# Coherent Plasma in a Lattice

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hamiltonian

## 3. Selection of the Quantum State for the Electromagnetic Field

## 4. Effective Hamiltonian

## 5. Lower Bound of the Energy Gap for the Coherent Phase

## 6. Second-Order Perturbation Theory for Composite Coherent States

- ${}_{n}\langle \sigma ,\alpha ,\mathcal{A}|{\sigma}^{\prime},\alpha ,\mathcal{A}\rangle {}_{{n}^{\prime}}={\delta}_{\sigma {\sigma}^{\prime}}{\delta}_{n{n}^{\prime}}$
- The expectation value of ${H}_{\mathrm{int}}$ (Equation (19)) on the states ${|\sigma ,\alpha ,\mathcal{A}\rangle}_{n}$ (see Appendix C) is provided by$${}_{n}\langle \sigma ,\alpha ,\mathcal{A}|{H}_{\mathrm{int}}|{\sigma}^{\prime},\alpha ,\mathcal{A}\rangle {}_{{n}^{\prime}}=-{\omega}_{p}\sqrt{\frac{8\pi}{3}}\left|\alpha \mathcal{A}\right|\frac{\sigma}{{R}^{2}}{\delta}_{\sigma {\sigma}^{\prime}}{\delta}_{n{n}^{\prime}}$$
- The expectation value of the unperturbed Hamiltonian (see Appendix C) is$$\langle \sigma ,\alpha ,\mathcal{A}|H+{\widehat{H}}_{\mathrm{photon}}|\sigma ,\alpha ,\mathcal{A}\rangle ={\omega}^{\prime}\left[N\left(\sigma \frac{{\left|\alpha \right|}^{2}}{{R}^{2}}+\frac{3}{2}+{\left|\mathcal{A}\right|}^{2}\frac{\sigma}{{R}^{2}}\right)+\frac{3}{2}\right]$$

## 7. The Second-Order Contribution

## 8. Spatial Dimension of the Coherent States and Concept of Natural Resonating Cavity

## 9. Discussion

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Jellium Crystal

## Appendix B. Renormalization of the Photon Field

## Appendix C. Detailed Calculation of the Expectation Values of the States${|\sigma ,\alpha ,\mathcal{A}\rangle}_{n}$

## Appendix D. Estimate of the Numerical Value of the Energy Gap

## Appendix E. Calculation of $\langle \mathrm{\Omega}|\delta \widehat{\overrightarrow{A}}(\overrightarrow{x},t)|\mathrm{\Omega}\rangle $

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**Figure 1.**Periodic harmonic potential in the 1—direction with square modulus of the wave functions of the oscillators in their ground state (

**left**) and elongated by $\xi $ due to their coherent oscillation (

**right**).

**Figure 2.**Normalized energy profile ${j}_{0}^{2}\left(\pi \frac{r}{{r}_{CD}}\right)$ as a function of the normalized radius of the coherence domain.

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**MDPI and ACS Style**

Gamberale, L.; Modanese, G.
Coherent Plasma in a Lattice. *Symmetry* **2023**, *15*, 454.
https://doi.org/10.3390/sym15020454

**AMA Style**

Gamberale L, Modanese G.
Coherent Plasma in a Lattice. *Symmetry*. 2023; 15(2):454.
https://doi.org/10.3390/sym15020454

**Chicago/Turabian Style**

Gamberale, Luca, and Giovanni Modanese.
2023. "Coherent Plasma in a Lattice" *Symmetry* 15, no. 2: 454.
https://doi.org/10.3390/sym15020454