Spin-flip quantum transition driven by the time-oscillating Rashba field

Focusing on the spin-flip quantum transition, we study the time-dependent phenomena by the oscillating Rashba spin–orbit interaction. An electron is confined by a harmonic potential surrounded by a cylindrical hard-wall in a two-dimensional (2D) quantum dot. The oscillating Rashba external field having a frequency w is applied perpendicular to the 2D plane. The projection and discrete Fourier transform analyses reveal that the interstate transition causes the characteristic spin-flip quantum transition when the Rashba field has a resonant frequency. Particularly in the cylindrical hard-wall confinement, a typical Rabi oscillation results with a spin flipping. The perturbation approach up to the fourth-order terms satisfactorily explains the origin of the oscillating components found in the spin density.


Introduction
The spin-electric coupling via the spin-orbit interaction (SOI) is the typical quantum phenomenon. Over a halfcentury ago, Rashba [1] theoretically predicted that the SOI couples an electron spin with an electric field. However, it was not until the end of the last century that this coupling has experimentally verified to be manipulated [2]. After the discovery of the Rashba SOI, the remarkable progress in the spin and SOI related phenomena has been sparked such as a single electron coherent oscillations [3], spin resonance [4], a spin fieldeffect transistor proposed by Datta and Das [5], the Aharonov-Casher effect as a function of Rashba SOI splitting strength [6,7], a spin filter and analyzer [8], the Rashba SOI coupling in the 2D magnetoexcitons [9], and the spin current and relating transport phenomena [10][11][12][13][14][15]. Furthermore, with an extreme development in the nano-fabrication technique of quantum dots (QDs) [16], the spin states in these QDs are considered to be promising for physical realization of the quantum computation algorithm [17][18][19]. Thus, spintronics [5,20] opens new avenues of exploration in both science and technology. Manchon et al have reviewed the new perspectives for the SOI coupling [21].
It is well known that an electron spin is a natural choice for creating a quantum bit (qubit) because the spin of a single electron forms a two-level system. Accordingly, the key element is an ability to induce transitions between the spin-up (α) and spin-down (β) states and to prepare arbitrary superpositions of these two basis states [22]. However, comparing with studies of the above static SOI, few works have been carried out for the dynamics with the time-dependent (TD) SOI couplings. Nowack et al [22] have achieved the pioneering experiment, by which they observed coherent transitions (Rabi oscillations) between α and β states and succeeded in the coherent control of the single electron spin in the QD using an oscillating electric field generated by a local gate. For the TD SOI coupling phenomena, Földi et al [23] have theoretically studied the similar Rabi flopping in the quantum ring system. They found that the spin components oscillate with Rabi frequency. Considering the future spin-based quantum information, van Berg et al [24] have demonstrated an electronically controlled spin-orbit qubit in an InSb nanowire. They have succeeded to observe the Rabi oscillation frequencies up to 104 MHz. Furthermore, Echeverría-Arrondo and Sherman [25] have studied the coupled spin and orbital dynamics of a spin-1/2 particle in a harmonic potential subject to ultrastrong SOI and external magnetic field. Walls [26] has studied the spin dynamics of a single electron under parametric modulation of a lateral QD's electrostatic potential in the presence of spin-orbit coupling.
In this work, we study TD phenomena caused by the spin-electric coupling via the Rashba SOI because the understanding of the TD quantum transitions between these spin α and β states is fundamental but crucial in the spintronics. We study how the Rashba external field time-modulate by the frequency w causes the quantum transition for an electron confined in the QD by solving the TD Schrödinger equation numerically. We then employ the projection and discrete Fourier transform (disFT) analyses, and study the interstate transitions by a comparison with the analytical results by the perturbation approach and investigate the possibility of the Rabi oscillation driven by the TD Rashba SOI. In order to realize the spintronics, the coherency of the quantum electronic spin states is crucial [27], but is influenced significantly by the spin-flips and/or spin-precession [28]. Therefore, the deep understanding of the spin-flip quantum transition driven by the time-modulate Rashba field is important not only in the elucidation of the spin dynamics but also in the design of the novel devices for the future quantum computation [22]. In section 2, we briefly summarize the influence on the electronic states by the static Rashba SOI. Then, we discuss the spin dynamics caused by the TD Rashba SOI in section 3. We mention how to solve the TD Schrödinger equation numerically (section 3.1) and show the resulting snapshots (section 3.2). We study the spin-flip quantum transitions by the projection analysis in section 3.3. The resulting state probabilities have been investigated by the perturbation approach of the TD expansion coefficients, taking into account up to the 4th order terms (section 3.4). In section 3.5, we discuss the difference in the spin-flip processes, in accordance with the cylindrical hard-wall and harmonic potentials and study the possibility of the Rabi oscillation caused by the TD Rashba SOI (section 3.6). In section 4, we further discuss the non-resonant Rashba SOI, and then conclude. We add the calculational details and others in Appendices.

