Topological quantization by controlled paths: Application to Cooper pairs pumps

Abstract

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.

Cooper-Pair Pump as a Quantized Current Source

“…proved elsewhere [11], generalize earlier results [14,15] to arbitrary three-dimensional closed paths in P. We first show that the geometrical contribution is quantized when averaged over the initial phase ' 0 of the helix. Taking two helices ÿ ' 0 and ÿ ' 0 ' shifted in ' by an infinitesimal increment ', a closed path ' 0 on the surface of the cylinder (shown in Fig.…”

“…The Josephson Hamiltonian H J , the sum of Josephson energies of the three junctions, depends on the phase bias ' and the phases s and d canonical conjugates ton s and n d as[11] …”

Merging diabolical points of a superconducting circuit

“…couples the neighboring states and generically lifts the degeneracies of H C . Explicitly, U (ϕ x ) = e iκ κ κ·nϕ x , with κ 1 = 1−C R (C −1 ) 12 and κ 2 = 2−C R (C −1 ) 22 . In the Coulomb blockade regime, H J is seen as a perturbation of H C .…”

“…Notably, superconducting circuits are widely used to engineer qubits [3][4][5][6][7][8]: the non-linear behavior of Josephson junctions serves to isolate couples of levels in a Hamiltonian spectrum. They are also used to perform the role of analogs of cavity quantum electrodynamics [9][10][11] (a qubit plays the role of an artificial atom while a transmission line carries artificial photon modes), (non-) Abelian holonomies [12][13][14][15][16], (non-) Abelian quantum charge pumpings [17][18][19][20][21][22][23][24], etc. In brief, they are good candidates for implementing quantum logic operations [7,25] as well as appear to be quite promising for applications in electrical metrology [26].…”

Cooper-pair current in the presence of flux noise

“…39 Analogous systems have been studied for the relations between pumping and topological phases. 40,41 The Hamiltonian of the sluiceĤ S is the sum of the charging HamiltonianĤ ch = E C (n − n g ) 2 and the Josephson Hamiltonian 34,42 …”