Microwave sensing of Andreev bound states in a gate-defined superconducting quantum point contact

We use a superconducting microresonator as a cavity to sense absorption of microwaves by a superconducting quantum point contact defined by surface gates over a proximitized two-dimensional electron gas. Renormalization of the cavity frequency with phase difference across the point contact is consistent with coupling to Andreev bound states. Near π phase difference, we observe random fluctuations in absorption with gate voltage, related to quantum interference-induced modulations in the electron transmission. Close to pinch-off, we identify features consistent with the presence of a single Andreev bound state and describe the Andreev-cavity interaction using a Jaynes-Cummings model. By fitting the weak Andreev-cavity coupling, we extract ∼GHz decoherence consistent with charge noise and the transmission dispersion associated with a localized state.

Figure 1

(a) Stack structure of an InAs 2DEG with split gates (red) used to define a SQPC. ABSs comprise right–left- (red) and left–right- (blue) propagating electron–hole pairs Andreev reflected at the interface. (b) Differential resistance $R=dV/d{I}_{SD}$ of a SQPC as a function of $V_{QPC}$ and $I_{SD}$. Regions: open (I), definition (II), depletion (III), and tunneling (IV). Lower panel shows $R$ over a narrower range of $I_{SD}$ in region IV, with $I_c$ resonances approaching $\sim 45\mathrm{nA}$ indicated by arrows. (c) Schematic ABS energy spectrum of the junction reconstructed as a function of phase difference $\delta ={\theta }_2-{\theta }_1$ (lower panels) and $V_{QPC}$ at $\delta =\pi$ (upper). In each region, the energy detuning between bare cavity photons ($f_R$) and ABS transitions ($f_A=2E_A/h$) determines the effective shift in the observed cavity frequency.

Figure 2

(a) Optical and scanning electron micrographs of the device. A serpentine microwave cavity (blue) is inductively coupled to a loop of Al/InAs (orange) interrupted by a SQPC (red). (b) Feedline transmission coefficient S21 as a function of feedline frequency $f$ and flux $\mathrm{\Phi }$. $f_0\sim 6.163\mathrm{GHz}$ is the observed resonance frequency at zero flux at zero gate voltage. (c) Resonant frequency shift extracted from (b). (d) S21 as a function of frequency and $V_{QPC}$ at $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.5$. (e) Number of cavity fluctuations per volt (black points) and resonant frequency (red line) extracted from (d).

Figure 3

(a) S21 as a function of $f$ and ${\mathrm{\Phi }}^{\prime }$ for $V_{QPC}$ values indicated by the dashed lines in Fig. 2. Measured and simulated S21 as a function of (b), (c) ${\mathrm{\Phi }}^{\prime }$ and (d), (e) $V_{QPC}$ and readout difference frequency. Lower panels show the transition frequency for three Andreev channels used in the simulations.${\tilde{V}}_{QPC}$ is the simulation sweep parameter scaled to the same range as $V_{QPC}$.

Figure 4

(a) S21 measured as a function of frequency and ${\mathrm{V}}_{QPC}$ at the paired crossings highlighted by the dashed box in Fig. 3. (b) Schematic of a quantum dot formed in the junction, coupled to the leads with tunneling rates ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$. The energy-level sketch (middle panel) shows a quantum-dot state with energy detuning $\varepsilon$ from the chemical potential $\mu$ modulated by ${\mathrm{V}}_{QPC}$. (Lower panel) Plot of Breit-Wigner transmission (${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=\mathrm{\Gamma }$) and ABS frequency. (c) Simulated S21 as a function of ${\tilde{\mathrm{V}}}_{QPC}$ using the $f_A\left(\varepsilon \right)$ shown in lower inset.

Figure 5

(a) S21 measured as a function of $\mathrm{\Delta }f$ and $\delta$ in the single-channel regime ($V_{QPC}\sim -6\mathrm{V}$). (b) Shift in cavity frequency extracted from the S21 data in (a) and plotted as a function of $\delta$ (green points). Fit to the experimental data using the JC model (orange dashed) and dispersive shift (blue dashed). (c) S21 measured as a function of frequency and $V_{QPC}$ at $\delta =\pi$. (d) Shift in cavity frequency shift with $V_{QPC}$ (green points), JC model (orange dashed), and dispersive shift (blue dashed). Insets in (b) and (d) illustrate the ABS-cavity hybridization.

Figure 6

Mode transmissions $\left\{{\tau }_i\right\}$ for the (a) hardwall (HW) and (b) parabolic (PB) potential as a function of QPC potential ${\mu }_{QPC}$. (c) The number of highly transmitting channels, defined with ${\tau }_i>0.5$, showing a progressive decrease with increasing ${\mu }_{QPC}$ as each channel is backscattered.

Figure 7

(a) Plot showing the Andreev transition frequency as a function of phase difference for HW (blue solid) and PB (red dashed) confinement at ${\mu }_{QPC}=4\mu \mathrm{eV}$. (b) Shift in cavity frequency as a function of phase difference for HW (blue solid) and PB (orange dashed), and the experimental data (black crosses).

Figure 8

(a) Plot showing the number of transmitting channels as a function of QPC potential for HW and PB confinement. Also shown is the transmission ${\tau }_{\max }$ of the lowest-energy mode. (b) Calculated frequency shift of the cavity as a function of QPC potential parameter ${\mu }_{QPC}$. (c) Experimental frequency shift with gate voltage $V_{QPC}$ along with a running average to show the overall frequency shift excluding fluctuations. Regions II–IV are indicative only and correspond to those indicated in Fig. 1.