A new charge quantization in a phase-polarized Cooper-pair pump is proposed, based on the topological properties of its Hamiltonian ground state over a three-dimensional parameter space P. The charge is quantized using a set of paths in P covering the surface of a torus, and is a multiple of the integer Chern index c1 of this surface. This quantization is asymptotic but the pumped charge converges rapidly to the quantized value with the increase in the path frequency. The topological nature of the current makes this Cooper-pair pump implementation an excellent candidate for a metrological current standard.
When physical systems are tunable by three classical parameters, level degeneracies may occur at isolated points in parameter space. A topological singularity in the phase of the degenerate eigenvectors exists at these points. When a path encloses such point, the accumulated geometrical phase is sensitive to its presence. Furthermore, surfaces in parameter space enclosing such point can be used to characterize the eigenvector singularities through their Chern indices, which are integers. They can be used to quantize a physical quantity of interest. This quantity changes continuously during an adiabatic evolution along a path in parameter space. Quantization requires to turn this path into a surface with a well defined Chern index. We analyze the conditions necessary to a Topological Quantization by Controlled Paths. It is applied to Cooper pair pumps. For more general problems, a set of four criteria are proposed to check if topological quantization is possible.
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