In these models, the preferred auxiliary state is not necessarily the coldest one. The theoretical study asks whether a target system can relax toward a prescribed thermal state more quickly when coupled to a separately prepared helper, or packet. It compares two goals: minimizing the target's slowest relaxation mode and minimizing its deviation at a chosen readout time. Those objectives can select different packet preparations, with the fixed-time choice generally colder.
The slowest mode sets the test
At the center is a spectral test, meaning the calculation separates relaxation into characteristic decay modes. The slowest mode leaves the longest-lasting tail. The packet is chosen by minimizing the target's coefficient a2 for that mode over allowed preparations. A zero minimum defines a perfect packet. In the examples, once that contribution is removed, a later mode determines the long-time rate.
No single temperature rule
That condition does not translate into a single temperature setting. Near thermal equilibrium, the paper reports a local curve of Boltzmann packet preparations that meet the perfect-packet condition. In the exactly solvable complete-graph Metropolis example, the contour passes through the bath-equilibrium point. The result describes a family of model-specific preparations, rather than a universal instruction to make the helper as cold as possible.
A two-qubit example
The minimal two-qubit example gives the condition in a compact form. If r is the bath population and pS the target population, a perfect packet has population pC* = 2r - pS. Under the same specified dynamics, the target excess has decay rate γ + 2κ instead of the bath rate γ, with the 2κ term representing exchange in this model. The equation and rate belong to the stated two-qubit model, not to a measured device.
Deadlines point to a different packet
Changing the objective changes the preferred packet. At a fixed readout time, the optimal packet is generally colder than the asymptotically perfect packet. If the analytic population selected by that calculation is not physically admissible, the cooling optimum lies at the boundary pC = 0. The criterion can also count transient arrival followed by overshoot because it judges the target at one selected time.
One larger model favors a colder packet
A larger illustration uses six target spins and four packet spins, with independent Ising Hamiltonians and bath-mediated exchanges at the boundaries. The joint Markov chain has 1,024 states and is analyzed by exact diagonalization. For a hot target with the paper's inverse-temperature parameter βs = 0.4, the slow-mode amplitude vanishes at a packet value βp* of about 1.442. The corresponding packet is colder than the bath and is termed a perfect ice cube. After slow-mode cancellation, the reported relaxation-rate ratio |λ3|/|λ2| is about 1.80.
Another model allows the opposite
Another modified coupled-qubit model points in the other direction. Its perfect-packet contour has a nontrivial common-temperature intersection away from bath equilibrium. Across different energy scales, the model switches between strong and strong inverse-Mpemba behavior, and it can select a packet hotter than the bath. Taken together, the examples make the model's rule clear: the relevant preparation depends on the zero-mode contour and the timing objective, not on temperature alone.
A theoretical result with costs left aside
The analysis remains a theoretical map. It covers exactly solvable Metropolis dynamics, a minimal two-qubit model and a boundary-coupled interacting Ising-spin system. The supplied manuscript is an arXiv preprint, version 1, posted on 25 August 2026. Packet preparation time and energetic cost are treated as sunk costs, so the optimization starts with the auxiliary state already prepared and concerns the relaxation that follows.
Paper data and sources
Original title: Thermalization packets and optimal ice cubes
Authors: Israel Klich, Marija Vucelja
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text