Beyond Hermiticity: Supercurrents Powered by Imaginary Andreev Levels

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Author: Denis Avetisyan

New research reveals how non-Hermitian physics can unlock unusual supercurrent behavior in Josephson junctions, offering a pathway to explore exotic quantum phenomena.

This review details the theoretical underpinnings and potential experimental signatures of supercurrents arising from the imaginary part of Andreev levels in non-Hermitian Josephson junctions.

Conventional descriptions of superconducting junctions often neglect the influence of non-Hermitian effects arising from open system couplings. This is addressed in ‘Supercurrent from the imaginary part of the Andreev levels in non-Hermitian Josephson junctions’, where we demonstrate that the imaginary component of Andreev levels-typically associated with broadening-directly contributes to the supercurrent via a phase-dependent term. We identify specific spectral configurations and an experimental protocol to highlight this novel current contribution, even in the absence of exceptional points. Could probing this phase dispersion of Andreev levels offer a new avenue for characterizing non-Hermiticity in open quantum systems?

The Whispers of Emergence: Beyond Simple Tunneling

Josephson junctions, while seemingly simple in concept, are far from basic components in superconducting circuits. Beyond the expected quantum tunneling of Cooper pairs, these junctions display a wealth of emergent phenomena arising from the nonlinear nature of superconductivity and the intricate interplay of quantum mechanics. This complexity manifests as a sensitivity to external fields, a rich harmonic spectrum in their current-voltage characteristics, and the existence of multiple stable states. These behaviors aren’t simply additions to a basic tunneling model; they represent qualitatively new physics, enabling functionalities impossible with classical electronic devices. The subtle balance between the superconducting energy gap, the charging energy, and the Josephson energy dictates these emergent properties, making the junction a fertile ground for exploring fundamental quantum effects and designing advanced quantum circuits.

Within Josephson junctions, the seemingly simple act of Cooper pair tunneling generates a more intricate phenomenon: Andreev levels. These subgap states arise when an electron from one superconducting electrode enters the junction and is converted into a hole in the other, effectively doubling the quasiparticle density. Critically, the energy of these Andreev levels is not fixed, but rather tunable via external parameters like gate voltage or magnetic flux. This tunability makes them exceptionally valuable for manipulating quantum information; Andreev levels can serve as quantized energy states for qubits, or as intermediaries in entanglement schemes. By carefully controlling these levels, researchers can design superconducting circuits capable of performing complex quantum computations and exploring novel quantum phenomena, paving the way for advanced technologies in areas like quantum sensing and communication.

Andreev levels, arising within Josephson junctions, are not merely theoretical constructs but rather the central determinants of a device’s operational characteristics. These energy levels, formed by the bound states of Cooper pairs reflecting off the superconducting barriers, fundamentally govern how the junction responds to applied voltages, magnetic fields, and electromagnetic radiation. A precise understanding of their formation and evolution is therefore critical for tailoring device behavior; manipulating these levels allows for the creation of highly sensitive detectors, tunable oscillators, and, crucially, the building blocks for robust quantum computation. The ability to control and utilize these Andreev states represents a pathway toward realizing novel superconducting devices with performance exceeding classical limitations, driving innovation in fields ranging from metrology to quantum information processing.

The integration of a quantum dot into a Josephson junction fundamentally alters the device’s energetic landscape, introducing discrete and tunable energy levels that are absent in conventional junctions. This arises because the quantum dot acts as a nanoscale reservoir for Cooper pairs, quantizing their allowed energies within the junction and creating what are known as Andreev subgap states. By controlling the size, shape, and material composition of the quantum dot – and applying external gate voltages – researchers can precisely tailor these energy scales. This degree of control unlocks the potential for sophisticated device functionality, including highly sensitive charge detectors, tunable microwave sources, and, crucially, the realization of topologically protected quantum bits for robust quantum computation. The ability to engineer these energy levels represents a significant step towards harnessing the full potential of Josephson junctions in advanced superconducting circuits.

Beyond Hermiticity: Confronting Dissipation

Conventional approaches to modeling superconducting junctions often rely on Hermitian Hamiltonians, which assume energy conservation and closed systems. However, these systems inherently exhibit dissipation due to quasi-particle leakage and complex interactions arising from the coupling between the superconducting condensate and the normal metal. These non-conservative effects introduce complex eigenvalues into the Hamiltonian, rendering standard perturbative techniques and diagonalization methods inaccurate or inapplicable. Specifically, the assumption of real energy levels, central to Hermitian approaches, breaks down, leading to an inability to correctly predict the junction’s behavior, particularly at higher bias voltages or in the presence of strong interfacial impedance. This limitation necessitates the development of alternative theoretical frameworks capable of handling non-Hermitian Hamiltonians and accurately capturing these dissipative and complex phenomena.

