High-Order Approximation to Two-Level Systems with Quasiresonant Control

In this paper, we focus on high-order approximate solutions to two-level systems with quasi-resonant control. Firstly, we develop a high-order renormalization group (RG) method for Schrödinger equations. By this method, we get the high-order RG approximate solution in both resonance case and out of resonance case directly. Secondly, we introduce a time transformation to avoid the invalid expansion and get the high-order RG approximate solution in near resonance case. Finally, some numerical simulations are presented to illustrate the e ﬀ ectiveness of our RG method. We aim to provide a mathematically rigorous framework for mathematicians and physicists to analyze the high-order approximate solutions of quasi-resonant control problems.


Introduction
The dynamics of a two-level system interacting with a weak electromagnetic field is a relevant field of study in quantum optics. Scientists have been working to search for a satisfactory long-time approximation for such systems. Rotating wave approximation (RWA), which neglects the fast oscillating terms, is the most commonly used method for physicists (see [1]). Amazingly, slow oscillating terms are often removed by RWA in some models. For example, 2 cos θ = e iθ + e −iθ = e iθ ð1 + e −2iθ Þ ≈ e iθ in RWA, where the neglected term e −iθ is only two times speed away from e iθ (see [2]). Is it fast enough to be neglected? Despite RWA is a "rude" method, it usually gives a good approximation for some important models in quantum optics. However, if we can construct a more reliable approximation is an important issue in quantum optics. Scientists are always looking for the methods without RWA (for example [3][4][5]). In this paper, we aim to provide a mathematically rigorous frame-work for mathematicians and physicists to get satisfactory long-time approximate solutions for such systems.
Suppose that the open quantum systems are composed of a stable states jgi and an excited state jei with hg, ei = 0. The free Hamiltonian H 0 and the coupling Hamiltonian H 1 are defined by H 0 = λ g g j i g +λ e j je h ie h j, where λ g and λ e are the eigenvalues of the free Hamiltonian H 0 with respect to jgi and jei, respectively. The evolution of the state of a quantum system with quasi-resonant control can be described by the following Schrödinger equations where the wave function ψ belongs to a finite-dimensional Hilbert space, and the quasi-resonant control Uðt ; εÞ is a weak electromagnetic field control. Here, the quasiresonant control includes the resonance case, near resonance case, and out of resonance case. Rotating wave approximation is an effective method to deal with the quantum system with resonant control (see [6]). However, to our knowledge, it is not clear how to get the high-order approximate solutions rigorously for the quasi-resonant control. The renormalization group (RG) method proposed by Chen, Goldenfeld and Oono in [7,8] is a unified asymptotic analysis tool which reduces a singular perturbation problem to a more simple equation called the RG equation. Ziane in [9], DeVille et al. in [10], and Chiba in [11] gave the error estimate of RG approximate solutions rigorously. Chiba in [11][12][13] defined the high-order RG equation and the RG transformation to improve error estimate. The RG method has been further developed in [14][15][16][17][18].
In the present paper, we develop a high-order RG method for Schrödinger equations with quasi-resonant control. Compared to the pioneering work in [12], we adjust the construction of the m-order RG approximate solution by getting rid of the m-order oscillation part, which is more reasonable to give an explicit form. Our proof is concise and rigorous. The selection of integral constant in our method is also the most suitable for numerical calculation. As a direct application, we apply the high-order RG method to two-level systems in and out of resonance. The numerical simulation shows that the first-order RG approximate solution is usually good enough in the case of resonance, but the second-order RG approximate solution plays a key role in the two-level systems out of resonance.
The dynamic behavior of two-level systems near resonance is the most concerned problem, which is also the original intention of developing the high-order RG method. On the one hand, we need to find out whether the approximate solutions of the two-level systems near resonance have a consistent representation when there are more than one "≫" relationships in a physics article; on the other hand, we hope to find out how to avoid an invalid expansion to appear in the case of near resonance. Here, we call the expansion with small parameters in coefficients the invalid expansion. This paper introduces a time transformation technique to avoid the invalid expansion near resonance. Near resonance problems with two-scale case, high-order near resonant case, and high-order weak driving case are discussed carefully. The numerical simulations are presented to illustrate that the first-order RG approximate solution is usually good enough in the high-order near resonant case, but the second-order RG approximate solution plays a fundamental role in the two-scale case and high-order weak driving case. This paper is organized as follows. In Section 2, we develop a high-order RG method for Schrödinger equations with quasi-resonant control. In Section 3, high-order RG approximate solutions both in and out of resonance are presented. In Section 4, we introduce a time transformation technique to avoid the invalid expansion near resonance. High-order RG approximate solutions in several types of near resonance are discussed in detail.

