# Suppression of Quantum-Mechanical Collapse in Bosonic Gases with Intrinsic Repulsion: A Brief Review

## Abstract

**:**

## 1. Introduction

## 2. The Basic Three- and Two-Dimensional Models

#### 2.1. The Quantum Collapse in the Linear Schrödinger Equation

#### 2.2. The Three-Dimensional Ground State (GS) Created by the Cubic Self-Repulsive Nonlinearity

#### 2.3. The Quantum Phase Transition Induced by the Lee–Huang–Yang (LHY) Correction to the Mean-Field Theory

#### 2.4. The Two-Dimensional Ground State Created by the Quintic Self-Repulsive Nonlinearity

#### 2.5. A Challenging Issue: The Fermi Gas Pulled to the Center

## 3. The Three-Dimensional Model with Cylindrical Symmetry

#### 3.1. Formulation of the Model

#### 3.2. The Linear Schrödinger Equation with Cylindrical Symmetry

#### 3.3. Suppression of the Quantum Collapse by the Repulsive Nonlinearity under the Cylindrical Symmetry

## 4. The Two-Component System in Three Dimensions: The Suppression of Quantum Collapse in Miscible and Immiscible Settings

#### 4.1. The Formulation of the Model and Analytical Considerations

#### 4.2. Numerical and Additional Analytical Results for Trapped Binary Modes

#### 4.2.1. Mixed Ground States

#### 4.2.2. The Immiscible Ground State

## 5. The Mean-Field Predictions versus the Many-Body Quantum Theory

#### 5.1. Introduction to the Section

#### 5.2. The Single-Particle Solution

#### 5.3. The Monte-Carlo Method

#### 5.4. Numerical Results for the Many-Body System

## 6. Discussion and Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

3D | three-dimensional |

BEC | Bose–Einstein condensate |

GPE | Gross–Pitaevskii equation |

LHY | Lee–Huang–Yang (correction to the mean-field theory) |

GS | ground state |

rms | root-mean-square (value) |

TFA | Thomas–Fermi approximation |

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**Figure 1.**Radial profiles of $|\chi (r,t)|\equiv \sqrt{r}|\psi \left(r\right)|$ at $t=0$, $0.005$, and $0.1$ (dotted, dashed, and solid curves, respectively), as originally produced in Ref. [5] by simulations of Equation (15) with ${\Omega}^{2}=0.1$ and ${U}_{0}=0.27$, which slightly exceeds the critical one, ${\left({U}_{0}\right)}_{\mathrm{cr}}^{\left(3\mathrm{D}\right)}=1/4$. The initial conditions are taken as ${\psi}_{0}\left(r\right)={r}^{-1/2}exp(-\Omega {r}^{2}/2)$, which is the exact stationary wave function for ${U}_{0}=1/4$—that is, precisely at the critical point, taken as per Equations (17) and (18) (for this reason, the evolution of the wave function is displayed here in terms of $\sqrt{r}|\psi \left(r\right)|$). The simulations demonstrate the onset of the quantum collapse in the linear Schrödinger equation.

**Figure 2.**(

**a**) A typical example of the 3D ground state, shown in terms of $\chi \left(r\right)\equiv r\left|\psi \left(r\right)\right|$, produced by the Gross–Pitaevskii equation (GPE) (22), as per Ref. [5], without the external trap ($\Omega =0$), for ${U}_{0}=0.8$ and $\mu =-0.225$. Panels (

**b**) and (

**c**) display curves $\mu \left(N\right)$ for the ground-state families with ${U}_{0}=0.8$ and $0.1$. These strengths of the attractive potential are, respectively, larger and smaller than the critical value $1/4$ (see Equation (6)) for the linear Schrödinger Equation (15). Here, solid and dashed curves, respectively, depict the numerical results and analytical approximation given by Equations (28) and (29). In panels (

**b**,

**c**), the curves follow scaling $\mu \sim {N}^{-2}$, which is an exact property of Equation (22). In particular, the analytical approximation predicts $N(\mu =-0.225)=5.30$ for ${U}_{0}=0.8$ (the solution shown in (

**a**)), while the numerically found counterpart of this value is ${N}_{\mathrm{num}}(\mu =-0.225)=6.26$. The convergence of the numerical and analytical curves for $N\left(\mu \right)$ at $\mu \to 0$ corresponds to the fact that Equation (28) gives exact solution (17) in this limit.

**Figure 3.**(

**a**) The radial profile of the ground state in the 2D model with the quintic nonlinearity for ${U}_{0}=0.05$ and $\mu =-0.1867$. (

**b**) Curves $\mu \left(N\right)$ for the ground states with ${U}_{0}=-0.18$ and ${U}_{0}=0.05$. In both panels (shown as per Ref. [5]), the numerical results and the respective analytical approximation (40) are depicted by the continuous and dashed curves, respectively. The convergence of the numerical and analytical curves for $N\left(\mu \right)$ at $\mu \to -0$ corresponds to the fact that Equation (40) gives the exact solution (41) in this limit.

