# Probing Many-Body Systems near Spectral Degeneracies

## Abstract

**:**

## 1. Introduction

## 2. Time Correlation Matrix

#### 2.1. Diagonal Elements of the TCM

#### 2.2. Effect of Spectral Degeneracies

## 3. Example: Bosonic Josephson Junction

#### 3.1. Non-Interacting Bosons

#### 3.2. Interacting Bosons

## 4. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Signatures of a qualitative change in the evolution of a bosonic Josephson junction upon increasing interaction strength u. The plots represent the dynamics of 20 bosons for $u=1,2$, where the top panel gives the return probability $|{u}_{r,k}{|}^{2}$ and the bottom panel the transition probability $|{u}_{t,k}{|}^{2}$.

**Figure 2.**A comparison of the return probability $|{u}_{r,k}{|}^{2}$ and the average return probability $\langle |{u}_{r,k}{{|}^{2}\rangle}_{\tau}$ (top panel) and of the corresponding transition probabilities (bottom panel). The average was taken with respect to the exponential distribution of Equation (16). The interaction parameter is $u=1$, and $\overline{\tau}=1/10$.

**Figure 3.**The critical regime of the Hilbert-space localization with ${u}_{c}\approx 1.89$ is visualized with $\langle |{u}_{r,k}{{|}^{2}\rangle}_{\tau}$ and $\langle |{u}_{t,k}{{|}^{2}\rangle}_{\tau}$ at $k=70$.

**Figure 4.**Decay of $\langle |{u}_{t,k}{{|}^{2}\rangle}_{\tau}$ for different interaction parameters $u=1.7,\dots ,2.2$.

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**MDPI and ACS Style**

Ziegler, K.
Probing Many-Body Systems near Spectral Degeneracies. *Symmetry* **2021**, *13*, 1796.
https://doi.org/10.3390/sym13101796

**AMA Style**

Ziegler K.
Probing Many-Body Systems near Spectral Degeneracies. *Symmetry*. 2021; 13(10):1796.
https://doi.org/10.3390/sym13101796

**Chicago/Turabian Style**

Ziegler, Klaus.
2021. "Probing Many-Body Systems near Spectral Degeneracies" *Symmetry* 13, no. 10: 1796.
https://doi.org/10.3390/sym13101796