# Quantum Beat of Excitons in the Prolate Ellipsoidal Quantum Dots

Authors: Le Thi Ngoc Bao; Duong Dinh Phuoc; Le Thi Dieu Hien; Dinh Nhu Thao
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

In this paper, renormalized wavefunction method was applied to study quantum beats of excitons in the InGaAs/InAlAs prolate ellipsoidal quantum dots (QDs). The obtained results show that, without the pump laser, the exciton absorption intensity is just a smooth curve. In contrast, when the system is illuminated by a strong pump laser resonating with two exciton levels, the oscillation behavior of exciton absorption intensity, which is known as quantum beats of excitons, is observed. That result can be interpreted as an indirect consequence of the Pauli exclusion principle leading to a splitting of the electron levels, which forms two close exciton levels, and if two excitons are excited coherently, the interference of these two excitons will finally form a quantum beat. The study also shows that the geometry shape of the QDs has strong influence on the properties of quantum beats. Changing the period of quantum beats in particular or optical properties in general in QDs thus becomes more flexible through changing their geometry shapes. This is one interested advantage of the ellipsoidal QDs and is expected to increase their applicability more than the spherical QDs.


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## Body

## 1. Introduction

Presently, low-dimensional semiconductors attract the attention of several researchers, thanks to their high applicability. In low-dimensional semiconductor systems, particles are confined in one dimension-quantum wells, two dimensions-quantum wires, and three dimensions-quantum dots (QDs). It is this quantum confinement that makes low-dimensional semiconductor systems, especially QDs, offer many effects with considerable potential in the manufacture of new optical devices [1]; in electronic and optoelectronic applications [2, 3]; for quantum-functional and memory devices [4, 5]; and many other research fields such as quantum computing, photovoltaics, infrared photodetectors, medical imaging, and biosensors [6–12].Recent studies on coupled optical properties in QDs with simple shapes, such as cubic, cylindrical, and spherical QDs have been carried out [13–16]. These works show that optical properties of QDs depend to a great extent on external fields and the size of QDs. Especially, the shape of QDs also makes a notable difference to their optical properties [17, 18]. This suggests that specially shaped QDs like ellipsoidal ones can possess interestingly different optical properties. In ellipsoidal QDs, the quantized energy levels of particles are highly dependent on structural parameters [19–23]. Therefore, the optical properties in these quantum dot structures are said to be easily modifiable by these structural parameters.Thanks to its high applicability, especially in manufacture of quantum computer, quantum beat in low-dimensional structures has attracted much attention. Using ultrashort laser pulses with various experiments [24–30], scientists have observed quantum beats of excitons in different semiconductor structures. Theoretically, scientists have applied many methods to study the quantum beats of excitons in quantum structures [25, 31–36], of which the most convenient is the renormalized wavefunction method [32, 35, 36].In this paper, we used the renormalized wavefunction method to study the existence of quantum beats of excitons in In0.53Ga0.47As/In0.52Al0.48As prolate ellipsoidal quantum dots. We chose these heterostructures because they have the large conduction-band discontinuity between In0.53Ga0.47As/In0.52Al0.48As layers that is ΔEc=500 meV [37]. This can be approximated as the infinite confinement potential for electrons in these quantum structures, which is consistent with our hypothesis. Besides, these heterostructures have many applications [37], as well as create modern infrared devices [38, 39]. This article focuses on investigating the dependence of the exciton absorption intensity on the external fields, sizes, and shapes of quantum dot. The article includes the following main sections: Section 2 is about the model and theory, Section 3 provides the main results and discussion, and Section 4 presents the conclusions.


## 2. Model and Theory

### 2.1. Wavefunctions and Energy Levels of Exciton

#### 2.1.1. The Case without the Pump Laser

In this case, we investigate quantum beats of excitons in prolate ellipsoidal QDs. We use a three-level diagram of exciton, including one ground levelEgroundex and two excited ones of exciton E100ex and E110ex (Figure 1). These exciton levels are originated from interband transitions between the lowest level of hole E100h and the two lowest ones of electron E100e and E110e in QDs (see Figure 10(a) in Appendix A). For simplicity, we assume the prolate ellipsoidal QDs lied in an infinite potential (see Appendix A).Figure 1
A three-level energy diagram and exciton transitions: (a) without the pump laser, the system contains three levels of exciton: the ground levelEgroundex and two excited ones E100ex, E110ex. The probe laser ℏωt only identifies the transition for the pair of levels Egroundex,E100ex (marked by a dotted arrow); (b) under the effect of the resonant strong pump laser coupling to the initial excited levels of exciton, E100ex is split into E100ex− and E100ex+; E110ex is separated to E110ex− and E110ex+. The probe laser ℏωt identifies two transitions for two pairs of levels Egroundex,E100ex− and Egroundex,E100ex+, obeying the selection rules for interband transitions (marked by thin dashed arrows).First, we find the stationary wavefunctions of the exciton and the corresponding energy levels in the absence of the pump laser. The wavefunctions of the exciton in the stationary states, in a strong confinement regime, are the combination of the one-particle total wavefunctions of the electron and the hole (see Equation (A.15)).
(1)Λ100exr⟶=Λ100er⟶.Λ100hr⟶,Λ110exr⟶=Λ110er⟶.Λ100hr⟶.The corresponding exciton energy levels are determined by the sum of the quantized levels of electronsEnlme and holes Enlmh (see Equation (A.16) and Equation (A.17) in Appendix A), minus the exciton binding energy Ebinding(2)E100ex=E100e+E100h−Ebinding,E110ex=E110e+E100h−Ebinding,where Ebinding is usually considerable smaller than the energy levels of electrons and holes. To investigate the characteristics of quantum beats with time, we utilize the time-dependent stationary wavefunctions given in the form
(3)Λ100exr⟶,t=Λ100exr⟶e−i/ℏE100ext,Λ110exr⟶,t=Λ110exr⟶e−i/ℏE110ext.


