# Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross–Pitaevskii Formalism with Form Factor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multipole Interactions

#### 2.1. D-D Interaction

#### 2.2. Q-Q Interaction

#### 2.3. O-O Interaction

#### 2.4. Comparative Analysis

#### 2.4.1. Interaction Energy of Multipoles in General Terms

#### 2.4.2. Discussion of Interactions in Momentum Representation

## 3. The Gross–Pitaevskii Equation

#### 3.1. Equations in Real-Space and Momentum-Space Representations

#### 3.2. Form Factor

#### 3.2.1. The Zoology of Form Factors

#### 3.2.2. Example of Translational Invariance Violation

#### 3.3. General Solution of the Gp Equation with a Quasilocal Form Factor

#### 3.4. The Choice of the Form Factor

#### Core and Split

#### 3.5. Condensate Density

#### 3.6. Quadrupoles in the Optical Lattice

#### 3.6.1. Where Do Small Concentrations Lead to?

#### 3.6.2. Schrödinger Equation in a Periodic Potential

#### 3.6.3. Weak Coupling Mode

#### 3.6.4. Strong Coupling Mode

#### 3.6.5. Comparative Analysis of Weak and Strong Coupling Modes

## 4. Analysis of Condensate Excitations

#### 4.1. Intermediate Results

#### 4.2. Derivation of the Gp Functional for a Form Factor of a Special Form

#### 4.3. Derivation of the Equation for the Critical Momentum

#### 4.4. Analysis of Condensate Stability

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BCS | Bardeen–Cooper–Schrieffer |

BEC | Bose–Einstein Condensation |

GP | Gross–Pitaevskii |

QFT | Quantum Field Theory |

## Appendix A

#### Appendix A.1. Calculation of the k-Representation of the D-D Interaction

**Figure A1.**The relation between the angles between the axis of symmetry of the multipole (the unit vector $\mathit{a}$) and the vectors $\mathit{r}$ and $\mathit{k}$

#### Appendix A.2. Calculation of the k-Representation of the Q-Q Interaction

#### Appendix A.3. Calculation of the r-Representation of the O-O Interaction

#### Appendix A.4. Calculation of the k-Representation of the O-O Interaction

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**Figure 2.**Interaction energy as a function of angles with a fixed momentum. For illustrative purposes the momentum in case of ${U}_{O}$ 10 times larger than the momentum ${U}_{D}$, ${U}_{Q}$.

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Alexandrov, A.A.; Badamshina, A.U.; Ogarkov, S.L. Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross–Pitaevskii Formalism with Form Factor. *Condens. Matter* **2020**, *5*, 61.
https://doi.org/10.3390/condmat5040061

**AMA Style**

Alexandrov AA, Badamshina AU, Ogarkov SL. Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross–Pitaevskii Formalism with Form Factor. *Condensed Matter*. 2020; 5(4):61.
https://doi.org/10.3390/condmat5040061

**Chicago/Turabian Style**

Alexandrov, Artem A., Alina U. Badamshina, and Stanislav L. Ogarkov. 2020. "Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross–Pitaevskii Formalism with Form Factor" *Condensed Matter* 5, no. 4: 61.
https://doi.org/10.3390/condmat5040061