# Effect of Transverse Confinement on a Quasi-One-Dimensional Dipolar Bose Gas

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## Abstract

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## 1. Introduction

## 2. Method: The Variational Approach for the Energy Functional

## 3. Results and Discussions

#### 3.1. Repulsive Dipolar Interaction

#### 3.2. Attractive Dipolar Interaction: Droplet Region

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**For a fixed ${l}_{0}=40$ nm we show how ${V}_{DDI}\left(x\right)/{\sigma}^{3}$ changes for different values of $\sigma $, namely $\sigma =0.8,1$ and $1.2$.

**Figure 2.**Optimal values for $\sigma $ obtained with the different approximations as a function of density n. As a point of reference we show $\sigma $ for the system without dipolar interaction, ${a}_{d}=0$ as a blue solid line. The solid lines show the results obtained minimizing the energy functional (9), ${\sigma}_{sma}$, while the dashed ones correspond to the minimization of the approximated Hamiltonian (11), ${\sigma}_{A}$. Red lines are for the repulsive case $\theta =\pi /2$ and black lines are for the attractive case $\theta =0$.

**Figure 3.**Ground state energy estimates, in ${\hslash}^{2}/2m{n}^{2}$ units, within variational ansatz for ${l}_{0}=57.3$ nm and ${a}_{1D}/{a}_{0}=-2000$ as a function of particles density. The solid black lines are the energies obtained using $\sigma =1$ [9] while solid red lines are using the optimal $\sigma $ obtained from variational minimization. We show results for $\theta =\pi /2$ in panel (

**a**) and $\theta =0$ in panel (

**b**). We subtracted the transverse energy for $\sigma =1$ for clarity. In panel (

**c**), we show the optimal values of $\sigma $ as a function of density for the cases reported in panel (

**a**), solid red line, and in panel (

**b**), solid blue line. The solid green line in panel (

**a**) is the limit of the energy in the Tonks–Girardeau limit.

**Figure 4.**Ground state energy estimates within variational ansatz for ${a}_{1D}/{a}_{0}=-6500$ for two selected values of ${l}_{0}$, namely ${l}_{0}=57.3$ nm as from Ref. [23] and ${l}_{0}=100$ nm. The solid black lines are the trial energies obtained using $\sigma =1$ while solid red lines are trial energies obtained using the optimal $\sigma $ from full minimization. The thick solid lines are for ${l}_{0}=57.3$ nm, while thin ones are for ${l}_{0}=100$ nm.

**Figure 5.**Ground state energy within the variational ansatz for ${l}_{0}=40$ nm and ${a}_{1D}/{a}_{0}=-2800$ as a function of particle density. The solid black lines are the trial energies computed using $\sigma =1$ [9] while the solid red lines are computed with the optimal $\sigma $ obtained from variational minimization.

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**MDPI and ACS Style**

De Palo, S.; Orignac, E.; Citro, R.; Salasnich, L.
Effect of Transverse Confinement on a Quasi-One-Dimensional Dipolar Bose Gas. *Condens. Matter* **2023**, *8*, 26.
https://doi.org/10.3390/condmat8010026

**AMA Style**

De Palo S, Orignac E, Citro R, Salasnich L.
Effect of Transverse Confinement on a Quasi-One-Dimensional Dipolar Bose Gas. *Condensed Matter*. 2023; 8(1):26.
https://doi.org/10.3390/condmat8010026

**Chicago/Turabian Style**

De Palo, Stefania, Edmond Orignac, Roberta Citro, and Luca Salasnich.
2023. "Effect of Transverse Confinement on a Quasi-One-Dimensional Dipolar Bose Gas" *Condensed Matter* 8, no. 1: 26.
https://doi.org/10.3390/condmat8010026