You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'The Raman interaction of a trapped ultracold ion with two travelling wave lasers is studied analytically with series solutions, in the absence of the rotating wave approximation (RWA) and the restriction of both the Lamb-Dicke limit and the weak excitation regime. The comparison is made between our solutions and those under the RWA to demonstrate the validity region of the RWA. As a practical example, the preparation of Schrödinger-cat states with our solutions is proposed beyond the weak excitation regime.'
address: |
    $^{1}$Max-Planck Institute for the Physics of Complex Systems,\
    N$\ddot{o}$thnitzer Street 38, 01187 Dresden, Germany\
    $^{2}$Laboratory of Magnetic Resonance and Atomic and Molecular Physics,\
    Wuhan Institute of Physics and Mathematics, Academia Sinica,\
    Wuhan, 430071, People’s Republic of China
author:
- 'Mang Feng [^1]'
title: 'Series solution to the laser-ion interaction in a Raman-type configuration'
---

introduction
============

The preparation of ultracold ions as well as the production of nonclassical motional states of these ions plays a central role in ion trap experiments$^{[1-3]}$. As the trapped ion system is in nearly perfect isolation, immune from the violation of the environment, and meanwhile the electric-dipole forbidden transition is usually introduced for avoiding the dissipation of the ion, the decoherence in the preparation and preservation of nonclassical states can be effectively suppressed. Consequently, the ion trap system is also being considered to be a promising tool for the quantum computing$^{[4-6]}$.

The various theoretical schemes for generating the nonclassical states of motion of the trapped ions and achieving quantum computing with trapped ions are based on the Jaynes-Cummings (JC) model$^{[7]}$, in which the ion is simplified to be of two levels, the trap’s potential is quantized as a harmonic oscillator, and the radiating lasers are supposed to be classical forms of standing or travelling waves. The general consideration is taken for the case of Lamb-Dicke limit (LDL) under the weak excitation regime (WER), which corresponds to the actual case in present ion trap experiments$^{[3]}$. The LDL means the ions moving within the region of the space much smaller than the effective wavelength of the laser wave, and the WER means the laser-ion interaction much smaller than the trap frequency. For this case, some techniques developed in the framework of cavity QED based on JC model can be immediately transcribed to the ion trap system by taking advantage of the analogy between the cavity QED and ion trap problem. In the case of motion of ions exceeding the LDL, we can use the technique proposed in Ref.\[8\] for a treatment. In a word, under the rotating-wave approximation (RWA), the problem of trapped ions interacting with lasers is solvable. However, the strength of the coupling between the ions and lasers can be conveniently adjusted simply by changing the intensity of lasers. If the coupling strength is much larger than the trap frequency, called the strong excitation regime (SER), then the RWA is no longer applicable since the rapidly oscillating (i.e., counter-rotating) terms also make significant contribution in the interaction, which is much different from the case in Cavity QED$^{[9]}$. It has been reported that the SER is useful for preparing nonclassical motional states of a trapped ion and implementing quantum computing with trapped ions$^{[6,10-12]}$. For example in Ref.\[10\], Schrödinger cat states could be prepared readily in simple way in the SER. Based on that work, a proposal was made for motional state engineering and endoscopy in the SER$^{[11]}$. As the quantum state is sensitive to decoherence, the manipulation of trapped ions in the SER is of importance to the experimental implementation and measurement because of the great reduction of the operation time. Due to the same reason, a scheme for the hot-ion quantum computing also requires the condition of the SER$^{[6]}$. In contrast to the theoretical studies, the ion-laser interaction in the SER has not yet been reported experimentally since the high-laser intensity produces increased off-resonant excitation of carrier transitions which limits the final ground state occupation of the cooling process$^{[13]}$, and the strong laser also modifies the original electronic levels of the ions$^{[14]}$, making the problem somewhat complicated. In a numerical study of the strong Raman sideband excitation driving the vibrational ground state of the trapped ion$^{[15]}$, it is shown that the non-RWA interaction terms produce some limitations for two strong laser beams detuned by the vibrational frequency to be used as Raman motional displacement beams, whereas the interaction is useful for the preparation of the Schrödinger cat state.