Renormalization Group Method
In this section, we generalize the high-order RG method in [12] to Schrödinger equations where H 0 is a real diagonal matrix with different eigenvalues, GðtÞ is a continuous quasiperiodic Hermitian matrix, p, q ∈ ℕ + is the order of near resonance and weak drive, respectively, and κ > 0 is the detuning parameter. Equation (3) can be rewritten as where G k ðtÞ is the quasiperiodic skew-Hermitian matrix with the same quasiperiodic frequency. Put ψ = ψ 0 + εψ 1 + ε 2 ψ 2 + ⋯. Equating the coefficients of each ε i , we have Define the resonant part and oscillating part by Assumption 1. Suppose that R k , e iH 0 t h ðkÞ t appearing in the finite step RG process are quasiperiodic skew-Hermitian matrices, whose Fourier exponent has no accumulation points.
Define the mth-order RG equation and the mth-order RG transformation Let ψðtÞ is the solution of (3) with initial value ψð0Þ = ξ.

Advances in Mathematical Physics
Then ψ app ðtÞ ≜ α ðmÞ t ðWðtÞÞ − ε m h ðmÞ t WðtÞ satisfying Wð0Þ = ξ is the mth-order approximate solution of (3) with the following error estimate. Theorem 2. Let Assumption 1 hold. There exist positive constants ε 0 , C, and T such that for any 0 < ε < ε 0 , Proof. Firstly, we calculate the differential of α ðmÞ t ðWðtÞÞ with respect to t, It is easy to know that we have where Secondly, we prove that α ðmÞ t ðWðtÞÞ is the m-order approximate solution satisfying (10). Define UðtÞ ≜ e iH 0 t ½ψ ðtÞ − α ðmÞ t ðWðtÞÞ, we have whereSðt, εÞ = ∑ ∞ j=1 ε j−1 G m+j ðtÞ. The convergence ofSðt, εÞ is guaranteed by the form of (3). Since Sðt, εÞ andSðt, εÞ are quasiperiodic matrices, e iH 0 t is the unitary matrix, and there exist L > 0 and M > 0 such that holds. By Grönwall inequality, we have with 0 ≤ t ≤ ðT/εÞ: Finally, ψ app ðtÞ is the mth-order approximate solution satisfying (10). Since e iH 0 t h ðmÞ t is a quasiperiodic matrix, it is easy to know that Remark 3. Chiba defined α ðmÞ t ðWðtÞÞ with arbitrary integral constant as the m-order RG approximate solution for a more general system (see [12]). Here, we use ψ app ðtÞ as the m-order approximate solution for the Schrödinger equation (3), which is easier to get the explicit expression for two-level systems. The selection of integral constant is suitable for numerical calculation.

Two-Level Systems in and out of Resonance
In this section, we show how to apply the RG method to twolevel systems in and out of resonance directly.

In Resonance. Let the quasi-resonant control
where ω = λ e − λ g is the transition frequency, and v ∈ ℂ is the complex amplitude. The two-level systems (2) becomes Let ψ = ψ 0 + εψ 1 + ε 2 ψ 2 + ⋯: Equating the coefficients 3 Advances in Mathematical Physics where G 1 ðtÞ = −iuH 1 . Firstly, we calculate the first-order RG approximate solution. Since the first-order resonant part is equal to The first-order RG approximate solution is where W is the solution of the first-order RG equation ðd/d tÞW = εR 1 W with Wð0Þ = ξ. Secondly, we calculate the second-order RG approximate solution. By (7), ð26Þ Now, the second-order resonant part is equal to by (6), Thus, the second-order RG approximate solution is where W is the solution of the second-order RG equation Example 4. Consider the two-level system (21) with initial value ξ = ð ffiffi ffi 2 p /2, ffiffi ffi 2 p /2Þ. Let λ e = 5, λ g = 1, v = 1 + i, and ε = 0:05. The simulations of Figure 1 illustrate that first-order RG approximate solution is as good as the second-order RG approximate solution.