#### 2.1.2. The Case with the Pump Laser

In this case, we examine a system subjected to a pump laser resonating with two exciton excited levels in the initial stationary statesE100ex and E110ex. Under the effect of the pump laser, the excitons are now no longer in the initial stationary states but in the nonstationary state which is described by the product of the wavefunctions of the electron in the mixed state Λmixer⟶,t (Equation (A.26)) and the initial wavefunction of hole Λ100hr⟶,t (see Equation (A.18))
(4)Λmixexr⟶,t=Λmixer⟶,t.Λ100hr⟶,t,The wavefunction in Equation (4) can be written in the explicit form
(5)Λmixexr⟶,t=12ΩRα1eiα2t+α2e−iα1te−i/ℏE100extΛ100exr⟶−V212ΩRℏeiα1t−e−iα2te−i/ℏE110extΛ110exr⟶,or it can be rewritten as
(6)Λmixexr⟶,t=12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+tΛ100exr⟶−V212ℏΩRe−i/ℏE110ex−t−e−i/ℏE110ex+tΛ110exr⟶,in which
(7)E100ex+=E100ex+ℏα1,E100ex−=E100ex−ℏα2,and
(8)E110ex+=E110ex+ℏα2,E110ex−=E110ex−ℏα1,and Λ100exr⟶ and Λ110exr⟶ are the stationary exciton wavefunctions when the pump laser does not turn on (Equation (1)). Equation (8) can be written as
(9)E110ex+=E100ex++ℏωp,E110ex−=E100ex−+ℏωp.From Equations (7) and (8), we have
(10)E100ex+−E100ex−=2ℏΩR,E110ex+−E110ex−=2ℏΩR.Under the effect of the pump laser, the two initial excited levels of exciton split into four new energy levels in which two levelsE100ex− and E100ex+ are separated from level E100ex, and two levels E110ex− and E110ex+ are separated from level E110ex (see Figure 1(b)). We see that the energy difference between two splitting levels E100ex− and E100ex+ or between two splitting ones E110ex− and E110ex+ is equal to 2ℏΩR (Equation (10)), which is much smaller than the photon energy of the pump laser ℏωp. This photon energy is also the energy distance between two pairs of levels E100ex−,E100ex+ and E110ex−,E110ex+ (Equation (9)). The Rabi frequency ΩR is proportional to the detuning of the pump laser as well as the transition matrix element for the intersubband transition and has the following form
(11)ΩR=Δω22+V212ℏ2,where ℏΔω is the detuning of the pump laser with two levels of electron E100e and E110e, and V21 is the matrix element for the intersubband transition between electron levels (see Equation (A.25) in Appendix A).


### 2.2. Absorption Intensity of Excitons

#### 2.2.1. The Case without the Pump Laser

The existence of the quantum beats of excitons is determined through the oscillatory behavior of the absorption intensity of excitons. On the other hand, the absorption intensity is a function of the transition matrix element among the levels of exciton. Therefore, we need to compute the dipole transition matrix element among the states of exciton, first among levels of exciton in the stationary states (Figure1(a)). Since there are two excited states of exciton in the system, when the system is illuminated by a probe laser, we would expect to obtain two transitions of exciton from the ground state 0 corresponding to level Egroundex to two initial excited states of exciton corresponding to two exciton levels E100ex and E110ex. However, according to the selection rule for the interband transition in QDs, only the exciton transition from the ground level Egroundex to the lowest excited level of exciton E100ex exists as described by the dotted arrow in Figure 1(a). Thus, in the absence of the effect of the pump laser, the permitted transition matrix element between levels Egroundex and E100ex under the action of a probe laser has the form
(12)T100ex=Λ100exr⟶,tH^int0=−eAte−iωttm0iωtΛ100exr⟶,tn⟶.p⟶^0,where At and ωt are, respectively, the amplitude and the frequency of the probe laser. Replace Equations (3) and (1) into Equation (12); we have
(13)T100ex=−eAtpcvm0iωtei/ℏE100e+E100h−ℏωttΨ100er⟶Ψ100hr⟶0,,where pcv is the polarization matrix element between conduction and valence bands
(14)pcv=ucn⟶.p⟶^uv.From that, we find the expression for the exciton absorption intensity when the pump laser does not turn on as(15)I100ext∝T100ex2=eAtpcvm0ωt2.Because the exciton lifetime on the excited levels is finite, the excited states of exciton will fade over time. To account for the decay ofI100ext, we phenomenologically provide a decay parameter γ=1/T1 in Equation (15). From that, we obtain the expression of the absorption intensity of excitons in the absence of the pump laser as follows
(16)I100ext∝eAtpcvm0ωt2exp−γt,where T1 is the lifetime of exciton on the energy level E100ex.


