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

Authors: Lin Wang; Jian Zu
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

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 effectiveness 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.


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## Body

## 1. 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, 2cosθ=eiθ+e−iθ=eiθ1+e−2iθ≈eiθ in RWA, where the neglected term e−iθ is only two times speed away from eiθ (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 [3–5]). In this paper, we aim to provide a mathematically rigorous framework 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 statesg and an excited state e with g,e=0. The free Hamiltonian H0 and the coupling Hamiltonian H1 are defined by(1)H0=λggg+λeee,H1=ge+eg,where λg and λe are the eigenvalues of the free Hamiltonian H0 with respect to g and e, respectively. The evolution of the state of a quantum system with quasi-resonant control can be described by the following Schrödinger equations(2)iddtψ=H0+Ut;εH1ψ,where the wave function ψ belongs to a finite-dimensional Hilbert space, and the quasi-resonant control Ut;ε is a weak electromagnetic field control. Here, the quasi-resonant 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 [11–13] defined the high-order RG equation and the RG transformation to improve error estimate. The RG method has been further developed in [14–18].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 Section2, 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.


## 2. Renormalization Group Method

In this section, we generalize the high-order RG method in [12] to Schrödinger equations(3)1−εpκ·iddtψ=H0ψ+εqGtψ,ψ∈ℂn,0<ε≪1,where H0 is a real diagonal matrix with different eigenvalues, Gt 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(4)ddtψ=−iH0ψ+εG1tψ+⋯+εkGktψ+⋯,where Gkt is the quasiperiodic skew-Hermitian matrix with the same quasiperiodic frequency. Put ψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(5)ε0:ddtψ0+iH0ψ0=0,εk:ddtψk+iH0ψk=∑j=0k−1Gk−jtψj,k≥1.Define the resonant part and oscillating part by(6)Rk≜limt→∞1t∫0teiH0sGkse−iH0s+∑j=1k−1eiH0sGk−jshsj−∑j=1k−1eiH0shsjRk−jds,(7)htk≜e−iH0t∫0teiH0sGkse−iH0s+∑j=1k−1eiH0sGk−jshsj−∑j=1k−1eiH0shsjRk−j−Rkds.Assumption 1.
Suppose thatRk, eiH0thtk appearing in the finite step RG process are quasiperiodic skew-Hermitian matrices, whose Fourier exponent has no accumulation points.
Define themth-order RG equation(8)dWtdt=εR1Wt+ε2R2Wt+⋯εmRmWt,Wt∈ℂn,and the mth-order RG transformation(9)αtmWt≜e−iH0tWt+εht1Wt+⋯+εmhtmWt.
Letψt is the solution of (3) with initial value ψ0=ξ. Then ψappt≜αtmWt−εmhtmWt satisfying W0=ξ is the mth-order approximate solution of (3) with the following error estimate.Theorem 2.
Let Assumption1 hold. There exist positive constants ε0, C, and T such that for any 0<ε<ε0,(10)ψt−ψappt≤Cεm,0≤t≤Tε.Proof.
Firstly, we calculate the differential ofαtmWt with respect to t,(11)ddtαtmWt=−iH0e−iH0tWt+e−iH0tddtWt+∑k=1mεkddthtkWt.
It is easy to know that(12)ddthtkWt=−iH0htkWt+Gkte−iH0tWt+∑j=1k−1Gk−jthtjWt−∑j=1k−1htjRk−jWt−e−iH0tRkWt+htk∑j=1mεjRjWt.
Since(13)∑k=2mεk∑j=1k−1Gk−jthtjWt=∑k=1m−1εkGkt∑j=1m−kεjhtjWt,∑k=2mεk∑j=1k−1htjRk−jWt=∑k=1m−1εkhtk∑j=1m−kεjRjWt,we have(14)ddtαtmWt=−iH0αtmWt+∑k=1mεkGktαtmWt+εm+1St,εWt,where(15)St,ε=∑k=1m∑j=1kεj−1htkRm−k+j−Gkthtm−k+j.
Secondly, we prove thatαtmWt is the m-order approximate solution satisfying (10). Define Ut≜eiH0tψt−αtmWt, we have(16)dUdt=∑k=1mεkeiH0tGkte−iH0tU−εm+1eiH0tSt,εWt+εm+1eiH0tS~t,εψt,where S~t,ε=∑j=1∞εj−1Gm+jt. The convergence of S~t,ε is guaranteed by the form of (3). Since St,ε and S~t,ε are quasiperiodic matrices, eiH0t is the unitary matrix, and there exist L>0 and M>0 such that(17)Ut≤εL∫0tUsds+εm+1Mt,holds. By Grönwall inequality, we have(18)Ut≤MLεmeεLt−1≤MLεmeLT−1,with 0≤t≤T/ε.
Finally,ψappt is the mth-order approximate solution satisfying (10). Since eiH0thtm is a quasiperiodic matrix, it is easy to know that(19)ψt−ψappt≤ψt−αtmWt+εmhtmWt≤Cεm,0≤t≤Tε.Remark 3.
Chiba definedαtmWt with arbitrary integral constant as the m-order RG approximate solution for a more general system (see [12]). Here, we use ψappt 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.


