🔬 mesoscale physics

Fermion parity of an Andreev molecule probed by nonlocal Josephson effect

This paper demonstrates that the global fermion parity of a delocalized superconducting state in a carbon nanotube-based Andreev molecule can be locally detected through characteristic $\pi$ -phase shifts in the nonlocal Josephson response, establishing parity as an experimentally accessible degree of freedom in hybrid superconducting circuits.

Original authors: S. Annabi, H. Riechert, K. Watanabe, T. Taniguchi, J. Griesmar, E. Arrighi, L. Bretheau, J. -D. Pillet

Published 2026-07-20

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Original authors: S. Annabi, H. Riechert, K. Watanabe, T. Taniguchi, J. Griesmar, E. Arrighi, L. Bretheau, J. -D. Pillet

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

Imagine the world of electricity not as a steady stream of water, but as a chaotic dance floor where electrons, the tiny particles carrying charge, usually bump into each other and lose their rhythm. In most materials, this chaos creates resistance, like trying to run through a crowded hallway. But in a special state of matter called a superconductor, these electrons pair up and glide together in perfect unison, a dance so smooth it encounters zero resistance. This is the realm of quantum physics, where the rules get weird and particles can act like waves.

One of the most fascinating things about these superconducting dances is a hidden "secret code" called fermion parity. Think of it like a cosmic coin flip: the total number of electrons in a superconducting pair is either even or odd. This simple fact—whether the count is even or odd—determines the entire personality of the supercurrent, deciding which direction it flows and how strong it is. Scientists have long known how to check this coin flip in a single, isolated spot. But what happens when you link two of these spots together? Does the secret code stay local, or does it become a shared, global property of the whole system? This is the big question that drives the research in this paper, because understanding how to read and control this global code is a crucial step toward building future quantum computers that can store information in these delicate, dancing pairs.

The researchers in this paper decided to build a tiny, custom-made playground to test this idea, creating what they call an "Andreev molecule." Imagine two quantum dots (which are like tiny, isolated cages for electrons) connected by a superconducting bridge. Usually, if you put a supercurrent through one cage, it stays there. But in this setup, the two cages are so close and so well-connected that their electron waves overlap and merge, forming a single, delocalized "molecule" that stretches across both junctions. The team used a single carbon nanotube—a microscopic tube made of carbon atoms, thinner than a human hair—as the highway for these electrons, connecting three superconducting electrodes. By carefully tuning the voltage on tiny gates, they could control the electrons in the left and right cages independently, essentially acting as conductors for this quantum orchestra.

What they discovered was a spectacular "nonlocal Josephson effect." In plain terms, this means they could control the supercurrent flowing through the left side of the device simply by changing the magnetic phase on the right side, even though the two sides were physically separated. It's as if turning a knob on the right side of a room instantly changed the brightness of a lightbulb on the left, without any wires connecting them directly. They found that this nonlocal connection was strongest exactly when the electron wavefunction was most "delocalized," meaning the electrons were truly sharing the space between both junctions, confirming that a genuine molecular state had formed.

But the real magic happened when they looked at the "secret code" of the system: the fermion parity. The team observed that whenever the total number of electrons in the molecule changed from even to odd (or vice versa), the nonlocal signal didn't just get stronger or weaker; it flipped completely. The current that was flowing "forward" suddenly started flowing "backward," a shift they describe as a $\pi$ -phase shift. It's like a dance that suddenly reverses direction the moment a new dancer joins the floor. By comparing their measurements to a theoretical model, they confirmed that these flips were directly tied to the global parity of the entire molecule.

This paper doesn't just suggest these things; they measured them directly. Using two different device setups—one with a global control knob and another with precise local controls—they observed these phase flips repeatedly. They ruled out the idea that this was just a random glitch or a simple local effect, showing instead that the global parity of the delocalized state is the key driver. The results are robust, supported by both experimental data and simulations that matched their findings perfectly.

In the end, this work shows that we can now "listen" to the global parity of a complex superconducting system just by looking at the current in one part of it. It proves that the mysterious "even or odd" nature of electrons isn't just a local quirk but a powerful, controllable property that can be read out non-locally. This opens the door to using these Andreev molecules as building blocks for more complex quantum networks, where information could be stored in these shared, delocalized states, potentially leading to more stable and powerful quantum technologies.

