# Single-Qubit Driving Fields and Mathieu Functions

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## Abstract

**:**

## 1. Introduction

## 2. Analytically Solvable Driving Fields

## 3. Dynamics in a Precessing Field with Oscillating Amplitude

#### 3.1. The Dynamics of Driving Fields and the Theory of Mathieu Functions

#### 3.1.1. Driving Fields in the Region $\mathcal{A}$

#### 3.1.2. Driving Fields in the Region ${\mathcal{A}}^{C}$

## 4. Evolution Loops, Cyclic Evolution and Phases

#### Dynamical and Geometric Phases

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The characteristic values ${a}_{r}\left(q\right)$ (blue-dashed curves) for $r=0,1,2,3,4,5$, and ${b}_{r}$ (red curves) for $r=1,2,3,4,5$, as functions of q, for several values of $r\in \mathbb{N}$. Note that ${a}_{r}\left(0\right)={b}_{r}\left(0\right)={r}^{2}$. (

**b**) The shaded region is produced by all the characteristic curves with non-integer characteristic values r, case in which ${a}_{r}={b}_{r}$.

**Figure 2.**The characteristic frequency ${\omega}_{0}({\omega}_{1},r)={a}_{r}\left({\omega}_{1}\right)+{\omega}_{1}/2$ as functions of ${\omega}_{1}$, for several rational values of r.

**Figure 3.**The driving field (23) in the complex plane for $g=1$, $\Delta =1$, (

**a**) $r=1.5$, ${\omega}_{1}=4$, ${\omega}_{0}=4.53718$, $\delta =0.7071$, (

**b**) $r=3.5$, ${\omega}_{1}=4$, ${\omega}_{0}=14.2946$, $\delta =0.28867$, (

**c**) $r=3.5$, ${\omega}_{1}=40$, ${\omega}_{0}=36.2607$, $\delta =0.28867$. In (

**d**–

**f**), the corresponding population inversion is shown. All of the previous values of ${\omega}_{0}$ correspond to characteristic values of the Mathieu functions (16) with rational values of r, that is, $({\omega}_{1},{\omega}_{0})\in \mathcal{A}$. Time flows from blue to red matching the scale between the types of plots.

**Figure 4.**The driving fields (23) in the complex plane (left column) and the associated population inversion (right column) for the cases of Figure 3, except for the values of g and $\delta $, which has been increased as follows: (

**a**,

**d**) $g=1$, $\delta =2$ (blue), and $g=5$, $\delta =2$ (red dashed), (

**b**,

**e**) $g=1$, $\delta =5$ (blue), and $g=7$, $\delta =5$ (red dashed), (

**c**,

**f**) $g=1$, $\delta =4$ (blue), and $g=7$, $\delta =4$ (red dashed). The driving field precesses counterclockwise.

**Figure 5.**(

**a**) The driving field (23) in the complex plane. (

**b**) Its square modulus as a function of time. The inset is a zooming of the plot for $t\in [0,10]$ to show the correspondent local maxima, and (

**c**) the associated population inversion with ${\omega}_{1}=1$, ${\omega}_{0}=1.5$, $g=1$, $\delta =0.2887$, where $({\omega}_{0},{\omega}_{1})\notin \mathcal{A}$. The driving field precesses counterclockwise.

**Figure 6.**Population inversion together with the dynamics represented on Bloch spheres for several prescriptions generating evolution loops. In any case, the color depicts the value of $P\left(t\right)$ in agreement with the left plot. (

**a**) $a=3.37813,q=2.6118,g=0.53635,\delta =3.30558,\Delta =0.190982,t=11.3718$. (

**b**) $a=3.13268,q=2.70972,g=1.93404,\delta =1.41305,\Delta =0.74138,t=18.7095$. (

**c**) $a=3.01588,q=0.022235,g=0.0123357,\delta =1.0189,\Delta =0.00100538,t=18.3134$ depicting a tiny regular evolution loop with a periodic field around of $|1\rangle $.

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**MDPI and ACS Style**

Enríquez, M.; Jaimes-Nájera, A.; Delgado, F.
Single-Qubit Driving Fields and Mathieu Functions. *Symmetry* **2019**, *11*, 1172.
https://doi.org/10.3390/sym11091172

**AMA Style**

Enríquez M, Jaimes-Nájera A, Delgado F.
Single-Qubit Driving Fields and Mathieu Functions. *Symmetry*. 2019; 11(9):1172.
https://doi.org/10.3390/sym11091172

**Chicago/Turabian Style**

Enríquez, Marco, Alfonso Jaimes-Nájera, and Francisco Delgado.
2019. "Single-Qubit Driving Fields and Mathieu Functions" *Symmetry* 11, no. 9: 1172.
https://doi.org/10.3390/sym11091172