Nano-welding of quantum spin-$1/2$ chains at minimal dissipation

We consider the optimal control of switching on a coupling term between two quantum many-body systems. Specifically, we (i) quantify the energetic cost of establishing a weak junction between two quantum spin-$1/2$ chains in finite time τ and (ii) identify the energetically optimal protocol to realize it. For linear driving protocols, we find that for long times the excess (irreversible) work scales as τ-η, where η=1, 2 or a nonuniversal number depending on the phase of the chains. Interestingly, increasing a Jz anisotropy in the chains suppresses the excess work thus promoting quasi-adiabaticity. The general optimal control problem is solved, employing a Chebyshev ansatz. We find that the optimal control protocol is intimately sensitive to the chain phases.