# Förster Blockade and Excitation Transfer Behaviors of Two-Photon Rabi Oscillations of Exciton–Biexciton Levels in Coupled Quantum Dots

Authors: Ameenah N. AlAhmadi
Journal: Journal of Nanomaterials
 (2023)

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## Abstract

We provide a theoretical study for the dynamical behavior of two-photon Rabi oscillations in two coupled quantum dots. Each dot is a three-level quantum system (ground, one-exciton, and biexciton states). We describe the optical dynamics within the density matrix formalism framework. The two-photon Rabi oscillations are found to be significantly affected by the strength of the interdot dipolar interaction. Using a detuned laser pulse with an exciton state of the driven quantum dot in our numerical simulations, a direct coherent excitation between the biexciton states can be established, but whether Rabi oscillations can be observed depend mainly on the strength of the dipolar interaction and bandwidth of the excitation laser. When the strength of the dipolar coupling is less than the excitation laser bandwidth, the time-dependent populations of the various two biexciton states of the system exhibit coherent oscillations with several frequencies. However, as we increase the dipolar coupling strength, the population transfer between the quantum dots levels amazingly becomes forbidden, and the excitation gets trapped at the initial state. The reduction of the influence of the dipolar interaction leads to a manipulation of two-photon Rabi oscillation in an ultrashort timescale (∼ps), opening possibilities to achieve ultrafast quantum state control and quantum logic gate generation.


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## Body

## 1. Introduction

Studying the interaction effects of light and quantum systems is one of the essential research contents of quantum optics. It plays a crucial role in developing future secure communication networks and quantum computers, where qubits are designed to process quantum information using single photon and photon pair sources. Semiconductor quantum dots (QDs) are the perfect optically active components for such applications, since they enable the control of the exciton and biexciton states in a solid-state environment [1–3]. Coherent optics control of exciton states in semiconductor QDs has important applications in quantum information, quantum computing, and laser technology [4–6]. This technological inspiration increases efforts to investigate coherent phenomena of semiconductor QDs. Coherent indicates the degree of phase regularity and amplitude invariance in a quantum system. A quantum coherence state is a superposition of multiple states taken from a reference basis. The generation of coherence is intimately connected to the operation of the quantum interaction [7]. In other words, the coherence pathway starts from an initial state of a quantum system, which is then converted to a state of single quantum coherence through a resonant excitation within a timescale that is small compared to any relaxation times of the system. Extensive experimental and theoretical investigations on coherence underlie phenomena, including atomic-like spectra [8–10], entanglement [11–13], modification of spontaneous emission rate by a microcavity [14, 15], coherent manipulation of wave function using quantum interference [16], and Rabi oscillation [17–19], have been verified in various types of semiconductor QDs. Semiconductor QDs exhibit many unique quantum physical characteristics under the effect of an external field, and these characteristics are strongly related to the characteristics of their energy level structure. QDs have atomic-like characteristics that make them potential candidates for use as parts of solid-state quantum computers [3]. The modulation of qubits in quantum logic gates and the control of the QD solid reaction pathways require coherent control of the quantum states of the system. Rabi oscillation plays a significant role in coherent monitoring. It is possible to directly control the excited state population of a quantum system through the pulse area of the excitation light by Rabi oscillation [20]. Since observing Rabi oscillations in InGaAs QDs [18], researchers have investigated the Rabi oscillations and quantum interference effects in QDs systems [17–19, 21, 22]. Similar coherent optical transients from the ground state to the biexciton state have also been achieved by using a coherent two-photon process. The two-photon absorption on the biexciton transition is created by setting the excitation laser to half of the biexciton generation energy, resulting in regulated Rabi oscillations similar to those seen in the previous studies on excitons [23, 24].Dipolar interactions have also been reported to have a noticeable impact on QD optical transfer. Numerous studies have coupled the optical excitation of an exciton with dipolar interactions, observing the coherent behavior of different dynamic processes of an exciton [25, 26]. The energy transfer among QDs is an important feature for developing technologies for quantum computing and understanding the transfer of quantum entanglement without losing the characteristic correlations among the components of a quantum system, which is lost by the decoherence phenomena. Two processes lead to the decay of optically induced coherence between quantum states. First, the amplitude of a state relaxes to other states in the system. The amplitude relaxation rate (denoted by Γk in this work) is determined by the lifetime of the state. The second process occurs when the interaction of the states with the environment destroys the phase correlations between states without decay of the individual probability amplitudes. The ability to control the coherent phenomena in QD systems has been intensively studied [27].In the present paper, we describe a method for coherently manipulating quantum information stored in biexcitonic states of semiconductor QDs. The dipole–dipole interaction provides a strong long-range coupling between QD systems. This interaction among closely-lying identical QDs can shift the resonance of the exciton and biexciton states out of the range of excitation laser bandwidth, inhibiting the excitations of all other dots while one is excited. When dipole–dipole solid interactions are added among the dots, they suppress the optical excitation of the system, analogous to the dipole blockade, an effect in quantum optics and one of the key concepts behind the promise of coherent quantum information at atomic levels [28].To the best of our knowledge, no dipole blockade effect has been investigated in the QD system, despite nearly a couple of decades of researching. The present work aims to study a model that yields quantitative results as a function of the essential experimental parameters, including the dipole–dipole interaction strength. The system investigated is two interacting QDs, and we will ignore some of the complications of actual systems. We model the QD as a three-level ladder-type quantum system, with one dot driven by an external laser field and transferring the excitation to the second dot. We report several characteristics of the dipole blockade-like, including tunable dynamics of the system and the dipolar coupling effect in strong and weak regime conditions.The biexciton and exciton dynamics equation is derived using the master equation. The equation is used to analyze the effect of the initial state conditions on the two QDs three-level systems interacting via dipolar coupling and after a pulse with a short-time duration that arrives to trigger the system interaction.Our findings extend our previous investigation that describes only the dynamic of exciton states by including biexciton states into the Hamiltonian; however, the calculation was carried out using different experimental parameters and different driven QD techniques, resulting in different conclusions [29].


