🔬 mesoscale physics

Phase engineering of odd parity state in a quantum dot Josephson junction

This paper demonstrates that the superconducting phase difference across a strongly coupled quantum dot Josephson junction can control the ground state fermion parity via phase-induced transitions near $\phi = \pi$ , offering a promising platform for exploring exotic phenomena like Majorana zero modes.

Original authors: Shohei Kobayashi, Tyler Lindemann, Sergei Gronin, Geoffrey Gardner, Peter Stano, Daniel Loss, Michael Manfra, Seigo Tarucha, Sadashige Matsuo

Published 2026-10-20

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CC BY 4.0

Original authors: Shohei Kobayashi, Tyler Lindemann, Sergei Gronin, Geoffrey Gardner, Peter Stano, Daniel Loss, Michael Manfra, Seigo Tarucha, Sadashige Matsuo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Superconductors are materials that conduct electricity with zero resistance, a property that allows them to carry current without losing any energy to heat. In many advanced experiments, scientists use these materials to create tiny circuits where electrons behave in strange, collective ways. One of the most fascinating behaviors occurs when a superconductor is connected to a quantum dot, a microscopic island so small that it can hold just a few electrons. In this setup, the electrons do not just sit still; they constantly exchange partners with the superconductor, forming special states of energy known as Andreev bound states. The most important feature of these states is a property called fermion parity, which simply describes whether the total number of electrons in the system is even or odd. While this sounds like a basic counting game, controlling whether the system prefers an even or odd number of electrons is a critical step toward building new types of quantum computers and exploring exotic physics that could revolutionize how we process information.

A team of researchers has now demonstrated a new way to control this even-or-odd state in a quantum dot Josephson junction, a device consisting of a quantum dot sandwiched between two superconducting leads. Instead of using the usual method of adjusting electrical voltages to change the electron count, the scientists discovered that they could flip the system's parity simply by changing the phase difference between the two superconducting leads. In superconductors, the phase is a property related to the wave-like nature of the electrons, and by threading a magnetic field through the loop formed by the device, the researchers could smoothly tune this phase. They found that when the phase difference reached a specific point, the energy levels of the electrons shifted in a way that forced the system to switch from an even-numbered ground state to an odd-numbered one. This transition was marked by the energy levels of the electrons crossing zero, a clear signal that the fundamental nature of the device had changed.

To observe this phenomenon, the team built a device using a thin layer of indium arsenide, a semiconductor material, covered with a film of aluminum. They used precise etching and electrical gates to carve out a tiny region where a quantum dot could form between two superconducting electrodes. By cooling the device to a temperature of 50 millikelvin, just above absolute zero, they ensured that thermal noise would not interfere with the delicate quantum effects. The researchers then measured the flow of electrons through the device while varying the magnetic field and the voltage applied to the gates. In their initial tests, they confirmed that the device was working as a strong-coupled quantum dot, meaning the electrons were interacting intensely with the superconducting leads. When they mapped out the energy levels of the electrons, they saw the expected oscillating patterns that change as the magnetic field alters the phase difference across the junction.

The key discovery came when the researchers adjusted the gate voltage to a specific setting and scanned the magnetic field again. They observed that at a phase difference near pi, the energy levels of the electrons crossed each other and inverted their order. This inversion is the signature of a parity transition, indicating that the ground state of the system had switched from even to odd. The researchers found that this odd-parity state existed only within a specific range of phase values, forming a loop-like region in their data where the system behaved differently than it did outside that range. This result was significant because, in the regime of strong coupling where the quantum dot is tightly connected to the superconductor, theory had previously suggested that the system would remain in an even state unless a magnetic field was applied to break the symmetry. However, by using the phase difference alone, the team showed that the odd state could be reached without needing to break the symmetry in the traditional way.

To understand exactly what was happening, the researchers compared their experimental data with a theoretical model that treated the quantum dot as a single energy level interacting with the superconductors. The model predicted that the energy of the system depends on the phase difference, and it showed that the ground state would indeed switch to an odd parity when the phase was close to pi, provided the coupling between the dot and the leads was strong enough. The experimental data matched this prediction remarkably well, confirming that the phase difference was the driving force behind the parity switch. The researchers also explored how an external magnetic field applied parallel to the device surface would affect this transition. They found that adding this magnetic field, which introduces a Zeeman energy that splits the energy levels of electrons with different spins, made the region of odd parity larger. As the magnetic field increased, the range of phase values over which the system remained in the odd state expanded, allowing for more robust control over the parity.

These findings suggest that arrays of such quantum dot Josephson junctions could serve as a powerful platform for engineering exotic quantum phenomena. By controlling the phase difference across each junction, scientists could potentially create a chain of devices that mimics the behavior of a topological superconductor, a state of matter that hosts Majorana zero modes. These modes are of great interest because they could be used to build qubits that are naturally protected from errors, a major hurdle in the development of practical quantum computers. The ability to toggle the parity of the ground state using only the superconducting phase, without relying solely on electrical gates or complex magnetic setups, offers a new degree of freedom for designing these circuits. The work demonstrates that even in the strong-coupling regime, where interactions are intense and complex, the fundamental parity of the system can be precisely manipulated, opening the door to exploring new frontiers in superconducting physics.

