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arXiv:2011.08160 · Physical review. B./Physical review. B

Majorana entropy revival via tunneling phases

Sergey Smirnov

Published Feb 25, 2021• 24 citations
arXivHTMLPDF

Abstract

Measuring the Majorana entropy ${S}_{M}={k}_{B}log({2}^{\frac{1}{2}})$ (where $log$ is the natural logarithm) may uniquely reveal whether an initial equilibrium state of a nanoscale device has a Majorana nature, and subsequent operations deal with an essentially nonlocal pair of non-Abelian Majorana bound states and not with trivial or other accidental non-Abelian states. However, in realistic setups both Majorana modes are inevitably involved in tunneling processes. We show that even when the tunneling amplitude of one Majorana mode is significantly suppressed, the Majorana entropy is ruined, and straightforward experiments will, in general, detect entropy $S\ensuremath{\ll}{S}_{M}$. To avoid this general problem we present a mechanism of the Majorana entropy revival via the tunneling phases of the Majorana modes and demonstrate that to successfully observe the universal Majorana plateau $S={S}_{M}$ one should intelligently tune the tunneling phases instead of leaving them uncontrolled. Practical feasibility of appropriate Majorana entropy measurements is supported by an example with parameters well achievable in modern labs.

DOI
10.1103/physrevb.103.075440

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Measuring the Majorana entropy SM=kBlog(212){S}_{M}={k}_{B}log({2}^{\frac{1}{2}})SM​=kB​log(221​) (where logloglog is the natural logarithm) may uniquely reveal whether an initial equilibrium state of a nanoscale device has a Majorana nature, and subsequent operations deal with an essentially nonlocal pair of non-Abelian Majorana bound states and not with trivial or other accidental non-Abelian states. However, in realistic setups both Majorana modes are inevitably involved in tunneling processes. We show that even when the tunneling amplitude of one Majorana mode is significantly suppressed, the Majorana entropy is ruined, and straightforward experiments will, in general, detect entropy S\ensuremath≪SMS\ensuremath{\ll}{S}_{M}S\ensuremath≪SM​. To avoid this general problem we present a mechanism of the Majorana entropy revival via the tunneling phases of the Majorana modes and demonstrate that to successfully observe the universal Majorana plateau S=SMS={S}_{M}S=SM​ one should intelligently tune the tunneling phases instead of leaving them uncontrolled. Practical feasibility of appropriate Majorana entropy measurements is supported by an example with parameters well achievable in modern labs.
A pair of Majorana zero modes (MZMs) constitutes a nonlocal qubit whose entropy is log2. Upon strongly coupling one of the constituent MZMs to a reservoir with a continuous density of states, a universal entropy change of 1/2log2 is expected to be observed across an intermediate temperature plateau. We adapt the entropy-measurement scheme that was the basis of a recent experiment by Hartman et al. [Nat. Phys. 14, 1083 (2018)10.1038/s41567-018-0250-5] to the case of a proximitized topological system hosting MZMs and propose a method to measure this 1/2log2 entropy change-an unambiguous signature of the nonlocal nature of the topological state. This approach offers an experimental strategy to distinguish MZMs from non topological states.
The authors employ entanglement entropy as the means to reliably predict the presence of Majorana zero modes (MZMs) in experimentally relevant setups. Connecting with quantum transport calculations of the conductance spectra, they present ways to reliably signal the presence of topological MZMs that possess true nonlocal correlations, thereby distinguishing topological phase transitions from trivial ones. The detection of MZMs is a contentious issue, and the models presented here set the stage for looking beyond conductance spectra for their reliable identification.
In a recent breakthrough experiment [Nature (London) 614, 445 (2023)], signatures of Majorana zero modes have been observed in tunnel spectroscopy for a minimal Kitaev chain constructed from coupled quantum dots. However, as Ising anyons, Majoranas' most fundamental property of non-Abelian statistics is yet to be detected. Moreover, the minimal Kitaev chain is qualitatively different from topological superconductors in that it supports Majoranas only at a sweet spot. Therefore, it is not obvious whether non-Abelian characteristics such as braiding and fusion can be demonstrated in this platform with a reasonable level of robustness. In this work, we theoretically propose a protocol for detecting the Majorana fusion rules in an artificial Kitaev chain consisting of four quantum dots. In contrast with the previous proposals for semiconductor-superconductor hybrid nanowire platforms, here we do not rely on mesoscopic superconducting islands, which are difficult to implement in quantum dot chains. To show the robustness of the fusion protocol, we discuss the effects of three types of realistic imperfections on the fusion outcomes, e.g., diabatic errors, dephasing errors, and calibration errors. We also propose a fermion parity readout scheme using quantum capacitance. Our work will shed light on future experiments on detecting the non-Abelian properties of Majorana modes in a quantum dot chain.
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Majorana zero modes are emergent zero-energy quasiparticle excitations in certain superconducting systems that can be viewed as fractionalized or “half” electrons. These quasiparticles obey non-Abelian braiding statistics which is one manifestation of such half-electron character. Due to such non-Abelian braiding property, Majorana zero mode pairs hold promise as potential qubits for topological quantum computation. It is somewhat surprising that, at least theoretically, ordinary one-dimensional semiconductor systems can be induced to host such esoteric Majorana modes as edge states if some precise experimental conditions are satisfied. Because of the relative simplicity of material and experimental requirements to host Majorana modes, there has been a flurry of experimental effort to realize them in semiconductor nanowire systems. While experimental efforts have produced preliminary evidence for the presence of Majorana zero modes in these systems, a thorough confirmation is lacking. The experimental signature in question is the presence of a zero-bias conductance-peak that, while necessary, is not a sufficient criterion to establish the presence of underlying Majorana modes. Given the importance of Majorana braiding for topological quantum computation and skepticism over the presence of Majorana modes in these experimental systems, it would seem natural to attempt braiding these putative Majorana modes in the near future. In that case, an observation of non-Abelian statistics would provide the necessary and sufficient condition in favor of Majorana presence in the studied experimental systems. This thesis has three distinct parts. First, we assume perfect Majorana modes as given that can be successfully braided. In this case, we calculate the diabatic error due to the finite speed of braiding when the system is coupled to a Bosonic bath. Next, we grant that the mechanism for zero-bias conductance-peak is indeed topological, albeit the underlying Majorana modes may be imperfect (the modes are not precisely at zero energy). We study the interplay of dissipation and finite energy splitting of the Majorana modes and study its consequence regarding the probability of successful braiding. Lastly, we propose studying the correlation between independent left and right conductance measurements as a means to distinguish between a topological versus a non-topological the mechanism underlying the observed zero-bias conductance-peak.
Majorana zero modes (MZMs) are non-Abelian quasiparticles with the potential to serve as topological qubits for fault-tolerant quantum computing due to their ability to encode quantum information nonlocally. In multi-Majorana systems configured into two separated subsystems, nontrivial quantum correlations persist, but the presence of trivial Andreev bound states (ABSs) can obscure this nonlocality if MZM preparation fails. To address this, we propose a protocol using an entanglement witness based solely on parity measurements to distinguish the nonlocal characteristics of MZM systems. Our framework, which is experimentally implementable, achieves a detection probability of approximately 18% in a 6-site system and demonstrates robustness under environmental noise, albeit with a reduced detection rate in the resence of quasiparticle contamination.