# Excitation Dynamics in Chain-Mapped Environments

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## Abstract

**:**

## 1. Introduction

## 2. Tedopa

## 3. Chain Dynamics

#### 3.1. Lorentzian Spectrum

#### 3.2. Ohmic Spectrum

## 4. Full Dynamics

## 5. Conclusions and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Lorentzian SD; in all frames ${\Omega}_{S}=10$, $T=0$; blue, green and red lines/marker refer respectively to $\gamma =0.001$, $\gamma =1$ and $\gamma =10$. (

**a**) The chain parameters ${\omega}_{n}$ (empty markers) and ${\kappa}_{n}$ (filled markers) for $\gamma =0.001$ (blue circles), $\gamma =1$ (green diamonds) and $\gamma =10$ (red squares); the couplings are shifted by $0.5$ to the right to lie between n and $n+1$. (

**b**) The population ${p}_{1}\left(t\right)$ of the first site as a function of time; the decay rates $exp(-2\gamma )$ are shown as dashed lines as a guide to the eye. (

**c**) The population of ${p}_{x}\left(\overline{t}\right)$ at $\overline{t}=0.2$ for $x=1,2,\dots ,120$.

**Figure 2.**(

**a**) The thermalized Lorentzian SD ${J}_{L,\beta}\left(\omega \right)$ (see Equations (13) and (14)) for $\Omega =100,\gamma =10$ at $T=0$ (blue solid line), $T=77$ (green dashed line) and $T=300$ (red dotted line). In all the remaining frames $\Omega =10,\gamma =0.001$. (

**b**) $T=77$; the first (dashed blue line) and the second (magenta dashed line) TEDOPA chain site populations ${p}_{1,2}\left(t\right)$ as a function of time. (

**c**) Same quantities and line styles as frame (

**b**) at $T=300$ (

**d**–

**f**): same quantities and styles as frames (

**a**–

**c**) of Figure 1 for $T=300$.

**Figure 3.**Lorentzian SD. The population ${p}_{1}\left(t\right)$ of the first TEDOPA chain as a function of time for $T=0$ (blue solid line), $T=77$ (green dotted line) and $T=300$ (red dashed line) for (

**a**) $\gamma =0.001$, (

**b**) $\gamma =1$ and (

**c**) $\gamma =10$. In all plots $exp(-2\gamma t)$ is show as a black dot-dashed line as a guide to the eye.

**Figure 4.**Ohmic SD. In all frames ${\omega}_{c}=100$, and red markers/solid lines, blue markers/dashed lines, green markers/dotted lines correspond, respectively, to the Ohmic ($s=1$), sub-Ohmic ($s=0.5$) and super-Ohmic ($s=2$) cases. (

**a**) $T=0$; the spectral density (15) for $s=0.5,\phantom{\rule{4pt}{0ex}}1,\phantom{\rule{4pt}{0ex}}2$. (

**b**) $T=0$; the chain coefficients ${\omega}_{n}$ (empty markers), ${\kappa}_{n}$ (filled markers). (

**c**) The population of the first chain site as a function of time; in the inset, the populations ${p}_{x}\left(\overline{t}\right)$ for $\overline{t}=0.1$ as a function of x. (

**d**) The thermalized SD ${J}_{O,\beta}^{s}\left(\omega \right)$ for $s=0.5,1,2$ at $T=300$. (

**e**,

**f**) Same quantities as frames (

**b**,

**c**) for $T=300$.

**Figure 5.**Ohmic SD. The population ${p}_{1}\left(t\right)$ of the first TEDOPA chain as a function of time for $T=0$ (blue solid line), $T=77$ (green dotted line) and $T=300$ (red dashed line) for (

**a**) $s=0.5$, (

**b**) $s=1$ and (

**c**) $s=2$. In the inset of all frames, the population ${p}_{x}\left(\overline{t}\right)$ at $\overline{t}=0.02$.

**Figure 6.**Lorentzian SD, full dynamics. $\gamma =0.001$ (

**a**) The expectation of ${\sigma}_{x}$ as a function of time for $T=0$ (purple dotted line) $T=77$ (orange dashed line) and $T=300$ (solid red line). (

**b**) The average occupation number ${p}_{1,2}\left(t\right)$ of the first (solid lines) and the second (dashed line) TEDOPA chain sites for $T=0$ (purple) $T=77$ (orange) and $T=300$ (red). (

**c**) The average occupation number of the chain sites $k,k=1,2,\dots ,20$ as a function of time for $T=300$.

**Figure 7.**Lorentzian SD, full dynamics. Same quantities as in Figure 6 for $\gamma =10$. (

**a**) The expectation of ${\sigma}_{x}$ as a function of time for $T=0$ (purple dotted line) $T=77$ (orange dashed line) and $T=300$ (solid red line). (

**b**) The average occupation number ${p}_{1,2}\left(t\right)$ of the first (solid lines) and the second (dashed line) TEDOPA chain sites for $T=0$ (purple) $T=77$ (orange) and $T=300$ (red). (

**c**) The average occupation number of the chain sites $k,k=1,2,\dots ,20$ as a function of time for $T=300$.

**Figure 8.**Sub-Ohmic SD ($s=0.5$). (

**a**) The expectation of ${\sigma}_{x}$ at different temperatures as a function of time (same line styles as in Figure 6a); in the inset, the average occupation number of the first and the second TEDOPA chain oscillators (same line styles as in Figure 6b). (

**b**) The average occupation number of the chain sites k, for $k=1,2,\dots ,20$ as a function of time at $T=0$. (

**c**) Same quantities as in frame (

**b**) for $T=300$.

**Figure 9.**Ohmic SD ($s=1$). Same quantities as in Figure 8. (

**a**) The expectation of ${\sigma}_{x}$ at different temperatures as a function of time (same line styles as in Figure 6a); in the inset, the average occupation number of the first and the second TEDOPA chain oscillators (same line styles as in Figure 6b). (

**b**) The average occupation number of the chain sites k, for $k=1,2,\dots ,20$ as a function of time at $T=0$. (

**c**) Same quantities as in frame (

**b**) for $T=300$.

**Figure 10.**Super -Ohmic SD ($s=2$). Same quantities as in Figure 8. (

**a**) The expectation of ${\sigma}_{x}$ at different temperatures as a function of time (same line styles as in Figure 6a); in the inset, the average occupation number of the first and the second TEDOPA chain oscillators (same line styles as in Figure 6b). (

**b**) The average occupation number of the chain sites k, for $k=1,2,\dots ,20$ as a function of time at $T=0$. (

**c**) Same quantities as in frame (

**b**) for $T=300$.

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**MDPI and ACS Style**

Tamascelli, D.
Excitation Dynamics in Chain-Mapped Environments. *Entropy* **2020**, *22*, 1320.
https://doi.org/10.3390/e22111320

**AMA Style**

Tamascelli D.
Excitation Dynamics in Chain-Mapped Environments. *Entropy*. 2020; 22(11):1320.
https://doi.org/10.3390/e22111320

**Chicago/Turabian Style**

Tamascelli, Dario.
2020. "Excitation Dynamics in Chain-Mapped Environments" *Entropy* 22, no. 11: 1320.
https://doi.org/10.3390/e22111320