#### 2.2.2. The Case with the Pump Laser

From Section2.1.2, we see that exciton will stay in the nonstationary state Λmixexr⟶,t when QDs are irradiated by a strong pump laser resonant with two initial exciton levels E100ex and E110ex. Now, to find the absorption intensity of excitons, we need to calculate the dipole transition matrix element between the ground state 0 and the nonstationary exciton one Λmixexr⟶,t. The matrix element in this case has the form
(17)Tmixex=Λmixexr⟶,tH^int0=−eAte−iωttm0iωtΛmixexr⟶,tn⟶.p⟶^0.Combining Equations (6) and (17), we get the following matrix element
(18)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100exr⟶n⟶.p⟶^0−V21∗2ΩRℏe−i/ℏE110ex−t−e−i/ℏE110ex+t∗×Λ110exr⟶n⟶.p⟶^0.Because of the selection rule for the interband transition in QDs, there are only the dipole transitions from the ground state to the pair of the lowest splitting levels of excitonE100ex−,E100ex+ that are split from the initial exciton level E100ex. So, we have
(19)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100exr⟶n⟶.p⟶^0,or
(20)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100er⟶Λ100hr⟶n⟶.p⟶^0.Equation (20) clearly shows that, after the effect of a strong pump laser resonating with two exciton excited levels in stationary state E100ex and E110ex, then in the system, there are two optical transitions from the ground state Egroundex to two lowest splitting level of exciton E100ex− and E100ex+ (illustrated by two thin dashed arrows in Figure 1(b)). Combining Equation (A.15) into Equation (20), we get the matrix element for the dipole transition between the ground level 0 and the nonstationary exciton one Λmixexr⟶,t as follows:
(21)Tmixex=−eAte−iωttpcvm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Ψ100er⟶Ψ100hr⟶0,or
(22)Tmixex=−eAte−iωttpcvm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗.Next, we investigate the time-resolved intensity of absorption under the effect of the resonant pump laserImixext. From Equation (10), we see that the probe laser needs a spectral width larger than 2ℏΩR, which is energy separation between two levels E100ex− and E100ex+, in order to excite coherently two excitons in the pair E100ex−,E100ex+. In order to fully observe at least a quantum beat oscillation, the period of the quantum beat needs to be less than or equal to coherence time T2, the time it takes for two excitons in the pair E100ex−,E100ex+ to oscillate in phase. Of course, the coherence time T2 is always less than or equal to the lifetime T1 of exciton in the state Λ100exr⟶,t (Equation (3)). Therefore, at any given time t<T2, the absorption intensity of excitons under the effect of the pump laser has the following form.
(23)Imixext∝Tmixex2=eAtpcvm0ωt212ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗2,or(24)Imixext∝eAtpcvm0ωt2α12ΩR2+α22ΩR2+2α12ΩRα22ΩRcosE100ex+−E100ex−ℏt.We put(25)β1=α12ΩR,β2=α22ΩR,and combined with Equation (10), we can rewrite the expression for the absorption intensity of excitons as
(26)Imixext∝eAtpcvm0ωt2β12+β22+2β1β2cos2ΩRt.As mentioned in Section2.2.1, in fact, the exciton lifetime on the excited states is finite, so the oscillation in Equation (6) decays with time. To account for damping in Imixext, we add phenomenologically the damped factors γ=1/T1 and τ=1/T2 in Equation (26). From this, we obtain the final expression of the absorption intensity of excitons in the presence of the resonant pump laser as follows
(27)Imixext∝eAtpcvm0ωt2β12+β22exp−γt+2β1β2exp−τtcos2ΩRt,where T2 is the coherent time of the state described in Equation (6).Equation (27) shows that in the case QDs are illuminated by a probe laser in the presence of a resonant strong pump laser, the absorption intensity has the form of a damped periodic oscillation that owns a frequency of twice the Rabi frequency of the electron 2ΩR. This oscillation indicates the existence of the quantum beats of excitons in the quantum dot structure, which we will examine and explain in detail in the next section.As we know, when there are two oscillations of similar frequencies in the system, they will interfere with each other to form a superposition wave of the frequency that is the average of two initial frequencies (i.e., exciton onesωex+ and ωex−) and its amplitude oscillates with the frequency equal to one half of the difference of two initial frequencies. Thence, the beat frequency is equal to the difference of two initial frequencies. Hence, from Equation (24) or Equation (27), we can formally deduce the expression of the total absorption intensity that oscillates with the effective frequency that equal to the sum of the two initial frequencies and have the following form
(28)It∝eAtpcvm0ωt2β12+β22exp−γt+2β1β2exp−τtcosE100ex+−E100ex−ℏtcosE100ex++E100ex−ℏt.We can rewrite Equation (28) as follows
(29)It∝Imixext⋅cosE100ex++E100ex−ℏt.


## 2.1. Wavefunctions and Energy Levels of Exciton

### 2.1.1. The Case without the Pump Laser

In this case, we investigate quantum beats of excitons in prolate ellipsoidal QDs. We use a three-level diagram of exciton, including one ground levelEgroundex and two excited ones of exciton E100ex and E110ex (Figure 1). These exciton levels are originated from interband transitions between the lowest level of hole E100h and the two lowest ones of electron E100e and E110e in QDs (see Figure 10(a) in Appendix A). For simplicity, we assume the prolate ellipsoidal QDs lied in an infinite potential (see Appendix A).Figure 1
A three-level energy diagram and exciton transitions: (a) without the pump laser, the system contains three levels of exciton: the ground levelEgroundex and two excited ones E100ex, E110ex. The probe laser ℏωt only identifies the transition for the pair of levels Egroundex,E100ex (marked by a dotted arrow); (b) under the effect of the resonant strong pump laser coupling to the initial excited levels of exciton, E100ex is split into E100ex− and E100ex+; E110ex is separated to E110ex− and E110ex+. The probe laser ℏωt identifies two transitions for two pairs of levels Egroundex,E100ex− and Egroundex,E100ex+, obeying the selection rules for interband transitions (marked by thin dashed arrows).First, we find the stationary wavefunctions of the exciton and the corresponding energy levels in the absence of the pump laser. The wavefunctions of the exciton in the stationary states, in a strong confinement regime, are the combination of the one-particle total wavefunctions of the electron and the hole (see Equation (A.15)).
(1)Λ100exr⟶=Λ100er⟶.Λ100hr⟶,Λ110exr⟶=Λ110er⟶.Λ100hr⟶.The corresponding exciton energy levels are determined by the sum of the quantized levels of electronsEnlme and holes Enlmh (see Equation (A.16) and Equation (A.17) in Appendix A), minus the exciton binding energy Ebinding(2)E100ex=E100e+E100h−Ebinding,E110ex=E110e+E100h−Ebinding,where Ebinding is usually considerable smaller than the energy levels of electrons and holes. To investigate the characteristics of quantum beats with time, we utilize the time-dependent stationary wavefunctions given in the form
(3)Λ100exr⟶,t=Λ100exr⟶e−i/ℏE100ext,Λ110exr⟶,t=Λ110exr⟶e−i/ℏE110ext.


### 2.1.2. The Case with the Pump Laser

In this case, we examine a system subjected to a pump laser resonating with two exciton excited levels in the initial stationary statesE100ex and E110ex. Under the effect of the pump laser, the excitons are now no longer in the initial stationary states but in the nonstationary state which is described by the product of the wavefunctions of the electron in the mixed state Λmixer⟶,t (Equation (A.26)) and the initial wavefunction of hole Λ100hr⟶,t (see Equation (A.18))
(4)Λmixexr⟶,t=Λmixer⟶,t.Λ100hr⟶,t,The wavefunction in Equation (4) can be written in the explicit form
(5)Λmixexr⟶,t=12ΩRα1eiα2t+α2e−iα1te−i/ℏE100extΛ100exr⟶−V212ΩRℏeiα1t−e−iα2te−i/ℏE110extΛ110exr⟶,or it can be rewritten as
(6)Λmixexr⟶,t=12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+tΛ100exr⟶−V212ℏΩRe−i/ℏE110ex−t−e−i/ℏE110ex+tΛ110exr⟶,in which
(7)E100ex+=E100ex+ℏα1,E100ex−=E100ex−ℏα2,and
(8)E110ex+=E110ex+ℏα2,E110ex−=E110ex−ℏα1,and Λ100exr⟶ and Λ110exr⟶ are the stationary exciton wavefunctions when the pump laser does not turn on (Equation (1)). Equation (8) can be written as
(9)E110ex+=E100ex++ℏωp,E110ex−=E100ex−+ℏωp.From Equations (7) and (8), we have
(10)E100ex+−E100ex−=2ℏΩR,E110ex+−E110ex−=2ℏΩR.Under the effect of the pump laser, the two initial excited levels of exciton split into four new energy levels in which two levelsE100ex− and E100ex+ are separated from level E100ex, and two levels E110ex− and E110ex+ are separated from level E110ex (see Figure 1(b)). We see that the energy difference between two splitting levels E100ex− and E100ex+ or between two splitting ones E110ex− and E110ex+ is equal to 2ℏΩR (Equation (10)), which is much smaller than the photon energy of the pump laser ℏωp. This photon energy is also the energy distance between two pairs of levels E100ex−,E100ex+ and E110ex−,E110ex+ (Equation (9)). The Rabi frequency ΩR is proportional to the detuning of the pump laser as well as the transition matrix element for the intersubband transition and has the following form
(11)ΩR=Δω22+V212ℏ2,where ℏΔω is the detuning of the pump laser with two levels of electron E100e and E110e, and V21 is the matrix element for the intersubband transition between electron levels (see Equation (A.25) in Appendix A).


