# Modulational Instability, Inter-Component Asymmetry, and Formation of Quantum Droplets in One-Dimensional Binary Bose Gases

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## Abstract

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## 1. Introduction

## 2. Model and Methods

## 3. Modulation Instability Versus QDs

#### 3.1. The Single-Component GP Model

#### 3.1.1. The Droplet Solution

#### 3.1.2. The Plane-Wave Solution

#### 3.1.3. Modulational Instability of the Plane Waves

#### 3.2. The Two-Component Gross–Pitaevskii Model

#### 3.2.1. Asymmetric QDs with Unequal Populations (${N}_{1}\ne {N}_{2}$) for ${g}_{1}={g}_{2}$ ($P=1$)

#### 3.2.2. Asymmetric QDs in the System with $P\ne 1$ (${g}_{1}\ne {g}_{2}$)

#### 3.2.3. The MI of the Asymmetric PW States

#### 3.2.4. The MI for $P=1$

#### 3.2.5. The MI for $P\ne 1$

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Other Exact Solutions for the Single-Component GP Equation

#### Appendix A.1. δg/g>0

#### Appendix A.2. δg/g<0

## Appendix B. Analytical Solutions for Strongly Asymmetric Fundamental and Dipole States

**Figure A1.**Comparison of the asymptotic analytical solutions, given by Equations (A8) and (A9), with their numerically obtained counterparts. The density of the first (${n}_{1}$) and second (${n}_{2}$) components are displayed in top and bottom panels, respectively. Solid blue lines represent the numerical results, while dashed red lines represent the analytical solution. Here, parameters are $\delta g/g=0.05$, ${N}_{1}=0.0001067$, ${N}_{2}=0.0148044$ and ${\left({\mu}_{2}\right)}_{\mathrm{GS}}={\mu}_{2}=-0.005$.

## Appendix C. Other Exact Solutions in the Case of **N**_{1} ≪ N_{2}

_{1}≪ N

_{2}

#### Appendix C.1. Solution of Equation (A7)

#### Appendix C.2. Solutions of Equation (A6)

**Solutions For ${P}^{2}=1/3$**

#### Appendix C.2.1. Solution I

#### Appendix C.2.2. Solution II

#### Appendix C.2.3. Solution III

**Solutions for ${P}^{2}=1$.**

#### Appendix C.2.4. Solution IV

#### Appendix C.2.5. Solution V

#### Appendix C.2.6. Solution VI

#### Appendix C.2.7. Solution VII

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**Figure 1.**The maximum density ${n}_{max}\equiv n(z=0)$ in the FT (flat-top) state, as per Equation (14), and the PW (plane-wave)/density are displayed as functions of $\mu $ by the red-solid and blue-dashed curves, respectively, for $\delta g/g=0.05$. In this case, Equation (11) yields ${n}_{0}=36.025$ and ${\mu}_{0}=-0.900633$. The PW solution includes upper and lower branches corresponding to ${n}^{\pm}$, as given by Equation (20), the lower one (marked by circles) being subject to the MI (modulational instability). The spinodal point is one with coordinates $\left({\mu}_{c},{n}_{c}\right)$. For other values of $\delta g/g$, the plot can be generated from the present one by rescaling. The inset shows the density profile of the FT solution for $\delta g/g=0.05$ and $\mu ={\mu}_{0}+0.00001$, very close to the delocalization limit (the transition to PW).

**Figure 2.**Color-coded values of the MI gain, $\sigma =\mathrm{Im}\left(\mathsf{\Omega}\right)$, are displayed for fixed $n=40$ in (

**a**), and for fixed $\delta g/g=0.05$ in (

**b**). Note that panel (

**a**) covers both signs of the cubic nonlinearity, $\delta g>0$ and $\delta g<0$. Solid and dashed white curves represent the MI boundary (Equation (26)) and the peak value of the MI gain (Equation (27)), respectively.

**Figure 3.**A typical example of the MI development, starting from an unstable PW state, with density $n=10$ and $\delta g/g=0.05$, which is subject to the MI, pursuant to Figure 2. In (

**a**), the spatiotemporal pattern of the evolution of the condensate density is shown. In the right-hand panels, cross sections of the density profiles are displayed at $t=100$ (

**b**), $t=120$ (

**c**), and $t=140$ (

**d**). The simulations were performed in domain $-50<z<+50$ with 2500 grid points and periodic boundary conditions.

**Figure 4.**(

**a**) Stationary weakly asymmetric (with respect to the two components) solutions of Equation (7), obtained for ${\mu}_{2}=-0.4$ with fixed ${N}_{1}=100$. Dashed and solid curves display density profiles of the first (${n}_{1}$) and second (${n}_{2}$) components, respectively. (

**b**) The semi-log plot of the density profiles of ${n}_{2}$ for ${\mu}_{2}=-0.4$, $-0.04$, and 0 at $z>0$. (

**c**) Dependences of ${N}_{2}$ (black dots: the left vertical axis) and asymmetry parameter ${\delta}_{21}$, defined as per Equation (28) (the red dashed line pertaining to the right vertical axis), on ${\mu}_{2}$ for fixed ${N}_{1}=100$. The parameters are $P=1$ $({g}_{1}={g}_{2})$ and $\delta g/g=0.05$. The symmetric point with ${N}_{1}={N}_{2}=100$ and ${\delta}_{21}=0$ corresponds to ${\mu}_{1}={\mu}_{2}=-0.88878$.

