† Corresponding author. E-mail:
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).
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.
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.
We consider the setup of the TQDs as composed of three QDs with different band structures and a triangular arrangement as shown in Fig.
![](cpb_29_3_034204/cpb_29_3_034204_ieqn2.gif)
![](cpb_29_3_034204/cpb_29_3_034204_ieqn3.gif)
![](cpb_29_3_034204/cpb_29_3_034204_ieqn4.gif)
To examine the dynamical evolution of the probe field, the following density matrix equations can be derived:
![](cpb_29_3_034204/cpb_29_3_034204_ieqn5.gif)
![](cpb_29_3_034204/cpb_29_3_034204_ieqn6.gif)
![](cpb_29_3_034204/cpb_29_3_034204_ieqn7.gif)
In the limit of a weak field, the steady-state solutions of ρ14 can be obtained, furthermore the probe susceptibility can be derived as
![](cpb_29_3_034204/cpb_29_3_034204_ieqn8.gif)
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.
![](cpb_29_3_034204/cpb_29_3_034204_ieqn9.gif)
![](cpb_29_3_034204/cpb_29_3_034204_ieqn10.gif)
While, when ω42 = ω43 = ω and T2 = T3 = T, the expressions of the dressed states are
![](cpb_29_3_034204/cpb_29_3_034204_ieqn11.gif)
![](cpb_29_3_034204/cpb_29_3_034204_ieqn12.gif)
In the transparency window region, there are steep varieties of the dispersion as shown in Fig.
![](cpb_29_3_034204/cpb_29_3_034204_ieqn13.gif)
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.
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