Two-level systems with periodic $N$-step driving fields: Exact dynamics and quantum state manipulations

In this work, we derive exact solutions of a dynamical equation, which can represent all two-level Hermitian systems driven by periodic $N$-step driving fields. For different physical parameters, this dynamical equation displays various phenomena for periodic $N$-step driven systems. The time-dependent transition probability can be expressed by a general formula that consists of cosine functions with discrete frequencies, and, remarkably, this formula is suitable for arbitrary parameter regimes. Moreover, only a few cosine functions (i.e., one to three main frequencies) are sufficient to describe the actual dynamics of the periodic $N$-step driven system. {Furthermore}, we find that a beating in the transition probability emerges when two (or three) main frequencies are similar. Some applications are also demonstrated in quantum state manipulations by periodic $N$-step driving fields.

This similar difficulty also exists in time-dependent periodically driven systems. To study the dynamics of periodically driven systems, one usually uses Floquet theory [39,40]. The basic idea is to deduce the timeindependent Floquet (effective) Hamiltonian and the corresponding micromotion operator, and then acquire * xia-208@163.com † yixx@nenu.edu.cn the effective dynamical evolution of the periodically driven systems. Therefore, some nontrivial physical properties (e.g., dynamical localization [11][12][13]) can be qualitatively derived. Nevertheless, it is worth mentioning that in the process of obtaining the analytical expressions of the Floquet Hamiltonian, one always makes certain approximations for the system parameters, e.g., the high-frequency limit, the weak-coupling regime, etc. When these approximations do not hold, the effective Hamiltonian of periodically driven systems cannot describe well the actual dynamics.
Recently, a general formalism [41], which extends the method introduced in Ref. [42], has been proposed to describe periodic-square-wave driven systems [43][44][45][46][47][48][49]. More specifically, the dynamics of periodic-squarewave driven systems can be divided into two parts: (i) the long-time-scale dynamics governed by an effective Hamiltonian and (ii) the short-time-scale dynamics governed by a micromotion operator. Nevertheless, this formalism is only suitable for the off-resonant regime (i.e., the frequency of the driving field is off resonant with any energy level separations of the static system) and applicable to a specific square wave that possesses the same time intervals.
Then, this method was generalized to the resonant regime [50] by performing a time-dependent unitary transformation. However, this approach might also be invalid when the frequency of the driving field is arbitrary.
Until now, for arbitrary parameter regimes, exact and analytical solutions of a periodically driven TLS are still difficult to obtain using nonperturbative approaches. Approximations are often applied in previous works to deal with periodically driven TLSs. For instance, in Ref. [51], approximations are needed to obtain the analytical expression of the Floquet Hamiltonian. Here, our goal is to obtain the accurate effective Hamiltonian and the micromotion operator for an arbitrary parameter regime without any approximations. As a result, we arXiv:2011.12473v2 [quant-ph] 26 Oct 2021 Stepwise evolution cos N1Θ = 0 Population swapping cos Θ = 0 Beat phenomenon |ω eff + ω − eff | |ω eff − ω − eff | manage to describe the exact dynamics at arbitrary time for periodic N -step driven (PNSD) systems. Moreover, we demonstrate that there are different phenomena in PNSD systems when physical parameters satisfy different conditions, as summarized in Table I. In this paper, by deriving the exact expression of the evolution operator, we systematically solve a class of dynamical equations, which can describe all two-level systems driven by periodic N -step driving fields. Note that the exact solution is valid for the full parameter range since we do not make any approximations. We derive general forms of the amplitudes and phases in the frequency domain. Moreover, we find that there always exist a beat-frequency-like phenomenon in the time-dependent transition probability (hereafter we call it the "beat phenomenon") in the PNSD system. In particular, the beat phenomenon can also emerge when adopting resonant pulses with different phases. We also demonstrate how to perform phase measurements by using this beat phenomenon.

