# Bose Polaron in a One-Dimensional Lattice with Power-Law Hopping

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Single-Particle Physics

## 4. Two-Body Scattering

## 5. Impurity in a Bose–Einstein Condensate

## 6. Conclusions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Landau, L.; Pekar, S. Effective mass of a polaron. J. Exp. Theor. Phys.
**1948**, 18, 419–423. [Google Scholar] - Pekar, S. Theory of electromagnetic waves in a crystal with excitations. J. Phys. Chem. Solids
**1958**, 5, 11–22. [Google Scholar] [CrossRef] - Dagotto, E.; Moreo, A.; Barnes, T. Hubbard model with one hole: Ground-state properties. Phys. Rev. B
**1989**, 40, 6721. [Google Scholar] [CrossRef] [PubMed] - Kane, C.L.; Lee, P.A.; Read, N. Motion of a single hole in a quantum antiferromagnet. Phys. Rev. B
**1989**, 39, 6880. [Google Scholar] [CrossRef] [PubMed] - Jakubczyk, T.; Nogajewski, K.; Molas, M.R.; Bartos, M.; Langbein, W.; Potemski, M.; Kasprzak, J. Impact of environment on dynamics of exciton complexes in a WS2 monolayer. 2D Mater.
**2018**, 5, 031007. [Google Scholar] [CrossRef] - Singh, A.; Moody, G.; Tran, K.; Scott, M.E.; Overbeck, V.; Berghäuser, G.; Schaibley, J.; Seifert, E.J.; Pleskot, D.; Gabor, N.M.; et al. Trion formation dynamics in monolayer transition metal dichalcogenides. Phys. Rev. B
**2016**, 93, 041401. [Google Scholar] [CrossRef] [Green Version] - Sidler, M.; Back, P.; Cotlet, O.; Srivastava, A.; Fink, T.; Kroner, M.; Demler, E.; Imamoglu, A. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors. Nat. Phys.
**2016**, 13, 255–261. [Google Scholar] [CrossRef] - Takemura, N.; Trebaol, S.; Wouters, M.; Portella-Oberli, M.T.; Deveaud, B. Polaritonic Feshbach resonance. Nat. Phys.
**2014**, 10, 500–504. [Google Scholar] [CrossRef] [Green Version] - Schirotzek, A.; Wu, C.-H.; Sommer, A.; Zwierlein, M.W. Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms. Phys. Rev. Lett.
**2009**, 102, 230402. [Google Scholar] [CrossRef] - Jørgensen, N.B.; Wacker, L.; Skalmstang, K.T.; Parish, M.M.; Levinsen, J.; Christensen, R.S.; Bruun, G.M.; Arlt, J.J. Observation of Attractive and Repulsive Polarons in a Bose–Einstein Condensate. Phys. Rev. Lett.
**2016**, 117, 055302. [Google Scholar] [CrossRef] [Green Version] - Scazza, F.; Valtolina, G.; Massignan, P.; Recati, A.; Amico, A.; Burchianti, A.; Fort, C.; Inguscio, M.; Zaccanti, M.; Roati, G. Repulsive Fermi Polarons in a Resonant Mixture of Ultracold
^{6}Li Atoms. Phys. Rev. Lett.**2017**, 118, 083602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hu, M.-G.; Van de Graaff, M.J.; Kedar, D.; Corson, J.P.; Cornell, E.A.; Jin, D.S. Bose Polarons in the Strongly Interacting Regime. Phys. Rev. Lett.
**2016**, 117, 055301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Naidon, P. Two Impurities in a Bose–Einstein Condensate: From Yukawa to Efimov Attracted Polarons. J. Phys. Soc. Jpn.
**2018**, 87, 043002. [Google Scholar] [CrossRef] - Camacho-Guardian, A.; Peña Ardila, L.A.; Pohl, T.; Bruun, G.M. Bipolarons in a Bose–Einstein Condensate. Phys. Rev. Lett.
**2018**, 121, 013401. [Google Scholar] [CrossRef] [Green Version] - Huber, D.; Hammer, H.W.; Volosniev, A.G. In-medium bound states of two bosonic impurities in a one-dimensional Fermi gas. Phys. Rev. Res.
