, T_{e2} = 10^{−1} Γ_{4}) is slightly stronger than the threshold coupling strength T_{ t } = Γ_{4}/2. Furthermore, these results also indicate that the coexistence of both TIT and AT doublet does not occur in such a tripleQD system, and the threshold of the tunneling coupling just corresponds to a transition point.
(2) The strongtunneling regime
In the strongtunneling regime with \({T}_{t} < \sqrt{{T}_{e1}^{2}+{T}_{e2}^{2}}\), \(\alpha =2\sqrt{{T}_{e1}^{2}+{T}_{e2}^{2}}\) is a real number, such that R_{±} = 1/2 and Δ_{±} = −iε_{2} ± α/2. The two resonances are located at ±α/2 and have the same linewidth ε_{2}. The imaginary part of ρ_{14} in this regime can be written as
This corresponds to an AT doublet because of Im(ρ_{14})_{ATD} is the sum of two identical Lorentz line profile peaked at ±α/2^{11}. It is interesting to see that the appearance of the AT doublet in the absorption spectrum of the probe field is equivalent to the case that there exist the double anticrossings in the eigenenergies of the system Hamiltonian [as shown in the red dashed loops in Fig. 2(b)], forming a transparency window between the pair of resonances. Now the positive value of the resonance pair is only responsible for the decreasing or even vanishing absorption of the probe field, instead of accounting for this phenomenon through the cause for that of the destructive interference in the TIT^{29,37}. This is because the pair of resonances is shifted relatively far away from each other, so that their overlap is insufficient to yield significant interference.
When the tunneling couplings are sufficiently strong (T_{e1}, T_{e2} ≥ Γ_{4}), there are prominent double anticrossings in the eigenenergies of the system Hamiltonian [as shown in the red dashed loops in Fig. 2(c)]. This also gives rise to an evident reduction of the overall absorption, displaying a wide transparency window (corresponding to vanishing absorption)^{37}. The pronounced double anticrossings in this situation can be used to deduce the positions and width of the transparency window of the AT doublet. Therefore, in this strongtunneling regime, the TIT evolves to the AT doublet, which results in a wellresolved doublet in the absorption spectrum and double anticrossings in the eigenenergy spectrum.
Double Tunnelinginduced transparency and AutlerTownes triplet
Next, we consider the case of an electron resonant tunneling between the left and central dots (i.e., ω_{42} = 0), while there is offresonant tunneling between the right and central dots (that is, ω_{43} ≠ 0) in the tripleQD system. The energy shift induced by the gate field is given by Δω_{43} = eFd^{38,42}, with F being the fixed electric field and d being the width of the barrier between the right and central dot. In this scenario, the optical absorption of the probe field can be manipulated by varying the fixed electric field F, which is a way of controlling the detuning of the right dot ω_{43}. It should be noted that, as shown in Fig. 3, the degenerate points of the eigenenergies of the system Hamiltonian disappear when ω_{43} is nonzero. For instance, when ω_{43} = ±Γ_{4}/5, there are not degenerate points and distinguishable anticrossing in the eigenenergies of the system Hamiltonian in the weaktunneling regime [see Fig. 3(a)], which is clearly distinct from the results shown in a doubleQD system^{37}. In the strongtunneling regime, one can also see that the eigenenergies of the system Hamiltonian exhibit triple anticrossings [see Fig. 3(b)], as compared to the case with double anticrossings being implied to denote AT doublet in Fig. 2(b) where ω_{43} = 0.