Static system 2.1. Confinement by hybrid potential
We confine an electron by the 2D isotropic harmonic potential of the strength 0 w . We further surround the harmonic potential by the cylindrical hard-wall to realize the practical 2D QD system. Figure 1 illustrates the potential profile of the present 2D QD fabricated by InSb [29]. The hybridized potential is produced on the 100×100 nm square substrate by the cylindrical potential having a radius r 50 nm 0 = . By employing the effective mass (m * ) approximation, the static and unperturbed Hamiltonian where we represent the hybrid potential by V x y , hyb ( ). As such, by varying 0 w , we can tune the eigenstate, in accordance with the change in the confinement from the cylindrical hard-wall ( 0 0 w = ) to the harmonic potential (larger 15 0 w > effective atomic unit (a.u.), appendix A) through the hybrid form.
We have the corresponding static Schrödinger equation of where symbols n and l are a radial and angular quantum number, respectively 3 . We solve this static Schrödinger equation (2) numerically by employing a finite difference technique where the wave function is discretized into square grid points of real space (i.e., numerically exact diagonalization). The second(2nd)-order differential in the kinetic energy is carried out by the central difference method. We employ a Cartesian x and y grid by dividing the real space into a mesh of 64×64 (1.5625 nm squares), and then carry out the numerical diagonalization in a twofold larger space due to the spin hybridization between the spin-up (α) and -down (β) states. Consequently, the eigenstates result being degenerate doubly by α and β. Since the present 2D QD is supposed to be fabricated by InSb [29], an effective mass (ratio) m m 0 * = 0.0138 [30] and a dielectric constant (ratio) 0 *   = 17.88 [31,32] of InSb are employed in the numerical calculation. We have reported the calculational details in the present numerical diagonalization in our previous works [33][34][35].
We give the calculated eigenstates for the unperturbed Hamiltonian 0  in figure 1. The solid lines indicate those eigenstates for the present 2D hybrid QD whereas the broken lines are those for the ideal harmonic confinement. The yellow line indicates an intercept potential between the harmonic and cylindrical confinements. According to figure 1, the confinement for the ground and lower excited states are well approximated by the harmonic potential when 0 w is larger than 15 a.u. (appendix A).

