Fermion parity measurement and control in Majorana circuit quantum electrodynamics

We investigate the quantum electrodynamics of a device based on a topological superconducting circuit embedded in a microwave resonator. The device stores its quantum information in coherent superpositions of fermion parity states originating from Majorana fermion hybridization. This generates a highly isolated qubit whose coherence time could be greatly enhanced. We extend the conventional semiclassical method and obtain analytical derivations for strong transmon-photon coupling. Using this formalism, we develop protocols to initialize, control, and measure the parity states. We show that, remarkably, the parity eigenvalue can be detected via dispersive shifts of the optical cavity in the strong-coupling regime and its state can be coherently manipulated via a second-order sideband transition.

Figure 1

Majorana-transmon circuit in a cavity. A topological superconductor (orange) bridges a Josephson junction nucleating Majorana fermions (yellow). Control gates (light blue and green) are used to drive the topological state and control the Majorana coupling. Together with its superconducting islands, this device is embedded inside a coplanar transmission-line resonator. Microwave pulses on the input port allow logical gates realizations, while the output port can be used for homodyne or heterodyne detection of the fermion parity state.

Figure 2

Comparison of the WKB result for the dipole matrix elements with numerics. Dipole transition strength is plotted vs $E_J/E_C$ for $n_g=0.2$ and $E_M=0.01E_C$. The ${\mathcal{G}}_o$ coupling corresponds to transitions which conserve $s=\pm$ of Eq. (3), while the ${\mathcal{G}}_x$ coupling describes transitions which flip $s$ [see also Fig. 3]. Within the transmon range, the anharmonic approximation is more accurate but should eventually coincide with the harmonic approximation for $E_J\to \infty$.

Figure 3

Qubit initialization and coherent control in the strong-dispersive regime. (a) Qubit initialization: Numerical calculation of the four levels population during the cooling process. A microwave pump with amplitude ${\xi }_p/h={10}^{-3}$ GHz and frequency $\omega /2\pi =5.34\mathrm{GHz}$ is admitted to the system with a preliminary temperature of $0.03\mathrm{K}$. We take the decay rate as $\gamma /2\pi ={10}^{-5}\mathrm{GHz}$. $E_M=0.025E_C, E_J=25E_C, E_C/h=0.4\mathrm{GHz}, n_g=0, g/\mathrm{\Delta }=0.3$. (b) Coherent control: The system prepared in its ground state and driven via two coherent microwave tones using a rectangular pulse with amplitude $\xi /h=8\times {10}^{-4}$ GHz and detuning $\delta =E_M$, resulting in transfer between the lower levels with a minimal higher-level population. (c) Level diagram: The qubit-cavity interaction (left) produces a shifted resonance frequency ${\omega }_c^{\prime }={\omega }_c+{\chi }_T$ accompanied by a parity-based shift $\pm {\chi }_M$. A process of coherent control (middle) in the double $\mathrm{\Lambda }$ system is preceded by the cooling process (right).