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arXiv:2507.12150 · Physical Review Letters

Local Control of Parity and Charge in Nanoscale Superconducting Lead Islands

Stefano Trivini, Jon Ortuzar, Katerina Vaxevani, Beatriz Viña-Bausá, F. S. Bergeret, José Ignacio Pascual

Published Nov 24, 2025• 1 citation
arXivHTMLPDF

Abstract

Small superconducting islands can exhibit charge quantization, where Coulomb interactions compete with Cooper pairing. Using scanning tunneling spectroscopy, we probe this interplay by measuring the charging energy (E_{C}) and the pairing energy (Δ) of individual Pb nanoislands. Below a critical island size, where E_{C}>Δ, we observe a crossover between even and odd parity ground states. By applying controlled voltage pulses, we continuously tune the island's electrostatic potential and map the full charge-parity landscape. These results demonstrate tunable superconducting ground states, offering a potential platform for qubit design and control.

DOI
10.1103/vkwr-m87v

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Small superconducting islands can exhibit charge quantization, where Coulomb interactions compete with Cooper pairing. Using scanning tunneling spectroscopy, we probe this interplay by measuring the charging energy (E_{C}) and the pairing energy (Δ) of individual Pb nanoislands. Below a critical island size, where E_{C}>Δ, we observe a crossover between even and odd parity ground states. By applying controlled voltage pulses, we continuously tune the island's electrostatic potential and map the full charge-parity landscape. These results demonstrate tunable superconducting ground states, offering a potential platform for qubit design and control.
The Cooper-pair transistor (CPT), a small superconducting island enclosed between two Josephson weak links, is the atomic building block of various superconducting quantum circuits. Utilizing gate-tunable semiconductor channels as weak links, the energy scale associated with the Josephson tunneling can be changed with respect to the charging energy of the island, tuning the extent of its charge fluctuations. Here, we directly demonstrate this control by mapping the energy level structure of a CPT made of an indium arsenide nanowire with a superconducting aluminum shell. We extract the device parameters based on the exhaustive modeling of the quantum dynamics of the phase-biased nanowire CPT and directly measure the even-odd parity occupation ratio as a function of the device temperature, relevant for superconducting and prospective topological qubits.
The number of electrons in small metallic or semiconducting islands is quantised. When tunnelling is enabled via opaque barriers this number can change by an integer. In superconductors the addition is in units of two electron charges (2e), reflecting that the Cooper pair condensate must have an even parity. This ground state (GS) is foundational for all superconducting qubit devices. Here, we study a hybrid superconducting-semiconducting island and find three typical GS evolutions in a parallel magnetic field: a robust 2e-periodic even-parity GS, a transition to a 2e-periodic odd-parity GS, and a transition from a 2e- to a 1e-periodic GS. The 2e-periodic odd-parity GS persistent in gate-voltage occurs when a spin-resolved subgap state crosses zero energy. For our 1e-periodic GSs we explicitly show the origin being a single zero-energy state gapped from the continuum, i.e., compatible with an Andreev bound states stabilized at zero energy or the presence of Majorana zero modes.
In a system where two small-area (100 nm x 100 nm) superconducting tunnel junctions are connected in series, the Josephson effect which tends to impose the phase competes with the charging effects in the central electrode (the of the device) which tend to set the charge of the island. By applying a voltage on an electrode capacitively coupled to the island (the electrode) one can arbitrate this competition in favour of either effect. This is can be seen in particular on the supercurrent which can flow through the junctions : it is modulated periodically by the gate voltage (hence the name of given to the system). This modulation shows sharp peaks when two charge states of the island are degenerate with respect to the electrostatic energy. The system is then in a state which has no classical description : it is in a coherent superposition of states. In this thesis we demonstrate experimentally the behaviour of this transistor and we analyse the theoretical and practical problems which can prevent the observation of the coherent superposition of states through the modulation of the supercurrent. In particular we show that the parity of the electron number in the island must be rigorously controlled and that the impedance connected to the circuit plays a crucial role in setting the magnitude of the measured supercurrent.
Superconducting quantum circuits derive their nonlinearity from the Josephson energy-phase relation. Besides the fundamental cos⁡φ\cosφcosφ term, this relation can also contain higher Fourier harmonics cos⁡(kφ)\cos(kφ)cos(kφ) corresponding to correlated tunneling of kkk Cooper pairs. The parity of the dominant tunneling process, i.e.~whether an odd or even number of Cooper pairs tunnel, results in qualitatively different properties, and controlling this opens up a wide range of applications in superconducting technology. However, access to even-dominated regimes has remained challenging and has so far relied on complex multi-junction or all-hybrid architectures. Here, we demonstrate a simple "harmonic parity qubit" (HPQ); an element that combines two aluminum-oxide tunnel junctions in parallel to a gate-tunable InAs/Al nanowire junction forming a SQUID, and use spectroscopy versus flux to reconstruct its energy-phase relation at 85 gate voltage points. At half flux quantum, the odd harmonics of the Josephson potential can be suppressed by up to two orders of magnitude relative to the even harmonics, producing a double-well potential dominated by even harmonics with minima near ±π/2\pmπ/2±π/2. The ability to control harmonic parity enables supercurrent carried by pairs of Cooper pairs and provides a new building block for Fourier engineering in superconducting circuits.
After a brief review of recent proposals for using small superconductors whose charging energy is comparable to their superconducting gap as physical qubits in quantum computation, the ‘Cooper-pair box’ experiments of Nakamura et al. are analysed. In these, the gate voltage on a small superconducting island is perturbed by a pulse of height δV and duration Δt and coherent oscillations between the charge states with n̄ and n̄ +1 Cooper pairs are detected. The effects of noise in δV and Δt on their measurements is studied and it is found that the two kinds of noise have qualitatively different signatures in the experimental data.
The work presented in this thesis investigates transitions between quantum states in superconducting double dots (SDDs), a nanoscale device consisting of two aluminium superconducting islands coupled together by a Josephson junction, with each dot connected to a normal state lead. The energy landscape consists of a two level manifold of even charge parity Cooper pair states, and continuous bands corresponding to charge states with single quasiparticles in one or both islands. These devices are fabricated using shadow mask evaporation, and are measured at sub Kelvin temperatures using a dilution refrigerator. We use radio frequency reflectometry to measure quantum capacitance, which is dependent on the quantum state of the device. We measure the quantum capacitance as a function of gate voltage, and observe capacitance maxima corresponding to the Josephson coupling between even parity states. We also perform charge sensing and detect odd parity states. These measurements support the theoretical model of the energy landscape of the SDD. By measuring the quantum capacitance in the time domain, we observe random switching of capacitance between two levels. We determine this to be the stochastic breaking and recombination of single Cooper pairs. By carrying out spectroscopy of the bath responsible for the pair breaking we attribute it to black-body radiation in the cryogenic environment. We also drive the breaking process with a continuous microwave signal, and find that the rate is linearly proportional to incident power. This suggests that a single photon process is responsible, and demonstrates the potential of the SDD as a single photon microwave detector. We investigate this mechanism further, and design an experiment in which the breaking rate is enhanced when the SDD is in the antisymmetric state rather than the symmetric state. We also measure the quantum capacitance of a charge isolated double dot. We observe 2e periodicity, indicating the tunnelling of Cooper pairs and the lack of occupation of quasiparticle states. This work is relevant to the range of experiments investigating the effect of non-equilibrium quasiparticles on the operation of superconducting qubits and other superconducting devices.