## 3. Two-Level Systems in and out of Resonance

In this section, we show how to apply the RG method to two-level systems in and out of resonance directly.


### 3.1. In Resonance

Let the quasi-resonant controlUt;ε=εut with(20)ut≜veiωt+v∗e−iωt,where ω=λe−λg is the transition frequency, and v∈ℂ is the complex amplitude. The two-level systems (2) becomes(21)iddtψ=H0+εuH1ψ.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(22)ε0:ddtψ0+iH0ψ0=0,εk:ddtψk+iH0ψk=G1tψk−1,k≥1,where G1t=−iuH1.Firstly, we calculate the first-order RG approximate solution. Since(23)eiH0τuH1e−iH0τ=v+v∗e−i2ωτge+vei2ωτ+v∗eg,the first-order resonant part is equal to(24)R1=limt→∞1t∫0teiH0τG1τe−iH0τdτ=−ivge+v∗eg.The first-order RG approximate solution is(25)ψ1stt=e−iH0tWt,0≤t≤Tε,where W is the solution of the first-order RG equation d/dtW=εR1W with W0=ξ.Secondly, we calculate the second-order RG approximate solution. By (7),(26)eiH0tht1=∫0teiH0τG1τe−iH0τ−R1dτ=v∗2ωe−i2ωt−1ge−v2ωei2ωt−1eg.Since(27)eiH0τG1τhτ1=−iv22ω1−ei2ωτ+v∗22ωe−i2ωτ−1ee+iv22ωei2ωτ−1+v22ω1−e−i2ωτgg,eiH0τhτ1R1=−iv∗22ωe−i2ωt−1gg−v22ωei2ωt−1ee.Now, the second-order resonant part is equal to by (6),(28)R2=limt→∞1t∫0teiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iv2−v2−v∗22ωee−gg.Thus, the second-order RG approximate solution is(29)ψ2ndt=e−iH0tWt+εht1Wt,0≤t≤Tε,where W is the solution of the second-order RG equation d/dtW=εR1W+ε2R2W with W0=ξ.Example 4.
Consider the two-level system (21) with initial value ξ=2/2,2/2. Let λe=5, λg=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.Figure 1
RG vs. Runge-Kutta approximate solutions of two-level systems in resonance.