Generally speaking, as the RWA is invalid, the ion trap problem in the SER can not be solved analytically with JC model. So some approximations have to be introduced in Refs.\[10,11\] and the studies in Refs.\[12,15\] are resorted to numerical calculations. In fact, as shown in Ref.\[16\], even if the ions are governed in the WER, in which the RWA is valid, the deletion of the RWA can also present some interesting results much different from those under the RWA. In that work, some particular analytical solutions have been obtained in the absence of the RWA for large detuning cases, and Schrödinger cat states could be readily prepared. However, that work is too simple. We hope to investigate the Raman configuration which has been extensively applied in actual ion trap experiments$^{[3]}$, instead of the simple case in Ref.\[16\]. We also hope to exclude the assumption related to both the WER and LDL. Therefore, in this contribution, we will try to undertake a more general and complete investigation on the trapped ion under the Raman process. Moreover, a comparison of our results with those under the RWA will be made. Besides, we will propose a scheme for preparing the Schrödinger cat state beyond the WER.

model and solution
==================

We study the single ultracold ion radiated by lasers in Raman-$\Lambda$-type configuration, as shown in Fig.1. The electronic structure is employed with two lower levels $|e>$ and $|g>$ coupled to a common upper state $|r>$, and the two lasers with frequencies $\omega_{1}$ and $\omega_{2}$ respectively are assumed to propagate along opposite directions. For a sufficiently large detuning to the level $|r>$, $|r>$ may be adiabatically eliminated, and what we have to treat is an effective two-level system, in which the lasers drive the electric-dipole forbidden transition $|g>$$\leftrightarrow |e>$. The dimensionless Hamiltonian of such a system in the frame rotating with the effective laser frequency $\omega_{l}$$(=\omega_{1}-\omega_{2})$ can be written as$^{[10]}$ $$H=\frac {\Delta}{2} \sigma_{z} + a^{+}a
+\frac {\Omega}{2}(\sigma_{+}e^{i\eta \hat{x}}+\sigma_{-}e^{-i\eta \hat{x}})$$ where the detuning $\Delta=(\omega_{0}-\omega_{l})/\nu$ with $\omega_{0}$ being the optical resonance frequency, i.e., the transition frequency of two levels of the ion, and $\nu$ the frequency of the trap. $\Omega$ is the dimensionless Rabi frequency and $\eta$ the effective Lamb-Dicke parameter given by $\eta=\eta_{1}+\eta_{2}$ with subscripts denoting the counterpropagating laser fields. $\sigma_{i}$ ($i=\pm, z$) are Pauli operators, $\hat{x}=a^{+}+a$ is the dimensionless position operator of the ion with $a^{+}$ and $a$ being operators of creation and annihilation of the phonon field, respectively. The notations ’+’ and ’-’ in front of $i\eta \hat{x}$ indicate the absorption of a photon from one beam followed by emission into the other beam and vice versa, respectively. $\nu$ is generally supposed to be much greater than the atomic decay rate, called the strong confinement limit, for neglecting the effect of the atomic decay. In general, one may expand $e^{\pm i\eta\hat{x}}$ in Eq.(1) to the first-order terms of $\eta\hat{x}$ and neglect other higher-order terms by supposing the ion governed under the WER ($\Omega\ll 1$) and within the LDL ($\eta\ll 1$). However, as we hope to treat the problem more exactly and generally, we first perform following unitary transformations on Eq.(1), that is $$H^{I}=V^{+}U^{+}HUV=\frac {\Omega}{2}\sigma_{z} + a^{+}a+ g(a^{+}+a)(\sigma_{+}+