The description of the superconducting junction utilizes a Non-Hermitian Hamiltonian to accurately model the effects of dissipation and complex interactions present within the system. Traditional Hermitian Hamiltonians are insufficient as they cannot represent processes where energy is not conserved, such as those involving quasiparticle relaxation. The Non-Hermitian formalism allows for complex eigenvalues E = E_R + iE_I, where E_R represents the real part of the energy and E_I corresponds to the imaginary part, quantifying the decay rate of the corresponding state. These complex energies are directly related to the lifetime of the Andreev bound states and their contribution to the junction’s current-voltage characteristics, providing a more realistic depiction of the system’s behavior than would be possible with a strictly Hermitian approach.

Andreev levels are calculated using Green’s Function techniques applied to the Non-Hermitian Hamiltonian describing the junction. Specifically, the poles of the Green’s Function, G(E) = \frac{1}{E - H}, directly correspond to the energies of the Andreev levels. By varying parameters within the Hamiltonian – such as the applied voltage, magnetic field, or interface transparency – we can systematically shift these poles and, therefore, predict the resulting changes in Andreev level positions and spectral characteristics. This allows for quantitative analysis of the junction’s behavior under different operational conditions and provides a means to model the system’s response to external stimuli without resorting to approximations that limit the accuracy of traditional methods.

Andreev levels, determined via Non-Hermitian Hamiltonian methods, directly correlate to the low-energy excitations within the superconducting junction. These levels represent the energy required to break an Andreev pair – an electron-hole pair formed at the interface – and thus dictate the junction’s response to external stimuli. Specifically, the spacing and energy of these levels influence the junction’s conductance, current-voltage characteristics, and sensitivity to external fields. Calculations of Andreev levels therefore provide a fundamental basis for predicting and interpreting a range of quantum phenomena observable in the system, including Josephson currents and the dynamics of quasiparticle tunneling. The derived level structure effectively maps the quantum states available for electron transport across the junction.

Exceptional Points: Where Theory Fractures

Calculations performed on the Josephson junction’s parameter space reveal the presence of Exceptional Points (EPs). These EPs are defined as points where eigenvalues and corresponding eigenvectors of the system’s Hamiltonian coalesce, leading to a breakdown of perturbative expansions and a qualitative change in system behavior. The location of these EPs is dependent on the junction’s physical parameters, including the critical current I_c and the applied magnetic flux. Numerical analysis confirms that EPs are not merely mathematical singularities but represent distinct, physically realizable configurations within the junction’s operational range, observable through changes in the supercurrent characteristics.

Exceptional Points (EPs) represent singularities in the parameter space of a non-Hermitian system, characterized by the coalescence of eigenvalues and eigenvectors. This coalescence signifies a breakdown in the diagonalizable basis, preventing the system from being described by a set of independent eigenstates. Consequently, at an EP, a small perturbation in system parameters can induce a substantial change in the system’s behavior, manifesting as a highly sensitive response. In the context of Josephson junctions, this altered behavior is reflected in the supercurrent and its dependence on the phase difference across the junction; standard perturbative approaches are invalid near EPs, requiring alternative theoretical treatments to accurately model the system’s response. The system’s sensitivity is quantified by the derivative of the eigenvalues with respect to the parameters at the EP, often diverging at these points.

The Current-Phase Relation (CPR) in Josephson junctions, typically described as I = I_c \sin(\phi), deviates from this sinusoidal form in the presence of Exceptional Points. These deviations manifest as non-sinusoidal contributions to the supercurrent, altering the junction’s response to applied voltages and magnetic fields. Specifically, the modified CPR introduces harmonic generation and asymmetry in the current flow, impacting the junction’s switching characteristics and potentially enabling novel functionalities. The extent of this modification is directly related to the proximity of the operating parameters to the Exceptional Point within the junction’s parameter space, resulting in a demonstrable shift in the supercurrent amplitude and phase.

The prominence of Exceptional Points in Josephson junction systems is strongly linked to the presence of Time-Reversal Symmetry (TRS). TRS, defined by the invariance of the system under the time reversal transformation t \rightarrow -t, dictates that the energy spectrum of the junction remains unchanged when the direction of time is reversed. When TRS is maintained, certain Hamiltonian terms do not contribute to the overall energy landscape, allowing for the coalescence of eigenvalues and eigenvectors characteristic of Exceptional Points. Deviations from TRS, typically introduced by applied magnetic fields or non-reciprocal elements, suppress the formation of these points and modify the system’s energy spectrum, thus influencing the supercurrent behavior. The strength of TRS directly correlates with the accessibility and stability of these Exceptional Points within the junction’s parameter space.

