Ring-exchange physics in a chain of three-level ions

Sourav Biswas; E. Rico; Tobias Grass

doi:10.22331/q-2025-04-01-1683

Ring-exchange physics in a chain of three-level ions

Sourav Biswas¹, E. Rico^1,2,3,4, and Tobias Grass^1,4

¹DIPC - Donostia International Physics Center, Paseo Manuel de Lardizábal 4, 20018 San Sebastián, Spain
²EHU Quantum Center and Department of Physical Chemistry, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain
³European Organization for Nuclear Research (CERN), Geneva 1211, Switzerland
⁴IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain

Published:	2025-04-01, volume 9, page 1683
Editor:	Tommaso Macrì
Eprint:	arXiv:2410.06655v2
Doi:	https://doi.org/10.22331/q-2025-04-01-1683
Citation:	Quantum 9, 1683 (2025).

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Abstract

In the presence of ring exchange interactions, bosons in a ladder-like lattice may form the bosonic analogon of a correlated metal, known as the d-wave Bose liquid (DBL). In this paper, we show that a chain of trapped ions with three internal levels can mimic a ladder-like system constrained to a maximum occupation of one boson per rung. The setup enables tunable ring exchange interactions, transitioning between a polarized regime with all bosons confined to one leg and the DBL regime. The latter state is characterized by a splitting of the peak in the momentum distribution and an oscillating pair correlation function.

Featured image: A proposal for simulating a synthetic ladder with the ring exchange mechanism using trapped ions. It could form the sought-after d-wave correlated Bose liquid (DBL) phase, a candidate for the elusive Bose metal.

Popular summary

A new proposal outlines how to realize metallic behavior in a system of frustrated bosons using three atomic levels of ions trapped in a chain. By tuning specific interactions, one can simulate quantum behavior in synthetic ladders and, in particular, the ring exchange mechanism. This approach could lead to the creation of the sought-after d-wave correlated Bose liquid (DBL) phase, which is a candidate for the elusive Bose metal. This exotic phase of gapless bosons neither condenses nor becomes an insulator. As a consequence, our results take the physics of interacting bosons beyond the well-studied “flow-no-flow" binary and its allies, in case of bosons. It also enables one to study higher-dimensional phenomena in lower-dimensional systems. The proposal provides a fresh approach to the search for exotic materials using modern quantum technologies and presents an opportunity to apply recent advances in trapped ion platforms.

► BibTeX data

@article{Biswas2025ringexchangephysics,
  doi = {10.22331/q-2025-04-01-1683},
  url = {https://doi.org/10.22331/q-2025-04-01-1683},
  title = {Ring-exchange physics in a chain of three-level ions},
  author = {Biswas, Sourav and Rico, E. and Grass, Tobias},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1683},
  month = apr,
  year = {2025}
}

► References

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[32] C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021b).
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https://doi.org/10.1103/PhysRevResearch.6.033038

[1] Matthew P. A. Fisher, Peter B. Weichman, G. Grinstein, and Daniel S. Fisher. ``Boson localization and the superfluid-insulator transition''. Phys. Rev. B 40, 546–570 (1989).
https://doi.org/10.1103/PhysRevB.40.546

[2] Philip Phillips and Denis Dalidovich. ``The elusive bose metal''. Science 302, 243–247 (2003).
https://doi.org/10.1126/science.1088253

[3] Chao Yang, Yi Liu, Yang Wang, Liu Feng, Qianmei He, Jian Sun, Yue Tang, Chunchun Wu, Jie Xiong, Wanli Zhang, Xi Lin, Hong Yao, Haiwen Liu, Gustavo Fernandes, Jimmy Xu, James M. Valles, Jian Wang, and Yanrong Li. ``Intermediate bosonic metallic state in the superconductor-insulator transition''. Science 366, 1505–1509 (2019).
https://doi.org/10.1126/science.aax5798

[4] Anthony Hegg, Jinning Hou, and Wei Ku. ``Geometric frustration produces long-sought bose metal phase of quantum matter''. Proceedings of the National Academy of Sciences 118, e2100545118 (2021).
https://doi.org/10.1073/pnas.2100545118

[5] Arun Paramekanti, Leon Balents, and Matthew P. A. Fisher. ``Ring exchange, the exciton bose liquid, and bosonization in two dimensions''. Phys. Rev. B 66, 054526 (2002).
https://doi.org/10.1103/PhysRevB.66.054526