### 3.2. Out of Resonance

We take the quasi-resonant controlUt;ε=εut with(30)ut≜veiω−Δt+v∗e−iω−Δt,where ω=λe−λg is the transition frequency, Δ is the detuning frequency, and v∈ℂ is the complex amplitude. The two-level system (2) becomes(31)iddtψ=H0+εuH1ψ.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(32)ε0:ddtψ0+iH0ψ0=0,εk:ddtψk+iH0ψk=G1tψk−1,k≥1,where G1t=−iuH1.Firstly, we calculate the first-order RG approximate solution. Since(33)eiH0τuH1e−iH0τ=ve−iΔτ+v∗e−i2ω−Δτge+vei2ω−Δτ+v∗eiΔτeg,the first-order resonant part is equal to(34)R1=limt→∞1t∫0teiH0τG1τe−iH0τdτ=0.The first-order RG approximate solution is(35)ψ1stt=e−iH0tWt,0≤t≤Tε,where W is the solution of the first-order RG equation d/dtW=εR1W with W0=ξ.Secondly, we calculate the second-order RG approximate solution. By (7),(36)eiH0tht1=∫0teiH0τG1τe−iH0τ−R1dτ=vΔe−iΔt−1+v∗2ω−Δe−i2ω−Δt−1ge−v2ω−Δei2ω−Δt−1+v∗ΔeiΔt−1eg.Since(37)eiH0τG1τhτ1=−iv2Δei2ω−Δτ−ei2ω−Δτ+v2Δ1−eiΔτ+v22ω−Δ1−ei2ω−Δτ+v∗22ω−Δe−i2ω−2Δτ−eiΔτee+iv22ω−Δei2ω−2Δτ−e−iΔτ+v22ω−Δ1−e−i2ω−Δτ+v2Δ1−e−iΔτ+v∗2Δe−i2ω−2Δτ−e−i2ω−Δτgg,the second-order resonant part is equal to(38)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iv2Δ+v22ω−Δee−gg.Thus, the second-order RG approximate solution is(39)ψ2ndt=e−iH0tWt+εht1Wt,0≤t≤Tε,where W is the solution of the second-order RG equation d/dtW=εR1W+ε2R2W with W0=ξ.Example 5.
Consider the two-level system (31) with initial value ξ=2/2,2/2. Let λe=5, λg=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.Figure 2
RG vs. Runge-Kutta approximate solutions of two-level systems out of resonance.


## 3.1. In Resonance

Let the quasi-resonant controlUt;ε=εut with(20)ut≜veiωt+v∗e−iωt,where ω=λe−λg is the transition frequency, and v∈ℂ is the complex amplitude. The two-level systems (2) becomes(21)iddtψ=H0+εuH1ψ.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(22)ε0:ddtψ0+iH0ψ0=0,εk:ddtψk+iH0ψk=G1tψk−1,k≥1,where G1t=−iuH1.Firstly, we calculate the first-order RG approximate solution. Since(23)eiH0τuH1e−iH0τ=v+v∗e−i2ωτge+vei2ωτ+v∗eg,the first-order resonant part is equal to(24)R1=limt→∞1t∫0teiH0τG1τe−iH0τdτ=−ivge+v∗eg.The first-order RG approximate solution is(25)ψ1stt=e−iH0tWt,0≤t≤Tε,where W is the solution of the first-order RG equation d/dtW=εR1W with W0=ξ.Secondly, we calculate the second-order RG approximate solution. By (7),(26)eiH0tht1=∫0teiH0τG1τe−iH0τ−R1dτ=v∗2ωe−i2ωt−1ge−v2ωei2ωt−1eg.Since(27)eiH0τG1τhτ1=−iv22ω1−ei2ωτ+v∗22ωe−i2ωτ−1ee+iv22ωei2ωτ−1+v22ω1−e−i2ωτgg,eiH0τhτ1R1=−iv∗22ωe−i2ωt−1gg−v22ωei2ωt−1ee.Now, the second-order resonant part is equal to by (6),(28)R2=limt→∞1t∫0teiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iv2−v2−v∗22ωee−gg.Thus, the second-order RG approximate solution is(29)ψ2ndt=e−iH0tWt+εht1Wt,0≤t≤Tε,where W is the solution of the second-order RG equation d/dtW=εR1W+ε2R2W with W0=ξ.Example 4.
Consider the two-level system (21) with initial value ξ=2/2,2/2. Let λe=5, λg=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.Figure 1
RG vs. Runge-Kutta approximate solutions of two-level systems in resonance.