\sigma_{-})+\epsilon (\sigma_{+}+\sigma_{-})+g^{2}$$ where $U=\pmatrix{D(\beta) & -D(\beta)\cr D^{+}(\beta)& D^{+}(\beta)}$ and $ V=\frac {1}{\sqrt{2}}e^{-i\pi a^{+}a/2}$ with $D(\beta)=e^{i\eta(a^{+}+a)/2}$, $g=\eta/2$ and $\epsilon=-\Delta/2$. Actually, the unitary transformation $U$ made above is identical to that in \[17\]. But for coinciding with the standard form of non-RWA model, $V$ is performed. Comparing with the standard non-RWA JC model$^{[9]}$, there exist an additional driven term and an additional constant term in Eq.(2), and the optical resonance frequency is replaced by the Rabi frequency. In what follows, we will adopt the coherent state representation$^{[18]}$, by which the non-RWA JC models have been analytically treated$^{[16,19]}$. Thus Eq.(2) is rewritten as $$H={\frac {\Omega}{2}}\sigma_{z}+  \alpha {\frac {d}{d \alpha}}+
g(\alpha+\frac {d}{d \alpha})(\sigma_{+}+\sigma_{-})
+\epsilon (\sigma_{+}+\sigma_{-})+g^{2}$$ where $\alpha$ is a complex number, and $\int \frac {d\alpha d\alpha^{*}}
{2\pi i} \exp (-|\alpha|^{2}) |\alpha^{*}><\alpha^{*}|=1$ with the relation between the coherent state and Fock state $$|n>=\int \frac {d\alpha d\alpha^{*}}{2\pi i} 
\exp (-|\alpha|^{2}) |\alpha^{*}>\frac {1}{\sqrt{n!}} \alpha^{n}.$$ Using the same idea as in Refs.\[16,19\], we assume the eigenfunction of the Schrödinger equation of Eq.(3) to be the series form $$\Psi(\alpha)= \pmatrix
{\Psi_{1}(\alpha) \cr \Psi_{2}(\alpha)}
= \pmatrix{exp(-z\alpha)\sum_{n=0}^{\infty} b_{n}\alpha^{n}\cr
exp(-z\alpha)\sum_{n=0}^{\infty} c_{n}\alpha^{n}}$$ where $z$, $b_{n}$ and $c_{n}$ are constants determined later. Taking $\Psi(\alpha)$ into the Schrödinger equation of Eq.(3) will yield two recurrence relations $$b_{n+1}={\frac {1}{g(n+1)}}[(E+{\frac {\Omega}{2}}
-n-g^{2})c_{n}+(gz-\epsilon)b_{n}-gb_{n-1}+ zc_{n-1}],$$ $$c_{n+1}={\frac {1}{g(n+1)}}[(E-{\frac {\Omega}{2}}
-n-g^{2})b_{n}+(gz-\epsilon)c_{n}-gc_{n-1}+ zb_{n-1}]$$ where $E$ is the trial solution of the eigenenergy of Eq.(3). As the procedure of solution is similar to that in Ref.\[16\], in what follows, we just list the results:

\(1) $b_{n}=c_{n}=0$ for $n\geq 2$, we have two solutions. One is $E_{1}=1+\epsilon$ with $E_{1}$ the eigenenergy of the system in this case, corresponding to $z= g$ and a restricted relation of $\Omega=2\sqrt{1+2\epsilon-4g^{2}}$. The other is $E^{'}_{1}=1-\epsilon$, with $z= -g$ and a restricted relation of $\Omega=2\sqrt{1-2\epsilon-4g^{2}}$. The eigenfunctions for above two cases are respectively $\Psi(\alpha)=e^{-g\alpha} \pmatrix{b_{0}+ b_{1}\alpha\cr
c_{0}+ b_{1}\alpha}$ and $\Psi^{'}(\alpha)=e^{g\alpha} 
\pmatrix{b_{0}^{'}+ b^{'}_{1}\alpha \cr
c^{'}_{0}- b^{'}_{1}\alpha}$, where $c_{0}=\frac {1-\Omega/2+2\epsilon-2g^{2}}
{1+\Omega/2+2\epsilon-2g^{2}}b_{0}$, $b_{1}=\frac {1}{g}
[(1+\frac {\Omega}{2}-g^{2}+\epsilon)c_{0}+(g^{2}
-\epsilon)b_{0}]$, $c^{'}_{0}=-\frac {1-\Omega/2-2\epsilon-2g^{2}}
{1+\Omega/2-2\epsilon-2g^{2}}b_{0}^{'}$, and $b^{'}_{1}=\frac {1}{g}
[(1-\epsilon-g^{2}+\frac {\Omega}{2})c^{'}_{0}-
(g^{2}+\epsilon)b^{'}_{0}]$. $b_{0}$ and $b^{'}_{0}$ can be set to be 1 for normalisation.

