# Clusters in Separated Tubes of Tilted Dipoles

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Binding Energies of Clusters

## 3. Abundances of Clusters

## 4. Results and Discussion

#### 4.1. One Particle per Tube

#### 4.2. Two Particles per Tube

**a**) and (

**b**), respectively. The lower curve in each panel shows the fraction of all clusters which contain at least one dimer, while the upper curve shows a related quantity: the fraction of systems with at least one bound cluster. In panel (

**a**), the lower curve is flat until ${k}_{B}T/{E}_{2}$ = 0.2, then rises rapidly before turning over and declining at the higher temperatures. The upper curve is unity until ${k}_{B}T/{E}_{2}$ = 0.4, then declines, and with the higher temperatures it approaches the lower curve. Thus, nearly all the bound systems contain a bound dimer systems.

**b**), as we saw in the previous results, there is little change in this two-particle-per-tube system, until higher temperatures than in the previous single particle case. The lower curve, shows nothing until about ${k}_{B}T/{E}_{2}$ = 0.7, then rises dramatically before turning over and declining gradually. The upper curve does not begin to decline rapidly around ${k}_{B}T/{E}_{2}$ = 0.8. Again, the curves approach each other, showing that all bound systems contain a bound dimer at high temperatures, which is even more clear in the single particle per layer graph.

