# Adiabatic Manipulation of a System Interacting with a Spin Bath

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## Abstract

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## 1. Introduction

## 2. Physical Model

## 3. Analysis of the Model

## 4. Numeric Results

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

STIRAP | Stimulated Raman Adiabatic Passage |

RWA | Rotating Wave Approximation |

NMR | Nuclear Magnetic Resonance |

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**Figure 1.**$\Lambda $-coupling scheme involving the three atomic states $\left(\right)$, $\left(\right)$ and $\left|e\right.\u232a$. Population is supposed to be transferred from state $\left(\right)$ to state $\left(\right)$, which is accomplished via two pulses in the so-called counterintuitive sequence, where the coupling between $\left(\right)$ and $\left|e\right.\u232a$ precedes the coupling between $\left|e\right.\u232a$ and $\left(\right)$. The inset represents the typical shape of the pulses.

**Figure 2.**$\Lambda $-coupling scheme in the presence of interaction with the spin bath. The state $\left|e\right.\u232a\left(\right)open="|"\; close="\rangle ">j,m$ is coupled to the states $\left(\right)open="|"\; close="\rangle ">{g}_{1}$ and $\left(\right)open="|"\; close="\rangle ">{g}_{2}$ via the two pulses (represented by the blue solid arrow and red solid arrow), and it is coupled also to the states $\left(\right)open="|"\; close="\rangle ">{g}_{1}$ and $\left(\right)open="|"\; close="\rangle ">{g}_{2}$ through the bath (represented by the green dashed arrows).

**Figure 3.**Dynamical properties for different bath sizes when the system is prepared in state $\left(\right)$: (

**a**) final population of state $\left(\right)$ ($P\left(T\right)\equiv \left(\right)open="\langle "\; close="|">{g}_{2}$); (

**b**) purity of the final state of the system; (

**c**) the variance in ${J}_{z}$ of the end of the process ($t=T$), denoted as ${\sigma}^{2}\left({J}_{z}\right)$. All such quantities are plotted as functions of the coupling constant $\eta $ (in units of $1/T$ and in logarithmic scale). In all the plots, different values of the number of spins in the bath are considered: $L=10$ (black dashed line), $L=20$ (red pointed line) and $L=40$ (green solid line). All the other parameters are always the same: $T{\Omega}_{0}=100$, $\tau /T=0.1$, $T{\Delta}_{S}=1$ and $T{\Delta}_{E}=1$.

**Figure 4.**Final population of state $\left(\right)$ ($P\left(T\right)\equiv \left(\right)open="\langle "\; close="|">{g}_{2}$) when the system is prepared in the state $\left(\right)$ as functions of the coupling constant $\eta $ (in units of $1/T$ and in logarithmic scale). Different values of the number of spins in the bath are considered along with relevant rescaling of the system–environment interaction term by the factor $\sqrt{10/L}$. The spin numbers are $L=10$ (black dashed line), $L=20$ (red pointed line) and $L=40$ (green solid line). All the other parameters are always the same: $T{\Omega}_{0}=100$, $\tau /T=0.1$, $T{\Delta}_{S}=1$ and $T{\Delta}_{E}=1$. According to the Holstein–Primakoff theory, the three situations are essentially equivalent and the relevant curves consequently coincide.

**Figure 5.**Final population of state $\left(\right)$ ($P\left(T\right)\equiv \left(\right)open="\langle "\; close="|">{g}_{2}$) when the system is prepared in the state $\left(\right)$ as functions of the coupling constant $\eta $ (in units of $1/T$ and in logarithmic scale). Different values of ${\Delta}_{E}$ are considered: $T{\Delta}_{E}=1$ (black dashed line), $T{\Delta}_{E}=50$ (red pointed line), $T{\Delta}_{E}=100$ (green solid line) and $T{\Delta}_{E}=500$ (blue bold solid line). All the other parameters are always the same: $T{\Omega}_{0}=100$, $\tau /T=0.1$ and $T{\Delta}_{S}=1$.

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**MDPI and ACS Style**

Militello, B.; Napoli, A.
Adiabatic Manipulation of a System Interacting with a Spin Bath. *Symmetry* **2023**, *15*, 2028.
https://doi.org/10.3390/sym15112028

**AMA Style**

Militello B, Napoli A.
Adiabatic Manipulation of a System Interacting with a Spin Bath. *Symmetry*. 2023; 15(11):2028.
https://doi.org/10.3390/sym15112028

**Chicago/Turabian Style**

Militello, Benedetto, and Anna Napoli.
2023. "Adiabatic Manipulation of a System Interacting with a Spin Bath" *Symmetry* 15, no. 11: 2028.
https://doi.org/10.3390/sym15112028