Revealing charge-tunneling processes between a quantum dot and a superconducting island through gate sensing

We report direct detection of charge tunneling between a quantum dot and a superconducting island through radio-frequency gate sensing. We are able to resolve spin-dependent quasiparticle tunneling as well as two-particle tunneling involving Cooper pairs. The quantum dot can act as an RF-only sensor to characterize the superconductor addition spectrum, enabling us to access subgap states without transport. Our results provide guidance for future dispersive parity measurements of Majorana modes, which can be realized by detecting the parity-dependent tunneling between dots and islands.

Figure 1

Experimental setup and sample characterization. (a) A gate sensor detects the tunneling of charges on/off the QD. Charge hybridization between QD and SC results in an anticrossing in the energy spectrum at zero detuning $\epsilon =0$ and shifts the $LC$ resonator, causing in a change in amplitude $A$ and phase $\varphi$ the response of a probe signal at the bare resonator frequency $f_0$. (b) False-colored electron micrograph of a nominally equivalent hybrid double dot. An additional dot, shown on the right of the device, is unused in this experiment. (c) Coulomb blockade measurement of the SC. Left: Device A, measured using RF reflectometry off the source (circuit not shown). The even-odd pattern indicates $E_0<E_{\mathrm{C}}^{\mathrm{S}}$. Right: Device B, measured using standard lock-in techniques. The doubling of the period at low bias $V_{\text{b}}$ illustrates that $E_0>E_{\mathrm{C}}^{\mathrm{S}}$. (d) Energy dispersion of the superconducting island for device A (left) and device B (right). The even (odd) energy levels are shown in blue (green). The odd parity sector consists of a discrete subgap state at $E_0$ and a continuum of states above $\mathrm{\Delta }$ (shaded).

Figure 2

Spin-dependent tunneling between a QD and a SC. (a), (b) Charge stability diagram of device A measured in phase (a) and amplitude (b). The charge states are labeled $\left(n^{\text{SC}},n^{\text{QD}}\right)$ with respect to the state $(N,M)$ with $N$ and $M$ even. Dashed pink lines: Expected locations of the lead-island transitions, in accordance with data obtained from a resonator connected to the SC gate, and simulations of the charge ground state of the system [28]. (c) Linecuts of the phase (green dots) and amplitude (blue dots) along two interdot transitions. Dashed line and left-pointing triangle marker, left panel: Transition between (0, 2) and (1, 1), representative of transitions between states of total even parity. Continuous line and right-pointing triangle marker, right panel: Transition between (1, 2) and (2, 1), representative of transitions between states of total odd parity. The lines are fits to a circuit QED model with $t_{\text{C}}^{\text{odd}}=t_{\text{C}}^{\text{even}}/\sqrt{2}$ [28]. We find a good fit with $t_{\text{C}}^{\text{even}}=20\mathrm{GHz}$ and $g_0=100\mathrm{MHz}$. (d) Full-frequency response of the resonator (symbols) together with fits (lines) obtained from a pair of interdot transitions that show the parity effect [outside the gate space shown in (a), (b)].

Figure 3

Cooper pair tunneling in a hybrid double dot. (a) Charge stability diagram measured in phase (left) and amplitude (middle) along with a simulation of the charge ground state (right) in the weakly coupled regime. The charge states are labeled $\left(n^{\text{SC}},n^{\text{QD}}\right)$ with respect to the state $(N,M)$ with $N$ even. Dashed pink lines: Locations of the transitions from the (0, 0) state as a guide to the eye. The gray scale in the simulation indicates the sum of the charge in the combined system. (b) Linecuts of the phase (green) and amplitude (blue) along the ($-2$, 0) to (0, $-1$) interdot transition. This transition involves a reservoir with a continuous spectrum, indicated by the shaded region above the lowest available energy state. The schematic shows how these states couple via crossed Andreev reflection. (c) Same as in (a) for the strongly coupled regime. Dashed pink lines: Locations of the lead transitions from the ($-2,-1$) and (0, $-1$) states as a guide to the eye. (d) Linecuts of the phase (green) and amplitude (blue) along the (2, $-2$) to (0, 0) interdot transition. These states couple via Cooper pair tunneling.

Figure 4

Spectroscopy of a subgap state in the floating regime. (a) Charge stability diagram measured in phase in device B. The diagonal lines indicate that the total charge in the system is conserved. (b) Calculated positions of the transitions in good agreement with the measured stability diagram. Inset: Energy spectrum with even states in black and odd states in green, showing that the even-odd pattern is caused by the parity effect even though $E_0>E_{\mathrm{C}}^{\mathrm{S}}$. (c) Temperature dependence of the even-odd pattern. (d) The evolution of the free energy difference with temperature (black dots), with a fit to the model described in Ref. [10] (green line). The free energy difference is extracted from the even-odd pattern via $F_{\text{o}}-F_{\text{e}}=\left(S_{\text{e}}-S_{\text{o}}\right)e{\alpha }_{\text{sc}}/4$ with ${\alpha }_{\text{sc}}=0.9$ the lever arm of the SC gate.