## 2.1.1. The Case without the Pump Laser

In this case, we investigate quantum beats of excitons in prolate ellipsoidal QDs. We use a three-level diagram of exciton, including one ground levelEgroundex and two excited ones of exciton E100ex and E110ex (Figure 1). These exciton levels are originated from interband transitions between the lowest level of hole E100h and the two lowest ones of electron E100e and E110e in QDs (see Figure 10(a) in Appendix A). For simplicity, we assume the prolate ellipsoidal QDs lied in an infinite potential (see Appendix A).Figure 1
A three-level energy diagram and exciton transitions: (a) without the pump laser, the system contains three levels of exciton: the ground levelEgroundex and two excited ones E100ex, E110ex. The probe laser ℏωt only identifies the transition for the pair of levels Egroundex,E100ex (marked by a dotted arrow); (b) under the effect of the resonant strong pump laser coupling to the initial excited levels of exciton, E100ex is split into E100ex− and E100ex+; E110ex is separated to E110ex− and E110ex+. The probe laser ℏωt identifies two transitions for two pairs of levels Egroundex,E100ex− and Egroundex,E100ex+, obeying the selection rules for interband transitions (marked by thin dashed arrows).First, we find the stationary wavefunctions of the exciton and the corresponding energy levels in the absence of the pump laser. The wavefunctions of the exciton in the stationary states, in a strong confinement regime, are the combination of the one-particle total wavefunctions of the electron and the hole (see Equation (A.15)).
(1)Λ100exr⟶=Λ100er⟶.Λ100hr⟶,Λ110exr⟶=Λ110er⟶.Λ100hr⟶.The corresponding exciton energy levels are determined by the sum of the quantized levels of electronsEnlme and holes Enlmh (see Equation (A.16) and Equation (A.17) in Appendix A), minus the exciton binding energy Ebinding(2)E100ex=E100e+E100h−Ebinding,E110ex=E110e+E100h−Ebinding,where Ebinding is usually considerable smaller than the energy levels of electrons and holes. To investigate the characteristics of quantum beats with time, we utilize the time-dependent stationary wavefunctions given in the form
(3)Λ100exr⟶,t=Λ100exr⟶e−i/ℏE100ext,Λ110exr⟶,t=Λ110exr⟶e−i/ℏE110ext.


## 2.1.2. The Case with the Pump Laser

In this case, we examine a system subjected to a pump laser resonating with two exciton excited levels in the initial stationary statesE100ex and E110ex. Under the effect of the pump laser, the excitons are now no longer in the initial stationary states but in the nonstationary state which is described by the product of the wavefunctions of the electron in the mixed state Λmixer⟶,t (Equation (A.26)) and the initial wavefunction of hole Λ100hr⟶,t (see Equation (A.18))
(4)Λmixexr⟶,t=Λmixer⟶,t.Λ100hr⟶,t,The wavefunction in Equation (4) can be written in the explicit form
(5)Λmixexr⟶,t=12ΩRα1eiα2t+α2e−iα1te−i/ℏE100extΛ100exr⟶−V212ΩRℏeiα1t−e−iα2te−i/ℏE110extΛ110exr⟶,or it can be rewritten as
(6)Λmixexr⟶,t=12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+tΛ100exr⟶−V212ℏΩRe−i/ℏE110ex−t−e−i/ℏE110ex+tΛ110exr⟶,in which
(7)E100ex+=E100ex+ℏα1,E100ex−=E100ex−ℏα2,and
(8)E110ex+=E110ex+ℏα2,E110ex−=E110ex−ℏα1,and Λ100exr⟶ and Λ110exr⟶ are the stationary exciton wavefunctions when the pump laser does not turn on (Equation (1)). Equation (8) can be written as
(9)E110ex+=E100ex++ℏωp,E110ex−=E100ex−+ℏωp.From Equations (7) and (8), we have
(10)E100ex+−E100ex−=2ℏΩR,E110ex+−E110ex−=2ℏΩR.Under the effect of the pump laser, the two initial excited levels of exciton split into four new energy levels in which two levelsE100ex− and E100ex+ are separated from level E100ex, and two levels E110ex− and E110ex+ are separated from level E110ex (see Figure 1(b)). We see that the energy difference between two splitting levels E100ex− and E100ex+ or between two splitting ones E110ex− and E110ex+ is equal to 2ℏΩR (Equation (10)), which is much smaller than the photon energy of the pump laser ℏωp. This photon energy is also the energy distance between two pairs of levels E100ex−,E100ex+ and E110ex−,E110ex+ (Equation (9)). The Rabi frequency ΩR is proportional to the detuning of the pump laser as well as the transition matrix element for the intersubband transition and has the following form
(11)ΩR=Δω22+V212ℏ2,where ℏΔω is the detuning of the pump laser with two levels of electron E100e and E110e, and V21 is the matrix element for the intersubband transition between electron levels (see Equation (A.25) in Appendix A).