## 3.2. Out of Resonance

We take the quasi-resonant controlUt;ε=εut with(30)ut≜veiω−Δt+v∗e−iω−Δt,where ω=λe−λg is the transition frequency, Δ is the detuning frequency, and v∈ℂ is the complex amplitude. The two-level system (2) becomes(31)iddtψ=H0+εuH1ψ.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(32)ε0:ddtψ0+iH0ψ0=0,εk:ddtψk+iH0ψk=G1tψk−1,k≥1,where G1t=−iuH1.Firstly, we calculate the first-order RG approximate solution. Since(33)eiH0τuH1e−iH0τ=ve−iΔτ+v∗e−i2ω−Δτge+vei2ω−Δτ+v∗eiΔτeg,the first-order resonant part is equal to(34)R1=limt→∞1t∫0teiH0τG1τe−iH0τdτ=0.The first-order RG approximate solution is(35)ψ1stt=e−iH0tWt,0≤t≤Tε,where W is the solution of the first-order RG equation d/dtW=εR1W with W0=ξ.Secondly, we calculate the second-order RG approximate solution. By (7),(36)eiH0tht1=∫0teiH0τG1τe−iH0τ−R1dτ=vΔe−iΔt−1+v∗2ω−Δe−i2ω−Δt−1ge−v2ω−Δei2ω−Δt−1+v∗ΔeiΔt−1eg.Since(37)eiH0τG1τhτ1=−iv2Δei2ω−Δτ−ei2ω−Δτ+v2Δ1−eiΔτ+v22ω−Δ1−ei2ω−Δτ+v∗22ω−Δe−i2ω−2Δτ−eiΔτee+iv22ω−Δei2ω−2Δτ−e−iΔτ+v22ω−Δ1−e−i2ω−Δτ+v2Δ1−e−iΔτ+v∗2Δe−i2ω−2Δτ−e−i2ω−Δτgg,the second-order resonant part is equal to(38)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iv2Δ+v22ω−Δee−gg.Thus, the second-order RG approximate solution is(39)ψ2ndt=e−iH0tWt+εht1Wt,0≤t≤Tε,where W is the solution of the second-order RG equation d/dtW=εR1W+ε2R2W with W0=ξ.Example 5.
Consider the two-level system (31) with initial value ξ=2/2,2/2. Let λe=5, λg=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.Figure 2
RG vs. Runge-Kutta approximate solutions of two-level systems out of resonance.


## 4. 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 controlUt;ε=εut;ε with(40)ut;ε≜veiω−εδt+v∗e−iω−εδt,where ω=λe−λg is the transition frequency, δ is the detuning frequency, and v∈ℂ is the complex amplitude. The two-level system (2) becomes(41)iddtψ=H0+εuH1ψ.


### 4.1. Invalid Expansion

gg+λggg An intuitive way to deal with such problems is to divide H0 into two parts:(42)H0=λeeeλggg+εδgg+−εδgg.LetH~0=λeee+λggg+εδgg, ω~=ω−εδ, and ϕ=ϕ0+εϕ1+ε2ϕ2+⋯. Formally, we have the following expansion(43)ε0:ddtϕ0+iH~0ϕ0=0,εk:ddtϕk+iH~0ϕk=G1tϕk−1,k≥1,with G1t=−i−δgg+uH1, where u=veiω~t+v∗e−iω~t. Because there are actually small parameters in H~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 partge+v∗eg(44)R1=limt→∞1s∫0teiH~0τG1τe−iH~0τdτ=−i−δgg+vge+v∗eg,which is exactly the result obtained by RWA (see [19]). Here, the first-order RG approximate solution can be expressed as follows.(45)ϕ1st=e−iH~0tWt,0≤t≤Tε,where W is the solution of the first-order RG equation d/dtW=εR1W with W0=ξ.