**Figure 5.**(

**a**) Stationary solutions of Equation (7), obtained for $\delta g/g=0.05$ and ${N}_{1}=100$. From the left panel to the right one, the parameter in Equation (8) is $P=1.25$, $1.67$, and $2.5$, and the chemical potential for the second component is ${\mu}_{2}=-0.018$, $-0.011$, and $-0.006$, respectively, just below the threshold above which the tails of ${\psi}_{2}$ extend to infinity. Dashed and solid curves represent the density of the first (${n}_{1}$) and second (${n}_{2}$) components. (

**b**) Dependences of ${N}_{2}$ (black dots: the left vertical axis) and asymmetry parameter ${\delta}_{21}$, defined as per Equation (28) (the red dashed line pertaining to the right vertical axis), on ${\mu}_{2}$ for fixed ${N}_{1}=100$ and $P=1.25$ or $P=1.67$.

**Figure 6.**The negative-pressure region in the $\left({n}_{1},{n}_{2}\right)$ plane for $\delta g/g=0.05$ and values of asymmetry parameter in Equation (8) $P=1$ (the solid curve), $1.25$ (dashed), $2.5$ (dashed-dotted), and 10 (dotted). Boundaries are determined by the zero-pressure condition, as given by Equation (31). The negative pressure, at which localized states may exist, occurs inside the boundaries. Thin lines represent relation ${n}_{2}=P{n}_{1}$.

**Figure 8.**Color-coded values of the MI gain, $\sigma =\mathrm{Im}\left(\mathsf{\Omega}\right)$, for asymmetric PWs, as calculated from Equation (36) in the plane of wave number $|k|$ and density ratio ${n}_{12}={n}_{1}/{n}_{2}$, are displayed for (

**a**) $P=1$ and (

**b**) $P=1.25$ with fixed $\delta g/g=0.05$ and $({n}_{1}+{n}_{2})/2=10$. The solid and dashed white curves represent the MI boundary $k={k}_{0}$ and the peak value of the MI gain at $k={k}_{\mathrm{max}}\phantom{\rule{4pt}{0ex}}={k}_{0}/\sqrt{2}$, respectively. In (

**c**), we plot $\sigma \left({k}_{\mathrm{max}}\right)$ (circles) and ${n}_{12}^{\mathrm{max}}$ (triangles) versus P.

**Figure 9.**Numerically simulated development of the MI of asymmetric PW states in the two-component system, with $P=1$ and $\delta g/g=0.05$. The initial PW states are taken with fixed total density, $({n}_{1}+{n}_{2})/2=10$. (

**a**) The evolution of the central density of the first component, ${n}_{1}(z=0)$, for different density ratios in the two components, ${n}_{12}={n}_{1}/{n}_{2}$. (

**b**,

**c**) Snapshots of density profiles for the cases of (

**b**) ${n}_{12}\equiv {n}_{1}/{n}_{2}=1$ at $t=200$ and (

**c**) ${n}_{12}=9$ at $t=400$. Panels (

**d**,

**e**) and (

**f**,

**g**) are top views of the spatiotemporal evolution of the densities, ${n}_{1}\left(z,t\right)$ and ${n}_{2}(z,t)$, for ${n}_{12}=1$ and ${n}_{12}=9$, respectively. Simulations were performed in the domain $-50\le z\le +50$ with 2048 grid points, subject to periodic boundary conditions. In this figure and in Figure 10, the scaled time unit corresponds to $\sim 1$ $\mathsf{\mu}$s in physical units.

**Figure 10.**Numerically simulated development of the modulational instability in the two-component system with $\delta g/g=0.05$ and $P=1.25$. The initial PW states are taken with a fixed total density, $({n}_{1}+{n}_{2})/2=10$. (

**a**) The evolution of the central density of the first component, ${n}_{1}(z=0)$, for different density ratios in the two components, ${n}_{12}={n}_{1}/{n}_{2}$. (

**b**–

**d**) Snapshots of density profiles for the cases of (

**b**) ${n}_{12}\equiv {n}_{1}/{n}_{2}\sim 0.1$ at $t=300$, (

**c**) ${n}_{12}=1$ at $t=200$ and (

**d**) ${n}_{12}=9$ at $t=600$. Panels (

**e**–

**g**) represent the top view of the spatiotemporal evolution of the densities, ${n}_{1}\left(z,t\right)$, corresponding to (

**b**–

**d**), respectively (the evolution of ${n}_{2}\left(z,t\right)$ shows similar patterns). Simulations were performed in the domain $-50\le z\le +50$ with 2048 grid points, subject to periodic boundary conditions.

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**MDPI and ACS Style**

Mithun, T.; Maluckov, A.; Kasamatsu, K.; Malomed, B.A.; Khare, A.
Modulational Instability, Inter-Component Asymmetry, and Formation of Quantum Droplets in One-Dimensional Binary Bose Gases. *Symmetry* **2020**, *12*, 174.
https://doi.org/10.3390/sym12010174

**AMA Style**

Mithun T, Maluckov A, Kasamatsu K, Malomed BA, Khare A.
Modulational Instability, Inter-Component Asymmetry, and Formation of Quantum Droplets in One-Dimensional Binary Bose Gases. *Symmetry*. 2020; 12(1):174.
https://doi.org/10.3390/sym12010174

**Chicago/Turabian Style**

Mithun, Thudiyangal, Aleksandra Maluckov, Kenichi Kasamatsu, Boris A. Malomed, and Avinash Khare.
2020. "Modulational Instability, Inter-Component Asymmetry, and Formation of Quantum Droplets in One-Dimensional Binary Bose Gases" *Symmetry* 12, no. 1: 174.
https://doi.org/10.3390/sym12010174