The Non-Hermitian Skin Effect: Boundaries Come Alive

The investigation demonstrates that employing a Non-Hermitian Hamiltonian-one where the operator describing the system is not equal to its adjoint-results in the Non-Hermitian Skin Effect. This phenomenon is characterized by the unusual accumulation of quantum states at the physical boundaries of the system, diverging from the expected uniform distribution observed in conventional Hermitian systems. Essentially, the wavefunction ‘piles up’ at the edges, leading to a localization drastically different from standard quantum mechanics. This boundary accumulation isn’t merely a surface phenomenon; it fundamentally alters the system’s bulk properties and dictates how it interacts with external influences, creating unique transport characteristics and observable signatures distinct from those of Hermitian counterparts. The strength of this effect is directly linked to the degree of non-Hermiticity introduced by the Hamiltonian, offering a tunable pathway to control the system’s behavior.

The non-Hermitian nature of the system gives rise to an unusual accumulation of states at the boundaries, resulting in extended Zero-Energy States that propagate across the entire junction. These states are not localized at specific points but rather delocalized, fundamentally altering the system’s electronic properties and influencing its response to external stimuli. The presence of these globally extended states creates pathways for dissipation and can dramatically modify the supercurrent behavior, leading to enhanced sensitivity to the system’s parameters. Consequently, the ability to control and manipulate these Zero-Energy States offers a pathway toward novel device functionalities, particularly in areas requiring precise control over energy transport and quantum interference effects; their broad spatial extent makes them particularly susceptible to external fields and interactions, offering opportunities for tailored responses and enhanced detection capabilities.

The presence of a Ferromagnetic Reservoir introduces significant alterations to the system’s electronic structure, notably impacting both the Andreev levels and the previously established Global Zero-Energy States. This interaction effectively lifts the degeneracy of the zero-energy modes, splitting them into distinct levels sensitive to the magnetization direction and strength of the reservoir. The resulting modifications aren’t merely quantitative; they fundamentally reshape the spatial distribution of these states, influencing their ability to mediate long-range transport and contributing to a measurable non-Hermitian current. Specifically, the coupling with the ferromagnetic leads induces a mixing between the conventional Andreev bound states and the emergent Global Zero-Energy States, leading to hybridized modes with unique properties and enhanced sensitivity to external parameters, potentially unlocking new avenues for device control and manipulation.

The research details precise conditions for amplifying the detection of a non-Hermitian current, a subtle contribution to supercurrent flow. Through careful parameter tuning – specifically establishing a dissipation imbalance of γ_N = 0.8 Γ_N – the phase dispersion of imaginary Andreev levels is maximized. This heightened dispersion significantly enhances the observability of the non-Hermitian current, offering a pathway to directly probe and characterize non-Hermitian effects within superconducting systems. The findings suggest that manipulating dissipation imbalances provides a crucial mechanism for controlling and detecting these previously elusive current contributions, potentially opening new avenues for exploring unconventional superconductivity and related phenomena.

The study of non-Hermitian Josephson junctions reveals a landscape where conventional understandings of symmetry and energy conservation begin to dissolve, not unlike observing matter seemingly laugh at established laws. This work, detailing the contribution of the imaginary part of Andreev levels to supercurrent, highlights how simplified models – pocket black holes, if one will – inevitably fall short when confronted with the full complexity of these systems. As John Locke observed, “All knowledge is ultimately based on recognition,” and this research expands that recognition to encompass a more nuanced understanding of quantum phenomena at the edge of predictability. The precise manipulation of phase dispersion, crucial for detecting these effects, demonstrates a humbling awareness of the limits of theoretical construction; a theory, like any signal, can vanish beyond the event horizon of experimental verification.

Where Do We Go From Here?

The exploration of non-Hermitian effects within Josephson junctions, as detailed in this work, reveals a landscape where the very notion of conserved probability begins to fray. It is a beautiful, if unsettling, reminder that the tools constructed to understand the universe are, ultimately, approximations. The demonstrated contribution of the imaginary part of Andreev levels to the supercurrent is not merely a technical detail; it’s an invitation to reconsider the foundations upon which these devices-and perhaps more fundamentally, the quantum world-are built. When light bends around a massive object, it’s a reminder of limitations.

Future investigations must confront the practical challenges of realizing and probing these non-Hermitian systems. The theoretical configurations that maximize the observable supercurrent are, as is often the case, delicately poised. Robustness against disorder and imperfections will be paramount. The search for experimental signatures of exceptional points within these junctions is also critical, but it requires an acceptance that these points might prove to be mirages, shimmering just beyond the reach of measurement.

Models are like maps that fail to reflect the ocean. This work offers a glimpse into a potentially vast territory where the boundaries between order and chaos, between real and imaginary, become increasingly blurred. It suggests that the true power of these investigations may not lie in building better devices, but in refining the questions asked of the universe itself.

Original article: https://arxiv.org/pdf/2512.24745.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-01 12:56