Technical Summary: Phase Engineering of Odd Parity State in a Quantum Dot Josephson Junction

Problem Statement
Controlling the fermion parity (even or odd) in quantum dot Josephson junctions (QDJJs) is a fundamental requirement for accessing novel superconducting phenomena, including the superconducting diode effect and the emulation of topological superconductivity (e.g., the Kitaev chain) for Majorana zero modes. While parity transitions are typically induced electrically by tuning gate voltages to adjust the quantum dot (QD) energy level ( $\epsilon$ ) or coupling strength ( $\Gamma$ ), achieving an odd-parity ground state in the strong-coupling regime ( $\Gamma > U$ , where $U$ is the charging energy) is challenging. In this regime, theoretical models suggest that without an external magnetic field, the ground state remains even. Furthermore, while phase-induced parity transitions have been predicted and observed in multiterminal junctions or transmon circuits, direct tunnel spectroscopy evidence of phase-controlled parity transitions in a strongly coupled QDJJ, and the specific role of the Zeeman effect in modifying these boundaries, requires further investigation.

Methodology
The authors fabricated QDJJ devices using an InAs quantum well covered by a thin epitaxial Al film. The device features a single QD formed between two superconducting (SC) leads by pinching off the 2D electron gas with a gate electrode ( $V_g$ ), while tunnel barriers are controlled by adjacent gates ( $V_{t1}, V_{t2}$ ). The system is embedded in a superconducting loop to control the phase difference ( $\phi$ ) across the junction via an out-of-plane magnetic field ( $B_\perp$ ). An in-plane magnetic field ( $B_\parallel$ ) is also applied to induce Zeeman splitting.

The study employs tunnel spectroscopy to measure the differential conductance ( $G$ ) as a function of bias voltage ( $V_{bias}$ ), gate voltage, and magnetic fields at a base temperature of 50 mK. The experimental procedure involves:

Device Characterization: Confirming QD formation and strong coupling to SC leads via Coulomb blockade measurements in the normal state ( $B_\perp > B_c$ ).
Phase Calibration: Mapping the Andreev bound state (ABS) spectrum against $B_\perp$ to calibrate the phase difference $\phi$ , identifying $\phi=0$ and $\phi=\pi$ based on the periodicity of the SC gap and ABS oscillations.
Parity Detection: Scanning $G$ versus $V_{bias}$ and $\phi$ (and later $B_\parallel$ ) to observe ABS energy crossings at zero bias. A parity transition is identified by the crossing and subsequent inversion of the lowest ABS pair energies.
Theoretical Modeling: The experimental data is compared against an effective Hamiltonian for a single-level QD proximitized by a superconductor in the large superconducting-gap limit. The model includes the QD energy ( $\epsilon$ ), charging energy ( $U$ ), coupling strength ( $\Gamma$ ), phase difference ( $\phi$ ), and Zeeman energy ( $E_Z$ ).

Key Results

Phase-Induced Parity Transition: The authors observed that varying the superconducting phase difference $\phi$ induces a transition in the ground-state parity. Specifically, as $\phi$ approaches $\pi$ , the lowest pair of Andreev bound states crosses zero energy. Between two critical phase values ( $\phi_{c1} < \phi < \phi_{c2}$ ), the ABS spectrum inverts, signaling a transition from an even-parity ground state to an odd-parity ground state. This loop-shaped transition boundary is centered around $\phi = \pi$ and specific gate voltages.
Strong-Coupling Regime: Unlike previous studies requiring magnetic fields to achieve odd parity in the strong-coupling regime, this work demonstrates that phase control alone is sufficient to stabilize the odd-parity state in a strongly coupled QDJJ ( $\Gamma > U$ ).
Zeeman Effect Enhancement: The application of an in-plane magnetic field ( $B_\parallel$ ) splits the degenerate odd-parity states via the Zeeman effect. Experimental data shows that increasing $B_\parallel$ linearly expands the phase interval ( $\phi_{c1} < \phi < \phi_{c2}$ ) over which the system maintains an odd-parity ground state.
Model Agreement: A simple theoretical model considering a single-level QD with finite charging energy and strong coupling reproduces the experimental observations. The model predicts that the odd-parity region exists when the ratio $\Gamma/U$ is below a phase-dependent threshold, which diverges as $\phi \to \pi$ . The inclusion of the Zeeman term in the model correctly predicts the linear expansion of the odd-parity region with increasing magnetic field.

Significance and Claims
The paper claims to demonstrate that an odd-parity ground state can be realized and controlled in a strongly coupled QDJJ through the superconducting phase difference, without relying solely on gate tuning or magnetic fields. The authors conclude that phase-controlled QDJJ arrays represent a promising platform for exploring exotic phenomena. Specifically, the ability to engineer the odd-parity state via phase control facilitates the study of the superconducting diode effect (which relies on odd-parity ground states) and the emulation of the Kitaev chain for topological superconductivity and Majorana zero modes. The work establishes phase engineering as a viable and complementary tool to electrical gating for manipulating fermion parity in hybrid superconductor-semiconductor devices.

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