# 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

- Modugno, G.; Modugno, M.; Riboli, F.; Roati, G.; Inguscio, M. Two Atomic Species Superfluid. Phys. Rev. Lett.
**2002**, 89, 190404. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chin, C.; Grimm, R.; Julienne, P.; Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys.
**2010**, 82, 1225–1286. [Google Scholar] [CrossRef] - Myatt, C.J.; Burt, E.A.; Ghrist, R.W.; Cornell, E.A.; Wieman, C.E. Production of Two Overlapping Bose-Einstein Condensates by Sympathetic Cooling. Phys. Rev. Lett.
**1997**, 78, 586–589. [Google Scholar] [CrossRef] - Catani, J.; De Sarlo, L.; Barontini, G.; Minardi, F.; Inguscio, M. Degenerate Bose-Bose mixture in a three-dimensional optical lattice. Phys. Rev. A
**2008**, 77, 011603. [Google Scholar] [CrossRef] [Green Version] - Mishra, T.; Pai, R.V.; Das, B.P. Phase separation in a two-species Bose mixture. Phys. Rev. A
**2007**, 76, 013604. [Google Scholar] [CrossRef] [Green Version] - Jain, P.; Boninsegni, M. Quantum demixing in binary mixtures of dipolar bosons. Phys. Rev. A
**2011**, 83, 023602. [Google Scholar] [CrossRef] [Green Version] - Lingua, F.; Guglielmino, M.; Penna, V.; Capogrosso Sansone, B. Demixing effects in mixtures of two bosonic species. Phys. Rev. A
**2015**, 92, 053610. [Google Scholar] [CrossRef] [Green Version] - Suthar, K.; Angom, D. Optical-lattice-influenced geometry of quasi-two-dimensional binary condensates and quasiparticle spectra. Phys. Rev. A
**2016**, 93, 063608. [Google Scholar] [CrossRef] [Green Version] - Suthar, K.; Roy, A.; Angom, D. Fluctuation-driven topological transition of binary condensates in optical lattices. Phys. Rev. A
**2015**, 91, 043615. [Google Scholar] [CrossRef] [Green Version] - Buonsante, P.; Giampaolo, S.M.; Illuminati, F.; Penna, V.; Vezzani, A. Mixtures of Strongly Interacting Bosons in Optical Lattices. Phys. Rev. Lett.
**2008**, 100, 240402. [Google Scholar] [CrossRef] [Green Version] - Roscilde, T.; Cirac, J.I. Quantum Emulsion: A Glassy Phase of Bosonic Mixtures in Optical Lattices. Phys. Rev. Lett.
**2007**, 98, 190402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Belemuk, A.; Chtchelkatchev, N.; Mikheyenkov, A.; Kugel, K. Quantum phase transitions and the degree of nonidentity in the system with two different species of vector bosons. New J. Phys.
**2018**, 20, 063039. [Google Scholar] [CrossRef] [Green Version] - Wang, W.; Penna, V.; Capogrosso-Sansone, B. Inter-species entanglement of Bose–Bose mixtures trapped in optical lattices. New J. Phys.
**2016**, 18, 063002. [Google Scholar] [CrossRef] [Green Version] - Mujal, P.; Juliá-Díaz, B.; Polls, A. Quantum properties of a binary bosonic mixture in a double well. Phys. Rev. A
**2016**, 93, 043619. [Google Scholar] [CrossRef] [Green Version] - Melé-Messeguer, M.; Julia-Diaz, B.; Guilleumas, M.; Polls, A.; Sanpera, A. Weakly linked binary mixtures of F = 1
^{87}Rb Bose–Einstein condensates. New J. Phys.**2011**, 13, 033012. [Google Scholar] [CrossRef] - Richaud, A.; Penna, V. Phase separation can be stronger than chaos. New J. Phys.
**2018**, 20, 105008. [Google Scholar] [CrossRef] - Lingua, F.; Richaud, A.; Penna, V. Residual entropy and critical behavior of two interacting boson species in a double well. Entropy
**2018**, 20, 84. [Google Scholar] [CrossRef] [Green Version] - Penna, V.; Richaud, A. The phase separation mechanism of a binary mixture in a ring trimer. Sci. Rep.
**2018**, 8, 10242. [Google Scholar] [CrossRef] - Richaud, A.; Zenesini, A.; Penna, V. The mixing-demixing phase diagram of ultracold heteronuclear mixtures in a ring trimer. Sci. Rep.
**2019**, 9, 6908. [Google Scholar] [CrossRef] [Green Version] - Penna, V.; Richaud, A. Spatial Phase Separation of a Binary Mixture in a Ring Trimer. J. Phys. Conf. Ser.
**2019**, 1206, 10242. [Google Scholar] [CrossRef] - Richaud, A.; Penna, V. Pathway toward the formation of supermixed states in ultracold boson mixtures loaded in ring lattices. Phys. Rev. A
**2019**, 100, 013609. [Google Scholar] [CrossRef] [Green Version] - Camesasca, M.; Kaufman, M.; Manas-Zloczower, I. Quantifying fluid mixing with the Shannon entropy. Macromol. Theory Simul.