[6] Olexei I. Motrunich and Matthew P. A. Fisher. ``$d$-wave correlated critical bose liquids in two dimensions''. Phys. Rev. B 75, 235116 (2007).
https://doi.org/10.1103/PhysRevB.75.235116

[7] D. N. Sheng, Olexei I. Motrunich, Simon Trebst, Emanuel Gull, and Matthew P. A. Fisher. ``Strong-coupling phases of frustrated bosons on a two-leg ladder with ring exchange''. Phys. Rev. B 78, 054520 (2008).
https://doi.org/10.1103/PhysRevB.78.054520

[8] Ryan V. Mishmash, Matthew S. Block, Ribhu K. Kaul, D. N. Sheng, Olexei I. Motrunich, and Matthew P. A. Fisher. ``Bose metals and insulators on multileg ladders with ring exchange''. Phys. Rev. B 84, 245127 (2011).
https://doi.org/10.1103/PhysRevB.84.245127

[9] Matthew S. Block, Ryan V. Mishmash, Ribhu K. Kaul, D. N. Sheng, Olexei I. Motrunich, and Matthew P. A. Fisher. ``Exotic gapless mott insulators of bosons on multileg ladders''. Phys. Rev. Lett. 106, 046402 (2011).
https://doi.org/10.1103/PhysRevLett.106.046402

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https://doi.org/10.1038/415039a

[11] Marcos Atala, Monika Aidelsburger, Michael Lohse, Julio T. Barreiro, Belén Paredes, and Immanuel Bloch. ``Observation of chiral currents with ultracold atoms in bosonic ladders''. Nature Physics 10, 588–593 (2014).
https://doi.org/10.1038/nphys2998

[12] Camille Lagoin, Stephan Suffit, Kirk Baldwin, Loren Pfeiffer, and François Dubin. ``Mott insulator of strongly interacting two-dimensional semiconductor excitons''. Nature Physics 18, 149–153 (2022).
https://doi.org/10.1038/s41567-021-01440-8

[13] C. Lagoin, U. Bhattacharya, T. Grass, R. W. Chhajlany, T. Salamon, K. Baldwin, L. Pfeiffer, M. Lewenstein, M. Holzmann, and F. Dubin. ``Extended bose–hubbard model with dipolar excitons''. Nature 609, 485–489 (2022).
https://doi.org/10.1038/s41586-022-05123-z

[14] C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao. ``Programmable quantum simulations of spin systems with trapped ions''. Rev. Mod. Phys. 93, 025001 (2021).
https://doi.org/10.1103/RevModPhys.93.025001

[15] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos. ``Quasiparticle engineering and entanglement propagation in a quantum many-body system''. Nature 511, 202–205 (2014).
https://doi.org/10.1038/nature13461

[16] Tobias Graß, Christine Muschik, Alessio Celi, Ravindra W. Chhajlany, and Maciej Lewenstein. ``Synthetic magnetic fluxes and topological order in one-dimensional spin systems''. Phys. Rev. A 91, 063612 (2015).
https://doi.org/10.1103/PhysRevA.91.063612

[17] Tobias Graß, Bruno Juliá-Díaz, Marek Kuś, and Maciej Lewenstein. ``Quantum chaos in su(3) models with trapped ions''. Phys. Rev. Lett. 111, 090404 (2013).
https://doi.org/10.1103/PhysRevLett.111.090404

[18] C. Senko, P. Richerme, J. Smith, A. Lee, I. Cohen, A. Retzker, and C. Monroe. ``Realization of a quantum integer-spin chain with controllable interactions''. Phys. Rev. X 5, 021026 (2015).
https://doi.org/10.1103/PhysRevX.5.021026

[19] Martin Ringbauer, Michael Meth, Lukas Postler, Roman Stricker, Rainer Blatt, Philipp Schindler, and Thomas Monz. ``A universal qudit quantum processor with trapped ions''. Nature Physics 18, 1053–1057 (2022).
https://doi.org/10.1038/s41567-022-01658-0

[20] Matthew Fishman, Steven R. White, and E. Miles Stoudenmire. ``The ITensor Software Library for Tensor Network Calculations''. SciPost Phys. CodebasesPage 4 (2022).
https://doi.org/10.21468/SciPostPhysCodeb.4