## 2. Model and Theory

Figure1 shows the energy level structure of two semiconductor QDs, QDa and QDb. Each dot is composed of three levels, where 0a, Xa, and XXa represent the ground state, exciton state, and biexciton state of QDa, respectively, and 0b,Xb, and XXb represent the ground state, exciton state, and biexciton state of QDb, respectively. The excitation of exciton states in QDa is carried out utilizing a picosecond pulse laser with a central photon energy of ℏω, equivalent to half of the biexciton energy. We assume that the QDs system is in close proximity (approximately few Å), allowing a molecular linker via dipolar interaction yet avoiding direct carrier tunneling. The relevant Hamiltonian of our system is given as follows:(1)H^=H^0+H^Ω+H^D,(2)H^0=∑j=a,bδ0j0j0+δXjXjX+δXXjXXjXX,(3)H^Ω=ΩXt2ℏ0aX+ΩXXt2ℏXaXX+H.c.,(4)H^D=VX−XXaXb+VXX−XXXXaXXb+H.c.,where δij=Eij−ℏω is the detuning of the exciton energy levels with respect to the laser excitation energy, H^Ω is the interaction with the laser excitation, and ΩXt=0μ→X⋅E→tX/ℏ and ΩXXt=Xμ→XX⋅E→tXX/ℏ are the couplings of the excitonic transition of exciton and biexciton states to the radiation field, where μ→X and μ→XX are the electric dipole moments of the exciton and biexciton states at QDa and E→t is the envelope of the pulse. We assume a hyperbolic secant squared pulse, Et=Θ/2πτpsecht−t0/τp, where Θ is the pulse area, τp is a pulse duration of the field envelope, and t0 is the delay time of the pulse. VX−X and VXX−XX are the dipolar interactions between the two dots. We consider VX−X to be the dipolar coupling between the exciton state in QDa and the exciton state in QDb, and VXX−XX is the dipolar interaction between the biexciton state in QDa and the biexciton state in QDb. We should point out that the overlap between the exciton state at QDa and the biexciton state of QDb, and vice versa, is negligible due to the large detuning between these states. Accordingly, the dipolar couplings between them are omitted from the Hamiltonian. The dynamic evolution of the system is described using the quantum master equation:(5)ddtρt=L^0ρt−L^dρt.Figure 1
Energy diagrams of three-level coupled quantum dots. Each dot consists of the ground state, the exciton state, and the biexciton state.The first term on the right-hand side of the formula represents the unitary evolution operator of the system and is given as follows:(6)L^0ρt=−iℏHt,ρt,where H(t) is the time-dependent Hamiltonian of the system given by Equation (1) and ρt is the density matrix. The second term on the right side represents the operator corresponding to the decoherence of the system caused by relaxation processes, including the dephasing time, which affects the decay of the amplitude (occupation). This phase relaxation time controls the coherence of the interaction. The nonunitary part of Equation (5) is often called the dissipator [30].(7)L^dρt=−12∑kΓkLk†Lkρt+ρtLk†Lk−2LkρtLk†,where the Lk and Lk† are collapse operators, through which the environment couples to the system and Γk is the corresponding rate. In this work, we consider that the dissipations due to the decay (relaxation) of the excitonic states and the dephasing rate are induced by the relaxation rate. The diagonal elements ρkk of the density matrix represent the populations in the state k, which decay by relaxation rate Γk. Meanwhile, the off-diagonal element ρlk describes the coherence between the l state and the k state, which decays by dephasing rate Γk/2.We first consider the dynamic evolution of a single-dot case, at which there are three-level excitonic states. We have numerically solved Equation (5) by considering QDa to be initialized in its exciton ground state, and then coherent Rabi oscillations, with frequency ΩXt, are generated between the ground state and exciton state, which decay out to the ground state via the dephasing rate ΓX. At the same time, the Rabi oscillations, with frequency ΩXXt, are created between the exciton state and biexciton state, which decay out to the exciton state via the dephasing rate ΓXX.Furthermore, the dynamic evolution of the two three-level dots case has been implemented to study the time evolution of the exciton and biexciton state population. Coherent Rabi oscillations in the six-level system for both strong and weak dipolar interaction cases are demonstrated. These oscillations are transferred to the exciton state and biexciton state in QDb via the dipolar coupling, as shown in Figure 1. We run our simulations using scientific python packages and some key libraries of QuTiP python [31].


