# Formation of Matter-Wave Droplet Lattices in Multi-Color Periodic Confinements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model and Analytical Framework

## 3. Results

#### 3.1. Potential Profiles and Corresponding Trap Parameters

#### 3.2. Periodic Lattice Density Patterns in QDs

#### 3.3. Double-Well Superlattice Density Patterns in QDs

#### 3.4. Stability of QDs in MOL Confinement

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Various potential profiles of the MOL by tuning parameters ${p}_{1}$ and ${p}_{2}$ of Equation (4) for $\gamma =0.05$, $k=0.84$, $E=-2/9$ in the interval $[-2\pi /k,2\pi /k]$: (

**a**) for fixed ${p}_{2}=1$, ${p}_{1}$ varying from $-5$ to $+5$; (

**b**) for fixed ${p}_{1}=1$, ${p}_{2}$ varying from $-5$ to $+5$; (

**c**) flipping of the OL phase by the half-wavelength: ${p}_{1}=0$, ${p}_{2}=0.1$ (black solid line) and ${p}_{1}=0$, ${p}_{2}=-0.1$ (dashed blue line); (

**d**) BOL: ${p}_{1}=0$, ${p}_{2}=3.5$ (black line); (

**e**) triple-well optical superlattice: ${p}_{1}=-5$, ${p}_{2}=1$ (red solid line); (

**f**) frustrated double-well optical superlattice: ${p}_{1}=1$, ${p}_{2}=-5$ (blue solid line); (

**g**) bi-periodic frustrated double-well optical superlattice: ${p}_{1}=1$, ${p}_{2}=4$ (red solid line). The spatial co-ordinate is scaled by the oscillator length.

**Figure 2.**Condensate density patterns for two-color BOL traps with ${p}_{1}<0$ and ${p}_{2}=0$: (

**a**) ${p}_{1}=0$ (free space); (

**b**) ${p}_{1}=-0.5$; (

**c**) ${p}_{1}=-1$; and (

**d**) ${p}_{1}=-1.50$. Each plot of (

**a**–

**d**) has three panels: the upper panel shows the density plot; the middle panel consists of a 2D plot of the density; the lower panel indicates the corresponding trap profile. Here, the magnitude of the physical parameters: $b=1$, $\gamma =1$, $k=0.84$, ${G}_{1}=-1$, ${G}_{2}=0.999999999$, $E=-2/9$. The spatial co-ordinate is scaled by the oscillator length.

**Figure 3.**Condensate density patterns for two-color BOL traps with ${p}_{1}>0$ and ${p}_{2}=0$ are depicted: (

**a**) ${p}_{1}=0$ (free space); (

**b**) ${p}_{1}=0.5$; (

**c**) ${p}_{1}=1$; and (

**d**) ${p}_{1}=1.5$. Each plot (

**a**–

**d**) has three panels: the upper panel shows the density plot; the middle panel consists of a 2D plot of the density; the last panel represents the corresponding trap profile. Here, the magnitude of the physical parameters: $b=1$, $\gamma =1$, $k=0.84$, ${G}_{1}=-1$, ${G}_{2}=0.999999999$, $E=-2/9$. The spatial co-ordinate is scaled by the oscillator length.

**Figure 4.**The profile of MF and BMF nonlinearities is plotted with respect to varying (

**a**) ${p}_{1}$ in the interval [$-1.5,1.5$] at $x=0$ for ${p}_{2}=0$ and (

**b**) ${p}_{2}$ in the interval [$-1.5,1.5$] at $x=0$ with ${p}_{1}=0.5$. Each inset plot depicts the variation of ${g}_{1}\left(x\right)$ (BMF, red line) and ${g}_{2}\left(x\right)$ (MF, blue line) for the indicated magnitude of ${p}_{1}$ and ${p}_{2}$, respectively. Here, the magnitude of the physical parameters: $b=1$, $\gamma =1$, $k=0.84$, ${G}_{1}=-1$, ${G}_{2}=0.999999999$, $E=-2/9$.