## 2.2. Absorption Intensity of Excitons

### 2.2.1. The Case without the Pump Laser

The existence of the quantum beats of excitons is determined through the oscillatory behavior of the absorption intensity of excitons. On the other hand, the absorption intensity is a function of the transition matrix element among the levels of exciton. Therefore, we need to compute the dipole transition matrix element among the states of exciton, first among levels of exciton in the stationary states (Figure1(a)). Since there are two excited states of exciton in the system, when the system is illuminated by a probe laser, we would expect to obtain two transitions of exciton from the ground state 0 corresponding to level Egroundex to two initial excited states of exciton corresponding to two exciton levels E100ex and E110ex. However, according to the selection rule for the interband transition in QDs, only the exciton transition from the ground level Egroundex to the lowest excited level of exciton E100ex exists as described by the dotted arrow in Figure 1(a). Thus, in the absence of the effect of the pump laser, the permitted transition matrix element between levels Egroundex and E100ex under the action of a probe laser has the form
(12)T100ex=Λ100exr⟶,tH^int0=−eAte−iωttm0iωtΛ100exr⟶,tn⟶.p⟶^0,where At and ωt are, respectively, the amplitude and the frequency of the probe laser. Replace Equations (3) and (1) into Equation (12); we have
(13)T100ex=−eAtpcvm0iωtei/ℏE100e+E100h−ℏωttΨ100er⟶Ψ100hr⟶0,,where pcv is the polarization matrix element between conduction and valence bands
(14)pcv=ucn⟶.p⟶^uv.From that, we find the expression for the exciton absorption intensity when the pump laser does not turn on as(15)I100ext∝T100ex2=eAtpcvm0ωt2.Because the exciton lifetime on the excited levels is finite, the excited states of exciton will fade over time. To account for the decay ofI100ext, we phenomenologically provide a decay parameter γ=1/T1 in Equation (15). From that, we obtain the expression of the absorption intensity of excitons in the absence of the pump laser as follows
(16)I100ext∝eAtpcvm0ωt2exp−γt,where T1 is the lifetime of exciton on the energy level E100ex.


### 2.2.2. The Case with the Pump Laser

From Section2.1.2, we see that exciton will stay in the nonstationary state Λmixexr⟶,t when QDs are irradiated by a strong pump laser resonant with two initial exciton levels E100ex and E110ex. Now, to find the absorption intensity of excitons, we need to calculate the dipole transition matrix element between the ground state 0 and the nonstationary exciton one Λmixexr⟶,t. The matrix element in this case has the form
(17)Tmixex=Λmixexr⟶,tH^int0=−eAte−iωttm0iωtΛmixexr⟶,tn⟶.p⟶^0.Combining Equations (6) and (17), we get the following matrix element
(18)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100exr⟶n⟶.p⟶^0−V21∗2ΩRℏe−i/ℏE110ex−t−e−i/ℏE110ex+t∗×Λ110exr⟶n⟶.p⟶^0.Because of the selection rule for the interband transition in QDs, there are only the dipole transitions from the ground state to the pair of the lowest splitting levels of excitonE100ex−,E100ex+ that are split from the initial exciton level E100ex. So, we have
(19)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100exr⟶n⟶.p⟶^0,or
(20)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100er⟶Λ100hr⟶n⟶.p⟶^0.Equation (20) clearly shows that, after the effect of a strong pump laser resonating with two exciton excited levels in stationary state E100ex and E110ex, then in the system, there are two optical transitions from the ground state Egroundex to two lowest splitting level of exciton E100ex− and E100ex+ (illustrated by two thin dashed arrows in Figure 1(b)). Combining Equation (A.15) into Equation (20), we get the matrix element for the dipole transition between the ground level 0 and the nonstationary exciton one Λmixexr⟶,t as follows:
(21)Tmixex=−eAte−iωttpcvm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Ψ100er⟶Ψ100hr⟶0,or
(22)Tmixex=−eAte−iωttpcvm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗.Next, we investigate the time-resolved intensity of absorption under the effect of the resonant pump laserImixext. From Equation (10), we see that the probe laser needs a spectral width larger than 2ℏΩR, which is energy separation between two levels E100ex− and E100ex+, in order to excite coherently two excitons in the pair E100ex−,E100ex+. In order to fully observe at least a quantum beat oscillation, the period of the quantum beat needs to be less than or equal to coherence time T2, the time it takes for two excitons in the pair E100ex−,E100ex+ to oscillate in phase. Of course, the coherence time T2 is always less than or equal to the lifetime T1 of exciton in the state Λ100exr⟶,t (Equation (3)). Therefore, at any given time t<T2, the absorption intensity of excitons under the effect of the pump laser has the following form.
(23)Imixext∝Tmixex2=eAtpcvm0ωt212ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗2,or(24)Imixext∝eAtpcvm0ωt2α12ΩR2+α22ΩR2+2α12ΩRα22ΩRcosE100ex+−E100ex−ℏt.We put(25)β1=α12ΩR,β2=α22ΩR,and combined with Equation (10), we can rewrite the expression for the absorption intensity of excitons as
(26)Imixext∝eAtpcvm0ωt2β12+β22+2β1β2cos2ΩRt.As mentioned in Section2.2.1, in fact, the exciton lifetime on the excited states is finite, so the oscillation in Equation (6) decays with time. To account for damping in Imixext, we add phenomenologically the damped factors γ=1/T1 and τ=1/T2 in Equation (26). From this, we obtain the final expression of the absorption intensity of excitons in the presence of the resonant pump laser as follows
(27)Imixext∝eAtpcvm0ωt2β12+β22exp−γt+2β1β2exp−τtcos2ΩRt,where T2 is the coherent time of the state described in Equation (6).Equation (27) shows that in the case QDs are illuminated by a probe laser in the presence of a resonant strong pump laser, the absorption intensity has the form of a damped periodic oscillation that owns a frequency of twice the Rabi frequency of the electron 2ΩR. This oscillation indicates the existence of the quantum beats of excitons in the quantum dot structure, which we will examine and explain in detail in the next section.As we know, when there are two oscillations of similar frequencies in the system, they will interfere with each other to form a superposition wave of the frequency that is the average of two initial frequencies (i.e., exciton onesωex+ and ωex−) and its amplitude oscillates with the frequency equal to one half of the difference of two initial frequencies. Thence, the beat frequency is equal to the difference of two initial frequencies. Hence, from Equation (24) or Equation (27), we can formally deduce the expression of the total absorption intensity that oscillates with the effective frequency that equal to the sum of the two initial frequencies and have the following form
(28)It∝eAtpcvm0ωt2β12+β22exp−γt+2β1β2exp−τtcosE100ex+−E100ex−ℏtcosE100ex++E100ex−ℏt.We can rewrite Equation (28) as follows
(29)It∝Imixext⋅cosE100ex++E100ex−ℏt.