II. PHYSICAL MODEL
We consider a two-level system interacting with a driving field. The dynamics is governed by the following Hamiltonian ( = 1) where Υ(t) = ( (t) cos θ(t), (t) sin θ(t), −∆(t)/2), σ = (σ x , σ y , σ z ) represents the Pauli matrices, and I is the identity matrix. The first and second terms of the second line of Eq. (1) can be regarded as the timedependent energy bias and tunneling amplitude of the TLS, respectively [2,26,30,36]. We do not restrict the values of ∆(t), (t), and θ(t) here, so this Hamiltonian can describe all Hermitian quantum systems with a two-level structure. Note that the coefficients ∆(t), (t), and θ(t) can represent different physical quantities in different systems. For example, ∆(t) and (t) are the quasiparticle momenta in condensed-matter systems [14,52,53], or the detuning and the Rabi frequency in an atomic system interacting with a laser field [54], or the dc and ac flux biases in superconductor qubits [55,56], etc. Without loss of generality, we assume that all coefficients are adjustable, and hereafter we denote ∆(t), (t), and θ(t) as the detuning, coupling strength, and phase.
The periodic driving field studied here has the form of a repeated N -step sequence S N : {H 1 , H 2 , . . . , H N }. For the nth step, the interaction time between the TLS and the driving field is τ n , and the Hamiltonian H n becomes (n = 1, . . . , N ) Thus, the period of the N -step sequence is Note that, for the sake of clarity, the time argument on the parameters are omitted hereafter if these parameters are time independent.
In this work, we also do not restrict the value of each duration τ n . As a result, because no quantities in H(t) can be treated as perturbations, both the Floquet theory [39,40] and the rotating-wave approximation [57][58][59][60][61] are invalid to calculate the analytical expression of the effective Hamiltonian for this class of systems.

III. EFFECTIVE HAMILTONIAN AND EVOLUTION OPERATOR
Different from the partitioning introduced in Refs. [41,42], we directly divide the evolution operator at arbitrary time t = t + N T (t < T , N = 0, 1, 2, . . . ) into two factors: Physically, the time-dependent micromotion operator M(t ) and the time-independent effective Hamiltonian H eff describe the short ("fast" part) and long ("slow" part) dynamics of the PNSD system, respectively. The eigenvalues of H eff are referred as the quasienergies of this system.
After some algebraic calculations, we obtain the expression for the micromotion operator M(t ) (see Appendix A for details), i.e., where The coefficients A n (t ), B n (t ), C n (t ), and D n (t ) satisfy the following relations: where τ n−1 < t < τ n , and
To derive the effective Hamiltonian H eff , we exploit the definition where U(T ) is the evolution operator within one period. It is found that H eff is a special case of the micromotion operator M(t), which satisfies H eff = M(T )/T . Thus, H eff is given by where From this effective Hamiltonian, one easily finds that the coherent destruction of tunneling (complete population transition) [1,2] would appear in the PNSD system when eff = 0 (∆ eff = 0); see Appendix B for details. Working with U(t ) and U(T ), we obtain the exact evolution operator U(t) for the PNSD system, which is expressed as (see Appendix A for details) The coefficients A(t), B(t), C(t), and D(t) are sin NΘ sin Θ , where Θ = arccos A N (T ) and the superscript T denotes the transposition. It is worth mentioning that we make no approximations in obtaining the micromotion operator and the effective Hamiltonian given by Eqs. (5) and (7). Hence, the evolution operator U(t) given by Eq. (8) is exact and valid for the full parameter range. Note that the evolution operator U(t) given by Eq. (8) is exact at any time, while it is only valid at some specific moments in Ref. [62]. This means that the protocol using rectangular pulses in Ref. [62] can be regarded as a special case of our results. Additionally, the phenomena of periodic evolution, stepwise evolution, and population swapping can also be observed in PNSD systems when the physical parameters satisfy cos N 1 Θ = 1, cos N 1 Θ = 0, and cos Θ = 0, respectively, where N 1 is an integer; see Appendix B for details.
In general situations, the first period of the pulse sequence may have a jump, e.g., from t 0 to t 1 in Fig. 1(a). When this jump is considered, an additional initial kick is needed to describe the dynamics, as discussed in Refs. [41,42], because this jump might affect the longtime-scale dynamics [see the solid curve in Fig. 1(b)]. In contrast, we integrate this jump into the effective Hamiltonian here, and do not require an additional kick. The result is that different starting times (phases) of the driving fields would lead to different effective Hamiltonians.