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Efimov, V. Energy levels arising from resonant two-body forces in a three-body system. Phys. Lett. B
**1970**, 33, 563. [Google Scholar] [CrossRef] - Jensen, A.S.; Riisager, K.; Fedorov, D.V.; Garrido, E. Structure and reactions of quantum halos. Rev. Mod. Phys.
**2004**, 76, 215. [Google Scholar] [CrossRef] - Braaten, E.; Hammer, H.W. Universality in few-body systems with large scattering length. Phys. Rep.
**2006**, 428, 259. [Google Scholar] [CrossRef] [Green Version] - Greene, C.H.; Giannakeas, P.; Pérez-Rios, J. Universal few-body physics and cluster formation. Rev. Mod. Phys.
**2017**, 89, 035006. [Google Scholar] [CrossRef] [Green Version] - Naidon, P.; Endo, S. Efimov physics: A review. Rep. Prog. Phys.
**2017**, 80, 056001. [Google Scholar] [CrossRef] [Green Version] - Kraemer, T.; Mark, M.; Waldburger, P.; Danzl, J.G.; Chin, C.; Engeser, B.; Lange, A.D.; Pilch, K.; Jaakkola, A.; Nägerl, H.-C.; et al. Evidence for Efimov quantum states in an ultracold gas of caesium atoms. Nature
**2006**, 440, 315. [Google Scholar] [CrossRef] [Green Version] - Lahaye, T.; Menotti, C.; Santos, L.; Lewenstein, M.; Pfau, T. The physics of dipolar bosonic quantum gases. Rep. Prog. Phys.
**2009**, 72, 126401. [Google Scholar] [CrossRef] - Baranov, M.A.; Dalmonte, M.; Pupillo, G.; Zoller, P. Condensed matter theory of dipolar quantum gases. Chem. Rev.
**2012**, 112, 5012. [Google Scholar] [CrossRef] [Green Version] - Bohn, J.L.; Rey, A.M.; Ye, J. Cold molecules: Progress in quantum engineering of chemistry and quantum matter. Science
**2017**, 357, 1002. [Google Scholar] [CrossRef] [Green Version] - Bloch, I.; Dalibard, J.; Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys.
**2008**, 80, 885. [Google Scholar] [CrossRef] [Green Version] - Wang, D.W.; Lukin, M.D.; Demler, E. Quantum fluids of self-assembled chains of polar molecules. Phys. Rev. Lett.
**2006**, 97, 180413. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Volosniev, A.G.; Armstrong, J.R.; Fedorov, D.V.; Jensen, A.S.; Valiente, M.; Zinner, N.T. Bound states of dipolar bosons in one-dimensional systems. New J. Phys.
**2013**, 15, 043046. [Google Scholar] [CrossRef] - Wunsch, B.; Zinner, N.T.; Mekhov, I.B.; Huang, S.J.; Wang, D.W.; Demler, E. Few-body bound states in dipolar gases and their detection. Phys. Rev. Lett.
**2011**, 107, 073201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zinner, N.T.; Wunsch, B.; Mekhov, I.B.; Huang, S.J.; Wang, D.W.; Demler, E. Few-body bound complexes in one-dimensional dipolar gases and nondestructive optical detection. Phys. Rev. A
**2011**, 84, 063606. [Google Scholar] [CrossRef] [Green Version] - Bjerlin, J.; Bengtsson, J.; Deuretzbacher, F.; Kristinsdóttir, L.H.; Reimann, S.M. Dipolar particles in a double-trap confinement: Response to tilting the dipolar orientation. Phys. Rev. A
**2018**, 97, 023634. [Google Scholar] [CrossRef] [Green Version] - Dalmonte, M.; Zoller, P.; Pupillo, G. Trimer liquids and crystals of polar molecules in coupled wires. Phys. Rev. Lett.
**2011**, 107, 163202. [Google Scholar] [CrossRef] [Green Version] - Pikovski, A.; Klawunn, M.; Shlyapnikov, G.V.; Santos, L. Interlayer superfluidity in bilayer systems of fermionic polar molecules. Phys. Rev. Lett.
**2010**, 105, 215302. [Google Scholar] [CrossRef] [Green Version] - Capogrosso-Sansone, B.; Kuklov, A.B. Superfluidity of flexible chains of polar molecules. J. Low Temp. Phys.
**2011**, 165, 213. [Google Scholar] [CrossRef] - Cinti, F.; Wang, D.W.; Boninsegni, M. Phases of dipolar bosons in a bilayer geometry. Phys. Rev. A
**2017**, 95, 023622. [Google Scholar] [CrossRef] [Green Version] - Sinha, S.; Santos, L. Cold dipolar gases in quasi-one-dimensional geometries. Phys. Rev. Lett.
**2007**, 99, 140406. [Google Scholar] [CrossRef] [Green Version] - Deuretzbacher, F.; Cremon, J.C.; Reimann, S.M. Ground-state properties of few dipolar bosons in a quasi-one-dimensional harmonic trap. Phys. Rev. A
**2010**, 81, 063616. [Google Scholar] [CrossRef] [Green Version] - Armstrong, J.R.; Zinner, N.T.; Fedorov, D.V.; Jensen, A.S. Analytic harmonic approach to the N-body problem. J. Phys. B: At. Molecular
**2011**, 44, 055303. [Google Scholar] [CrossRef] [Green Version] - Landau, L.D.; Lifschitz, E.M. Quantum Mechanics: Non-Relativistic Theory, 3rd ed.; Elsevier Butterworth-Heinemann: Amsterdam, The Netherlands, 1977. [Google Scholar]
- Simon, B. The bound state of weakly coupled Schrodinger operators in one and two dimensions. Ann. Phys.
**1976**, 97, 279. [Google Scholar] [CrossRef] - Armstrong, J.R.; Zinner, N.T.; Fedorov, D.V.; Jensen, A.S. Layers of cold dipolar molecules in the harmonic approximation. Eur. Phys. J. D
**2012**, 66, 85. [Google Scholar] [CrossRef] - McGuire, J.B. Study of exactly soluble one-dimensional N-body problems. J. Math. Phys.
**1964**, 5, 622. [Google Scholar] [CrossRef] - De Palo, S.; Citro, R.; Orignac, E. Variational Bethe ansatz approach for dipolar one-dimensional bosons. Phys. Rev. B
**2020**, 101, 045102. [Google Scholar] [CrossRef] [Green Version] - Kora, Y.; Boninsegni, M. Dipolar bosons in one dimension: The case of longitudinal dipole alignment. Phys. Rev. A
**2020**, 101, 023602. [Google Scholar] [CrossRef] [Green Version] - Oldziejewski, R.; Górecki, W.; Pawłowski, K.; Rzazewski, K. Strongly correlated quantum droplets in quasi-1D dipolar Bose gas. Phys. Rev. Lett.
**2020**, 124, 090401. [Google Scholar] [CrossRef] [Green Version] - Tanzi, L.; Lucioni, E.; Fama, F.; Catani, J.; Fioretti, A.; Gabbanini, C.; Bisset, R.N.; Santos, L.; Modugno, G. Observation of a dipolar quantum gas with metastable supersolid properties. Phys. Rev. Lett.
**2019**, 122, 130405. [Google Scholar] [CrossRef] [Green Version] - Böttcher, F.; Schmidt, J.N.; Wenzel, M.; Hertkorn, J.; Guo, M.; Langen, T.; Pfau, T. Transient supersolid properties in an array of dipolar quantum droplets. Phys. Rev. X
**2019**, 9, 011051. [Google Scholar] [CrossRef] [Green Version] - Chomaz, L.; Petter, D.; Ilzhöfer, P.; Natale, G.; Trautmann, A.; Politi, C.; Durastante, G.; van Bijnen, R.M.W.; Patscheider, A.; Sohmen, M.; et al. Long-lived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X
**2019**, 9, 021012. [Google Scholar] [CrossRef] [Green Version] - Böttcher, F.; Wenzel, M.; Schmidt, J.N.; Guo, M.; Langen, T.; Ferrier-Barbut, I.; Pfau, T.; Bombin, R.; Sanchez-Baena, J.; Boronat, J.; et al. Dilute dipolar quantum droplets beyond the extended Gross-Pitaevskii equation. Phys. Rev. Res.
**2019**, 1, 033088. [Google Scholar] [CrossRef] [Green Version] - Karwowski, J. A separable model of N interacting particles. Int. J. Quantum Chem.
**2008**, 108, 2253. [Google Scholar] [CrossRef] - Karwowski, J.; Szewc, K. Separable N-particle Hookean models. J. Phys. Conf. Ser.
**2010**, 213, 012016. [Google Scholar] [CrossRef] - Volosniev, A.G.; Jensen, A.S.; Harshman, N.L.; Armstrong, J.R.; Zinner, N.T. A solvable model for decoupling of interacting clusters. EPL (Europhys. Lett.)
**2019**, 125, 20003. [Google Scholar] [CrossRef] [Green Version] - Armstrong, J.R.; Volosniev, A.G.; Fedorov, D.V.; Jensen, A.S.; Zinner, N.T. Analytic solutions of topologically disjoint systems. J. Phys. A Math. Theor.
**2015**, 48, 085301. [Google Scholar] [CrossRef] [Green Version] - Suzuki, Y.; Varga, K. Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems; Springer: Berlin, Germany, 1998. [Google Scholar]
- Mitroy, J.; Bubin, S.; Horiuchi, W.; Suzuki, Y.; Adamowicz, L.; Cencek, W.; Szalewicz, K.; Komasa, J.; Blume, D.; Varga, K. Theory and application of explicitly correlated Gaussians. Rev. Mod. Phys.
**2013**, 85, 693. [Google Scholar] [CrossRef] [Green Version] - Busch, T.; Englert, B.-G.; Rzaźewski, K.; Wilkens, M. Two cold atoms in a harmonic trap. Found. Phys.
**1998**, 28, 549. [Google Scholar] [CrossRef] - Dehkharghani, A.S.; Volosniev, A.G.; Zinner, N.T. Impenetrable mass-imbalanced particles in one-dimensional harmonic traps. J. Phys. B At. Mol. Opt. Phys.
**2016**, 49, 085301. [Google Scholar] [CrossRef] - Klawunn, M.; Pikovski, A.; Santos, L. Two-dimensional scattering and bound states of polar molecules in bilayers. Phys. Rev. A
**2010**, 82, 044701. [Google Scholar] [CrossRef] [Green Version] - Volosniev, A.G.; Fedorov, D.V.; Jensen, A.S.; Zinner, N.T. Model independence in two dimensions and polarized cold dipolar molecules. Phys. Rev. Lett.
**2011**, 106, 250401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Guijarro, G.; Astrakharchik, G.E.; Boronat, J.; Bazak, B.; Petrov, D.S. Few-body bound states of two-dimensional bosons. arXiv
**2019**, arXiv:1911.01701. [Google Scholar]