\(2) $b_{n}=c_{n}=0$ for $n \geq 3$, the two solutions are $E_{2}=2 \pm\epsilon$ corresponding to $z= \pm g$ and two independent restricted conditions, which can be solved from the following equation, respectively $$\frac {2g^{2}-(1-2g^{2}\pm 2\epsilon)(2+\Omega/2-2g^{2}\pm 2\epsilon)
+\Omega+\Omega^{2}/4}
{2g^{2}-(1-2g^{2}\pm 2\epsilon)(2-\Omega/2-2g^{2}\pm 2\epsilon)-\Omega+\Omega^{2}/4}=$$ $$\frac {4g^{2}(2+3\Omega /4-2g^{2}\pm 2\epsilon)-\Omega(g^{2}\mp\epsilon)(3\pm 2\epsilon+\Omega-2g^{2})}
{4g^{2}(2-3\Omega /4-2g^{2}\pm 2\epsilon)+\Omega[(1+\Omega/2)(2-\Omega/2)+ 2(g^{2}\mp \epsilon)^{2}
-3(g^{2}\mp \epsilon)]}$$ The eigenfunctions at this moment in the unit of $b_{0}=b^{'}_{0}=1$ are, respectively, $$\Psi(\alpha)=e^{-g\alpha}\pmatrix{1+ b_{1}\alpha + b_{2}\alpha^{2} \cr
c_{0}+ c_{1}\alpha +b_{2}\alpha^{2}}$$ and $$\Psi^{'}(\alpha)=e^{g\alpha} 
\pmatrix{1+ b^{'}_{1}\alpha +b^{'}_{2}\alpha^{2}\cr
c^{'}_{0}+ c^{'}_{1}\alpha -b^{'}_{2}\alpha^{2}}$$ where $c_{0}=\frac {(1-2g^{2}+ 2\epsilon)(2-\Omega/2-2g^{2}+ 2\epsilon)
+\Omega-\Omega^{2}/4-2g^{2}}
{(1-2g^{2}+2\epsilon)(2+\Omega/2-2g^{2}+ 2\epsilon)-\Omega-\Omega^{2}/4
-2g^{2}}$, $c_{1}=\frac {1}{g}[2+\epsilon-g^{2}-\frac {\Omega}{2}+
(g^{2}-\epsilon)c_{0}]$, $b_{1}=\frac {1}{g}[(2+\epsilon-g^{2}
+\frac {\Omega}{2})c_{0}+g^{2}-\epsilon]$, $b_{2}=\frac {1}{2g^{2}}\{(g^{2}-\epsilon)(3-\Omega+2\epsilon-2g^{2})+
g^{2}+[(1+\epsilon-g^{2})^{2}-1-\frac {\Omega^{2}}{4}-\frac
{\Omega}{2}-g^{2}-(g^{2}-\epsilon)^{2}]c_{0} \}$, $c^{'}_{0}=\frac {2g^{2}-(1-2g^{2}- 2\epsilon)(2-\Omega/2-2g^{2}-2\epsilon)
-\Omega+\Omega^{2}/4}
{(1-2g^{2}-2\epsilon)(2+\Omega/2-2g^{2}-2\epsilon)-\Omega-\Omega^{2}/4
-2g^{2}}$, $c^{'}_{1}=\frac {1}{g}[2-\epsilon-g^{2}-\frac {\Omega}{2}-
(g^{2}+\epsilon)c^{'}_{0}]$, $b^{'}_{1}=\frac {1}{g}[(2-\epsilon-g^{2}+
\frac {\Omega}{2})c^{'}_{0}- g^{2}+\epsilon]$, and $b^{'}_{2}=\frac {1}{2g^{2}}\{(g^{2}+\epsilon)(3-\Omega-
2\epsilon-2g^{2})+g^{2}+[(1-\epsilon-g^{2})^{2}-1-\frac {\Omega^{2}}{4}-\frac
{\Omega}{2}-g^{2}-(g^{2}+\epsilon)^{2}]c^{'}_{0} \}$;

