# Exactly Solvable One-Qubit Driving Fields Generated via Nonlinear Equations

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## Abstract

**:**

## 1. Introduction

## 2. The Direct Approach

- ${X}^{i,j}{X}^{k,m}={\delta}_{j,k}{X}^{i,m}$ (multiplication rule),
- $\sum _{k=1}^{n}{X}^{k,k}=\mathbb{I}$ (completeness),
- ${\left({X}^{i,j}\right)}^{\u2020}={X}^{j,i}$ (non-hermiticity),
- $[{X}^{i,j},{X}^{k,m}]={\delta}_{j,k}{X}^{i,m}-{\delta}_{m,i}{X}^{k,j}$ (commutation rule),

## 3. The Inverse Approach

#### Periodic Interactions

## 4. A Physical Model

## 5. Some Simple Applications

#### 5.1. Case $\mathsf{\Omega}\left(t\right)=0$: A Decaying Driving Field

#### 5.2. Case $\mathsf{\Omega}\left(t\right)={\mathsf{\Omega}}_{1}$: A Precessing Field with Oscillating Amplitude

#### 5.3. Case $\mathsf{\Omega}\left(t\right)=i\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Omega}}_{1}$: A Precessing Decaying Driving Field

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Function η(t)

## References

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**Figure 1.**Schematic representation of the magnetic field $\mathbf{B}$. The transversal component ${\mathbf{B}}_{12}$ rotates around the axis ${\mathbf{e}}_{3}$ in the plane $z={B}_{3}$. The amplitude of the driving field is time-dependent and describes a trajectory that in general is not closed. The circularly polarized field is a particular case of this expression when the amplitude is constant.

**Figure 2.**(

**Left**) the interaction term (37) in the complex plane; (

**Right**) population inversion as a function of time. The parameters in both cases are: $g=0.5$, $\delta =0.01$ (red), $\delta =1$ (black) and $\delta =2$ (blue).

**Figure 3.**The driving field (46) with $g=\sqrt{5}$ and $\delta =4$ for (

**a**) $\kappa =0.6$, $\mathsf{\Delta}=10$ and (

**b**) $\kappa =3.1$, $\mathsf{\Delta}=1.9$. In the lower graphics $g=\sqrt{160}$ and $\delta =6$, and (

**c**) $\kappa =0.8$, $\mathsf{\Delta}=32.5$ and (

**d**) $\kappa =2.5$, $\mathsf{\Delta}=10.4$. The choice of the parameters grants the periodicity of V and R. The blue circle in all the cases corresponds to the case of a circularly polarized field and the gray one is the maximum (minimum) amplitude for $\kappa <1$ ($\kappa >1$).

**Figure 4.**The driving field (46) with $g=\sqrt{\pi}$, $\delta =2\sqrt{\pi}$ and (

**a**) $\kappa =0.6$ and (

**b**) $\kappa =2.5$. These parameters do not fulfill the periodicity condition and the corresponding trajectories are not closed. In both cases, the time interval is $[0,10\kappa \pi /{\mathsf{\Omega}}_{0}]$.

**Figure 5.**Population inversion (51) as function of time. In the upper panels, the dashed line corresponds to the case of a qubit interacting with a circularly polarized field. In these plots, we have used (

**a**) $\kappa =0.6$ and (

**b**) $\kappa =3.1$. In the lower plots, the function P shows a collapse-and-revival-like behavior for (

**c**) $\kappa =0.005$ and (

**d**) $\kappa =50$. In all cases, we have taken $g=\sqrt{5}$ and $\delta =4$.

**Figure 6.**Time-evolution of population inversion (51) as a function of the parameters (

**a**) g and (

**b**) $\delta $, which can be used to manipulate the amplitude and period of the oscillations.

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Enríquez, M.; Cruz y Cruz, S.
Exactly Solvable One-Qubit Driving Fields Generated via Nonlinear Equations. *Symmetry* **2018**, *10*, 567.
https://doi.org/10.3390/sym10110567

**AMA Style**

Enríquez M, Cruz y Cruz S.
Exactly Solvable One-Qubit Driving Fields Generated via Nonlinear Equations. *Symmetry*. 2018; 10(11):567.
https://doi.org/10.3390/sym10110567

**Chicago/Turabian Style**

Enríquez, Marco, and Sara Cruz y Cruz.
2018. "Exactly Solvable One-Qubit Driving Fields Generated via Nonlinear Equations" *Symmetry* 10, no. 11: 567.
https://doi.org/10.3390/sym10110567