Dephasing-Assisted Macrospin Transport
<p>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).</p> "> Figure 2
<p>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.</p> "> Figure 3
<p>Total SW power <span class="html-italic">P</span> vs time for different values of the dephasing noise amplitude <math display="inline"><semantics> <mi>θ</mi> </semantics></math> (<b>a</b>) and of the bath temperature <span class="html-italic">T</span> (<b>b</b>). One can see that in the first case <span class="html-italic">P</span> drops to zero and the magnetization aligns with the <span class="html-italic">z</span> axis, since the dephasing conserves the total power. In the second case, the bath temperature excites the dynamics and the system thermalizes with <span class="html-italic">P</span> increasing with the bath temperature <span class="html-italic">T</span>.</p> "> Figure 4
<p>Effect of the dephasing noise on the dynamics of the macrospin chain: time evolution of the local SW power <math display="inline"><semantics> <msub> <mi>p</mi> <mn>10</mn> </msub> </semantics></math> of the last disk for different values of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. Transmitted power is maximized for an optimal value around <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>≈</mo> <mn>4</mn> </mrow> </semantics></math>. Simulation parameters as given in the text.</p> "> Figure 5
<p>(<b>a</b>) Efficiency <span class="html-italic">E</span> and (<b>b</b>) Kuramoto parameter <span class="html-italic">K</span> versus <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. <span class="html-italic">E</span> increases of a factor 3 until <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> and then decreases again, showing that transport can be effectively promoted by dephasing. On the other hand, <span class="html-italic">K</span> decreases monotonically with <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. Thus, in the present case, transport is not related to phase synchronization.</p> "> Figure 6
<p>Wavelet analysis of the complex spin amplitudes <math display="inline"><semantics> <msub> <mi>ψ</mi> <mi>n</mi> </msub> </semantics></math> on the central disk (<math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, left panels) and on the last disk (<math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, right panels) for different values of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. Each plot shows the density map of the average square modulus <math display="inline"><semantics> <mrow> <mo>〈</mo> <mo>|</mo> <msub> <mi>G</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>〉</mo> </mrow> </semantics></math> of the Gabor transform (<a href="#FD7-entropy-22-00210" class="html-disp-formula">7</a>) averaged over a sample of 32 independent realizations of the dyanamics. The parameter <span class="html-italic">a</span> has been set equal to <math display="inline"><semantics> <mrow> <mn>7.5</mn> </mrow> </semantics></math> ns<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, optimized so as to maximize the resolution in both the time and frequency domains. Notice the difference in the density scales.</p> "> Figure 7
<p>Average amplitude contributions <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mover accent="true"> <mi>ω</mi> <mo>˜</mo> </mover> <mo>)</mo> </mrow> </mrow> </semantics></math> on site <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (<b>a</b>) and site <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> (<b>b</b>) for <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>ω</mi> <mo>˜</mo> </mover> <mo>=</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mn>4</mn> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mn>5</mn> </msub> </mrow> </semantics></math> computed from data of <a href="#entropy-22-00210-f006" class="html-fig">Figure 6</a>. Solid curves are obtained with <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>ω</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> GHz, while the black dashed curve refers to <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>ω</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> GHz.</p> "> Figure 8
<p>Transport efficiency versus dephasing noise strength of a chain of <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> damped discrete nonlinear Schrödinger equation (DNLS) oscillators evolving according to Equation (<a href="#FD9-entropy-22-00210" class="html-disp-formula">9</a>). Simulations refer to <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.008</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and linear frequencies <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>ω</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>,</mo> <msubsup> <mi>ω</mi> <mn>2</mn> <mn>0</mn> </msubsup> <mo>,</mo> <msubsup> <mi>ω</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mo>,</mo> <msubsup> <mi>ω</mi> <mn>4</mn> <mn>0</mn> </msubsup> <mo>,</mo> <msubsup> <mi>ω</mi> <mn>5</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1.09</mn> <mo>,</mo> <mn>1.15</mn> <mo>,</mo> <mn>1.21</mn> <mo>,</mo> <mn>1.33</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. For each value of <math display="inline"><semantics> <msup> <mi>θ</mi> <mo>′</mo> </msup> </semantics></math>, data are averaged over a set of 100 independent realizations of the dynamics.</p> ">
Abstract
: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 |
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Iubini, S.; Borlenghi, S.; Delin, A.; Lepri, S.; Piazza, F. Dephasing-Assisted Macrospin Transport. Entropy 2020, 22, 210. https://doi.org/10.3390/e22020210
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 StyleIubini, 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