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arXiv:2004.02123 · Physical Review Research

Parity-to-charge conversion in Majorana qubit readout

Morten I. K. Munk, Jens Schulenborg, Reinhold Egger, Karsten Flensberg

Published Aug 17, 2020• 44 citations
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Abstract

The paper studies the readout processes of Majorana qubits using coupling to a quantum dot and a charge sensor. The dynamics of the system after switching on the measurement device is calculated by an effective Lindblad master equation going beyond the rotating-wave approximation.
DOI
10.1103/physrevresearch.2.033254

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Reading out the parity degree of freedom of Majorana bound states is key to demonstrating their non-Abelian exchange properties. Here, we present a low-energy model describing localized edge states in a two-arm device. We study parity-to-charge conversion based on coupling the superconductor bound states to a quantum dot whose charge is read out by a sensor. The dynamics of the system, including the readout device, is analyzed in full using a quantum-jump approach. We show how the resulting signal and signal-to-noise ratio differentiates between local Majorana and Andreev bound states.
We study theoretical models for charge reflectometry and capacitive readout of the Majorana parity degree of freedom in Majorana box qubits, considering decoherence channels within the framework of the Lindblad master equation. Noting that a parity-dependent dynamical susceptibility χ z ( ω ) governs both readout schemes, we provide a general expression for χ z ( ω ) which covers the full crossover from the resonant regime to the off-resonant dispersive regime. In addition, we reexamine previous results which were obtained under a semiclassical factorization assumption. Using three different error measures, we show that this approximation is quantitatively justified in the dispersive regime. In the resonant regime, however, we find deviations from exact reference data, obtained by numerical solution for the steady state of the full Lindblad equation. These deviations are typically of the order of a few percent in the considered error measures.
We theoretically study a scheme to distinguish the two ground states of a one-dimensional topological superconductor, which could serve as a basis for the readout of Majorana qubits. The scheme is based on parity-to-charge conversion, i.e., the ground-state parity of the superconductor is converted to the charge occupation on a tunnel-coupled auxiliary quantum dot. We describe how certain error mechanisms degrade the quality of the parity-to-charge conversion process. We consider (i) leakage due to a strong readout tunnel pulse, (ii) incomplete charge Rabi oscillations due to slow charge noise, and (iii) charge relaxation due to phonon emission and absorption. To describe these effects, we use simple model Hamiltonians based on the ideal Kitaev chain, and draw conclusions to generic one-dimensional topological superconductors wherever possible. In general, the effects of the error mechanisms can be minimized by choosing a smooth shape and an optimal strength for the readout tunnel pulse. In a case study based on InAs heterostructure device parameters, we estimate that the parity-to-charge conversion error is mainly due to slow charge noise for weak tunnel pulses and leakage for strong tunnel pulses.
We present a theoretical analysis of a Majorana-based qubit consisting of two topological superconducting islands connected via a Josephson junction. The qubit is operated by electrostatic gates which control the coupling of two of the four Majorana zero modes. At the end of the operation, readout is performed in the charge basis. Even though the operations are not topologically protected, the proposed experiment can potentially shed light on the coherence of the parity degree of freedom in Majorana devices and serve as a first step towards topological Majorana qubits. We discuss in detail the charge-stability diagram and its use for characterizing the parameters of the devices, including the overlap of the Majorana edge states. We describe the multilevel spectral properties of the system and present a detailed study of its controlled coherent oscillations, as well as decoherence resulting from coupling to a non-Markovian environment. In particular, we study a gate-controlled protocol where conversion between Coulomb-blockade and transmon regimes generates coherent oscillations of the qubit state due to the overlap of Majorana modes. We show that, in addition to fluctuations of the Majorana coupling, considerable measurement errors may be accumulated during the conversion intervals when electrostatic fluctuations in the superconducting islands are present. These results are also relevant for several proposed implementations of topological qubits which rely on readout based on charge detection.
A sensitive technique for the readout of the state of a qubit is based on the measurement of the conductance through a proximal sensor quantum dot (SQD). Here, we theoretically study the coherent backaction of such a measurement on a coupled SQD-charge-qubit system. We derive Markovian kinetic equations for the ensemble-averaged state of the SQD-qubit system, expressed in the coupled dynamics of two charge-state occupations of the SQD and two qubit isospin vectors, one for each SQD charge state. We find that aside from introducing dissipation, the detection also renormalizes the coherent evolution of the SQD-qubit system. Basically, if the electron on the detector has time to probe the qubit, then it also has time to fluctuate and thereby renormalize the system parameters. In particular, this induces torques on the qubit isospins, similar to the spin torque generated by the spintronic exchange field in noncollinear spin-valve structures. Second, we show that for a consistent description of the detection, one must also include the renormalization effects in the next-to-leading order in the electron tunneling rates, especially at the point of maximal sensitivity of the detector. Although we focus on a charge-qubit model, our findings are generic for qubit readout schemes that are based on spin-to-charge conversion using a quantum dot detector. Furthermore, our study of the stationary current through the SQD, a test measurement verifying that the qubit couples to the detector current, already reveals various significant effects of the isospin torques on the qubit. Our kinetic equations provide a starting point for further studies of the time evolution in charge-based qubit readout. Finally, we provide a rigorous sum rule that constrains such approximate descriptions of the qubit isospin dynamics and show that it is obeyed by our kinetic equations.