Revisiting 2π phase slip suppression in topological Josephson junctions

Abstract

Current state of the art devices for detecting and manipulating Majorana fermions commonly consist of networks of Majorana wires and tunnel junctions. We study a key ingredient of these networks -a topological Josephson junction with charging energy -and pinpoint crucial features for device implementation. The phase dependent tunneling term contains both the usual 2π-periodic Josephson term and a 4π-periodic Majorana tunneling term representing the coupling between Majoranas on both sides of the junction. In non-topological junctions when the charging energy is small compared to the Josephson tunneling scale the low energy physics is described by 2π phase slips. By contrast, in a topological junction, due to the 4π periodicity of the tunneling term it is usually expected that only 4π phase slips are possible while 2π phase slips are suppressed. However, we find that if the ratio between the strengths of the Majorana assisted tunneling and the Josephson tunneling is small, as is likely to be the case for many setups, 2π phase slips occur and may even dominate the low energy physics. In this limit one can view the 4π phase slips as a pair of 2π phase slips with arbitrarily large separation. We provide an effective descriptions of the system in terms of 2π and 4π phase slips valid for all values of the tunneling ratio. Comparing the spectrum of the effective models with numerical simulations we determine the cross-over between the 4π phase slip regime to 2π phase slip dominated regime. We also discuss the role of the charging energy as well as the implications of our results on the dissipative phase transitions expected in such a system. arXiv:1805.10601v1 [cond-mat.supr-con]

Building a bigger Hilbert space for superconducting devices, one Bloch state at a time

“…(29) qualitatively comparable to the aperiodic potential in Eq. (22): the two potential functions have a global minimum at ϕ = 0, local minima at or near ϕ = ±2π , and with a potential difference between these local minima and the global minimum ≈ E J ). We define the time for a coherent oscillation between the lowest and first excited states at the center of the extended Brillouin zone (i.e., at κ = 0), t 2π = 2π/(E 4π 0,1 − E 4π 0,0 ).…”

“…An alternative approach to coupling different k states is to break the 2π -periodic φ-translation symmetry by shunting the transmon with a 4π -periodic element [18,19]. For the purposes of this section, we do not describe the implementation of such an element [13,[20][21][22], but begin with a phenomenological Hamiltonian [22]…”

“…III B, would be ideally performed by a 4π -periodic Josephson junction. This element is a current experimental challenge and is being investigated in the context of the Majorana qubit [13,[20][21][22]. A proposal for a 4π -periodic element has been put forward by Bell et al [29].…”

“…Thus a secondary hurdle in future experiments is the development of robust 4π -periodic elements. While we have adopted a purely phenomenological description of such an element, realistic implementations are the subject of recent theoretical and experimental attention [13,[18][19][20][21][22]. Understanding the performance of the switched system using realistic models for practical 4π -periodic elements will be the subject of future research.…”

Report on scipost_201906_00002v1

“…In the case of Andreev qubits, detailed models which are amenable to an analytical approach are available in simple limits, such as that of a short Josephson junction with a single conducting channel [37,[46][47][48]. On the other hand, most of the theory literature treating the presence of MZMs in superconducting circuits [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66] relies on simple toy-models with phenomenological terms representing Majorana couplings, bypassing a microscopic description of the topological phase.…”