To address this issue more clearly, we employ a quantity λ to describe this jump [see Fig. 1(a)]. By using another partition of the sequence, we can regard the jump sequence as a complete (N + 1)-step sequence S N +1 : {H m , H m+1 , . . . , H N , H 1 , . . . , H m }. Here, the final step Hamiltonian is the same to the mth step Hamiltonian with the interaction time (1 − λ)τ 1 . Then, we readily obtain the effective Hamiltonian H eff (λ) for the jump sequences. It is shown in Fig. 1(b) that the long-time-scale dynamics (i.e., the envelope of the solid curve) caused by different starting times of the driving fields are well described by the effective Hamiltonian H eff (λ).

IV. ANALYTICAL EXPRESSIONS FOR THE TRANSITION PROBABILITY
Note that there are two fundamental frequencies in this system: the frequency of the N -step driving field ω T = 2π/T , (9) and the Rabi-like frequency of the effective Hamiltonian With U(t) in Eq. (8), we can express the time-dependent transition probability P 12 (t) from the state |1 to |2 in terms of cosine functions, yielding From a physical point of view, due to the highfrequency oscillations, most amplitudes {b l , b l } have extremely small values, and thus can be ignored. Therefore, only a few terms in Eq. (11) are kept, but these are sufficient to describe the actual dynamics of the PNSD system at arbitrary times.
A. Example: Two-step sequence

Expression for the transition probability
Taking the two-step sequence as an example, we can empirically write the expression of the transition probability as where we set ∆ eff = 0, and Here, p is the maximum transition probability for every Hamiltonian H n , and the dynamical phases are ϑ n = E n τ n /π. Note that the focus here is on the dynamical behaviors at arbitrary time rather than some specific moments (e.g., t = N T ) [62,63]. To quantify the validity of Eq. (13) in more detail, we adopt the following definition, Namely, ε m (t s ) represents the average error in the time interval [0, t s ] when adopting P m 12 (t) to describe the actual dynamics.
We plot ε m (t s ) as a function of the detuning ∆ 1 and ∆ 2 in Fig. 2(b), which confirms the classification in Fig. 2(a). To be more specific, in region I of Fig. 2(a), i.e., ∆ n / n 1, the actual dynamics of the PNSD system can be well described by the effective Hamiltonian (7) and shows Rabi-like oscillations with the frequency ω eff , as shown in Fig. 2(c). Note that most of the previously studied periodically driven systems [64][65][66] 7) is valid in region I for an arbitrary τ1. The beat phenomenon can emerge in regions II and III, and is more obvious on the light blue axis or the origin of coordinates (∆1 = 0 or ∆2 = 0). (b) ε m (ts) vs ∆n, where {θn} = {0, π/3}, 2 = 2 1, τ1 = 0.2/ 1, and τ2 is chosen to be ∆ eff = 0. The yellow solid curves, red dotted curves, and blue solid curves correspond to ε m (ts) = 0.05, 0.08, 0.1, respectively. The results verify the classification in Fig. 2 . The appreciable overlap between these two curves implies that the actual dynamics at arbitrary time can be well described by P m 12 (t). Note that P m 12 (t) contains three main frequencies in (e). regime. But, in contrast to those systems, the validity of H eff does not depend on the frequency of the driving field in this model [67,68]. In region II of Fig. 2(a), the effective Hamiltonian (7) is invalid but the actual dynamics can still be well described by Eq. (13), as shown in Fig. 2

(d).
Note that Eq. (13) might be invalid in some areas of region III, because ε m (t s ) is sufficiently large. However, it can be easily solved by including more cosine functions in Eq. (13). For instance, the expression of P m 12 (t) keeps three main frequencies (i.e., ω eff and ω ± eff = ω T /2 ± ω eff ) in Fig. 2(e), where the amplitudes and the phases are achieved by Eq. (12). More examples can be found in Table II of Appendix B.
