A 2012 quantum gas microscope experiment (Cheneau et al., Nature 481, 484 (2012)) observed light-cone-like spreading of correlations after a sudden quench of a one-dimensional Mott-insulating Bose-Hubbard gas, confirming the quasiparticle picture of correlation propagation deep in the Mott phase. The paper's closing outlook identified open directions: revealing the entanglement carried by the propagating quasiparticle pairs, extending to two dimensions, treating the non-perturbative dynamics close to the superfluid-Mott critical point, and exploring a quantum analog of the Kibble-Zurek mechanism by ramping the interaction slowly across the critical point. The unresolved directions are three linked parts: (i) direct measurement of the entanglement carried by the doublon-holon pairs launched by the original quench protocol; (ii) a quantitative theory of correlation spreading near the critical point where the quasiparticle and fermionized descriptions break down, including finite temperature and trap inhomogeneity; and (iii) an experimental test of quantum Kibble-Zurek scaling of excitation production and domain formation through the superfluid-Mott transition of a lattice Bose gas.

Background

The Bose-Hubbard model describes bosons hopping between neighboring sites of an optical lattice with amplitude JJ and repelling each other on the same site with energy UU. For large U/JU/J at integer filling one boson per site it realizes a Mott insulator, a phase with fixed site occupation and suppressed number fluctuations; for small U/JU/J it realizes a superfluid. Its superfluid-Mott insulator quantum phase transition is a central benchmark of ultracold-atom quantum simulation. A quantum quench is a sudden change of a Hamiltonian parameter, here a rapid jump of U/JU/J from deep in the Mott regime toward the critical value (U/J)c(U/J)_c, after which an isolated many-body state evolves far from equilibrium. Calabrese and Cardy (2006) predicted that correlations after a quench spread because pairs of energetically opposite quasiparticles are created at the same point and propagate ballistically in opposite directions: two-point functions remain at equilibrium values until each point is reached by a quasiparticle, producing a ballistic causal cone. Rigorous counterparts of this picture are Lieb-Robinson bounds (Lieb and Robinson, 1972), which establish a finite maximal velocity for the propagation of operators and correlations in lattice systems with short-range interactions. In the Mott phase of the Bose-Hubbard model the relevant quasiparticles are doublon-holon pairs: a quench creates, at the same point, a site with two bosons (a doublon) and an empty site (a holon), which then propagate in opposite directions. Cheneau et al. (2012) quenched a one-dimensional chain of bosons held in an optical lattice and, using a quantum gas microscope (a device that images individual atoms on individual lattice sites with single-site resolution), measured parity correlations (site-occupation correlations after projecting each site to even or odd occupation) between site pairs as a function of distance and evolution time. They observed an explicit causal cone with a propagation velocity in agreement with a quasiparticle calculation in which each site is restricted to at most two atoms (a Gutzwiller-projected ansatz) and the doublons and holons are then treated as free fermions (a 'fermionized' description, obtained via a Jordan-Wigner transformation), valid deep in the Mott-insulating regime; the data closest to the critical point could only be matched by ab-initio numerical simulations. The Kibble-Zurek (KZ) mechanism addresses the complementary slow protocol: when a parameter is ramped across a continuous phase transition, the diverging relaxation time forces the system to fall out of equilibrium at a freeze-out point, leaving excitations and domains whose density and size scale with the ramp rate through universal critical exponents; the quantum version replaces thermal fluctuations with quantum ones, relevant when the crossing occurs at zero or low temperature relative to the many-body gap. In the one-dimensional Bose-Hubbard chain the transition at the tip of the Mott lobe is of Berezinskii-Kosterlitz-Thouless type (a transition with exponentially diverging equilibrium correlation length), while in two dimensions it is in the (2+1)-dimensional XY universality class (the critical behavior of the classical three-dimensional XY model). For the present problem the relevant context from later literature is: the two-dimensional extension proposed by the source paper has since been realized (single-particle correlation velocities in a quenched 2D Bose-Hubbard gas, Takasu et al. 2020) and described by quantitative tensor-network and strong-coupling methods (Kaneko and Danshita 2022; Mokhtari-Jazi et al. 2021), and a rigorous Lieb-Robinson bound with an almost-linear light cone for interacting boson systems now exists (Kuwahara and Saito 2021); entanglement propagation has been measured in adjacent platforms (a spin-entanglement wave in the same Bose-Hubbard chain platform by Fukuhara et al. 2015; trapped-ion and tweezer experiments by Jurcevic et al. 2014 and Kaufman et al. 2016); and the quantum Kibble-Zurek quench from the Mott insulator to the superfluid has been worked out theoretically in detail for one and two dimensions, at zero temperature, finite temperature, and in finite trapped systems, but tested experimentally only on other platforms such as Rydberg Ising simulators and spin-1 condensates.