## 2.2.1. The Case without the Pump Laser

The existence of the quantum beats of excitons is determined through the oscillatory behavior of the absorption intensity of excitons. On the other hand, the absorption intensity is a function of the transition matrix element among the levels of exciton. Therefore, we need to compute the dipole transition matrix element among the states of exciton, first among levels of exciton in the stationary states (Figure1(a)). Since there are two excited states of exciton in the system, when the system is illuminated by a probe laser, we would expect to obtain two transitions of exciton from the ground state 0 corresponding to level Egroundex to two initial excited states of exciton corresponding to two exciton levels E100ex and E110ex. However, according to the selection rule for the interband transition in QDs, only the exciton transition from the ground level Egroundex to the lowest excited level of exciton E100ex exists as described by the dotted arrow in Figure 1(a). Thus, in the absence of the effect of the pump laser, the permitted transition matrix element between levels Egroundex and E100ex under the action of a probe laser has the form
(12)T100ex=Λ100exr⟶,tH^int0=−eAte−iωttm0iωtΛ100exr⟶,tn⟶.p⟶^0,where At and ωt are, respectively, the amplitude and the frequency of the probe laser. Replace Equations (3) and (1) into Equation (12); we have
(13)T100ex=−eAtpcvm0iωtei/ℏE100e+E100h−ℏωttΨ100er⟶Ψ100hr⟶0,,where pcv is the polarization matrix element between conduction and valence bands
(14)pcv=ucn⟶.p⟶^uv.From that, we find the expression for the exciton absorption intensity when the pump laser does not turn on as(15)I100ext∝T100ex2=eAtpcvm0ωt2.Because the exciton lifetime on the excited levels is finite, the excited states of exciton will fade over time. To account for the decay ofI100ext, we phenomenologically provide a decay parameter γ=1/T1 in Equation (15). From that, we obtain the expression of the absorption intensity of excitons in the absence of the pump laser as follows
(16)I100ext∝eAtpcvm0ωt2exp−γt,where T1 is the lifetime of exciton on the energy level E100ex.


## 2.2.2. The Case with the Pump Laser

From Section2.1.2, we see that exciton will stay in the nonstationary state Λmixexr⟶,t when QDs are irradiated by a strong pump laser resonant with two initial exciton levels E100ex and E110ex. Now, to find the absorption intensity of excitons, we need to calculate the dipole transition matrix element between the ground state 0 and the nonstationary exciton one Λmixexr⟶,t. The matrix element in this case has the form
(17)Tmixex=Λmixexr⟶,tH^int0=−eAte−iωttm0iωtΛmixexr⟶,tn⟶.p⟶^0.Combining Equations (6) and (17), we get the following matrix element
(18)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100exr⟶n⟶.p⟶^0−V21∗2ΩRℏe−i/ℏE110ex−t−e−i/ℏE110ex+t∗×Λ110exr⟶n⟶.p⟶^0.Because of the selection rule for the interband transition in QDs, there are only the dipole transitions from the ground state to the pair of the lowest splitting levels of excitonE100ex−,E100ex+ that are split from the initial exciton level E100ex. So, we have
(19)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100exr⟶n⟶.p⟶^0,or
(20)Tmixex=−eAte−iωttm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Λ100er⟶Λ100hr⟶n⟶.p⟶^0.Equation (20) clearly shows that, after the effect of a strong pump laser resonating with two exciton excited levels in stationary state E100ex and E110ex, then in the system, there are two optical transitions from the ground state Egroundex to two lowest splitting level of exciton E100ex− and E100ex+ (illustrated by two thin dashed arrows in Figure 1(b)). Combining Equation (A.15) into Equation (20), we get the matrix element for the dipole transition between the ground level 0 and the nonstationary exciton one Λmixexr⟶,t as follows:
(21)Tmixex=−eAte−iωttpcvm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗×Ψ100er⟶Ψ100hr⟶0,or
(22)Tmixex=−eAte−iωttpcvm0iωt12ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗.Next, we investigate the time-resolved intensity of absorption under the effect of the resonant pump laserImixext. From Equation (10), we see that the probe laser needs a spectral width larger than 2ℏΩR, which is energy separation between two levels E100ex− and E100ex+, in order to excite coherently two excitons in the pair E100ex−,E100ex+. In order to fully observe at least a quantum beat oscillation, the period of the quantum beat needs to be less than or equal to coherence time T2, the time it takes for two excitons in the pair E100ex−,E100ex+ to oscillate in phase. Of course, the coherence time T2 is always less than or equal to the lifetime T1 of exciton in the state Λ100exr⟶,t (Equation (3)). Therefore, at any given time t<T2, the absorption intensity of excitons under the effect of the pump laser has the following form.
(23)Imixext∝Tmixex2=eAtpcvm0ωt212ΩRα1e−i/ℏE100ex−t+α2e−i/ℏE100ex+t∗2,or(24)Imixext∝eAtpcvm0ωt2α12ΩR2+α22ΩR2+2α12ΩRα22ΩRcosE100ex+−E100ex−ℏt.We put(25)β1=α12ΩR,β2=α22ΩR,and combined with Equation (10), we can rewrite the expression for the absorption intensity of excitons as
(26)Imixext∝eAtpcvm0ωt2β12+β22+2β1β2cos2ΩRt.As mentioned in Section2.2.1, in fact, the exciton lifetime on the excited states is finite, so the oscillation in Equation (6) decays with time. To account for damping in Imixext, we add phenomenologically the damped factors γ=1/T1 and τ=1/T2 in Equation (26). From this, we obtain the final expression of the absorption intensity of excitons in the presence of the resonant pump laser as follows
(27)Imixext∝eAtpcvm0ωt2β12+β22exp−γt+2β1β2exp−τtcos2ΩRt,where T2 is the coherent time of the state described in Equation (6).Equation (27) shows that in the case QDs are illuminated by a probe laser in the presence of a resonant strong pump laser, the absorption intensity has the form of a damped periodic oscillation that owns a frequency of twice the Rabi frequency of the electron 2ΩR. This oscillation indicates the existence of the quantum beats of excitons in the quantum dot structure, which we will examine and explain in detail in the next section.As we know, when there are two oscillations of similar frequencies in the system, they will interfere with each other to form a superposition wave of the frequency that is the average of two initial frequencies (i.e., exciton onesωex+ and ωex−) and its amplitude oscillates with the frequency equal to one half of the difference of two initial frequencies. Thence, the beat frequency is equal to the difference of two initial frequencies. Hence, from Equation (24) or Equation (27), we can formally deduce the expression of the total absorption intensity that oscillates with the effective frequency that equal to the sum of the two initial frequencies and have the following form
(28)It∝eAtpcvm0ωt2β12+β22exp−γt+2β1β2exp−τtcosE100ex+−E100ex−ℏtcosE100ex++E100ex−ℏt.We can rewrite Equation (28) as follows
(29)It∝Imixext⋅cosE100ex++E100ex−ℏt.


