Quantum Beat of Excitons in the Prolate Ellipsoidal Quantum Dots

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 ﬁ nally form a quantum beat. The study also shows that the geometry shape of the QDs has strong in ﬂ uence on the properties of quantum beats. Changing the period of quantum beats in particular or optical properties in general in QDs thus becomes more ﬂ exible 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.


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 dimensionsquantum 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][7][8][9][10][11][12].
Recent studies on coupled optical properties in QDs with simple shapes, such as cubic, cylindrical, and spherical QDs have been carried out [13][14][15][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][20][21][22][23]. Therefore, the optical properties in these quantum dot structures are said to be easily modifiable by these structural parameters.
In this paper, we used the renormalized wavefunction method to study the existence of quantum beats of excitons in In 0.53 Ga 0.47 As/In 0.52 Al 0.48 As prolate ellipsoidal quantum dots. We chose these heterostructures because they have the large conduction-band discontinuity between In 0.53 Ga 0.47 As/In 0.52 Al 0.48 As layers that is ΔE c = 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. In this case, we investigate quantum beats of excitons in prolate ellipsoidal QDs. We use a three-level diagram of exciton, including one ground level E ex ground and two excited ones of exciton E ex 100 and E ex 110 (Figure 1). These exciton levels are originated from interband transitions between the lowest level of hole E h 100 and the two lowest ones of electron E e 100 and E e 110 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).

Model and Theory
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 station-ary 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)).  Journal of Nanomaterials two exciton excited levels in the initial stationary states E ex 100 and E ex 110 . 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 Λ e mix ð r ! , tÞ (Equation (A.26)) and the initial wavefunction The wavefunction in Equation (4) can be written in the explicit form or it can be rewritten as in which  (1)). Equation (8) can be written as From Equations (7) and (8), we have Under the effect of the pump laser, the two initial excited levels of exciton split into four new energy levels in which two levels E ex− 100 and E ex+ 100 are separated from level E ex 100 , and two levels E ex− 110 and E ex+ 110 are separated from level E ex 110 (see Figure 1(b)). We see that the energy difference between two splitting levels E ex− 100 and E ex+ 100 or between two splitting ones E ex− 110 and E ex+ 110 is equal to 2ℏΩ R (Equa-tion (10)), which is much smaller than the photon energy of the pump laser ℏω p . This photon energy is also the energy distance between two pairs of levels ðE ex− 100 , E ex+ 100 Þ and ðE ex− 110 , E ex+ 110 Þ (Equation (9)). The Rabi frequency Ω R 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 where ℏΔω is the detuning of the pump laser with two levels of electron E e 100 and E e 110 , and V 21 is the matrix element for the intersubband transition between electron levels (see Equation (A.25) in Appendix A). 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 (Figure 1(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 j0i corresponding to level E ex ground to two initial excited states of exciton corresponding to two exciton levels E ex 100 and E ex 110 . However, according to the selection rule for the interband transition in QDs, only the exciton transition from the ground level E ex ground to the lowest excited level of exciton E ex 100 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 E ex ground and E ex 100 under the action of a probe laser has the form where A t and ω t are, respectively, the amplitude and the frequency of the probe laser. Replace Equations (3) and (1) into Equation (12); we have where p cv is the polarization matrix element between conduction and valence bands From that, we find the expression for the exciton absorption intensity when the pump laser does not turn on as Because the exciton lifetime on the excited levels is finite, the excited states of exciton will fade over time. To account for the decay of I ex 100 ðtÞ, we phenomenologically provide a decay parameter γ = 1/T 1 in Equation (15). From that, we obtain the expression of the absorption intensity of excitons in the absence of the pump laser as follows where T 1 is the lifetime of exciton on the energy level E ex 100 .