### 4.2. 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 techniques≜1−εκt with κ=δ/ω. Now, the system (41) is equal to(46)1−εκddsψ=−iH0+εuH1ψ,where u is the resonant control with respect to s:(47)us=veiωs+v∗e−iωs.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(48)ε0:ddsψ0+iH0ψ0=0,ε1:ddsψ1+iH0ψ1=G1sψ0,ε2:ddsψ2+iH0ψ2=G1sψ1+G2sψ0,⋯where Gks=−iκk−1κH0+uH1.Firstly, we calculate the first-order RG approximate solution.(49)R1=lims→∞1s∫0seiH0τG1τe−iH0τdτ=−iκH0+H2,where H2≜vge+v∗eg. Thus, the first-order RG equation and first-order RG approximate solution are(50)ddsW=εR1Wandψ1st=e−iH0sWs,0≤s≤Tε,respectively, where w0=ξ, and s=1−εκt.Secondly, we calculate the second-order RG approximate solution. By (7),(51)eiH0shs1=∫0seiH0τG1τe−iH0τ−R1dτ=v∗2ωe−i2ωs−1gev2ωei2ωs−1eg..Since(52)eiH0τG1τhτ1=−iv22ω1−ei2ωτee−v22ω1−e−i2ωτgg+v∗22ωe−i2ωτ−1ee−v22ωei2ωτ−1gg+κλgv∗2ωe−i2ωτ−1ge−κλev2ωei2ωτ−1eg,eiH0τhτ1R1=−iκλev∗2ωe−i2ωτ−1ge−κλgv2ωei2ωτ−1eg+v∗22ωe−i2ωτ−1gg−v22ωei2ωτ−1ee,the second-order resonant part is equal to(53)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iκ2H0+κH2+κv∗2ge+κv2eg+v2−v2−v∗22ωee−gg.Thus, the second-order RG equation and the second-order RG approximate solution are(54)ddsW=εR1W+ε2R2Wandψ2nd=e−iH0sWs+εhs1Ws,0≤s≤Tε,respectively, where W0=ξ, and s=1−εκt.Example 6.
Consider the two-level system (41) with initial value ξ=2/2,2/2. Let λe=5, λg=1, v=1+i, ε=0.05, and δ=1. The simulations of Figure 3 illustrate that the first-order approximate solution is not good enough in two-level systems near resonance. But the second-order approximate solution is a more reliable one.Example 7.
Consider a spin-half system(55)ddtψ=−iωeg2σz+v2eiωeg−Δt+e−iωeg−Δtσxψ,where v~Oε and Δ~Oε are real numbers, σz=ee−gg, σx=ge+eg, and ωeg is the energy difference between e and g. Let λe=ωeg/2, and λg=−ωeg/2, we have(56)H0=ωeg2ee−gg=ωeg2σz.
The first-order RG equation is equal to(57)dWds=−iΔ2σz+v2σxW,where s=1−Δ/ωegt, v, and Δ are Rabi frequency and detuning, respectively. The first-order generalized Rabi frequency is(58)Ω1st=v2+Δ2.
Furthermore, we have the second-order RG equation(59)dWds=−iΔ2+Δ22ωeg−v28ωegσz+v2+3Δv4ωegσxW.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.Figure 3
Valid and invalid RG approximate solutions of two-level systems near resonance.


## 4.1. Invalid Expansion

gg+λggg An intuitive way to deal with such problems is to divide H0 into two parts:(42)H0=λeeeλggg+εδgg+−εδgg.LetH~0=λeee+λggg+εδgg, ω~=ω−εδ, and ϕ=ϕ0+εϕ1+ε2ϕ2+⋯. Formally, we have the following expansion(43)ε0:ddtϕ0+iH~0ϕ0=0,εk:ddtϕk+iH~0ϕk=G1tϕk−1,k≥1,with G1t=−i−δgg+uH1, where u=veiω~t+v∗e−iω~t. Because there are actually small parameters in H~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 partge+v∗eg(44)R1=limt→∞1s∫0teiH~0τG1τe−iH~0τdτ=−i−δgg+vge+v∗eg,which is exactly the result obtained by RWA (see [19]). Here, the first-order RG approximate solution can be expressed as follows.(45)ϕ1st=e−iH~0tWt,0≤t≤Tε,where W is the solution of the first-order RG equation d/dtW=εR1W with W0=ξ.