(1) The weaktunneling regime with ω _{43} ≠ 0
As shown in Fig. 4(a), in the weaktunneling regime, it is revealed that double TIT can be realized by manipulating the energylevel detuning Δ_{3} to achieve slight offresonance. Narrow double transparency windows arise, when the tunneling couplings are weaker than or equal to the threshold value \({T^{\prime} }_{t}\) [as shown in Eq. (8b), there is indeed threshold value in this case but no explicit expression], in the case of ω_{43} ≠ 0. In particular, the new TIT dip [see the blue dashed curve in Fig. 4(a)] is redshifted for a bluedetuned Δ_{3} (e.g., ω_{43} = Γ_{4}/5) in the probe field absorption spectrum. However, the new TIT dip [see the red dotted curve in Fig. 4(a)] becomes blueshifted at a reddetuned Δ_{3} (e.g., ω_{43} = −Γ_{4}/5) in the absorption spectrum. Furthermore, both the undistinguished anticrossing in the eigenenergy spectrum [see Fig. 3(a)] and the new detuningdependent dip presenting within the scope of the fulllinewidth [see the black curve in Fig. 4(a)] in the absorption spectrum demonstrate that the double TIT is implemented. Hence, the double TIT is significantly different from AT doublet, where the peaks of the pair resonances are far enough separated apart and the double anticrossings occur in the eigenenergy spectrum. Moreover, as shown in Fig. 4(b), one of the absorption minima in the probe field absorption spectrum obeys the condition Δ_{ p } = ω_{42} = 0, and the other absorption minimum satisfies the condition Δ_{ p } = −ω_{43}. Therefore, we are able to realize double TIT without forming wellresolved anticrossing in such a weaktunneling regime.
(2) The strongtunneling regime with ω _{43} ≠ 0
In the strongtunneling regime, the transparency window of the AT doublet [see the black solid curve in Fig. 4(c)] exhibits a new peak, turning the probe field absorption profile into three peaks. This yields two transparency windows [as shown in the blue dashed and red dotted curves in Fig. 4(c)], where a new dip arises in the blue (red) side of the probe detuning accompanied with the other two peaks of the transparency window being blueshifted (redshifted) when there is a reddetuned (bluedetuned) Δ_{3}, e.g., \({\omega }_{43}=\mp {{\rm{\Gamma }}}_{4}\mathrm{/5}\). Also, one can see that the triple anticrossings in the eigenenergy spectrum [see Fig. 3(b)] make sure that the AT triplet is fulfilled at this moment. In Fig. 4(d), similar features can be observed, where two absorption minima in the absorption spectrum locate at Δ_{ p } = ω_{42} = 0 and Δ_{ p } = −ω_{43}, respectively. It should be noted that the width of the central peak increases when raising the bluedetuning (reddetuning) Δ_{3}, but the width of the peak on the reddetuned (bluedetuned) side decreases. In particular, for a bluedetuned (reddetuned) Δ_{3}, the decrease of the width of the reddetuned (bluedetuned) sideband is compensated by the increase of the width of the central peak. Therefore, in the strongtunneling regime, the AT triplet can be realized by manipulating the detuning Δ_{3}.
Discussion
In this work, we have presented a theoretical study of the eigenenergy spectrum and optical absorption properties of a tripleQD system with four effective energy levels. The results show that, in the case of the electron resonanttunneling in the tripleQD system, there are degenerate points in the eigenenergy spectrum of the system Hamiltonian. In the weaktunneling regime, the presence of the TIT does not show an obvious anticrossing in the eigenenergy spectrum. However, in the strongtunneling regime, the emergence of double anticrossings in the eigenenergy spectrum indicates that the TIT evolves to the AT doublet which includes two wellresolved peaks in the probe field absorption spectrum. The pronounced double anticrossings in the eigenenergy spectrum of the system Hamiltonian that can be used to deduce the positions and width of the transparency window of the AT doublet. For the offresonance case, that is, the right dot is not resonant with the central dot, we demonstrate that the degenerate points in the eigenenergy spectrum disappear. The realization of double TIT in the weaktunneling regime, where the distinguishable anticrossing does not appear in the eigenenergy spectrum, exhibits a new transparency dip in the absorption spectrum which can be controlled by manipulating one of the energylevel detunings. However, in the strongtunneling regime, the presence of triple anticrossings in the eigenenergy spectrum illustrates the realization of AT triplet in the absorption spectrum. More importantly, there is a new dip in the blue (red) side of the probe detuning along with the other two peaks of the transparency window being blueshifted (redshifted) at a reddetuned (bluedetuned) Δ_{3}. The linewidth narrowing in one of the side peaks could be compensated for the linewidth broadening in the central peak.
