Rigorous population bounds in the simultaneous-coupling model at strong coupling

Prove rigorous upper and lower bounds on the system-qubit ground-state population in the limit-cycle state of the simultaneous-coupling repeated-interaction model for strong interactions and long collision durations, specifically establish that 1 − p_C ≤ p_S^{(∞)} ≤ p_C holds for arbitrary interaction parameters and τ, where p_C is the ground-state population of the cold ancilla qubit prepared in a Gibbs state.

Background

For the alternating-coupling model, the authors derive rigorous bounds showing that the system’s limit-cycle ground-state population is always bounded above by the cold bath population and bounded below by its complement.

In the simultaneous-coupling model, they only provide numerical support and short-time (stroboscopic) arguments implying identical behavior, but explicitly state that a rigorous proof at strong coupling and long interaction times is missing.

References

While for the alternating-coupling model we derived rigorous upper and lower bounds on the population in Appendices \ref{AppA} and \ref{AppB}, respectively, in the simultaneous model we did not obtain such proofs for the general case, but only in the stroboscopic limit, which trivially reduces to the alternating case. For the simultaneous model, we thus miss such a rigorous proof for the case with strong interactions and long collisions.

— Impossibility of Refrigeration and Engine Operation in Minimal Qubit Repeated-Interaction Models  (2602.18300 - Barsky-Giles et al., 20 Feb 2026) in Appendix C: Numerical support for population bounds in both alternating-coupling and simultaneous-coupling thermal machines