Static Rashba SOI
We here apply the Rashba external electric field , , x y z X = X X X ( )in this 2D system with a time-modulation. Before the discussion of the time-developing phenomena, we summarize the static Rashba phenomena in the 2D QD system. By employing the spin Pauli operator ŝ , we have the Rashba SOI Hamiltonian R  for an electron confined in the 2D QD by, In equation (3), we employ the spin raising (+) and lowering (−) operators s  , defined by, where x ŝ and y ŝ are the spin Pauli matrices. Accordingly, we decompose the Rashba Hamiltonian of equation (3) into three parts by, Potential profile of the present hybrid confinement; a harmonic potential is surrounded by a cylindrical hard-wall [33][34][35]   , , cos sin , sin sin , cos , where 0 X is the so-called Rashba coupling constant [36][37][38]. We show how the Rashba SOI changes the unperturbed ground state 00 00 0 j = ñ ( ) | ( ) 4 against the applied direction θ in figure 2(a). The Rashba SOI causes the largest stabilization when the Rashba electric field is applied perpendicular to the 2D plane ( 0 q = ) and the smallest stabilization when 2 q p = . One should further note that the Rashba field of 0 q = hybridizes the opposite spin most strongly, and the perturbed ground state 00a ( ) results (figure 2(a)). Also in the first (1st) excited states 01 ( )and 01 (¯), the Rashba field applied perpendicular to the 2D plane ( 0 q = ) resolves the spin degeneration most largely whereas the Rashba field of 2 q p = restores the degeneration. The completely complementary energetics is found both in the perturbed ground and 1st excited states so as to conserve the Kramers degeneracy between the two perturbed states; e.g., 00a ( ) and 00b ( ), and 01a ( ) and 01b (¯). These features are caused by the ls-like components of the Rashba SOI ( R  + and R  -) in *X . We also show the spin densities r a (left) and r b (right) for the Rashba SOI perturbed state at 0 q = . Hereafter, we expediently use the unperturbed notation n l , ( ) for the classification of the perturbed eigenstates. In the present InSb QD, the confinement strength of 1w a.u. corresponds to 11.9 meV. 4 We denote the eigenstate by the isotropic confinement of the harmonic potential by n l , ( ) and that by the anisotropic case by n n , ¢  [ ]. The corresponding eigenfunction is also denoted by n l j (Laguerre polynomial) for the isotropic harmonic potential and by nn f ¢ (Hermitian polynomial) for the elliptic harmonic potential. equation (3) and their contributions become most significant at 0 q = . Accordingly, we here apply the timemodulated Rashba field perpendicular to the 2D plane.

Suzuki-Trotter decomposition
We now study the dynamical properties of the confined electron with the application of the time-modulated Rashba electric field t R  ( ) by solving the following TD Schrödinger equation: Here, the TD Hamiltonian x y t , ;  ( ) is given by the sum of the static/unperturbed Hamiltonian 0  of equation (1) and the time-modulated Rashba Hamiltonian x y t , ; x y t x y x y t , ; , , ; where we tune the Rashba electric field periodically having a frequency w by, We solve the TD Schrödinger equation (5) numerically by employing the real-space and real-time grid approach via Crank-Nicolson method. For this purpose, we rewrite the time-evolution propagator by separating the potential operator from the kinetic one based on Suzuki-Trotter exponential product theory [39][40][41][42] with taking into account the decomposition up to the second order of t D . One should note that the time-modulated Rashba field also causes the oscillating scalar potential V z (t) by which the other type of the quantum transition is generated. We can here exclude this type of the quantum transition, as discussed in appendix B.
The general solution of this TD Schrödinger equation (5) is approximately given by, where t D is a time-grid interval (appendix C). By employing the Suzuki-Trotter exponential decomposition approach, the time-development operator of equation (9) Consequently, we can determine the TD wave function by changing the time-grid interval t D through equation (10).
The exponential operators for the kinetic term (spatial gradient) are commutative between the x and y components when the Cartesian coordinate system is employed. These components are, then, rewritten using Cayley's approach into the second order approximation and are further represented in terms of the Crank-Nicolson matrix form [43]; As such, a pair of simultaneous linear equations is produced, independently toward the directions of x and y. Each simultaneous linear equation is easily solved because the matrix appearing on the left-hand side is straightforwardly decomposed into the LU form. Thus, this procedure enables us to solve the TD Schrödinger equation directly and quickly with precision [43]. The Suzuki-Trotter exponential approach decompose the present time-evolution operator (equation (10)) into thirteen exponential products and causes a slight complexity in the numerical calculation. However, this decomposition is effective for the CPU time with an order N ( )of the division number N. In order to confirm the numerical accuracy in the Suzuki-Trotter exponential decomposition approach, we also solve the TD Schrödinger equation (5) directly by the real-time and real-space finite difference method with the Crank-Nicolson diagonal algorithm. This direct difference method is simple to solve the TD Schrödinger equation but consumes lots of CPU time with an order of N 3 ( ). We compare both results through the projection analysis of the calculated TD wave functions and confirm the numerical accuracy in the Suzuki-Trotter exponential decomposition approach (appendix D).