## 3. Result and Discussion

The numerical results are obtained by simulating the master equation (Equations (5)–(7)), where ρijt represents the population of the state ij where i = 0, X, XX state, at the dot j = a, b. The dephasing time ΓX and ΓXX for two optical transitions, Xa⟶0a and XXa⟶Xa, are taken to be 630 ps, equivalent to only ΓX=2μeV homogeneous broadening, we consider ΓXX≈2ΓX=4μeV [32, 33].The excitation toXXa is accomplished by a two-step optical excitation with the pump and coupling laser Rabi frequencies ΩXt, and ΩXXt, ΩXXt=1.4ΩXt, detuned by half of the binding energy of the biexciton EB, δXa=EB/2=1meV from Xa and δXXa=0 from XXa. The input pulse area of Rabi oscillation is assumed to be Θ=∫Etdt=7π, with τp=4ps and t0=20ps.First, a single dot is studied as a reference case for studying coupled QDs. A sustained oscillation can be observed during the time evolution of all populations within the pulse duration range. Here, the pulse spectrum, having a width estimated by≈1/2τp, which is 12 times narrower than the detuning away from the one-exciton resonance, δX=1meV. In this case, we are well within the adiabatic range where the ground state 0 is effectively decoupled from the one-exciton state X and, hence, the transition to ρX is almost prohibited by energy conservation [23, 34].As a result, effective population transfer between the ground and biexciton states occurs via two-photon absorption, suggesting that the system can be reduced to a two-level scheme, in which two levels0 and XX are coupled by a Rabi frequency that is effectively off-resonant with an effective detuning [35].As shown in Figure2, we display the system dependency of ρX and ρXX on the pulse area for the population that initially resides in state 0 for the single dot. The Rabi oscillations between ground and biexciton states are significantly influenced by the pulse area, as opposed to conventional pulse area-dependent Rabi oscillations between ground and single exciton states. This periodic Rabi oscillation of the biexciton state has been reported experimentally [23]. The universal function that describes the Rabi oscillations in a two-level system is given by sin2Θ/2, depending only on the pulse area. Despite that the Rabi oscillation of the biexciton state has some qualitative analogies to the Rabi oscillations in a two-level system, as defined by the universal function, there are several important differences. In the case of two-photon Rabi oscillations, the biexciton occupation increases slowly at low excitation resulting in nonprecisely periodic oscillation. This lack of following the universal behavior of Rabi frequencies is predicted theoretically, where it is controlled by the effective pulse shape of nonlinear dependency [22, 31].Figure 2
The pulse area dependence ofρ0, ρX, and ρXX in a single quantum dot for the region of the adiabatic limit, calculated for the incident pulse with area Θ=7π, duration τp=4ps, and delay time td=20ps.It is interesting to investigate the dynamical property of two-photon Rabi oscillations when a single QD is coupled to another via Förster coupling. Figure3 displays the population dynamics of the ground state, the biexciton state at QDa, and the biexciton state at QDb, when the QDa is coupled to QDb via the dipolar interaction for different values of dipolar interaction. The relevant parameters are chosen from the measured values for the transition dipole moment μ of the exciton state to estimate the magnitude of VX−X between the two stacked dots. In InGaAs/GaAs, QD values for μ of ∼5–7 eÅ have been reported [36]. The nonradiative interdot dipolar coupling between exciton states using the relation VX−X=μ10μ01/4π∈0ϵrκ/R3 with values for μ10=μ01=6eÅ, ϵr=12 (for InGaAs/GaAs), interdot spacing R≈12nm, and the average value over all orientations of the exciton dipole moments in which κ2=2/3, we obtain an estimate of 0.048 meV for the Förster coupling energy VX−X. We consider the coupling strength VXX−XX≈1.4VX−X [37]. Other parameters are the same, as shown in Figure 2.Figure 3