**Figure 1.**The system of interest is five one-dimensional tubes filled with two dipolar particles with dipole moments aligned at the so-called “magic angle”. The system is in a thermal equilibrium with a bath at temperature T. At high temperatures, the system will consist of a gas of independent particles, and at zero temperature the attraction between the layers will lead to a certain bound structure. At intermediate temperatures, various few-body clusters will form.

**Figure 2.**The interaction potential between adjacent tubes at the “magic angle” $\varphi ={\varphi}_{m}$ (see the text for the definition of the angle). The potential is given by Equation (3) with $n=1$ and $\varphi ={\varphi}_{m}$: ${V}_{dip}=-U({d}^{2}+2\sqrt{2}xd)/{({d}^{2}+{x}^{2})}^{5/2}.$

**Figure 3.**Comparison between the harmonic oscillator (HO) model and the stochastic variational method (SVM) in obtaining energies for few-body dipolar clusters. The solid curves represent the results of the HO model: the upper curve is for the 12 system, the lower is for the 111 system. The dots are the corresponding SVM results. The 111 system is slightly more bound than 12 system due to an additional attraction between the outer layers. For comparison, we also plot the energy of the 11 system (see the upper dotted curve) whose energy, by construction, is the same in the HO and SVM calculations. The lower dotted curve presents two times the energy of the 11 system.

**Figure 4.**A figure to illustrate Equation (16). This specific configuration has five free particles, and two clusters. The energy of each cluster consists of two parts: The binding energy, which is calculated as in Section 2, and the center-of-mass part, which is determined by the confining harmonic oscillator.

**Figure 5.**This plot shows the energy of the one-particle-per-layer system, $\langle E\rangle $, and the difference of the energy with the energy of the system of completely free particles, $\langle E\rangle -{\langle E\rangle}_{free}$. Panel (

**a**) shows curves for different U with oscillator length values being twice the inter-layer distance, and in panel (

**b**) the oscillator length is $\sqrt{10}$ the inter-layer distance. The energy curves all start at large negative energies because of the finite binding energy at low temperatures, then approach the high-temperature limit of 5 ${k}_{B}T$ (the equipartition of energy limit for this system). The curves showing the difference of energies start to separate from the energy curve at ${k}_{B}T/{E}_{2}\approx $0.25 to 0.4 as the free-particle state becomes populated before rapidly turning over and descending towards the high-temperature limit of 0.

