# Quantum-Granularity Effect in the Formation of Supermixed Solitons in Ring Lattices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Two Species Model

## 3. A Continuous-Variable Picture to Investigate the Formation of Supermixed Solitons

#### 3.1. The System Phase Diagram

#### 3.2. Some Quantum Indicators to Characterize the Different Phases

## 4. Beyond the Continuous-Variable Picture: Emergence of the Quantum-Granularity

#### 4.1. Exact Numerical Results

#### 4.2. Analytic Treatment

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BH | Bose–Hubbard |

CV | Continuous-variable |

SM | Supermixed |

PL | Partially localized |

M | Mixed |

EE | Entanglement entropy |

## References

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**Figure 1.**Pictorial representation of some states belonging to phases M, PL, and SM, respectively. Labels $1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}3$ correspond to site numbers, while the vertical axis corresponds to (normalized) boson populations ${x}_{\ast ,j}$ and ${y}_{\ast ,j}$ characterizing the ground-state configuration. The majority (minority) species is depicted in green (yellow) and corresponds to the left (right) columns of the histograms in each panel. (

**a**) In phase M, the two bosonic species are mixed and uniformly distributed in the ring trimer; (

**b**) in phase PL, the minority species is highly localized, while the majority species occupies all the sites (although in a non-uniform manner); (

**c**) phase SM is characterized by supermixed solitons.

**Figure 2.**Phase diagram of a (possibly asymmetric) two species bosonic mixture confined in a three-well potential and featuring repulsive intraspecies and attractive interspecies interactions. Each phase is characterized by a specific functional dependence of the energy minimum (5) on effective model parameters (4). Along the red dashed ($\alpha =-1$) and the red solid ($\beta =-1/\alpha $) lines, ${V}_{\ast}$ is not analytic, a circumstance that strongly suggests the occurrence of phase transitions. In the former (latter) case, it is the first (second) derivative of ${V}_{\ast}$ with respect to control parameter $\alpha $ that is discontinuous.

**Figure 3.**Each row illustrates the behavior of a genuinely quantum indicator as a function of model parameters $\alpha $ and $\beta $. Going from left to right, plots correspond to $T/{U}_{a}=0$, $0.02$, and $0.50$, where $T:={T}_{a}={T}_{b}$. (

**a**–

**c**): ground-state energy ${E}_{0}/{U}_{a}$ (11). (

**d**–

**f**): quantum version of the entropy of mixing, ${\tilde{S}}_{mix}$ (8). (

**g**–

**i**): quantum version of the entropy of location ${\tilde{S}}_{loc}$ (9). (

**j**–

**l**): entanglement between the two condensed species, $EE$ (13). Model parameters ${N}_{a}={N}_{b}=15$, ${U}_{a}=1$, ${U}_{b}\in [0,1]\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\beta \in [0,1]$, and $\alpha \in [-3,0]$ were used. Each plot includes more than 20k points [41], corresponding to as many numerical diagonalizations of Hamiltonian (1).

**Figure 4.**Each row illustrates the behavior of a genuinely quantum indicator as a function of model parameters $\alpha $ and $\beta $. Going from left to right, plots correspond to $T/{U}_{a}=0$, $0.02$, and $0.50$, where $T:={T}_{a}={T}_{b}$. (

**a**–

**c**): second derivative of the ground-state energy ${E}_{0}$. (

**d**–

**f**): second derivative of the quantum version of the entropy of mixing, ${\tilde{S}}_{mix}$. (

**g**–

**i**): second derivative of the quantum version of the entropy of location ${\tilde{S}}_{loc}$. (

**j**–

**l**): second derivative of the entanglement between the two condensed species, $EE$. Model parameters ${N}_{a}={N}_{b}=15$, ${U}_{a}=1$, ${U}_{b}\in [0,1]\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\beta \in [0,1]$, and $\alpha \in [-3,0]$ were used. Each plot includes more than 20k points [41], corresponding to as many numerical diagonalizations of Hamiltonian (1).

**Figure 5.**Pictorial representation of the discrete character of the interwell boson exchange. (

**a**): macroscopic configuration of the system for a certain choice of model parameters. A small variation of control parameters $\alpha $ and $\beta $ may [panel (

**b**)] or may not modify it. The fact that the supermixed soliton can gain or loose a boson at a time upon varying a control parameter is what we mean with the term “quantum-granularity”. Green and yellow circles represent species-a and species-b bosons respectively. The gray circle represent a species-a boson which is being transferred to the supermixed soliton.

**Figure 6.**First eight excited energy levels, obtained by means of an exact numerical diagonalization of Hamiltonian (1), for $T:={T}_{a}={T}_{b}=0,\phantom{\rule{0.166667em}{0ex}}0.02,\phantom{\rule{0.166667em}{0ex}}0.50$ in panel (

**a**–

**c**), respectively. Model parameters ${N}_{a}={N}_{b}=15$, ${U}_{a}=1$, ${U}_{b}=0.36\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\beta =0.6$, and $W\in [-1.8,0]\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\alpha \in [-3,0]$ were chosen. Each color corresponds to a different energy level.

**Figure 7.**Map of the system’s minimum energy configurations. It corresponds to the graphical representation of the set of inequalities derived in Section 4.2. More specifically: the solid black (dashed) line corresponds to the condition (23) and (28), while the set of purple (yellow) stripes is given by the condition (24) and (25). Model parameters ${N}_{a}={N}_{b}=15$, ${U}_{a}=1$, ${U}_{b}\in [0,1]\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\beta \in [0,1]$, $\alpha \in [-3,0]$, and ${T}_{a}={T}_{b}=0$ were used.