Problem Statement

Within the post-quench dynamics of the Bose-Hubbard gas quenched from the deep Mott-insulating regime, and taking the two-dimensional extension as resolved by later work (References 4-7), the problem is to address three linked parts, which share one objective — establishing what propagates and what is produced when the Mott insulator is driven out of equilibrium in and beyond the quasiparticle picture: (1) Entanglement of the propagating pairs: directly measure, in the original particle-sector quench protocol (a rapid jump of U/JU/J in the Mott phase, with site-resolved detection), the entanglement carried by the doublon-holon quasiparticle pairs (measured at a fixed post-quench evolution time as the two-site entanglement between the pair's two components — the doublon, i.e. the doubly occupied site, and the holon, i.e. the empty site — quantified by a standard bipartite entanglement measure such as the negativity or the mutual information of the reduced two-site density matrix), whose ballistic propagation produces the observed correlation light cone — going beyond the parity correlations already measured, and beyond the magnon-pair, ion-chain, tweezer and attractive-interaction incarnations in which pair entanglement has been observed. (2) Near-critical correlation spreading: determine the dynamics of correlation propagation in the regime approaching the critical point (U/J)c(U/J)_c, where the Gutzwiller-projected fermionized quasiparticle description valid deep in the Mott phase breaks down and only ab-initio numerics are currently available — specifically, a quantitative analytic or numerical treatment that delimits the regime of applicability of the quasiparticle description and incorporates finite temperature and the inhomogeneity of the harmonic trap. (3) Quantum Kibble-Zurek experiment in the superfluid-Mott transition: when the effective interaction strength is ramped slowly through the critical point, measure the production rate of excitations and the formation of domains (the post-quench correlation length and defect density) as a function of ramp time, and test the quantum Kibble-Zurek scaling predictions that theory has worked out for the one- and two-dimensional Bose-Hubbard model, including their finite-temperature and inhomogeneous modifications — an experimental test on the Bose-Hubbard superfluid-Mott transition itself, which currently exists only on other platforms.