2. Beat phenomenon in the transition probability Figure 2(b) clearly demonstrates that the actual dynamics of the PNSD two-level system can be well described by a superposition of two cosine functions in region II. An interesting finding is that when the sum of the frequencies of these two cosine functions is much greater than their difference, i.e., where P m 12 (t) contains three cosine functions (i.e., three similar main frequencies ω eff and ω ± eff ). Physically, this beat phenomenon results from the interference of the phases (ϑ n or θ n ), which is different from previous works, where the quantum beat originates from the interference between different transition channels [69][70][71]. The beat phenomenon cannot be explained by only using the effective Hamiltonian, because the micromotion operator M(t ) plays a key role in the transition process. Furthermore, the beat phenomenon only emerges in some special physical regimes [e.g., regions II and III in Fig. 2(a)], where the traditional methods, such as the Floquet theory [2,39,40] and the rotating-wave approximation [57][58][59][60][61], might not hold.

B. Beat phenomenon in periodic N -step driven systems
According to Eq. (13), the beat phenomenon is more obvious when λ = 1/2, as shown in Fig. 3(b). Actually, this is the case where the beating always exists in the PNSD system, when at least one Hamiltonian is in the largely detuned (resonant) regime; see Appendix D for details. By modulating the dynamical phases, the transition probability of the PNSD system can be   Fig. 7 3. Time-dependent transition probabilities P12(t) and P m 12 (t), which show distinct beat phenomena in different regions of Fig. 2(a). The red solid (blue dashed) curves are plotted by P12(t) [P m 12 (t)], and the appreciable overlap between these two curves shows the validity of Eq. (13). Note that P m 12 (t) contains three main frequencies in (e) and (f).
approximately given by where two similar main frequencies 1 and 1 are closely related to the effective quantity ω eff , and t p is the time-shifting factor. Moreover, the parity number of N determines the values of the frequencies 1 and 1 .
To be specific, when there are n 1 Hamiltonians in the resonant regime, we have

V. APPLICATIONS IN QUANTUM STATE MANIPULATIONS
The beat phenomenon observed in the PNSD system could find potential applications in homodyne detection, such as phase measurements. Taking the two-step sequence as an example, we assume that the physical parameters {∆ n , n } are given and the phase θ n is unknown. Then, each duration can be chosen to be τ n = π/2E n . In this situation, the beating in the transition probability would appear in this system, as shown in Fig. 4. According to Eq. (15), the physical parameters are closely interconnected, and have the relations (N = 2) Thus, we can map the phase θ n onto the beat frequency, enabling it to be easily measured by detecting the envelope waveform of the beat signal. In Fig. 4, we illustrate the time-dependent transition probability in the case of an unknown phase. These results demonstrate that the dynamical behaviors are different and we can acquire the beat frequency ω b by population measurements. Then, by inversely solving Eq. (16), one can obtain the phase of the system.
The periodic N -step driving field can be widely used in quantum coherent control, since all physical quantities can be exploited to manipulate quantum states. For example, to implement the complete transition from |1 to |2 , we can adopt multiple choices to design physical parameters in the PNSD system, such as the detuning ∆(t) [see the bottom panel of Fig. 5(a) This method is particularly useful for complicated quantum systems. Furthermore, for resonant pulses with different coupling strengths, the total evolution time of the complete transition reads , τ n < π 2 n , n = 1, 2. (17) Here, π/2 1 ≤ T ≤ π/2 2 , when 2 < 1 . By modulating the interaction time τ n or the coupling strength n , we can control the transition time from |1 to |2 , as shown in Fig. 5(d).
Finally, the coherent destruction of tunneling phe-   nomenon in PNSD systems offers us a possible way to implement the forbidden transition by resonant pulses. The physical parameters should satisfy θ 2 = θ 1 + π and 2 τ 2 = 1 τ 1 π/2.

VI. CONCLUSION
We have presented the exact solution of the evolution operator for the full parameter range in the periodic N -step driven TLS. Then, we display different physical parameter regimes for various phenomena in this system, including the well-known coherent destruction of tunneling, complete population transition, and beat phenomenon.
The time-dependent transition probability of the PNSD system can be expanded by a few cosine functions with discrete frequencies. Generally speaking, no more than three main frequencies are needed to describe the actual dynamics of the PNSD system. In addition, we derive the exact expression of the effective Hamiltonian of the PNSD system, which is valid in the largely detuned regime (i.e., ∆ n / n 1) or the high-frequency limit. When the micromotion operator contributes to the evolution, the effective Hamiltonian would be invalid. For this situation, we introduce a modified transition probability to describe the actual dynamics.