**Figure 5.**Condensate density patterns for four-color BOL traps with ${p}_{1}=0.5$ and: (

**a**) ${p}_{2}=0$ (BOL); (

**b**) ${p}_{2}=-0.5$; (

**c**) ${p}_{2}=-1$; and (

**d**) ${p}_{2}=-1.5$. Each plot (

**a**–

**d**) has three panels: the upper panel shows the density plot; the middle panel consists of a 2D plot of the density; the lower panel indicates the corresponding trap profile. Here, the magnitude of the physical parameters: $b=1$, $\gamma =1$, $k=0.84$, ${G}_{1}=-1$, ${G}_{2}=0.999999999$, $E=-2/9$. The spatial co-ordinate is scaled by the oscillator length.

**Figure 6.**Condensate density patterns for four-color BOL traps with ${p}_{1}=0.5$ and: (

**a**) ${p}_{2}=0$; (

**b**) ${p}_{2}=0.5$; (

**c**) ${p}_{2}=1$; and (

**d**) ${p}_{2}=1.5$. Each plot (

**a**–

**d**) has three panels: the upper panel shows the density plot; the middle panel consists of a 2D plot of the density; the lower panel indicates the corresponding trap profile. Here, the magnitude of the physical parameters: $b=1$, $\gamma =1$, $k=0.84$, ${G}_{1}=-1$, ${G}_{2}=0.999999999$, $E=-2/9$. The spatial co-ordinate is scaled by the oscillator length.

**Figure 7.**(Color online) (

**a**) For the VK stability criterion, the slope of normalization with the chemical potential (${N}_{E}$) is plotted with respect to a varying chemical potential (E); (

**b**) $Im(\Omega )$ is depicted as a function of ${p}_{1}$ and the wavenumber (l) keeping ${p}_{2}=0$; (

**c**) with ${p}_{1}=0.5$, $Im(\Omega )$ is depicted as a function of ${p}_{2}$ and the wavenumber (l). Here, the magnitude of the physical parameters: $b=1$, $\gamma =0.05$, $k=0.84$, ${G}_{1}=-1$, ${G}_{2}=0.999999999$, $E=-2/9$ with l varying from 0 to 3.

**Table 1.**Various shapes of the MOL potential by tuning the magnitude of the power of the laser beam, i.e., ${p}_{1}$ and ${p}_{2}$.

Multi-Color OL (for $\mathit{\mu}=0$) | ||
---|---|---|

${p}_{1}$ | ${p}_{2}$ | Trap form |

0 | 0 | Free space |

<1 | 0 | OL (k) |

0 | ≠0 | BOL ($2k$, $4k$) |

>1 | 0 | BOL (k, $2k$) |

8 | −2 | BOL ($3k$, $4k$) |

8 | 2 | BOL ($3k$, $4k$) |

≠8 | −2 | TOL ($2k$, $3k$, $4k$) |

≠8 | 2 | TOL ($2k$, $3k$, $4k$) |

Other points | Other points | FOL (k, $2k$, $3k$, $4k$) |

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**MDPI and ACS Style**

Pathak, M.R.; Nath, A.
Formation of Matter-Wave Droplet Lattices in Multi-Color Periodic Confinements. *Symmetry* **2022**, *14*, 963.
https://doi.org/10.3390/sym14050963

**AMA Style**

Pathak MR, Nath A.
Formation of Matter-Wave Droplet Lattices in Multi-Color Periodic Confinements. *Symmetry*. 2022; 14(5):963.
https://doi.org/10.3390/sym14050963

**Chicago/Turabian Style**

Pathak, Maitri R., and Ajay Nath.
2022. "Formation of Matter-Wave Droplet Lattices in Multi-Color Periodic Confinements" *Symmetry* 14, no. 5: 963.
https://doi.org/10.3390/sym14050963