\(3) when above technique is extended to the case of $b_{n}=c_{n}=0$ for $n \geq N+1$, the series solution becomes very complicated. However the eigenenergy of the system is still of the simple form, that is, $E_{N}=N \pm\epsilon$, corresponding to $z= \pm g$ and two independent sets of much complicated expressions of restricted conditions which we have to omit here, respectively.

discussion
==========

As limitations for both the Rabi frequency and Lamb-Dicke parameter as well as the RWA are excluded in the present treatment, the results we obtained above are more exact and general than those depending on the RWA or the approximate expansion of small values of $\eta$ and $\Omega$. However, as the problem involving the counter-rotating terms is non-integrable$^{[16,18,19]}$, the present solutions obtained by the termination of the series of the trial solution $\Psi(\alpha)$ are some particular ones, along with some restricted conditions related to some parameters of the system. Nevertheless, with the present solutions, we can investigate some cases in a wide range of parameters, particularly for the case beyond the WER and LDL. It is also of interest to make a comparison of our results with those under the RWA. Let us first consider a special case with both $\Omega \sim \epsilon \ge 1$ and $\eta\rightarrow 0$. In such a case, Eq.(3) is reduced to $$H^{s}={\frac {\Omega}{2}}\sigma_{z}+  \alpha {\frac {d}{d \alpha}}
+\epsilon (\sigma_{+}+\sigma_{-}).$$ Repeating the above procedure of the series solution for the simplest case, i.e., $b_{n}=c_{n}=0$ for $n\geq 2$, we can find $z=0$ and $\Psi^{s}=\pmatrix{b_{0}+b_{1}\alpha \cr c_{0}+c_{1}\alpha}$ with $c_{0}=\frac {E^{s}-\Omega/2-1}{\epsilon}b_{0}$, $c_{1}=\frac {E^{s}-\Omega/2-1}{\epsilon}b_{1}$ and $E^{s}=1\pm\sqrt{\epsilon^{2}+\frac {\Omega^{2}}{4}}$. We can easily find that the eigenenergy $E^{s}$ is in good agreement with the solution in a Fock state representation for $n=1$. However, to make a comparison for eigenfunctions with former solutions, we have to return to the original representation before Eq.(2). The eigenfunction in the original representation is $$\Psi^{os} = UV \Psi^{s} \approx \frac {1}{\sqrt{2}}
\pmatrix{1& -1 \cr 1& 1}\Psi^{s}(\alpha)e^{-i\pi a^{+}a/2}$$ $$=\frac {1}{\sqrt{2}}\pmatrix{(b_{0}-c_{0})|0>-i(b_{1}+c_{1})|1> \cr 
(b_{0}+c_{0})|0>-i(b_{1}+c_{1})|1>}$$ where we have used the fact$^{[16,18]}$ that $e^{g\alpha}\alpha^{n}$ in a coherent state representation corresponds to the displaced Fock states $|n,g>$ by means of Eq.(4) and Baker-Campbell-Hausdorff formula. Therefore, in the case of $\Omega \ll 1$ but $\epsilon\gg 1$, we have from Eq.(8) $\Psi^{os}\sim \pmatrix{1 \cr 0}$ or $\pmatrix{0 \cr 1}$, which means no transitions existing in the large detuning case. If we assume $\epsilon \rightarrow 0$ and $\Omega$ being arbitrary positive real number, we have $\Psi^{os}\sim \pmatrix{1 \cr 1}$, which corresponds to the carrier excitation. Both solutions above are also in good agreement with the solutions in a Fock state representation$^{[12]}$. However, from Eq.(8) we can know that, different from former solutions for the WER case, the probability amplitudes of up and down states of the ion are different and complicated in the case of the SER with large detunings. This is a new result which has not been obtained before by general approaches made in a Fock state or dressed state representation.