Snapshots
We show the snapshots of the α and β spin densities against time in and z 0,1 are zero points of the Bessel function, mentioned in appendix A. Figure 3(a) demonstrates that the α spin decreases whereas the β spin increases (t 3 a.u.). The density distribution of the induced β spin has a single node toward the radial direction. At t 6 , the α spin 'fully' disappears whereas the β spin becomes tangible. After that, the β spin decreases but the α spin increases, and the system at t 9 provides the same spin-distributions found at t 3 . The system then restores completely the initial spin distribution at t 12 . These TD features repeat periodically. . The induced β spin shows the distribution with a single radial node, as found in the cylindrical hard-wall confinement. In the cylindrical hardwall confinement, we also find the 'full' exchange from the α spin to the β one at t 6 . However, we cannot find such a full spin-exchange in the harmonic confinement of 15 0 w = . At t 4 , a characteristic distribution of the double radial nodes is found in the α spin. This feature indicates that the excitation into higher states results. Furthermore, we cannot find the peculiar time at which the system completely restores the initial state within the short time-period. A comparable localization of both spin densities to the center is also characteristics of the eigenfunctions in the harmonic confinement (b).

State probability
We discuss the above TD features by employing the projection analysis. We project the numerically calculated TD wave function t S Y ñ | ( ) into the eigenstate nl 0 j ñ s | ( ) of the unperturbed Hamiltonian 0  of equation (1), and calculate the state probability P t t r; against time by varying the eigenstate nls ( ). For the electron confined in the cylindrical hard-wall, we find the meaningful two state-probabilities P t ( ), transits into the first excited state with changing its spin oppositely, 01b ( ). At t 3 , the electron is stochastically equally in both the ground and 1st excited states having the opposite spin. Then, at t 6 , the electron is resonantly excited into the 1st excited state by changing its spin completely. After that, the electron excited in the state 01b ( ) returns to the ground state with recovering the original α spin and the system restores the initial state at t 12 . Thus, the electron repeats the resonant interstate transition between the ground and 1st excited states periodically with the characteristic spin exchanging/flopping. Consequently, the electron confined in the cylindrical hard-wall results in the interstate (Rabi-like) oscillation. One can find the small amount of the state probability of the 2nd excited state 10a ( ) around t 6 and also note the beating due to higher frequencies in the state probabilities of 00a ( ) and 01b ( ). 5 The present cylindrical hard-wall confinement gives the minimum excitation frequency We give the state probability for the harmonic confinement against time in figure 4(b). The Rashba field is also time-modulated by having the resonant frequency of w 15 i.e., the minimum excitation is from the ground state to the 1st excited one. Similarly to the cylindrical hard-wall confinement, the electron is initially in the ground state 00a ( ), and transits into the first excited state 01b ( ). However, different from the cylindrical hard-wall confinement, the state probability of 01b ( ) amounts at most to 62% even in its maximum (t 3.8 ). The remaining 38% is shared by the higher excited states of 10a ( ) and 11b ( ), so on. Around t 9.3 , most of the dissipated electron return to the original ground state. Nevertheless, the system does not recover its initial state completely. Parts of the electron still remains the other higher states. The Rashba SOI in the harmonic confinement causes the multiple resonant interstate transitions to the higher excited states.

Expansion coefficients
In order to deepen our understanding on the present TD features, we here study the oscillation components included in each projection t . By employing the disFT analysis [44], we extract the oscillation components for the harmonic confinement  and 3rd 11b ( ) (d) excited states. The disFT amplitudes of the former three states are in the same order but that of the 3rd excited state is smaller than these three formers by an order of magnitude. The disFT analysis further demonstrates that each state has the multiple oscillation components and all those components are interestingly an integral multiplication of the resonant frequency 0 w . Figures 5 demonstrates that the ground (a) and 2nd (b)  ). We assign the oscillation components (peaks) by the 4th order perturbation approach based on table 1. excited states have the same oscillation components of 0 w  , 3 0 w -, and 5 0 w whereas the 1st (b) and 3rd (d) excited states have those of 2 0 w  and 4 0 w -. That is, the former two states have the odd numbered multiplication of 0 w whereas the latter has the even numbered multiplication of 0 w .
