Diamond Defects Reveal New Quantum States with Infinite Sensitivity to Change

Researchers are increasingly interested in Dirac exceptional points (EPs), a unique type of non-Hermitian singularity exhibiting linear energy dispersion, and Chia-Yi Ju, Gunnar Möller (University of Kent and National Yang Ming Chiao Tung University), and Yu-Chin Tzeng (National Yang Ming Chiao Tung University) et al. have now explored their geometric properties within diamond nitrogen-vacancy centres. Their theoretical investigation, utilising fidelity susceptibility as a key probe, demonstrates that these Dirac EPs induce a significant geometric singularity even without a typical symmetry-breaking phase transition, validating fidelity’s effectiveness in identifying such non-Hermitian features. The team’s findings reveal a directional divergence in fidelity susceptibility, diverging along the coupling direction but remaining finite along the detuning axis, a behaviour markedly different from conventional EPs and offering new insights into exploiting these points for quantum control and applications.

Unlike conventional exceptional points, these Dirac EPs exist entirely within a stable, parity-time unbroken phase and exhibit a linear energy dispersion.

This research, utilising fidelity susceptibility as a key probe, demonstrates that despite the absence of a traditional phase transition, the Dirac EP induces a pronounced geometric effect, validating the use of fidelity in characterising these non-Hermitian singularities. Specifically, the real component of the fidelity susceptibility diverges to negative infinity, signalling non-Hermitian criticality.
Crucially, this divergence isn’t uniform; it exhibits a marked anisotropy, diverging sharply along the direction of non-reciprocal coupling but remaining finite along the detuning axis. This behaviour sharply contrasts with the omnidirectional divergence typically observed in conventional exceptional points, highlighting a fundamental difference in their geometric properties.

The work provides a detailed picture of the fidelity probe near the Dirac EP, emphasising the critical role of parameter directionality in harnessing these points for advanced quantum control and applications. Researchers theoretically investigated the quantum geometry of Dirac EPs, focusing on nitrogen-vacancy centres in diamond as a physically relevant platform.

Their analysis reveals that the fidelity susceptibility serves as a sensitive indicator of the Dirac EP’s unique characteristics, confirming its validity as a non-Hermitian singularity. The team demonstrated that the divergence of fidelity susceptibility is not merely a consequence of approaching a phase transition, but an intrinsic property of the Dirac EP’s geometry.

This finding challenges existing theoretical frameworks developed for conventional exceptional points, which often rely on the presence of a symmetry-breaking transition. Furthermore, the observed anisotropy in the divergence, directional dependence, suggests a novel mechanism for controlling and manipulating quantum states near the Dirac EP.

By carefully tuning the parameters governing the non-reciprocal coupling, it may be possible to selectively enhance sensitivity along specific directions, opening up possibilities for high-resolution quantum sensing and precision measurements. This research establishes a foundation for exploiting Dirac EPs in solid-state quantum systems, paving the way for innovative quantum technologies.

Effective Hamiltonian derivation and Dirac exceptional point characterisation are presented

A 6 × 6 Hermitian Hamiltonian underpins the study, originating from a truncated tight-binding model with non-reciprocal couplings to explore Dirac exceptional points in nitrogen-vacancy centres in diamond. Coupling the spin-1 electronic qutrit to an ancillary nuclear spin allows for conditional measurements and post-selection, projecting an effective 3 × 3 non-Hermitian Hamiltonian crucial for engineering the structure hosting these Dirac EPs.

Leveraging this controllable platform, researchers calculated fidelity susceptibility across a synthetic parameter space to characterise the geometric properties of the Dirac EP. The effective Hamiltonian governing the system dynamics is defined as H(q1, q2) = 3S2z + 2q1Sz + √2(Sx −iq2Sy), where Sx,y,z represent the standard spin-1 operators and q1 and q2 denote the effective momentum and degree of non-reciprocal coupling respectively.

Eigenstates |Ln⟩ and |Rn⟩ are defined by eigenvalue equations ⟨Ln|H = En⟨Ln| and H|Rn⟩= En|Rn⟩, with band index n taking values of 0 and ±1. A completeness relation, Σn |Rn⟩⟨Ln| = 11, ensures a biorthogonal basis away from the exceptional points. The research demonstrates that the Dirac EP induces a geometric singularity, validating fidelity as a probe even within the parity-time unbroken phase.