## 3. Results and Discussion

To further study and demonstrate the obtained results, in this part, we perform numerical calculations for the time-resolved intensity of absorption in In0.53Ga0.47As/In0.52Al0.48As prolate ellipsoidal QDs. We utilize those parameters for the calculation: the effective mass of the electron and the hole in the dot material In0.53Ga0.47As is me=0.042m0 and mh=0.052m0; the bandgap of the dot material is Eg=750 meV [37, 40]; the linewidth and the amplitude of pump laser are Γ=0.1 meV and Ap=4×104 V/cm, respectively; the lifetime of excitons is chosen as T1=40 ps; and the coherent time T2 is assumed to be less than the lifetime T1 as discussed above and is chosen as T2=20 ps.In order to clearly study the properties of the quantum beat before the decoherence of excitons happens, laser pulses must have the pulse duration less than the decoherence time of excitons. In addition, the paper studies the prolate ellipsoidal QDs, which is a quantum structure so the major axis2c and the minor axis 2a must have the length smaller than the bulk exciton Bohr radius in the dot material In0.53Ga0.47As, which has a value aBex≃308 Å. Therefore, in order to calculate, we have chosen the length of the semi-minor axis to be a=25 Å; and the length of the semi-major axis will vary depending on the ellipsoid aspect ratio χ (c=χ⋅a).First, we plot the dependence of the absorption intensity on time when the value of the ellipsoid aspect ratio of QDs isχ=3 in two cases without and with the effect of the pump laser (Figure 2). From the figure, we realize that, without the pump laser, the absorption intensity of excitons is just a smooth curve gradually going to zero in time (dashed line), that is, there is no quantum beat of excitons. However, if we irradiate a resonant strong pump laser to connect two initial exciton levels ℏΔω=0meV, we find the absorption intensity oscillating with the constant frequency, which is equal to twice the electron Rabi frequency 2ΩR, and fading over time (the solid line). This is the quantum beat of excitons in prolate ellipsoidal QDs, like the similar to quantum beat of excitons obtained in the previously studied spherical quantum dot, quantum well, and quantum wire structures [35, 36, 41]. We find that the results obtained in prolate ellipsoidal QDs are similar to ones of spherical QDs [35], it is explained as both spherical and prolate ellipsoidal QDs belong to quasi-zero-dimensional systems.Figure 2
The time-dependent absorption intensity in prolate ellipsoidal QDs of the ellipsoid aspect ratioχ=3 (in consistent with the length of the semi-major axis c=75 Å, satisfying the condition 2c<aBex) in two cases: in the absence (dashed line) and in the presence of the pump laser (solid lines) with the detuning ℏΔω=0 meV.The existence of the quantum beat can be interpreted as follows. Initially, when the system is not irradiated by the pump laser, in the system, there existed two electron quantization energy levels, and according to the Pauli exclusion principle, these energy levels existed four permitted states of electron. Afterward, if the system is subjected to a strong pump laser resonant with two electron-quantized levels, those levels couple to each other and form a unique level. That new one, to obey the Pauli exclusion principle, allows only two electron states while the system needs four ones, leading to the lack of allowed electron states. To have enough number of the allowed states, each initial electron level must separate into two splitting levels as illustrated in Figure1(b). Consequently, under the influence of the probe laser, in the absorption spectrum, we observed two interband transitions between the hole level and two splitting levels of electron (marked by thin dashed arrows in Figure 1(b)). Those transitions result in two closely spaced exciton levels, similar to the results obtained before in other quantum structures [16, 18]. Since these two exciton levels have roughly the same frequency, when they oscillate in phase, they will interfere with each other to form a quantum beat [35, 36], as showed in Figure 3 below.Figure 3
(a) The dependence of the total absorption intensity on time in the case of the ellipsoid aspect ratioχ=3. (b) An enlarged version of the curve limited by red lines.Starting from Equation (28), we plot the total absorption intensity with the effective frequency that equal to the sum of the two exciton frequencies ωex+ and ωex− as shown in Figure 3. Here, we add the absorption intensity of quantum beat to the total absorption intensity as described by Equation (29). In addition, the absorption intensity of quantum beat changes with time and oscillates with the frequency being twice the electron Rabi frequency, as described by the solid lines ở Figure 2.Next, to find out the characteristics of the quantum beat of excitons, we examine the dependence of the absorption intensity of excitons on the ellipsoid aspect ratio. Figure4 shows the dependence of the absorption intensity over time for different values of the ellipsoid aspect ratio χ. In all three cases, we observe a damped oscillation of the absorption intensity, confirming the existence of the quantum beat of excitons. Besides, investigation shows that the oscillation of the absorption intensity strongly depends on the ellipsoid aspect ratio χ. As the value of the ellipsoid aspect ratio χ increases, so does the amplitude and frequency of the oscillation. This means that as the value χ increases, the stronger quantum beat phenomenon appears, and the greater oscillation frequency is. This can be explained as follows.Figure 4
The time dependence of the absorption intensity in the prolate ellipsoidal quantum dot with different valuesχ: χ=1.8 (corresponding to c=45 Å) is denoted by solid line, χ=3 (corresponding to c=75 Å) is denoted by dashed line, and χ=4 (corresponding to c=100 Å) is denoted by dotted line in the case of the effect of the pump laser with the detuning ℏΔω=0 meV.According to Equation (A.16) and Equations (A.6), (A.8), and (A.11), electron quantization energy levels E100e and E110e are inversely proportional to the ellipsoid aspect ratio χ. Therefore, when we increase the value of the ellipsoid aspect ratio χ, then the separation between the two electron levels becomes smaller or, in other words, the electron energy levels shift closer, that is, the transition probability between these two levels will increase (or V21 increases). Therefore, when a resonant pump laser is turned on, each initial electron level must immediately split into two new levels in order to comply with the Pauli exclusion principle. Then, if we irradiate a suitable probe laser into the system, we will see the transitions from the hole energy level to these new energy levels of the electron. As a result, excitons are rapidly generated, and since they are at roughly equal levels, a quantum beat of excitons is quickly formed. Otherwise, according to the quantum size effects, we have ωt being a decreasing function of χ, and from Equation (27), we again have the beat amplitude inversely proportional with frequency of the probe photon ωt. Therefore, as the ellipsoid aspect ratio χ increases, so does the beat amplitude. In brief, the more we increase the value of χ, the closer the initial energy levels of the electron are to each other and the easier it is for the corresponding states to couple to each other. Consequently, the more likely the splitting of the electron levels is to occur, and the more rapidly the quantum beat of excitons forms, the higher the amplitude of quantum beat as a result. Besides, according to Equation (11), the electron Rabi frequency is proportional to the transition matrix element V21. Also, as mentioned above, as we increase the value of χ, the transition probability between two electron quantized levels increases (or V21 increases). As a result, when we increase the value of χ, the electron Rabi frequency increases, so the oscillatory frequency of quantum beat also increases because the oscillatory frequency of beat is twice the electron Rabi frequency.To study more clearly the feature of the quantum beat period, in Figure5, we plot the quantum beat period versus the ellipsoid aspect ratio χ with different detuning values. We see that in all three cases when increasing χ, the quantum beat period decreases (or the quantum beat frequency increases as argued above) and approaches the same value which is said to be the period of quantum beat in the bulk semiconductor. In addition, we see that as the detuning increases, the quantum beat period decreases accordingly. This relationship will be discussed in detail in the scenario of Figure 6.Figure 5
The period of quantum beat versus the ellipsoid aspect ratioχ with various detuning values: ℏΔω=0.1 meV (dotted line), ℏΔω=0.3 meV (dashed line), and ℏΔω=0.5 meV (solid line).Figure 6
The time dependence of the absorption intensity in the prolate ellipsoidal quantum dot in the case of the ellipsoid aspect ratioχ =1.8 with various detuning values: ℏΔω=0 meV (thick solid line), ℏΔω=0.5 meV (thin solid line), ℏΔω=0.9 meV (dashed line), and ℏΔω=15 meV (dotted line).Figure6 examines the dependence of the feature of quantum beat of excitons on the pump field detuning. We see that as the detuning increases, both the period and the amplitude of quantum beat decrease. It is clear that as the detuning increases, the coupling probability of two electron energy level lowers, leading to a decrease in the splitting of the electron energy levels as well as a decrease in the ability to generate two closely spaced excitons. As a result, the probability of generating quantum beats decreases, leading to a smaller amplitude of quantum beat. Besides, according to Equation (11), the oscillation frequency of beat is proportional to the detuning so when we increase the detuning, the oscillation frequency of beat increases, or the oscillatory period of quantum beat decreases. Notably, when the detuning is too large, we cannot observe the oscillation of the absorption intensity over time, that is, the quantum beat of excitons does not appear (corresponding to ℏΔω=15 meV represented by the dotted line in Figure 7).Figure 7
The period of quantum beat versus the laser detuning for variousχ values: χ=1.8 (dotted line), χ=3 (dashed line), and χ=4 (solid line).Next, we examine the relation between the period of quantum beat and the detuning of the pump wave (Figure7). The graph in Figure 7 reveals that when the detuning increases, the period of quantum beat decreases as mentioned above and that periods of various detuning values approach the same value that is said to be the oscillatory period of the initial excitons. In addition, we see that the period of quantum beat decreases with the increasing value of the ellipsoid aspect ratio χ as explained in the discussion of Figure 6. Compared to spherical quantum dots, which have only radius to work with, ellipsoidal QDs have more adjustable geometrical parameters such as semi-minor axis and semi-major axis. It makes easier for one to adjust and obtain desired optical properties in more detailed exciton energy spectrum [42, 43].Finally, we compare the absorption intensity of excitons in the prolate ellipsoidal QD Figure8(a)) and the spherical one (Figure 8(b)) [35] with the same volume. Starting from Equation (A.14), we consider the prolate ellipsoidal QD with a=25 Å and χ=4.096 that has the same volume as the spherical one with radius R =40 Å. The graph in Figure 8 shows that the characteristics of the quantum beat of excitons in two QDs of different shapes but having the same volume are completely different. This means that the quantum beat of excitons depends not only on the pump laser detuning, size of QDs but also on their geometric shapes. The reason for the difference between the feature of the quantum beat in those two kinds of QDs can be explained as follows. As we know, the shape of the QDs strongly influences the wavefunctions as well as the energy spectrum of particles. So, for two QDs with different shapes, the wavefunctions and the corresponding energy spectra are completely different, even though the QDs have the same volume. This results in the photon energy of the probe laser necessary to excite interband transitions in the prolate ellipsoidal QDs being different from that in the spherical ones, that is, the absorption intensity of excitons in these QDs is different. In addition, in this paper, we use the renormalized wavefunction method based on the theory of quantum mechanics, so our formulation can be applied to other quantum structures as long as we can define the wavefunctions and energy spectrum of particles. In fact, we have applied our theory to similar problems in the spherical QDs as well as in quantum wires and quantum wells [16, 18, 35, 36, 41].Figure 8
The dependence of the absorption intensity of excitons over time: (a) in the prolate ellipsoidal QD with the length of the semi-minor axisa=25Å, χ=4.096 (in consistent with the length of the semi-major axis c=102.4 Å) and (b) in the spherical QD with radius R=40Å of the same volume.
(a)(b)