## 4.2. 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 techniques≜1−εκt with κ=δ/ω. Now, the system (41) is equal to(46)1−εκddsψ=−iH0+εuH1ψ,where u is the resonant control with respect to s:(47)us=veiωs+v∗e−iωs.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(48)ε0:ddsψ0+iH0ψ0=0,ε1:ddsψ1+iH0ψ1=G1sψ0,ε2:ddsψ2+iH0ψ2=G1sψ1+G2sψ0,⋯where Gks=−iκk−1κH0+uH1.Firstly, we calculate the first-order RG approximate solution.(49)R1=lims→∞1s∫0seiH0τG1τe−iH0τdτ=−iκH0+H2,where H2≜vge+v∗eg. Thus, the first-order RG equation and first-order RG approximate solution are(50)ddsW=εR1Wandψ1st=e−iH0sWs,0≤s≤Tε,respectively, where w0=ξ, and s=1−εκt.Secondly, we calculate the second-order RG approximate solution. By (7),(51)eiH0shs1=∫0seiH0τG1τe−iH0τ−R1dτ=v∗2ωe−i2ωs−1gev2ωei2ωs−1eg..Since(52)eiH0τG1τhτ1=−iv22ω1−ei2ωτee−v22ω1−e−i2ωτgg+v∗22ωe−i2ωτ−1ee−v22ωei2ωτ−1gg+κλgv∗2ωe−i2ωτ−1ge−κλev2ωei2ωτ−1eg,eiH0τhτ1R1=−iκλev∗2ωe−i2ωτ−1ge−κλgv2ωei2ωτ−1eg+v∗22ωe−i2ωτ−1gg−v22ωei2ωτ−1ee,the second-order resonant part is equal to(53)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iκ2H0+κH2+κv∗2ge+κv2eg+v2−v2−v∗22ωee−gg.Thus, the second-order RG equation and the second-order RG approximate solution are(54)ddsW=εR1W+ε2R2Wandψ2nd=e−iH0sWs+εhs1Ws,0≤s≤Tε,respectively, where W0=ξ, and s=1−εκt.Example 6.
Consider the two-level system (41) with initial value ξ=2/2,2/2. Let λe=5, λg=1, v=1+i, ε=0.05, and δ=1. The simulations of Figure 3 illustrate that the first-order approximate solution is not good enough in two-level systems near resonance. But the second-order approximate solution is a more reliable one.Example 7.
Consider a spin-half system(55)ddtψ=−iωeg2σz+v2eiωeg−Δt+e−iωeg−Δtσxψ,where v~Oε and Δ~Oε are real numbers, σz=ee−gg, σx=ge+eg, and ωeg is the energy difference between e and g. Let λe=ωeg/2, and λg=−ωeg/2, we have(56)H0=ωeg2ee−gg=ωeg2σz.
The first-order RG equation is equal to(57)dWds=−iΔ2σz+v2σxW,where s=1−Δ/ωegt, v, and Δ are Rabi frequency and detuning, respectively. The first-order generalized Rabi frequency is(58)Ω1st=v2+Δ2.
Furthermore, we have the second-order RG equation(59)dWds=−iΔ2+Δ22ωeg−v28ωegσz+v2+3Δv4ωegσxW.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.Figure 3
Valid and invalid RG approximate solutions of two-level systems near resonance.


