# Accurate Truncations of Chain Mapping Models for Open Quantum Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Fitted Chain

#### 3.2. Next-Nearest Neighbor Coupled Chain

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Quantum system coupled to a discrete set of environment modes; (

**b**) quantum system coupled to a reaction mode (a collective environment mode), with this mode coupled to a residual bath of modes; (

**c**) chain mapping for the environment modes after n steps, with a residual bath of $N-n$ modes at the end of the chain.

**Figure 2.**(

**a**) Spectral density for an emitter in the combined sphere-bowtie antenna (illustrated in the inset). The emitter is situated at the center of the bowtie (indicated by the red arrow), with its frequency chosen close to the maximum of the spectral density (indicated by the thin vertical line); (

**b**) chain mapping frequencies ${\tilde{\mathsf{\Omega}}}_{n}$, ${\mathsf{\Omega}}_{n}$ and coupling parameters ${\tilde{D}}_{n}$, ${\lambda}_{n}$ for the phonon and particle mapping.

**Figure 3.**Excited-state population of an initially excited two-level quantum emitter with frequency ${\omega}_{e}=3.6$ eV and dipole moment ${\mu}_{e}=0.6$ e nm coupled to the antenna shown in Figure 2a, calculated using chains truncated after different numbers of sites ${N}_{C}$.

**Figure 4.**Population of the same emitter as in Figure 3 for two different chain lengths: ${N}_{C}=30$ (orange) and ${N}_{C}=25$ (blue) compared with a long chain (${N}_{C}=300$), which is considered exact. Comparison between the chains without any absorbing terms (dashed lines) and chains with a single absorbing term at their end, given by $\gamma =2\pi {J}_{{N}_{C}}\left({\mathsf{\Omega}}_{{N}_{C}}\right)$.

**Figure 5.**(

**a**) Population for the quantum emitter for three different chain lengths: ${N}_{C}=30$ (orange), ${N}_{C}=25$ (blue) and ${N}_{C}=15$ (green) compared to the “exact” one (black). An absorbing function has been added to the first three chains. The parameters are chosen to give a good description of the dynamics, but a wide range is valid for the first to lengths. When the chain is very short, as for ${N}_{C}=15$, the prediction of the dynamics starts to break down; (

**b**) effective spectral densities when the absorbing terms are added in the Hamiltonian for the same lengths as in (

**a**). For ${N}_{C}=30$ and ${N}_{C}=25$, the effective spectral density is similar to ${J}_{0}\left(\omega \right)$, although neither of them describes it in detail. For ${N}_{C}=15$, the effective spectral density presents a shift in the frequency of the main peak of ${J}_{0}\left(\omega \right)$, affecting the dynamics.

**Figure 6.**Population of the quantum emitter for ${N}_{C}=15$ after the fit to the spectral density (orange) and the fit to the correlation function (blue).

**Figure 7.**(

**a**) Fit of the spectral density of Figure 2a when allowing next-nearest neighbor interactions in the chain, compared to a fit with only nearest neighbor interactions with the same number of modes ($N=15$); (

**b**) fit of the spectral density of Ref. [38] using a chain with next-nearest neighbor interactions. Black line: Original fit using a full matrix ${\omega}_{ij}$, with all modes coupled to the emitter. Yellow line: Fit of the same data using a chain with only next-nearest neighbor interactions, and the emitter only coupled to the first two sites.

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Sánchez-Barquilla, M.; Feist, J.
Accurate Truncations of Chain Mapping Models for Open Quantum Systems. *Nanomaterials* **2021**, *11*, 2104.
https://doi.org/10.3390/nano11082104

**AMA Style**

Sánchez-Barquilla M, Feist J.
Accurate Truncations of Chain Mapping Models for Open Quantum Systems. *Nanomaterials*. 2021; 11(8):2104.
https://doi.org/10.3390/nano11082104

**Chicago/Turabian Style**

Sánchez-Barquilla, Mónica, and Johannes Feist.
2021. "Accurate Truncations of Chain Mapping Models for Open Quantum Systems" *Nanomaterials* 11, no. 8: 2104.
https://doi.org/10.3390/nano11082104