The population dynamics of the ground state at QDa, the exciton biexciton state at QDa, and the exciton biexciton state at QDb, for four different values of VXX−XX; other parameters are shown in Figure 2.As shown in Figure3(a) when the QDa is connected to QDb via the dipolar coupling, the population of the biexciton state at QDb progressively grows as fewer plateaus during the pulse action. This number of plateaus is pronouncedly equal to the number of damping Rabi cycles at both the ground and biexciton states of QDa. After the pulse has passed, the Förster contribution to the Hamiltonian of the QD system produces dynamical oscillations between the biexciton states XXa and XXb, which coherently oscillate with time approximately equal to τXX−XX=h/2VXX−XX. As shown in Figure 3, a fraction of the optical excitation gets blockaded at the exciton ground state ρ0a, and the occupation of biexciton states XXa and the dipolar energy transfer to the state XXb are suppressed by increasing the strength of dipole–dipole interactions between these states. The fraction of excitation blockade at the exciton ground state ρ0a of dot a as a function of time depends on the relative strength of the dipolar interactions VXX−XX and excitation laser bandwidth. Here, the pulse spectrum, having a width estimated by Υ≈ℏ/2τp=0.082meV. The strength of excitation blockade can be classified according to the ratio between dipolar interaction VXX−XX and two times the bandwidth of the excitation laser 2Υ. It is possible to control the dynamics of the population of the system by adjusting the ratio between the timescale of the dipolar coupling and the timescale of the incident pulse VXX−XX/2Υ. According to this ratio, there are two significant limits for the time evolution of the considered states. The first one is, when VXX−XX≪2Υ (Figure 3(a)) there is no blockade effect, and the population of the biexciton state XXa decreases and shows oscillation that exponential decay during the pulse action due to the dipolar transfer channel to the state XXb. After the laser pulse has been passed, the transferred population oscillates back and forth between two biexciton states XXa and XXb. The second limit is when VXX−XX≫2Υ; in this case, a full blockade takes place, at which the off-resonant two-photon process in QDa will suffer from large decoherence due to the effect of the strong dipolar interaction and block the trigger of the system interaction. As a consequence, the population becomes trapped in the ground state of QDa (see Figure 3(d)). If VXX−XX and 2Υ are comparable, the dynamics of the system behave more complexly, being sensitive to the exact value of VXX−XX (Figures 3(b) and 3(c)).In addition to time domain dynamics, we also consider the population of the biexciton states at QDa and QDb as a function of pulse area Θ for constant pulse duration and different values of VXX−XX. The results shown in Figure 4 were obtained for four different values of the dipolar interaction VXX−XX=0.1,0.16,0.2,and0.66meV. The occupation probability of the biexciton state ρXXb at QDb shows an explicit Rabi oscillatory behavior between 0 and 1/2, with a period of about 3π/2 in the pulse area similar to the case of the isolated dot, as shown in Figure 2.Figure 4
Two-photon Rabi oscillations of (a) the biexciton state in QDaρXXa and (b) the biexciton state in QDaρXXb as a function of the pulse area for three different timescale ratio of τXX−XX/τp.
(a)(b)The oscillation amplitudes of the coupled QDs decrease gradually as the dipolar interaction strength increases in comparison with the isolated QDs.To present a comprehensive overview of the excitation behavior among the QDs states, Figure5 displays the contour plots of the averaged occupation probabilities of biexciton state at QDb, as a function of the dipolar interaction and the pulse area. The occupation probability of the biexciton state in QDb displays Rabi oscillation of a period 3π/2. The amplitude of Rabi cycles decreases with increasing VXX−XX. As a result, the pulse areas Θ at which the Rabi oscillations attain their successive maxima grow within the first timescale limit.Figure 5