By employing the spinor representation, the TD wave function t S Y ñ | ( ) of the TD Schrödinger equation (5) is decomposed into, Each spinor function can be further expressed in the expansion form for the unperturbed eigenstates by, Consequently, the projection t . Since the present Rashba external field is applied perpendicular to the 2D plane with the frequency w, the corresponding TD perturbation potential V t ( ) is given by Here, the time-independent operator Fˆin equation (17) is defined by Owing to the feature of the Rashba SOI, one should note the following relation; Let us now estimate the expansion coefficients for the present case, where the electron is initially in the ground state 00 0 j ñ | ( ) having an α spin, 00a ( ). Namely, the initial condition is given by For example, we calculate the expansion coefficient C t 00a ( ) for the ground state 00a ( ) based on the approximation up to the fourth-order perturbation theory (appendix E); 2 1 00 00 00 1 00 2 00 3 00 These TD coefficients by the perturbation expansion up to the 4th order terms are explicitly given by, We now extract the oscillation components from each TD coefficient.
One should note that the Rashba SOI results in the non-zero transition matrix elements of F 00 ,01 a b and F 01 ,00 b a . Accordingly, the 2nd order term multiply causes the oscillation components of w and 0 because of w 0 w = and the harmonicity of the confinement We cannot find the appropriate states k 1 ñ | and k 2 ñ | which cause the non-zero 3rd order Rashba SOI matrix elements for the initial condition of equation (20). Consequently, c t 0 Consequently, the remaining virtual time integration in equation (28),    (16)). As such, the Rashba external field having the resonant frequency w 0 w = causes the oscillation components of in the ground state projection. These peaks completely coincide with those found by the disFT analysis as shown in figure 5(a). The perturbation approach further elucidates that the components of 0 w  and 3 0 w are caused by the second order perturbation term whereas the component 5 0 w is caused by fourth order term (equation (21)). We summarize the possible oscillation components for the ground 00a ( ), 1st 01b ( ), 2nd 10a ( ) and 3rd 11b ( ) excited states in table 1, by which we can assign the disFT peaks found in figures 5(a)-(d) consistently.
Eventually, one should note that the Rashba field applied perpendicular to the 2D plane polarizes the electron spin confined in the harmonic potential and the resonant oscillation of the Rashba field separates the polarized spin state into α and β completely by the different frequencies. The α spin state oscillates with the odd multiple harmonic frequencies whereas the β spin state does with the even ones when the electron is initially polarized by the α state.
We also show the oscillation components for the cylindrical hard-wall confinement 0 0 w = in figure 6. The disFT analysis reveals that the present time-modulated Rashba field causes the significant interstate transition among three states of 00a ( ), 01b ( ), and 10a ( ). Figure 6 further demonstrates that each state has the multiple frequencies, similarly to the harmonic confinement. Although the lack of the equivalency in the energy differences among the neighboring eigenstates causes a complicatedness to represent the disFT peak explicitly in figure 6, the present fourth-order perturbation approach gives the consistent assignment in the disFT peaks of the cylindrical confinement (table E1 in appendix E).