## 5. 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 high-order weak driving case.


### 5.1. High-Order Near Resonant Case

LetUt;ε=εut;ε with(60)ut;ε≜veiω−ε2δt+v∗e−iω−ε2δt.Introduce a quasi-resonant time transformations≜1−ε2κt with κ=δ/ω. The two-level system (2) becomes(61)1−ε2κddsψ=−iH0+εuH1ψ,where u is the resonant control with respect to s:(62)us=veiωs+v∗e−iωs.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(63)ε0:ddsψ0+iH0ψ0=0,ε1:ddsψ1+iH0ψ1=G1sψ0,ε2:ddsψ2+iH0ψ2=G1sψ1+G2sψ0,⋯where G1s=−iuH1, and G2s=−iκH0.Firstly, we calculate the first-order RG approximate solution.(64)R1=lims→∞1s∫0seiH0τG1τe−iH0τdτ=−iH2.Thus, the first-order RG equation and the first-order RG approximate solution are(65)ddsW=εR1Wandψ1st=e−iH0sWs,0≤s≤Tε,respectively, where W0=ξ, and s=1−ε2κt.Secondly, we calculate the second-order RG approximate solution. By (7)(66)eiH0shs1=∫0seiH0τG1τe−iH0τ−R1dτ=v∗2ωe−i2ωs−1ge−v2ωei2ωs−1eg..The second-order resonant part is equal to(67)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iκH0+v2−v2−v∗22ωee−gg.Thus, the second-order RG equation and the second-order RG approximate solution are(68)ddsW=εR1W+ε2R2Wandψ2nd=e−iH0sWs+εhs1Ws,0≤s≤Tε,respectively, where W0=ξ, and s=1−ε2κt.Example 9.
Consider the two-level system in high-order near resonance case with initial valueξ=2/2,2/2. Let λe=5, λg=1, v=1+i, ε=0.05, and δ=1. The simulations of Figure 4 illustrate that the first-order approximate solution is reliable in two-level systems with high-order near resonant control.Figure 4
RG vs. Runge-Kutta approximate solutions of two-level systems in high-order quasiresonant case.


### 5.2. High-Order Weak Driving Case

LetUt;ε=ε2ut;ε with(69)ut;ε≜veiω−εδt+v∗e−iω−εδt.Introduce a quasi-resonant time transformations≜1−εκt with κ=δ/ω. The two-level system (2) becomes(70)1−εκddsψ=−iH0+ε2uH1ψ.Nowu is the resonant control with respect to s:(71)us=veiωs+v∗e−iωs.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(72)ε0:ddsψ0+iH0ψ0=0,ε1:ddsψ1+iH0ψ1=G1sψ0,ε2:ddsψ2+iH0ψ2=G1sψ1+G2sψ0,⋯where G1s=−iκH0, and G2s=−iκ2H0+uH1.Firstly, we calculate the first-order RG approximate solution.(73)R1=lims→∞1s∫0seiH0τG1τe−iH0τdτ=−iκH0.Thus, the first-order RG equation and the first-order RG approximate solution are(74)ddsW=εR1Wandψ1st=e−iH0sWs,0≤s≤Tε,respectively, where W0=ξ, and s=1−εκt.Secondly, we calculate the second-order RG approximate solution. By (7),(75)eiH0shs1=∫0seiH0τG1τe−iH0τ−R1dτ=0,the second-order resonant part is equal to(76)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iκ2H0+H2.Thus, the second-order RG equation and the second-order RG approximate solution are(77)ddsW=εR1W+ε2R2Wandψ2nd=e−iH0sWs+εhs1Ws,0≤s≤Tεrespectively, where W0=ξ, and s=1−εκt.Example 10.
Consider the two-level system in high-order weak driving case with initial valueξ=2/2,2/2. Let λe=5, λg=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.Figure 5
RG vs. Runge-Kutta approximate solutions of two-level systems in high-order weak driving case.