**2019**, 1, 033177. [Google Scholar] [CrossRef] [Green Version] - Deng, F.L.; Shi, T.; Yi, S. Effective interactions between two impurities in quasi-two-dimensional dipolar Bose–Einstein condensates. Commun. Theor. Phys.
**2020**, 72, 075501. [Google Scholar] [CrossRef] - Ding, S.; Domínguez-Castro, G.A.; Julku, A.; Camacho-Guardian, A.; Bruun, G.M. Polarons and bipolarons in a two-dimensional square lattice. SciPost Phys.
**2023**, 14, 143. [Google Scholar] [CrossRef] - Ardila, L.A.; Pohl, T. Ground-state properties of dipolar Bose polarons. J. Phys. B At. Mol. Opt. Phys.
**2018**, 52, 015004. [Google Scholar] [CrossRef] [Green Version] - Kain, B.; Ling, H.Y. Polarons in a dipolar condensate. Phys. Rev. A
**2014**, 89, 023612. [Google Scholar] [CrossRef] [Green Version] - Nishimura, K.; Nakano, E.; Iida, K.; Tajima, H.; Miyakawa, T.; Yabu, H. Ground state of the polaron in an ultracold dipolar Fermi gas. Phys. Rev. A
**2021**, 103, 033324. [Google Scholar] [CrossRef] - Guebli, N.; Boudjemâa, A. Effects of quantum fluctuations on the dynamics of dipolar Bose polarons. J. Phys. B At. Mol. Opt. Phys.
**2019**, 52, 185303. [Google Scholar] [CrossRef] - Astrakharchik, G.E.; Ardila, L.A.P.; Schmidt, R.; Jachymski, K.; Negretti, A. Ionic polaron in a Bose–Einstein condensate. Commun. Phys.
**2021**, 4, 94. [Google Scholar] [CrossRef] - Christensen, E.R.; Camacho-Guardian, A.; Bruun, G.M. Charged Polarons and Molecules in a Bose–Einstein Condensate. Phys. Rev. Lett.
**2021**, 126, 243001. [Google Scholar] [CrossRef] [PubMed] - Astrakharchik, G.E.; Peña Ardila, L.A.; Jachymski, K.; Negretti, A. Many-body bound states and induced interactions of charged impurities in a bosonic bath. Nat. Commun.
**2023**, 14, 1647. [Google Scholar] [CrossRef] [PubMed] - Ding, S.; Drewsen, M.; Arlt, J.J.; Bruun, G.M. Mediated Interaction between Ions in Quantum Degenerate Gases. Phys. Rev. Lett.
**2022**, 129, 153401. [Google Scholar] [CrossRef] - Bruderer, M.; Klein, A.; Clark, S.R.; Jaksch, D. Polaron physics in optical lattices. Phys. Rev. A
**2007**, 76, 011605(R). [Google Scholar] [CrossRef] [Green Version] - Bruderer, M.; Klein, A.; Clark, S.R.; Jaksch, D. Transport of strong-coupling polarons in optical lattices. New J. Phys.
**2008**, 10, 033015. [Google Scholar] [CrossRef] - Privitera, A.; Hofstetter, W. Polaronic slowing of fermionic impurities in lattice Bose–Fermi mixtures. Phys. Rev. A
**2010**, 82, 063614. [Google Scholar] [CrossRef] [Green Version] - Massel, F.; Kantian, A.; Daley, A.J.; Giamarchi, T.; Törmä, P. Dynamics of an impurity in a one-dimensional lattice. New J. Phys.
**2013**, 15, 045018. [Google Scholar] [CrossRef] [Green Version] - Sarkar, S.; McEndoo, S.; Schneble, D.; Daley, A.J. Interspecies entanglement with impurity atoms in a lattice gas. New J. Phys.
**2020**, 22, 083017. [Google Scholar] [CrossRef] - Keiler, K.; Mistakidis, S.I.; Schmelcher, P. Doping a lattice-trapped bosonic species with impurities: From ground state properties to correlated tunneling dynamics. New J. Phys.
**2020**, 22, 083003. [Google Scholar] [CrossRef] - Hu, H.; Wang, A.-B.; Yi, S.; Liu, X.-J. Fermi polaron in a one-dimensional quasiperiodic optical lattice: The simplest many-body localization challenge. Phys. Rev. A
**2016**, 93, 053601. [Google Scholar] [CrossRef] - Dutta, S.; Mueller, E.J. Variational study of polarons and bipolarons in a one-dimensional Bose lattice gas in both the superfluid and the Mott-insulator regimes. Phys. Rev. A
**2013**, 88, 053601. [Google Scholar] [CrossRef] [Green Version] - Colussi, V.E.; Caleffi, F.; Menotti, C.; Recati, A. Lattice polarons across the superfluid to mott insulator transition. Phys. Rev. Lett.