Multiple transitions
We here discuss the selection rule for the Rashba SOI coupling for an electron confined by the ideal cylindrical hardwall or harmonic potential, respectively. When the confinement is achieved by the 2D central force field, the interstate SOI transition basically occurs between those states having the different angular momenta (l and l¢) due to the in-plane momentum components. The present Rashba field is applied perpendicular to the 2D plane, and the ls-like SOI coupling result [33,34]. Consequently, the conservation of the total angular momentum j l s z z = + governs the present Rashba SOI selection rule. Eventually, the interstate Rashba SOI transition matrix element n l nlm is given by, where R  is the static part of the Rashba SOI Hamiltonian of equation (7) and is given by, Since the eigenstate in the cylindrical hard-wall confinement is expressed by the Bessel function, the selection rule is fully given by equation (30). For example, an electron having a spin α is initially in the ground state; i.e., l=0 and s z In the harmonic confinement, the eigenstates are represented by the Laguerre polynomial, and the condition for the radial quantum number n n 1 ¢ =  is added to the selection rule. Accordingly, the interstate Rashba SOI coupling is given by,  Figure 7 illustrates the possible Rashba SOI interstate couplings in the harmonic confinement. The harmonic confinement causes the multiple but inter-neighboring-state couplings, and the conservation of j z changes the spin state alternately. Eventually, the 'multiple but successive' interstate couplings results with the spin flipflopping, as illustrated in figure 7. The meaning of the 'successive' couplings is naturally symbolic and has no actual time-delays, as found in the cylindrical hard-wall confinement. These different selection rules and energy eigenvalues between the cylindrical hard-wall and harmonic confinements produce the characteristic difference in the Rashba SOI interstate couplings. In the harmonic confinement of 0 w , the Rashba external field having the oscillating frequency w 0 w = produces the infinite number of the interstate resonant couplings. Contrary, the cylindrical hard-wall confinement does not have the equivalency in the energy difference between the neighboring eigenstates. As such, the resonant interstate coupling is specified uniquely, in accordance with the Rashba excitation frequency w. One should, then, remember that the numerical calculations based on the Suzuki-Trotter method result in the not infinite but finite number of the four interstate transitions in the harmonic confinement ( figure 4(b)). This inconsistency is caused by the incompleteness of the harmonic confinement in the present hybrid potential where the cylindrical hard-wall terminates the infinite expansion of the harmonic eigenfunctions and breaks the equivalency in the energy difference between the neighboring eigenstates.
In order to understand the multiple resonant-couplings in the ideal harmonic confinement, we rewrite the TD Schrödinger equation (5) into the simultaneous (rate) equations for the expansion coefficients C n (t), and solve them numerically but directly by supposing the ideal harmonic confinement. The corresponding simultaneous equations is given by Here, C n (t) is the TD expansion coefficient defined in equation (13) and V t ( ) is the TD perturbation potential. We also suppose the minimum excitation from the ground state to the 1st excited state of the 2D harmonic potential of 15 0 w = . Accordingly, the Rashba field applied perpendicular to the 2D plane has the oscillation frequency w 0 w = . In the calculation of equation (33), we further estimate the interstate Rashba SOI matrix elements analytically by using the Laguerre polynomial eigenfunctions. We tabulate several of those results in table A1 in appendix A.
In figure 8, we show the resulting state-probability obtained by equation (33) against time. Figure 7 and equation (32) predict that the TD Rashba SOI with the frequency 15 0 w = causes the multiple and successive interstate couplings between the neighboring eigenstates with changing the spin alternately. The solution of the simultaneous equation (33) surely reproduces all the possible interstate couplings among the employed 30 eigenstates, as shown in figure 8. One should, however, note the monotonous reduction in each stateprobability. This reduction prevents the restitution of the initial state, and causes no periodicity in the interstate oscillation 6 . As such, the interstate Rabi-like oscillation disappears in the ideal harmonic confinement. This feature is very contrast to the initial-state restoring in the present hybrid confinement. Particularly, in the cylindrical hard-wall confinement ( figure 4(a)), we can find the 'full' restitution of the initial state. Namely, in the ideal harmonic (and non-dissipative) confinement system, the number of the interstate resonant couplings is principally infinite, and the infinitesimal state-probability results in all the eigenstates after the infinite time passes, i.e., in the static state.

Rabi oscillation
The Rabi oscillation is the typical dynamics found in the interstate resonant transition caused by the oscillating TD external field. We here discuss whether the TD Rashba field causes the Rabi oscillation or not. We focus on the hybrid confinement system of 0 0 w = (a) and 15 (b), and the external Rashba field is periodically oscillating with the resonant frequency R w of the minimal excitation from the ground state to the 1st excited state. By When the interstate transition occurs resonantly between the two states of the initial (ground state nñ | ) and the final (excited state kñ | ) only, the rotational wave approximation (RWA) determines the state probability at time t by, Here, Ω is the Rabi frequency of the inter-two-state resonant transition. By using the interstate transition matrix element F kn defined by equation (19), we have  (18), this matrix element F kn is given by,   .