Finally, our proposed schemes for these spectral features are naturally inherited and open up new ways for physicists and chemists to work in the field of laser spectroscopy, quantum measurement, nonlinear optics and quantumoptics devices in solid systems.
Methods
Quantum dynamics behavior of the triple QD system
By applying the BornMarkov approximation, the coupled differential equations for the density matrix ρ_{ mn } in the interaction picture can be obtained as
with \({{\rm{\Gamma }}}_{m}={{\rm{\Gamma }}}_{m1}\mathrm{/2}+{\gamma }_{m}^{\varphi }\) (m = 2, 3 and 4), and \({\gamma }_{mn}=({{\rm{\Gamma }}}_{m1}+{{\rm{\Gamma }}}_{n1}\mathrm{)/2}+{\gamma }_{m}^{\varphi }+{\gamma }_{n}^{\varphi }\) (m = 2, 3; n = 3, 4).
The derivation of the solution of the density matrix ρ _{14}
From Eq. (12), when the time is explicitly shown in the equations of ∂_{ t }ρ_{12}, ∂_{ t }ρ_{13}, and ∂_{ t }ρ_{14}, it follows that
The absorption and dispersion coefficients are respectively proportional to the imaginary and real parts of the density matrix element ρ_{14} in the steady state^{37,45}. With a weak probe field (\({{\rm{\Omega }}}_{p}\ll {T}_{e1},{T}_{e2}\)) acting on the considered tripleQD system, the term \({{\rm{\Omega }}}_{p}^{\ast }{\rho }_{42}(t)\) and \({{\rm{\Omega }}}_{p}^{\ast }{\rho }_{43}(t)\) in Eq. (13) can be respectively approximated by \({{\rm{\Omega }}}_{p}^{\ast }{\rho }_{42}\mathrm{(0)}\) and \({{\rm{\Omega }}}_{p}^{\ast }{\rho }_{43}\mathrm{(0)}\), and the term \({{\rm{\Omega }}}_{p}^{\ast }[{\rho }_{11}(t){\rho }_{44}(t)]\) can also be approximated by \({{\rm{\Omega }}}_{p}^{\ast }[{\rho }_{11}\mathrm{(0)}{\rho }_{44}\mathrm{(0)]}\). Moreover, the system is assumed to be initially in the ground state 1〉, so ρ_{11}(0) = 1 and ρ_{42}(0) = ρ_{43}(0) = ρ_{44}(0) = 0. Then, Eq. (13) becomes
In the steadystate, ∂_{ t }ρ_{12} = ∂_{ t }ρ_{13} = ∂_{ t }ρ_{14} = 0, so we have
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Acknowledgements
We acknowledge valuable discussions with Prof. J. Q. You and thank Dr. Rui Li for his timely help. This work is supported by the National Key Research and Development Program of China (Grant No. 2016YFA0301200), National Basic Research Program of China Grant Nos 2014CB921401 and 2014CB848700, National Natural Science Foundation of China Grant Nos 91121015, 11575071, 11404019, 11404020, U1330201 and U1530401. T.F.L. is partially supported by Science Challenge Project, No. TZ2018003.
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All authors have made substantial intellectual contributions to the research work. X.Q.L. performed the calculations, and Z.Z.L. also participated in the discussions. All authors contributed to the interpretation of the work and the writing of the manuscript.
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Luo, XQ., Li, ZZ., Jing, J. et al. Spectral features of the tunnelinginduced transparency and the AutlerTownes doublet and triplet in a triple quantum dot. Sci Rep 8, 3107 (2018). https://doi.org/10.1038/s41598018212213
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DOI: https://doi.org/10.1038/s41598018212213
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