**2006**, 15, 595–607. [Google Scholar] [CrossRef] [Green Version] - Lingua, F.; Mazzarella, G.; Penna, V. Delocalization effects, entanglement entropy and spectral collapse of boson mixtures in a double well. J. Phys. B At. Mol. Opt. Phys.
**2016**, 49, 205005. [Google Scholar] [CrossRef] [Green Version] - Fisher, M.P.A.; Weichman, P.B.; Grinstein, G.; Fisher, D.S. Boson localization and the superfluid-insulator transition. Phys. Rev. B
**1989**, 40, 546. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jaksch, D.; Bruder, C.; Cirac, J.I.; Gardiner, C.W.; Zoller, P. Cold Bosonic Atoms in Optical Lattices. Phys. Rev. Lett.
**1998**, 81, 3108. [Google Scholar] [CrossRef] - Greiner, M.; Mandel, O.; Esslinger, T.; Hänsch, T.W.; Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature
**2002**, 415, 39. [Google Scholar] [CrossRef] [PubMed] - Caleffi, F.; Capone, M.; Menotti, C.; Carusotto, I.; Recati, A. Quantum fluctuations beyond the Gutzwiller approximation in the Bose-Hubbard model. arXiv
**2019**, arXiv:1908.03470. [Google Scholar] - Sowiński, T.; García-March, M.Á. One-dimensional mixtures of several ultracold atoms: A review. Rep. Prog. Phys.
**2019**, 82, 104401. [Google Scholar] [CrossRef] [Green Version] - Amico, L.; Osterloh, A.; Cataliotti, F. Quantum Many Particle Systems in Ring-Shaped Optical Lattices. Phys. Rev. Lett.
**2005**, 95, 063201. [Google Scholar] [CrossRef] [Green Version] - Amico, L.; Aghamalyan, D.; Auksztol, F.; Crepaz, H.; Dumke, R.; Kwek, L.C. Superfluid qubit systems with ring shaped optical lattices. Sci. Rep.
**2014**, 4, 4298. [Google Scholar] [CrossRef] [Green Version] - Penna, V.; Richaud, A. Two-species boson mixture on a ring: A group-theoretic approach to the quantum dynamics of low-energy excitations. Phys. Rev. A
**2017**, 96, 053631. [Google Scholar] [CrossRef] [Green Version] - Spekkens, R.W.; Sipe, J.E. Spatial fragmentation of a Bose-Einstein condensate in a double-well potential. Phys. Rev. A
**1999**, 59, 3868–3877. [Google Scholar] [CrossRef] [Green Version] - Javanainen, J. Phonon approach to an array of traps containing Bose-Einstein condensates. Phys. Rev. A
**1999**, 60, 4902–4909. [Google Scholar] [CrossRef] [Green Version] - Ho, T.L.; Ciobanu, C.V. The Schrödinger Cat Family in Attractive Bose Gases. J. Low Temp. Phys.
**2004**, 135, 257–266. [Google Scholar] [CrossRef] - Ziń, P.; Chwedeńczuk, J.; Oleś, B.; Sacha, K.; Trippenbach, M. Critical fluctuations of an attractive Bose gas in a double-well potential. EPL
**2008**, 83, 64007. [Google Scholar] - Buonsante, P.; Burioni, R.; Vescovi, E.; Vezzani, A. Quantum criticality in a bosonic Josephson junction. Phys. Rev. A
**2012**, 85, 043625. [Google Scholar] [CrossRef] [Green Version] - Buonsante, P.; Penna, V.; Vezzani, A. Dynamical bifurcation as a semiclassical counterpart of a quantum phase transition. Phys. Rev. A
**2011**, 84, 061601. [Google Scholar] [CrossRef] [Green Version] - Oelkers, N.; Links, J. Ground-state properties of the attractive one-dimensional Bose-Hubbard model. Phys. Rev. B
**2007**, 75, 115119. [Google Scholar] [CrossRef] [Green Version] - Sachdev, S. Quantum Phase Transitions; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Brandani, G.B.; Schor, M.; MacPhee, C.E.; Grubmüller, H.; Zachariae, U.; Marenduzzo, D. Quantifying disorder through conditional entropy: an application to fluid mixing. PLoS ONE
**2013**, 8, e65617. [Google Scholar] [CrossRef] [PubMed] - Computational Resources Provided by HPC@POLITO. Available online: http://hpc.polito.it/ (accessed on 3 January 2020).
- Wang, W.; Penna, V.; Capogrosso-Sansone, B. Analysis and resolution of the ground-state degeneracy of the two-component Bose-Hubbard model. Phys. Rev. E
**2014**, 90, 022116. [Google Scholar] [CrossRef] [Green Version] - Kordas, G.; Wimberger, S.; Witthaut, D. Decay and fragmentation in an open Bose-Hubbard chain. Phys. Rev. A
**2013**, 87, 043618. [Google Scholar] [CrossRef] [Green Version] - Ferrini, G.; Minguzzi, A.; Hekking, F.W.J. Number squeezing, quantum fluctuations, and oscillations in mesoscopic Bose Josephson junctions. Phys. Rev. A
**2008**, 78, 023606. [Google Scholar] [CrossRef] [Green Version]

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

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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