The population dynamics average of the biexciton stateρXXb as a function of pulse area Θ and the dipolar interaction VXX−XX with the fixed pulse area. The upper panel represents the result for VXX−XX=0.06meV, and the right panel displays the result for Θ=7π.The upper panel of Figure5 represents the result for VXX−XX=0.06meV, where Rabi oscillations are clearly shown. The right panel of Figure 5 shows the result for Θ=7π, where the occupation average reaches a maximum at a particular value of dipolar interaction strength VXX−XX=0.06meV. We can attain this maximum value by varying the pulse area of the excitation energy.About 40% of the biexciton population is adiabatically pumped into stateXXa from state 0a by the off-resonant two-photon technique and then is transferred into state XXb via the dipolar interaction VXX−XX. Accordingly, the maximal coherence is generated between state 0a and state XXb. It is worth noting that this transferred population is significantly reduced as we increase the dipolar coupling strength. This reduction is due to the large induced detuning from the resonance frequency of the two-photon excitation with the biexciton state XXa. The energy of the biexciton state XXa is shifted out of resonance with the excitation laser. The dipolar interactions effectively decouple the initial state 0a from any multiple excitation eigenstates, blocking the excitation at the ground state. More specifically, excitation blockade is expected to occur at interdot distances for which the dipolar interaction VXX−XX exceeds the excitation bandwidth 2Υ. In close analogy to the pair interaction of the atomic system, we specify a blockade sphere surrounding the dots with a radius given by Rb=C/2γ3, where C=μ10μ01/4π∈0ϵr2κ. This radius is defined as the distance between two dots at which the dipolar interaction energy VXX−XX equals double the excitation bandwidth 2γ. In our system Rb = 9.6 nm, the excitation is coherently shared among dots when the interdot distance is significantly larger than the blockade radius. However, when the interdot distance is less than the blockade radius, the excitation is blockaded at the exciton ground state of dot a. Our findings provide promising possibilities for real-time control of interactions in QD systems, particularly the ability to turn on and off dipole blockade on nanosecond timescales. It is important to note that the present analysis ignores population transfer dynamics associated with phonon coupling, and future work will address the phonon coupling effect.


## 4. Conclusion

In summary, we have proposed and demonstrated a new method to control the population transfer between two coupled QDs via dipolar interactions with two-photon transitions driven by pulses detuned from the exciton state and in resonance with the biexciton state. We have found that the dipolar interaction between the two dots plays a prominent role during the time evolution of quantum systems. This dipole–dipole coupling has a strong effect on the population transfer dynamics of coupled QDs compared to the dynamics of the single QD. It is only upon the condition that the dipolar interaction timescale is sufficiently long relative to the duration of the excitation pulse that the population transfer between the two dots acquires real significance. In this condition, the excitation is transferred from the adiabatically populated biexciton state at QDa to the biexciton state at QDb, resulting in coherence oscillation between the two biexciton levels.On the other hand, the adiabaticity deteriorates when the timescale of the dipolar interaction is much shorter than the pulse duration of the excitation laser. Consequently, the transfer efficiency ultimately vanishes for strong dipolar coupling. This behavior contrasts with our expectation of enhancing the coherence transfer between the two dots. Increasing the dipolar coupling strength between the two dots trapped the populations at their initial coherently distributing in the ground level of QDa and resulted in what we have called “Förster blockade.”



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*Source: 4662648-2023-02-12.xml*