Moreover, we have demonstrated that the beat phenomenon always exists in the N -step driven TLS, which cannot be explained by only the effective Hamiltonian.
We have also described the beat phenomenon by the resonant pulses with different phases in the PNSD system, and given the general expressions of two similar frequencies when N > 2. The beat phenomenon originates from the interference of the phases and can be helpful for a driven quantum TLS used in quantum state manipulations. The physical model we consider here is a two-level system (TLS) driven by a periodic N -step driving field. The general form of its Hamiltonian reads ( = 1) where ζ(t) is the Heaviside function, τ 0 = 0, τ n (n = 1, . . . , N ) is the interaction time between the TLS and the nth-step driving field, and T = τ N + τ N −1 + · · · + τ 1 is the period of the N -step sequence. Generally speaking, the evolution operator of periodically driven systems can be divided into two parts: where the unitary operator P(t) has the same period as the system Hamiltonian. The eigenvalues of the effective Hamiltonian H eff are referred to as the quasienergies of the system. Different from both this partitioning and the partition introduced in Refs. [41,42], we directly divide the evolution operator of a periodic N -step driven system at arbitrary final time t = t + N T (t < T , N = 1, 2, . . . ) into two parts: where is the evolution operator within one period. U n (t ) = exp(−iH n t ) represents the evolution operator for the nth Hamiltonian, n = 1, . . . , N . Physically, the time-dependent micromotion operator M(t ) describes the short-timescale ("fast" part) dynamics of the system, and the time-independent effective Hamiltonian H eff describes the longtime-scale ("slow" part) dynamics of the system. In order to derive the expression of the evolution operator U(t) for the periodic N -step driven system, we need to calculate both the micromotion operator M(t ) and the effective Hamiltonian H eff . To be specific, the micromotion operator M(t ) = ∆ M (t )|2 2| + M (t ) exp[iθ M (t )]|1 2| + H.c. is defined by the evolution operator U(t ), such that where U n (t ) = exp (−iH n t ).
When t ∈ [0, τ 1 ], the evolution operator becomes Comparing the left-hand side and the right-hand side of Eq. (A5), we find Similarly, when t ∈ (τ 1 , τ 2 + τ 1 ], the evolution operator is According to Eq. (A6), we obtain the coefficients A 2 (t ), B 2 (t ), C 2 (t ), and D 2 (t ) As a result, we have It is easily found from Eqs. (A5) and (A6) that the matrix U(t ) has definite symmetry: the real (imaginary) parts of the diagonal (off-diagonal) elements are identical, while the imaginary (real) parts of the diagonal (off-diagonal) elements are opposite. Thus, we assume further that the evolution operator U(τ n−1 ) after the (n − 1)-step sequence can be written as where τ n = n k=1 τ k . When τ n−1 < t < τ n , the evolution operator U(t ) is According to Eq. (A8), we obtain the following recursive equations for the coefficients A n (t ), B n (t ), C n (t ), and D n (t ): where the vectors are A n−1 (t ) = (D n−1 (t ), C n−1 (t ), B n−1 (t )), E n = ( n cos θ n , n sin θ n , ∆ n /2)/E n , and E n = 2 n + ∆ 2 n /4. As a result, we have These are the exact solutions for the micromotion operator M(t ) when τ n−1 < t < τ n , n = 1, . . . , N − 1.
To derive the effective Hamiltonian, we need to use the definition We further find that the effective Hamiltonian H eff is a special case of the micromotion operator M(t), and it satisfies the relation: H eff = M(T )/T . Therefore, the general form of the time-independent effective Hamiltonian H eff for the periodic N -step driven system reads where the effective quantities are Note that the coefficient A N (T ) should be positive when the transition probability P 12 (t) from the state |1 to |2 during the time interval [0, T ] is less than P 12 π 2ω eff , where ω eff = 2 eff + ∆ 2 eff /4. Now, we can calculate the evolution operator U(t), which reads The coefficients A(t), B(t), C(t), and D(t) are given by where Θ = arccos A N (T ) and the superscript T denotes the transposition. It is worth mentioning that both the effective Hamiltonian in Eq. (A11) and the evolution operator in Eq. (A12) are exact, since we do not make any approximations.