The
Case with the Pump Laser. From Section 2.1.2, we see that exciton will stay in the nonstationary state Λ ex mix ð r ! , tÞ when QDs are irradiated by a strong pump laser resonant with two initial exciton levels E ex 100 and E ex 110 . Now, to find the absorption intensity of excitons, we need to calculate the dipole transition matrix element between the ground state j0i and the nonstationary exciton one Λ ex mix ð r ! , tÞ. The matrix element in this case has the form Combining Equations (6) and (17), we get the following matrix element 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 exciton ðE ex− 100 , E ex+ 100 Þ that are split from the initial exciton level E ex 100 . So, we have or Equation (20) clearly shows that, after the effect of a strong pump laser resonating with two exciton excited levels in stationary state E ex 100 and E ex 110 , then in the system, there are two optical transitions from the ground state E ex ground to two lowest splitting level of exciton E ex− 100 and E ex+ 100 (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 j0i and the nonstationary exciton one Λ ex mix ð r ! , tÞ as follows: or Next, we investigate the time-resolved intensity of absorption under the effect of the resonant pump laser I ex mix ðtÞ. From Equation (10), we see that the probe laser needs a spectral width larger than 2ℏΩ R , which is energy separation between two levels E ex− 100 and E ex+ 100 , in order to excite coherently two excitons in the pair ðE ex− 100 , E ex+ 100 Þ. 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 T 2 , the time it takes for two excitons in the pair ðE ex− 100 , E ex+ 100 Þ to oscillate in phase. Of course, the coherence time T 2 is always less than or equal to the lifetime T 1 of exciton in the state Λ ex 100 ð r ! , tÞ (Equation (3)). Therefore, at any given time t < T 2 , the absorption intensity of excitons under the effect of the pump laser has the following form. or Journal of Nanomaterials We put and combined with Equation (10), we can rewrite the expression for the absorption intensity of excitons as As mentioned in Section 2.2.1, 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 I ex mix ðtÞ, we add phenomenologically the damped factors γ = 1/T 1 and τ = 1/ T 2 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 where T 2 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    Journal of Nanomaterials damped periodic oscillation that owns a frequency of twice the Rabi frequency of the electron 2Ω R . 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 ones ω + ex and ω − ex ) 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 We can rewrite Equation (28) as follows

Results and Discussion
To further study and demonstrate the obtained results, in this part, we perform numerical calculations for the timeresolved intensity of absorption in In 0.53 Ga 0.47 As/In 0.52 Al 0.48 As prolate ellipsoidal QDs. We utilize those parameters for the calculation: the effective mass of the electron and the hole in the dot material In 0.53 Ga 0.47 As is m e = 0:042m 0 and m h = 0:052m 0 ; the bandgap of the dot material is E g = 750 meV [37,40]; the linewidth and the amplitude of pump laser are Γ = 0:1 meV and A p = 4 × 10 4 V/cm, respectively; the lifetime of excitons is chosen as T 1 = 40 ps; and the coherent time T 2 is assumed to be less than the lifetime T 1 as discussed above and is chosen as T 2 = 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 axis 2c and the minor axis 2a must have the length smaller than the bulk exciton Bohr radius in the dot material In 0.53 Ga 0.47 As, which has a value a ex B ≃ 308 Å. Therefore, in order to calculate, we have chosen the length of the semiminor axis to be a = 25 Å; and the length of the semimajor axis will vary depending on the ellipsoid aspect ratio First, we plot the dependence of the absorption intensity on time when the value of the ellipsoid aspect ratio of QDs is χ = 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  Journal of Nanomaterials 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 ðℏΔω = 0meVÞ, we find the absorption intensity oscillating with the constant frequency, which is equal to twice the electron Rabi frequency 2Ω R , 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.
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 Figure 1(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.
Starting from Equation (28), we plot the total absorption intensity with the effective frequency that equal to the sum of the two exciton frequencies ω + ex and ω − ex 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. Figure 4 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    7 Journal of Nanomaterials 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.
According and E e 110 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 V 21 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 ω t being a decreasing function of χ, and from Equation (27), we again have the beat amplitude inversely proportional with frequency of the probe photon ω t . 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 V 21 . Also, as mentioned above, as we increase the value of χ, the transition probability between two electron quantized levels increases (or V 21 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 Figure 5, 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.   Journal of Nanomaterials This relationship will be discussed in detail in the scenario of Figure 6. Figure 6 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).
Next, we examine the relation between the period of quantum beat and the detuning of the pump wave ( Figure 7). 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 Figure 8(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].