**Figure 8.**Degeneracy of the ground-state level ${E}_{0}$, obtained by means of an exact numerical diagonalization of Hamiltonian (1), for $T:={T}_{a}={T}_{b}=0$. Model parameters ${N}_{a}={N}_{b}=15$, ${U}_{a}=1$, ${U}_{b}=0.16\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\beta =0.4$, and $W\in [-1.2,0]\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\alpha \in [-3,0]$ were chosen. Each jump discontinuity corresponds to a change in the ground state structure of the type illustrated in Figure 5.

**Figure 9.**The mechanism of jerky interwell boson transfer is present provided that tunneling T is small enough. (

**a**): second derivative of the ground-state energy (11) with respect to control parameter $\alpha $. (

**b**): entropy (15) of the probability distribution associated with coefficients $|c(\overrightarrow{N},\phantom{\rule{0.166667em}{0ex}}\overrightarrow{M}){|}^{2}$ (see Formula (10)). The black line corresponds to the border of the stability region (20) of the mixed configuration. Notice that, unlike Figure 3 and Figure 4, these plots are referred to the $(\alpha ,\phantom{\rule{0.166667em}{0ex}}T/{U}_{a})$ plane, instead of the $(\alpha ,\phantom{\rule{0.166667em}{0ex}}\beta )$ plane. Model parameters ${N}_{a}={N}_{b}=15$, ${U}_{a}=1$, ${U}_{b}=0.16\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\beta =0.4$, ${T}_{a}={T}_{b}=:T\in [0,0.5]$, and $\alpha \in [-3,0]$ were used. Each plot includes more than 75k points [41], corresponding to as many numerical diagonalizations of Hamiltonian (1).

**Figure 10.**Expectation value $\langle \Delta n\rangle =\langle {\psi}_{0}|\Delta n|{\psi}_{0}\rangle $ of operator imbalance operator $\Delta n$ (see Formula (19)) as a function of $\alpha $ and $T/{U}_{a}$. The mechanism of jerky interwell boson transfer is present provided that tunneling T is small enough (compare the staircase-like structure for $T/{U}_{a}\to 0$ with the slide-like appearance for $T/{U}_{a}\approx 0.2$). Notice that, unlike Figure 3 and Figure 4, these plots are referred to the $(\alpha ,\phantom{\rule{0.166667em}{0ex}}T/{U}_{a})$ plane, instead of the $(\alpha ,\phantom{\rule{0.166667em}{0ex}}\beta )$ plane. Notice also that, unlike Figure 9, the range of $T/{U}_{a}$ is $[0,\phantom{\rule{0.166667em}{0ex}}0.2]$ in order to better appreciate the presence of lobe-like regions. Model parameters ${N}_{a}={N}_{b}=15$, ${U}_{a}=1$, ${U}_{b}=0.16\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\beta =0.4$, ${T}_{a}={T}_{b}=:T\in [0,0.5]$, and $\alpha \in [-3,0]$ were used. The plot includes more than 60k points [41], corresponding to as many numerical diagonalizations of Hamiltonian (1). Color is used to better emphasize the features of the surface: blue (red) corresponds to $\langle \Delta n\rangle =0$ ($\langle \Delta n\rangle =15$).

**Table 1.**Summary of the typical minimum energy configuration and of the associated energy in each phase.

Phase | $({\overrightarrow{\mathit{x}}}_{\ast},{\overrightarrow{\mathit{y}}}_{\ast})$ | $\mathit{V}}_{\ast$ |
---|---|---|

M | ${x}_{\ast ,j}=1/3\phantom{\rule{1.em}{0ex}}\forall j$ ${y}_{\ast ,j}=1/3\phantom{\rule{1.em}{0ex}}\forall j$ | ${V}_{\ast}^{\mathrm{M}}=\frac{1}{6}({\beta}^{2}+2\alpha \beta +1)$ |

PL | ${x}_{\ast ,i}=[1-2\alpha \beta ]/3$ ${x}_{\ast ,j}=[1+\alpha \beta ]/3\phantom{\rule{1.em}{0ex}}\forall j\ne i$ ${y}_{\ast ,i}=1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{y}_{\ast ,j}=0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall j\ne i$ | ${V}_{\ast}^{\mathrm{PL}}=\frac{1}{6}[1+2\alpha \beta$$+{\beta}^{2}(3-2{\alpha}^{2})]$ |

SM | ${x}_{\ast ,i}=1$ ${x}_{\ast ,j}=0\phantom{\rule{1.em}{0ex}}\forall j\ne i$ ${y}_{\ast ,i}=1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{y}_{\ast ,j}=0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall j\ne i$ | ${V}_{\ast}^{\mathrm{SM}}=\frac{1}{2}({\beta}^{2}+2\alpha \beta +1)$ |

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**MDPI and ACS Style**

Richaud, A.; Penna, V. Quantum-Granularity Effect in the Formation of Supermixed Solitons in Ring Lattices. *Condens. Matter* **2020**, *5*, 2.
https://doi.org/10.3390/condmat5010002

**AMA Style**

Richaud A, Penna V. Quantum-Granularity Effect in the Formation of Supermixed Solitons in Ring Lattices. *Condensed Matter*. 2020; 5(1):2.
https://doi.org/10.3390/condmat5010002

**Chicago/Turabian Style**

Richaud, Andrea, and Vittorio Penna. 2020. "Quantum-Granularity Effect in the Formation of Supermixed Solitons in Ring Lattices" *Condensed Matter* 5, no. 1: 2.
https://doi.org/10.3390/condmat5010002