Example: Two-step sequence
As an example, let us calculate the expression (A9) for the two-step sequence, where the Hamiltonian is Here, the period is T = τ 1 + τ 2 , and n = 0, 1, . . . , N . The coefficients A 2 (T ), B 2 (T ), C 2 (T ), and D 2 (T ) are Thus, the effective Hamiltonian H eff for the two-step sequence becomes where the effective quantities are For the two-step sequence with one jump, the coefficients A 3 (T ), B 3 (T ), C 3 (T ), and D 3 (T ) are Thus, the effective Hamiltonian H eff for the two-step sequence with one jump becomes where the effective quantities are Appendix B: Some other well-known phenomena in the periodic N -step driven system In this section, apart from the beat phenomenon mentioned in the main text, we demonstrate that some other well-known phenomena, such as coherent destruction of tunneling, population complete transition, periodic evolution, stepwise evolution, and population swapping, can also be recovered in the periodic N -step driven (PNSD) system.
Note that we only employ the effective Hamiltonian to study this issue, and thus investigate the dynamical behaviors at the time t = N T , where N is an integer. In addition, whether the transition probability can reach unity or not is determined by the quantity P l = 2 eff /ω 2 eff . The physical parameters should satisfy different conditions for different phenomena, as explained in the following. 3. Periodic evolution: cos N 1 Θ = 1, namely, N 1 Θ = π, where N 1 is an integer. 4. Stepwise evolution: cos N 1 Θ = 0, namely, N 1 Θ = π/2, where N 1 is the steps.
In Figs. 6-8, we plot the dynamical evolutions to demonstrate different phenomena by the two-step sequence. Furthermore, the results again demonstrate that the actual dynamics of the PNSD system can be well described by only employing P m 12 (t) with a few main frequencies, where the expressions can be found in Table II.  Table II. The red solid curves represent the actual dynamics and the blue dashed curves correspond to the predictions given by P m 12 (t). The appreciable overlap between the red solid and the blue dashed curves verifies that the actual dynamics of the PNSD system can be well described by only employing P m 12 (t) with a few main frequencies.  Table II. The appreciable overlap between the red solid and the blue dashed curves verifies that the actual dynamics of the PNSD system can be well described by only employing P m 12 (t) with a few main frequencies.
FIG. 8. Examples of the step evolution in the PNSD system. The parameters used here are found in Table II. The appreciable overlap between the red solid and the blue dashed curves verifies that the actual dynamics of the PNSD system can be well described by only employing P m 12 (t) with a few main frequencies.
Appendix C: Fourier transform of the time-dependent transition probability P12(t) In this section, we consider a two-step sequence as an example to demonstrate the Fourier transform of the timedependent transition probability P 12 (t). Note that it is easy to generalize to the case of N > 2. With the evolution operator U(t) in Eq. (A12), the time-dependent transition probability P 12 (t) from state |1 to |2 reads According to Eq. (A13), the expressions of C(t) and D(t) can be written as (k = 0, 1, 2, . . . ) where the time-independent coefficients are sin kΘ sin Θ , As a result, the exact solution of the time-dependent transition probability P 12 (t) from |1 to |2 becomes where the coefficients are Although P 12 (t) in Eqs. (C3) is a quasiperiodic function rather than a periodic function, the frequency spectrum is still discrete and the fundamental frequency is ω T = 2π/T . Moreover, the frequency spectrum is also associated with the effective Rabi frequency ω eff = 2 eff + ∆ 2 eff /4. By the Fourier transform, the expression of P 12 (t) can be expanded in terms of the cosine functions, yielding   In order to derive those coefficients, one needs to calculate the following integrals: Then, the coefficients arē one can obtain the limits for concrete physical parameters.
