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Register a new userCoulomb stability of the 4pi-periodic Josephson effect of Majorana fermions
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The Josephson energy of two superconducting islands containing Majorana fermions is a 4pi-periodic function of the superconducting phase difference. If the islands have a small capacitance, their ground state energy is governed by the competition of Josephson and charging energies. We calculate this ground state energy in a ring geometry, as a function of the flux -Phi- enclosed by the ring, and show that the dependence on the Aharonov-Bohm phase 2ePhi/hbar remains 4pi-periodic regardless of the ratio of charging and Josephson energies - provided that the entire ring is in a topologically nontrivial state. If part of the ring is topologically trivial, then the charging energy induces quantum phase slips that restore the usual 2pi-periodicity.
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The $4pi$-periodic Josephson effect is an indicator of Majorana zero modes and a ground-state degeneracy which are central to topological quantum computation. However, the observability of a $4pi$-periodic Josephson current-phase relation (CPR) is hindered by the necessity to fix the fermionic parity. As an alternative to a $4pi$-periodic CPR, this paper proposes a chiral CPR for the $4pi$-periodic Josephson effect. This is a CPR of the form $J(phi) propto C , |sin(phi/2)|$, describing a unidirectional supercurrent with the chirality $C= pm 1$. Its non-analytic dependence on the Josephson phase difference $phi$ translates into the $4pi$-periodic CPR $J(phi) propto sin(phi/2)$. The proposal requires a spin-polarized topological Josephson junction which is modeled here as a short link between spin-split superconducting channels at the edge of a two-dimensional topological insulator. In this case, $C$ coincides with the Chern number of the occupied spin band of the topological insulator. The paper details three scenarios of achieving a chiral CPR: By only Zeeman-like splitting, by Zeeman splitting combined with bias currents, and by an external out-of-plane magnetic field.
We show how to exchange (braid) Majorana fermions in a network of superconducting nanowires by control over Coulomb interactions rather than tunneling. Even though Majorana fermions are charge-neutral quasiparticles (equal to their own antiparticle), they have an effective long-range interaction through the even-odd electron number dependence of the superconducting ground state. The flux through a split Josephson junction controls this interaction via the ratio of Josephson and charging energies, with exponential sensitivity. By switching the interaction on and off in neighboring segments of a Josephson junction array, the non-Abelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes. This is a solution to the problem how to operate on topological qubits when gate voltages are screened by the superconductor.
We investigate theoretically the dynamics of a Josephson junction in the framework of the RSJ model. We consider a junction that hosts two supercurrrent contributions: a $2pi$- and a $4pi$-periodic in phase, with intensities $I_{2pi}$ and $I_{4pi}$ respectively. We study the size of the Shapiro steps as a function of the ratio of the intensity of the mentioned contributions, i.e. $I_{4pi}/I_{2pi}$. We provide detailed explanations where to expect clear signatures of the presence of the $4pi$-periodic contribution as a function of the external parameters: the intensity AC-bias $I_text{ac}$ and frequency $omega_text{ac}$. On the one hand, in the low AC-intensity regime (where $I_text{ac}$ is much smaller than the critical current, $I_text{c}$), we find that the non-linear dynamics of the junction allows the observation of only even Shapiro steps even in the unfavorable situation where $I_{4pi}/I_{2pi}ll 1$. On the other hand, in the opposite limit ($I_text{ac}gg I_text{c}$), even and odd Shapiro steps are present. Nevertheless, even in this regime, we find signatures of the $4pi$-supercurrent in the beating pattern of the even step sizes as a function of $I_text{ac}$.
We propose a scheme to detect the Majorana bound states (MBSs) by a thermodynamically stable D.C. Josephson current with $4pi$-periodicity in the superconducting phase difference, which is distinct from the previous A.C. $4pi$-periodicity found in topological superconducting Josephson junctions. The scheme, consisting of a quantum dot coupled to two s-wave superconducting leads and a floating topological superconductor supporting two MBSs at its ends, only exploits the interplay of a local Zeeman field and the exotic helical and self-Hermitian properties of MBSs, without requiring the conservation of fermion parity and not relying on the zero-energy property of MBSs. Our D.C. $4pi$-periodicity is thus robust against the overlap between the two MBSs and various system parameters, including the local Coulomb interaction, the tunneling asymmetry, and the width of superconducting gap, which facilitates experimentally detection of the MBSs.
We propose a topological qubit in which braiding and readout are mediated by the $4pi$ Majorana-Josephson effect. The braidonium device consists of three Majorana nanowires that come together to make a tri-junction; in order to control the superconducting phase differences at the tri-junction the nanowires are enclosed in a ring made of a conventional superconductor; and in order to perform initialization/readout one of the nanowires is coupled to a fluxonium qubit through a topological Josephson junction. We analyze how flux-based control and readout protocols can be used to demonstrate braiding and qubit operation for realistic materials and circuit parameters.
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