We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the onedimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

No abstract

Ongoing development of quantum simulators allows for a progressively finer degree of control of quantum many-body systems. This motivates the development of efficient approaches to facilitate the control of such systems and enable the preparation of non-trivial quantum states. Here we formulate an approach to control quantum systems based on matrix product states (MPS). We compare counter-diabatic and leakage minimization approaches to the so-called local steering problem, that consists in finding the best value of the control parameters for generating a unitary evolution of the specific MPS state in a given direction. In order to benchmark the different approaches, we apply them to the generalization of the PXP model known to exhibit coherent quantum dynamics due to quantum many-body scars. We find that the leakage-based approach generally outperforms the counter-diabatic framework and use it to construct a Floquet model with quantum scars. Finally, we perform the first steps towards global trajectory optimization and demonstrate entanglement steering capabilities in the generalized PXP model.

It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Ξ and the critical temperature T c is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density, and high density. The goal of this short note is to extend the universal behavior to lower dimensions d = 1, 2 and give an exemplary proof in the weak coupling limit.