For a more general comparison, we should first present the solutions under the RWA. As referred to in Sec.I, Eq.(1) can be treated to be a nonlinear JC model with the approach proposed in Ref.\[8\] under the RWA. However, to make the comparison more easy and clear, we start our RWA treatment from Eq.(2). By performing a unitary transformation $\exp [-i(\frac {\Omega}{2}\sigma_{z}+\frac {a^{+}a}{2^{M}})t]$ on Eq.(2) with $M=1,2,3,\cdots$, we can obtain $$H_{M}=(1-2^{-M})a^{+}a+g(a^{+}\sigma_{-}+a\sigma_{+})+g^{2}$$ corresponding to the resonance conditions of $\Omega=2^{-M}$ under the RWA, and the eigenenergy in a Fock state representation is of the form $$E^{\pm}_{M}=(1-2^{-M})(n+\frac {1}{2})+\frac {\eta^{2}}{4}\pm\frac {1}{2}\sqrt{\eta^{2}(n+1)+(1-2^{-M})^{2}}$$ with $n=0, 1, 2, \cdots$. Meanwhile, performing a unitary transformation of $\exp [-i(\frac {\Omega}{2K}\sigma_{z}+\frac {a^{+}a}{2})t]$ on Eq.(2) with $K=1,2,3, \cdots$, under the RWA resonance conditions of $\Omega=K$, we have $$H_{K}=\frac {K-1}{2K}\Omega\sigma_{z}+g(a^{+}\sigma_{-}+a\sigma_{+})+g^{2}$$ whose eigenenergy in a Fock state representation is of the form $$E^{\pm}_{K}=\frac {\eta^{2}}{4}\pm\frac {1}{2}\sqrt{\eta^{2}(n+1)+(K-1)^{2}}$$ with $n=0, 1, 2, \cdots$.

As shown in the last section, although our method can in principle present all series solutions, we merely present the specific forms of the non-RWA solution for two simplest cases. Here we will use those two cases for a comparison with Eqs.(10) and (12). For the case (1), we find that the two solutions for $E=1\pm\epsilon$, along with two different restricted relations respectively, actually correspond to the same expression $$E=\frac {1}{2}+ \frac {1}{2}\eta^{2}+\frac {1}{8}\Omega^{2}.$$ However, for the case (2), the situation is somewhat complicated. The direct algebra shown in Appendix presents us four solutions of E for this case, whereas the specific calculation shows that the solutions of Eqs.(A3) and (A4) are actually identical.