**2023**, 130, 173002. [Google Scholar] [CrossRef] [PubMed] - Zähringer, F.; Kirchmair, G.; Gerritsma, R.; Solano, E.; Blatt, R.; Roos, C.F. Realization of a Quantum Walk with One and Two Trapped Ions. Phys. Rev. Lett.
**2010**, 104, 100503. [Google Scholar] [CrossRef] [PubMed] - Schmitz, H.; Matjeschk, R.; Schneider, C.; Glueckert, J.; Enderlein, M.; Huber, T.; Schaetz, T. Quantum Walk of a Trapped Ion in Phase Space. Phys. Rev. Lett.
**2009**, 103, 090504. [Google Scholar] [CrossRef] - Yan, B.; Moses, S.A.; Gadway, B.; Covey, J.P.; Hazzard, K.R.A.; Rey, A.M.; Jin, D.S.; Ye, J. Observation of dipolar spin-exchange interactions with lattice-confined polar molecules. Nature
**2013**, 501, 521–525. [Google Scholar] [CrossRef] [Green Version] - Yan, B.; Moses, S.A.; Gadway, B.; Covey, J.P.; Hazzard, K.R.A.; Rey, A.M.; Jin, D.S.; Ye, J. A degenerate Fermi gas of polar molecules. Science
**2019**, 363, 853–856. [Google Scholar] - Browaeys, A.; Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys.
**2020**, 16, 132–142. [Google Scholar] [CrossRef] - Álvarez, G.A.; Suter, D.; Kaiser, R. Localization-delocalization transition in the dynamics of dipolar-coupled nuclear spins. Science
**2015**, 349, 846–848. [Google Scholar] [CrossRef] [Green Version] - Hung, C.-L.; González-Tuleda, A.; Cirac, J.I.; Kimble, H.J. Quantum spin dynamics with pairwise-tunable, long-range interactions. Proc. Natl. Acad. Sci. USA
**2016**, 113, E4946–E4955. [Google Scholar] [CrossRef] [PubMed] - Defenu, N.; Donner, T.; Macrì, T.; Pagano, G.; Ruffo, S.; Trombettoni, A. Long-range interacting quantum systems. arXiv
**2021**, arXiv:2109.01063. [Google Scholar] - Tran, M.C.; Guo, A.Y.; Baldwin, C.L.; Ehrenberg, A.; Gorshkov, A.V.; Lucas, A. Lieb-Robinson Light Cone for Power-Law Interactions. Phys. Rev. Lett.
**2021**, 127, 160401. [Google Scholar] [CrossRef] - Safavi-Naini, A.; Wall, M.L.; Acevedo, O.L.; Rey, A.M.; Nandkishore, R.M. Quantum dynamics of disordered spin chains with power-law interactions. Phys. Rev. A
**2019**, 99, 033610. [Google Scholar] [CrossRef] [Green Version] - Macrì, T.; Lepori, L.; Pagano, G.; Lewenstein, M.; Barbiero, L. Bound state dynamics in the long-range spin-1/2 XXZ model. Phys. Rev. B
**2021**, 104, 214309. [Google Scholar] [CrossRef] - Hermes, S.; Apollaro, T.J.G.; Paganelli, S.; Macrì, T. Dimensionality-enhanced quantum state transfer in long-range-interacting spin systems. Phys. Rev. A
**2020**, 101, 053607. [Google Scholar] [CrossRef] - Roy, N.; Sharma, A. Fraction of delocalized eigenstates in the long-range Aubry-André-Harper model. Phys. Rev. B
**2021**, 103, 075124. [Google Scholar] [CrossRef] - Domínguez-Castro, G.A.; Paredes, R. Enhanced transport of two interacting quantum walkers in a one-dimensional quasicrystal with power-law hopping. Phys. Rev. A
**2021**, 104, 033306. [Google Scholar] [CrossRef] - Deng, X.; Kravtsov, V.E.; Shlyapnikov, G.V.; Santos, L. Duality in Power-Law Localization in Disordered One-Dimensional Systems. Phys. Rev. Lett.