Advances in Mathematical Physics
where ω = λ e − λ g is the transition frequency, Δ is the detuning frequency, and v ∈ ℂ is the complex amplitude. The two-level system (2) becomes Let ψ = ψ 0 + εψ 1 + ε 2 ψ 2 + ⋯: Equating the coefficients of each ε i , we have where G 1 ðtÞ = −iuH 1 . Firstly, we calculate the first-order RG approximate solution. Since the first-order resonant part is equal to The first-order RG approximate solution is where W is the solution of the first-order RG equation ðd/d tÞW = εR 1 W with Wð0Þ = ξ. Secondly, we calculate the second-order RG approximate solution. By (7), Since the second-order resonant part is equal to Thus, the second-order RG approximate solution is where W is the solution of the second-order RG equation ðd/dtÞW = εR 1 W + ε 2 R 2 W with Wð0Þ = ξ.
Example 5. Consider the two-level system (31) with initial value ξ = ð ffiffi ffi 2 p /2, ffiffi ffi 2 p /2Þ. Let λ e = 5, λ g = 1, v = 1 + i, ε = 0:05, and Δ = 1. The simulations of Figure 2 illustrate that the first-order approximate solution is not a reliable one in the two-level system out of resonance, while the second-order approximate solution is a better one.

Near Resonance Problems with Two Scales
In this section, we will discuss how to apply RG method reasonably to near resonance problem with two scales. Take the quasi-resonant control Uðt ; εÞ = εuðt ; εÞ with where ω = λ e − λ g is the transition frequency, δ is the detuning frequency, and v ∈ ℂ is the complex amplitude. The twolevel system (2) becomes LetH 0 = λ e jeihej+λ g jgihgj+εδjgihgj,ω = ω − εδ, and ϕ = ϕ 0 + εϕ 1 + ε 2 ϕ 2 + ⋯: Formally, we have the following expansion with Because there are actually small parameters inH 0 and u, we call the above expression invalid expansion. If we calculate the first-order RG approximate solution formally, we have the first-order resonant part jgihej+v * j eihgj which is exactly the result obtained by RWA (see [19]).
Here, the first-order RG approximate solution can be expressed as follows.
where W is the solution of the first-order RG equation

Valid Expansion.
Although the so-called invalid expansion is sometimes a good approximation, shall we construct a valid expansion and find a more reliable long-time higher-order approximate solution? Here, we introduce a quasi-resonant time transformation technique s ≜ ð1 − εκÞt with κ = δ/ω. Now, the system (41) is equal to where u is the resonant control with respect to s: Let ψ = ψ 0 + εψ 1 + ε 2 ψ 2 + ⋯: Equating the coefficients of each ε i , we have where G k ðsÞ = −iκ k−1 ðκH 0 + uH 1 Þ. Firstly, we calculate the first-order RG approximate solution.

Advances in Mathematical Physics
Since the second-order resonant part is equal to Thus, the second-order RG equation and the secondorder RG approximate solution are respectively, where Wð0Þ = ξ, and s = ð1 − εκÞt.
Example 7. Consider a spin-half system where v~OðεÞ and Δ~OðεÞ are real numbers, σ z = jeihej−j gihgj, σ x = jgihej+jeihgj, and ω eg is the energy difference between jei and jgi. Let λ e = ω eg /2, and λ g = −ω eg /2, we have The first-order RG equation is equal to where s = ð1 − ðΔ/ω eg ÞÞt, v, and Δ are Rabi frequency and detuning, respectively. The first-order generalized Rabi frequency is Furthermore, we have the second-order RG equation Remark 8. Our method can be applied to the spin-half system directly. In this article, we only take near resonance problems with two scales as an example.

Near Resonance Problems with Three Scales
Physicists like to use "≫" to indicate different time scales. If there are more than one "≫" relationships in near resonance problems, how can we analyze the long-time dynamic behavior reasonably? In this section, we shall apply our method to two-level systems in high-order near resonant case and highorder weak driving case.
Thus, the first-order RG equation and the first-order RG approximate solution are respectively, where Wð0Þ = ξ, and s = ð1 − ε 2 κÞt. Secondly, we calculate the second-order RG approximate solution. By (7) The second-order resonant part is equal to Thus, the second-order RG equation and the secondorder RG approximate solution are respectively, where Wð0Þ = ξ, and s = ð1 − ε 2 κÞt.
respectively, where Wð0Þ = ξ, and s = ð1 − εκÞt. Secondly, we calculate the second-order RG approximate solution. By (7), the second-order resonant part is equal to Example 10. Consider the two-level system in high-order weak driving case with initial value ξ = ð ffiffi ffi 2 p /2, ffiffi ffi 2 p /2Þ. Let λ e = 5, λ g = 1, v = 1 + i, ε = 0:05, and δ = 1. The simulations of Figure 5 illustrate that the first-order approximate solution is not reliable in two-level systems with the high-order weak driving control. But the second-order approximate solution is a more accurate one.

Data Availability
The 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.