Appendix D: Beat phenomenon in periodic N -step driven systems In the main text, we demonstrate that the beat phenomenon can emerge in the periodic N -step driven system. Without loss of generality, here we set θ n = 0 and study the situation where the beat phenomenon is most obvious. That is, there exists a complete transition in the periodic N -step driven system (∆ eff = 0), and the transition probability can be approximately given by where the frequencies 1 and 1 are closely related to the physical quantities of the periodic N -step driven system, and t p is a time-shifting factor. First of all, we need to estimate the sum of all dynamical phases. If all Hamiltonians are in the largely detuned regime (|∆ n / n | 1, n = 1, 2, . . . , N ), the following relations hold: Substituting into the equation B N (T ) = 0, we have N n=1 E n τ n kπ, k = 1, 2, . . . . As a result, to implement a complete transition between levels, the accumulation of all dynamical phases approximately equal to an integer multiple of π if all Hamiltonians are in the largely detuned regime. When N is an even number, the choice of physical quantities is quite obvious. That is, we can set each dynamical phase to be E n τ n = π/2. Then we have N n=1 E n τ n = N π/2, which satisfies the complete transition condition. At the same time, in order for the beating to emerge, one should guarantee that at least one Hamiltonian is in the resonant regime and the remaining Hamiltonians are in the largely detuned regime. Then, the expressions of the frequencies 1 and 1 are empirically given by where n 1 represents the number of Hamiltonians that are in the resonant regime. Indeed, the beat phenomenon always exists in the periodic N -step driven system. Taking the two-step sequence as an example, let us estimate the values of the frequencies 1 and 1 . Assume that the Hamiltonian H 1 is in the resonant regime and the Hamiltonian H 2 is in the largely detuned regime. We have ∆ 1 = 0 and |∆ 2 / 2 | 1. Substituting those parameters into expression (A14), we find that the value of 1 can be approximated by since 2 / 2 2 + ∆ 2 2 /4 ≈ 0. Then, the value of 1 can be approximated by As a result, once cos(θ 1 − θ 2 ) = 0, the frequencies 1 and 1 are different but similar. Moreover, we have derived the expression of the transition probability in Eq. (C8), which demonstrates that the amplitudes are approximately equal to each other. Thus, the beating always exists in this system. Using a similar derivation, one can obtain the same results when N > 2. We plot in Fig. 9(a) the average error ε m (t s ) as a function of ∆ 6 for different n 1 . These results demonstrate that Eq. (D1) with the frequencies 1 and 1 given by Eqs. (D4) and (D5) can be used to describe well the actual dynamics of the periodic N -step driven system, because the maximum value of ε m (t s ) is not larger than 0.066. As examples, we plot in Figs. 9(b) and 9(c) the actual and predicted dynamical evolution of the system when n 1 = 1 and n 1 = 2, respectively.
When N is an odd number, the choice of physical quantities is slightly different from the even N case. That is, the dynamical phase caused by one Hamiltonian H k0 should satisfy E k0 τ k0 = π, while the remaining Hamiltonians still satisfy E n τ n = π/2 (n = k 0 ). Thus we can consider two cases according to the Hamiltonian H k0 : (i) The Hamiltonian H k0 is in the largely detuned regime. One easily finds that the evolution operator of the Hamiltonian H k0 is the identity operator; thus it does not affect the system dynamics. On the other hand, due to the largely detuned regime, the transition probability caused by this Hamiltonian is small enough. Thus, we can ignore this Hamiltonian and regard the N -step driven system as an (N − 1)-step driven system. As a result, the expressions of the coefficients 1 and 1 in this case are the same as in the situation where N is an odd number, which is governed by Eqs. (D4) and (D5).
(ii) The Hamiltonian H k0 is in the resonant regime. In this case, the dynamical phase caused by the Hamiltonian H k0 is twice the one caused by the Hamiltonian in the situation where N is an odd number. Therefore, when the number of Hamiltonians that are in the resonant regime is assumed to be n 1 , we should substitute (n 1 + 1) into Eqs. (D4) and (D5) to achieve the expressions of the frequencies 1 and 1 . Consequently, in this case the expressions can be written as We also plot in Fig. 10(a) the average error ε m (t s ) as a function of ∆ 7 with different n 1 . It demonstrates that Eq. (D1) with the frequencies 1 and 1 given by Eqs. (D8) and (D9) can be used to describe well the actual dynamics of the periodic N -step driven system, since the maximum value of ε m (t s ) is not larger than 0.082. As examples, we plot in Figs. 10(b) and 10(c) the actual and predicted dynamical evolution of the system when n 1 = 2 and n 1 = 3, respectively.