**2018**, 120, 110602. [Google Scholar] [CrossRef] [Green Version] - Domínguez-Castro, G.A.; Paredes, R. Localization of pairs in one-dimensional quasicrystals with power-law hopping. Phys. Rev. B
**2022**, 106, 134208. [Google Scholar] [CrossRef] - Ferraretto, M.; Salasnich, L. Effects of long-range hopping in the Bose-Hubbard model. Phys. Rev. A
**2019**, 99, 013618. [Google Scholar] [CrossRef] [Green Version] - Giachetti, G.; Defenu, N.; Ruffo, S.; Trombettoni, A. Berezinskii-Kosterlitz-Thouless Phase Transitions with Long-Range Couplings. Phys. Rev. Lett.
**2021**, 127, 156801. [Google Scholar] [CrossRef] [PubMed] - Dias, W.S.; Bertrand, D.; Lyra, M.L. Bose–Einstein condensation in chains with power-law hoppings: Exact mapping on the critical behavior in d-dimensional regular lattices. Phys. Rev. E
**2017**, 95, 062105. [Google Scholar] [CrossRef] - Jaouadi, A.; Telmini, M.; Charron, E. Bose–Einstein condensation with a finite number of particles in a power-law trap. Phys. Rev. A
**2011**, 83, 023616. [Google Scholar] [CrossRef] [Green Version] - Storch, D.-M.; Worm, M.; Kastner, M. Interplay of soundcone and supersonic propagation in lattice models with power law interactions. New J. Phys.
**2015**, 17, 063021. [Google Scholar] [CrossRef] - Winkler, K.; Thalhammer, G.; Lang, F.; Grimm, R.; Denschlag, J.H.; Daley, A.J.; Kantian, A.; Büchler, H.P.; Zoller, P. Repulsively bound atom pairs in an optical lattice. Nature
**2006**, 441, 853–856. [Google Scholar] [CrossRef] [Green Version] - Bruus, H.; Flensberg, K. Many-Body Quantum Theory in Condensed Matter Physics: An Introduction; Oxford Graduate Texts; Oxford University Press: Oxford, UK, 2016. [Google Scholar]
- Ardila, L.A.; Jørgensen, N.B.; Pohl, T.; Giorgini, S.; Bruun, G.M.; Arlt, J.J. Analyzing a Bose polaron across resonant interactions. Phys. Rev. A
**2019**, 99, 063607. [Google Scholar] [CrossRef] [Green Version] - Skou, M.G.; Skov, T.G.; Jørgensen, N.B.; Nielsen, K.K.; Camacho-Guardian, A.; Pohl, T.; Bruun, G.M.; Arlt, J.J. Non-equilibrium quantum dynamics and formation of the Bose polaron. Nat. Phys.
**2021**, 17, 731–735. [Google Scholar] [CrossRef] - Yan, Z.Z.; Ni, Y.; Robens, C.; Zwierlein, M.W. Bose polarons near quantum criticality. Science
**2020**, 368, 190–194. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rath, S.P.; Schmidt, R. Field-theoretical study of the Bose polaron. Phys. Rev. A
**2013**, 88, 053632. [Google Scholar] [CrossRef] [Green Version] - Bruun, G.M.; Massignan, P. Decay of Polarons and Molecules in a Strongly Polarized Fermi Gas. Phys. Rev. Lett.
**2010**, 105, 020403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Massignan, P.; Bruun, G.M. Repulsive polarons and itinerant ferromagnetism in strongly polarized Fermi gases. Eur. Phys. J. D
**2011**, 65, 83–89. [Google Scholar] [CrossRef] - Schollwöck, U. The density-matrix renormalization group. Rev. Mod. Phys.
**2005**, 77, 259. [Google Scholar] [CrossRef] [Green Version] - Danshita, I.; Polkovnikov, A. Superfluid-to-Mott-insulator transition in the one-dimensional Bose-Hubbard model for arbitrary integer filling factors. Phys. Rev. A
**2011**, 84, 063637. [Google Scholar] [CrossRef] [Green Version] - Boéris, G.; Gori, L.; Hoogerland, M.D.; Kumar, A.; Lucioni, E.; Tanzi, L.; Inguscio, M.; Giamarchi, T.; D’Errico, C.; Carleo, G.; et al. Mott transition for strongly interacting one-dimensional bosons in a shallow periodic potential. Phys. Rev. A
**2016**, 93, 011601(R). [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Panels (