In Figs.2 and 3, the solutions under the RWA are compared with those for the non-RWA treatment in the case of $\Omega=$0.5 and 3.0. We can find that, in the case of small values of $\Omega$ and $\eta$, some of the solutions under the RWA can approach those without the RWA. It is physically reasonable because the RWA is only valid for the WER and LDL. However, although the solutions without the RWA are exact, only a few of these solutions can be obtained for a certain termination of the series solution. We can not obtain all the particular solutions of the system unless we make the termination of the series solution for almost infinite times up to the case with infinite series terms. In contrast, the RWA treatment can present general solutions, like Eqs.(10) and (12). Nevertheless, as it merely retains some resonance terms in the Hamiltonian, a specific solution under the RWA only corresponds to a specific value of $\Omega$. Moreover, the figures tell us that, only when the value of $\eta$ is not taken to be larger than 0.1, can we consider that the solutions under the RWA and without the RWA are in good agreement in the case of the WER. The RWA description is obviously inaccurate for the case beyond the LDL although $\Omega$ is less than 1.0 in this case. When the value of $\Omega$ is larger, the difference between the RWA case and non-RWA one becomes great, as shown in Figs.3. It is difficult to find any correspondence for the two cases. However, there are some crossing points between the RWA solutions and non-RWA ones. It means at certain values of $\eta$, the solutions for eigenenergies in these two cases can be the same. In physics, it can be considered that the non-RWA solutions are some isolated ones, merely corresponding to those of up internal levels of the ion. One may not expect to obtain the general eigenenergies with our non-RWA treatment. However, the demonstration of above series solutions means the existence of the finite dimensional invariant subspace of some operators. Particularly from the simple forms of the eigenenergies, one may expect that the quantum KAM theorem should be applicable for the SER problem$^{[20]}$.

Before ending our discussion of the usefulness of non-RWA solutions, it is also of interest to present a practical example of preparing the nonclassical motional states of the ion with our solutions. As referred to in Sec.I, there is a considerable interest in investigating the SER, particularly for rapidly preparing nonclassical motional states of the ion. In fact, with the results in the present paper, one can also generate some particular nonclassical states through suitable adjustment of the Rabi frequencies and proper measurements. For example, consider the simplest case, i.e., the case (1) in Sec II. For the case of the resonance ($\epsilon=0$), we have $E_{1}=E_{1}^{'}$ with the same $\Omega$ and double degenerate eigenfunctions $\Psi (\alpha)$ and $\Psi^{'}(\alpha)$. Transforming our results into the Schrödinger representation will yield the states $\Phi (t)=\exp{(-iE_{1}t+
i\omega_{l}\sigma_{z}t/2)}UV \Psi(\alpha)$ and $\Phi^{'} (t)=
\exp{(-iE_{1}t+i\omega_{l}\sigma_{z}t/2)}UV \Psi^{'}(\alpha)$, respectively. In the case of the LDL($g\ll 1$), the Rabi frequency $\Omega$ would be $2\sqrt{1-4g^{2}}\sim 2$. So we have $c_{0}\sim \frac {g^{2}}{g^{2}-1}b_{0}$, $b_{1}\sim \frac {g}{g^{2}-1}b_{0}$, $c^{'}_{0}\sim \frac {g^{2}}{g^{2}-1}b^{'}_{0}$ and $b^{'}_{1}\sim \frac {g}{g^{2}-1}b^{'}_{0}$. By controlling the evolution time $t=4\pi/\omega_{l}$, and measuring the excited state of the ion, we can obtain a displaced even coherent state, a kind of Schrödinger-cat states $$\Phi^{M}\sim \frac {1}{\sqrt{2}}D(\beta)(|i\frac {\eta}{2}>+|-i\frac {\eta}{2}>)=\frac {1}{\sqrt{2}}(|i\eta>+|0>)$$ where the relation between the coherent state and Fock state has been used. As the Rabi frequency approaches 2, such a preparation is obviously outside the WER.

conclusion
==========

As $e^{g\alpha}\alpha^{n}$ corresponds to the displaced Fock states $|n,g>$, the eigenfunctions of the system we obtained are infinite superposition of displaced Fock states. Therefore, it is understandable that we can not obtain similar analytical results by general Fock state expansion. The obvious advantage of our approach is the possibility to obtain some relations between parameters of the system corresponding to certain nonclassical states by truncating the series expansion step by step. So as long as we can reach these conditions experimentally, the different nonclassical states predicted by our theory would be obtained.