Consequently, under the two-state transition approach with the RWA, the interstate Rabi frequency Ω is proportional to the Rashba SOI coupling constant 0 X and given by, We can estimate this constant D by the two approaches; the numerical estimation D LSM by the LSM application, and the direct calculation D by We then compare these two values D LSM and D.
We return the change of the state probability against time in the cylindrical hard-wall confinement. (red dots line in figure 9(a)). The value of D = 7.84 (green line in figure 9(a)) is also calculated by equation (38), where the interstate Rashba SOI transition matrix element is directly obtained by the Bessel eigenfunctions of the states 00a ( ) and 01b ( ) (appendix A). We can find the consistent coincidence between these two constants D 6.87 LSM = and D = 7.84. A slight discrepancy from the relation 2 0 2 W µ X is found when the Rashba coupling 0 X increases ( figure 9(a)). With an increase in the number of the possible transition states, the slope of the relationship a Wdecreases ( figure 9(b)). The ideal harmonic confinement has the infinite number of the possible states by the resonant transition. Eventually in the static state, an infinitesimal state-probability is distributed in each state, and the interstate oscillation of Ω is not generated. Namely, the finite number of the possible interstate transitions is crucial for the Rabi oscillation for the non-dissipated system. When the interstate transitions occur infinitely, the system cannot return its initial state.

Non-resonant Rashba SOI
In this section, we study the non-resonant TD phenomena when the time-modulated Rashba field has a detuning frequency δ from the resonant one R w . We here set δ by a quarter of R w . Accordingly, the practical detuning is   The disFT analysis extracts the oscillation components in the non-resonant SOI couplings in figure 10(b). We show those of the ground 00a ( ), 1st 01b ( ), 2nd 10a ( ) and 3rd 11b ( ) excited states separately in figures 11(a)-(d), respectively. We should remember that the projection to the ground state includes the five harmonic sounds in the resonant excitation (fourth order perturbation treatment). They are the odd-number multiplied frequencies of 0 w  , 3 0 w  and 5 0 w -( figure 5(a)). The disFT analysis for the non-resonant excitation demonstrates that these four resonant peaks split multiply as shown in figure 11(a). Similar peak-splittings are further found in the other states of 01b ( ), 10a ( ) and 11b ( ) (figures 11(b)-(d)). We calculate the non-resonant oscillation components in the harmonic confinement up to the 4th order perturbed terms, and summarize them in table 2 with the comparison of those resonant components. As mentioned in section 3.4, the 2nd order perturbation approach elucidates that the expansion coefficient C t 00a ( ) for the ground state includes the two oscillation components of 0 w and 3 0 w by the minimal resonant excitation. The 4th order perturbation approach further adds the other three components of 0 w , 3 0 w and 5 0 w -. In the non-resonant excitation, the perturbation approach predicts that these frequency components are splitted multiply into 3 2 w -peaks in the other excited states of 01b ( ), 10a ( ) and 11b ( ) (appendix E). As such, the characteristic peaksplittings in the non-resonant Rashba SOI coupling is well described by the discrepant frequency δ and its integer multiples. Thus, in figures 11(a)-(d), we can give the peak position and separation explicitly based on table 2. The assignment by employing the perturbation approach up to the 4th order terms completely coincides with the resulting disFT peaks and explains the characteristic peak-splittings even in the non-resonant excitation.
