Exact analytical limit-cycle solution for the simultaneous-coupling model at arbitrary coupling and collision time

Derive an exact analytical expression for the limit-cycle state of the simultaneous-coupling repeated-interaction qubit thermal machine, valid for arbitrary coupling strengths and finite collision duration τ, beyond the Dyson-series short-collision-time approximation.

Background

The simultaneous-coupling setup involves a three-body collision unitary that the authors analyze perturbatively via a Dyson expansion for short collision times. They emphasize that, unlike the alternating model, a general exact solution at arbitrary coupling and interaction duration is not available.

They explicitly note the absence of a general analytical expression for the limit cycle at strong coupling and long collisions, motivating a precise analytical treatment that would parallel the exact alternating-model solution.