In summary, the Raman interaction between a trapped ultracold ion and two travelling wave lasers has been treated analytically in an interaction representation, without consideration of the RWA and the limitation for both the LDL and WER. Although the coherent state expansion technique was used previously for treating a similar problem, and the solutions we presented here are some isolated ones under certain conditions related to $\Omega$, $\eta$ and $\Delta$, the present investigation is more general and exact so that we can study the ion trap problem in a wide range of parameters and compare our results with other approximate works or numerical solutions in this respect as we did in the present paper and for the cavity-atom problem$^{[21]}$. More importantly, the Raman process has been widely used for the laser-cooling in current ion trap experiments, and the work outside the WER is attracting much interest of experimentalists. One possible way$^{[10]}$ to experimentally realizing the SER is that the ion is first cooled within the LDL and under the WER, and then the trap frequency is decreased by opening the trap adiabatically so that the ratio of the Rabi frequency to the trap frequency is increased to a large number. Therefore, we believe that the difficulties in realizing the SER will soon be overcome, and our work would be helpful for any possible future exploration of the quantum properties of the ion-trap system outside the WER.

The author sincerely thanks M.A.Kornberg for his carefully reading the manuscript. The work is partly supported by National Natural Science Foundation of China.

appendix
========

For obtaining a more specific expression of the eigenenergy for the case (2) in Sec.II, we can suppose $X=\epsilon - g^{2}$ for the case of $z=g$ and $Y=\epsilon +g^{2}$ for $z=-g$. Direct algebra on Eq.(7) can present us following two second-order differential equations $$AX^{2}+BX+C=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A1)$$ and $$AY^{2}-BY+C=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A2)$$ with $A=8(1-g^{2})$, $B=(3+\frac {\Omega}{2})(2+\Omega)(2-\frac {\Omega}{2})-3(\frac {\Omega}{2}-2
+\frac {\Omega^{2}}{4}+2g^{2})-28g^{2}-(3+\Omega)(2+\frac {\Omega}{2}-\frac {\Omega^{2}}{4}-2g^{2})$ and $C=-[(\frac {\Omega}{2}-2+\frac {\Omega^{2}}{4}+2g^{2})(1+\frac {\Omega}{2})(2-\frac {\Omega}{2})
+20g^{2}-6g^{2}(\frac {\Omega^{2}}{4}+2g^{2})]$. So we can obtain the eigenenergies $$E^{\pm}_{1}=2+g^{2}+\frac {-B\pm\sqrt{B^{2}-4AC}}{2A}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A3)$$ for the case of $z=g$ and $$E^{\pm}_{2}=2+g^{2}-\frac {B\pm\sqrt{B^{2}-4AC}}{2A}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A4)$$ for the case of $z=-g$.

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[**Captions of the figures**]{}

Fig.1   Level scheme of the internal structure of the trapped ultracold ion, where $|g>\leftrightarrow |e>$ is dipole forbidden.

Fig.2   Variation of E with respect to $\eta$ in the case of $\Omega=0.5$, where solid curves from the bottom to top are RWA solutions of Eq.(10), corresponding to $n=$0, 1, 2, 3, 4, 5 and 6, respectively, dot-dashed curve is the solution of Eq(13) and dashed curves are from Eq.(A3) or Eq.(A4) (because the solutions of Eqs.(A3) and (A4) are identical). (a) demonstrates $E^{+}$, and (b) is for $E^{-}$.

Fig.3   Variation of E with respect to $\eta$ in the case of $\Omega=3.0$, where solid curves from the bottom to top are RWA solutions of Eq.(12) for the case of $E^{+}$, corresponding to $n=$0, 1, 2, 3, 4, 5 and 6, respectively, the dot-dashed curve is the solution of Eq(13) and dashed curves are from Eq.(A3) for cases of $E^{\pm}$ respectively.

[^1]: Present address: Institute for Scientific Interchange Foundation, Villa Gualino, Viale Settimio Severo 65, I-10131, Torino, Italy. Electronic address: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>