Andreev bound states in a superconducting qubit at odd parity

TL;DR

This work analyzes how odd-parity Andreev bound states modify the Josephson quantum mechanics of a single Josephson junction with a purely capacitive environment. It develops a low-energy description by integrating out fermionic degrees of freedom, yielding a scalar phase-based eigenproblem that depends on the binding energy $\\Omega$ and the phase variable $\\varphi$. Across the Cooper-pair box and transmon regimes, the authors show that multiple discrete bound states can exist per channel, with energies controlled by $E_J^*$, $E_C$, and gate charge $\\mathcal{N}$, summarized by $N_{ m ch}=E_J^*/E_J$, and accessible via microwave spectroscopy. The results reveal a rich odd-parity bound-state spectrum that is markedly different from the even-parity case and offer a route to observe these states in forthcoming superconductor/semiconductor/superconductor junction experiments, where the bound states exhibit an $e$-periodic dispersion with gate charge.

Abstract

The quantum mechanics of the Josephson effect is the core ingredient for quantum technologies with superconducting circuits. A new avenue was recently opened in this field by predicting that the Josephson quantum mechanics in the odd parity sector, when a quasiparticle in trapped in an Andreev bound state, is fundamentally different from the conventional one in the even sector. The focus was then on a Josephson junction surrounded by an electromagnetic environment formed of a collection of bosonic modes, including the case of an ohmic environment. Here we consider the distinct case of a superconducting qubit made of a single Josephson junction whose environment reduces to a capacitance. We find a novel structure for the low-lying discrete states in the odd sector, which is altogether different from the one that appears in the even sector. Our study of the bound-state spectrum ranges from the Coulomb-dominated (Cooper pair box) to the Josephson-dominated (transmon) regime. Our prediction could be tested in forthcoming experiments with superconductor/semiconductor/superconductor junctions, which have been studied intensively in recent years, both using nanowires as well as two-dimensional electron gases.

Figures (3)

• Figure 1: The circuit consists of a superconducting island contacted to a superconducting lead through a tunnel junction (hatched region) characterized by Josephson energy $E_J^*$ and capacitance $C_J$. The external gate charge ${\@fontswitch\mathcal{N}}=-C_gV_g/e$ is controlled by the voltage $V_g$ applied to an electrostatic gate with capacitance $C_g$. The Coulomb energy of the island is $E_C=e^2/2C$ with $C=C_J+C_g$.
• Figure 2: Bound state energies in the odd sector of a transmon, $E^*_J\gg E_C$. Each channel of the junction accommodates a finite number of bound states whose energy ${\@fontswitch\mathcal{E}}_n=1-2\Omega_n/(\hbar \omega_0)$ is given as a function of $N_{\rm ch}=E_J^*/E_J$ (plain lines). The dashed and dotted lines give the asymptotes at $N_{\rm ch}\gg 1$ [Eq. \ref{['eq:asymptote1']}] and $0<N_{\rm ch}-1\ll 1$ [Eq. \ref{['eq:asymptote2']}], respectively.
• Figure 3: Bound state energies in the odd sector of a single-channel superconducting qubit as a function of $E_J^*/E_C$ at $N_{\rm ch}=1$ (plain line). The left panel zooms in the boxed region of the right panel to resolve the gate dependence of the spectrum between ${\@fontswitch\mathcal{N}}=0$ (plain line) and ${\@fontswitch\mathcal{N}}=1/2$ (dash-dotted line). The dotted lines give the asymptotes in a transmon at $E_J^*\gg E_C$ [Eq. \ref{['eq:asymptote3']}] and the dashed lines give the edge of the continuum spectrum at ${\@fontswitch\mathcal{N}}=0$ and ${\@fontswitch\mathcal{N}}=1/2$, respectively.