**a**–

**d**) show the lattice dispersion ${\tilde{\u03f5}}_{k}/t$ as a function of the quasi-momentum in the first Brillouin zone $k\in [-\pi ,\pi ]$. Panels (

**e**–

**h**) illustrate the density of states $\rho $ as function of $\omega /{\mathsf{\Delta}}_{\alpha}$ with ${\mathsf{\Delta}}_{\alpha}={\u03f5}_{k=\pi}^{\alpha}={\tilde{\u03f5}}_{k=\pi}^{\alpha}-{\tilde{\u03f5}}_{k=0}^{\alpha}$ the lattice bandwidth. Panels (

**a**,

**e**) are associated with a lattice with nearest-neighbor hopping.

**Figure 2.**Spectral function ${\mathcal{A}}_{\mathcal{T}}=-2\Im m\mathcal{T}(k=0,\omega )$ as a function of the interaction strength ${U}_{BI}$ and the energy $\omega $ for vanishing quasi-momentum k. In each panel, the dashed white lines enclose the two-body scattering continuum. (

**a**) Spectral function for nearest-neighbor hopping, (

**b**) $\alpha =3$, (

**c**) $\alpha =2$, and (

**d**) $\alpha =3/2$.

**Figure 3.**Square modulus of the zero center-of-mass momentum wave function vs. the relative distance r. Upper panels are associated with repulsively bound pairs ${U}_{BI}/t=4$, whereas lower panels correspond to attractively bound dimers ${U}_{BI}/t=-4$. (

**a**,

**e**) consider nearest-neighbor hopping, (

**b**,

**f**) $\alpha =3$, (

**c**,

**g**) $\alpha =2$, and (

**d**,

**h**) $\alpha =3/2$.

**Figure 4.**Spectral function of the impurity ${A}_{I}=-2\Im m{G}_{I}(k=0,\omega )$ as a function of the interaction strength ${U}_{BI}$ and the energy $\omega $. We consider ${n}_{0}=1$ and ${U}_{B}/t=0.02$. The dashed white lines enclose the Bogoliubov continuum, the yellow curve is associated with the energy of the dimer states. (

**a**) Spectral function for nearest-neighbor hopping, (

**b**) $\alpha =3$, (

**c**) $\alpha =2$, and (

**d**) $\alpha =3/2$.

**Figure 5.**(

**a**) Quasiparticle residue as a function of the impurity-boson interaction for vanishing crystal momentum $k=0$ and several values of the power hopping $\alpha $. (

**b**) Damping rate as a function of the impurity-boson interaction for vanishing crystal momentum $k=0$ and several values of the power hopping $\alpha $.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Domínguez-Castro, G.A.
Bose Polaron in a One-Dimensional Lattice with Power-Law Hopping. *Atoms* **2023**, *11*, 110.
https://doi.org/10.3390/atoms11080110

**AMA Style**

Domínguez-Castro GA.
Bose Polaron in a One-Dimensional Lattice with Power-Law Hopping. *Atoms*. 2023; 11(8):110.
https://doi.org/10.3390/atoms11080110

**Chicago/Turabian Style**

Domínguez-Castro, G. A.
2023. "Bose Polaron in a One-Dimensional Lattice with Power-Law Hopping" *Atoms* 11, no. 8: 110.
https://doi.org/10.3390/atoms11080110