# Non-Markovian Quantum Dynamics in a Squeezed Reservoir

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Perturbative Master Equations

## 4. Exact Method

## 5. Dynamics

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NMQSD | Non-Markovian quantum state diffusion; |

HEOM | Hierarchical equations Of motion; |

HOPS | Hierarchy of pure states; |

BCF | Bath correlation function. |

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**Figure 1.**(

**Left**): Linear rate ${\Gamma}_{M}$ for short time dynamics due to the Markov master equation divided by the overall coupling strength $\gamma $. (

**Right**): Quadratic rate for short time dynamics for the total system ${\Gamma}_{F}$ divided by $\gamma \Gamma $, where $\Gamma $ is the inverse bath correlation time scale. ${\Gamma}_{F}$ is proportional to ${\left|u-v\right|}^{2}$, which has a maximum when $\phi =\pi $. ${\Gamma}_{M}$ is proportional to $u(v+{v}^{*})$, which has a minimum at $\phi =\pi $. Here, we consider the spin boson model (14) in a squeezed bath (9) with $u=cosh\left(r\right)$ and $v=sinh\left(r\right){e}^{i\phi}$ and initial state $|{\psi}_{0}\rangle =|+\rangle :=\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$, where $|0\rangle $ and $|1\rangle $ are the eigenstates of ${\sigma}_{z}$. We have chosen the following parameters: $r=1/2$, $\Gamma =5\gamma $, and $\Omega ={\omega}_{B}={\omega}_{0}=\gamma $.

**Figure 2.**Short time dynamics of the spin boson model (14) in the squeezed bath (9) with $\Gamma =5\gamma $, $\Omega ={\omega}_{B}={\omega}_{0}=\gamma $, $r=0.5$, and $|{\psi}_{0}\rangle =|+\rangle $. The logarithm of the fidelity ${F}_{t}$ is displayed for short times within different approximate and exact descriptions. The quadratic short time dynamics holds up to $\gamma t=0.002$ for $\phi =\pi $ as the HEOM and the Redfield theory start to deviate from the short time expansion. For $\phi =0$, the quadratic short time dynamics, HEOM, and the Redfield agree well in the range of the plot. This is explained by the effectively strong system environment coupling for $\phi =\pi $, which sets a different regime of validity for the short time expansion. The parameters are chosen as in Figure 1.

**Figure 3.**Dynamics of the fidelity as in Figure 2 but for long times and for two squeezing directions $\phi =0$ (

**top**) and for $\phi =\pi $ (

**bottom**). The decay of the fidelity is faster for $\phi =\pi $ for HEOM and Redfield as expected from the microscopic model. Predictions from the quantum optical master equation (Markov) show an opposite behavior and the fidelity even has a positive slope at intermediate times.

**Figure 4.**Long time dynamics of the spin boson model (14) in the squeezed bath (9) as in Figure 2 with $\phi =0$. Both approximate methods describe correctly that the x and y Bloch sphere components decay to zero. The ${\sigma}_{z}$ expectation value asymptotically acquires a finite value modulated by weak oscillations which are captured properly in the Redfield theory.

**Figure 6.**Dynamics of the spin boson model (14) in the squeezed bath (9) with $\Gamma =3\gamma $, $\Omega =10\gamma $, ${\omega}_{0}=\Omega $, $r=0.5$, $\phi =0$, and initial state $|{\psi}_{0}\rangle =|+\rangle $. The Markovian master Equation (18) agrees well with the exact dynamics but does not capture small asymptotic oscillations.

**Figure 8.**Highly non-Markovian dynamics in the spin boson model. We have chosen $|{\psi}_{0}\rangle =|+\rangle $, $\Gamma =\gamma /2$, $\Omega ={\omega}_{B}={\omega}_{0}=\gamma /2$, and $\phi =0$. We are in the strong system–envinronment coupling regime since $\gamma >\Gamma $. This means that system–environment dynamics occurs on a faster time scale than the bath correlation function decay time. The failure of weak coupling master equations is expected.

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**MDPI and ACS Style**

Link, V.; Strunz, W.T.; Luoma, K.
Non-Markovian Quantum Dynamics in a Squeezed Reservoir. *Entropy* **2022**, *24*, 352.
https://doi.org/10.3390/e24030352

**AMA Style**

Link V, Strunz WT, Luoma K.
Non-Markovian Quantum Dynamics in a Squeezed Reservoir. *Entropy*. 2022; 24(3):352.
https://doi.org/10.3390/e24030352

**Chicago/Turabian Style**

Link, Valentin, Walter T. Strunz, and Kimmo Luoma.
2022. "Non-Markovian Quantum Dynamics in a Squeezed Reservoir" *Entropy* 24, no. 3: 352.
https://doi.org/10.3390/e24030352