# Dephasing-Assisted Macrospin Transport

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## Abstract

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## 1. Introduction

## 2. The System: Microscopic and Macroscopic Descriptions

## 3. Micromagnetics Simulations with Dephasing Noise

## 4. Comparison with Coupled-Oscillators Model

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SW | Spin Wave |

DNLS | Discrete Nonlinear Schrödinger |

LLG | Landau Lifshitz Gilbert |

## References

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**Figure 1.**Sketch of the system, consisting of 10 Py nano-disks coupled through the magneto-dipolar interaction. The magnetization in the first disk (input) is tilted away from equilibrium. The transport efficiency is the integrated Spin Wave (SW) power in the last disk (output).

**Figure 2.**Time-average of the SW powers of each collective mode, obtained by exciting the dynamics with a uniform time-dependent magnetic field with the frequencies of the modes until the system reaches a steady state. Simulations are performed at zero temperature. The total SW power of each profile is normalized to one for better comparison.

**Figure 3.**Total SW power P vs time for different values of the dephasing noise amplitude $\theta $ (

**a**) and of the bath temperature T (

**b**). One can see that in the first case P drops to zero and the magnetization aligns with the z axis, since the dephasing conserves the total power. In the second case, the bath temperature excites the dynamics and the system thermalizes with P increasing with the bath temperature T.

**Figure 4.**Effect of the dephasing noise on the dynamics of the macrospin chain: time evolution of the local SW power ${p}_{10}$ of the last disk for different values of $\theta $. Transmitted power is maximized for an optimal value around $\theta \approx 4$. Simulation parameters as given in the text.

**Figure 5.**(

**a**) Efficiency E and (

**b**) Kuramoto parameter K versus $\theta $. E increases of a factor 3 until $\theta =6$ and then decreases again, showing that transport can be effectively promoted by dephasing. On the other hand, K decreases monotonically with $\theta $. Thus, in the present case, transport is not related to phase synchronization.

**Figure 6.**Wavelet analysis of the complex spin amplitudes ${\psi}_{n}$ on the central disk ($n=5$, left panels) and on the last disk ($n=10$, right panels) for different values of $\theta $. Each plot shows the density map of the average square modulus $\langle |{G}_{n}(\omega ,t){|}^{2}\rangle $ of the Gabor transform (7) averaged over a sample of 32 independent realizations of the dyanamics. The parameter a has been set equal to $7.5$ ns${}^{2}$, optimized so as to maximize the resolution in both the time and frequency domains. Notice the difference in the density scales.

**Figure 7.**Average amplitude contributions ${g}_{n}\left(\tilde{\omega}\right)$ on site $n=5$ (

**a**) and site $n=10$ (

**b**) for $\tilde{\omega}={\omega}_{1},{\omega}_{2},{\omega}_{3},{\omega}_{4},{\omega}_{5}$ computed from data of Figure 6. Solid curves are obtained with $\delta \omega =0.25$ GHz, while the black dashed curve refers to $\delta \omega =2$ GHz.

**Figure 8.**Transport efficiency versus dephasing noise strength of a chain of $N=10$ damped discrete nonlinear Schrödinger equation (DNLS) oscillators evolving according to Equation (9). Simulations refer to $J=0.1$, $\alpha =0.008$, $\nu =1$ and linear frequencies $({\omega}_{1}^{0},{\omega}_{2}^{0},{\omega}_{3}^{0},{\omega}_{4}^{0},{\omega}_{5}^{0})=(1,1.09,1.15,1.21,1.33)$. For each value of ${\theta}^{\prime}$, data are averaged over a set of 100 independent realizations of the dynamics.

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**MDPI and ACS Style**

Iubini, S.; Borlenghi, S.; Delin, A.; Lepri, S.; Piazza, F.
Dephasing-Assisted Macrospin Transport. *Entropy* **2020**, *22*, 210.
https://doi.org/10.3390/e22020210

**AMA Style**

Iubini S, Borlenghi S, Delin A, Lepri S, Piazza F.
Dephasing-Assisted Macrospin Transport. *Entropy*. 2020; 22(2):210.
https://doi.org/10.3390/e22020210

**Chicago/Turabian Style**

Iubini, Stefano, Simone Borlenghi, Anna Delin, Stefano Lepri, and Francesco Piazza.
2020. "Dephasing-Assisted Macrospin Transport" *Entropy* 22, no. 2: 210.
https://doi.org/10.3390/e22020210