We also give the disFT oscillation components in the cylindrical hard-wall confinement under the nonresonant excitation w figure 12. The perturbation approach predicts accurately the characteristic splittings, and one can assign the peak position and separation fully by table E1. Nevertheless, we do not give the resulting formulae of the position and separation explicitly, because the cylindrical hard-wall confinement breaks the equivalency in the energy difference between the neighboring eigenstates. Accordingly, the resulting expression for the peak position and separation are too complicated to be indicated in figure 12. We apply the two-state transition approach with the RWA to the non-resonant excitation w   In figure 13(b), we also plot 2 W against 0 2 X (blue squares) in the harmonic confinement ( 15 0 w = ) under the non-resonant excitation. The applied Rashba field similarly has the non-resonant frequency of w where R w is the minimal resonant frequency 0 w and 4 0 d w = . Figure 10(b) elucidates that the inter-two-state coupling occurs between the ground and 1st excited states but the transition is non-resonant. Nevertheless, we can find the relationship 2 0 2 W -X (blue squares) in figure 13(b). As such, the proportional constant estimated by the LSM is D 4.68 LSM = (blue dots line), being significantly different from D = 15.0 predicted by the two-state approach with the RWA (green line). Thus, the two-state transition with the RWA (green line) hardly explains the non-resonant interstate oscillation in the harmonic confinement. For the resonant excitation in the harmonic confinement, one should remember that the three-state approach based on the simultaneous equation (33) consistently explains the interstate oscillation ( figure 9(b)). However, even the three-state transition approach cannot explain the interstate non-resonant oscillation (purple dots line in figure 13(b)). This discrepancy is caused because the harmonicity of the confinement causes the complicated multiple transitions even in the non-resonant SOI coupling. Only the parallel displacement 2 d is the common feature in the nonresonant excitation, irrespective of the difference in the confinements.

Conclusion
We computationally study the TD phenomena caused by the oscillating Rashba SOI. When the Rashba field is applied to the cylindrical hard-wall confinement system with having the resonant frequency, the typical Rabi oscillation results with causing the spin flipping. The TD Rashba field applied to the harmonic confinement system causes a characteristic beating, and separates the polarized spin state into α and β completely by the different frequencies. The perturbation approach up to the 4th order terms satisfactorily explains the characteristic beating found in the spin density.

Appendix B. Transition by TD scalar potential
The Rashba external oscillating field of t w t sin z 0 X = X ( ) applied perpendicular to the 2D quantum plane induces the oscillating scalar potential V z (t).
Here, A is the transformation constant of the Rashba external field to the electric one.
The present system as given in figure 1 is the 2D (or quasi-2D) system. For the later discussion, we give the thickness perpendicular to the 2D plane (z direction) by L z . An electron is, then, confined toward z direction by the hard-wall potential. Accordingly, we can represent the eigenfunction x y z , ,   One should note the transition matrix element is proportional to the thickness L z . Here, we suppose the 2D and/ or quasi-2D systems whose atomic layer thickness L z is extremely small. Therefore, we can conclude that the TD transition caused by the TD scalar potential V z (t) of equation (B.1) is negligible in the present TD phenomena. Figure B1. Illustration of the eigenstates confined by 2D and quasi-2D harmonic potential.
Furthermore, one should consider the following evidence. The transition by the TD scalar potential V z (t) occurs between the two quantum states in the different subband groups having the different n z values ( figure B1). Accordingly, the corresponding energy difference is inversely proportional to L z 2 . The small value of L z in the present 2D and/or quasi-2D system results in the energy difference, being extremely larger than the energy difference caused by the harmonic potential 0 w as shown in figure B1. This evidence also strengthens the negligibility of the transition by the TD scalar potential V z (t).

Appendix C. Time-propagator by Suzuki-Trotter decomposition
A general solution of the TD Schrödinger equation (5) is given as, where symbol T is a time-ordering operator and t D is a time-grid interval. Then, the time-propagator in equation (9) is further reduced into the following exponential operator; Appendix F. Three-state approach F.1. Rate equation We suppose that an electron is confined by the ideal harmonic potential 0 w and the external filed is applied having an oscillation frequency w 0 w d = + (δ is a detuning from the resonant frequency 0 w and 0 d w  ). We consider that the interstate transition occurs among the three states of nñ | , mñ | and kñ | in an energy order. Accordingly, we have the following three simultaneous differential equations, When the excitation energy is given by 0 w d + , the interstate matrix elements of equation (F.1) are given by,     We employ the unitary transformation and rewrite these coefficients C n (t), C m (t), C k (t) into b n (t), b m (t), b k (t) by, Consequently, equation (F.4) is rewritten by W -X is still conserve in the three-state approach with the RWA. Therefore, the three-state approximation with the RWA also results the linear relation