## 5.1. High-Order Near Resonant Case

LetUt;ε=εut;ε with(60)ut;ε≜veiω−ε2δt+v∗e−iω−ε2δt.Introduce a quasi-resonant time transformations≜1−ε2κt with κ=δ/ω. The two-level system (2) becomes(61)1−ε2κddsψ=−iH0+εuH1ψ,where u is the resonant control with respect to s:(62)us=veiωs+v∗e−iωs.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(63)ε0:ddsψ0+iH0ψ0=0,ε1:ddsψ1+iH0ψ1=G1sψ0,ε2:ddsψ2+iH0ψ2=G1sψ1+G2sψ0,⋯where G1s=−iuH1, and G2s=−iκH0.Firstly, we calculate the first-order RG approximate solution.(64)R1=lims→∞1s∫0seiH0τG1τe−iH0τdτ=−iH2.Thus, the first-order RG equation and the first-order RG approximate solution are(65)ddsW=εR1Wandψ1st=e−iH0sWs,0≤s≤Tε,respectively, where W0=ξ, and s=1−ε2κt.Secondly, we calculate the second-order RG approximate solution. By (7)(66)eiH0shs1=∫0seiH0τG1τe−iH0τ−R1dτ=v∗2ωe−i2ωs−1ge−v2ωei2ωs−1eg..The second-order resonant part is equal to(67)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iκH0+v2−v2−v∗22ωee−gg.Thus, the second-order RG equation and the second-order RG approximate solution are(68)ddsW=εR1W+ε2R2Wandψ2nd=e−iH0sWs+εhs1Ws,0≤s≤Tε,respectively, where W0=ξ, and s=1−ε2κt.Example 9.
Consider the two-level system in high-order near resonance case with initial valueξ=2/2,2/2. Let λe=5, λg=1, v=1+i, ε=0.05, and δ=1. The simulations of Figure 4 illustrate that the first-order approximate solution is reliable in two-level systems with high-order near resonant control.Figure 4
RG vs. Runge-Kutta approximate solutions of two-level systems in high-order quasiresonant case.


## 5.2. High-Order Weak Driving Case

LetUt;ε=ε2ut;ε with(69)ut;ε≜veiω−εδt+v∗e−iω−εδt.Introduce a quasi-resonant time transformations≜1−εκt with κ=δ/ω. The two-level system (2) becomes(70)1−εκddsψ=−iH0+ε2uH1ψ.Nowu is the resonant control with respect to s:(71)us=veiωs+v∗e−iωs.Letψ=ψ0+εψ1+ε2ψ2+⋯. Equating the coefficients of each εi, we have(72)ε0:ddsψ0+iH0ψ0=0,ε1:ddsψ1+iH0ψ1=G1sψ0,ε2:ddsψ2+iH0ψ2=G1sψ1+G2sψ0,⋯where G1s=−iκH0, and G2s=−iκ2H0+uH1.Firstly, we calculate the first-order RG approximate solution.(73)R1=lims→∞1s∫0seiH0τG1τe−iH0τdτ=−iκH0.Thus, the first-order RG equation and the first-order RG approximate solution are(74)ddsW=εR1Wandψ1st=e−iH0sWs,0≤s≤Tε,respectively, where W0=ξ, and s=1−εκt.Secondly, we calculate the second-order RG approximate solution. By (7),(75)eiH0shs1=∫0seiH0τG1τe−iH0τ−R1dτ=0,the second-order resonant part is equal to(76)R2=lims→∞1s∫0seiH0τG2τe−iH0τ+eiH0τG1τhτ1−eiH0τhτ1R1dτ=−iκ2H0+H2.Thus, the second-order RG equation and the second-order RG approximate solution are(77)ddsW=εR1W+ε2R2Wandψ2nd=e−iH0sWs+εhs1Ws,0≤s≤Tεrespectively, where W0=ξ, and s=1−εκt.Example 10.
Consider the two-level system in high-order weak driving case with initial valueξ=2/2,2/2. Let λe=5, λg=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.Figure 5
RG vs. Runge-Kutta approximate solutions of two-level systems in high-order weak driving case.



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