Propagation of locally injected entropy in strongly correlated quantum lattice gases

Sherson et al. (Nature 467, 68, 2010) introduced single-atom- and single-site-resolved fluorescence imaging (the quantum gas microscope) of a bosonic Mott insulator, and closed their paper by proposing as future work to investigate 'how entropy propagates in strongly correlated systems, after injecting it on a local scale into the system'. The technical challenges named in the same concluding paragraph — parity-projected detection, spin-resolved and number-resolved site-by-site imaging, single-site manipulation, and cooling schemes that exploit access to high-entropy regions — have since been substantially met or advanced by quantum gas microscopy. The entropy-propagation question itself, however, remains without a direct answer: no experiment has locally injected entropy into a Mott-insulating or otherwise strongly correlated lattice gas and tracked its site-resolved spreading, and the existing theory and neighboring experiments treat only adjacent configurations (heat equilibration between Mott regions at different temperatures, correlation and quasiparticle spreading after local quenches, and entropy transport in a quench-driven spinor Bose gas). The problem is to inject entropy on a locally controlled scale into a strongly correlated lattice gas, follow the space- and time-resolved evolution of the local entropy distribution, and establish the resulting propagation law (ballistic, diffusive, or anomalous) in terms of the microscopic parameters, including the roles of the excitation gap, quasiparticles, and superexchange.

Background

Ultracold atoms trapped in optical lattices reproduce condensed-matter Hamiltonians with tunable parameters; the paradigmatic example is the Bose-Hubbard model, in which bosons hop between lattice sites with amplitude $t$ and repel each other on a site with energy $U$. For $U/t$ large, the ground state at integer filling is a Mott insulator: interaction energy forbids double occupation, so each site holds a fixed number of atoms and the gas loses compressibility. In 2009–2010 two groups realized the quantum gas microscope, an apparatus that images such a gas by freezing the atom distribution and collecting laser-induced fluorescence through a high-numerical-aperture objective, so that individual atoms on individual lattice sites can be counted. Sherson and co-workers' version of this microscope produced the first full reconstruction of the atom distribution in a Mott insulating cloud of rubidium-87, resolved individual thermal defects (particle–hole excitations, that is, a missing atom or an extra atom on a site relative to the ideal integer filling) with high fidelity, and, by comparing measured radial density and number-variance profiles with finite-temperature Bose-Hubbard calculations, extracted an in-situ temperature and entropy from single images. They observed Mott plateaus of near-zero entropy separated by high-entropy rings only about one lattice site wide. Two caveats of the detection method matter here: light-induced collisions during imaging eject atom pairs, so the recorded on-site occupation is its parity (occupation modulo two); and some apparent defects may be 'collateral damage' of the imaging pulses rather than intrinsic thermal excitations. In their concluding paragraph the authors note that one should be able to observe quantum critical phenomena and spin-resolved images with such a microscope, that focusing a laser onto a single atom enables single-site manipulation, that access to high-entropy regions could serve novel cooling schemes, and — as the explicitly posed open question — that it would be interesting to investigate how entropy propagates in strongly correlated systems after injecting it on a local scale. The subsequent fate of the neighboring items frames the surviving question. Single-site and single-spin addressing was demonstrated within a year (Weitenberg et al., Nature 471, 319, 2011). Number-resolved microscopy that overcomes the parity limitation — counting $n=0$ to $n=3$ atoms per site by occupation-dependent transport into a coupled bilayer, explicitly circumventing light-assisted pair loss — was achieved by Preiss et al. (Phys. Rev. A 91, 041602(R), 2015), who also used the bilayer for spin-resolved readout; fermionic quantum gas microscopes with Raman-sideband cooling (a continuous laser-cooling technique compatible with repeated photon scattering during imaging) and spin resolution followed (Parsons et al., Phys. Rev. Lett. 114, 213002, 2015; Cheuk et al., Phys. Rev. Lett. 114, 193001, 2015; Boll et al., Science 353, 1257, 2016, spin- and density-resolved microscopy of Fermi-Hubbard chains). Cooling by entropy redistribution — reshaping the trap so entropy flows adiabatically from a gapped, low-heat-capacity Mott region into a surrounding metallic reservoir — was developed into a working technique (Yang et al., Science 369, 550, 2020; see the lecture notes of Bakr, Ba and Prichard, arXiv:2507.04042, 2025, for the current state). In-situ entropy measurement in a strongly correlated lattice gas is thus established in equilibrium (Cocchi et al., Phys. Rev. X 7, 031025, 2017, for the two-dimensional Hubbard model). On the dynamical side, the same Garching microscope lineage observed the light-cone-like spreading of correlations after a local quench, that is, a sudden parameter change confined to a small region (Cheneau et al., Nature 481, 484, 2012), and hole-injection experiments in Fermi-Hubbard chains and planes track spinons and holons (the separate spin-carrying and charge-carrying fragments into which an injected hole separates, a phenomenon called spin-charge separation) and magnetic polarons (an injected hole dressed by a local distortion of the surrounding antiferromagnetic order) — but these track quasiparticles and correlation functions, not entropy. Theory has treated heat transport between two Bose-Hubbard Mott insulators initially at different temperatures, simulated in real time with the Minimally Entangled Typical Thermal States (METTS) algorithm (a quantum Monte Carlo sampling method that represents a thermal ensemble through typical pure states) and described by a quasiparticle transport theory (Łącki, Delande and Zakrzewski, Phys. Rev. B 91, 134304, 2015); related work analyzes real-space thermalization of locally driven quantum magnets (Melendrez et al., Phys. Rev. B 110, 104304, 2024). Most recently, Marijan, Strobel, Oberthaler and Berges (arXiv:2510.26873, 2025) measured entropy transport with spatial resolution in a closed, far-from-equilibrium spinor Bose gas (a Bose gas whose atoms carry an internal spin degree of freedom) — but there the entropy was generated by the quench itself, the gas is not a strongly correlated lattice system, and no local injection was performed. A subtle conceptual point that any answer must confront: an isolated quantum system conserves its von Neumann entropy, so 'entropy' here means the coarse-grained thermal entropy (the quantity measured in the Sherson thermometry, inferred from local occupation statistics against a finite-temperature model), and 'injecting entropy locally' is an operational notion — for example locally heating or locally creating defects in a region of sites — whose calibration is part of the problem. What is missing is precisely the experiment and theory the source paper called for: a localized entropy disturbance in a strongly correlated lattice and the space-time law of its spreading.