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  Journal of Nanomaterials 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.

Appendix
For the sake of convenience, we recall here the wavefunctions and the energy spectra of electron and hole in prolate ellipsoidal quantum dots (QDs) [18][19][20][21][22]. Consider the pro-late ellipsoidal QD with rotational symmetry around the z axis. Let a and c be the length of semi-axes of the ellipse in the xOy plane and z-direction, respectively, where x, y, z are the coordinates in Cartesian coordinate system with its origin at the ellipsoid symmetry center. For simplicity, we have assumed the prolate ellipsoidal QD is in an infinite potential and has the form [18][19][20][21][22] U r where Sð r ! i Þ depends on parameters a and c which are the semi-axes of the ellipsoidal QD; we have S r with c > a, we have the prolate ellipsoidal QD as shown in Figure 9.
The envelope wavefunctions of electron (hole) in prolate ellipsoidal QD have the form [19][20][21][22][23]  The energy of electron (hole) is given as The values of the parameter h depend on the values of the indices n, l, m. When h ⟶ 0 (or f ⟶ 0, the prolate ellipsoidal QD will become the spherical one and χ ⟶ 1. Then, the wavefunctions of electron (hole) in QD will have the following form [44] Ψ S e,h ð Þ where Y lm ðθ, φÞ is the spherical harmonic function; j l ðrÞ is the spherical Bessel function with χ nl is its zeros. The energy levels of electron (hole) being consistent with wavefunctions in Equation

A. The Case without the Pump Laser
In the effective mass envelope-function approximation, the total wavefunction of electron (hole) in a prolate ellipsoidal QD with infinite potential is given as where E g is the bandgap of the semiconductor. We examine a three-level energy model where E h 100 is the lowest quantized energy level of the hole correspond-ing to the state j0i; E e 100 and E e 110 are the first quantized levels of the electron corresponding to the states j1i and j2i, see Figure 10(a).
Here, we need to use the time-dependent wavefunctions of the particles to find the time-dependent properties of the quantum beats. The time-dependent wavefunctions of the particles are now defined with

B. The Case with the Pump Laser
To search for the quantum beats in three-level model, we used two different laser beams concurrently. A strong pump laser resonant with two electron energy levels is irradiated to support the intersubband transition between these levels. A weak probe laser is utilized to search for the excitonic transitions between ground state and excited ones of exciton. The lasers can be described as follows: ðA:19Þ where n ! is the unit vector along the wave propagation direction; A x and ω x are the amplitude and frequency of lasers with x indicating which laser is pump or probe laser. In case the electromagnetic field is not too strong, we can omit the higher-order term, and by applying some gauges and approximations, the expression for the Hamiltonian interaction between the electron and the electromagnetic field can be written as follows [44,45].
where q, m 0 , and p ! are the charge, the bare mass, and the momentum of the electron, respectively. When there is the effect of strong pump laser resonating with the energy distance between the two quantized levels of the electron, the wavefunctions of the electron are renormalized under the effect of the pump laser and have the form where coefficients c l ðtÞ (l = 0, 1) are determined from the time-dependent Schrödinger equation and has the following expression [16]: It should also note that ℏΔω = ℏω p − ℏω 21 is the detuning between the pump laser and two initial levels of electron E e 100 and E e 110 . We find that those two initial levels are all split under the effect of a resonant strong pump laser, Figure 10

Data Availability
The material data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.