Large impedances and Majorana bound states in superconducting circuits

  • Große Impedanzen und gebundene Majorana-Zustände in supraleitenden Schaltkreisen

Ulrich, Jascha; Hassler, Fabian (Thesis advisor); Schoeller, Herbert (Thesis advisor)

Aachen (2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017


Superconducting circuits offer the opportunity to study quantum mechanics on mesoscopic scales unimpeded by dissipation. This fact and the nonlinearity of the Josephson inductance make it possible to use superconducting circuits as artificial atoms whose long-lived states can be selectively addressed and studied. A pronounced nonlinearity of the energy spectrum, however, requires quantum fluctuations of the flux across the Josephson junction which are large on the scale of the superconducting flux quantum $\Phi_Q = h/2e$. This implies charge fluctuations below the single Cooper-pair limit via flux-charge duality. The localization of charge leads to a strong susceptibility to interactions with charges in the environment which has motivated the search for schemes to decouple charges from their environment. This thesis is concerned with theoretical challenges arising from two complementary approaches to this problem: the realization of large impedances and the fractionalization of electrons by means of Majorana bound states.In recent years, the decoupling of charges from the environment through reactive large impedances, so-called "superinductances" $L$, has attracted much interest. These inductances feature small parasitic capacitance $C$ such that the characteristic impedance $\sqrt{L/C}$ is much larger than the superconducting resistance quantum $R_Q = h/4e^2$. Superinductances have various applications ranging from qubit designs such as the $0$-$\pi$ qubit or the fluxonium to impedance matching, Bloch oscillations and the stabilization of phase slips in superconducting nanowires. Although there exists a well-established formalism for the quantization of superconducting circuits in terms of node fluxes, this formalism is ill-suited for the description of fast flux transport with localized charges in large-impedance environments. In particular, the nonlinear capacitive behavior of phase slip junctions cannot be modeled in a straightforward way using node fluxes. In view of the ever growing interest in superinductances, in the first part of the thesis, we present a recipe for quantization of planar circuits in terms of loop charges. As we will show, the loop charge approach is dual to the usual node flux formalism and well-adapted to a large impedance setting.In the second part of the thesis, we turn to a complementary approach of charge decoupling by means of Majorana bound states (MBS). MBS solve the decoupling problem by encoding a fermionic mode nonlocally into two bound states with large spatial separation such that a local coupling to the stored charge is no longer possible. It has been shown that despite the apparent nonlocality of the fermionic mode, transport through the mode remains local unless the MBS are coupled by a global perturbation like a finite charging energy. Here we show that, even in absence of charging energy, decoupling the superconductor from the ground plane achieves subtle coupling of the MBS that leads to nonlocal transport.Finally, in the last part of the thesis, we turn to two mesoscopic applications related to supersymmetric quantum mechanics. The simultaneous presence of a fermionic mode due to the MBS and a bosonic mode due to the Cooper-pair condensate makes systems involving MBS appealing candidates for the realization of supersymmetric quantum mechanics. For a Majorana Cooper-pair box, we discuss an unusual "bosonic" supersymmetry and its experimental signatures. Since MBS remain challenging to realize experimentally, we show that a similar supersymmetry can even be realized in a setup using standard superconducting circuitry without MBS.


  • Junior Professorship of Theoretical Physics (Condensed Matter) [137230]
  • Department of Physics [130000]