Problem Statement

How does entropy propagate through a strongly correlated quantum lattice system after it is injected on a local scale — the question posed at the end of Sherson et al. (Nature 467, 68, 2010), enabled by their demonstration of single-atom- and single-site-resolved imaging and in-situ entropy measurement in a bosonic Mott insulator? Concretely, the problem is to prepare a strongly correlated lattice gas (for example, a Bose- or Fermi-Hubbard Mott insulator in a quantum gas microscope), inject entropy within a controlled local region of an initially low-entropy Mott state, and follow the subsequent space- and time-resolved evolution of the local entropy distribution, where the local entropy density is the coarse-grained thermal entropy measured from site-resolved occupation statistics against finite-temperature model calculations, as in the source experiment. For a specified strongly correlated lattice gas, the question of how the injected entropy spreads comprises three jointly posed parts: (i) an experimental protocol that realizes controlled local entropy injection in a strongly correlated system without destroying the surrounding correlations, together with the site-resolved measurement of the entropy-density dynamics; (ii) the characterization of the propagation law — ballistic, diffusive, or anomalous, with its characteristic velocity or diffusion constant expressed in terms of the microscopic parameters (tunneling $t$, interaction $U$, the superexchange coupling $J=4t^2/U$ generated by virtual hopping processes, and the Mott excitation gap) — including how entropy transport relates to, and differs from, the propagation of the underlying quasiparticles, particles, and energy; and (iii) the corresponding theoretical description or simulation of entropy transport in the strongly correlated system out of equilibrium, against which the measurements are compared.

References

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