**Figure 6.**Fractional occupancies of different clusters as a function of temperature for a system of five 1D layers each with one particle. (

**a**): The confining frequency is chosen such that the oscillator length in the tubes is equal to twice the inter-layer distance. The dipole strength is $U=5$. (

**b**): The confining frequency is chosen such that the oscillator length in the tubes is equal to twice the inter-layer distance. The dipole strength is $U=8$. (

**c**) The dipole strength is $U=5$, and the confining frequency is chosen such that the oscillator length in the tubes is equal to $\sqrt{10}$ the inter-layer distance. (

**d**): The dipole strength is $U=8$ and the confining frequency is that same as in panel (

**c**).

**Figure 7.**This plot shows the same as in Figure 5, but for two-particle-per-layer system. The energy curve again starts at large negative energies, then approaches the high-temperature limit of 10 ${k}_{B}T$ (the equipartition limit for 10 total particles in 1D harmonic potentials). The curve showing the difference of energies starts to separate from the energy curve at ${k}_{B}T/{E}_{2}\approx $1 as the free-particle state becomes populated before rapidly turning over and descending towards the high-temperature limit of 0. In contrast with the single particle per layer system, the high-temperature limits are achieved much more slowly. The interaction strength for this plot was $U=5$.

**Figure 8.**(

**a**) Fractional occupancies of all the clusters as a function of temperature for a system of five 1D layers each with two particles. The dipole strength is $U=5$, and the confining frequency is chosen such that the oscillator length in the tubes is equal to twice inter-layer distance. The clusters are labeled by their layer occupancy, so a cluster consisting of one particle each in adjacent layers would be labeled ‘11’. (

**b**) Fractional occupancies of the most populated of the most bound clusters as a function of temperature for a system of five 1D layers each with two particles. The dipole strength is $U=5$, and the confining frequency is chosen such that the oscillator length in the tubes is equal to twice the inter-layer distance. Pictures of the clusters can be seen in (

**c**). (

**d**) Fractional occupancies of the least bound clusters as a function of temperature for a system of five 1D layers each with two particles. The dipole strength is $U=5$, and the confining frequency is chosen such that the oscillator length in the tubes is equal to twice the inter-layer distance. (

**e**) Pictures of the different smaller clusters with the lowest binding energies.

**Figure 9.**This figure shows the fraction of states with at least one bound cluster of any kind (‘all bound clusters’) or the sum of all clusters that contain at least one 11-cluster (all 11 systems). The dipole strength is $U=5$, and the confining frequency is chosen such that the oscillator length in the tubes is equal to twice the inter-layer distance. Panel (

**a**) shows the one-particle-per-layer system and panel (

**b**) shows the two-particle-per-layer system.

**Table 1.**Comparison between the harmonic oscillator (HO) model and the stochastic variational method (SVM) in obtaining energies for few-body dipolar clusters. The different clusters are three and four particle chains (labeled 111 and 1111, respectively), and a system with two particles in one layer and a single particle in the adjacent layer (labeled 12). The units of energy are ${\u0127}^{2}/\left(m{d}^{2}\right)$, and the units of U are ${\u0127}^{2}d/m$.

U | 111 (HO) | 111 (SVM) | 1111 (HO) | 1111 (SVM) | 12 (HO) | 12 (SVM) |
---|---|---|---|---|---|---|

3 | −4.35 | −4.45 | −6.88 | −7.11 | −4.02 | −4.01 |

5 | −8.67 | −8.81 | −13.63 | −13.95 | −8.04 | −8.02 |

10 | −20.67 | −20.97 | −32.33 | −32.96 | −19.31 | −19.30 |

© 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**

Armstrong, J.R.; Jensen, A.S.; Volosniev, A.G.; Zinner, N.T.
Clusters in Separated Tubes of Tilted Dipoles. *Mathematics* **2020**, *8*, 484.
https://doi.org/10.3390/math8040484

**AMA Style**

Armstrong JR, Jensen AS, Volosniev AG, Zinner NT.
Clusters in Separated Tubes of Tilted Dipoles. *Mathematics*. 2020; 8(4):484.
https://doi.org/10.3390/math8040484

**Chicago/Turabian Style**

Armstrong, Jeremy R., Aksel S. Jensen, Artem G. Volosniev, and Nikolaj T. Zinner.
2020. "Clusters in Separated Tubes of Tilted Dipoles" *Mathematics* 8, no. 4: 484.
https://doi.org/10.3390/math8040484