**Figure 4.**Eigenvalue $\sigma $ of the singular eigenmode (49) (generated as per Ref. [6] by the numerical solution of linear Equation (50)) vs. strength ${U}_{0}$ of the attractive potential, for three values of the azimuthal quantum number: (

**a**) $m=0$, (

**b**) $m=1$, (

**c**) $m=2$. The eigenmode does not exist, signaling the onset of the quantum collapse at ${U}_{0}>{\left({U}_{0}\right)}_{\mathrm{cr}}\left(m\right)$. See Equation (52), where ${U}_{0}={\left({U}_{0}\right)}_{\mathrm{cr}}\left(m\right)$ corresponds to $\sigma =1/2$.

**Figure 5.**Typical profiles of real function $\chi (r,\xi )$, produced in Ref. [6] by the numerical solution of Equation (55), which determines the shape of the bound state with the reduced (cylindrical) symmetry, as per Equation (54): (

**a**) $m=0,{U}_{0}=3$; (

**b**) $m=1,{U}_{0}=8.5$; (

**c**) $m=2,{U}_{0}=20$. The solutions are subject to normalization $N=2\pi $, see Equation (56).

**Figure 6.**Panels (

**a**–

**c**) display, respectively, the chemical potential of the bound states with azimuthal quantum numbers $m=0,1,2$ vs. the strength of the attractive potential, ${U}_{0}$, of the potential (45), with reduced (cylindrical) symmetry, and for the fixed norm, $N=2\pi $, as obtained in Ref. [6]. The dashed curve in (a) additionally shows dependence (60) predicted by the Thomas–Fermi approximation (TFA).

**Figure 7.**(

**a**) The numerically found profile of wave functions ${\chi}_{1}\left(r\right)={\chi}_{2}\left(r\right)$ of the GS in the miscible binary system at ${V}_{0}=1$, $\gamma =0.9$, and ${N}_{1}={N}_{2}=4\pi $, as found in Ref. [7], and its comparison with the analytical approximation given by Equation (72) (short-dashed line), and TFA based on Equation (75) (long-dashed line). (

**b**) The chain of rhombuses depicts the numerically found relation between $|\mu |{N}^{2}$ and ${V}_{0}$ at $\gamma =0.9$. The short- and long-dashed lines represent the approximations provided by Equations (73) and (76), respectively.

**Figure 8.**(

**a**) ${\chi}_{1}$ (continuous line) and ${\chi}_{2}$ (dashed line) components of the imbalanced mixed GS of the binary system at ${V}_{0}=2$ and $\gamma =0.9$, with ${N}_{1}=4\pi $ and ${N}_{2}=2\pi $, as found in Ref. [7]. (

**b**,

**c**) Comparison of the numerical result (continuous lines) with the two-layer TFA (dashed lines, see Equations (79) and (80)) for ${\chi}_{1}\left(r\right)$ and ${\chi}_{2}\left(r\right)$.

**Figure 9.**(

**a**,

**b**) Comparison of the numerically found profiles for components ${\chi}_{1}\left(r\right)$ and ${\chi}_{2}\left(r\right)$ of the immiscible GS (solid lines) in the binary condensate (${V}_{0}=1$, $\gamma =1.2$) with equal norms of both components (${N}_{1}={N}_{2}=0.8\pi $), and the corresponding TFA, given by Equations (81) and (82), respectively (dashed lines), as per Ref. [7]. The numerical solution gives widely different values of chemical potentials of the two components in this case: ${\mu}_{1}=-14.2$, ${\mu}_{2}=-0.84$.

**Figure 10.**The energy per particle in the many-body system for the soft-sphere interaction potential, as a function of the inverse-Gaussian-width parameter, $\alpha $ (see Equation (84)), for ${U}_{0}=1$, ${a}_{s}=0.1$, $R=1.3{a}_{s}$ and the number of particles $N=2,3,4,5,10,100,1000,10,000$ (larger numbers of particles correspond to larger values at the maximum), as obtained in Ref. [21]. Solid lines: the variational result; dashed lines: the asymptotic energy of the fully-collapsed state, as per Equation (96); the dash-dotted line: typical energy associated with the Gaussian localization, as given by Equation (95).

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**MDPI and ACS Style**

Malomed, B.A.
Suppression of Quantum-Mechanical Collapse in Bosonic Gases with Intrinsic Repulsion: A Brief Review. *Condens. Matter* **2018**, *3*, 15.
https://doi.org/10.3390/condmat3020015

**AMA Style**

Malomed BA.
Suppression of Quantum-Mechanical Collapse in Bosonic Gases with Intrinsic Repulsion: A Brief Review. *Condensed Matter*. 2018; 3(2):15.
https://doi.org/10.3390/condmat3020015

**Chicago/Turabian Style**

Malomed, Boris A.
2018. "Suppression of Quantum-Mechanical Collapse in Bosonic Gases with Intrinsic Repulsion: A Brief Review" *Condensed Matter* 3, no. 2: 15.
https://doi.org/10.3390/condmat3020015