## 4. Conclusion

In this work, we have studied quantum beats of excitons in InGaAs/InAlAs prolate ellipsoidal QDs using a three-level model by the renormalized wavefunctions method. We have found the form of the renormalized wavefunction of exciton and calculated the absorption intensity of excitons in two cases without and with the effect of the pump laser. In the presence of a resonant pump laser, the time-dependent spectrum of the exciton absorption intensity has the form of a damped periodic oscillation with a frequency being twice the electron Rabi frequency. That oscillatory behavior of the exciton absorption intensity reveals the existence of quantum beats in these QDs. The amplitude and frequency (or period) of quantum beats depend very sensitively on the detuning as well as the ellipsoid aspect ratio. The semi-minor axis and semi-major one have made controlling optical properties in ellipsoidal QDs easier and more flexible than in the spherical QDs. This is one interesting advantage of the ellipsoidal QDs over spherical ones. Moreover, the features of quantum beats of excitons also depend sensitively on geometric shapes of QDs. Specifically, with the same volume, the frequency and amplitude of quantum beats in the two spherical and prolate ellipsoidal QDs are completely different. We believe that the interesting features in optical absorption of QDs when the quantum beat of excitons occurs will have great potential for application in manufacturing quantum-computing devices. We expect our findings to be confirmed by further appropriate experiments.



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