References

  1. Cheneau, M.; Barmettler, P.; Poletti, D.; Endres, M.; Schauß, P.; Fukuhara, T.; Gross, C.; Bloch, I.; Kollath, C.; Kuhr, S. Light-cone-like spreading of correlations in a quantum many-body system. Nature 481, 484-487 (2012). DOI: 10.1038/nature10748; arXiv:1111.0776. https://doi.org/10.1038/nature10748
  2. Calabrese, P.; Cardy, J. Time Dependence of Correlation Functions Following a Quantum Quench. Physical Review Letters 96, 136801 (2006). DOI: 10.1103/PhysRevLett.96.136801. https://doi.org/10.1103/PhysRevLett.96.136801
  3. Lieb, E. H.; Robinson, D. W. The finite group velocity of quantum spin systems. Communications in Mathematical Physics 28, 251-257 (1972). DOI: 10.1007/bf01645779. https://doi.org/10.1007/bf01645779
  4. Takasu, Y.; Yagami, T.; Asaka, H.; Fukushima, Y.; Nagao, K.; Goto, S.; Danshita, I.; Takahashi, Y. Energy redistribution and spatiotemporal evolution of correlations after a sudden quench of the Bose-Hubbard model. Science Advances 6, eaba9255 (2020). DOI: 10.1126/sciadv.aba9255. https://doi.org/10.1126/sciadv.aba9255
  5. Kaneko, R.; Danshita, I. Tensor-network study of correlation-spreading dynamics in the two-dimensional Bose-Hubbard model. Communications Physics 5 (2022). DOI: 10.1038/s42005-022-00848-9; arXiv:2108.11051. https://doi.org/10.1038/s42005-022-00848-9
  6. Mokhtari-Jazi, A.; Fitzpatrick, M. R. C.; Kennett, M. P. Phase and group velocities for correlation spreading in the Mott phase of the Bose-Hubbard model in dimensions greater than one. Physical Review A 103, 023334 (2021). DOI: 10.1103/PhysRevA.103.023334. https://doi.org/10.1103/PhysRevA.103.023334
  7. Kuwahara, T.; Saito, K. Lieb-Robinson Bound and Almost-Linear Light Cone in Interacting Boson Systems. Physical Review Letters 127, 070403 (2021). DOI: 10.1103/PhysRevLett.127.070403. https://doi.org/10.1103/PhysRevLett.127.070403
  8. Fukuhara, T.; Hild, S.; Zeiher, J.; Schauß, P.; Bloch, I.; Endres, M.; Gross, C. Spatially Resolved Detection of a Spin-Entanglement Wave in a Bose-Hubbard Chain. Physical Review Letters 115, 035302 (2015). DOI: 10.1103/PhysRevLett.115.035302. https://doi.org/10.1103/PhysRevLett.115.035302
  9. Jurcevic, P.; Lanyon, B. P.; Hauke, P.; Hempel, C.; Zoller, P.; Blatt, R.; Roos, C. F. Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature 511, 202-205 (2014). DOI: 10.1038/nature13461. https://doi.org/10.1038/nature13461
  10. Kaufman, A. M.; Tai, M. E.; Lukin, A.; Rispoli, M.; Schittko, R.; Preiss, P. M.; Greiner, M. Quantum thermalization through entanglement in an isolated many-body system. Science 353, 794-800 (2016). DOI: 10.1126/science.aaf6725. https://doi.org/10.1126/science.aaf6725
  11. Chen, C.-A.; Khlebnikov, S.; Hung, C.-L. Observation of Quasiparticle Pair Production and Quantum Entanglement in Atomic Quantum Gases Quenched to an Attractive Interaction. Physical Review Letters 127, 060404 (2021). DOI: 10.1103/PhysRevLett.127.060404. https://doi.org/10.1103/PhysRevLett.127.060404
  12. Fitzpatrick, M. R. C.; Kennett, M. P. Light-cone-like spreading of single-particle correlations in the Bose-Hubbard model after a quantum quench in the strong-coupling regime. Physical Review A 98, 053618 (2018). DOI: 10.1103/PhysRevA.98.053618. https://doi.org/10.1103/PhysRevA.98.053618
  13. Pelayo, J. C.; Danshita, I. Suppressed correlation-spreading in a one-dimensional Bose-Hubbard model with strong interactions (2026). arXiv:2602.20780. https://arxiv.org/abs/2602.20780
  14. Gardas, B.; Dziarmaga, J.; Zurek, W. H. Dynamics of the quantum phase transition in the one-dimensional Bose-Hubbard model: Excitations and correlations induced by a quench. Physical Review B 95, 104306 (2017). DOI: 10.1103/PhysRevB.95.104306. https://doi.org/10.1103/PhysRevB.95.104306
  15. Dziarmaga, J.; Mazur, J. M. Tensor network simulation of the quantum Kibble-Zurek quench from the Mott to the superfluid phase in the two-dimensional Bose-Hubbard model. Physical Review B 107, 144510 (2023). DOI: 10.1103/PhysRevB.107.144510. https://doi.org/10.1103/PhysRevB.107.144510
  16. Shimizu, K.; Kuno, Y.; Hirano, T.; Ichinose, I. Dynamics of a quantum phase transition in the Bose-Hubbard model: Kibble-Zurek mechanism and beyond. Physical Review A 97, 033626 (2018). DOI: 10.1103/PhysRevA.97.033626. https://doi.org/10.1103/PhysRevA.97.033626
  17. Weiss, W.; Gerster, M.; Jaschke, D.; Silvi, P.; Montangero, S. Kibble-Zurek scaling of the one-dimensional Bose-Hubbard model at finite temperatures. Physical Review A 98, 063601 (2018). DOI: 10.1103/PhysRevA.98.063601. https://doi.org/10.1103/PhysRevA.98.063601
  18. Machida, Y.; Kasamatsu, K. Application of the inhomogeneous Kibble-Zurek mechanism to quench dynamics in the transition from a Mott insulator to a superfluid in a finite system. Physical Review A 103, 013310 (2021). DOI: 10.1103/PhysRevA.103.013310. https://doi.org/10.1103/PhysRevA.103.013310
  19. Bernier, J.-S.; Roux, G.; Kollath, C. Slow Quench Dynamics of a One-Dimensional Bose Gas Confined to an Optical Lattice. Physical Review Letters 106, 200601 (2011). DOI: 10.1103/PhysRevLett.106.200601. https://doi.org/10.1103/PhysRevLett.106.200601
  20. Keesling, A.; Omran, A.; Levine, H.; Bernien, H.; Pichler, H.; Choi, S.; Samajdar, R.; Schwartz, S.; Silvi, P.; Sachdev, S.; Zoller, P.; Endres, M.; Greiner, M.; Vuletić, V.; Lukin, M. D. Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator. Nature 568, 207-211 (2019). DOI: 10.1038/s41586-019-1070-1. https://doi.org/10.1038/s41586-019-1070-1
  21. Anquez, M.; Robbins, B. A.; Bharath, H. M.; Boguslawski, M.; Hoang, T. M.; Chapman, M. S. Quantum Kibble-Zurek Mechanism in a Spin-1 Bose-Einstein Condensate. Physical Review Letters 116, 155301 (2016). DOI: 10.1103/PhysRevLett.116.155301. https://doi.org/10.1103/PhysRevLett.116.155301