Effect of interatomic repulsion and quasi-degenerate states on a Kitaev-transmon qubit based on double quantum dots

TL;DR

This paper addresses the robustness of Majorana-like PMMs in a Kitaev-transmon built from two spinless double quantum dots (DQDs) in the presence of interdot Coulomb repulsion $V$ and previously neglected states. It develops analytic expressions for the DQD ground states in parity sectors, derives sweet-spot conditions that yield degenerate ground states, and extends the analysis to two DQDs connected by a Josephson junction, revealing degeneracy of subspaces when $E_{oo}=E_{ee}$ and $\phi_{ext}=0$. The full Kitaev-transmon is then analyzed by including charging and Josephson terms, showing that the Hilbert space splits into two decoupled parity subspaces that become spectrally identical at the sweet spots, leading to doubly degenerate eigenstates; the microwave spectrum is computed and shown to reflect these degeneracies and their crossings. The findings demonstrate the resilience (within parameter ranges) of PMMs under $V$-induced renormalizations and provide spectroscopic signatures to identify sweet spots, with implications for designing robust topological qubits in superconducting circuits.

Abstract

We investigate the effect of interatomic Coulomb repulsion $V$ and particular states disregarded previously on the Kitaev-transmon system proposed by Pino \textit{et al.} \cite{pino} which consists of a Josephson junction between two double quantum dots (DQDs) modeled by the spinless Kitaev Hamiltonian. For an isolated DQD, we demonstrate that a ``sweet spot'' hosting ``poor man's Majorana'' states persist in the presence of $V$, provided that system parameters are appropriately tuned. For the full system, we demonstrate that at the sweet spots of both DQDs, all eigenstates are doubly degenerate. This degeneracy arises from the existence of an operator that maps between two decoupled Hilbert subspaces. Away from the sweet spots, the microwave spectrum becomes sensitive to the choice of initial state. In our study, we consider transitions from the ground state (which depending on the flux alternates between the above mentioned subspaces) to all possible excited states. This scenario corresponds to a system initially in thermal equilibrium at low temperature.

Figures (5)

• Figure 1: Scheme of the LEFT and RIGHT DQDs with CAR (ECT) amplitudes $\Delta$ ($t$) coupled through a Josephson junction with energy $t_J$. The chemical potential of each dot is denoted by $\mu_{\nu j}$.
• Figure 2: Scheme of the Kitaev-transmon qubit. A capacitance with energy $E_C$ and a Josephson junction with an energy $E_J$ are added to the two DQDs closing the circuit.
• Figure 3: Energies in units of the plasma frequency $\omega_{pl}=\sqrt{8 E_J E_C}$, as a function of the offset charge for $\phi _{ext}=0$ and $E_{ee}=E_{oo}$. Parameters are $t_J=E_J=E_c=1$, $\mu_1=\mu_2=V/2$, $E_O=-t-V/2=E_{oo}/2=1$, and $E_E=-\Delta =E_{ee}/2$.
• Figure 4: Energies as a function of the offset charge for $\phi _{ext}=0$ and $E_{ee}=E_{oo}/2$ (top) and $E_{ee}=3E_{oo}/2$ (bottom). Other parameters as in Fig. \ref{['sweet']}
• Figure 5: Microwave spectrum for $\phi _{ext}=0$ and $E_{ee}=E_{oo}/2$ (top) and $E_{ee}=3E_{oo}/2$ (bottom). The vertical lines indicate crossings of the ground state. Other parameters as in Fig. \ref{['sweet']}.