Transmission lines and resonators based on quantum Hall plasmonics: Electromagnetic field, attenuation, and coupling to qubits

Abstract

Quantum Hall edge states have some characteristic features that can prove useful to measure and control solid state qubits. For example, their high voltage to current ratio and their dissipationless nature can be exploited to manufacture low-loss microwave transmission lines and resonators with a characteristic impedance of the order of the quantum of resistance h/e 2 ∼ 25kΩ. The high value of the impedance guarantees that the voltage per photon is high and for this reason high impedance resonators can be exploited to obtain larger values of coupling to systems with a small charge dipole, e.g. spin qubits. In this paper, we provide a microscopic analysis of the physics of quantum Hall effect devices capacitively coupled to external electrodes. The electrical current in these devices is carried by edge magnetoplasmonic excitations and by using a semiclassical model, valid for a wide range of quantum Hall materials, we discuss the spatial profile of the electromagnetic field in a variety of situations of interest. Also, we perform a numerical analysis to estimate the lifetime of these excitations and, from the numerics, we extrapolate a simple fitting formula which quantifies the Q factor in quantum Hall resonators. We then explore the possibility of reaching the strong photonqubit coupling regime, where the strength of the interaction is higher than the losses in the system. We compute the Coulomb coupling strength between the edge magnetoplasmons and singlet-triplet qubits, and we obtain values of the coupling parameter of the order 100MHz; comparing these values to the estimated attenuation in the resonator, we find that for realistic qubit designs the coupling can indeed be strong. * bosco@physik.rwth-aachen.de † d.divincenzo@fz-juelich.de A. Far-field analysis 1. EMPs in the half-plane

Transmission Lines and Metamaterials Based on Quantum Hall Plasmonics

“…A detailed analysis of dissipation in these devices and a quantitative analysis of the coupling to semiconductor qubits are issues that are not addressed here. More insights into these aspects can be found in [26].…”

“…If we use a piecewise approximation for the velocity and assume a symmetric configuration, we can divide each droplet into two propagation regions T, B of length L p at the top and bottom of the droplet, where the excitation moves at a constant velocity v 0 and two coupling regions R, L of length a at the right and left of the droplet, characterized by intra-and inter-edge velocities, v F and v 1 respectively. Using the results from [26] and rearranging the coordinate system to have the same (clockwise) direction in each droplet and the same origin (fixed conventionally at the boundary between region T and L), it is quite simple to find that the general solutions for the differential equations in the four regions of the ith droplet are…”

“…iωρ(y, ω) = ∂ y (v(y)ρ(y, ω)) + σ xy ∂ y V a (y, ω), (1) where V a (y, ω) is the time dependent drive applied, σ xy is the off-diagonal conductivity of the QH material and v is the propagation velocity of the EMP as function of position y, possibly accounting for different screening of Coulomb interactions due to the presence of the electrodes. A detailed discussion of the validity of this model can be found in [16,26].…”

“…The total current flowing into the ith electrode is obtained by integrating the displacement current density over its area. In the model presented here, we neglect fringing fields and, because the EMP charge density is assumed to be localized in an infinitesimally narrow stripe along the edge, the integral over the area of the electrode simplifies into the one dimensional integral [16,26]:…”

“…Note that in the setup chosen, all parts of the perimeter of the droplet are coupled to some external electrode. This choice guarantees that the EMP velocity has no logarithmic singularity in the long-wavelength limit [11], and, when d i /L i 1, it allows one to use a simple piecewise decomposition for the velocity function v(y), with constant velocity v i in the region coupled to the ith electrode [26]. The velocity v i can then be estimated by using…”