Microwave Andreev bound state spectroscopy in a semiconductor-based Planar Josephson junction

TL;DR

Together, the paper demonstrates microwave spectroscopy of Andreev bound states (ABS) in a wide planar Al–InAs Josephson junction embedded in a superconducting circuit. By coupling the ABS to a superconducting resonator and using two-tone spectroscopy, the authors map gate- and flux-tunable ABS spectra, observing near-unity transparency in a gate-defined constriction and clear avoided crossings indicative of resonator–ABS photon exchange. Tight-binding Bogoliubov–de Gennes calculations with Rashba spin-orbit coupling and a Jaynes–Cummings model explain the non-monotonic ABS evolution with confinement and the resonator response, linking microscopic parameters to the observed spectra. The results provide a high-resolution probe of ABS in superconductor–semiconductor hybrids with implications for Andreev-spin qubits and emergent topological superconductivity in hybrid devices.

Abstract

By coupling a semiconductor-based planar Josephson junction to a superconducting resonator, we investigate the Andreev bound states in the junction using dispersive readout techniques. Using electrostatic gating to create a narrow constriction in the junction, our measurements unveil a strong coupling interaction between the resonator and the Andreev bound states. This enables the mapping of isolated tunable Andreev bound states, with an observed transparency of up to 99.94\% along with an average induced superconducting gap of $\sim 150 μ$eV. Exploring the gate parameter space further elucidates a non-monotonic evolution of multiple Andreev bound states with varying gate voltage. Complimentary tight-binding calculations of an Al-InAs planar Josephson junction with strong Rashba spin-orbit coupling provide insight into possible mechanisms responsible for such behavior. Our findings highlight the subtleties of the Andreev spectrum of Josephson junctions fabricated on superconductor-semiconductor heterostructures and offering potential applications in probing topological states in these hybrid platforms.

Figures (16)

• Figure 1: (a) Schematic of the device design where a superconducting loop with an Al-InAs junction is inductively coupled to a coplanar waveguide resonator. (b) A gated Al-InAs planar Josephson junction embedded in a superconducting loop. We confine the width of the loop in the bottom side of the superconducting loop to limit the amount of supercurrent flowing in the loop. The superconducting loop length, width and loop-resonator separation are designed to be $L_{l} = 300µ m$, $W_{l} = 40µ m$ and $d = 5µ m$, respectively. (c), (d) Split gate designed to electrostatically define and tune a narrow constriction in the junction. The microwave circuit made of Nb is presented in grey, Al forming the superconducting loop and the junction electrodes in light blue, junction gap in green and the Nb used for the split gate in orange. (e) Energy of the positive (blue) and negative (green) branches of an ABS corresponding to a mode with $\Delta=150µ eV$ and $\tau = 0.98$. A transition between the negative and positive branches, represented with a purple arrow, requires an energy $hf_{A}(\phi,\tau)$. The energy space highlighted in red represents the accessible drive frequency range limited by the range of our microwave generator. (f) The excitation spectrum represented as the transition frequency $f_{A}$ corresponding to a transition between the negative and positive branches of an Andreev bound state with different transparencies and $\Delta=150µ eV$ as a function of applied flux $\Phi/\Phi_{0}$ where $\Phi/\Phi_{0} = \phi/2$. The grey dashed line correspond to the resonant frequency of the resonator $f_{r}$. The intersection of $f_{A}$ for a near-unity transparency mode with $f_{r}$ is represented by orange circles.
• Figure 2: Resonator response to gate-tunability of the junction shown as the magnitude of the transmission coefficient $|S_{21}|$ as a function of the readout frequency $f$ and applied gate voltage $V_{g}$ at (a)$\Phi/\Phi_{0} = 0.0$ and (b)$\Phi/\Phi_{0} = 0.5$. Inset: Extracted change in resonant frequency $\Delta f_{r} = f_{r}(V_{g}) - f_{r}(V_{g} = 0V)$ (blue) and internal quality factor $Q_{int}$ (red) as a function of $V_{g}$. (c)-(h) Flux modulation of the resonant frequency at different $V_{g}$.
• Figure 3: (a) Shift in resonant frequency normalized by the coupling rate $g_{A}$ as a function of $\Phi/\Phi_{0}$ for two cases: 1) a mode with transparency $\tau = 0.993$ corresponding to $f_{A}>f_{r}$ for all values of $\Phi/\Phi_{0}$ and 2) a mode with $\tau = 0.999$ corresponding $f_{A}<f_{r}$ for a range of values of $\Phi/\Phi_{0}$. The superconducting induced gap is taken to be $\Delta = 150µ eV$ and the resonant frequency to be $f_{r} = 4.777G Hz$. (b)-(l) Transmission coefficient $|S_{21}|$ as a function of the readout frequency $f$ and applied flux $\Phi$ around $\Phi/\Phi_{0} = 0.5$ for different $V_{g}$ values.
• Figure 4: Transmission coefficient $|S_{21}|$ at a set readout frequency slightly offset $f_{r}$ as a function of drive tone frequency $f_{d}$ and applied flux $\Phi/\Phi_{0}$ for specific $V_{g}$ values where a single isolated ABS is observed. Fits of the parabola to $f_{A} = 2|E_{A}|/h$, where $E_{A}$ follows \ref{['eq:abs_pm']}, are shown as dashed lines along with the extracted values for the superconducting gap $\Delta$ and the transparency $\tau$.
• Figure 5: Positive (blue) and negative (green) energy branches of two Andreev bound states with $\Delta=150µ eV$ and $\tau = 0.98$ (dark) and $\tau = 0.88$ (light). The allowed transitions between the modes are represented by purple arrows. (b) Excitation spectra of $f_{A}$ corresponding to each transition outlined in (a). (c)-(i)$|S_{21}|$ as a function of drive tone frequency $f_{d}$ and flux $\Phi/\Phi_{0}$ for specific $V_{g}$ values where several transitions are observed.