[21] Steven R. White. ``Density matrix formulation for quantum renormalization groups''. Phys. Rev. Lett. 69, 2863–2866 (1992).
https://doi.org/10.1103/PhysRevLett.69.2863

[22] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96–192 (2011).
https://doi.org/10.1016/j.aop.2010.09.012

[23] Arya Dhar, Maheswar Maji, Tapan Mishra, R. V. Pai, Subroto Mukerjee, and Arun Paramekanti. ``Bose-hubbard model in a strong effective magnetic field: Emergence of a chiral mott insulator ground state''. Phys. Rev. A 85, 041602 (2012).
https://doi.org/10.1103/PhysRevA.85.041602

[24] Tapan Mishra, Ramesh V. Pai, Subroto Mukerjee, and Arun Paramekanti. ``Quantum phases and phase transitions of frustrated hard-core bosons on a triangular ladder''. Phys. Rev. B 87, 174504 (2013).
https://doi.org/10.1103/PhysRevB.87.174504

[25] Tapan Mishra, Ramesh V. Pai, and Subroto Mukerjee. ``Supersolid in a one-dimensional model of hard-core bosons''. Phys. Rev. A 89, 013615 (2014).
https://doi.org/10.1103/PhysRevA.89.013615

[26] Zeno Bacciconi, Gian M. Andolina, Titas Chanda, Giuliano Chiriacò, Marco Schirò, and Marcello Dalmonte. ``First-order photon condensation in magnetic cavities: A two-leg ladder model''. SciPost Phys. 15, 113 (2023).
https://doi.org/10.21468/SciPostPhys.15.3.113

[27] Catalin-Mihai Halati and Thierry Giamarchi. ``Bose-hubbard triangular ladder in an artificial gauge field''. Phys. Rev. Res. 5, 013126 (2023).
https://doi.org/10.1103/PhysRevResearch.5.013126

[28] Luca Barbiero, Josep Cabedo, Maciej Lewenstein, Leticia Tarruell, and Alessio Celi. ``Frustrated magnets without geometrical frustration in bosonic flux ladders''. Phys. Rev. Res. 5, L042008 (2023).
https://doi.org/10.1103/PhysRevResearch.5.L042008

[29] Murray Gell-Mann. ``Symmetries of baryons and mesons''. Phys. Rev. 125, 1067–1084 (1962).
https://doi.org/10.1103/PhysRev.125.1067

[30] D. Porras and J. I. Cirac. ``Effective quantum spin systems with trapped ions''. Phys. Rev. Lett. 92, 207901 (2004).
https://doi.org/10.1103/PhysRevLett.92.207901

[31] Shi-Liang Zhu, C. Monroe, and L.-M. Duan. ``Trapped ion quantum computation with transverse phonon modes''. Phys. Rev. Lett. 97, 050505 (2006).
https://doi.org/10.1103/PhysRevLett.97.050505

[32] C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao. ``Programmable quantum simulations of spin systems with trapped ions''. Rev. Mod. Phys. 93, 025001 (2021).
https://doi.org/10.1103/RevModPhys.93.025001

[33] Florian Kranzl, Stefan Birnkammer, Manoj K. Joshi, Alvise Bastianello, Rainer Blatt, Michael Knap, and Christian F. Roos. ``Observation of magnon bound states in the long-range, anisotropic heisenberg model''. Phys. Rev. X 13, 031017 (2023).
https://doi.org/10.1103/PhysRevX.13.031017

[34] Nikhil Kotibhaskar, Chung-You Shih, Sainath Motlakunta, Anthony Vogliano, Lewis Hahn, Yu-Ting Chen, and Rajibul Islam. ``Programmable xy-type couplings through parallel spin-dependent forces on the same trapped ion motional modes''. Phys. Rev. Res. 6, 033038 (2024).
https://doi.org/10.1103/PhysRevResearch.6.033038

Cited by

[1] Sourav Biswas, E. Rico, and Tobias Grass, "Frustrated Bose ladder with extended range density-density interaction", Physical Review B 112 11, 115122 (2025).

[2] João P. Mendonça, S. Biswas, M. Dziurawiec, U. Bhattacharya, K. Jachymski, M. Aidelsburger, M. Lewenstein, M. M. Maśka, and T. Grass, "Controlled pairing symmetries in a Fermi-Hubbard ladder with band flattening", Physical Review B 113 24, L241119 (2026).

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