Technical Summary: Fermion Parity of an Andreev Molecule Probed by Nonlocal Josephson Effect

Problem Statement
Fermion parity is a fundamental quantum number in superconductivity that dictates the microscopic nature of the many-body ground state. In single Josephson junctions containing interacting quantum dots, parity is a local property encoded in the occupation of Andreev bound states (ABS), governing the supercurrent magnitude and direction. This manifests experimentally as the 0– $\pi$ transition, where a change in ground-state parity reverses the local supercurrent. However, in complex superconducting networks, ABS can hybridize and delocalize across multiple coupled junctions, forming "Andreev molecules." In this regime, global fermion parity emerges as a natural quantum number. A significant challenge remains in experimentally establishing a direct connection between the nonlocal transport signatures of these delocalized states and their underlying spatial structure and global parity. Previous implementations using extended weak links often introduce multiple conduction channels that obscure these relationships, necessitating a minimal, highly controllable platform to unambiguously identify delocalized fermionic states and their parity.

Methodology
The authors realized a minimal Andreev molecule using a single carbon nanotube (CNT) hosting two strongly interacting quantum-dot Josephson junctions coupled through a common superconducting electrode (niobium/gold bilayer). The ballistic and quasi-one-dimensional nature of the CNT ensures well-defined quantum confinement and efficient electrostatic tunability.

Two complementary devices were fabricated and measured at 10 mK:

Device B: Controlled solely by a global back-gate electrode. This device allowed for the measurement of the switching current ( $I_{sw,L}$ ) of the left junction while varying the superconducting phase difference ( $\delta_R$ ) across the right junction via a magnetic flux loop.
Device T: Incorporated two additional local top-gates (Gate L and Gate R) to independently control the energy levels of the left and right quantum dots. In this device, the Josephson response was probed via zero-bias conductance ( $G_L$ ).

The nonlocal Josephson effect (NLJE) was probed by measuring the transport through one junction (left) while periodically modulating the phase bias of the other (right). The strength of the nonlocal response was quantified using a modulation contrast ( $C_I$ or $C_G$ ).

Key Contributions and Results

Observation of Pronounced Nonlocal Josephson Effect:
In Device B, the switching current of the left junction exhibited a strong periodic modulation as a function of the phase difference $\delta_R$ applied to the right junction. The modulation period corresponded to one superconducting flux quantum ( $\Phi_0$ ), confirming that the supercurrent in one junction is directly sensitive to the phase of the other. The modulation contrast reached values approaching unity, indicating that the supercurrent could be nearly suppressed by changing only the phase of the remote junction.
Correlation with Molecular Hybridization:
Using Device T, the authors mapped the zero-bias conductance as a function of local gate voltages, revealing a stability diagram characteristic of a double quantum dot. They identified "avoided crossings" where the energy levels of the two dots resonated and hybridized. The nonlocal response was found to be maximized precisely at these avoided crossings. This systematic enhancement demonstrates that the strength of the nonlocal Josephson coupling is directly linked to the spatial delocalization of the Andreev molecular wavefunction across both junctions.
Detection of Global Fermion Parity via $\pi$ -Phase Shifts:
The study demonstrated that changes in the global ground-state parity of the Andreev molecule manifest as characteristic $\pi$ -phase shifts in the nonlocal response.
- In Device T, crossing parity boundaries (where charge states differing by two electrons hybridize) resulted in an abrupt shift of the conductance modulation phase. At zero flux, the conductance was maximized for one parity configuration and minimized for the other.
- In Device B, varying the global back-gate voltage induced similar abrupt $\pi$ -phase shifts in the switching current modulation.
- These experimental observations were supported by a minimal theoretical model of two interacting quantum dots. The model, which included induced superconducting correlations, interdot hybridization, and Coulomb interactions, reproduced the sequence of $\pi$ -phase shifts. Crucially, the model confirmed that each phase shift coincided with a change in the total ground-state charge by one electron, thereby verifying that the phase shifts track changes in global fermion parity.

Significance and Claims
The paper claims to establish the nonlocal Josephson effect as a direct transport probe for two defining properties of delocalized superconducting states:

Spatial Delocalization: The magnitude of the nonlocal response serves as a signature of the formation of delocalized Andreev molecular states.
Global Fermion Parity: The phase of the nonlocal response serves as a signature of the global fermion parity.

The authors assert that this work demonstrates that global fermion parity, typically considered a property of the entire network, can be inferred locally through transport measurements in a single junction. This behavior mirrors the 0– $\pi$ transition of a single quantum-dot junction but appears here in the phase of a nonlocal signal. The authors position these results as a foundational step for hybrid superconducting circuits, identifying global fermion parity as an experimentally accessible degree of freedom that can be revealed through the nonlocal Josephson effect. They suggest that such minimal Andreev molecules serve as building blocks for investigating how parity evolves from a local to a delocalized degree of freedom, with potential relevance for future devices encoding quantum information in delocalized fermionic states.

Technical Summary: Fermion Parity of an Andreev Molecule Probed by Nonlocal Josephson Effect

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