Notably, the fidelity susceptibility diverges to negative infinity along the coupling parameter direction, while remaining finite along the detuning axis, a behaviour contrasting conventional EPs which exhibit omnidirectional divergence. This anisotropy highlights the critical role of parameter directionality in exploiting Dirac EPs for control and applications.

The study maintains rigorous validity by operating within a non-defective regime where the spectrum remains real and non-degenerate, ensuring the biorthogonal formalism and post-selection measurement scheme are consistently applicable. Fidelity is calculated using a specific formulation to emphasise facets of the underlying quantum geometry, providing a comprehensive picture of fidelity-based detection across the spectrum of non-Hermitian singularities.

Anisotropic Fidelity Susceptibility Reveals Non-Hermitian Criticality in Diamond Nitrogen-Vacancy Centers via symmetry breaking

Researchers demonstrate a pronounced geometric singularity using fidelity susceptibility as a probe for Dirac exceptional points (EPs) in nitrogen-vacancy centers in diamond. The real part of the fidelity susceptibility diverges to negative infinity, serving as a signature of non-Hermitian criticality near these points.

This divergence, however, exhibits a distinct anisotropy, diverging along the non-reciprocal coupling direction while remaining finite along the detuning axis. This anisotropic behavior contrasts sharply with the omnidirectional divergence observed in conventional EPs. The study focuses on an effective non-Hermitian Hamiltonian describing NV centers in diamond, a solid-state system where Dirac EPs have been experimentally observed.

By coupling the spin-1 electronic qutrit to an ancillary nuclear spin, the composite system evolves under a dilated 6 × 6 Hermitian Hamiltonian, ultimately projecting an effective 3 × 3 non-Hermitian Hamiltonian. Calculations of the fidelity susceptibility across the synthetic parameter space confirm the universality of fidelity as a probe even entirely within the parity-time unbroken phase.

Specifically, the analysis reveals that the susceptibility diverges to negative infinity along the coupling parameter direction, but remains finite along the detuning axis. This behavior is fundamentally different from conventional EPs, where the geometric singularity is robust regardless of the direction of approach.

The effective Hamiltonian governing the system dynamics is defined as H(q1, q2) = 3S2 z + 2q1Sz + √ 2(Sx −iq2Sy), where q1 and q2 represent the effective momentum and degree of non-reciprocal coupling, respectively. The Dirac EP is located at (q1, q2) = (0, 1), where the bands exhibit a linear crossing characteristic of a Dirac cone, remaining within the PT-unbroken phase where eigenvalues are purely real. The completeness relation, ⟨Ln|Rm⟩= δnm, holds strictly away from the EPs, ensuring a well-defined biorthogonal basis.

Anisotropic Fidelity Susceptibility Reveals Geometric Criticality in Dirac Exceptional Points and their universal scaling

Researchers have demonstrated a comprehensive geometric understanding of Dirac exceptional points (EPs) within nitrogen-vacancy centers in diamond, utilising fidelity susceptibility as a key analytical tool. Their investigation reveals that, despite existing within a parity-time unbroken phase, Dirac EPs induce a geometric singularity characterised by a divergence of the real part of fidelity susceptibility to negative infinity, mirroring a feature found in conventional EPs.

However, this divergence exhibits a marked anisotropy, diverging along the non-reciprocal coupling direction but remaining finite along the detuning axis, a behaviour distinctly different from the omnidirectional divergence seen in standard EPs. This study establishes that geometric criticality is a robust characteristic of non-Hermitian degeneracies, persisting even without a phase transition.

The anisotropy observed in the Dirac EP’s response stems not from spectral gaps, but from the defective eigenstate structure inherent to these points, where the primary deformation of the eigenstate is limited to a single direction in Hilbert space. The authors acknowledge that their model achieves high numerical precision, but further refinement may be necessary for broader applicability. Future work could focus on exploiting this directional sensitivity for controlled manipulation of Dirac EPs, aligning parameter variations with directions of maximal geometric sensitivity to enhance predictable responses.